{"text": "[STATEMENT]\nlemma \"bool_fun_threshold_2_3\n          (vec 4 (\\<lambda>i. if i = 0 \\<or> i = 1 then True else False)) = True\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bool_fun_threshold_2_3 (vec 4 (\\<lambda>i. if i = 0 \\<or> i = 1 then True else False)) = True\n[PROOF STEP]\nunfolding bool_fun_threshold_2_3_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (if 2 \\<le> count_true (vec 4 (\\<lambda>i. if i = 0 \\<or> i = 1 then True else False)) then True else False) = True\n[PROOF STEP]\nunfolding count_true_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (if 2 \\<le> (\\<Sum>i = 0..<dim_vec (vec 4 (\\<lambda>i. if i = 0 \\<or> i = 1 then True else False)). if vec 4 (\\<lambda>i. if i = 0 \\<or> i = 1 then True else False) $ i then 1 else 0) then True else False) = True\n[PROOF STEP]\nunfolding dim_vec\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (if 2 \\<le> (\\<Sum>i = 0..<4. if vec 4 (\\<lambda>i. if i = 0 \\<or> i = 1 then True else False) $ i then 1 else 0) then True else False) = True\n[PROOF STEP]\nunfolding sum.eq_fold\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (if 2 \\<le> Finite_Set.fold ((+) \\<circ> (\\<lambda>i. if vec 4 (\\<lambda>i. if i = 0 \\<or> i = 1 then True else False) $ i then 1 else 0)) 0 {0..<4} then True else False) = True\n[PROOF STEP]\nusing index_vec [of _ 4]\n[PROOF STATE]\nproof (prove)\nusing this:\n?i < 4 \\<Longrightarrow> vec 4 ?f $ ?i = ?f ?i\n\ngoal (1 subgoal):\n 1. (if 2 \\<le> Finite_Set.fold ((+) \\<circ> (\\<lambda>i. if vec 4 (\\<lambda>i. if i = 0 \\<or> i = 1 then True else False) $ i then 1 else 0)) 0 {0..<4} then True else False) = True\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 \\<le> Finite_Set.fold ((+) \\<circ> (\\<lambda>i. if vec 4 (\\<lambda>i. i = 0 \\<or> i = Suc 0) $ i then 1 else 0)) 0 {0..<4}\n[PROOF STEP]\nunfolding set_list_four\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 \\<le> Finite_Set.fold ((+) \\<circ> (\\<lambda>i. if vec 4 (\\<lambda>i. i = 0 \\<or> i = Suc 0) $ i then 1 else 0)) 0 (set [0, 1, 2, 3])\n[PROOF STEP]\nunfolding comp_fun_commute_on.fold_set_fold_remdups [OF comp_fun_commute_lambda, simplified]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 \\<le> fold ((+) \\<circ> (\\<lambda>i. if vec 4 (\\<lambda>i. i = 0 \\<or> i = Suc 0) $ i then 1 else 0)) (remdups [0, 1, 2, 3]) 0\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 1036, "file": "Simplicial_complexes_and_boolean_functions_Simplicial_complex", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9111797051879431, "lm_q2_score": 0.8774767986961401, "lm_q1q2_score": 0.799539050745209}}
{"text": "[STATEMENT]\nlemma card_bijections_range_permutation_eq_1:\n  assumes \"finite A\" \"finite B\"\n  assumes \"card A = card B\"\n  shows \"card ({f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // range_permutation A B) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ({f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // range_permutation A B) = 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card ({f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // range_permutation A B) = 1\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\nfinite B\ncard A = card B\n[PROOF STEP]\nhave \"{f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // range_permutation A B =\n    {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} // range_permutation A B\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nfinite B\ncard A = card B\n\ngoal (1 subgoal):\n 1. {f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // range_permutation A B = {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} // range_permutation A B\n[PROOF STEP]\nby (metis (no_types, lifting) PiE_cong bij_betw_implies_surj_on_and_card_eq)\n[PROOF STATE]\nproof (state)\nthis:\n{f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // range_permutation A B = {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} // range_permutation A B\n\ngoal (1 subgoal):\n 1. card ({f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // range_permutation A B) = 1\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\n{f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // range_permutation A B = {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} // range_permutation A B\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n{f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // range_permutation A B = {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} // range_permutation A B\n\ngoal (1 subgoal):\n 1. card ({f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // range_permutation A B) = 1\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n{f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // range_permutation A B = {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} // range_permutation A B\nfinite A\nfinite B\ncard A = card B\n\ngoal (1 subgoal):\n 1. card ({f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // range_permutation A B) = 1\n[PROOF STEP]\nby (simp add: card_surjective_functions_range_permutation)\n[PROOF STATE]\nproof (state)\nthis:\ncard ({f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // range_permutation A B) = 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1060, "file": "Twelvefold_Way_Card_Bijections", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9111797172476385, "lm_q2_score": 0.8757869851639066, "lm_q1q2_score": 0.7979993375108102}}
{"text": "[STATEMENT]\nlemma card_triangle_triples_rotate: \"card (triangle_triples X Y Z) = card (triangle_triples Y Z X)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (triangle_triples X Y Z) = card (triangle_triples Y Z X)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (triangle_triples X Y Z) = card (triangle_triples Y Z X)\n[PROOF STEP]\nhave \"triangle_triples Y Z X = (\\<lambda>(x,y,z). (y,z,x)) ` triangle_triples X Y Z\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. triangle_triples Y Z X = (\\<lambda>(x, y, z). (y, z, x)) ` triangle_triples X Y Z\n[PROOF STEP]\nby (auto simp: triangle_triples_def case_prod_unfold image_iff insert_commute triangle_in_graph_def)\n[PROOF STATE]\nproof (state)\nthis:\ntriangle_triples Y Z X = (\\<lambda>(x, y, z). (y, z, x)) ` triangle_triples X Y Z\n\ngoal (1 subgoal):\n 1. card (triangle_triples X Y Z) = card (triangle_triples Y Z X)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ntriangle_triples Y Z X = (\\<lambda>(x, y, z). (y, z, x)) ` triangle_triples X Y Z\n\ngoal (1 subgoal):\n 1. card (triangle_triples X Y Z) = card (triangle_triples Y Z X)\n[PROOF STEP]\nhave \"inj_on (\\<lambda>(x, y, z). (y, z, x)) (triangle_triples X Y Z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inj_on (\\<lambda>(x, y, z). (y, z, x)) (triangle_triples X Y Z)\n[PROOF STEP]\nby (auto simp: inj_on_def)\n[PROOF STATE]\nproof (state)\nthis:\ninj_on (\\<lambda>(x, y, z). (y, z, x)) (triangle_triples X Y Z)\n\ngoal (1 subgoal):\n 1. card (triangle_triples X Y Z) = card (triangle_triples Y Z X)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ntriangle_triples Y Z X = (\\<lambda>(x, y, z). (y, z, x)) ` triangle_triples X Y Z\ninj_on (\\<lambda>(x, y, z). (y, z, x)) (triangle_triples X Y Z)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ntriangle_triples Y Z X = (\\<lambda>(x, y, z). (y, z, x)) ` triangle_triples X Y Z\ninj_on (\\<lambda>(x, y, z). (y, z, x)) (triangle_triples X Y Z)\n\ngoal (1 subgoal):\n 1. card (triangle_triples X Y Z) = card (triangle_triples Y Z X)\n[PROOF STEP]\nby (simp add: card_image)\n[PROOF STATE]\nproof (state)\nthis:\ncard (triangle_triples X Y Z) = card (triangle_triples Y Z X)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 933, "file": "Undirected_Graph_Theory_Graph_Triangles", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9005297807787537, "lm_q2_score": 0.8856314723088733, "lm_q1q2_score": 0.7975375156090745}}
{"text": "[STATEMENT]\nlemma  trace_add: \n  assumes \"square_mat A\"\n  and \"square_mat B\"\n  and \"dim_row A = dim_row B\"\n  shows \"Complex_Matrix.trace (A + B) = Complex_Matrix.trace A + Complex_Matrix.trace B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.trace (A + B) = Complex_Matrix.trace A + Complex_Matrix.trace B\n[PROOF STEP]\nusing  assms\n[PROOF STATE]\nproof (prove)\nusing this:\nsquare_mat A\nsquare_mat B\ndim_row A = dim_row B\n\ngoal (1 subgoal):\n 1. Complex_Matrix.trace (A + B) = Complex_Matrix.trace A + Complex_Matrix.trace B\n[PROOF STEP]\nby (simp add: Complex_Matrix.trace_def sum.distrib)", "meta": {"llama_tokens": 234, "file": "Projective_Measurements_Linear_Algebra_Complements", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.919642528975397, "lm_q2_score": 0.867035763237924, "lm_q1q2_score": 0.7973629620162379}}
{"text": "[STATEMENT]\nlemma le_powr_half_mult:\n  fixes x y z:: real\n  assumes \"x ^ 2 \\<le> y * z\" and \"0 \\<le> y\" and \"0 \\<le> z\"\n  shows \"x \\<le> y powr(1/2) * z powr (1/2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\<le> y powr (1 / 2) * z powr (1 / 2)\n[PROOF STEP]\nusing assms power2_eq_square\n[PROOF STATE]\nproof (prove)\nusing this:\nx\\<^sup>2 \\<le> y * z\n0 \\<le> y\n0 \\<le> z\n?a\\<^sup>2 = ?a * ?a\n\ngoal (1 subgoal):\n 1. x \\<le> y powr (1 / 2) * z powr (1 / 2)\n[PROOF STEP]\nby (metis dual_order.trans linorder_linear powr_ge_pzero powr_half_sqrt powr_mult real_le_rsqrt \n    real_sqrt_le_0_iff)", "meta": {"llama_tokens": 292, "file": "Balog_Szemeredi_Gowers_Miscellaneous_Lemmas", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9433475683211323, "lm_q2_score": 0.8438950966654772, "lm_q1q2_score": 0.7960863873575048}}
{"text": "[STATEMENT]\nlemma inner_prod_with_itself_Re:\n  \"Re (\\<langle>u|u\\<rangle>) \\<ge> 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\<le> Re \\<langle>u|u\\<rangle>\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 0 \\<le> Re \\<langle>u|u\\<rangle>\n[PROOF STEP]\nhave \"Re (\\<langle>u|u\\<rangle>) = (\\<Sum>i<dim_vec u. Re (cnj(u $ i) * (u $ i)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Re \\<langle>u|u\\<rangle> = (\\<Sum>i<dim_vec u. Re (cnj (u $ i) * u $ i))\n[PROOF STEP]\nby (simp add: inner_prod_def lessThan_atLeast0)\n[PROOF STATE]\nproof (state)\nthis:\nRe \\<langle>u|u\\<rangle> = (\\<Sum>i<dim_vec u. Re (cnj (u $ i) * u $ i))\n\ngoal (1 subgoal):\n 1. 0 \\<le> Re \\<langle>u|u\\<rangle>\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nRe \\<langle>u|u\\<rangle> = (\\<Sum>i<dim_vec u. Re (cnj (u $ i) * u $ i))\n\ngoal (1 subgoal):\n 1. 0 \\<le> Re \\<langle>u|u\\<rangle>\n[PROOF STEP]\nhave \"\\<dots> = (\\<Sum>i<dim_vec u. (Re (u $ i))\\<^sup>2 + (Im (u $ i))\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>i<dim_vec u. Re (cnj (u $ i) * u $ i)) = (\\<Sum>i<dim_vec u. (Re (u $ i))\\<^sup>2 + (Im (u $ i))\\<^sup>2)\n[PROOF STEP]\nusing complex_mult_cnj\n[PROOF STATE]\nproof (prove)\nusing this:\n?z * cnj ?z = complex_of_real ((Re ?z)\\<^sup>2 + (Im ?z)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. (\\<Sum>i<dim_vec u. Re (cnj (u $ i) * u $ i)) = (\\<Sum>i<dim_vec u. (Re (u $ i))\\<^sup>2 + (Im (u $ i))\\<^sup>2)\n[PROOF STEP]\nby (metis (no_types, lifting) Re_complex_of_real semiring_normalization_rules(7))\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>i<dim_vec u. Re (cnj (u $ i) * u $ i)) = (\\<Sum>i<dim_vec u. (Re (u $ i))\\<^sup>2 + (Im (u $ i))\\<^sup>2)\n\ngoal (1 subgoal):\n 1. 0 \\<le> Re \\<langle>u|u\\<rangle>\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nRe \\<langle>u|u\\<rangle> = (\\<Sum>i<dim_vec u. Re (cnj (u $ i) * u $ i))\n(\\<Sum>i<dim_vec u. Re (cnj (u $ i) * u $ i)) = (\\<Sum>i<dim_vec u. (Re (u $ i))\\<^sup>2 + (Im (u $ i))\\<^sup>2)\n[PROOF STEP]\nshow \"Re (\\<langle>u|u\\<rangle>) \\<ge> 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nRe \\<langle>u|u\\<rangle> = (\\<Sum>i<dim_vec u. Re (cnj (u $ i) * u $ i))\n(\\<Sum>i<dim_vec u. Re (cnj (u $ i) * u $ i)) = (\\<Sum>i<dim_vec u. (Re (u $ i))\\<^sup>2 + (Im (u $ i))\\<^sup>2)\n\ngoal (1 subgoal):\n 1. 0 \\<le> Re \\<langle>u|u\\<rangle>\n[PROOF STEP]\nby (simp add: sum_nonneg)\n[PROOF STATE]\nproof (state)\nthis:\n0 \\<le> Re \\<langle>u|u\\<rangle>\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1203, "file": "Isabelle_Marries_Dirac_Quantum", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8976952866333483, "lm_q2_score": 0.8856314647623016, "lm_q1q2_score": 0.7950271916113064}}
{"text": "[STATEMENT]\nlemma det_interchange_rows:\nshows \"det (interchange_rows A i j) = of_int (if i = j then 1 else -1) * det A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (interchange_rows A i j) = of_int (if i = j then 1 else - 1) * det A\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. det (interchange_rows A i j) = of_int (if i = j then 1 else - 1) * det A\n[PROOF STEP]\nhave \"(interchange_rows A i j) = (\\<chi> a. A $ (Transposition.transpose i j) a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. interchange_rows A i j = (\\<chi>a. A $ Transposition.transpose i j a)\n[PROOF STEP]\nunfolding interchange_rows_def Transposition.transpose_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<chi>ia ja. if ia = i then A $ j $ ja else if ia = j then A $ i $ ja else A $ ia $ ja) = (\\<chi>a. A $ (if a = i then j else if a = j then i else a))\n[PROOF STEP]\nby vector\n[PROOF STATE]\nproof (state)\nthis:\ninterchange_rows A i j = (\\<chi>a. A $ Transposition.transpose i j a)\n\ngoal (1 subgoal):\n 1. det (interchange_rows A i j) = of_int (if i = j then 1 else - 1) * det A\n[PROOF STEP]\nhence \"det(interchange_rows A i j) = det(\\<chi> a. A$(Transposition.transpose i j) a)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ninterchange_rows A i j = (\\<chi>a. A $ Transposition.transpose i j a)\n\ngoal (1 subgoal):\n 1. det (interchange_rows A i j) = det (\\<chi>a. A $ Transposition.transpose i j a)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndet (interchange_rows A i j) = det (\\<chi>a. A $ Transposition.transpose i j a)\n\ngoal (1 subgoal):\n 1. det (interchange_rows A i j) = of_int (if i = j then 1 else - 1) * det A\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndet (interchange_rows A i j) = det (\\<chi>a. A $ Transposition.transpose i j a)\n\ngoal (1 subgoal):\n 1. det (interchange_rows A i j) = of_int (if i = j then 1 else - 1) * det A\n[PROOF STEP]\nhave \"... = of_int (sign (Transposition.transpose i j)) * det A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (\\<chi>a. A $ Transposition.transpose i j a) = of_int (sign (Transposition.transpose i j)) * det A\n[PROOF STEP]\nby (rule det_permute_rows[of \"Transposition.transpose i j\" A], simp add: permutes_swap_id)\n[PROOF STATE]\nproof (state)\nthis:\ndet (\\<chi>a. A $ Transposition.transpose i j a) = of_int (sign (Transposition.transpose i j)) * det A\n\ngoal (1 subgoal):\n 1. det (interchange_rows A i j) = of_int (if i = j then 1 else - 1) * det A\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ndet (interchange_rows A i j) = of_int (sign (Transposition.transpose i j)) * det A\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndet (interchange_rows A i j) = of_int (sign (Transposition.transpose i j)) * det A\n\ngoal (1 subgoal):\n 1. det (interchange_rows A i j) = of_int (if i = j then 1 else - 1) * det A\n[PROOF STEP]\nunfolding sign_swap_id\n[PROOF STATE]\nproof (prove)\nusing this:\ndet (interchange_rows A i j) = of_int (if i = j then 1 else - 1) * det A\n\ngoal (1 subgoal):\n 1. det (interchange_rows A i j) = of_int (if i = j then 1 else - 1) * det A\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ndet (interchange_rows A i j) = of_int (if i = j then 1 else - 1) * det A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1336, "file": "Gauss_Jordan_Determinants2", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9073122113355091, "lm_q2_score": 0.8757869948899665, "lm_q1q2_score": 0.7946122349924957}}
{"text": "[STATEMENT]\nlemma unity_root_eq_1_iff:\n  fixes k n :: nat\n  assumes \"k > 0\" \n  shows \"unity_root k n = 1 \\<longleftrightarrow> k dvd n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (unity_root k (int n) = 1) = (k dvd n)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (unity_root k (int n) = 1) = (k dvd n)\n[PROOF STEP]\nhave \"unity_root k n = exp ((2*pi*n/k) * \\<i>)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. unity_root k (int n) = exp (complex_of_real (2 * pi * real n / real k) * \\<i>)\n[PROOF STEP]\nby (simp add: unity_root_conv_exp)\n[PROOF STATE]\nproof (state)\nthis:\nunity_root k (int n) = exp (complex_of_real (2 * pi * real n / real k) * \\<i>)\n\ngoal (1 subgoal):\n 1. (unity_root k (int n) = 1) = (k dvd n)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nunity_root k (int n) = exp (complex_of_real (2 * pi * real n / real k) * \\<i>)\n\ngoal (1 subgoal):\n 1. (unity_root k (int n) = 1) = (k dvd n)\n[PROOF STEP]\nhave \"exp ((2*pi*n/k)* \\<i>) = 1 \\<longleftrightarrow> k dvd n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (exp (complex_of_real (2 * pi * real n / real k) * \\<i>) = 1) = (k dvd n)\n[PROOF STEP]\nusing complex_root_unity_eq_1[of k n] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n1 \\<le> k \\<Longrightarrow> (exp (2 * complex_of_real pi * \\<i> * of_nat n / of_nat k) = 1) = (k dvd n)\n0 < k\n\ngoal (1 subgoal):\n 1. (exp (complex_of_real (2 * pi * real n / real k) * \\<i>) = 1) = (k dvd n)\n[PROOF STEP]\nby (auto simp add: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(exp (complex_of_real (2 * pi * real n / real k) * \\<i>) = 1) = (k dvd n)\n\ngoal (1 subgoal):\n 1. (unity_root k (int n) = 1) = (k dvd n)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(unity_root k (int n) = 1) = (k dvd n)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(unity_root k (int n) = 1) = (k dvd n)\n\ngoal (1 subgoal):\n 1. (unity_root k (int n) = 1) = (k dvd n)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(unity_root k (int n) = 1) = (k dvd n)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 965, "file": "Gauss_Sums_Complex_Roots_Of_Unity", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110569397307, "lm_q2_score": 0.8902942203004185, "lm_q1q2_score": 0.7939742295934497}}
{"text": "[STATEMENT]\nlemma dest_segment:\n  fixes x b::real\n  assumes \"(x, b) \\<in> closed_segment (x0, y0) (x1, y1)\"\n  assumes \"x0 \\<noteq> x1\"\n  shows \"b = (y1 - y0) * (x - x0) / (x1 - x0) + y0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. b = (y1 - y0) * (x - x0) / (x1 - x0) + y0\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. b = (y1 - y0) * (x - x0) / (x1 - x0) + y0\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n(x, b) \\<in> closed_segment (x0, y0) (x1, y1)\nx0 \\<noteq> x1\n[PROOF STEP]\nobtain u where u: \"x = x0 *\\<^sub>R (1 - u) + u * x1\" \"b = y0 *\\<^sub>R (1 - u) + u * y1\" \"0 \\<le> u\" \"u \\<le> 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(x, b) \\<in> closed_segment (x0, y0) (x1, y1)\nx0 \\<noteq> x1\n\ngoal (1 subgoal):\n 1. (\\<And>u. \\<lbrakk>x = x0 *\\<^sub>R (1 - u) + u * x1; b = y0 *\\<^sub>R (1 - u) + u * y1; 0 \\<le> u; u \\<le> 1\\<rbrakk> \\<Longrightarrow> thesis) \\<Longrightarrow> thesis\n[PROOF STEP]\nby (auto simp: closed_segment_def algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nx = x0 *\\<^sub>R (1 - u) + u * x1\nb = y0 *\\<^sub>R (1 - u) + u * y1\n0 \\<le> u\nu \\<le> 1\n\ngoal (1 subgoal):\n 1. b = (y1 - y0) * (x - x0) / (x1 - x0) + y0\n[PROOF STEP]\nshow \"b = (y1 - y0) * (x - x0) / (x1 - x0) + y0 \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. b = (y1 - y0) * (x - x0) / (x1 - x0) + y0\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n(x, b) \\<in> closed_segment (x0, y0) (x1, y1)\nx0 \\<noteq> x1\n\ngoal (1 subgoal):\n 1. b = (y1 - y0) * (x - x0) / (x1 - x0) + y0\n[PROOF STEP]\nby (auto simp: closed_segment_def field_simps u)\n[PROOF STATE]\nproof (state)\nthis:\nb = (y1 - y0) * (x - x0) / (x1 - x0) + y0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 894, "file": "Affine_Arithmetic_Intersection", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887587964389112, "lm_q2_score": 0.8933094103149354, "lm_q1q2_score": 0.7939365963590554}}
{"text": "[STATEMENT]\nlemma setdist_eq_0_compact_closed:\n  assumes S: \"compact S\" and T: \"closed T\"\n    shows \"setdist S T = 0 \\<longleftrightarrow> S = {} \\<or> T = {} \\<or> S \\<inter> T \\<noteq> {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (setdist S T = 0) = (S = {} \\<or> T = {} \\<or> S \\<inter> T \\<noteq> {})\n[PROOF STEP]\nproof (cases \"S = {} \\<or> T = {}\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. S = {} \\<or> T = {} \\<Longrightarrow> (setdist S T = 0) = (S = {} \\<or> T = {} \\<or> S \\<inter> T \\<noteq> {})\n 2. \\<not> (S = {} \\<or> T = {}) \\<Longrightarrow> (setdist S T = 0) = (S = {} \\<or> T = {} \\<or> S \\<inter> T \\<noteq> {})\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nS = {} \\<or> T = {}\n\ngoal (2 subgoals):\n 1. S = {} \\<or> T = {} \\<Longrightarrow> (setdist S T = 0) = (S = {} \\<or> T = {} \\<or> S \\<inter> T \\<noteq> {})\n 2. \\<not> (S = {} \\<or> T = {}) \\<Longrightarrow> (setdist S T = 0) = (S = {} \\<or> T = {} \\<or> S \\<inter> T \\<noteq> {})\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nS = {} \\<or> T = {}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nS = {} \\<or> T = {}\n\ngoal (1 subgoal):\n 1. (setdist S T = 0) = (S = {} \\<or> T = {} \\<or> S \\<inter> T \\<noteq> {})\n[PROOF STEP]\nby force\n[PROOF STATE]\nproof (state)\nthis:\n(setdist S T = 0) = (S = {} \\<or> T = {} \\<or> S \\<inter> T \\<noteq> {})\n\ngoal (1 subgoal):\n 1. \\<not> (S = {} \\<or> T = {}) \\<Longrightarrow> (setdist S T = 0) = (S = {} \\<or> T = {} \\<or> S \\<inter> T \\<noteq> {})\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<not> (S = {} \\<or> T = {}) \\<Longrightarrow> (setdist S T = 0) = (S = {} \\<or> T = {} \\<or> S \\<inter> T \\<noteq> {})\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\<not> (S = {} \\<or> T = {})\n\ngoal (1 subgoal):\n 1. \\<not> (S = {} \\<or> T = {}) \\<Longrightarrow> (setdist S T = 0) = (S = {} \\<or> T = {} \\<or> S \\<inter> T \\<noteq> {})\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\<not> (S = {} \\<or> T = {})\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<not> (S = {} \\<or> T = {})\n\ngoal (1 subgoal):\n 1. (setdist S T = 0) = (S = {} \\<or> T = {} \\<or> S \\<inter> T \\<noteq> {})\n[PROOF STEP]\nby (metis S T disjoint_iff_not_equal in_closed_iff_infdist_zero setdist_attains_inf setdist_eq_0I setdist_sym)\n[PROOF STATE]\nproof (state)\nthis:\n(setdist S T = 0) = (S = {} \\<or> T = {} \\<or> S \\<inter> T \\<noteq> {})\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1119, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887587817066391, "lm_q2_score": 0.8933094174159129, "lm_q1q2_score": 0.7939365895096343}}
{"text": "[STATEMENT]\nlemma sin_times_sin: \"sin w * sin z = (cos (w - z) - cos (w + z)) / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin w * sin z = (cos (w - z) - cos (w + z)) / (2::'a)\n[PROOF STEP]\nby (simp add: cos_diff cos_add)", "meta": {"llama_tokens": 109, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9473810466522862, "lm_q2_score": 0.8376199552262967, "lm_q1q2_score": 0.7935452698791301}}
{"text": "[STATEMENT]\nlemma frechet_derivative_pair:\n  \"frechet_derivative (\\<lambda>x. (f x, g x)) (at x) = (\\<lambda>v. (frechet_derivative f (at x) v, frechet_derivative g (at x) v))\"\n  if \"f differentiable (at x)\" \"g differentiable (at x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. frechet_derivative (\\<lambda>x. (f x, g x)) (at x) = (\\<lambda>v. (frechet_derivative f (at x) v, frechet_derivative g (at x) v))\n[PROOF STEP]\napply (rule frechet_derivative_at')\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\<lambda>x. (f x, g x)) has_derivative (\\<lambda>v. (frechet_derivative f (at x) v, frechet_derivative g (at x) v))) (at x)\n[PROOF STEP]\napply (rule derivative_eq_intros)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. (f has_derivative ?f'3) (at x)\n 2. (g has_derivative ?g'3) (at x)\n 3. (\\<lambda>h. (?f'3 h, ?g'3 h)) = (\\<lambda>v. (frechet_derivative f (at x) v, frechet_derivative g (at x) v))\n[PROOF STEP]\napply (rule frechet_derivative_worksI)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. f differentiable at x\n 2. (g has_derivative ?g'3) (at x)\n 3. (\\<lambda>h. (frechet_derivative f (at x) h, ?g'3 h)) = (\\<lambda>v. (frechet_derivative f (at x) v, frechet_derivative g (at x) v))\n[PROOF STEP]\napply fact\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. (g has_derivative ?g'3) (at x)\n 2. (\\<lambda>h. (frechet_derivative f (at x) h, ?g'3 h)) = (\\<lambda>v. (frechet_derivative f (at x) v, frechet_derivative g (at x) v))\n[PROOF STEP]\napply (rule frechet_derivative_worksI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. g differentiable at x\n 2. (\\<lambda>h. (frechet_derivative f (at x) h, frechet_derivative g (at x) h)) = (\\<lambda>v. (frechet_derivative f (at x) v, frechet_derivative g (at x) v))\n[PROOF STEP]\napply fact\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<lambda>h. (frechet_derivative f (at x) h, frechet_derivative g (at x) h)) = (\\<lambda>v. (frechet_derivative f (at x) v, frechet_derivative g (at x) v))\n[PROOF STEP]\n..", "meta": {"llama_tokens": 913, "file": "Smooth_Manifolds_Analysis_More", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9230391685381605, "lm_q2_score": 0.8596637469145054, "lm_q1q2_score": 0.7935033101743647}}
{"text": "[STATEMENT]\nlemma kernel_code[code]: \n  \\<comment> \\<open>Computes the kernel of an operator \\<^term>\\<open>A\\<close>.\n      This is implemented using the existing functions \n      for transforming a matrix into row echelon form (\\<^term>\\<open>gauss_jordan_single\\<close>)\n      and for computing a basis of the kernel of such a matrix\n      (\\<^term>\\<open>find_base_vectors\\<close>)\\<close>\n  \"kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\" \n  for A::\"('a::onb_enum,'b::onb_enum) cblinfun\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\ndefine dA dB Am Ag base\n    where \"dA = length (canonical_basis :: 'a list)\"\n      and \"dB = length (canonical_basis :: 'b list)\"\n      and \"Am = mat_of_cblinfun A\"\n      and \"Ag = gauss_jordan_single Am\"\n      and \"base = find_base_vectors Ag\"\n[PROOF STATE]\nproof (state)\nthis:\ndA = length canonical_basis\ndB = length canonical_basis\nAm = mat_of_cblinfun A\nAg = gauss_jordan_single Am\nbase = find_base_vectors Ag\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\ninterpret complex_vec_space dA\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nhave Am_carrier: \"Am \\<in> carrier_mat dB dA\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Am \\<in> carrier_mat dB dA\n[PROOF STEP]\nunfolding Am_def mat_of_cblinfun_def dA_def dB_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat (length canonical_basis) (length canonical_basis) (\\<lambda>(i, j). crepresentation (set canonical_basis) (A *\\<^sub>V canonical_basis ! j) (canonical_basis ! i)) \\<in> carrier_mat (length canonical_basis) (length canonical_basis)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nAm \\<in> carrier_mat dB dA\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nhave row_echelon: \"row_echelon_form Ag\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. row_echelon_form Ag\n[PROOF STEP]\nunfolding Ag_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. row_echelon_form (gauss_jordan_single Am)\n[PROOF STEP]\nusing Am_carrier refl\n[PROOF STATE]\nproof (prove)\nusing this:\nAm \\<in> carrier_mat dB dA\n?t = ?t\n\ngoal (1 subgoal):\n 1. row_echelon_form (gauss_jordan_single Am)\n[PROOF STEP]\nby (rule gauss_jordan_single)\n[PROOF STATE]\nproof (state)\nthis:\nrow_echelon_form Ag\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nhave Ag_carrier: \"Ag \\<in> carrier_mat dB dA\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Ag \\<in> carrier_mat dB dA\n[PROOF STEP]\nunfolding Ag_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. gauss_jordan_single Am \\<in> carrier_mat dB dA\n[PROOF STEP]\nusing Am_carrier refl\n[PROOF STATE]\nproof (prove)\nusing this:\nAm \\<in> carrier_mat dB dA\n?t = ?t\n\ngoal (1 subgoal):\n 1. gauss_jordan_single Am \\<in> carrier_mat dB dA\n[PROOF STEP]\nby (rule gauss_jordan_single(2))\n[PROOF STATE]\nproof (state)\nthis:\nAg \\<in> carrier_mat dB dA\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nhave base_carrier: \"set base \\<subseteq> carrier_vec dA\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set base \\<subseteq> carrier_vec dA\n[PROOF STEP]\nunfolding base_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set (find_base_vectors Ag) \\<subseteq> carrier_vec dA\n[PROOF STEP]\nusing find_base_vectors(1)[OF row_echelon Ag_carrier]\n[PROOF STATE]\nproof (prove)\nusing this:\nset (find_base_vectors Ag) \\<subseteq> mat_kernel Ag\n\ngoal (1 subgoal):\n 1. set (find_base_vectors Ag) \\<subseteq> carrier_vec dA\n[PROOF STEP]\nusing Ag_carrier mat_kernel_def\n[PROOF STATE]\nproof (prove)\nusing this:\nset (find_base_vectors Ag) \\<subseteq> mat_kernel Ag\nAg \\<in> carrier_mat dB dA\nmat_kernel ?A = {v \\<in> carrier_vec (dim_col ?A). ?A *\\<^sub>v v = 0\\<^sub>v (dim_row ?A)}\n\ngoal (1 subgoal):\n 1. set (find_base_vectors Ag) \\<subseteq> carrier_vec dA\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nset base \\<subseteq> carrier_vec dA\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\ninterpret k: kernel dB dA Ag\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Matrix_Kernel.kernel dB dA Ag\n[PROOF STEP]\napply standard\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Ag \\<in> carrier_mat dB dA\n[PROOF STEP]\nusing Ag_carrier\n[PROOF STATE]\nproof (prove)\nusing this:\nAg \\<in> carrier_mat dB dA\n\ngoal (1 subgoal):\n 1. Ag \\<in> carrier_mat dB dA\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nhave basis_base: \"kernel.basis dA Ag (set base)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. k.basis (set base)\n[PROOF STEP]\nusing row_echelon Ag_carrier\n[PROOF STATE]\nproof (prove)\nusing this:\nrow_echelon_form Ag\nAg \\<in> carrier_mat dB dA\n\ngoal (1 subgoal):\n 1. k.basis (set base)\n[PROOF STEP]\nunfolding base_def\n[PROOF STATE]\nproof (prove)\nusing this:\nrow_echelon_form Ag\nAg \\<in> carrier_mat dB dA\n\ngoal (1 subgoal):\n 1. k.basis (set (find_base_vectors Ag))\n[PROOF STEP]\nby (rule find_base_vectors(3))\n[PROOF STATE]\nproof (state)\nthis:\nk.basis (set base)\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nhave \"space_as_set (SPAN base)\n       = space_as_set (ccspan (basis_enum_of_vec ` set base :: 'a set))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. space_as_set (SPAN base) = space_as_set (ccspan (basis_enum_of_vec ` set base))\n[PROOF STEP]\nunfolding SPAN_def dA_def[symmetric] Let_def filter_set\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. space_as_set (ccspan (basis_enum_of_vec ` set (filter (\\<lambda>v. dim_vec v = dA) base))) = space_as_set (ccspan (basis_enum_of_vec ` set base))\n[PROOF STEP]\napply (subst filter_True)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\<forall>x\\<in>set base. dim_vec x = dA\n 2. space_as_set (ccspan (basis_enum_of_vec ` set base)) = space_as_set (ccspan (basis_enum_of_vec ` set base))\n[PROOF STEP]\nusing base_carrier\n[PROOF STATE]\nproof (prove)\nusing this:\nset base \\<subseteq> carrier_vec dA\n\ngoal (2 subgoals):\n 1. \\<forall>x\\<in>set base. dim_vec x = dA\n 2. space_as_set (ccspan (basis_enum_of_vec ` set base)) = space_as_set (ccspan (basis_enum_of_vec ` set base))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nspace_as_set (SPAN base) = space_as_set (ccspan (basis_enum_of_vec ` set base))\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nspace_as_set (SPAN base) = space_as_set (ccspan (basis_enum_of_vec ` set base))\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nhave \"\\<dots> = cspan (basis_enum_of_vec ` set base)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. space_as_set (ccspan (basis_enum_of_vec ` set base)) = cspan (basis_enum_of_vec ` set base)\n[PROOF STEP]\napply transfer\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>base. closure (cspan (basis_enum_of_vec ` set base)) = cspan (basis_enum_of_vec ` set base)\n[PROOF STEP]\napply (subst closure_finite_cspan)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\<And>base. finite (basis_enum_of_vec ` set base)\n 2. \\<And>base. cspan (basis_enum_of_vec ` set base) = cspan (basis_enum_of_vec ` set base)\n[PROOF STEP]\nby simp_all\n[PROOF STATE]\nproof (state)\nthis:\nspace_as_set (ccspan (basis_enum_of_vec ` set base)) = cspan (basis_enum_of_vec ` set base)\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nspace_as_set (ccspan (basis_enum_of_vec ` set base)) = cspan (basis_enum_of_vec ` set base)\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nhave \"\\<dots> = basis_enum_of_vec ` span (set base)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cspan (basis_enum_of_vec ` set base) = basis_enum_of_vec ` local.span (set base)\n[PROOF STEP]\napply (subst basis_enum_of_vec_span)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. length canonical_basis = dA\n 2. set base \\<subseteq> carrier_vec dA\n 3. cspan (basis_enum_of_vec ` set base) = cspan (basis_enum_of_vec ` set base)\n[PROOF STEP]\nusing base_carrier dA_def\n[PROOF STATE]\nproof (prove)\nusing this:\nset base \\<subseteq> carrier_vec dA\ndA = length canonical_basis\n\ngoal (3 subgoals):\n 1. length canonical_basis = dA\n 2. set base \\<subseteq> carrier_vec dA\n 3. cspan (basis_enum_of_vec ` set base) = cspan (basis_enum_of_vec ` set base)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncspan (basis_enum_of_vec ` set base) = basis_enum_of_vec ` local.span (set base)\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncspan (basis_enum_of_vec ` set base) = basis_enum_of_vec ` local.span (set base)\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nhave \"\\<dots> = basis_enum_of_vec ` mat_kernel Ag\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. basis_enum_of_vec ` local.span (set base) = basis_enum_of_vec ` mat_kernel Ag\n[PROOF STEP]\nusing basis_base k.Ker.basis_def k.span_same\n[PROOF STATE]\nproof (prove)\nusing this:\nk.basis (set base)\nk.basis ?A = (k.lin_indpt ?A \\<and> k.span ?A = mat_kernel Ag \\<and> ?A \\<subseteq> mat_kernel Ag)\n?S \\<subseteq> mat_kernel Ag \\<Longrightarrow> k.span ?S = local.span ?S\n\ngoal (1 subgoal):\n 1. basis_enum_of_vec ` local.span (set base) = basis_enum_of_vec ` mat_kernel Ag\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nbasis_enum_of_vec ` local.span (set base) = basis_enum_of_vec ` mat_kernel Ag\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nbasis_enum_of_vec ` local.span (set base) = basis_enum_of_vec ` mat_kernel Ag\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nhave \"\\<dots> = basis_enum_of_vec ` {v \\<in> carrier_vec dA. Ag *\\<^sub>v v = 0\\<^sub>v dB}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. basis_enum_of_vec ` mat_kernel Ag = basis_enum_of_vec ` {v \\<in> carrier_vec dA. Ag *\\<^sub>v v = 0\\<^sub>v dB}\n[PROOF STEP]\napply (rule arg_cong[where f=\"\\<lambda>x. basis_enum_of_vec ` x\"])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat_kernel Ag = {v \\<in> carrier_vec dA. Ag *\\<^sub>v v = 0\\<^sub>v dB}\n[PROOF STEP]\nunfolding mat_kernel_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {v \\<in> carrier_vec (dim_col Ag). Ag *\\<^sub>v v = 0\\<^sub>v (dim_row Ag)} = {v \\<in> carrier_vec dA. Ag *\\<^sub>v v = 0\\<^sub>v dB}\n[PROOF STEP]\nusing Ag_carrier\n[PROOF STATE]\nproof (prove)\nusing this:\nAg \\<in> carrier_mat dB dA\n\ngoal (1 subgoal):\n 1. {v \\<in> carrier_vec (dim_col Ag). Ag *\\<^sub>v v = 0\\<^sub>v (dim_row Ag)} = {v \\<in> carrier_vec dA. Ag *\\<^sub>v v = 0\\<^sub>v dB}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nbasis_enum_of_vec ` mat_kernel Ag = basis_enum_of_vec ` {v \\<in> carrier_vec dA. Ag *\\<^sub>v v = 0\\<^sub>v dB}\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nbasis_enum_of_vec ` mat_kernel Ag = basis_enum_of_vec ` {v \\<in> carrier_vec dA. Ag *\\<^sub>v v = 0\\<^sub>v dB}\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nhave \"\\<dots> = basis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. basis_enum_of_vec ` {v \\<in> carrier_vec dA. Ag *\\<^sub>v v = 0\\<^sub>v dB} = basis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB}\n[PROOF STEP]\nusing gauss_jordan_single(1)[OF Am_carrier Ag_def[symmetric]]\n[PROOF STATE]\nproof (prove)\nusing this:\n?x \\<in> carrier_vec dA \\<Longrightarrow> (Am *\\<^sub>v ?x = 0\\<^sub>v dB) = (Ag *\\<^sub>v ?x = 0\\<^sub>v dB)\n\ngoal (1 subgoal):\n 1. basis_enum_of_vec ` {v \\<in> carrier_vec dA. Ag *\\<^sub>v v = 0\\<^sub>v dB} = basis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nbasis_enum_of_vec ` {v \\<in> carrier_vec dA. Ag *\\<^sub>v v = 0\\<^sub>v dB} = basis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB}\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nbasis_enum_of_vec ` {v \\<in> carrier_vec dA. Ag *\\<^sub>v v = 0\\<^sub>v dB} = basis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB}\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nhave \"\\<dots> = {w. A *\\<^sub>V w = 0}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. basis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB} = {w. A *\\<^sub>V w = (0::'b)}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. basis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB} = {w. A *\\<^sub>V w = (0::'b)}\n[PROOF STEP]\nhave \"basis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB}\n        = basis_enum_of_vec ` {v \\<in> carrier_vec dA. A *\\<^sub>V basis_enum_of_vec v = 0}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. basis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB} = basis_enum_of_vec ` {v \\<in> carrier_vec dA. A *\\<^sub>V basis_enum_of_vec v = (0::'b)}\n[PROOF STEP]\napply (rule arg_cong[where f=\"\\<lambda>t. basis_enum_of_vec ` t\"])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB} = {v \\<in> carrier_vec dA. A *\\<^sub>V basis_enum_of_vec v = (0::'b)}\n[PROOF STEP]\napply (rule Collect_cong)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>v. (v \\<in> carrier_vec dA \\<and> Am *\\<^sub>v v = 0\\<^sub>v dB) = (v \\<in> carrier_vec dA \\<and> A *\\<^sub>V basis_enum_of_vec v = (0::'b))\n[PROOF STEP]\napply (simp add: Am_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>v. (v \\<in> carrier_vec dA \\<and> mat_of_cblinfun A *\\<^sub>v v = 0\\<^sub>v dB) = (v \\<in> carrier_vec dA \\<and> A *\\<^sub>V basis_enum_of_vec v = (0::'b))\n[PROOF STEP]\nby (metis Am_carrier Am_def carrier_matD(2) carrier_vecD dB_def mat_carrier \n          mat_of_cblinfun_def mat_of_cblinfun_cblinfun_apply vec_of_basis_enum_inverse \n          basis_enum_of_vec_inverse vec_of_basis_enum_zero)\n[PROOF STATE]\nproof (state)\nthis:\nbasis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB} = basis_enum_of_vec ` {v \\<in> carrier_vec dA. A *\\<^sub>V basis_enum_of_vec v = (0::'b)}\n\ngoal (1 subgoal):\n 1. basis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB} = {w. A *\\<^sub>V w = (0::'b)}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nbasis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB} = basis_enum_of_vec ` {v \\<in> carrier_vec dA. A *\\<^sub>V basis_enum_of_vec v = (0::'b)}\n\ngoal (1 subgoal):\n 1. basis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB} = {w. A *\\<^sub>V w = (0::'b)}\n[PROOF STEP]\nhave \"\\<dots> = {w \\<in> basis_enum_of_vec ` carrier_vec dA. A *\\<^sub>V w = 0}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. basis_enum_of_vec ` {v \\<in> carrier_vec dA. A *\\<^sub>V basis_enum_of_vec v = (0::'b)} = {w \\<in> basis_enum_of_vec ` carrier_vec dA. A *\\<^sub>V w = (0::'b)}\n[PROOF STEP]\napply (subst Compr_image_eq[symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {x \\<in> basis_enum_of_vec ` carrier_vec dA. A *\\<^sub>V x = (0::'b)} = {w \\<in> basis_enum_of_vec ` carrier_vec dA. A *\\<^sub>V w = (0::'b)}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nbasis_enum_of_vec ` {v \\<in> carrier_vec dA. A *\\<^sub>V basis_enum_of_vec v = (0::'b)} = {w \\<in> basis_enum_of_vec ` carrier_vec dA. A *\\<^sub>V w = (0::'b)}\n\ngoal (1 subgoal):\n 1. basis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB} = {w. A *\\<^sub>V w = (0::'b)}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nbasis_enum_of_vec ` {v \\<in> carrier_vec dA. A *\\<^sub>V basis_enum_of_vec v = (0::'b)} = {w \\<in> basis_enum_of_vec ` carrier_vec dA. A *\\<^sub>V w = (0::'b)}\n\ngoal (1 subgoal):\n 1. basis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB} = {w. A *\\<^sub>V w = (0::'b)}\n[PROOF STEP]\nhave \"\\<dots> = {w. A *\\<^sub>V w = 0}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {w \\<in> basis_enum_of_vec ` carrier_vec dA. A *\\<^sub>V w = (0::'b)} = {w. A *\\<^sub>V w = (0::'b)}\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>x. A *\\<^sub>V x = (0::'b) \\<Longrightarrow> x \\<in> basis_enum_of_vec ` carrier_vec dA\n[PROOF STEP]\nby (metis (no_types, lifting) Am_carrier Am_def carrier_matD(2) carrier_vec_dim_vec dim_vec_of_basis_enum' image_iff mat_carrier mat_of_cblinfun_def vec_of_basis_enum_inverse)\n[PROOF STATE]\nproof (state)\nthis:\n{w \\<in> basis_enum_of_vec ` carrier_vec dA. A *\\<^sub>V w = (0::'b)} = {w. A *\\<^sub>V w = (0::'b)}\n\ngoal (1 subgoal):\n 1. basis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB} = {w. A *\\<^sub>V w = (0::'b)}\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nbasis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB} = {w. A *\\<^sub>V w = (0::'b)}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nbasis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB} = {w. A *\\<^sub>V w = (0::'b)}\n\ngoal (1 subgoal):\n 1. basis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB} = {w. A *\\<^sub>V w = (0::'b)}\n[PROOF STEP]\nby -\n[PROOF STATE]\nproof (state)\nthis:\nbasis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB} = {w. A *\\<^sub>V w = (0::'b)}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nbasis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB} = {w. A *\\<^sub>V w = (0::'b)}\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nbasis_enum_of_vec ` {v \\<in> carrier_vec dA. Am *\\<^sub>v v = 0\\<^sub>v dB} = {w. A *\\<^sub>V w = (0::'b)}\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nhave \"\\<dots> = space_as_set (kernel A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {w. A *\\<^sub>V w = (0::'b)} = space_as_set (kernel A)\n[PROOF STEP]\napply transfer\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>A. bounded_clinear A \\<Longrightarrow> {w. A w = (0::'b)} = A -` {0::'b}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{w. A *\\<^sub>V w = (0::'b)} = space_as_set (kernel A)\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nspace_as_set (SPAN base) = space_as_set (kernel A)\n[PROOF STEP]\nhave \"SPAN base = kernel A\"\n[PROOF STATE]\nproof (prove)\nusing this:\nspace_as_set (SPAN base) = space_as_set (kernel A)\n\ngoal (1 subgoal):\n 1. SPAN base = kernel A\n[PROOF STEP]\nby (simp add: space_as_set_inject)\n[PROOF STATE]\nproof (state)\nthis:\nSPAN base = kernel A\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nSPAN base = kernel A\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nSPAN base = kernel A\n\ngoal (1 subgoal):\n 1. kernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n[PROOF STEP]\nby (simp add: base_def Ag_def Am_def)\n[PROOF STATE]\nproof (state)\nthis:\nkernel A = SPAN (find_base_vectors (gauss_jordan_single (mat_of_cblinfun A)))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 9269, "file": "Complex_Bounded_Operators_Cblinfun_Code", "length": 88, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9241418116217418, "lm_q2_score": 0.857768108626046, "lm_q1q2_score": 0.7926993738570292}}
{"text": "[STATEMENT]\nlemma L2_set_eq_0_iff: \"finite A \\<Longrightarrow> L2_set f A = 0 \\<longleftrightarrow> (\\<forall>x\\<in>A. f x = 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite A \\<Longrightarrow> (L2_set f A = 0) = (\\<forall>x\\<in>A. f x = 0)\n[PROOF STEP]\nunfolding L2_set_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite A \\<Longrightarrow> (sqrt (\\<Sum>i\\<in>A. (f i)\\<^sup>2) = 0) = (\\<forall>x\\<in>A. f x = 0)\n[PROOF STEP]\nby (simp add: sum_nonneg sum_nonneg_eq_0_iff)", "meta": {"llama_tokens": 227, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9149009457116781, "lm_q2_score": 0.865224091265267, "lm_q1q2_score": 0.7915943393511201}}
{"text": "[STATEMENT]\nlemma Inf_ccsubspace_code[code]: \n  \\<comment> \\<open>Infimum (intersection) of a set of subspaces. \n      Implemented by the orthogonal complement of the supremum.\\<close>\n  \"Inf (Set_Monad l :: 'a::onb_enum ccsubspace set)\n  = - Sup (Set_Monad (map uminus l))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<Sqinter> Set_Monad l = - \\<Squnion> Set_Monad (map uminus l)\n[PROOF STEP]\nunfolding Set_Monad_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<Sqinter> set l = - \\<Squnion> set (map uminus l)\n[PROOF STEP]\napply (induction l)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\<Sqinter> set [] = - \\<Squnion> set (map uminus [])\n 2. \\<And>a l. \\<Sqinter> set l = - \\<Squnion> set (map uminus l) \\<Longrightarrow> \\<Sqinter> set (a # l) = - \\<Squnion> set (map uminus (a # l))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 350, "file": "Complex_Bounded_Operators_Cblinfun_Code", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9449947055100816, "lm_q2_score": 0.8376199633332891, "lm_q1q2_score": 0.7915464305795069}}
{"text": "[STATEMENT]\nlemma sum_swap: \"(\\<Sum>i=0..<(x::nat). \\<Sum>j=0..<(y::nat). f i j) = \n                 (\\<Sum>j=0..<(y::nat). \\<Sum>i=0..<(x::nat). f i j ) \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<x. sum (f i) {0..<y}) = (\\<Sum>j = 0..<y. \\<Sum>i = 0..<x. f i j)\n[PROOF STEP]\nusing Groups_Big.comm_monoid_add_class.sum.swap\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<Sum>i\\<in>?A. sum (?g i) ?B) = (\\<Sum>j\\<in>?B. \\<Sum>i\\<in>?A. ?g i j)\n\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<x. sum (f i) {0..<y}) = (\\<Sum>j = 0..<y. \\<Sum>i = 0..<x. f i j)\n[PROOF STEP]\nby fast", "meta": {"llama_tokens": 313, "file": "Number_Theoretic_Transform_Preliminary_Lemmas", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9433475699138559, "lm_q2_score": 0.8376199673867852, "lm_q1q2_score": 0.7901667607456471}}
{"text": "[STATEMENT]\nlemma main_real:\n  assumes \"b < a\"\n  shows \"valid_countings a b = (a - b) / (a + b) * all_countings a b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (valid_countings a b) = (real a - real b) / (real a + real b) * real (all_countings a b)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nb < a\n\ngoal (1 subgoal):\n 1. real (valid_countings a b) = (real a - real b) / (real a + real b) * real (all_countings a b)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. b < a \\<Longrightarrow> real (valid_countings a b) = (real a - real b) / (real a + real b) * real (all_countings a b)\n[PROOF STEP]\nfrom main_nat[of a b] \\<open>b < a\\<close>\n[PROOF STATE]\nproof (chain)\npicking this:\n(a + b) * valid_countings a b = (a - b) * all_countings a b\nb < a\n[PROOF STEP]\nhave\n    \"(real a + real b) * real (valid_countings a b) = (real a - real b) * real (all_countings a b)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(a + b) * valid_countings a b = (a - b) * all_countings a b\nb < a\n\ngoal (1 subgoal):\n 1. (real a + real b) * real (valid_countings a b) = (real a - real b) * real (all_countings a b)\n[PROOF STEP]\nby (simp only: of_nat_add[symmetric] of_nat_mult[symmetric]) auto\n[PROOF STATE]\nproof (state)\nthis:\n(real a + real b) * real (valid_countings a b) = (real a - real b) * real (all_countings a b)\n\ngoal (1 subgoal):\n 1. b < a \\<Longrightarrow> real (valid_countings a b) = (real a - real b) / (real a + real b) * real (all_countings a b)\n[PROOF STEP]\nfrom this \\<open>b < a\\<close>\n[PROOF STATE]\nproof (chain)\npicking this:\n(real a + real b) * real (valid_countings a b) = (real a - real b) * real (all_countings a b)\nb < a\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(real a + real b) * real (valid_countings a b) = (real a - real b) * real (all_countings a b)\nb < a\n\ngoal (1 subgoal):\n 1. real (valid_countings a b) = (real a - real b) / (real a + real b) * real (all_countings a b)\n[PROOF STEP]\nby (subst mult_left_cancel[of \"real a + real b\", symmetric]) auto\n[PROOF STATE]\nproof (state)\nthis:\nreal (valid_countings a b) = (real a - real b) / (real a + real b) * real (all_countings a b)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 908, "file": null, "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8976953003183443, "lm_q2_score": 0.8791467564270271, "lm_q1q2_score": 0.7892059115346584}}
{"text": "[STATEMENT]\nlemma finite_k_words: \"finite (k_words k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite {w. length w = k \\<and> set w \\<subseteq> L}\n[PROOF STEP]\nproof (induct k)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. finite {w. length w = 0 \\<and> set w \\<subseteq> L}\n 2. \\<And>k. finite {w. length w = k \\<and> set w \\<subseteq> L} \\<Longrightarrow> finite {w. length w = Suc k \\<and> set w \\<subseteq> L}\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. finite {w. length w = 0 \\<and> set w \\<subseteq> L}\n 2. \\<And>k. finite {w. length w = k \\<and> set w \\<subseteq> L} \\<Longrightarrow> finite {w. length w = Suc k \\<and> set w \\<subseteq> L}\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite {w. length w = 0 \\<and> set w \\<subseteq> L}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfinite {w. length w = 0 \\<and> set w \\<subseteq> L}\n\ngoal (1 subgoal):\n 1. \\<And>k. finite {w. length w = k \\<and> set w \\<subseteq> L} \\<Longrightarrow> finite {w. length w = Suc k \\<and> set w \\<subseteq> L}\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\nfinite {w. length w = n \\<and> set w \\<subseteq> L}\n\ngoal (1 subgoal):\n 1. \\<And>k. finite {w. length w = k \\<and> set w \\<subseteq> L} \\<Longrightarrow> finite {w. length w = Suc k \\<and> set w \\<subseteq> L}\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite {w. length w = n \\<and> set w \\<subseteq> L}\n\ngoal (1 subgoal):\n 1. finite {w. length w = Suc n \\<and> set w \\<subseteq> L}\n[PROOF STEP]\nusing bij_k_words bij_betw_finite fin_L\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite {w. length w = n \\<and> set w \\<subseteq> L}\nbij_betw (\\<lambda>wi. fst wi # snd wi) (L \\<times> {w. length w = ?k \\<and> set w \\<subseteq> L}) {w. length w = Suc ?k \\<and> set w \\<subseteq> L}\nbij_betw ?f ?A ?B \\<Longrightarrow> finite ?A = finite ?B\nfinite L\n\ngoal (1 subgoal):\n 1. finite {w. length w = Suc n \\<and> set w \\<subseteq> L}\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nfinite {w. length w = Suc n \\<and> set w \\<subseteq> L}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 875, "file": "Source_Coding_Theorem_Source_Coding_Theorem", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213799730775, "lm_q2_score": 0.8774767778695834, "lm_q1q2_score": 0.7889581314124294}}
{"text": "[STATEMENT]\nlemma tanh_real_altdef: \"tanh (x::real) = (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. tanh x = (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. tanh x = (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))\n[PROOF STEP]\nhave [simp]: \"exp (2 * x) = exp x * exp x\" \"exp (x * 2) = exp x * exp x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exp (2 * x) = exp x * exp x &&& exp (x * 2) = exp x * exp x\n[PROOF STEP]\nby (subst exp_add [symmetric]; simp)+\n[PROOF STATE]\nproof (state)\nthis:\nexp (2 * x) = exp x * exp x\nexp (x * 2) = exp x * exp x\n\ngoal (1 subgoal):\n 1. tanh x = (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))\n[PROOF STEP]\nhave \"tanh x = (2 * exp (-x) * sinh x) / (2 * exp (-x) * cosh x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. tanh x = 2 * exp (- x) * sinh x / (2 * exp (- x) * cosh x)\n[PROOF STEP]\nby (simp add: tanh_def)\n[PROOF STATE]\nproof (state)\nthis:\ntanh x = 2 * exp (- x) * sinh x / (2 * exp (- x) * cosh x)\n\ngoal (1 subgoal):\n 1. tanh x = (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ntanh x = 2 * exp (- x) * sinh x / (2 * exp (- x) * cosh x)\n\ngoal (1 subgoal):\n 1. tanh x = (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))\n[PROOF STEP]\nhave \"2 * exp (-x) * sinh x = 1 - exp (-2*x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * exp (- x) * sinh x = 1 - exp (- 2 * x)\n[PROOF STEP]\nby (simp add: exp_minus field_simps sinh_def)\n[PROOF STATE]\nproof (state)\nthis:\n2 * exp (- x) * sinh x = 1 - exp (- 2 * x)\n\ngoal (1 subgoal):\n 1. tanh x = (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n2 * exp (- x) * sinh x = 1 - exp (- 2 * x)\n\ngoal (1 subgoal):\n 1. tanh x = (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))\n[PROOF STEP]\nhave \"2 * exp (-x) * cosh x = 1 + exp (-2*x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * exp (- x) * cosh x = 1 + exp (- 2 * x)\n[PROOF STEP]\nby (simp add: exp_minus field_simps cosh_def)\n[PROOF STATE]\nproof (state)\nthis:\n2 * exp (- x) * cosh x = 1 + exp (- 2 * x)\n\ngoal (1 subgoal):\n 1. tanh x = (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ntanh x = (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ntanh x = (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))\n\ngoal (1 subgoal):\n 1. tanh x = (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ntanh x = (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1285, "file": null, "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9059898153067649, "lm_q2_score": 0.8705972667296309, "lm_q1q2_score": 0.7887522568909526}}
{"text": "[STATEMENT]\nlemma Rats_solution_QE_converse:\n  assumes \"a \\<in> \\<rat>\" \"b \\<in> \\<rat>\"\n  and \"a*x^2 + b*x + c = 0\"\n  and \"x \\<in> \\<rat>\"\n  shows \"sqrt (discrim a b c) \\<in> \\<rat>\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt (discrim a b c) \\<in> \\<rat>\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sqrt (discrim a b c) \\<in> \\<rat>\n[PROOF STEP]\nfrom assms(3)\n[PROOF STATE]\nproof (chain)\npicking this:\na * x\\<^sup>2 + b * x + c = 0\n[PROOF STEP]\nhave \"discrim a b c = (2*a*x+b)^2\"\n[PROOF STATE]\nproof (prove)\nusing this:\na * x\\<^sup>2 + b * x + c = 0\n\ngoal (1 subgoal):\n 1. discrim a b c = (2 * a * x + b)\\<^sup>2\n[PROOF STEP]\nunfolding discrim_def\n[PROOF STATE]\nproof (prove)\nusing this:\na * x\\<^sup>2 + b * x + c = 0\n\ngoal (1 subgoal):\n 1. b\\<^sup>2 - 4 * a * c = (2 * a * x + b)\\<^sup>2\n[PROOF STEP]\nby algebra\n[PROOF STATE]\nproof (state)\nthis:\ndiscrim a b c = (2 * a * x + b)\\<^sup>2\n\ngoal (1 subgoal):\n 1. sqrt (discrim a b c) \\<in> \\<rat>\n[PROOF STEP]\nhence \"sqrt (discrim a b c) = \\<bar>2*a*x+b\\<bar>\"\n[PROOF STATE]\nproof (prove)\nusing this:\ndiscrim a b c = (2 * a * x + b)\\<^sup>2\n\ngoal (1 subgoal):\n 1. sqrt (discrim a b c) = \\<bar>2 * a * x + b\\<bar>\n[PROOF STEP]\nby (simp)\n[PROOF STATE]\nproof (state)\nthis:\nsqrt (discrim a b c) = \\<bar>2 * a * x + b\\<bar>\n\ngoal (1 subgoal):\n 1. sqrt (discrim a b c) \\<in> \\<rat>\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsqrt (discrim a b c) = \\<bar>2 * a * x + b\\<bar>\n\ngoal (1 subgoal):\n 1. sqrt (discrim a b c) \\<in> \\<rat>\n[PROOF STEP]\nusing \\<open>a \\<in> \\<rat>\\<close> \\<open>b \\<in> \\<rat>\\<close> \\<open>x \\<in> \\<rat>\\<close>\n[PROOF STATE]\nproof (prove)\nusing this:\nsqrt (discrim a b c) = \\<bar>2 * a * x + b\\<bar>\na \\<in> \\<rat>\nb \\<in> \\<rat>\nx \\<in> \\<rat>\n\ngoal (1 subgoal):\n 1. sqrt (discrim a b c) \\<in> \\<rat>\n[PROOF STEP]\nby (simp)\n[PROOF STATE]\nproof (state)\nthis:\nsqrt (discrim a b c) \\<in> \\<rat>\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 961, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942203004186, "lm_q2_score": 0.8856314768368161, "lm_q1q2_score": 0.7884725851439415}}
{"text": "[STATEMENT]\ntheorem catalan_Suc':\n  \"catalan (Suc n) = (catalan n * (2*(2*n+1))) div (n+2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. catalan (Suc n) = catalan n * (2 * (2 * n + 1)) div (n + 2)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. catalan (Suc n) = catalan n * (2 * (2 * n + 1)) div (n + 2)\n[PROOF STEP]\nfrom catalan_Suc_aux[of n]\n[PROOF STATE]\nproof (chain)\npicking this:\n(n + 2) * catalan (Suc n) = 2 * (2 * n + 1) * catalan n\n[PROOF STEP]\nhave \"catalan n * (2*(2*n+1)) = catalan (Suc n) * (n+2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(n + 2) * catalan (Suc n) = 2 * (2 * n + 1) * catalan n\n\ngoal (1 subgoal):\n 1. catalan n * (2 * (2 * n + 1)) = catalan (Suc n) * (n + 2)\n[PROOF STEP]\nby (simp add: algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\ncatalan n * (2 * (2 * n + 1)) = catalan (Suc n) * (n + 2)\n\ngoal (1 subgoal):\n 1. catalan (Suc n) = catalan n * (2 * (2 * n + 1)) div (n + 2)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncatalan n * (2 * (2 * n + 1)) = catalan (Suc n) * (n + 2)\n\ngoal (1 subgoal):\n 1. catalan (Suc n) = catalan n * (2 * (2 * n + 1)) div (n + 2)\n[PROOF STEP]\nhave \"\\<dots> div (n+2) = catalan (Suc n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. catalan (Suc n) * (n + 2) div (n + 2) = catalan (Suc n)\n[PROOF STEP]\nby (simp del: mult_Suc mult_Suc_right)\n[PROOF STATE]\nproof (state)\nthis:\ncatalan (Suc n) * (n + 2) div (n + 2) = catalan (Suc n)\n\ngoal (1 subgoal):\n 1. catalan (Suc n) = catalan n * (2 * (2 * n + 1)) div (n + 2)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncatalan n * (2 * (2 * n + 1)) div (n + 2) = catalan (Suc n)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncatalan n * (2 * (2 * n + 1)) div (n + 2) = catalan (Suc n)\n\ngoal (1 subgoal):\n 1. catalan (Suc n) = catalan n * (2 * (2 * n + 1)) div (n + 2)\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\ncatalan (Suc n) = catalan n * (2 * (2 * n + 1)) div (n + 2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 993, "file": "Catalan_Numbers_Catalan_Numbers", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9207896693699845, "lm_q2_score": 0.8558511469672594, "lm_q1q2_score": 0.7880588946459048}}
{"text": "[STATEMENT]\nlemma cos_multiple: \"cos (n * x) = 2 * cos x * cos ((n - 1) * x) - cos ((n - 2) * x)\"\n  for x :: \"'a :: {banach,real_normed_field}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos (n * x) = (2::'a) * cos x * cos ((n - (1::'a)) * x) - cos ((n - (2::'a)) * x)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. cos (n * x) = (2::'a) * cos x * cos ((n - (1::'a)) * x) - cos ((n - (2::'a)) * x)\n[PROOF STEP]\nhave \"cos ((n - 1) * x + x) + cos ((n - 1) * x - x) = 2 * cos ((n - 1) * x) * cos x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos ((n - (1::'a)) * x + x) + cos ((n - (1::'a)) * x - x) = (2::'a) * cos ((n - (1::'a)) * x) * cos x\n[PROOF STEP]\nby (simp add: cos_add cos_diff)\n[PROOF STATE]\nproof (state)\nthis:\ncos ((n - (1::'a)) * x + x) + cos ((n - (1::'a)) * x - x) = (2::'a) * cos ((n - (1::'a)) * x) * cos x\n\ngoal (1 subgoal):\n 1. cos (n * x) = (2::'a) * cos x * cos ((n - (1::'a)) * x) - cos ((n - (2::'a)) * x)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ncos ((n - (1::'a)) * x + x) + cos ((n - (1::'a)) * x - x) = (2::'a) * cos ((n - (1::'a)) * x) * cos x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncos ((n - (1::'a)) * x + x) + cos ((n - (1::'a)) * x - x) = (2::'a) * cos ((n - (1::'a)) * x) * cos x\n\ngoal (1 subgoal):\n 1. cos (n * x) = (2::'a) * cos x * cos ((n - (1::'a)) * x) - cos ((n - (2::'a)) * x)\n[PROOF STEP]\nby (simp add: left_diff_distrib' eq_diff_eq)\n[PROOF STATE]\nproof (state)\nthis:\ncos (n * x) = (2::'a) * cos x * cos ((n - (1::'a)) * x) - cos ((n - (2::'a)) * x)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 823, "file": "Hyperdual_AnalyticTestFunction", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9032942067038785, "lm_q2_score": 0.8723473796562744, "lm_q1q2_score": 0.7879863342768215}}
{"text": "[STATEMENT]\nlemma matrix_sum_trace_le:\n  fixes f :: \"nat \\<Rightarrow> complex mat\" and g :: \"nat \\<Rightarrow> complex mat\"\n  assumes \"(\\<And>k. k < n \\<Longrightarrow> f k \\<in> carrier_mat d d)\" \n    \"(\\<And>k. k < n \\<Longrightarrow> g k \\<in> carrier_mat d d)\"\n    \"(\\<And>k. k < n \\<Longrightarrow> trace (f k) \\<le> trace (g k))\"\n  shows \"trace (matrix_sum d f n) \\<le> trace (matrix_sum d g n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trace (matrix_sum d f n) \\<le> trace (matrix_sum d g n)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. trace (matrix_sum d f n) \\<le> trace (matrix_sum d g n)\n[PROOF STEP]\nhave \"sum (\\<lambda>k. trace (f k)) {0..<n} \\<le>  sum (\\<lambda>k. trace (g k)) {0..<n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>k = 0..<n. trace (f k)) \\<le> (\\<Sum>k = 0..<n. trace (g k))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n?k < n \\<Longrightarrow> f ?k \\<in> carrier_mat d d\n?k < n \\<Longrightarrow> g ?k \\<in> carrier_mat d d\n?k < n \\<Longrightarrow> trace (f ?k) \\<le> trace (g ?k)\n\ngoal (1 subgoal):\n 1. (\\<Sum>k = 0..<n. trace (f k)) \\<le> (\\<Sum>k = 0..<n. trace (g k))\n[PROOF STEP]\nby (meson atLeastLessThan_iff sum_mono)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>k = 0..<n. trace (f k)) \\<le> (\\<Sum>k = 0..<n. trace (g k))\n\ngoal (1 subgoal):\n 1. trace (matrix_sum d f n) \\<le> trace (matrix_sum d g n)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\<Sum>k = 0..<n. trace (f k)) \\<le> (\\<Sum>k = 0..<n. trace (g k))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<Sum>k = 0..<n. trace (f k)) \\<le> (\\<Sum>k = 0..<n. trace (g k))\n\ngoal (1 subgoal):\n 1. trace (matrix_sum d f n) \\<le> trace (matrix_sum d g n)\n[PROOF STEP]\nusing trace_matrix_sum_linear assms\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<Sum>k = 0..<n. trace (f k)) \\<le> (\\<Sum>k = 0..<n. trace (g k))\n(\\<And>k. k < ?n \\<Longrightarrow> ?f k \\<in> carrier_mat ?d ?d) \\<Longrightarrow> trace (matrix_sum ?d ?f ?n) = (\\<Sum>k = 0..<?n. trace (?f k))\n?k < n \\<Longrightarrow> f ?k \\<in> carrier_mat d d\n?k < n \\<Longrightarrow> g ?k \\<in> carrier_mat d d\n?k < n \\<Longrightarrow> trace (f ?k) \\<le> trace (g ?k)\n\ngoal (1 subgoal):\n 1. trace (matrix_sum d f n) \\<le> trace (matrix_sum d g n)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ntrace (matrix_sum d f n) \\<le> trace (matrix_sum d g n)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1069, "file": "QHLProver_Quantum_Program", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9124361676202372, "lm_q2_score": 0.8633916152464016, "lm_q1q2_score": 0.787789736570873}}
{"text": "[STATEMENT]\ntheorem of_nat_catalan_Suc':\n  \"of_nat (catalan (Suc n)) =\n     (of_nat (2*(2*n+1)) / of_nat (n+2) * of_nat (catalan n) :: 'a :: field_char_0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_nat (catalan (Suc n)) = of_nat (2 * (2 * n + 1)) / of_nat (n + 2) * of_nat (catalan n)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. of_nat (catalan (Suc n)) = of_nat (2 * (2 * n + 1)) / of_nat (n + 2) * of_nat (catalan n)\n[PROOF STEP]\nhave \"(of_nat (2*(2*n+1)) / of_nat (n+2) * of_nat (catalan n) :: 'a) =\n          of_nat (2*(2*n + 1) * catalan n) / of_nat (n+2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_nat (2 * (2 * n + 1)) / of_nat (n + 2) * of_nat (catalan n) = of_nat (2 * (2 * n + 1) * catalan n) / of_nat (n + 2)\n[PROOF STEP]\nby (simp add: divide_simps mult_ac del: mult_Suc mult_Suc_right)\n[PROOF STATE]\nproof (state)\nthis:\nof_nat (2 * (2 * n + 1)) / of_nat (n + 2) * of_nat (catalan n) = of_nat (2 * (2 * n + 1) * catalan n) / of_nat (n + 2)\n\ngoal (1 subgoal):\n 1. of_nat (catalan (Suc n)) = of_nat (2 * (2 * n + 1)) / of_nat (n + 2) * of_nat (catalan n)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nof_nat (2 * (2 * n + 1)) / of_nat (n + 2) * of_nat (catalan n) = of_nat (2 * (2 * n + 1) * catalan n) / of_nat (n + 2)\n\ngoal (1 subgoal):\n 1. of_nat (catalan (Suc n)) = of_nat (2 * (2 * n + 1)) / of_nat (n + 2) * of_nat (catalan n)\n[PROOF STEP]\nnote catalan_Suc_aux[of n, symmetric]\n[PROOF STATE]\nproof (state)\nthis:\n2 * (2 * n + 1) * catalan n = (n + 2) * catalan (Suc n)\n\ngoal (1 subgoal):\n 1. of_nat (catalan (Suc n)) = of_nat (2 * (2 * n + 1)) / of_nat (n + 2) * of_nat (catalan n)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n2 * (2 * n + 1) * catalan n = (n + 2) * catalan (Suc n)\n\ngoal (1 subgoal):\n 1. of_nat (catalan (Suc n)) = of_nat (2 * (2 * n + 1)) / of_nat (n + 2) * of_nat (catalan n)\n[PROOF STEP]\nhave \"of_nat ((n + 2) * catalan (Suc n)) / of_nat (n + 2) = (of_nat (catalan (Suc n)) :: 'a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_nat ((n + 2) * catalan (Suc n)) / of_nat (n + 2) = of_nat (catalan (Suc n))\n[PROOF STEP]\nby (simp del: of_nat_Suc mult_Suc_right mult_Suc)\n[PROOF STATE]\nproof (state)\nthis:\nof_nat ((n + 2) * catalan (Suc n)) / of_nat (n + 2) = of_nat (catalan (Suc n))\n\ngoal (1 subgoal):\n 1. of_nat (catalan (Suc n)) = of_nat (2 * (2 * n + 1)) / of_nat (n + 2) * of_nat (catalan n)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nof_nat (2 * (2 * n + 1)) / of_nat (n + 2) * of_nat (catalan n) = of_nat (catalan (Suc n))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nof_nat (2 * (2 * n + 1)) / of_nat (n + 2) * of_nat (catalan n) = of_nat (catalan (Suc n))\n\ngoal (1 subgoal):\n 1. of_nat (catalan (Suc n)) = of_nat (2 * (2 * n + 1)) / of_nat (n + 2) * of_nat (catalan n)\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\nof_nat (catalan (Suc n)) = of_nat (2 * (2 * n + 1)) / of_nat (n + 2) * of_nat (catalan n)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1496, "file": "Catalan_Numbers_Catalan_Numbers", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9304582593509315, "lm_q2_score": 0.8459424334245617, "lm_q1q2_score": 0.7871141241153089}}
{"text": "[STATEMENT]\ntheorem pi_series: \"pi / 4 = (\\<Sum>k. (-1)^k * 1 / real (k * 2 + 1))\"\n  (is \"_ = ?SUM\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pi / 4 = (\\<Sum>k. (- 1) ^ k * 1 / real (k * 2 + 1))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. pi / 4 = (\\<Sum>k. (- 1) ^ k * 1 / real (k * 2 + 1))\n[PROOF STEP]\nhave \"pi / 4 = arctan 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pi / 4 = arctan 1\n[PROOF STEP]\nusing arctan_one\n[PROOF STATE]\nproof (prove)\nusing this:\narctan 1 = pi / 4\n\ngoal (1 subgoal):\n 1. pi / 4 = arctan 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\npi / 4 = arctan 1\n\ngoal (1 subgoal):\n 1. pi / 4 = (\\<Sum>k. (- 1) ^ k * 1 / real (k * 2 + 1))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\npi / 4 = arctan 1\n\ngoal (1 subgoal):\n 1. pi / 4 = (\\<Sum>k. (- 1) ^ k * 1 / real (k * 2 + 1))\n[PROOF STEP]\nhave \"\\<dots> = ?SUM\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. arctan 1 = (\\<Sum>k. (- 1) ^ k * 1 / real (k * 2 + 1))\n[PROOF STEP]\nusing arctan_series[of 1]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<bar>1\\<bar> \\<le> 1 \\<Longrightarrow> arctan 1 = (\\<Sum>k. (- 1) ^ k * (1 / real (k * 2 + 1) * 1 ^ (k * 2 + 1)))\n\ngoal (1 subgoal):\n 1. arctan 1 = (\\<Sum>k. (- 1) ^ k * 1 / real (k * 2 + 1))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\narctan 1 = (\\<Sum>k. (- 1) ^ k * 1 / real (k * 2 + 1))\n\ngoal (1 subgoal):\n 1. pi / 4 = (\\<Sum>k. (- 1) ^ k * 1 / real (k * 2 + 1))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\npi / 4 = (\\<Sum>k. (- 1) ^ k * 1 / real (k * 2 + 1))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\npi / 4 = (\\<Sum>k. (- 1) ^ k * 1 / real (k * 2 + 1))\n\ngoal (1 subgoal):\n 1. pi / 4 = (\\<Sum>k. (- 1) ^ k * 1 / real (k * 2 + 1))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\npi / 4 = (\\<Sum>k. (- 1) ^ k * 1 / real (k * 2 + 1))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 994, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9161096204605946, "lm_q2_score": 0.8577681049901037, "lm_q1q2_score": 0.7858096131056874}}
{"text": "[STATEMENT]\nlemma card_cartesian_product_6: \"card (A \\<times> A \\<times> A \\<times> A \\<times> A \\<times> A) = (card A) ^ 6\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (A \\<times> A \\<times> A \\<times> A \\<times> A \\<times> A) = card A ^ 6\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (A \\<times> A \\<times> A \\<times> A \\<times> A \\<times> A) = card A ^ 6\n[PROOF STEP]\nhave \"card (A \\<times> A \\<times> A \\<times> A \\<times> A \\<times> A) = \n    card A * card A * card A * card A * card A * card A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (A \\<times> A \\<times> A \\<times> A \\<times> A \\<times> A) = card A * card A * card A * card A * card A * card A\n[PROOF STEP]\nusing card_cartesian_product mult.commute\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (?A \\<times> ?B) = card ?A * card ?B\n?a * ?b = ?b * ?a\n\ngoal (1 subgoal):\n 1. card (A \\<times> A \\<times> A \\<times> A \\<times> A \\<times> A) = card A * card A * card A * card A * card A * card A\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\ncard (A \\<times> A \\<times> A \\<times> A \\<times> A \\<times> A) = card A * card A * card A * card A * card A * card A\n\ngoal (1 subgoal):\n 1. card (A \\<times> A \\<times> A \\<times> A \\<times> A \\<times> A) = card A ^ 6\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (A \\<times> A \\<times> A \\<times> A \\<times> A \\<times> A) = card A * card A * card A * card A * card A * card A\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (A \\<times> A \\<times> A \\<times> A \\<times> A \\<times> A) = card A * card A * card A * card A * card A * card A\n\ngoal (1 subgoal):\n 1. card (A \\<times> A \\<times> A \\<times> A \\<times> A \\<times> A) = card A ^ 6\n[PROOF STEP]\nby algebra\n[PROOF STATE]\nproof (state)\nthis:\ncard (A \\<times> A \\<times> A \\<times> A \\<times> A \\<times> A) = card A ^ 6\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 785, "file": "Balog_Szemeredi_Gowers_Miscellaneous_Lemmas", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213691605412, "lm_q2_score": 0.8723473713594992, "lm_q1q2_score": 0.7843461629203521}}
{"text": "[STATEMENT]\nlemma Euclid_GCD: \"VARS a b\n {0<A & 0<B}\n a := A; b := B;\n WHILE  a \\<noteq> b\n INV {0<a & 0<b & gcd A B = gcd a b}\n DO IF a<b THEN b := b-a ELSE a := a-b FI OD\n {a = gcd A B}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {0 < A \\<and> 0 < B}\na := A; b := B; WHILE a \\<noteq> b INV {0 < a \\<and> 0 < b \\<and> Arith2.gcd A B = Arith2.gcd a b}  VAR {0} \nDO IF a < b THEN b := b - a  ELSE a := a - b FI OD\n{a = Arith2.gcd A B}\n[PROOF STEP]\napply vcg\n  \\<comment> \\<open>Now prove the verification conditions\\<close>\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\<And>a b. 0 < A \\<and> 0 < B \\<Longrightarrow> 0 < A \\<and> 0 < B \\<and> Arith2.gcd A B = Arith2.gcd A B\n 2. \\<And>a b. (0 < a \\<and> 0 < b \\<and> Arith2.gcd A B = Arith2.gcd a b) \\<and> a \\<noteq> b \\<Longrightarrow> (a < b \\<longrightarrow> 0 < a \\<and> 0 < b - a \\<and> Arith2.gcd A B = Arith2.gcd a (b - a)) \\<and> (\\<not> a < b \\<longrightarrow> 0 < a - b \\<and> 0 < b \\<and> Arith2.gcd A B = Arith2.gcd (a - b) b)\n 3. \\<And>a b. (0 < a \\<and> 0 < b \\<and> Arith2.gcd A B = Arith2.gcd a b) \\<and> \\<not> a \\<noteq> b \\<Longrightarrow> a = Arith2.gcd A B\n[PROOF STEP]\napply auto\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\<And>a b. \\<lbrakk>0 < a; Arith2.gcd A B = Arith2.gcd a b; a < b\\<rbrakk> \\<Longrightarrow> Arith2.gcd a b = Arith2.gcd a (b - a)\n 2. \\<And>a b. \\<lbrakk>0 < b; Arith2.gcd A B = Arith2.gcd a b; a \\<noteq> b; \\<not> a < b\\<rbrakk> \\<Longrightarrow> Arith2.gcd a b = Arith2.gcd (a - b) b\n 3. \\<And>a. \\<lbrakk>0 < a; Arith2.gcd A B = Arith2.gcd a a\\<rbrakk> \\<Longrightarrow> a = Arith2.gcd a a\n[PROOF STEP]\napply(simp add: gcd_diff_r less_imp_le)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\<And>a b. \\<lbrakk>0 < b; Arith2.gcd A B = Arith2.gcd a b; a \\<noteq> b; \\<not> a < b\\<rbrakk> \\<Longrightarrow> Arith2.gcd a b = Arith2.gcd (a - b) b\n 2. \\<And>a. \\<lbrakk>0 < a; Arith2.gcd A B = Arith2.gcd a a\\<rbrakk> \\<Longrightarrow> a = Arith2.gcd a a\n[PROOF STEP]\napply(simp add: linorder_not_less gcd_diff_l)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>a. \\<lbrakk>0 < a; Arith2.gcd A B = Arith2.gcd a a\\<rbrakk> \\<Longrightarrow> a = Arith2.gcd a a\n[PROOF STEP]\napply(erule gcd_nnn)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1116, "file": null, "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.918480252950991, "lm_q2_score": 0.8539127492339909, "lm_q1q2_score": 0.784301997914512}}
{"text": "[STATEMENT]\nlemma col_space_eq_row_space_transpose:\n  fixes A::\"'a::{field}^'n^'m\"\n  shows \"col_space A = row_space (transpose A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. col_space A = row_space (Finite_Cartesian_Product.transpose A)\n[PROOF STEP]\nunfolding col_space_def row_space_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.span (columns A) = vec.span (rows (Finite_Cartesian_Product.transpose A))\n[PROOF STEP]\nunfolding rows_transpose[of A]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.span (columns A) = vec.span (columns A)\n[PROOF STEP]\n..", "meta": {"llama_tokens": 240, "file": "Rank_Nullity_Theorem_Fundamental_Subspaces", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9073122163480667, "lm_q2_score": 0.8633916047011595, "lm_q1q2_score": 0.7833657504377229}}
{"text": "[STATEMENT]\nlemma Cauchy_product:\n  fixes a b :: \"nat \\<Rightarrow> 'a::{real_normed_algebra,banach}\"\n  assumes \"summable (\\<lambda>k. norm (a k))\"\n    and \"summable (\\<lambda>k. norm (b k))\"\n  shows \"(\\<Sum>k. a k) * (\\<Sum>k. b k) = (\\<Sum>k. \\<Sum>i\\<le>k. a i * b (k - i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>k. a k) * (\\<Sum>k. b k) = (\\<Sum>k. \\<Sum>i\\<le>k. a i * b (k - i))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nsummable (\\<lambda>k. norm (a k))\nsummable (\\<lambda>k. norm (b k))\n\ngoal (1 subgoal):\n 1. (\\<Sum>k. a k) * (\\<Sum>k. b k) = (\\<Sum>k. \\<Sum>i\\<le>k. a i * b (k - i))\n[PROOF STEP]\nby (rule Cauchy_product_sums [THEN sums_unique])", "meta": {"llama_tokens": 322, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9324533144915912, "lm_q2_score": 0.83973396967765, "lm_q1q2_score": 0.7830127233171061}}
{"text": "[STATEMENT]\nlemma odd_div2_plus_1_eq: \"(n::nat) mod 2 \\<noteq> 0 \\<Longrightarrow> (n div 2) + 1 = (n+1) div 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n mod 2 \\<noteq> 0 \\<Longrightarrow> n div 2 + 1 = (n + 1) div 2\n[PROOF STEP]\nproof (induct n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. 0 mod 2 \\<noteq> 0 \\<Longrightarrow> 0 div 2 + 1 = (0 + 1) div 2\n 2. \\<And>n. \\<lbrakk>n mod 2 \\<noteq> 0 \\<Longrightarrow> n div 2 + 1 = (n + 1) div 2; Suc n mod 2 \\<noteq> 0\\<rbrakk> \\<Longrightarrow> Suc n div 2 + 1 = (Suc n + 1) div 2\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n0 mod 2 \\<noteq> 0\n\ngoal (2 subgoals):\n 1. 0 mod 2 \\<noteq> 0 \\<Longrightarrow> 0 div 2 + 1 = (0 + 1) div 2\n 2. \\<And>n. \\<lbrakk>n mod 2 \\<noteq> 0 \\<Longrightarrow> n div 2 + 1 = (n + 1) div 2; Suc n mod 2 \\<noteq> 0\\<rbrakk> \\<Longrightarrow> Suc n div 2 + 1 = (Suc n + 1) div 2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n0 mod 2 \\<noteq> 0\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n0 mod 2 \\<noteq> 0\n\ngoal (1 subgoal):\n 1. 0 div 2 + 1 = (0 + 1) div 2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 div 2 + 1 = (0 + 1) div 2\n\ngoal (1 subgoal):\n 1. \\<And>n. \\<lbrakk>n mod 2 \\<noteq> 0 \\<Longrightarrow> n div 2 + 1 = (n + 1) div 2; Suc n mod 2 \\<noteq> 0\\<rbrakk> \\<Longrightarrow> Suc n div 2 + 1 = (Suc n + 1) div 2\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>n. \\<lbrakk>n mod 2 \\<noteq> 0 \\<Longrightarrow> n div 2 + 1 = (n + 1) div 2; Suc n mod 2 \\<noteq> 0\\<rbrakk> \\<Longrightarrow> Suc n div 2 + 1 = (Suc n + 1) div 2\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\nn mod 2 \\<noteq> 0 \\<Longrightarrow> n div 2 + 1 = (n + 1) div 2\nSuc n mod 2 \\<noteq> 0\n\ngoal (1 subgoal):\n 1. \\<And>n. \\<lbrakk>n mod 2 \\<noteq> 0 \\<Longrightarrow> n div 2 + 1 = (n + 1) div 2; Suc n mod 2 \\<noteq> 0\\<rbrakk> \\<Longrightarrow> Suc n div 2 + 1 = (Suc n + 1) div 2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nn mod 2 \\<noteq> 0 \\<Longrightarrow> n div 2 + 1 = (n + 1) div 2\nSuc n mod 2 \\<noteq> 0\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nn mod 2 \\<noteq> 0 \\<Longrightarrow> n div 2 + 1 = (n + 1) div 2\nSuc n mod 2 \\<noteq> 0\n\ngoal (1 subgoal):\n 1. Suc n div 2 + 1 = (Suc n + 1) div 2\n[PROOF STEP]\nby arith\n[PROOF STATE]\nproof (state)\nthis:\nSuc n div 2 + 1 = (Suc n + 1) div 2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1209, "file": "Universal_Turing_Machine_Turing", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392817460332, "lm_q2_score": 0.8856314632529871, "lm_q1q2_score": 0.7829330026658592}}
{"text": "[STATEMENT]\nlemma tanh_altdef:\n  \"tanh x = (exp x - exp (-x)) / (exp x + exp (-x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. tanh x = (exp x - exp (- x)) / (exp x + exp (- x))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. tanh x = (exp x - exp (- x)) / (exp x + exp (- x))\n[PROOF STEP]\nhave \"tanh x = (2 *\\<^sub>R sinh x) / (2 *\\<^sub>R cosh x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. tanh x = 2 *\\<^sub>R sinh x / 2 *\\<^sub>R cosh x\n[PROOF STEP]\nby (simp add: tanh_def scaleR_conv_of_real)\n[PROOF STATE]\nproof (state)\nthis:\ntanh x = 2 *\\<^sub>R sinh x / 2 *\\<^sub>R cosh x\n\ngoal (1 subgoal):\n 1. tanh x = (exp x - exp (- x)) / (exp x + exp (- x))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ntanh x = 2 *\\<^sub>R sinh x / 2 *\\<^sub>R cosh x\n\ngoal (1 subgoal):\n 1. tanh x = (exp x - exp (- x)) / (exp x + exp (- x))\n[PROOF STEP]\nhave \"2 *\\<^sub>R sinh x = exp x - exp (-x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 *\\<^sub>R sinh x = exp x - exp (- x)\n[PROOF STEP]\nby (simp add: sinh_def)\n[PROOF STATE]\nproof (state)\nthis:\n2 *\\<^sub>R sinh x = exp x - exp (- x)\n\ngoal (1 subgoal):\n 1. tanh x = (exp x - exp (- x)) / (exp x + exp (- x))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n2 *\\<^sub>R sinh x = exp x - exp (- x)\n\ngoal (1 subgoal):\n 1. tanh x = (exp x - exp (- x)) / (exp x + exp (- x))\n[PROOF STEP]\nhave \"2 *\\<^sub>R cosh x = exp x + exp (-x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 *\\<^sub>R cosh x = exp x + exp (- x)\n[PROOF STEP]\nby (simp add: cosh_def)\n[PROOF STATE]\nproof (state)\nthis:\n2 *\\<^sub>R cosh x = exp x + exp (- x)\n\ngoal (1 subgoal):\n 1. tanh x = (exp x - exp (- x)) / (exp x + exp (- x))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ntanh x = (exp x - exp (- x)) / (exp x + exp (- x))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ntanh x = (exp x - exp (- x)) / (exp x + exp (- x))\n\ngoal (1 subgoal):\n 1. tanh x = (exp x - exp (- x)) / (exp x + exp (- x))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ntanh x = (exp x - exp (- x)) / (exp x + exp (- x))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 997, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.885631470799559, "lm_q2_score": 0.8840392695254319, "lm_q1q2_score": 0.7829329985143759}}
{"text": "[STATEMENT]\ntheorem of_nat_catalan_closed_form:\n  \"of_nat (catalan n) = (of_nat ((2*n) choose n) / of_nat (Suc n) :: 'a :: field_char_0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_nat (catalan n) = of_nat (2 * n choose n) / of_nat (Suc n)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. of_nat (catalan n) = of_nat (2 * n choose n) / of_nat (Suc n)\n[PROOF STEP]\nhave \"of_nat (catalan n * Suc n) = of_nat ((2*n) choose n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_nat (catalan n * Suc n) = of_nat (2 * n choose n)\n[PROOF STEP]\nby (subst catalan_closed_form_aux) (rule refl)\n[PROOF STATE]\nproof (state)\nthis:\nof_nat (catalan n * Suc n) = of_nat (2 * n choose n)\n\ngoal (1 subgoal):\n 1. of_nat (catalan n) = of_nat (2 * n choose n) / of_nat (Suc n)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nof_nat (catalan n * Suc n) = of_nat (2 * n choose n)\n\ngoal (1 subgoal):\n 1. of_nat (catalan n) = of_nat (2 * n choose n) / of_nat (Suc n)\n[PROOF STEP]\nhave \"of_nat (catalan n * Suc n) = of_nat (catalan n) * of_nat (Suc n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_nat (catalan n * Suc n) = of_nat (catalan n) * of_nat (Suc n)\n[PROOF STEP]\nby (simp only: of_nat_mult)\n[PROOF STATE]\nproof (state)\nthis:\nof_nat (catalan n * Suc n) = of_nat (catalan n) * of_nat (Suc n)\n\ngoal (1 subgoal):\n 1. of_nat (catalan n) = of_nat (2 * n choose n) / of_nat (Suc n)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nof_nat (catalan n) * of_nat (Suc n) = of_nat (2 * n choose n)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nof_nat (catalan n) * of_nat (Suc n) = of_nat (2 * n choose n)\n\ngoal (1 subgoal):\n 1. of_nat (catalan n) = of_nat (2 * n choose n) / of_nat (Suc n)\n[PROOF STEP]\nby (simp add: divide_simps del: of_nat_Suc)\n[PROOF STATE]\nproof (state)\nthis:\nof_nat (catalan n) = of_nat (2 * n choose n) / of_nat (Suc n)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 898, "file": "Catalan_Numbers_Catalan_Numbers", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9086178994073576, "lm_q2_score": 0.8615382147637196, "lm_q1q2_score": 0.7828090429577759}}
{"text": "[STATEMENT]\nlemma inverse_matrix_code[code_unfold]:\nfixes A::\"'a::{field}^'n::{mod_type}^'n::{mod_type}\"\nshows \"inverse_matrix A = (let GJ = Gauss_Jordan_PA A;\n                                rank_A = (if A = 0 then 0 else to_nat (GREATEST a. row a (snd GJ) \\<noteq> 0) + 1) in \n                                if nrows A = rank_A then Some (fst(GJ)) else None)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse_matrix A = (let GJ = Gauss_Jordan_PA A; rank_A = if A = 0 then 0 else mod_type_class.to_nat (GREATEST a. row a (snd GJ) \\<noteq> 0) + 1 in if nrows A = rank_A then Some (fst GJ) else None)\n[PROOF STEP]\nunfolding inverse_matrix\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (if invertible A then Some (P_Gauss_Jordan A) else None) = (let GJ = Gauss_Jordan_PA A; rank_A = if A = 0 then 0 else mod_type_class.to_nat (GREATEST a. row a (snd GJ) \\<noteq> 0) + 1 in if nrows A = rank_A then Some (fst GJ) else None)\n[PROOF STEP]\nunfolding invertible_eq_full_rank\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (if rank A = nrows A then Some (P_Gauss_Jordan A) else None) = (let GJ = Gauss_Jordan_PA A; rank_A = if A = 0 then 0 else mod_type_class.to_nat (GREATEST a. row a (snd GJ) \\<noteq> 0) + 1 in if nrows A = rank_A then Some (fst GJ) else None)\n[PROOF STEP]\nunfolding rank_Gauss_Jordan_code\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (if (if A = 0 then 0 else let A' = Gauss_Jordan A in mod_type_class.to_nat (GREATEST a. row a A' \\<noteq> 0) + 1) = nrows A then Some (P_Gauss_Jordan A) else None) = (let GJ = Gauss_Jordan_PA A; rank_A = if A = 0 then 0 else mod_type_class.to_nat (GREATEST a. row a (snd GJ) \\<noteq> 0) + 1 in if nrows A = rank_A then Some (fst GJ) else None)\n[PROOF STEP]\nunfolding P_Gauss_Jordan_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (if (if A = 0 then 0 else let A' = Gauss_Jordan A in mod_type_class.to_nat (GREATEST a. row a A' \\<noteq> 0) + 1) = nrows A then Some (fst (Gauss_Jordan_PA A)) else None) = (let GJ = Gauss_Jordan_PA A; rank_A = if A = 0 then 0 else mod_type_class.to_nat (GREATEST a. row a (snd GJ) \\<noteq> 0) + 1 in if nrows A = rank_A then Some (fst GJ) else None)\n[PROOF STEP]\nunfolding Let_def Gauss_Jordan_PA_eq\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (if (if A = 0 then 0 else mod_type_class.to_nat (GREATEST a. row a (Gauss_Jordan A) \\<noteq> 0) + 1) = nrows A then Some (fst (Gauss_Jordan_PA A)) else None) = (if nrows A = (if A = 0 then 0 else mod_type_class.to_nat (GREATEST a. row a (Gauss_Jordan A) \\<noteq> 0) + 1) then Some (fst (Gauss_Jordan_PA A)) else None)\n[PROOF STEP]\nby presburger", "meta": {"llama_tokens": 1155, "file": "Gauss_Jordan_Inverse", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9252299488452012, "lm_q2_score": 0.8459424334245617, "lm_q1q2_score": 0.7826912744033921}}
{"text": "[STATEMENT]\nlemma C_msort_log: \"length xs = 2^k \\<Longrightarrow> C_msort xs \\<le> length xs * log 2 (length xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length xs = 2 ^ k \\<Longrightarrow> real (C_msort xs) \\<le> real (length xs) * log 2 (real (length xs))\n[PROOF STEP]\nusing C_msort_le[of xs k]\n[PROOF STATE]\nproof (prove)\nusing this:\nlength xs = 2 ^ k \\<Longrightarrow> C_msort xs \\<le> k * 2 ^ k\n\ngoal (1 subgoal):\n 1. length xs = 2 ^ k \\<Longrightarrow> real (C_msort xs) \\<le> real (length xs) * log 2 (real (length xs))\n[PROOF STEP]\napply (simp add: log_nat_power algebra_simps)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>length xs = 2 ^ k; C_msort xs \\<le> k * 2 ^ k\\<rbrakk> \\<Longrightarrow> real (C_msort xs) \\<le> real k * 2 ^ k\n[PROOF STEP]\nby (metis (mono_tags) numeral_power_eq_of_nat_cancel_iff of_nat_le_iff of_nat_mult)", "meta": {"llama_tokens": 360, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9273632976542184, "lm_q2_score": 0.8438951025545426, "lm_q1q2_score": 0.7825973451792254}}
{"text": "[STATEMENT]\nlemma card_ordered_pairs:\n  fixes M :: \"('a ::linorder) set\" \n  assumes \"finite M\"\n  shows \"2 * card {(x,y) \\<in> M \\<times> M. x < y} = card M * (card M - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n[PROOF STEP]\nhave a: \"finite (M \\<times> M)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite (M \\<times> M)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite M\n\ngoal (1 subgoal):\n 1. finite (M \\<times> M)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nfinite (M \\<times> M)\n\ngoal (1 subgoal):\n 1. 2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n[PROOF STEP]\nhave inj_swap: \"inj (\\<lambda>x. (snd x, fst x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inj (\\<lambda>x. (snd x, fst x))\n[PROOF STEP]\nby (rule inj_onI, simp add: prod_eq_iff)\n[PROOF STATE]\nproof (state)\nthis:\ninj (\\<lambda>x. (snd x, fst x))\n\ngoal (1 subgoal):\n 1. 2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n[PROOF STEP]\nhave \"2 * card {(x,y) \\<in> M \\<times> M. x < y} =\n    card {(x,y) \\<in> M \\<times> M. x < y} + card ((\\<lambda>x. (snd x, fst x))`{(x,y) \\<in> M \\<times> M. x < y})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} + card ((\\<lambda>x. (snd x, fst x)) ` {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y})\n[PROOF STEP]\nby (simp add: card_image[OF inj_on_subset[OF inj_swap]])\n[PROOF STATE]\nproof (state)\nthis:\n2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} + card ((\\<lambda>x. (snd x, fst x)) ` {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y})\n\ngoal (1 subgoal):\n 1. 2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} + card ((\\<lambda>x. (snd x, fst x)) ` {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y})\n\ngoal (1 subgoal):\n 1. 2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n[PROOF STEP]\nhave \"... = card {(x,y) \\<in> M \\<times> M. x < y} + card {(x,y) \\<in> M \\<times> M. y < x}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} + card ((\\<lambda>x. (snd x, fst x)) ` {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y}) = card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} + card {(x, y). (x, y) \\<in> M \\<times> M \\<and> y < x}\n[PROOF STEP]\nby (auto intro: arg_cong[where f=\"card\"] simp add:set_eq_iff image_iff)\n[PROOF STATE]\nproof (state)\nthis:\ncard {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} + card ((\\<lambda>x. (snd x, fst x)) ` {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y}) = card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} + card {(x, y). (x, y) \\<in> M \\<times> M \\<and> y < x}\n\ngoal (1 subgoal):\n 1. 2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} + card ((\\<lambda>x. (snd x, fst x)) ` {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y}) = card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} + card {(x, y). (x, y) \\<in> M \\<times> M \\<and> y < x}\n\ngoal (1 subgoal):\n 1. 2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n[PROOF STEP]\nhave \"... = card ({(x,y) \\<in> M \\<times> M. x < y} \\<union> {(x,y) \\<in> M \\<times> M. y < x})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} + card {(x, y). (x, y) \\<in> M \\<times> M \\<and> y < x} = card ({(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} \\<union> {(x, y). (x, y) \\<in> M \\<times> M \\<and> y < x})\n[PROOF STEP]\nby (intro card_Un_disjoint[symmetric] a finite_subset[where B=\"M \\<times> M\"] subsetI) auto\n[PROOF STATE]\nproof (state)\nthis:\ncard {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} + card {(x, y). (x, y) \\<in> M \\<times> M \\<and> y < x} = card ({(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} \\<union> {(x, y). (x, y) \\<in> M \\<times> M \\<and> y < x})\n\ngoal (1 subgoal):\n 1. 2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} + card {(x, y). (x, y) \\<in> M \\<times> M \\<and> y < x} = card ({(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} \\<union> {(x, y). (x, y) \\<in> M \\<times> M \\<and> y < x})\n\ngoal (1 subgoal):\n 1. 2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n[PROOF STEP]\nhave \"... = card ((M \\<times> M) - {(x,y) \\<in> M \\<times> M. x = y})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ({(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} \\<union> {(x, y). (x, y) \\<in> M \\<times> M \\<and> y < x}) = card (M \\<times> M - {(x, y). (x, y) \\<in> M \\<times> M \\<and> x = y})\n[PROOF STEP]\nby (auto intro: arg_cong[where f=\"card\"] simp add:set_eq_iff)\n[PROOF STATE]\nproof (state)\nthis:\ncard ({(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} \\<union> {(x, y). (x, y) \\<in> M \\<times> M \\<and> y < x}) = card (M \\<times> M - {(x, y). (x, y) \\<in> M \\<times> M \\<and> x = y})\n\ngoal (1 subgoal):\n 1. 2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ({(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} \\<union> {(x, y). (x, y) \\<in> M \\<times> M \\<and> y < x}) = card (M \\<times> M - {(x, y). (x, y) \\<in> M \\<times> M \\<and> x = y})\n\ngoal (1 subgoal):\n 1. 2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n[PROOF STEP]\nhave \"... = card (M \\<times> M) - card {(x,y) \\<in> M \\<times> M. x = y}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (M \\<times> M - {(x, y). (x, y) \\<in> M \\<times> M \\<and> x = y}) = card (M \\<times> M) - card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x = y}\n[PROOF STEP]\nby (intro card_Diff_subset a finite_subset[where B=\"M \\<times> M\"] subsetI) auto\n[PROOF STATE]\nproof (state)\nthis:\ncard (M \\<times> M - {(x, y). (x, y) \\<in> M \\<times> M \\<and> x = y}) = card (M \\<times> M) - card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x = y}\n\ngoal (1 subgoal):\n 1. 2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (M \\<times> M - {(x, y). (x, y) \\<in> M \\<times> M \\<and> x = y}) = card (M \\<times> M) - card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x = y}\n\ngoal (1 subgoal):\n 1. 2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n[PROOF STEP]\nhave \"... = card M ^ 2 - card ((\\<lambda>x. (x,x)) ` M)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (M \\<times> M) - card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x = y} = (card M)\\<^sup>2 - card ((\\<lambda>x. (x, x)) ` M)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite M\n\ngoal (1 subgoal):\n 1. card (M \\<times> M) - card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x = y} = (card M)\\<^sup>2 - card ((\\<lambda>x. (x, x)) ` M)\n[PROOF STEP]\nby (intro arg_cong2[where f=\"(-)\"] arg_cong[where f=\"card\"])\n      (auto simp:power2_eq_square set_eq_iff image_iff)\n[PROOF STATE]\nproof (state)\nthis:\ncard (M \\<times> M) - card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x = y} = (card M)\\<^sup>2 - card ((\\<lambda>x. (x, x)) ` M)\n\ngoal (1 subgoal):\n 1. 2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (M \\<times> M) - card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x = y} = (card M)\\<^sup>2 - card ((\\<lambda>x. (x, x)) ` M)\n\ngoal (1 subgoal):\n 1. 2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n[PROOF STEP]\nhave \"... = card M ^ 2 - card M\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (card M)\\<^sup>2 - card ((\\<lambda>x. (x, x)) ` M) = (card M)\\<^sup>2 - card M\n[PROOF STEP]\nby (intro arg_cong2[where f=\"(-)\"] card_image inj_onI, auto)\n[PROOF STATE]\nproof (state)\nthis:\n(card M)\\<^sup>2 - card ((\\<lambda>x. (x, x)) ` M) = (card M)\\<^sup>2 - card M\n\ngoal (1 subgoal):\n 1. 2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(card M)\\<^sup>2 - card ((\\<lambda>x. (x, x)) ` M) = (card M)\\<^sup>2 - card M\n\ngoal (1 subgoal):\n 1. 2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n[PROOF STEP]\nhave \"... = card M * (card M - 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (card M)\\<^sup>2 - card M = card M * (card M - 1)\n[PROOF STEP]\nby (cases \"card M \\<ge> 0\", auto simp:power2_eq_square algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(card M)\\<^sup>2 - card M = card M * (card M - 1)\n\ngoal (1 subgoal):\n 1. 2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n\ngoal (1 subgoal):\n 1. 2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n2 * card {(x, y). (x, y) \\<in> M \\<times> M \\<and> x < y} = card M * (card M - 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 4789, "file": "Frequency_Moments_Frequency_Moments_Preliminary_Results", "length": 34, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9073122238669025, "lm_q2_score": 0.8615382058759128, "lm_q1q2_score": 0.7816841455195757}}
{"text": "[STATEMENT]\nlemma Cauchy_product:\n  fixes a b :: \"nat \\<Rightarrow> 'a::{real_normed_algebra,banach}\"\n  assumes \"summable (\\<lambda>k. norm (a k))\"\n    and \"summable (\\<lambda>k. norm (b k))\"\n  shows \"(\\<Sum>k. a k) * (\\<Sum>k. b k) = (\\<Sum>k. \\<Sum>i\\<le>k. a i * b (k - i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>k. a k) * (\\<Sum>k. b k) = (\\<Sum>k. \\<Sum>i\\<le>k. a i * b (k - i))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nsummable (\\<lambda>k. norm (a k))\nsummable (\\<lambda>k. norm (b k))\n\ngoal (1 subgoal):\n 1. (\\<Sum>k. a k) * (\\<Sum>k. b k) = (\\<Sum>k. \\<Sum>i\\<le>k. a i * b (k - i))\n[PROOF STEP]\nby (rule Cauchy_product_sums [THEN sums_unique])", "meta": {"llama_tokens": 322, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9324533144915912, "lm_q2_score": 0.8376199714402813, "lm_q1q2_score": 0.7810415186538423}}
{"text": "[STATEMENT]\nlemma det_map_matrix:\n  fixes A :: \"int^'n::mod_type^'n::mod_type\"\n  shows \"det (map_matrix rat_of_int A) = rat_of_int (det A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (map_matrix rat_of_int A) = rat_of_int (det A)\n[PROOF STEP]\nunfolding map_matrix_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (\\<chi>i j. rat_of_int (A $ i $ j)) = rat_of_int (det A)\n[PROOF STEP]\nunfolding Determinants.det_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>p | p permutes UNIV. rat_of_int (sign p) * (\\<Prod>i\\<in>UNIV. (\\<chi>i j. rat_of_int (A $ i $ j)) $ i $ p i)) = rat_of_int (\\<Sum>p | p permutes UNIV. of_int (sign p) * (\\<Prod>i\\<in>UNIV. A $ i $ p i))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 330, "file": "Modular_arithmetic_LLL_and_HNF_algorithms_Uniqueness_Hermite", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9324533163686646, "lm_q2_score": 0.8376199673867852, "lm_q1q2_score": 0.7810415164464205}}
{"text": "[STATEMENT]\nlemma Cauchy_product:\n  fixes a b :: \"nat \\<Rightarrow> 'a::{real_normed_algebra,banach}\"\n  assumes \"summable (\\<lambda>k. norm (a k))\"\n    and \"summable (\\<lambda>k. norm (b k))\"\n  shows \"(\\<Sum>k. a k) * (\\<Sum>k. b k) = (\\<Sum>k. \\<Sum>i\\<le>k. a i * b (k - i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>k. a k) * (\\<Sum>k. b k) = (\\<Sum>k. \\<Sum>i\\<le>k. a i * b (k - i))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nsummable (\\<lambda>k. norm (a k))\nsummable (\\<lambda>k. norm (b k))\n\ngoal (1 subgoal):\n 1. (\\<Sum>k. a k) * (\\<Sum>k. b k) = (\\<Sum>k. \\<Sum>i\\<le>k. a i * b (k - i))\n[PROOF STEP]\nby (rule Cauchy_product_sums [THEN sums_unique])", "meta": {"llama_tokens": 322, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9324533144915912, "lm_q2_score": 0.8376199673867852, "lm_q1q2_score": 0.7810415148741464}}
{"text": "[STATEMENT]\nlemma Cauchy_product:\n  fixes a b :: \"nat \\<Rightarrow> 'a::{real_normed_algebra,banach}\"\n  assumes \"summable (\\<lambda>k. norm (a k))\"\n    and \"summable (\\<lambda>k. norm (b k))\"\n  shows \"(\\<Sum>k. a k) * (\\<Sum>k. b k) = (\\<Sum>k. \\<Sum>i\\<le>k. a i * b (k - i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>k. a k) * (\\<Sum>k. b k) = (\\<Sum>k. \\<Sum>i\\<le>k. a i * b (k - i))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nsummable (\\<lambda>k. norm (a k))\nsummable (\\<lambda>k. norm (b k))\n\ngoal (1 subgoal):\n 1. (\\<Sum>k. a k) * (\\<Sum>k. b k) = (\\<Sum>k. \\<Sum>i\\<le>k. a i * b (k - i))\n[PROOF STEP]\nby (rule Cauchy_product_sums [THEN sums_unique])", "meta": {"llama_tokens": 322, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9324533144915912, "lm_q2_score": 0.8376199633332891, "lm_q1q2_score": 0.7810415110944505}}
{"text": "[STATEMENT]\nlemma real_of_nat_div_aux: \"real x / real d = real (x div d) + real (x mod d) / real d\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real x / real d = real (x div d) + real (x mod d) / real d\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. real x / real d = real (x div d) + real (x mod d) / real d\n[PROOF STEP]\nhave \"x = (x div d) * d + x mod d\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x = x div d * d + x mod d\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nx = x div d * d + x mod d\n\ngoal (1 subgoal):\n 1. real x / real d = real (x div d) + real (x mod d) / real d\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx = x div d * d + x mod d\n[PROOF STEP]\nhave \"real x = real (x div d) * real d + real(x mod d)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx = x div d * d + x mod d\n\ngoal (1 subgoal):\n 1. real x = real (x div d) * real d + real (x mod d)\n[PROOF STEP]\nby (metis of_nat_add of_nat_mult)\n[PROOF STATE]\nproof (state)\nthis:\nreal x = real (x div d) * real d + real (x mod d)\n\ngoal (1 subgoal):\n 1. real x / real d = real (x div d) + real (x mod d) / real d\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nreal x = real (x div d) * real d + real (x mod d)\n[PROOF STEP]\nhave \"real x / real d = \\<dots> / real d\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal x = real (x div d) * real d + real (x mod d)\n\ngoal (1 subgoal):\n 1. real x / real d = (real (x div d) * real d + real (x mod d)) / real d\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nreal x / real d = (real (x div d) * real d + real (x mod d)) / real d\n\ngoal (1 subgoal):\n 1. real x / real d = real (x div d) + real (x mod d) / real d\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nreal x / real d = (real (x div d) * real d + real (x mod d)) / real d\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nreal x / real d = (real (x div d) * real d + real (x mod d)) / real d\n\ngoal (1 subgoal):\n 1. real x / real d = real (x div d) + real (x mod d) / real d\n[PROOF STEP]\nby (auto simp add: add_divide_distrib algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nreal x / real d = real (x div d) + real (x mod d) / real d\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 959, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933094032139577, "lm_q2_score": 0.87407724336544, "lm_q1q2_score": 0.7808214206336824}}
{"text": "[STATEMENT]\nlemma real_of_nat_div_aux: \"real x / real d = real (x div d) + real (x mod d) / real d\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real x / real d = real (x div d) + real (x mod d) / real d\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. real x / real d = real (x div d) + real (x mod d) / real d\n[PROOF STEP]\nhave \"x = (x div d) * d + x mod d\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x = x div d * d + x mod d\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nx = x div d * d + x mod d\n\ngoal (1 subgoal):\n 1. real x / real d = real (x div d) + real (x mod d) / real d\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx = x div d * d + x mod d\n[PROOF STEP]\nhave \"real x = real (x div d) * real d + real(x mod d)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx = x div d * d + x mod d\n\ngoal (1 subgoal):\n 1. real x = real (x div d) * real d + real (x mod d)\n[PROOF STEP]\nby (metis of_nat_add of_nat_mult)\n[PROOF STATE]\nproof (state)\nthis:\nreal x = real (x div d) * real d + real (x mod d)\n\ngoal (1 subgoal):\n 1. real x / real d = real (x div d) + real (x mod d) / real d\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nreal x = real (x div d) * real d + real (x mod d)\n[PROOF STEP]\nhave \"real x / real d = \\<dots> / real d\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal x = real (x div d) * real d + real (x mod d)\n\ngoal (1 subgoal):\n 1. real x / real d = (real (x div d) * real d + real (x mod d)) / real d\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nreal x / real d = (real (x div d) * real d + real (x mod d)) / real d\n\ngoal (1 subgoal):\n 1. real x / real d = real (x div d) + real (x mod d) / real d\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nreal x / real d = (real (x div d) * real d + real (x mod d)) / real d\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nreal x / real d = (real (x div d) * real d + real (x mod d)) / real d\n\ngoal (1 subgoal):\n 1. real x / real d = real (x div d) + real (x mod d) / real d\n[PROOF STEP]\nby (auto simp add: add_divide_distrib algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nreal x / real d = real (x div d) + real (x mod d) / real d\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 959, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933094032139577, "lm_q2_score": 0.8740772253241802, "lm_q1q2_score": 0.7808214045172555}}
{"text": "[STATEMENT]\nlemma Schwartz_inequality_strong:\n  assumes \"f square_integrable S\" \"g square_integrable S\"\n  shows \"l2product S (\\<lambda>x. \\<bar>f x\\<bar>) (\\<lambda>x. \\<bar>g x\\<bar>) \\<le> l2norm S f * l2norm S g\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. l2product S (\\<lambda>x. \\<bar>f x\\<bar>) (\\<lambda>x. \\<bar>g x\\<bar>) \\<le> l2norm S f * l2norm S g\n[PROOF STEP]\nusing Holder_inequality_lnorm [of 2 2 f \"lebesgue_on S\" g] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>0 < 2; 0 < 2; 1 / 2 + 1 / 2 = 1; f \\<in> borel_measurable (lebesgue_on S); g \\<in> borel_measurable (lebesgue_on S); integrable (lebesgue_on S) (\\<lambda>x. \\<bar>f x\\<bar> powr 2); integrable (lebesgue_on S) (\\<lambda>x. \\<bar>g x\\<bar> powr 2)\\<rbrakk> \\<Longrightarrow> LINT x|lebesgue_on S. \\<bar>f x * g x\\<bar> \\<le> lnorm (lebesgue_on S) 2 f * lnorm (lebesgue_on S) 2 g\n\\<lbrakk>0 < 2; 0 < 2; 1 / 2 + 1 / 2 = 1; f \\<in> borel_measurable (lebesgue_on S); g \\<in> borel_measurable (lebesgue_on S); integrable (lebesgue_on S) (\\<lambda>x. \\<bar>f x\\<bar> powr 2); integrable (lebesgue_on S) (\\<lambda>x. \\<bar>g x\\<bar> powr 2)\\<rbrakk> \\<Longrightarrow> \\<bar>LINT x|lebesgue_on S. f x * g x\\<bar> \\<le> lnorm (lebesgue_on S) 2 f * lnorm (lebesgue_on S) 2 g\nf square_integrable S\ng square_integrable S\n\ngoal (1 subgoal):\n 1. l2product S (\\<lambda>x. \\<bar>f x\\<bar>) (\\<lambda>x. \\<bar>g x\\<bar>) \\<le> l2norm S f * l2norm S g\n[PROOF STEP]\nby (simp add: square_integrable_def l2product_def abs_mult flip: l2norm_lnorm)", "meta": {"llama_tokens": 686, "file": "Fourier_Square_Integrable", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9184802350995703, "lm_q2_score": 0.8499711832583695, "lm_q1q2_score": 0.7806817322270072}}
{"text": "[STATEMENT]\nlemma cos_times_sin: \"cos w * sin z = (sin (w + z) - sin (w - z)) / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos w * sin z = (sin (w + z) - sin (w - z)) / (2::'a)\n[PROOF STEP]\nby (simp add: sin_diff sin_add)", "meta": {"llama_tokens": 109, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.936285002192296, "lm_q2_score": 0.8333245994514084, "lm_q1q2_score": 0.7802293244242561}}
{"text": "[STATEMENT]\ntheorem descartes_sign:\n  fixes p::\"real poly\"\n  assumes \"p\\<noteq>0\"\n  shows \" changes (coeffs p) \\<ge> proots_count p {x. 0 < x} \\<and> \n          even (changes (coeffs p) - proots_count p {x. 0< x})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. int (proots_count p {x. 0 < x}) \\<le> changes (coeffs p) \\<and> even (changes (coeffs p) - int (proots_count p {x. 0 < x}))\n[PROOF STEP]\nusing budan_fourier_gt[OF \\<open>p\\<noteq>0\\<close>,of 0]\n[PROOF STATE]\nproof (prove)\nusing this:\nint (proots_count p {x. 0 < x}) \\<le> changes_gt_der 0 p \\<and> even (changes_gt_der 0 p - int (proots_count p {x. 0 < x}))\n\ngoal (1 subgoal):\n 1. int (proots_count p {x. 0 < x}) \\<le> changes (coeffs p) \\<and> even (changes (coeffs p) - int (proots_count p {x. 0 < x}))\n[PROOF STEP]\nunfolding changes_gt_der_def\n[PROOF STATE]\nproof (prove)\nusing this:\nint (proots_count p {x. 0 < x}) \\<le> changes_poly_at (pders p) 0 \\<and> even (changes_poly_at (pders p) 0 - int (proots_count p {x. 0 < x}))\n\ngoal (1 subgoal):\n 1. int (proots_count p {x. 0 < x}) \\<le> changes (coeffs p) \\<and> even (changes (coeffs p) - int (proots_count p {x. 0 < x}))\n[PROOF STEP]\nby (simp add:changes_poly_at_pders_0)", "meta": {"llama_tokens": 530, "file": "Budan_Fourier_Budan_Fourier", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9314625069680097, "lm_q2_score": 0.8376199552262967, "lm_q1q2_score": 0.7802115833815184}}
{"text": "[STATEMENT]\nlemma \"filterlim (\\<lambda>x::real. x^2 / (sqrt (x^2 + 12) - sqrt (12))) (nhds (12 / sqrt 3)) (at 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<lambda>x. x\\<^sup>2 / (sqrt (x\\<^sup>2 + 12) - sqrt 12)) \\<midarrow>0\\<rightarrow> 12 / sqrt 3\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\<lambda>x. x\\<^sup>2 / (sqrt (x\\<^sup>2 + 12) - sqrt 12)) \\<midarrow>0\\<rightarrow> 12 / sqrt 3\n[PROOF STEP]\nnote [simp] = powr_half_sqrt sqrt_def\n[PROOF STATE]\nproof (state)\nthis:\n0 \\<le> ?x \\<Longrightarrow> ?x powr (1 / 2) = sqrt ?x\nsqrt = root 2\n\ngoal (1 subgoal):\n 1. (\\<lambda>x. x\\<^sup>2 / (sqrt (x\\<^sup>2 + 12) - sqrt 12)) \\<midarrow>0\\<rightarrow> 12 / sqrt 3\n[PROOF STEP]\nhave \"sqrt (12 :: real) = sqrt (4 * 3)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt 12 = sqrt (4 * 3)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsqrt 12 = sqrt (4 * 3)\n\ngoal (1 subgoal):\n 1. (\\<lambda>x. x\\<^sup>2 / (sqrt (x\\<^sup>2 + 12) - sqrt 12)) \\<midarrow>0\\<rightarrow> 12 / sqrt 3\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsqrt 12 = sqrt (4 * 3)\n\ngoal (1 subgoal):\n 1. (\\<lambda>x. x\\<^sup>2 / (sqrt (x\\<^sup>2 + 12) - sqrt 12)) \\<midarrow>0\\<rightarrow> 12 / sqrt 3\n[PROOF STEP]\nhave \"\\<dots> = 2 * sqrt 3\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt (4 * 3) = 2 * sqrt 3\n[PROOF STEP]\nby (subst real_sqrt_mult) simp\n[PROOF STATE]\nproof (state)\nthis:\nsqrt (4 * 3) = 2 * sqrt 3\n\ngoal (1 subgoal):\n 1. (\\<lambda>x. x\\<^sup>2 / (sqrt (x\\<^sup>2 + 12) - sqrt 12)) \\<midarrow>0\\<rightarrow> 12 / sqrt 3\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nsqrt 12 = 2 * sqrt 3\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsqrt 12 = 2 * sqrt 3\n\ngoal (1 subgoal):\n 1. (\\<lambda>x. x\\<^sup>2 / (sqrt (x\\<^sup>2 + 12) - sqrt 12)) \\<midarrow>0\\<rightarrow> 12 / sqrt 3\n[PROOF STEP]\nby real_asymp\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>x. x\\<^sup>2 / (sqrt (x\\<^sup>2 + 12) - sqrt 12)) \\<midarrow>0\\<rightarrow> 12 / sqrt 3\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1012, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9263037363973294, "lm_q2_score": 0.8418256432832333, "lm_q1q2_score": 0.7797862387683444}}
{"text": "[STATEMENT]\nlemma homeomorphic_scaling:\n  fixes S :: \"'a::real_normed_vector set\"\n  assumes \"c \\<noteq> 0\"\n  shows \"S homeomorphic ((\\<lambda>x. c *\\<^sub>R x) ` S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. S homeomorphic (*\\<^sub>R) c ` S\n[PROOF STEP]\nunfolding homeomorphic_minimal\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<exists>f g. (\\<forall>x\\<in>S. f x \\<in> (*\\<^sub>R) c ` S \\<and> g (f x) = x) \\<and> (\\<forall>y\\<in>(*\\<^sub>R) c ` S. g y \\<in> S \\<and> f (g y) = y) \\<and> continuous_on S f \\<and> continuous_on ((*\\<^sub>R) c ` S) g\n[PROOF STEP]\napply (rule_tac x=\"\\<lambda>x. c *\\<^sub>R x\" in exI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<exists>g. (\\<forall>x\\<in>S. c *\\<^sub>R x \\<in> (*\\<^sub>R) c ` S \\<and> g (c *\\<^sub>R x) = x) \\<and> (\\<forall>y\\<in>(*\\<^sub>R) c ` S. g y \\<in> S \\<and> c *\\<^sub>R g y = y) \\<and> continuous_on S ((*\\<^sub>R) c) \\<and> continuous_on ((*\\<^sub>R) c ` S) g\n[PROOF STEP]\napply (rule_tac x=\"\\<lambda>x. (1 / c) *\\<^sub>R x\" in exI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<forall>x\\<in>S. c *\\<^sub>R x \\<in> (*\\<^sub>R) c ` S \\<and> (1 / c) *\\<^sub>R c *\\<^sub>R x = x) \\<and> (\\<forall>y\\<in>(*\\<^sub>R) c ` S. (1 / c) *\\<^sub>R y \\<in> S \\<and> c *\\<^sub>R (1 / c) *\\<^sub>R y = y) \\<and> continuous_on S ((*\\<^sub>R) c) \\<and> continuous_on ((*\\<^sub>R) c ` S) ((*\\<^sub>R) (1 / c))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nc \\<noteq> 0\n\ngoal (1 subgoal):\n 1. (\\<forall>x\\<in>S. c *\\<^sub>R x \\<in> (*\\<^sub>R) c ` S \\<and> (1 / c) *\\<^sub>R c *\\<^sub>R x = x) \\<and> (\\<forall>y\\<in>(*\\<^sub>R) c ` S. (1 / c) *\\<^sub>R y \\<in> S \\<and> c *\\<^sub>R (1 / c) *\\<^sub>R y = y) \\<and> continuous_on S ((*\\<^sub>R) c) \\<and> continuous_on ((*\\<^sub>R) c ` S) ((*\\<^sub>R) (1 / c))\n[PROOF STEP]\nby (auto simp: continuous_intros)", "meta": {"llama_tokens": 862, "file": null, "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8991213799730775, "lm_q2_score": 0.8670357683915538, "lm_q1q2_score": 0.7795703965622315}}
{"text": "[STATEMENT]\nlemma alt_group_card_carrier:\n  assumes \"n \\<ge> 2\" shows \"2 * card (carrier (alt_group n)) = fact n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * card (carrier (alt_group n)) = fact n\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 2 * card (carrier (alt_group n)) = fact n\n[PROOF STEP]\nhave \"card (rcosets\\<^bsub>sym_group n\\<^esub> (carrier (alt_group n))) = 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (rcosets\\<^bsub>sym_group n\\<^esub> carrier (alt_group n)) = 2\n[PROOF STEP]\nusing iso_same_card[OF sign_iso[OF assms]]\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (carrier (sym_group n Mod carrier (alt_group n))) = card (carrier sign_img)\n\ngoal (1 subgoal):\n 1. card (rcosets\\<^bsub>sym_group n\\<^esub> carrier (alt_group n)) = 2\n[PROOF STEP]\nunfolding FactGroup_def sign_img_def\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (carrier \\<lparr>carrier = rcosets\\<^bsub>sym_group n\\<^esub> carrier (alt_group n), monoid.mult = (<#>\\<^bsub>sym_group n\\<^esub>), one = carrier (alt_group n)\\<rparr>) = card (carrier \\<lparr>carrier = {- 1, 1}, monoid.mult = (*), one = 1\\<rparr>)\n\ngoal (1 subgoal):\n 1. card (rcosets\\<^bsub>sym_group n\\<^esub> carrier (alt_group n)) = 2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncard (rcosets\\<^bsub>sym_group n\\<^esub> carrier (alt_group n)) = 2\n\ngoal (1 subgoal):\n 1. 2 * card (carrier (alt_group n)) = fact n\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (rcosets\\<^bsub>sym_group n\\<^esub> carrier (alt_group n)) = 2\n\ngoal (1 subgoal):\n 1. 2 * card (carrier (alt_group n)) = fact n\n[PROOF STEP]\nusing group.lagrange[OF sym_group_is_group alt_group_is_subgroup, of n]\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (rcosets\\<^bsub>sym_group n\\<^esub> carrier (alt_group n)) = 2\ncard (rcosets\\<^bsub>sym_group n\\<^esub> carrier (alt_group n)) * card (carrier (alt_group n)) = order (sym_group n)\n\ngoal (1 subgoal):\n 1. 2 * card (carrier (alt_group n)) = fact n\n[PROOF STEP]\nunfolding order_def sym_group_card_carrier\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (rcosets\\<^bsub>sym_group n\\<^esub> carrier (alt_group n)) = 2\ncard (rcosets\\<^bsub>sym_group n\\<^esub> carrier (alt_group n)) * card (carrier (alt_group n)) = fact n\n\ngoal (1 subgoal):\n 1. 2 * card (carrier (alt_group n)) = fact n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n2 * card (carrier (alt_group n)) = fact n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1048, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9032942119105696, "lm_q2_score": 0.861538211208597, "lm_q1q2_score": 0.7782224795245114}}
{"text": "[STATEMENT]\nlemma M_cong_7_mod_8: \"[M = 7] (mod 8)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [M = 7] (mod 8)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. [M = 7] (mod 8)\n[PROOF STEP]\nhave \"2 ^ 3 dvd (2 ^ p :: int)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 ^ 3 dvd 2 ^ p\n[PROOF STEP]\nusing p_gt_2\n[PROOF STATE]\nproof (prove)\nusing this:\n2 < p\n\ngoal (1 subgoal):\n 1. 2 ^ 3 dvd 2 ^ p\n[PROOF STEP]\nby (intro le_imp_power_dvd) auto\n[PROOF STATE]\nproof (state)\nthis:\n2 ^ 3 dvd 2 ^ p\n\ngoal (1 subgoal):\n 1. [M = 7] (mod 8)\n[PROOF STEP]\nhence \"[2 ^ p - 1 = 0 - 1] (mod (8 :: int))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n2 ^ 3 dvd 2 ^ p\n\ngoal (1 subgoal):\n 1. [2 ^ p - 1 = 0 - 1] (mod 8)\n[PROOF STEP]\nby (intro cong_diff) (auto simp: cong_def)\n[PROOF STATE]\nproof (state)\nthis:\n[2 ^ p - 1 = 0 - 1] (mod 8)\n\ngoal (1 subgoal):\n 1. [M = 7] (mod 8)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n[2 ^ p - 1 = 0 - 1] (mod 8)\n\ngoal (1 subgoal):\n 1. [M = 7] (mod 8)\n[PROOF STEP]\nhave \"2 ^ p - 1 = int M\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 ^ p - 1 = int M\n[PROOF STEP]\nby (simp add: M_def of_nat_diff)\n[PROOF STATE]\nproof (state)\nthis:\n2 ^ p - 1 = int M\n\ngoal (1 subgoal):\n 1. [M = 7] (mod 8)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n[int M = 0 - 1] (mod 8)\n[PROOF STEP]\nhave \"int M mod int 8 = 7\"\n[PROOF STATE]\nproof (prove)\nusing this:\n[int M = 0 - 1] (mod 8)\n\ngoal (1 subgoal):\n 1. int M mod int 8 = 7\n[PROOF STEP]\nby (simp add: cong_def)\n[PROOF STATE]\nproof (state)\nthis:\nint M mod int 8 = 7\n\ngoal (1 subgoal):\n 1. [M = 7] (mod 8)\n[PROOF STEP]\nthus \"[M = 7] (mod 8)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nint M mod int 8 = 7\n\ngoal (1 subgoal):\n 1. [M = 7] (mod 8)\n[PROOF STEP]\nby (subst (asm) zmod_int [symmetric]) (auto simp: cong_def)\n[PROOF STATE]\nproof (state)\nthis:\n[M = 7] (mod 8)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1004, "file": "Mersenne_Primes_Lucas_Lehmer", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942319436397, "lm_q2_score": 0.8740772335247532, "lm_q1q2_score": 0.7781859192803415}}
{"text": "[STATEMENT]\nlemma append_split_vec_distrib_scalar_prod:\n  assumes \"dim_vec (u @\\<^sub>v w) = dim_vec x\" \n  shows \"(u @\\<^sub>v w) \\<bullet> x = u \\<bullet> (vec_first x (dim_vec u)) + w \\<bullet> (vec_last x (dim_vec w))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (u @\\<^sub>v w) \\<bullet> x = u \\<bullet> vec_first x (dim_vec u) + w \\<bullet> vec_last x (dim_vec w)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (u @\\<^sub>v w) \\<bullet> x = u \\<bullet> vec_first x (dim_vec u) + w \\<bullet> vec_last x (dim_vec w)\n[PROOF STEP]\nhave \"(u @\\<^sub>v w) \\<bullet> (vec_first x (dim_vec u) @\\<^sub>v vec_last x (dim_vec w)) = \n              u \\<bullet> vec_first x (dim_vec u) + w \\<bullet> vec_last x (dim_vec w)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (u @\\<^sub>v w) \\<bullet> (vec_first x (dim_vec u) @\\<^sub>v vec_last x (dim_vec w)) = u \\<bullet> vec_first x (dim_vec u) + w \\<bullet> vec_last x (dim_vec w)\n[PROOF STEP]\nby (meson carrier_vec_dim_vec scalar_prod_append vec_first_carrier vec_last_carrier)\n[PROOF STATE]\nproof (state)\nthis:\n(u @\\<^sub>v w) \\<bullet> (vec_first x (dim_vec u) @\\<^sub>v vec_last x (dim_vec w)) = u \\<bullet> vec_first x (dim_vec u) + w \\<bullet> vec_last x (dim_vec w)\n\ngoal (1 subgoal):\n 1. (u @\\<^sub>v w) \\<bullet> x = u \\<bullet> vec_first x (dim_vec u) + w \\<bullet> vec_last x (dim_vec w)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(u @\\<^sub>v w) \\<bullet> (vec_first x (dim_vec u) @\\<^sub>v vec_last x (dim_vec w)) = u \\<bullet> vec_first x (dim_vec u) + w \\<bullet> vec_last x (dim_vec w)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(u @\\<^sub>v w) \\<bullet> (vec_first x (dim_vec u) @\\<^sub>v vec_last x (dim_vec w)) = u \\<bullet> vec_first x (dim_vec u) + w \\<bullet> vec_last x (dim_vec w)\n\ngoal (1 subgoal):\n 1. (u @\\<^sub>v w) \\<bullet> x = u \\<bullet> vec_first x (dim_vec u) + w \\<bullet> vec_last x (dim_vec w)\n[PROOF STEP]\nby (metis assms carrier_vec_dim_vec index_append_vec(2) vec_first_last_append)\n[PROOF STATE]\nproof (state)\nthis:\n(u @\\<^sub>v w) \\<bullet> x = u \\<bullet> vec_first x (dim_vec u) + w \\<bullet> vec_last x (dim_vec w)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 955, "file": "Linear_Programming_More_Jordan_Normal_Forms", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.899121366457407, "lm_q2_score": 0.865224091265267, "lm_q1q2_score": 0.7779414672302951}}
{"text": "[STATEMENT]\nlemma asymp_equiv_at_top_imp_at:\n  \"(\\<lambda>x. f (a - inverse x)) \\<sim>[at_top] (\\<lambda>x. g (a - inverse x)) \\<Longrightarrow>\n   (\\<lambda>x. f (a + inverse x)) \\<sim>[at_top] (\\<lambda>x. g (a + inverse x)) \\<Longrightarrow> f \\<sim>[at a] g\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>(\\<lambda>x. f (a - inverse x)) \\<sim>[at_top] (\\<lambda>x. g (a - inverse x)); (\\<lambda>x. f (a + inverse x)) \\<sim>[at_top] (\\<lambda>x. g (a + inverse x))\\<rbrakk> \\<Longrightarrow> f \\<sim>[at a] g\n[PROOF STEP]\nunfolding asymp_equiv_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>((\\<lambda>x. if f (a - inverse x) = 0 \\<and> g (a - inverse x) = 0 then 1 else f (a - inverse x) / g (a - inverse x)) \\<longlongrightarrow> 1) at_top; ((\\<lambda>x. if f (a + inverse x) = 0 \\<and> g (a + inverse x) = 0 then 1 else f (a + inverse x) / g (a + inverse x)) \\<longlongrightarrow> 1) at_top\\<rbrakk> \\<Longrightarrow> (\\<lambda>x. if f x = 0 \\<and> g x = 0 then 1 else f x / g x) \\<midarrow>a\\<rightarrow> 1\n[PROOF STEP]\nusing asymp_equiv_at_top_imp_at_left[of a] asymp_equiv_at_top_imp_at_right[of a]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<lambda>x. f (a - inverse x)) \\<sim>[at_top] (\\<lambda>x. g (a - inverse x)) \\<Longrightarrow> f \\<sim>[at_left a] g\n(\\<lambda>x. f (a + inverse x)) \\<sim>[at_top] (\\<lambda>x. g (a + inverse x)) \\<Longrightarrow> f \\<sim>[at_right a] g\n\ngoal (1 subgoal):\n 1. \\<lbrakk>((\\<lambda>x. if f (a - inverse x) = 0 \\<and> g (a - inverse x) = 0 then 1 else f (a - inverse x) / g (a - inverse x)) \\<longlongrightarrow> 1) at_top; ((\\<lambda>x. if f (a + inverse x) = 0 \\<and> g (a + inverse x) = 0 then 1 else f (a + inverse x) / g (a + inverse x)) \\<longlongrightarrow> 1) at_top\\<rbrakk> \\<Longrightarrow> (\\<lambda>x. if f x = 0 \\<and> g x = 0 then 1 else f x / g x) \\<midarrow>a\\<rightarrow> 1\n[PROOF STEP]\nby (intro filterlim_split_at) (auto simp: asymp_equiv_def)", "meta": {"llama_tokens": 793, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9263037262250327, "lm_q2_score": 0.839733963661418, "lm_q1q2_score": 0.7778486995772876}}
{"text": "[STATEMENT]\nlemma frechet_derivative_sqrt: \"frechet_derivative (\\<lambda>x. sqrt (f x)) (at x) =\n  (\\<lambda>v. (if f x > 0 then 1 else -1) / (2 * sqrt (f x)) * frechet_derivative f (at x) v)\"\n  if \"f differentiable at x\" \"f x \\<noteq> 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. frechet_derivative (\\<lambda>x. sqrt (f x)) (at x) = (\\<lambda>v. (if 0 < f x then 1 else - 1) / (2 * sqrt (f x)) * frechet_derivative f (at x) v)\n[PROOF STEP]\napply (rule frechet_derivative_at')\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\<lambda>x. sqrt (f x)) has_derivative (\\<lambda>v. (if 0 < f x then 1 else - 1) / (2 * sqrt (f x)) * frechet_derivative f (at x) v)) (at x)\n[PROOF STEP]\napply (rule sqrt_has_derivative[THEN has_derivative_eq_rhs])\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. (f has_derivative ?f'3) (at x)\n 2. f x \\<noteq> 0\n 3. (\\<lambda>xa. (if 0 < f x then 1 else - 1) / (2 * sqrt (f x)) * ?f'3 xa) = (\\<lambda>v. (if 0 < f x then 1 else - 1) / (2 * sqrt (f x)) * frechet_derivative f (at x) v)\n[PROOF STEP]\nby (auto intro!: frechet_derivative_worksI that simp: divide_simps)", "meta": {"llama_tokens": 519, "file": "Smooth_Manifolds_Analysis_More", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9173026528034426, "lm_q2_score": 0.847967764140929, "lm_q1q2_score": 0.7778430795382781}}
{"text": "[STATEMENT]\nlemma mat_of_vec_mult_vec:\n  fixes a b c :: \"'a::conjugatable_field vec\"\n  assumes a: \"a \\<in> carrier_vec d\" and b: \"b \\<in> carrier_vec d\"\n  shows \"mat 1 d (\\<lambda>(i, j). (conjugate a) $ j) *\\<^sub>v b = vec 1 (\\<lambda>k. inner_prod a b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mat 1 d (\\<lambda>(i, j). conjugate a $ j) *\\<^sub>v b = vec 1 (\\<lambda>k. inner_prod a b)\n[PROOF STEP]\napply (rule eq_vecI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\<And>i. i < dim_vec (vec 1 (\\<lambda>k. inner_prod a b)) \\<Longrightarrow> (mat 1 d (\\<lambda>(i, j). conjugate a $ j) *\\<^sub>v b) $ i = vec 1 (\\<lambda>k. inner_prod a b) $ i\n 2. dim_vec (mat 1 d (\\<lambda>(i, j). conjugate a $ j) *\\<^sub>v b) = dim_vec (vec 1 (\\<lambda>k. inner_prod a b))\n[PROOF STEP]\napply (simp add: scalar_prod_def carrier_vecD[OF a] carrier_vecD[OF b])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. (\\<Sum>i = 0..<d. conjugate (a $ i) * b $ i) = (\\<Sum>i = 0..<d. b $ i * conjugate (a $ i))\n 2. dim_vec (mat 1 d (\\<lambda>(i, j). conjugate a $ j) *\\<^sub>v b) = dim_vec (vec 1 (\\<lambda>k. inner_prod a b))\n[PROOF STEP]\napply (rule sum.cong)\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. {0..<d} = {0..<d}\n 2. \\<And>x. x \\<in> {0..<d} \\<Longrightarrow> conjugate (a $ x) * b $ x = b $ x * conjugate (a $ x)\n 3. dim_vec (mat 1 d (\\<lambda>(i, j). conjugate a $ j) *\\<^sub>v b) = dim_vec (vec 1 (\\<lambda>k. inner_prod a b))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 657, "file": "QHLProver_Complex_Matrix", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505376715775, "lm_q2_score": 0.8596637523076225, "lm_q1q2_score": 0.7776952757418566}}
{"text": "[STATEMENT]\nlemma conjugate_vec_sprod_comm:\n  fixes v w :: \"'a :: {conjugatable_ring, comm_ring} vec\"\n  assumes \"v : carrier_vec n\" and \"w : carrier_vec n\"\n  shows \"v \\<bullet>c w = (conjugate w \\<bullet> v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. v \\<bullet>c w = conjugate w \\<bullet> v\n[PROOF STEP]\nunfolding scalar_prod_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<dim_vec (conjugate w). v $ i * conjugate w $ i) = (\\<Sum>i = 0..<dim_vec v. conjugate w $ i * v $ i)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nv \\<in> carrier_vec n\nw \\<in> carrier_vec n\n\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<dim_vec (conjugate w). v $ i * conjugate w $ i) = (\\<Sum>i = 0..<dim_vec v. conjugate w $ i * v $ i)\n[PROOF STEP]\nby(subst sum.ivl_cong, auto simp: ac_simps)", "meta": {"llama_tokens": 358, "file": "Jordan_Normal_Form_Matrix", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505248181417, "lm_q2_score": 0.8596637505099167, "lm_q1q2_score": 0.7776952630659282}}
{"text": "[STATEMENT]\nlemma cos_times_cos: \"cos w * cos z = (cos (w - z) + cos (w + z)) / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos w * cos z = (cos (w - z) + cos (w + z)) / (2::'a)\n[PROOF STEP]\nby (simp add: cos_diff cos_add)", "meta": {"llama_tokens": 109, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9381240177362488, "lm_q2_score": 0.8289388146603365, "lm_q1q2_score": 0.7776474112666785}}
{"text": "[STATEMENT]\nlemma cube_minus2:\nfixes p q\nshows \"(((p::rat_poly) - (q::rat_poly))^3) \n                = (p^3) - 3*(p^2)*(q) + 3*(p)*(q^2) - (q^3)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (p - q) ^ 3 = rat_poly_plus (p ^ 3 - rat_poly_times (rat_poly_times 3 (p\\<^sup>2)) q) (rat_poly_times (rat_poly_times 3 p) (q\\<^sup>2)) - q ^ 3\n[PROOF STEP]\nusing cube_minus\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<forall>p q. (p - q) ^ 3 = rat_poly_plus (p ^ 3 - rat_poly_times (rat_poly_times 3 (p\\<^sup>2)) q) (rat_poly_times (rat_poly_times 3 p) (q\\<^sup>2)) - q ^ 3\n\ngoal (1 subgoal):\n 1. (p - q) ^ 3 = rat_poly_plus (p ^ 3 - rat_poly_times (rat_poly_times 3 (p\\<^sup>2)) q) (rat_poly_times (rat_poly_times 3 p) (q\\<^sup>2)) - q ^ 3\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 370, "file": "Knot_Theory_Computations", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.914900950352329, "lm_q2_score": 0.8499711775577736, "lm_q1q2_score": 0.7776394381196953}}
{"text": "[STATEMENT]\nlemma outer_minus_minus:\n  fixes a::\"complex Matrix.vec\" \n  assumes \"dim_vec a = dim_vec b\"\n  and \"dim_vec u = dim_vec v\"\n  shows \"outer_prod (a - b) (u - v) = outer_prod a u - outer_prod a v -\n      outer_prod b u +  outer_prod b v\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. outer_prod (a - b) (u - v) = outer_prod a u - outer_prod a v - outer_prod b u + outer_prod b v\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. outer_prod (a - b) (u - v) = outer_prod a u - outer_prod a v - outer_prod b u + outer_prod b v\n[PROOF STEP]\nhave \"outer_prod (a - b) (u - v) = outer_prod a (u - v)\n    - outer_prod b (u - v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. outer_prod (a - b) (u - v) = outer_prod a (u - v) - outer_prod b (u - v)\n[PROOF STEP]\nusing  outer_prod_minus_left assms\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec ?v = dim_vec ?x \\<Longrightarrow> outer_prod (?v - ?x) ?w = outer_prod ?v ?w - outer_prod ?x ?w\ndim_vec a = dim_vec b\ndim_vec u = dim_vec v\n\ngoal (1 subgoal):\n 1. outer_prod (a - b) (u - v) = outer_prod a (u - v) - outer_prod b (u - v)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nouter_prod (a - b) (u - v) = outer_prod a (u - v) - outer_prod b (u - v)\n\ngoal (1 subgoal):\n 1. outer_prod (a - b) (u - v) = outer_prod a u - outer_prod a v - outer_prod b u + outer_prod b v\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nouter_prod (a - b) (u - v) = outer_prod a (u - v) - outer_prod b (u - v)\n\ngoal (1 subgoal):\n 1. outer_prod (a - b) (u - v) = outer_prod a u - outer_prod a v - outer_prod b u + outer_prod b v\n[PROOF STEP]\nhave \"... = outer_prod a u - outer_prod a v -\n    outer_prod b (u - v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. outer_prod a (u - v) - outer_prod b (u - v) = outer_prod a u - outer_prod a v - outer_prod b (u - v)\n[PROOF STEP]\nusing assms outer_prod_minus_right\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec a = dim_vec b\ndim_vec u = dim_vec v\ndim_vec ?w = dim_vec ?x \\<Longrightarrow> outer_prod ?v (?w - ?x) = outer_prod ?v ?w - outer_prod ?v ?x\n\ngoal (1 subgoal):\n 1. outer_prod a (u - v) - outer_prod b (u - v) = outer_prod a u - outer_prod a v - outer_prod b (u - v)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nouter_prod a (u - v) - outer_prod b (u - v) = outer_prod a u - outer_prod a v - outer_prod b (u - v)\n\ngoal (1 subgoal):\n 1. outer_prod (a - b) (u - v) = outer_prod a u - outer_prod a v - outer_prod b u + outer_prod b v\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nouter_prod a (u - v) - outer_prod b (u - v) = outer_prod a u - outer_prod a v - outer_prod b (u - v)\n\ngoal (1 subgoal):\n 1. outer_prod (a - b) (u - v) = outer_prod a u - outer_prod a v - outer_prod b u + outer_prod b v\n[PROOF STEP]\nhave \"... = outer_prod a u - outer_prod a v -\n    (outer_prod b u - outer_prod b v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. outer_prod a u - outer_prod a v - outer_prod b (u - v) = outer_prod a u - outer_prod a v - (outer_prod b u - outer_prod b v)\n[PROOF STEP]\nusing assms outer_prod_minus_right\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec a = dim_vec b\ndim_vec u = dim_vec v\ndim_vec ?w = dim_vec ?x \\<Longrightarrow> outer_prod ?v (?w - ?x) = outer_prod ?v ?w - outer_prod ?v ?x\n\ngoal (1 subgoal):\n 1. outer_prod a u - outer_prod a v - outer_prod b (u - v) = outer_prod a u - outer_prod a v - (outer_prod b u - outer_prod b v)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nouter_prod a u - outer_prod a v - outer_prod b (u - v) = outer_prod a u - outer_prod a v - (outer_prod b u - outer_prod b v)\n\ngoal (1 subgoal):\n 1. outer_prod (a - b) (u - v) = outer_prod a u - outer_prod a v - outer_prod b u + outer_prod b v\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nouter_prod a u - outer_prod a v - outer_prod b (u - v) = outer_prod a u - outer_prod a v - (outer_prod b u - outer_prod b v)\n\ngoal (1 subgoal):\n 1. outer_prod (a - b) (u - v) = outer_prod a u - outer_prod a v - outer_prod b u + outer_prod b v\n[PROOF STEP]\nhave \"...  = outer_prod a u - outer_prod a v -\n    outer_prod b u +  outer_prod b v\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. outer_prod a u - outer_prod a v - (outer_prod b u - outer_prod b v) = outer_prod a u - outer_prod a v - outer_prod b u + outer_prod b v\n[PROOF STEP]\nproof (rule mat_minus_minus)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. outer_prod a u - outer_prod a v \\<in> carrier_mat ?n ?m\n 2. outer_prod b u \\<in> carrier_mat ?n ?m\n 3. outer_prod b v \\<in> carrier_mat ?n ?m\n[PROOF STEP]\nshow \"outer_prod b u \\<in> carrier_mat (dim_vec b) (dim_vec u)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. outer_prod b u \\<in> carrier_mat (dim_vec b) (dim_vec u)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nouter_prod b u \\<in> carrier_mat (dim_vec b) (dim_vec u)\n\ngoal (2 subgoals):\n 1. outer_prod a u - outer_prod a v \\<in> carrier_mat (dim_vec b) (dim_vec u)\n 2. outer_prod b v \\<in> carrier_mat (dim_vec b) (dim_vec u)\n[PROOF STEP]\nshow \"outer_prod b v \\<in> carrier_mat (dim_vec b) (dim_vec u)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. outer_prod b v \\<in> carrier_mat (dim_vec b) (dim_vec u)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec a = dim_vec b\ndim_vec u = dim_vec v\n\ngoal (1 subgoal):\n 1. outer_prod b v \\<in> carrier_mat (dim_vec b) (dim_vec u)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nouter_prod b v \\<in> carrier_mat (dim_vec b) (dim_vec u)\n\ngoal (1 subgoal):\n 1. outer_prod a u - outer_prod a v \\<in> carrier_mat (dim_vec b) (dim_vec u)\n[PROOF STEP]\nshow \"outer_prod a u - outer_prod a v \\<in> carrier_mat (dim_vec b) (dim_vec u)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. outer_prod a u - outer_prod a v \\<in> carrier_mat (dim_vec b) (dim_vec u)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec a = dim_vec b\ndim_vec u = dim_vec v\n\ngoal (1 subgoal):\n 1. outer_prod a u - outer_prod a v \\<in> carrier_mat (dim_vec b) (dim_vec u)\n[PROOF STEP]\nby (metis carrier_vecI minus_carrier_mat outer_prod_dim)\n[PROOF STATE]\nproof (state)\nthis:\nouter_prod a u - outer_prod a v \\<in> carrier_mat (dim_vec b) (dim_vec u)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nouter_prod a u - outer_prod a v - (outer_prod b u - outer_prod b v) = outer_prod a u - outer_prod a v - outer_prod b u + outer_prod b v\n\ngoal (1 subgoal):\n 1. outer_prod (a - b) (u - v) = outer_prod a u - outer_prod a v - outer_prod b u + outer_prod b v\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nouter_prod (a - b) (u - v) = outer_prod a u - outer_prod a v - outer_prod b u + outer_prod b v\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nouter_prod (a - b) (u - v) = outer_prod a u - outer_prod a v - outer_prod b u + outer_prod b v\n\ngoal (1 subgoal):\n 1. outer_prod (a - b) (u - v) = outer_prod a u - outer_prod a v - outer_prod b u + outer_prod b v\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nouter_prod (a - b) (u - v) = outer_prod a u - outer_prod a v - outer_prod b u + outer_prod b v\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3035, "file": "Projective_Measurements_Linear_Algebra_Complements", "length": 28, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.894789454880027, "lm_q2_score": 0.8688267711434708, "lm_q1q2_score": 0.7774170329366403}}
{"text": "[STATEMENT]\nlemma convex_hull_set_plus:\n  \"convex hull (S + T) = convex hull S + convex hull T\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. convex hull (S + T) = convex hull S + convex hull T\n[PROOF STEP]\nunfolding set_plus_image\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. convex hull (\\<lambda>(x, y). x + y) ` (S \\<times> T) = (\\<lambda>(x, y). x + y) ` ((convex hull S) \\<times> (convex hull T))\n[PROOF STEP]\napply (subst convex_hull_linear_image [symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. linear (\\<lambda>(x, y). x + y)\n 2. (\\<lambda>(x, y). x + y) ` (convex hull S \\<times> T) = (\\<lambda>(x, y). x + y) ` ((convex hull S) \\<times> (convex hull T))\n[PROOF STEP]\napply (simp add: linear_iff scaleR_right_distrib)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<lambda>(x, y). x + y) ` (convex hull S \\<times> T) = (\\<lambda>(x, y). x + y) ` ((convex hull S) \\<times> (convex hull T))\n[PROOF STEP]\napply (simp add: convex_hull_Times)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 446, "file": null, "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9353465152482724, "lm_q2_score": 0.8311430394931456, "lm_q1q2_score": 0.777406745662771}}
{"text": "[STATEMENT]\nlemma sin_times_cos: \"sin w * cos z = (sin (w + z) + sin (w - z)) / 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin w * cos z = (sin (w + z) + sin (w - z)) / (2::'a)\n[PROOF STEP]\nby (simp add: sin_diff sin_add)", "meta": {"llama_tokens": 109, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.935346504434783, "lm_q2_score": 0.8311430478583168, "lm_q1q2_score": 0.7774067444995482}}
{"text": "[STATEMENT]\ntheorem (in group) group_right_one: \"x * 1 = x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x * (1::'a) = x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x * (1::'a) = x\n[PROOF STEP]\nhave \"x * 1 = x * (inverse x * x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x * (1::'a) = x * (inverse x * x)\n[PROOF STEP]\nby (simp only: group_left_inverse)\n[PROOF STATE]\nproof (state)\nthis:\nx * (1::'a) = x * (inverse x * x)\n\ngoal (1 subgoal):\n 1. x * (1::'a) = x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nx * (1::'a) = x * (inverse x * x)\n\ngoal (1 subgoal):\n 1. x * (1::'a) = x\n[PROOF STEP]\nhave \"\\<dots> = x * inverse x * x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x * (inverse x * x) = x * inverse x * x\n[PROOF STEP]\nby (simp only: group_assoc)\n[PROOF STATE]\nproof (state)\nthis:\nx * (inverse x * x) = x * inverse x * x\n\ngoal (1 subgoal):\n 1. x * (1::'a) = x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nx * (inverse x * x) = x * inverse x * x\n\ngoal (1 subgoal):\n 1. x * (1::'a) = x\n[PROOF STEP]\nhave \"\\<dots> = 1 * x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x * inverse x * x = (1::'a) * x\n[PROOF STEP]\nby (simp only: group_right_inverse)\n[PROOF STATE]\nproof (state)\nthis:\nx * inverse x * x = (1::'a) * x\n\ngoal (1 subgoal):\n 1. x * (1::'a) = x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nx * inverse x * x = (1::'a) * x\n\ngoal (1 subgoal):\n 1. x * (1::'a) = x\n[PROOF STEP]\nhave \"\\<dots> = x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (1::'a) * x = x\n[PROOF STEP]\nby (simp only: group_left_one)\n[PROOF STATE]\nproof (state)\nthis:\n(1::'a) * x = x\n\ngoal (1 subgoal):\n 1. x * (1::'a) = x\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nx * (1::'a) = x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nx * (1::'a) = x\n\ngoal (1 subgoal):\n 1. x * (1::'a) = x\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nx * (1::'a) = x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 980, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278664544911, "lm_q2_score": 0.880797071719777, "lm_q1q2_score": 0.7772398807770462}}
{"text": "[STATEMENT]\nlemma det_interchange_columns:\nshows \"det (interchange_columns A i j) = of_int (if i = j then 1 else -1) * det A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (interchange_columns A i j) = of_int (if i = j then 1 else - 1) * det A\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. det (interchange_columns A i j) = of_int (if i = j then 1 else - 1) * det A\n[PROOF STEP]\nhave \"(interchange_columns A i j) = (\\<chi> a b. A $ a $ (Transposition.transpose i j) b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. interchange_columns A i j = (\\<chi>a b. A $ a $ Transposition.transpose i j b)\n[PROOF STEP]\nunfolding interchange_columns_def Transposition.transpose_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<chi>ia ja. if ja = i then A $ ia $ j else if ja = j then A $ ia $ i else A $ ia $ ja) = (\\<chi>a b. A $ a $ (if b = i then j else if b = j then i else b))\n[PROOF STEP]\nby vector\n[PROOF STATE]\nproof (state)\nthis:\ninterchange_columns A i j = (\\<chi>a b. A $ a $ Transposition.transpose i j b)\n\ngoal (1 subgoal):\n 1. det (interchange_columns A i j) = of_int (if i = j then 1 else - 1) * det A\n[PROOF STEP]\nhence \"det(interchange_columns A i j) = det(\\<chi> a b. A $ a $ (Transposition.transpose i j) b)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ninterchange_columns A i j = (\\<chi>a b. A $ a $ Transposition.transpose i j b)\n\ngoal (1 subgoal):\n 1. det (interchange_columns A i j) = det (\\<chi>a b. A $ a $ Transposition.transpose i j b)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndet (interchange_columns A i j) = det (\\<chi>a b. A $ a $ Transposition.transpose i j b)\n\ngoal (1 subgoal):\n 1. det (interchange_columns A i j) = of_int (if i = j then 1 else - 1) * det A\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndet (interchange_columns A i j) = det (\\<chi>a b. A $ a $ Transposition.transpose i j b)\n\ngoal (1 subgoal):\n 1. det (interchange_columns A i j) = of_int (if i = j then 1 else - 1) * det A\n[PROOF STEP]\nhave \"... = of_int (sign (Transposition.transpose i j)) * det A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det (\\<chi>a b. A $ a $ Transposition.transpose i j b) = of_int (sign (Transposition.transpose i j)) * det A\n[PROOF STEP]\nby (rule det_permute_columns[of \"Transposition.transpose i j\" A], simp add: permutes_swap_id)\n[PROOF STATE]\nproof (state)\nthis:\ndet (\\<chi>a b. A $ a $ Transposition.transpose i j b) = of_int (sign (Transposition.transpose i j)) * det A\n\ngoal (1 subgoal):\n 1. det (interchange_columns A i j) = of_int (if i = j then 1 else - 1) * det A\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ndet (interchange_columns A i j) = of_int (sign (Transposition.transpose i j)) * det A\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndet (interchange_columns A i j) = of_int (sign (Transposition.transpose i j)) * det A\n\ngoal (1 subgoal):\n 1. det (interchange_columns A i j) = of_int (if i = j then 1 else - 1) * det A\n[PROOF STEP]\nunfolding sign_swap_id\n[PROOF STATE]\nproof (prove)\nusing this:\ndet (interchange_columns A i j) = of_int (if i = j then 1 else - 1) * det A\n\ngoal (1 subgoal):\n 1. det (interchange_columns A i j) = of_int (if i = j then 1 else - 1) * det A\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ndet (interchange_columns A i j) = of_int (if i = j then 1 else - 1) * det A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1370, "file": "Gauss_Jordan_Determinants2", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513620489619, "lm_q2_score": 0.8670357666736773, "lm_q1q2_score": 0.7770819868264492}}
{"text": "[STATEMENT]\nlemma card_extensional_funcset_not_inj_on:\n  assumes \"finite S\" \"finite T\" \"card S \\<le> card T\"\n  shows \"card {f \\<in> extensional_funcset S T. \\<not> inj_on f S} = (card T) ^ (card S) - (fact (card T)) div (fact (card T - card S))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {f \\<in> S \\<rightarrow>\\<^sub>E T. \\<not> inj_on f S} = card T ^ card S - fact (card T) div fact (card T - card S)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {f \\<in> S \\<rightarrow>\\<^sub>E T. \\<not> inj_on f S} = card T ^ card S - fact (card T) div fact (card T - card S)\n[PROOF STEP]\nhave subset: \"{f : extensional_funcset S T. inj_on f S} <= extensional_funcset S T\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {f \\<in> S \\<rightarrow>\\<^sub>E T. inj_on f S} \\<subseteq> S \\<rightarrow>\\<^sub>E T\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{f \\<in> S \\<rightarrow>\\<^sub>E T. inj_on f S} \\<subseteq> S \\<rightarrow>\\<^sub>E T\n\ngoal (1 subgoal):\n 1. card {f \\<in> S \\<rightarrow>\\<^sub>E T. \\<not> inj_on f S} = card T ^ card S - fact (card T) div fact (card T - card S)\n[PROOF STEP]\nfrom finite_subset[OF subset] assms\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite (S \\<rightarrow>\\<^sub>E T) \\<Longrightarrow> finite {f \\<in> S \\<rightarrow>\\<^sub>E T. inj_on f S}\nfinite S\nfinite T\ncard S \\<le> card T\n[PROOF STEP]\nhave finite: \"finite {f : extensional_funcset S T. inj_on f S}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite (S \\<rightarrow>\\<^sub>E T) \\<Longrightarrow> finite {f \\<in> S \\<rightarrow>\\<^sub>E T. inj_on f S}\nfinite S\nfinite T\ncard S \\<le> card T\n\ngoal (1 subgoal):\n 1. finite {f \\<in> S \\<rightarrow>\\<^sub>E T. inj_on f S}\n[PROOF STEP]\nby (auto intro!: finite_PiE)\n[PROOF STATE]\nproof (state)\nthis:\nfinite {f \\<in> S \\<rightarrow>\\<^sub>E T. inj_on f S}\n\ngoal (1 subgoal):\n 1. card {f \\<in> S \\<rightarrow>\\<^sub>E T. \\<not> inj_on f S} = card T ^ card S - fact (card T) div fact (card T - card S)\n[PROOF STEP]\nhave \"{f \\<in> extensional_funcset S T. \\<not> inj_on f S} = extensional_funcset S T - {f \\<in> extensional_funcset S T. inj_on f S}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {f \\<in> S \\<rightarrow>\\<^sub>E T. \\<not> inj_on f S} = (S \\<rightarrow>\\<^sub>E T) - {f \\<in> S \\<rightarrow>\\<^sub>E T. inj_on f S}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{f \\<in> S \\<rightarrow>\\<^sub>E T. \\<not> inj_on f S} = (S \\<rightarrow>\\<^sub>E T) - {f \\<in> S \\<rightarrow>\\<^sub>E T. inj_on f S}\n\ngoal (1 subgoal):\n 1. card {f \\<in> S \\<rightarrow>\\<^sub>E T. \\<not> inj_on f S} = card T ^ card S - fact (card T) div fact (card T - card S)\n[PROOF STEP]\nfrom assms this finite subset\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite S\nfinite T\ncard S \\<le> card T\n{f \\<in> S \\<rightarrow>\\<^sub>E T. \\<not> inj_on f S} = (S \\<rightarrow>\\<^sub>E T) - {f \\<in> S \\<rightarrow>\\<^sub>E T. inj_on f S}\nfinite {f \\<in> S \\<rightarrow>\\<^sub>E T. inj_on f S}\n{f \\<in> S \\<rightarrow>\\<^sub>E T. inj_on f S} \\<subseteq> S \\<rightarrow>\\<^sub>E T\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite S\nfinite T\ncard S \\<le> card T\n{f \\<in> S \\<rightarrow>\\<^sub>E T. \\<not> inj_on f S} = (S \\<rightarrow>\\<^sub>E T) - {f \\<in> S \\<rightarrow>\\<^sub>E T. inj_on f S}\nfinite {f \\<in> S \\<rightarrow>\\<^sub>E T. inj_on f S}\n{f \\<in> S \\<rightarrow>\\<^sub>E T. inj_on f S} \\<subseteq> S \\<rightarrow>\\<^sub>E T\n\ngoal (1 subgoal):\n 1. card {f \\<in> S \\<rightarrow>\\<^sub>E T. \\<not> inj_on f S} = card T ^ card S - fact (card T) div fact (card T - card S)\n[PROOF STEP]\nby (simp add: card_Diff_subset card_PiE card_extensional_funcset_inj_on prod_constant)\n[PROOF STATE]\nproof (state)\nthis:\ncard {f \\<in> S \\<rightarrow>\\<^sub>E T. \\<not> inj_on f S} = card T ^ card S - fact (card T) div fact (card T - card S)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1548, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513703624557, "lm_q2_score": 0.8670357494949105, "lm_q1q2_score": 0.7770819786380524}}
{"text": "[STATEMENT]\nlemma iso_same_card: \"G \\<cong> H \\<Longrightarrow> card (carrier G) = card (carrier H)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. G \\<cong> H \\<Longrightarrow> card (carrier G) = card (carrier H)\n[PROOF STEP]\nusing bij_betw_same_card\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw ?f ?A ?B \\<Longrightarrow> card ?A = card ?B\n\ngoal (1 subgoal):\n 1. G \\<cong> H \\<Longrightarrow> card (carrier G) = card (carrier H)\n[PROOF STEP]\nunfolding is_iso_def iso_def\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw ?f ?A ?B \\<Longrightarrow> card ?A = card ?B\n\ngoal (1 subgoal):\n 1. {h \\<in> hom G H. bij_betw h (carrier G) (carrier H)} \\<noteq> {} \\<Longrightarrow> card (carrier G) = card (carrier H)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 296, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9099070035949657, "lm_q2_score": 0.853912760387131, "lm_q1q2_score": 0.7769812011353603}}
{"text": "[STATEMENT]\nlemma card_bijections_domain_and_range_permutation_eq_1:\n  assumes \"finite A\" \"finite B\"\n  assumes \"card A = card B\"\n  shows \"card ({f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ({f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B) = 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card ({f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B) = 1\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\nfinite B\ncard A = card B\n[PROOF STEP]\nhave \"{f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B =\n    {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} // domain_and_range_permutation A B\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nfinite B\ncard A = card B\n\ngoal (1 subgoal):\n 1. {f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B = {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} // domain_and_range_permutation A B\n[PROOF STEP]\nby (metis (no_types, lifting) PiE_cong bij_betw_implies_surj_on_and_card_eq)\n[PROOF STATE]\nproof (state)\nthis:\n{f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B = {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} // domain_and_range_permutation A B\n\ngoal (1 subgoal):\n 1. card ({f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B) = 1\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\n{f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B = {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} // domain_and_range_permutation A B\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n{f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B = {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} // domain_and_range_permutation A B\n\ngoal (1 subgoal):\n 1. card ({f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B) = 1\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n{f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B = {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} // domain_and_range_permutation A B\nfinite A\nfinite B\ncard A = card B\n\ngoal (1 subgoal):\n 1. card ({f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B) = 1\n[PROOF STEP]\nby (simp add: card_surjective_functions_domain_and_range_permutation Partition_diag)\n[PROOF STATE]\nproof (state)\nthis:\ncard ({f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B) = 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1149, "file": "Twelvefold_Way_Card_Bijections", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9032942014971872, "lm_q2_score": 0.8596637433190939, "lm_q1q2_score": 0.7765292745775039}}
{"text": "[STATEMENT]\nlemma op_norm_eq_0: \"(\\<parallel>A\\<parallel>\\<^sub>o\\<^sub>p = 0) = (A = 0)\" for A :: \"('a::real_normed_field)^'n^'m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<parallel>A\\<parallel>\\<^sub>o\\<^sub>p = 0) = (A = 0)\n[PROOF STEP]\nunfolding onorm_eq_0[OF blin_matrix_vector_mult]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<forall>x. A *v x = 0) = (A = 0)\n[PROOF STEP]\nusing matrix_axis_0[of 1 A]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>(1::'a) \\<noteq> (0::'a); \\<forall>i. A *v \\<e> i = 0\\<rbrakk> \\<Longrightarrow> A = 0\n\ngoal (1 subgoal):\n 1. (\\<forall>x. A *v x = 0) = (A = 0)\n[PROOF STEP]\nby fastforce", "meta": {"llama_tokens": 313, "file": "Matrices_for_ODEs_MTX_Norms", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9314625107731765, "lm_q2_score": 0.8333246015211009, "lm_q1q2_score": 0.7762106256219015}}
{"text": "[STATEMENT]\nlemma euclid_ext2_works:\n  assumes \"euclid_ext2 a b = (p,q,u,v,d)\"\n  shows \"p*a+q*b = d\" and \"d = gcd a b\" and \"gcd a b * u = -b\" and \"gcd a b * v = a\"\n  and \"u = -b div gcd a b\" and \"v = a div gcd a b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (p * a + q * b = d &&& d = gcd a b &&& gcd a b * u = - b) &&& gcd a b * v = a &&& u = - b div gcd a b &&& v = a div gcd a b\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\neuclid_ext2 a b = (p, q, u, v, d)\n\ngoal (1 subgoal):\n 1. (p * a + q * b = d &&& d = gcd a b &&& gcd a b * u = - b) &&& gcd a b * v = a &&& u = - b div gcd a b &&& v = a div gcd a b\n[PROOF STEP]\nunfolding euclid_ext2_def\n[PROOF STATE]\nproof (prove)\nusing this:\n(fst (bezout_coefficients a b), snd (bezout_coefficients a b), - b div gcd a b, a div gcd a b, gcd a b) = (p, q, u, v, d)\n\ngoal (1 subgoal):\n 1. (p * a + q * b = d &&& d = gcd a b &&& gcd a b * u = - b) &&& gcd a b * v = a &&& u = - b div gcd a b &&& v = a div gcd a b\n[PROOF STEP]\nby (auto simp add: bezout_coefficients_fst_snd)", "meta": {"llama_tokens": 514, "file": "Modular_arithmetic_LLL_and_HNF_algorithms_HNF_Mod_Det_Soundness", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9314624993576758, "lm_q2_score": 0.8333246015211008, "lm_q1q2_score": 0.7762106161090837}}
{"text": "[STATEMENT]\nlemma cosine_law_triangle:\n  \"dist b c ^ 2 = dist a b ^ 2 + dist a c ^ 2 - 2 * dist a b * dist a c * cos (angle b a c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (dist b c)\\<^sup>2 = (dist a b)\\<^sup>2 + (dist a c)\\<^sup>2 - 2 * dist a b * dist a c * cos (angle b a c)\n[PROOF STEP]\nusing cosine_law_vector[of \"b - a\" \"c - a\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n(norm (b - a - (c - a)))\\<^sup>2 = (norm (b - a))\\<^sup>2 + (norm (c - a))\\<^sup>2 - 2 * norm (b - a) * norm (c - a) * cos (vangle (b - a) (c - a))\n\ngoal (1 subgoal):\n 1. (dist b c)\\<^sup>2 = (dist a b)\\<^sup>2 + (dist a c)\\<^sup>2 - 2 * dist a b * dist a c * cos (angle b a c)\n[PROOF STEP]\nby (simp add: dist_norm angle_def vangle_commute norm_minus_commute)", "meta": {"llama_tokens": 345, "file": "Triangle_Triangle", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9362850057480347, "lm_q2_score": 0.8289388146603364, "lm_q1q2_score": 0.7761229828490221}}
{"text": "[STATEMENT]\nlemma inner_prod_adjoint_comp:\n  assumes \"(U::'a::conjugatable_field Matrix.mat) \\<in> carrier_mat n n\"\nand \"(V::'a::conjugatable_field Matrix.mat) \\<in> carrier_mat n n\"\nand \"i < n\"\nand \"j < n\"\nshows \"Complex_Matrix.inner_prod  (Matrix.col V i) (Matrix.col U j) = \n  ((Complex_Matrix.adjoint V) * U) $$ (i, j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (Matrix.col V i) (Matrix.col U j) = (Complex_Matrix.adjoint V * U) $$ (i, j)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (Matrix.col V i) (Matrix.col U j) = (Complex_Matrix.adjoint V * U) $$ (i, j)\n[PROOF STEP]\nhave \"Complex_Matrix.inner_prod (Matrix.col V i) (Matrix.col U j) = \n    Matrix.scalar_prod (Matrix.col U j) (Matrix.row (Complex_Matrix.adjoint V) i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (Matrix.col V i) (Matrix.col U j) = Matrix.col U j \\<bullet> Matrix.row (Complex_Matrix.adjoint V) i\n[PROOF STEP]\nusing adjoint_row[of i V] assms\n[PROOF STATE]\nproof (prove)\nusing this:\ni < dim_col V \\<Longrightarrow> Matrix.row (Complex_Matrix.adjoint V) i = conjugate (Matrix.col V i)\nU \\<in> carrier_mat n n\nV \\<in> carrier_mat n n\ni < n\nj < n\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (Matrix.col V i) (Matrix.col U j) = Matrix.col U j \\<bullet> Matrix.row (Complex_Matrix.adjoint V) i\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod (Matrix.col V i) (Matrix.col U j) = Matrix.col U j \\<bullet> Matrix.row (Complex_Matrix.adjoint V) i\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (Matrix.col V i) (Matrix.col U j) = (Complex_Matrix.adjoint V * U) $$ (i, j)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod (Matrix.col V i) (Matrix.col U j) = Matrix.col U j \\<bullet> Matrix.row (Complex_Matrix.adjoint V) i\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (Matrix.col V i) (Matrix.col U j) = (Complex_Matrix.adjoint V * U) $$ (i, j)\n[PROOF STEP]\nhave \"... = Matrix.scalar_prod (Matrix.row (Complex_Matrix.adjoint V) i) (Matrix.col U j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Matrix.col U j \\<bullet> Matrix.row (Complex_Matrix.adjoint V) i = Matrix.row (Complex_Matrix.adjoint V) i \\<bullet> Matrix.col U j\n[PROOF STEP]\nby (metis adjoint_row assms(1) assms(2) assms(3) carrier_matD(1) carrier_matD(2) Matrix.col_dim \n        conjugate_vec_sprod_comm)\n[PROOF STATE]\nproof (state)\nthis:\nMatrix.col U j \\<bullet> Matrix.row (Complex_Matrix.adjoint V) i = Matrix.row (Complex_Matrix.adjoint V) i \\<bullet> Matrix.col U j\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (Matrix.col V i) (Matrix.col U j) = (Complex_Matrix.adjoint V * U) $$ (i, j)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nMatrix.col U j \\<bullet> Matrix.row (Complex_Matrix.adjoint V) i = Matrix.row (Complex_Matrix.adjoint V) i \\<bullet> Matrix.col U j\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (Matrix.col V i) (Matrix.col U j) = (Complex_Matrix.adjoint V * U) $$ (i, j)\n[PROOF STEP]\nhave \"... = ((Complex_Matrix.adjoint V) * U) $$ (i, j)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Matrix.row (Complex_Matrix.adjoint V) i \\<bullet> Matrix.col U j = (Complex_Matrix.adjoint V * U) $$ (i, j)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nU \\<in> carrier_mat n n\nV \\<in> carrier_mat n n\ni < n\nj < n\n\ngoal (1 subgoal):\n 1. Matrix.row (Complex_Matrix.adjoint V) i \\<bullet> Matrix.col U j = (Complex_Matrix.adjoint V * U) $$ (i, j)\n[PROOF STEP]\nby (simp add:times_mat_def)\n[PROOF STATE]\nproof (state)\nthis:\nMatrix.row (Complex_Matrix.adjoint V) i \\<bullet> Matrix.col U j = (Complex_Matrix.adjoint V * U) $$ (i, j)\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (Matrix.col V i) (Matrix.col U j) = (Complex_Matrix.adjoint V * U) $$ (i, j)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nComplex_Matrix.inner_prod (Matrix.col V i) (Matrix.col U j) = (Complex_Matrix.adjoint V * U) $$ (i, j)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nComplex_Matrix.inner_prod (Matrix.col V i) (Matrix.col U j) = (Complex_Matrix.adjoint V * U) $$ (i, j)\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod (Matrix.col V i) (Matrix.col U j) = (Complex_Matrix.adjoint V * U) $$ (i, j)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod (Matrix.col V i) (Matrix.col U j) = (Complex_Matrix.adjoint V * U) $$ (i, j)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1843, "file": "Projective_Measurements_Linear_Algebra_Complements", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9173026550642019, "lm_q2_score": 0.84594244507642, "lm_q1q2_score": 0.775985250900103}}
{"text": "[STATEMENT]\nlemma bst_of_list_map: \n  fixes f :: \"'a :: linorder \\<Rightarrow> 'b :: linorder\"\n  assumes \"strict_mono_on A f\" \"set xs \\<subseteq> A\"\n  shows   \"bst_of_list (map f xs) = map_tree f (bst_of_list xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bst_of_list (map f xs) = map_tree f (bst_of_list xs)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nstrict_mono_on A f\nset xs \\<subseteq> A\n\ngoal (1 subgoal):\n 1. bst_of_list (map f xs) = map_tree f (bst_of_list xs)\n[PROOF STEP]\nproof (induction xs rule: bst_of_list.induct)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\<lbrakk>strict_mono_on A f; set [] \\<subseteq> A\\<rbrakk> \\<Longrightarrow> bst_of_list (map f []) = map_tree f (bst_of_list [])\n 2. \\<And>x xs. \\<lbrakk>\\<lbrakk>strict_mono_on A f; set (filter (\\<lambda>y. y < x) xs) \\<subseteq> A\\<rbrakk> \\<Longrightarrow> bst_of_list (map f (filter (\\<lambda>y. y < x) xs)) = map_tree f (bst_of_list (filter (\\<lambda>y. y < x) xs)); \\<lbrakk>strict_mono_on A f; set (filter ((<) x) xs) \\<subseteq> A\\<rbrakk> \\<Longrightarrow> bst_of_list (map f (filter ((<) x) xs)) = map_tree f (bst_of_list (filter ((<) x) xs)); strict_mono_on A f; set (x # xs) \\<subseteq> A\\<rbrakk> \\<Longrightarrow> bst_of_list (map f (x # xs)) = map_tree f (bst_of_list (x # xs))\n[PROOF STEP]\ncase (2 x xs)\n[PROOF STATE]\nproof (state)\nthis:\n\\<lbrakk>strict_mono_on A f; set (filter (\\<lambda>y. y < x) xs) \\<subseteq> A\\<rbrakk> \\<Longrightarrow> bst_of_list (map f (filter (\\<lambda>y. y < x) xs)) = map_tree f (bst_of_list (filter (\\<lambda>y. y < x) xs))\n\\<lbrakk>strict_mono_on A f; set (filter ((<) x) xs) \\<subseteq> A\\<rbrakk> \\<Longrightarrow> bst_of_list (map f (filter ((<) x) xs)) = map_tree f (bst_of_list (filter ((<) x) xs))\nstrict_mono_on A f\nset (x # xs) \\<subseteq> A\n\ngoal (2 subgoals):\n 1. \\<lbrakk>strict_mono_on A f; set [] \\<subseteq> A\\<rbrakk> \\<Longrightarrow> bst_of_list (map f []) = map_tree f (bst_of_list [])\n 2. \\<And>x xs. \\<lbrakk>\\<lbrakk>strict_mono_on A f; set (filter (\\<lambda>y. y < x) xs) \\<subseteq> A\\<rbrakk> \\<Longrightarrow> bst_of_list (map f (filter (\\<lambda>y. y < x) xs)) = map_tree f (bst_of_list (filter (\\<lambda>y. y < x) xs)); \\<lbrakk>strict_mono_on A f; set (filter ((<) x) xs) \\<subseteq> A\\<rbrakk> \\<Longrightarrow> bst_of_list (map f (filter ((<) x) xs)) = map_tree f (bst_of_list (filter ((<) x) xs)); strict_mono_on A f; set (x # xs) \\<subseteq> A\\<rbrakk> \\<Longrightarrow> bst_of_list (map f (x # xs)) = map_tree f (bst_of_list (x # xs))\n[PROOF STEP]\nhave \"[xa\\<leftarrow>xs . f xa < f x] = [xa\\<leftarrow>xs . xa < x]\" and \"[xa\\<leftarrow>xs . f xa > f x] = [xa\\<leftarrow>xs . xa > x]\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. filter (\\<lambda>xa. f xa < f x) xs = filter (\\<lambda>xa. xa < x) xs &&& filter (\\<lambda>xa. f x < f xa) xs = filter ((<) x) xs\n[PROOF STEP]\nusing \"2.prems\"\n[PROOF STATE]\nproof (prove)\nusing this:\nstrict_mono_on A f\nset (x # xs) \\<subseteq> A\n\ngoal (1 subgoal):\n 1. filter (\\<lambda>xa. f xa < f x) xs = filter (\\<lambda>xa. xa < x) xs &&& filter (\\<lambda>xa. f x < f xa) xs = filter ((<) x) xs\n[PROOF STEP]\nby (auto simp: strict_mono_on_imp_less_iff intro!: filter_cong)\n[PROOF STATE]\nproof (state)\nthis:\nfilter (\\<lambda>xa. f xa < f x) xs = filter (\\<lambda>xa. xa < x) xs\nfilter (\\<lambda>xa. f x < f xa) xs = filter ((<) x) xs\n\ngoal (2 subgoals):\n 1. \\<lbrakk>strict_mono_on A f; set [] \\<subseteq> A\\<rbrakk> \\<Longrightarrow> bst_of_list (map f []) = map_tree f (bst_of_list [])\n 2. \\<And>x xs. \\<lbrakk>\\<lbrakk>strict_mono_on A f; set (filter (\\<lambda>y. y < x) xs) \\<subseteq> A\\<rbrakk> \\<Longrightarrow> bst_of_list (map f (filter (\\<lambda>y. y < x) xs)) = map_tree f (bst_of_list (filter (\\<lambda>y. y < x) xs)); \\<lbrakk>strict_mono_on A f; set (filter ((<) x) xs) \\<subseteq> A\\<rbrakk> \\<Longrightarrow> bst_of_list (map f (filter ((<) x) xs)) = map_tree f (bst_of_list (filter ((<) x) xs)); strict_mono_on A f; set (x # xs) \\<subseteq> A\\<rbrakk> \\<Longrightarrow> bst_of_list (map f (x # xs)) = map_tree f (bst_of_list (x # xs))\n[PROOF STEP]\nwith 2\n[PROOF STATE]\nproof (chain)\npicking this:\n\\<lbrakk>strict_mono_on A f; set (filter (\\<lambda>y. y < x) xs) \\<subseteq> A\\<rbrakk> \\<Longrightarrow> bst_of_list (map f (filter (\\<lambda>y. y < x) xs)) = map_tree f (bst_of_list (filter (\\<lambda>y. y < x) xs))\n\\<lbrakk>strict_mono_on A f; set (filter ((<) x) xs) \\<subseteq> A\\<rbrakk> \\<Longrightarrow> bst_of_list (map f (filter ((<) x) xs)) = map_tree f (bst_of_list (filter ((<) x) xs))\nstrict_mono_on A f\nset (x # xs) \\<subseteq> A\nfilter (\\<lambda>xa. f xa < f x) xs = filter (\\<lambda>xa. xa < x) xs\nfilter (\\<lambda>xa. f x < f xa) xs = filter ((<) x) xs\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>strict_mono_on A f; set (filter (\\<lambda>y. y < x) xs) \\<subseteq> A\\<rbrakk> \\<Longrightarrow> bst_of_list (map f (filter (\\<lambda>y. y < x) xs)) = map_tree f (bst_of_list (filter (\\<lambda>y. y < x) xs))\n\\<lbrakk>strict_mono_on A f; set (filter ((<) x) xs) \\<subseteq> A\\<rbrakk> \\<Longrightarrow> bst_of_list (map f (filter ((<) x) xs)) = map_tree f (bst_of_list (filter ((<) x) xs))\nstrict_mono_on A f\nset (x # xs) \\<subseteq> A\nfilter (\\<lambda>xa. f xa < f x) xs = filter (\\<lambda>xa. xa < x) xs\nfilter (\\<lambda>xa. f x < f xa) xs = filter ((<) x) xs\n\ngoal (1 subgoal):\n 1. bst_of_list (map f (x # xs)) = map_tree f (bst_of_list (x # xs))\n[PROOF STEP]\nby (auto simp: filter_map o_def)\n[PROOF STATE]\nproof (state)\nthis:\nbst_of_list (map f (x # xs)) = map_tree f (bst_of_list (x # xs))\n\ngoal (1 subgoal):\n 1. \\<lbrakk>strict_mono_on A f; set [] \\<subseteq> A\\<rbrakk> \\<Longrightarrow> bst_of_list (map f []) = map_tree f (bst_of_list [])\n[PROOF STEP]\nqed auto", "meta": {"llama_tokens": 2439, "file": "Random_BSTs_Random_BSTs", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9086178895092414, "lm_q2_score": 0.8539127548105611, "lm_q1q2_score": 0.7758804051009943}}
{"text": "[STATEMENT]\nlemma subseteq_list_Union_mset:\n  assumes \"length Ci = n\"\n  assumes \"length CAi = n\"\n  assumes \"\\<forall>i<n.  Ci ! i \\<subseteq># CAi ! i \"\n  shows \"\\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nlength Ci = n\nlength CAi = n\n\\<forall>i<n. Ci ! i \\<subseteq># CAi ! i\n\ngoal (1 subgoal):\n 1. \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi)\n[PROOF STEP]\nproof (induction n arbitrary: Ci CAi)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\<And>Ci CAi. \\<lbrakk>length Ci = 0; length CAi = 0; \\<forall>i<0. Ci ! i \\<subseteq># CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi)\n 2. \\<And>n Ci CAi. \\<lbrakk>\\<And>Ci CAi. \\<lbrakk>length Ci = n; length CAi = n; \\<forall>i<n. Ci ! i \\<subseteq># CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi); length Ci = Suc n; length CAi = Suc n; \\<forall>i<Suc n. Ci ! i \\<subseteq># CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\nlength Ci = 0\nlength CAi = 0\n\\<forall>i<0. Ci ! i \\<subseteq># CAi ! i\n\ngoal (2 subgoals):\n 1. \\<And>Ci CAi. \\<lbrakk>length Ci = 0; length CAi = 0; \\<forall>i<0. Ci ! i \\<subseteq># CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi)\n 2. \\<And>n Ci CAi. \\<lbrakk>\\<And>Ci CAi. \\<lbrakk>length Ci = n; length CAi = n; \\<forall>i<n. Ci ! i \\<subseteq># CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi); length Ci = Suc n; length CAi = Suc n; \\<forall>i<Suc n. Ci ! i \\<subseteq># CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nlength Ci = 0\nlength CAi = 0\n\\<forall>i<0. Ci ! i \\<subseteq># CAi ! i\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nlength Ci = 0\nlength CAi = 0\n\\<forall>i<0. Ci ! i \\<subseteq># CAi ! i\n\ngoal (1 subgoal):\n 1. \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi)\n\ngoal (1 subgoal):\n 1. \\<And>n Ci CAi. \\<lbrakk>\\<And>Ci CAi. \\<lbrakk>length Ci = n; length CAi = n; \\<forall>i<n. Ci ! i \\<subseteq># CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi); length Ci = Suc n; length CAi = Suc n; \\<forall>i<Suc n. Ci ! i \\<subseteq># CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>n Ci CAi. \\<lbrakk>\\<And>Ci CAi. \\<lbrakk>length Ci = n; length CAi = n; \\<forall>i<n. Ci ! i \\<subseteq># CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi); length Ci = Suc n; length CAi = Suc n; \\<forall>i<Suc n. Ci ! i \\<subseteq># CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi)\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\n\\<lbrakk>length ?Ci = n; length ?CAi = n; \\<forall>i<n. ?Ci ! i \\<subseteq># ?CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset ?Ci) \\<subseteq># \\<Sum>\\<^sub># (mset ?CAi)\nlength Ci = Suc n\nlength CAi = Suc n\n\\<forall>i<Suc n. Ci ! i \\<subseteq># CAi ! i\n\ngoal (1 subgoal):\n 1. \\<And>n Ci CAi. \\<lbrakk>\\<And>Ci CAi. \\<lbrakk>length Ci = n; length CAi = n; \\<forall>i<n. Ci ! i \\<subseteq># CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi); length Ci = Suc n; length CAi = Suc n; \\<forall>i<Suc n. Ci ! i \\<subseteq># CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi)\n[PROOF STEP]\nfrom Suc\n[PROOF STATE]\nproof (chain)\npicking this:\n\\<lbrakk>length ?Ci = n; length ?CAi = n; \\<forall>i<n. ?Ci ! i \\<subseteq># ?CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset ?Ci) \\<subseteq># \\<Sum>\\<^sub># (mset ?CAi)\nlength Ci = Suc n\nlength CAi = Suc n\n\\<forall>i<Suc n. Ci ! i \\<subseteq># CAi ! i\n[PROOF STEP]\nhave \"\\<forall>i<n. tl Ci ! i \\<subseteq># tl CAi ! i\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>length ?Ci = n; length ?CAi = n; \\<forall>i<n. ?Ci ! i \\<subseteq># ?CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset ?Ci) \\<subseteq># \\<Sum>\\<^sub># (mset ?CAi)\nlength Ci = Suc n\nlength CAi = Suc n\n\\<forall>i<Suc n. Ci ! i \\<subseteq># CAi ! i\n\ngoal (1 subgoal):\n 1. \\<forall>i<n. tl Ci ! i \\<subseteq># tl CAi ! i\n[PROOF STEP]\nby (simp add: nth_tl)\n[PROOF STATE]\nproof (state)\nthis:\n\\<forall>i<n. tl Ci ! i \\<subseteq># tl CAi ! i\n\ngoal (1 subgoal):\n 1. \\<And>n Ci CAi. \\<lbrakk>\\<And>Ci CAi. \\<lbrakk>length Ci = n; length CAi = n; \\<forall>i<n. Ci ! i \\<subseteq># CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi); length Ci = Suc n; length CAi = Suc n; \\<forall>i<Suc n. Ci ! i \\<subseteq># CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi)\n[PROOF STEP]\nhence \"\\<Sum>\\<^sub>#(mset (tl Ci)) \\<subseteq># \\<Sum>\\<^sub>#(mset (tl CAi))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<forall>i<n. tl Ci ! i \\<subseteq># tl CAi ! i\n\ngoal (1 subgoal):\n 1. \\<Sum>\\<^sub># (mset (tl Ci)) \\<subseteq># \\<Sum>\\<^sub># (mset (tl CAi))\n[PROOF STEP]\nusing Suc\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<forall>i<n. tl Ci ! i \\<subseteq># tl CAi ! i\n\\<lbrakk>length ?Ci = n; length ?CAi = n; \\<forall>i<n. ?Ci ! i \\<subseteq># ?CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset ?Ci) \\<subseteq># \\<Sum>\\<^sub># (mset ?CAi)\nlength Ci = Suc n\nlength CAi = Suc n\n\\<forall>i<Suc n. Ci ! i \\<subseteq># CAi ! i\n\ngoal (1 subgoal):\n 1. \\<Sum>\\<^sub># (mset (tl Ci)) \\<subseteq># \\<Sum>\\<^sub># (mset (tl CAi))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\<Sum>\\<^sub># (mset (tl Ci)) \\<subseteq># \\<Sum>\\<^sub># (mset (tl CAi))\n\ngoal (1 subgoal):\n 1. \\<And>n Ci CAi. \\<lbrakk>\\<And>Ci CAi. \\<lbrakk>length Ci = n; length CAi = n; \\<forall>i<n. Ci ! i \\<subseteq># CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi); length Ci = Suc n; length CAi = Suc n; \\<forall>i<Suc n. Ci ! i \\<subseteq># CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\<Sum>\\<^sub># (mset (tl Ci)) \\<subseteq># \\<Sum>\\<^sub># (mset (tl CAi))\n\ngoal (1 subgoal):\n 1. \\<And>n Ci CAi. \\<lbrakk>\\<And>Ci CAi. \\<lbrakk>length Ci = n; length CAi = n; \\<forall>i<n. Ci ! i \\<subseteq># CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi); length Ci = Suc n; length CAi = Suc n; \\<forall>i<Suc n. Ci ! i \\<subseteq># CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi)\n[PROOF STEP]\nhave \"hd Ci \\<subseteq># hd CAi\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. hd Ci \\<subseteq># hd CAi\n[PROOF STEP]\nusing Suc\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>length ?Ci = n; length ?CAi = n; \\<forall>i<n. ?Ci ! i \\<subseteq># ?CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset ?Ci) \\<subseteq># \\<Sum>\\<^sub># (mset ?CAi)\nlength Ci = Suc n\nlength CAi = Suc n\n\\<forall>i<Suc n. Ci ! i \\<subseteq># CAi ! i\n\ngoal (1 subgoal):\n 1. hd Ci \\<subseteq># hd CAi\n[PROOF STEP]\nby (metis hd_conv_nth length_greater_0_conv zero_less_Suc)\n[PROOF STATE]\nproof (state)\nthis:\nhd Ci \\<subseteq># hd CAi\n\ngoal (1 subgoal):\n 1. \\<And>n Ci CAi. \\<lbrakk>\\<And>Ci CAi. \\<lbrakk>length Ci = n; length CAi = n; \\<forall>i<n. Ci ! i \\<subseteq># CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi); length Ci = Suc n; length CAi = Suc n; \\<forall>i<Suc n. Ci ! i \\<subseteq># CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n\\<Sum>\\<^sub># (mset (tl Ci)) \\<subseteq># \\<Sum>\\<^sub># (mset (tl CAi))\nhd Ci \\<subseteq># hd CAi\n[PROOF STEP]\nshow \"\\<Sum>\\<^sub>#(mset Ci) \\<subseteq># \\<Sum>\\<^sub>#(mset CAi)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<Sum>\\<^sub># (mset (tl Ci)) \\<subseteq># \\<Sum>\\<^sub># (mset (tl CAi))\nhd Ci \\<subseteq># hd CAi\n\ngoal (1 subgoal):\n 1. \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi)\n[PROOF STEP]\nusing Suc\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<Sum>\\<^sub># (mset (tl Ci)) \\<subseteq># \\<Sum>\\<^sub># (mset (tl CAi))\nhd Ci \\<subseteq># hd CAi\n\\<lbrakk>length ?Ci = n; length ?CAi = n; \\<forall>i<n. ?Ci ! i \\<subseteq># ?CAi ! i\\<rbrakk> \\<Longrightarrow> \\<Sum>\\<^sub># (mset ?Ci) \\<subseteq># \\<Sum>\\<^sub># (mset ?CAi)\nlength Ci = Suc n\nlength CAi = Suc n\n\\<forall>i<Suc n. Ci ! i \\<subseteq># CAi ! i\n\ngoal (1 subgoal):\n 1. \\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi)\n[PROOF STEP]\nby (cases Ci; cases CAi) (auto intro: subset_mset.add_mono)\n[PROOF STATE]\nproof (state)\nthis:\n\\<Sum>\\<^sub># (mset Ci) \\<subseteq># \\<Sum>\\<^sub># (mset CAi)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 4078, "file": "Nested_Multisets_Ordinals_Multiset_More", "length": 23, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9124361557147439, "lm_q2_score": 0.8499711718571774, "lm_q1q2_score": 0.7755444285177189}}
{"text": "[STATEMENT]\nlemma congruent_triangleI_sas:\n  assumes \"dist a1 b1 = dist a2 b2\"\n  assumes \"dist b1 c1 = dist b2 c2\"\n  assumes \"angle a1 b1 c1 = angle a2 b2 c2\"\n  shows   \"congruent_triangle a1 b1 c1 a2 b2 c2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. congruent_triangle a1 b1 c1 a2 b2 c2\n[PROOF STEP]\nproof (rule congruent_triangleI_sss)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. dist a1 b1 = dist a2 b2\n 2. dist b1 c1 = dist b2 c2\n 3. dist a1 c1 = dist a2 c2\n[PROOF STEP]\nshow \"dist a1 c1 = dist a2 c2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist a1 c1 = dist a2 c2\n[PROOF STEP]\nproof (rule power2_eq_imp_eq)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. (dist a1 c1)\\<^sup>2 = (dist a2 c2)\\<^sup>2\n 2. 0 \\<le> dist a1 c1\n 3. 0 \\<le> dist a2 c2\n[PROOF STEP]\nfrom cosine_law_triangle[of a1 c1 b1] cosine_law_triangle[of a2 c2 b2] assms\n[PROOF STATE]\nproof (chain)\npicking this:\n(dist a1 c1)\\<^sup>2 = (dist b1 a1)\\<^sup>2 + (dist b1 c1)\\<^sup>2 - 2 * dist b1 a1 * dist b1 c1 * cos (angle a1 b1 c1)\n(dist a2 c2)\\<^sup>2 = (dist b2 a2)\\<^sup>2 + (dist b2 c2)\\<^sup>2 - 2 * dist b2 a2 * dist b2 c2 * cos (angle a2 b2 c2)\ndist a1 b1 = dist a2 b2\ndist b1 c1 = dist b2 c2\nangle a1 b1 c1 = angle a2 b2 c2\n[PROOF STEP]\nshow \"(dist a1 c1)\\<^sup>2 = (dist a2 c2)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(dist a1 c1)\\<^sup>2 = (dist b1 a1)\\<^sup>2 + (dist b1 c1)\\<^sup>2 - 2 * dist b1 a1 * dist b1 c1 * cos (angle a1 b1 c1)\n(dist a2 c2)\\<^sup>2 = (dist b2 a2)\\<^sup>2 + (dist b2 c2)\\<^sup>2 - 2 * dist b2 a2 * dist b2 c2 * cos (angle a2 b2 c2)\ndist a1 b1 = dist a2 b2\ndist b1 c1 = dist b2 c2\nangle a1 b1 c1 = angle a2 b2 c2\n\ngoal (1 subgoal):\n 1. (dist a1 c1)\\<^sup>2 = (dist a2 c2)\\<^sup>2\n[PROOF STEP]\nby (simp add: dist_commute)\n[PROOF STATE]\nproof (state)\nthis:\n(dist a1 c1)\\<^sup>2 = (dist a2 c2)\\<^sup>2\n\ngoal (2 subgoals):\n 1. 0 \\<le> dist a1 c1\n 2. 0 \\<le> dist a2 c2\n[PROOF STEP]\nqed simp_all\n[PROOF STATE]\nproof (state)\nthis:\ndist a1 c1 = dist a2 c2\n\ngoal (2 subgoals):\n 1. dist a1 b1 = dist a2 b2\n 2. dist b1 c1 = dist b2 c2\n[PROOF STEP]\nqed fact+", "meta": {"llama_tokens": 1054, "file": "Triangle_Triangle", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9019206712569267, "lm_q2_score": 0.8596637505099168, "lm_q1q2_score": 0.7753485069151513}}
{"text": "[STATEMENT]\ntheorem catalan_closed_form_gbinomial:\n  \"real (catalan n) = 2 * (- 4) ^ n * (1/2 gchoose Suc n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (catalan n) = 2 * (- 4) ^ n * (1 / 2 gchoose Suc n)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. real (catalan n) = 2 * (- 4) ^ n * (1 / 2 gchoose Suc n)\n[PROOF STEP]\nhave \"(catalan n :: real) = fps_nth fps_catalan n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (catalan n) = fps_nth fps_catalan n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nreal (catalan n) = fps_nth fps_catalan n\n\ngoal (1 subgoal):\n 1. real (catalan n) = 2 * (- 4) ^ n * (1 / 2 gchoose Suc n)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreal (catalan n) = fps_nth fps_catalan n\n\ngoal (1 subgoal):\n 1. real (catalan n) = 2 * (- 4) ^ n * (1 / 2 gchoose Suc n)\n[PROOF STEP]\nhave \"\\<dots> = 2 * (- 4) ^ n * (1/2 gchoose Suc n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fps_nth fps_catalan n = 2 * (- 4) ^ n * (1 / 2 gchoose Suc n)\n[PROOF STEP]\nby (subst fps_catalan_fps_binomial)\n       (simp add: fps_div_fps_X_nth numeral_fps_const fps_compose_linear)\n[PROOF STATE]\nproof (state)\nthis:\nfps_nth fps_catalan n = 2 * (- 4) ^ n * (1 / 2 gchoose Suc n)\n\ngoal (1 subgoal):\n 1. real (catalan n) = 2 * (- 4) ^ n * (1 / 2 gchoose Suc n)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nreal (catalan n) = 2 * (- 4) ^ n * (1 / 2 gchoose Suc n)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nreal (catalan n) = 2 * (- 4) ^ n * (1 / 2 gchoose Suc n)\n\ngoal (1 subgoal):\n 1. real (catalan n) = 2 * (- 4) ^ n * (1 / 2 gchoose Suc n)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nreal (catalan n) = 2 * (- 4) ^ n * (1 / 2 gchoose Suc n)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 861, "file": "Catalan_Numbers_Catalan_Numbers", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9184802484881361, "lm_q2_score": 0.8438950947024555, "lm_q1q2_score": 0.7751009762802306}}
{"text": "[STATEMENT]\ntheorem card_equiv_rel_eq_card_partitions:\n  \"card {R. equiv A R} = card {P. partition_on A P}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {R. equiv A R} = card {P. partition_on A P}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {R. equiv A R} = card {P. partition_on A P}\n[PROOF STEP]\nhave \"bij_betw (\\<lambda>R. A // R) {R. equiv A R} {P. partition_on A P}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bij_betw ((//) A) {R. equiv A R} {P. partition_on A P}\n[PROOF STEP]\nby (rule bij_betw_partition_of)\n[PROOF STATE]\nproof (state)\nthis:\nbij_betw ((//) A) {R. equiv A R} {P. partition_on A P}\n\ngoal (1 subgoal):\n 1. card {R. equiv A R} = card {P. partition_on A P}\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\nbij_betw ((//) A) {R. equiv A R} {P. partition_on A P}\n[PROOF STEP]\nshow \"card {R. equiv A R} = card {P. partition_on A P}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw ((//) A) {R. equiv A R} {P. partition_on A P}\n\ngoal (1 subgoal):\n 1. card {R. equiv A R} = card {P. partition_on A P}\n[PROOF STEP]\nby (rule bij_betw_same_card)\n[PROOF STATE]\nproof (state)\nthis:\ncard {R. equiv A R} = card {P. partition_on A P}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 566, "file": "Card_Equiv_Relations_Card_Equiv_Relations", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942348544447, "lm_q2_score": 0.8705972784807408, "lm_q1q2_score": 0.775087737911373}}
{"text": "[STATEMENT]\nlemma inner_prod_smult_left_right:\n  fixes u v :: \"complex vec\"\n  assumes dimu: \"u \\<in> carrier_vec n\" and dimv: \"v \\<in> carrier_vec n\" \n  shows \"inner_prod (a \\<cdot>\\<^sub>v u) (b \\<cdot>\\<^sub>v v) = conjugate a * b  * inner_prod u v\" (is \"?lhs = ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inner_prod (a \\<cdot>\\<^sub>v u) (b \\<cdot>\\<^sub>v v) = conjugate a * b * inner_prod u v\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nu \\<in> carrier_vec n\nv \\<in> carrier_vec n\n\ngoal (1 subgoal):\n 1. inner_prod (a \\<cdot>\\<^sub>v u) (b \\<cdot>\\<^sub>v v) = conjugate a * b * inner_prod u v\n[PROOF STEP]\napply (simp add: scalar_prod_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>u \\<in> carrier_vec n; v \\<in> carrier_vec n\\<rbrakk> \\<Longrightarrow> (\\<Sum>i = 0..<n. b * v $ i * (cnj a * cnj (u $ i))) = cnj a * b * (\\<Sum>i = 0..<n. v $ i * cnj (u $ i))\n[PROOF STEP]\napply (subst sum_distrib_left)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>u \\<in> carrier_vec n; v \\<in> carrier_vec n\\<rbrakk> \\<Longrightarrow> (\\<Sum>i = 0..<n. b * v $ i * (cnj a * cnj (u $ i))) = (\\<Sum>n = 0..<n. cnj a * b * (v $ n * cnj (u $ n)))\n[PROOF STEP]\nby (rule sum.cong, auto)", "meta": {"llama_tokens": 559, "file": "QHLProver_Complex_Matrix", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032942067038785, "lm_q2_score": 0.8577681049901037, "lm_q1q2_score": 0.7748169599329249}}
{"text": "[STATEMENT]\nlemma onorm_cinner_right:\n  assumes \"bounded_linear r\"\n  shows \"onorm (\\<lambda>x. f \\<bullet>\\<^sub>C r x) \\<le> norm f * onorm r\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. onorm (\\<lambda>x. f \\<bullet>\\<^sub>C r x) \\<le> norm f * onorm r\n[PROOF STEP]\nproof (rule onorm_bound)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. 0 \\<le> norm f * onorm r\n 2. \\<And>x. cmod (f \\<bullet>\\<^sub>C r x) \\<le> norm f * onorm r * norm x\n[PROOF STEP]\nfix x\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. 0 \\<le> norm f * onorm r\n 2. \\<And>x. cmod (f \\<bullet>\\<^sub>C r x) \\<le> norm f * onorm r * norm x\n[PROOF STEP]\nhave \"norm (f \\<bullet>\\<^sub>C r x) \\<le> norm f * norm (r x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cmod (f \\<bullet>\\<^sub>C r x) \\<le> norm f * norm (r x)\n[PROOF STEP]\nby (simp add: Cauchy_Schwarz_ineq2)\n[PROOF STATE]\nproof (state)\nthis:\ncmod (f \\<bullet>\\<^sub>C r x) \\<le> norm f * norm (r x)\n\ngoal (2 subgoals):\n 1. 0 \\<le> norm f * onorm r\n 2. \\<And>x. cmod (f \\<bullet>\\<^sub>C r x) \\<le> norm f * onorm r * norm x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncmod (f \\<bullet>\\<^sub>C r x) \\<le> norm f * norm (r x)\n\ngoal (2 subgoals):\n 1. 0 \\<le> norm f * onorm r\n 2. \\<And>x. cmod (f \\<bullet>\\<^sub>C r x) \\<le> norm f * onorm r * norm x\n[PROOF STEP]\nhave \"\\<dots> \\<le> onorm r * norm x * norm f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm f * norm (r x) \\<le> onorm r * norm x * norm f\n[PROOF STEP]\nby (simp add: assms mult.commute mult_left_mono onorm)\n[PROOF STATE]\nproof (state)\nthis:\nnorm f * norm (r x) \\<le> onorm r * norm x * norm f\n\ngoal (2 subgoals):\n 1. 0 \\<le> norm f * onorm r\n 2. \\<And>x. cmod (f \\<bullet>\\<^sub>C r x) \\<le> norm f * onorm r * norm x\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncmod (f \\<bullet>\\<^sub>C r x) \\<le> onorm r * norm x * norm f\n[PROOF STEP]\nshow \"norm (f \\<bullet>\\<^sub>C r x) \\<le> norm f * onorm r * norm x\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncmod (f \\<bullet>\\<^sub>C r x) \\<le> onorm r * norm x * norm f\n\ngoal (1 subgoal):\n 1. cmod (f \\<bullet>\\<^sub>C r x) \\<le> norm f * onorm r * norm x\n[PROOF STEP]\nby (simp add: ac_simps)\n[PROOF STATE]\nproof (state)\nthis:\ncmod (f \\<bullet>\\<^sub>C r x) \\<le> norm f * onorm r * norm x\n\ngoal (1 subgoal):\n 1. 0 \\<le> norm f * onorm r\n[PROOF STEP]\nqed (intro mult_nonneg_nonneg norm_ge_zero onorm_pos_le assms)", "meta": {"llama_tokens": 1066, "file": "Complex_Bounded_Operators_Complex_Bounded_Linear_Function0", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8840392725805822, "lm_q2_score": 0.8757869932689566, "lm_q1q2_score": 0.7742300964650236}}
{"text": "[STATEMENT]\nlemma dist_point_def:\n  fixes p\\<^sub>0 :: \"('k::finite) point\"\n  shows \"dist p\\<^sub>0 p\\<^sub>1 = sqrt (\\<Sum>k \\<in> UNIV. (p\\<^sub>0$k - p\\<^sub>1$k)\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dist p\\<^sub>0 p\\<^sub>1 = sqrt (\\<Sum>k\\<in>UNIV. (p\\<^sub>0 $ k - p\\<^sub>1 $ k)\\<^sup>2)\n[PROOF STEP]\nunfolding dist_vec_def L2_set_def dist_real_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt (\\<Sum>i\\<in>UNIV. \\<bar>p\\<^sub>0 $ i - p\\<^sub>1 $ i\\<bar>\\<^sup>2) = sqrt (\\<Sum>k\\<in>UNIV. (p\\<^sub>0 $ k - p\\<^sub>1 $ k)\\<^sup>2)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 287, "file": "KD_Tree_KD_Tree", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.92414182206801, "lm_q2_score": 0.837619961306541, "lm_q1q2_score": 0.7740796372423628}}
{"text": "[STATEMENT]\nlemma fps_compose_mult_distrib_lemma:\n  assumes c0: \"c$0 = (0::'a::idom)\"\n  shows \"((a oo c) * (b oo c))$n = sum (\\<lambda>s. sum (\\<lambda>i. a$i * b$(s - i) * (c^s) $ n) {0..s}) {0..n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((a oo c) * (b oo c)) $ n = (\\<Sum>s = 0..n. \\<Sum>i = 0..s. a $ i * b $ (s - i) * c ^ s $ n)\n[PROOF STEP]\nunfolding product_composition_lemma[OF c0 c0] power_add[symmetric]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>(k, m)\\<in>{(k, m). k + m \\<le> n}. a $ k * b $ m * c ^ (k + m) $ n) = (\\<Sum>s = 0..n. \\<Sum>i = 0..s. a $ i * b $ (s - i) * c ^ s $ n)\n[PROOF STEP]\nunfolding sum_pair_less_iff[where a = \"\\<lambda>k. a$k\" and b=\"\\<lambda>m. b$m\" and c=\"\\<lambda>s. (c ^ s)$n\" and n = n]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>s = 0..n. \\<Sum>i = 0..s. a $ i * b $ (s - i) * c ^ s $ n) = (\\<Sum>s = 0..n. \\<Sum>i = 0..s. a $ i * b $ (s - i) * c ^ s $ n)\n[PROOF STEP]\n..", "meta": {"llama_tokens": 487, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.925229959153748, "lm_q2_score": 0.8354835411997897, "lm_q1q2_score": 0.7730144026979101}}
{"text": "[STATEMENT]\nlemma card_extensional_funcset_surj_on:\n  assumes \"finite A\" \"finite B\"\n  shows \"card {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} = fact (card B) * Stirling (card A) (card B)\" (is \"card ?F = _\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} = fact (card B) * Stirling (card A) (card B)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} = fact (card B) * Stirling (card A) (card B)\n[PROOF STEP]\nhave \"card ?F = fact (card B) * card (?F // range_permutation A B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} = fact (card B) * card ({f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} // range_permutation A B)\n[PROOF STEP]\nusing \\<open>finite B\\<close>\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite B\n\ngoal (1 subgoal):\n 1. card {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} = fact (card B) * card ({f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} // range_permutation A B)\n[PROOF STEP]\nby (simp only: card_equiv_class_restricted_same_size[OF equiv_range_permutation surj_on_respects_range_permutation card_of_equiv_class])\n[PROOF STATE]\nproof (state)\nthis:\ncard {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} = fact (card B) * card ({f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} // range_permutation A B)\n\ngoal (1 subgoal):\n 1. card {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} = fact (card B) * Stirling (card A) (card B)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} = fact (card B) * card ({f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} // range_permutation A B)\n\ngoal (1 subgoal):\n 1. card {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} = fact (card B) * Stirling (card A) (card B)\n[PROOF STEP]\nhave \"\\<dots> = fact (card B) * Stirling (card A) (card B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. fact (card B) * card ({f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} // range_permutation A B) = fact (card B) * Stirling (card A) (card B)\n[PROOF STEP]\nusing \\<open>finite A\\<close> \\<open>finite B\\<close>\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nfinite B\n\ngoal (1 subgoal):\n 1. fact (card B) * card ({f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} // range_permutation A B) = fact (card B) * Stirling (card A) (card B)\n[PROOF STEP]\nby (simp only: card_surjective_functions_range_permutation)\n[PROOF STATE]\nproof (state)\nthis:\nfact (card B) * card ({f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} // range_permutation A B) = fact (card B) * Stirling (card A) (card B)\n\ngoal (1 subgoal):\n 1. card {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} = fact (card B) * Stirling (card A) (card B)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} = fact (card B) * Stirling (card A) (card B)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} = fact (card B) * Stirling (card A) (card B)\n\ngoal (1 subgoal):\n 1. card {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} = fact (card B) * Stirling (card A) (card B)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard {f \\<in> A \\<rightarrow>\\<^sub>E B. f ` A = B} = fact (card B) * Stirling (card A) (card B)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1406, "file": "Twelvefold_Way_Twelvefold_Way_Entry3", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9073122263731811, "lm_q2_score": 0.8519528094861981, "lm_q1q2_score": 0.772987200339809}}
{"text": "[STATEMENT]\nlemma sin_multiple: \"sin (n * x) = 2 * cos x * sin ((n - 1) * x) - sin ((n - 2) * x)\"\n  for x :: \"'a :: {banach,real_normed_field}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin (n * x) = (2::'a) * cos x * sin ((n - (1::'a)) * x) - sin ((n - (2::'a)) * x)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sin (n * x) = (2::'a) * cos x * sin ((n - (1::'a)) * x) - sin ((n - (2::'a)) * x)\n[PROOF STEP]\nhave \"sin ((n - 1) * x + x) + sin ((n - 1) * x - x) = 2 * cos x * sin ((n - 1) * x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin ((n - (1::'a)) * x + x) + sin ((n - (1::'a)) * x - x) = (2::'a) * cos x * sin ((n - (1::'a)) * x)\n[PROOF STEP]\nby (simp add: sin_add sin_diff)\n[PROOF STATE]\nproof (state)\nthis:\nsin ((n - (1::'a)) * x + x) + sin ((n - (1::'a)) * x - x) = (2::'a) * cos x * sin ((n - (1::'a)) * x)\n\ngoal (1 subgoal):\n 1. sin (n * x) = (2::'a) * cos x * sin ((n - (1::'a)) * x) - sin ((n - (2::'a)) * x)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nsin ((n - (1::'a)) * x + x) + sin ((n - (1::'a)) * x - x) = (2::'a) * cos x * sin ((n - (1::'a)) * x)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsin ((n - (1::'a)) * x + x) + sin ((n - (1::'a)) * x - x) = (2::'a) * cos x * sin ((n - (1::'a)) * x)\n\ngoal (1 subgoal):\n 1. sin (n * x) = (2::'a) * cos x * sin ((n - (1::'a)) * x) - sin ((n - (2::'a)) * x)\n[PROOF STEP]\nby (simp add: left_diff_distrib' eq_diff_eq)\n[PROOF STATE]\nproof (state)\nthis:\nsin (n * x) = (2::'a) * cos x * sin ((n - (1::'a)) * x) - sin ((n - (2::'a)) * x)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 822, "file": "Hyperdual_AnalyticTestFunction", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.899121388082479, "lm_q2_score": 0.8596637559030338, "lm_q1q2_score": 0.7729420694917332}}
{"text": "[STATEMENT]\nlemma matrix_mult_transpose_dot_column:\n  shows \"transpose A ** A = (\\<chi> i j. inner (column i A) (column j A))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Finite_Cartesian_Product.transpose A ** A = (\\<chi>i j. inner (column i A) (column j A))\n[PROOF STEP]\nby (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def column_def inner_vec_def)", "meta": {"llama_tokens": 139, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9136765281148513, "lm_q2_score": 0.8459424353665381, "lm_q1q2_score": 0.7729177473307206}}
{"text": "[STATEMENT]\nlemma cos_treble_cos: \"cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x\"\n  for x :: \"'a::{real_normed_field,banach}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos ((3::'a) * x) = (4::'a) * cos x ^ 3 - (3::'a) * cos x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. cos ((3::'a) * x) = (4::'a) * cos x ^ 3 - (3::'a) * cos x\n[PROOF STEP]\nhave *: \"(sin x * (sin x * 3)) = 3 - (cos x * (cos x * 3))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin x * (sin x * (3::'a)) = (3::'a) - cos x * (cos x * (3::'a))\n[PROOF STEP]\nby (simp add: mult.assoc [symmetric] sin_squared_eq [unfolded power2_eq_square])\n[PROOF STATE]\nproof (state)\nthis:\nsin x * (sin x * (3::'a)) = (3::'a) - cos x * (cos x * (3::'a))\n\ngoal (1 subgoal):\n 1. cos ((3::'a) * x) = (4::'a) * cos x ^ 3 - (3::'a) * cos x\n[PROOF STEP]\nhave \"cos(3 * x) = cos(2*x + x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos ((3::'a) * x) = cos ((2::'a) * x + x)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncos ((3::'a) * x) = cos ((2::'a) * x + x)\n\ngoal (1 subgoal):\n 1. cos ((3::'a) * x) = (4::'a) * cos x ^ 3 - (3::'a) * cos x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncos ((3::'a) * x) = cos ((2::'a) * x + x)\n\ngoal (1 subgoal):\n 1. cos ((3::'a) * x) = (4::'a) * cos x ^ 3 - (3::'a) * cos x\n[PROOF STEP]\nhave \"\\<dots> = 4 * cos x ^ 3 - 3 * cos x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos ((2::'a) * x + x) = (4::'a) * cos x ^ 3 - (3::'a) * cos x\n[PROOF STEP]\nunfolding cos_add cos_double sin_double\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((cos x)\\<^sup>2 - (sin x)\\<^sup>2) * cos x - (2::'a) * sin x * cos x * sin x = (4::'a) * cos x ^ 3 - (3::'a) * cos x\n[PROOF STEP]\nby (simp add: * field_simps power2_eq_square power3_eq_cube)\n[PROOF STATE]\nproof (state)\nthis:\ncos ((2::'a) * x + x) = (4::'a) * cos x ^ 3 - (3::'a) * cos x\n\ngoal (1 subgoal):\n 1. cos ((3::'a) * x) = (4::'a) * cos x ^ 3 - (3::'a) * cos x\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncos ((3::'a) * x) = (4::'a) * cos x ^ 3 - (3::'a) * cos x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncos ((3::'a) * x) = (4::'a) * cos x ^ 3 - (3::'a) * cos x\n\ngoal (1 subgoal):\n 1. cos ((3::'a) * x) = (4::'a) * cos x ^ 3 - (3::'a) * cos x\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncos ((3::'a) * x) = (4::'a) * cos x ^ 3 - (3::'a) * cos x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1245, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9005297861178929, "lm_q2_score": 0.8577681122619885, "lm_q1q2_score": 0.7724457346740372}}
{"text": "[STATEMENT]\nlemma proj2_incident_abs:\n  assumes \"v \\<noteq> 0\" and \"w \\<noteq> 0\"\n  shows \"proj2_incident (proj2_abs v) (proj2_line_abs w) \\<longleftrightarrow> v \\<bullet> w = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. proj2_incident (proj2_abs v) (proj2_line_abs w) = (v \\<bullet> w = 0)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. proj2_incident (proj2_abs v) (proj2_line_abs w) = (v \\<bullet> w = 0)\n[PROOF STEP]\nfrom \\<open>v \\<noteq> 0\\<close> and proj2_rep_abs2\n[PROOF STATE]\nproof (chain)\npicking this:\nv \\<noteq> 0\n?v \\<noteq> 0 \\<Longrightarrow> \\<exists>k. k \\<noteq> 0 \\<and> proj2_rep (proj2_abs ?v) = k *\\<^sub>R ?v\n[PROOF STEP]\nobtain j where \"j \\<noteq> 0\" and \"proj2_rep (proj2_abs v) = j *\\<^sub>R v\"\n[PROOF STATE]\nproof (prove)\nusing this:\nv \\<noteq> 0\n?v \\<noteq> 0 \\<Longrightarrow> \\<exists>k. k \\<noteq> 0 \\<and> proj2_rep (proj2_abs ?v) = k *\\<^sub>R ?v\n\ngoal (1 subgoal):\n 1. (\\<And>j. \\<lbrakk>j \\<noteq> 0; proj2_rep (proj2_abs v) = j *\\<^sub>R v\\<rbrakk> \\<Longrightarrow> thesis) \\<Longrightarrow> thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nj \\<noteq> 0\nproj2_rep (proj2_abs v) = j *\\<^sub>R v\n\ngoal (1 subgoal):\n 1. proj2_incident (proj2_abs v) (proj2_line_abs w) = (v \\<bullet> w = 0)\n[PROOF STEP]\nfrom \\<open>w \\<noteq> 0\\<close> and proj2_line_rep_abs\n[PROOF STATE]\nproof (chain)\npicking this:\nw \\<noteq> 0\n?v \\<noteq> 0 \\<Longrightarrow> \\<exists>k. k \\<noteq> 0 \\<and> proj2_line_rep (proj2_line_abs ?v) = k *\\<^sub>R ?v\n[PROOF STEP]\nobtain k where \"k \\<noteq> 0\"\n    and \"proj2_line_rep (proj2_line_abs w) = k *\\<^sub>R w\"\n[PROOF STATE]\nproof (prove)\nusing this:\nw \\<noteq> 0\n?v \\<noteq> 0 \\<Longrightarrow> \\<exists>k. k \\<noteq> 0 \\<and> proj2_line_rep (proj2_line_abs ?v) = k *\\<^sub>R ?v\n\ngoal (1 subgoal):\n 1. (\\<And>k. \\<lbrakk>k \\<noteq> 0; proj2_line_rep (proj2_line_abs w) = k *\\<^sub>R w\\<rbrakk> \\<Longrightarrow> thesis) \\<Longrightarrow> thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nk \\<noteq> 0\nproj2_line_rep (proj2_line_abs w) = k *\\<^sub>R w\n\ngoal (1 subgoal):\n 1. proj2_incident (proj2_abs v) (proj2_line_abs w) = (v \\<bullet> w = 0)\n[PROOF STEP]\nwith \\<open>j \\<noteq> 0\\<close> and \\<open>proj2_rep (proj2_abs v) = j *\\<^sub>R v\\<close>\n[PROOF STATE]\nproof (chain)\npicking this:\nj \\<noteq> 0\nproj2_rep (proj2_abs v) = j *\\<^sub>R v\nk \\<noteq> 0\nproj2_line_rep (proj2_line_abs w) = k *\\<^sub>R w\n[PROOF STEP]\nshow \"proj2_incident (proj2_abs v) (proj2_line_abs w) \\<longleftrightarrow> v \\<bullet> w = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nj \\<noteq> 0\nproj2_rep (proj2_abs v) = j *\\<^sub>R v\nk \\<noteq> 0\nproj2_line_rep (proj2_line_abs w) = k *\\<^sub>R w\n\ngoal (1 subgoal):\n 1. proj2_incident (proj2_abs v) (proj2_line_abs w) = (v \\<bullet> w = 0)\n[PROOF STEP]\nunfolding proj2_incident_def\n[PROOF STATE]\nproof (prove)\nusing this:\nj \\<noteq> 0\nproj2_rep (proj2_abs v) = j *\\<^sub>R v\nk \\<noteq> 0\nproj2_line_rep (proj2_line_abs w) = k *\\<^sub>R w\n\ngoal (1 subgoal):\n 1. (proj2_rep (proj2_abs v) \\<bullet> proj2_line_rep (proj2_line_abs w) = 0) = (v \\<bullet> w = 0)\n[PROOF STEP]\nby (simp add: dot_scaleR_mult)\n[PROOF STATE]\nproof (state)\nthis:\nproj2_incident (proj2_abs v) (proj2_line_abs w) = (v \\<bullet> w = 0)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1540, "file": "Tarskis_Geometry_Projective", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942319436395, "lm_q2_score": 0.8670357460591569, "lm_q1q2_score": 0.7719169236054176}}
{"text": "[STATEMENT]\nlemma sum_list_map2_plus:\n  assumes \"length xs = length ys\"\n  shows \"sum_list (map2 (+) xs ys) = sum_list xs + sum_list (ys::'a::comm_monoid_add list)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum_list (map2 (+) xs ys) = sum_list xs + sum_list ys\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nlength xs = length ys\n\ngoal (1 subgoal):\n 1. sum_list (map2 (+) xs ys) = sum_list xs + sum_list ys\n[PROOF STEP]\nproof (induct rule: list_induct2)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. sum_list (map2 (+) [] []) = sum_list [] + sum_list []\n 2. \\<And>x xs y ys. \\<lbrakk>length xs = length ys; sum_list (map2 (+) xs ys) = sum_list xs + sum_list ys\\<rbrakk> \\<Longrightarrow> sum_list (map2 (+) (x # xs) (y # ys)) = sum_list (x # xs) + sum_list (y # ys)\n[PROOF STEP]\ncase Nil\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. sum_list (map2 (+) [] []) = sum_list [] + sum_list []\n 2. \\<And>x xs y ys. \\<lbrakk>length xs = length ys; sum_list (map2 (+) xs ys) = sum_list xs + sum_list ys\\<rbrakk> \\<Longrightarrow> sum_list (map2 (+) (x # xs) (y # ys)) = sum_list (x # xs) + sum_list (y # ys)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum_list (map2 (+) [] []) = sum_list [] + sum_list []\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsum_list (map2 (+) [] []) = sum_list [] + sum_list []\n\ngoal (1 subgoal):\n 1. \\<And>x xs y ys. \\<lbrakk>length xs = length ys; sum_list (map2 (+) xs ys) = sum_list xs + sum_list ys\\<rbrakk> \\<Longrightarrow> sum_list (map2 (+) (x # xs) (y # ys)) = sum_list (x # xs) + sum_list (y # ys)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>x xs y ys. \\<lbrakk>length xs = length ys; sum_list (map2 (+) xs ys) = sum_list xs + sum_list ys\\<rbrakk> \\<Longrightarrow> sum_list (map2 (+) (x # xs) (y # ys)) = sum_list (x # xs) + sum_list (y # ys)\n[PROOF STEP]\ncase (Cons x xs y ys)\n[PROOF STATE]\nproof (state)\nthis:\nlength xs = length ys\nsum_list (map2 (+) xs ys) = sum_list xs + sum_list ys\n\ngoal (1 subgoal):\n 1. \\<And>x xs y ys. \\<lbrakk>length xs = length ys; sum_list (map2 (+) xs ys) = sum_list xs + sum_list ys\\<rbrakk> \\<Longrightarrow> sum_list (map2 (+) (x # xs) (y # ys)) = sum_list (x # xs) + sum_list (y # ys)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum_list (map2 (+) (x # xs) (y # ys)) = sum_list (x # xs) + sum_list (y # ys)\n[PROOF STEP]\nby (simp add: Cons(2) ac_simps)\n[PROOF STATE]\nproof (state)\nthis:\nsum_list (map2 (+) (x # xs) (y # ys)) = sum_list (x # xs) + sum_list (y # ys)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1167, "file": "Groebner_Bases_General", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942173896132, "lm_q2_score": 0.8670357477770337, "lm_q1q2_score": 0.7719169125159724}}
{"text": "[STATEMENT]\nlemma exp_first_terms:\n  fixes x :: \"'a::{real_normed_algebra_1,banach}\"\n  shows \"exp x = (\\<Sum>n<k. inverse(fact n) *\\<^sub>R (x ^ n)) + (\\<Sum>n. inverse(fact (n + k)) *\\<^sub>R (x ^ (n + k)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exp x = (\\<Sum>n<k. x ^ n /\\<^sub>R fact n) + (\\<Sum>n. x ^ (n + k) /\\<^sub>R fact (n + k))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. exp x = (\\<Sum>n<k. x ^ n /\\<^sub>R fact n) + (\\<Sum>n. x ^ (n + k) /\\<^sub>R fact (n + k))\n[PROOF STEP]\nhave \"exp x = suminf (\\<lambda>n. inverse(fact n) *\\<^sub>R (x^n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exp x = (\\<Sum>n. x ^ n /\\<^sub>R fact n)\n[PROOF STEP]\nby (simp add: exp_def)\n[PROOF STATE]\nproof (state)\nthis:\nexp x = (\\<Sum>n. x ^ n /\\<^sub>R fact n)\n\ngoal (1 subgoal):\n 1. exp x = (\\<Sum>n<k. x ^ n /\\<^sub>R fact n) + (\\<Sum>n. x ^ (n + k) /\\<^sub>R fact (n + k))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nexp x = (\\<Sum>n. x ^ n /\\<^sub>R fact n)\n\ngoal (1 subgoal):\n 1. exp x = (\\<Sum>n<k. x ^ n /\\<^sub>R fact n) + (\\<Sum>n. x ^ (n + k) /\\<^sub>R fact (n + k))\n[PROOF STEP]\nfrom summable_exp_generic\n[PROOF STATE]\nproof (chain)\npicking this:\nsummable (\\<lambda>n. ?x ^ n /\\<^sub>R fact n)\n[PROOF STEP]\nhave \"\\<dots> = (\\<Sum> n. inverse(fact(n+k)) *\\<^sub>R (x ^ (n + k))) +\n    (\\<Sum> n::nat<k. inverse(fact n) *\\<^sub>R (x^n))\" (is \"_ = _ + ?a\")\n[PROOF STATE]\nproof (prove)\nusing this:\nsummable (\\<lambda>n. ?x ^ n /\\<^sub>R fact n)\n\ngoal (1 subgoal):\n 1. (\\<Sum>n. x ^ n /\\<^sub>R fact n) = (\\<Sum>n. x ^ (n + k) /\\<^sub>R fact (n + k)) + (\\<Sum>n<k. x ^ n /\\<^sub>R fact n)\n[PROOF STEP]\nby (rule suminf_split_initial_segment)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>n. x ^ n /\\<^sub>R fact n) = (\\<Sum>n. x ^ (n + k) /\\<^sub>R fact (n + k)) + (\\<Sum>n<k. x ^ n /\\<^sub>R fact n)\n\ngoal (1 subgoal):\n 1. exp x = (\\<Sum>n<k. x ^ n /\\<^sub>R fact n) + (\\<Sum>n. x ^ (n + k) /\\<^sub>R fact (n + k))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nexp x = (\\<Sum>n. x ^ (n + k) /\\<^sub>R fact (n + k)) + (\\<Sum>n<k. x ^ n /\\<^sub>R fact n)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nexp x = (\\<Sum>n. x ^ (n + k) /\\<^sub>R fact (n + k)) + (\\<Sum>n<k. x ^ n /\\<^sub>R fact n)\n\ngoal (1 subgoal):\n 1. exp x = (\\<Sum>n<k. x ^ n /\\<^sub>R fact n) + (\\<Sum>n. x ^ (n + k) /\\<^sub>R fact (n + k))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nexp x = (\\<Sum>n<k. x ^ n /\\<^sub>R fact n) + (\\<Sum>n. x ^ (n + k) /\\<^sub>R fact (n + k))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1224, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9124361580958426, "lm_q2_score": 0.8459424431344437, "lm_q1q2_score": 0.7718684727838026}}
{"text": "[STATEMENT]\nlemma per_col_mult_unitary:\n  fixes A::\"complex Matrix.mat\"\n  assumes \"A \\<in> carrier_mat n n\"\n  and \"unitary A\"\n  and \"D\\<in> carrier_mat n n\"\n  and \"diagonal_mat D\"\n  and \"0 < n\"\n  and \"bij_betw f {..< n} {..< n}\"\nshows \"A * D * (Complex_Matrix.adjoint A) = \n  (per_col A f) * (per_diag D f) * (Complex_Matrix.adjoint (per_col A f))\" \n  (is \"?L = ?R\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nproof  -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nhave row: \"dim_row ?L = dim_row ?R\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_row (A * D * Complex_Matrix.adjoint A) = dim_row (per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f))\n[PROOF STEP]\nusing per_col_carrier assms\n[PROOF STATE]\nproof (prove)\nusing this:\n?A \\<in> carrier_mat ?n ?m \\<Longrightarrow> per_col ?A ?f \\<in> carrier_mat ?n ?m\nA \\<in> carrier_mat n n\nComplex_Matrix.unitary A\nD \\<in> carrier_mat n n\ndiagonal_mat D\n0 < n\nbij_betw f {..<n} {..<n}\n\ngoal (1 subgoal):\n 1. dim_row (A * D * Complex_Matrix.adjoint A) = dim_row (per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f))\n[PROOF STEP]\nby (metis carrier_matD(1) index_mult_mat(2))\n[PROOF STATE]\nproof (state)\nthis:\ndim_row (A * D * Complex_Matrix.adjoint A) = dim_row (per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f))\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nhave col: \"dim_col ?L = dim_col ?R\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_col (A * D * Complex_Matrix.adjoint A) = dim_col (per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f))\n[PROOF STEP]\nusing per_col_carrier assms\n[PROOF STATE]\nproof (prove)\nusing this:\n?A \\<in> carrier_mat ?n ?m \\<Longrightarrow> per_col ?A ?f \\<in> carrier_mat ?n ?m\nA \\<in> carrier_mat n n\nComplex_Matrix.unitary A\nD \\<in> carrier_mat n n\ndiagonal_mat D\n0 < n\nbij_betw f {..<n} {..<n}\n\ngoal (1 subgoal):\n 1. dim_col (A * D * Complex_Matrix.adjoint A) = dim_col (per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f))\n[PROOF STEP]\nby (metis adjoint_dim carrier_matD(2) index_mult_mat(3))\n[PROOF STATE]\nproof (state)\nthis:\ndim_col (A * D * Complex_Matrix.adjoint A) = dim_col (per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f))\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\ndefine fc::\"complex Matrix.mat set\" where \"fc = carrier_mat n n\"\n[PROOF STATE]\nproof (state)\nthis:\nfc = carrier_mat n n\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\ninterpret cpx_sq_mat n n fc\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cpx_sq_mat n n fc\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. fc = carrier_mat n n\n 2. n = n\n 3. 0 < n\n[PROOF STEP]\nshow \"0 < n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < n\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\<in> carrier_mat n n\nComplex_Matrix.unitary A\nD \\<in> carrier_mat n n\ndiagonal_mat D\n0 < n\nbij_betw f {..<n} {..<n}\n\ngoal (1 subgoal):\n 1. 0 < n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 < n\n\ngoal (2 subgoals):\n 1. fc = carrier_mat n n\n 2. n = n\n[PROOF STEP]\nqed (auto simp add: fc_def)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\ndefine h where \n    \"h = (\\<lambda>i. (if i < n then diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i) \n      else (0\\<^sub>m n n)))\"\n[PROOF STATE]\nproof (state)\nthis:\nh = (\\<lambda>i. if i < n then diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i) else 0\\<^sub>m n n)\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\ndefine g where\n    \"g = (\\<lambda>i. (if i < n \n      then diag_mat D ! (f i) \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i)) \n      else (0\\<^sub>m n n)))\"\n[PROOF STATE]\nproof (state)\nthis:\ng = (\\<lambda>i. if i < n then diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i)) else 0\\<^sub>m n n)\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nhave \"f ` {..<n} = {..<n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f ` {..<n} = {..<n}\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\<in> carrier_mat n n\nComplex_Matrix.unitary A\nD \\<in> carrier_mat n n\ndiagonal_mat D\n0 < n\nbij_betw f {..<n} {..<n}\n\ngoal (1 subgoal):\n 1. f ` {..<n} = {..<n}\n[PROOF STEP]\nby (simp add: bij_betw_imp_surj_on)\n[PROOF STATE]\nproof (state)\nthis:\nf ` {..<n} = {..<n}\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nhave g: \"\\<forall>i. g i \\<in> fc\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<forall>i. g i \\<in> fc\n[PROOF STEP]\nunfolding g_def fc_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<forall>i. (if i < n then diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i)) else 0\\<^sub>m n n) \\<in> carrier_mat n n\n[PROOF STEP]\nby (metis adjoint_dim_col assms(1) carrier_matD(1) carrier_matI \n        dim_col fc_mats_carrier rank_1_proj_adjoint rank_1_proj_dim \n        smult_carrier_mat zero_mem)\n[PROOF STATE]\nproof (state)\nthis:\n\\<forall>i. g i \\<in> fc\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\<forall>i. g i \\<in> fc\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nhave h:\"\\<forall>i. h i \\<in> fc\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<forall>i. h i \\<in> fc\n[PROOF STEP]\nunfolding h_def fc_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<forall>i. (if i < n then diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i) else 0\\<^sub>m n n) \\<in> carrier_mat n n\n[PROOF STEP]\nby (metis assms(1) carrier_matD(1) dim_col fc_mats_carrier \n        rank_1_proj_carrier smult_carrier_mat zero_mem)\n[PROOF STATE]\nproof (state)\nthis:\n\\<forall>i. h i \\<in> fc\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\<forall>i. h i \\<in> fc\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nhave \"inj_on f {..<n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inj_on f {..<n}\n[PROOF STEP]\nusing assms(6) bij_betw_def\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw f {..<n} {..<n}\nbij_betw ?f ?A ?B = (inj_on ?f ?A \\<and> ?f ` ?A = ?B)\n\ngoal (1 subgoal):\n 1. inj_on f {..<n}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ninj_on f {..<n}\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ninj_on f {..<n}\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nhave \"\\<And>x. x \\<in> {..<n} \\<Longrightarrow> h (f x) = g x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>x. x \\<in> {..<n} \\<Longrightarrow> h (f x) = g x\n[PROOF STEP]\nunfolding h_def g_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>x. x \\<in> {..<n} \\<Longrightarrow> (if f x < n then diag_mat D ! f x \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f x)) else 0\\<^sub>m n n) = (if x < n then diag_mat D ! f x \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f x)) else 0\\<^sub>m n n)\n[PROOF STEP]\nby (meson assms(6) bij_betwE lessThan_iff)\n[PROOF STATE]\nproof (state)\nthis:\n?x1 \\<in> {..<n} \\<Longrightarrow> h (f ?x1) = g ?x1\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n\\<forall>i. g i \\<in> fc\n\\<forall>i. h i \\<in> fc\ninj_on f {..<n}\n?x1 \\<in> {..<n} \\<Longrightarrow> h (f ?x1) = g ?x1\n[PROOF STEP]\nhave \"sum_mat g {..<n} = sum_mat h (f`{..<n})\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<forall>i. g i \\<in> fc\n\\<forall>i. h i \\<in> fc\ninj_on f {..<n}\n?x1 \\<in> {..<n} \\<Longrightarrow> h (f ?x1) = g ?x1\n\ngoal (1 subgoal):\n 1. local.sum_mat g {..<n} = local.sum_mat h (f ` {..<n})\n[PROOF STEP]\nusing sum_with_reindex_cong[of h g f \"{..<n}\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<forall>i. g i \\<in> fc\n\\<forall>i. h i \\<in> fc\ninj_on f {..<n}\n?x1 \\<in> {..<n} \\<Longrightarrow> h (f ?x1) = g ?x1\n\\<lbrakk>\\<forall>x. h x \\<in> fc; \\<forall>x. g x \\<in> fc; inj_on f {..<n}; ?A = f ` {..<n}; \\<And>x. x \\<in> {..<n} \\<Longrightarrow> h (f x) = g x\\<rbrakk> \\<Longrightarrow> sum_with (+) (0\\<^sub>m n n) h (f ` {..<n}) = sum_with (+) (0\\<^sub>m n n) g {..<n}\n\ngoal (1 subgoal):\n 1. local.sum_mat g {..<n} = local.sum_mat h (f ` {..<n})\n[PROOF STEP]\nunfolding  sum_mat_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<forall>i. g i \\<in> fc\n\\<forall>i. h i \\<in> fc\ninj_on f {..<n}\n?x1 \\<in> {..<n} \\<Longrightarrow> h (f ?x1) = g ?x1\n\\<lbrakk>\\<forall>x. h x \\<in> fc; \\<forall>x. g x \\<in> fc; inj_on f {..<n}; ?A = f ` {..<n}; \\<And>x. x \\<in> {..<n} \\<Longrightarrow> h (f x) = g x\\<rbrakk> \\<Longrightarrow> sum_with (+) (0\\<^sub>m n n) h (f ` {..<n}) = sum_with (+) (0\\<^sub>m n n) g {..<n}\n\ngoal (1 subgoal):\n 1. sum_with (+) (0\\<^sub>m n n) g {..<n} = sum_with (+) (0\\<^sub>m n n) h (f ` {..<n})\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlocal.sum_mat g {..<n} = local.sum_mat h (f ` {..<n})\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nlocal.sum_mat g {..<n} = local.sum_mat h (f ` {..<n})\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nhave \"... = sum_mat h {..<n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. local.sum_mat h (f ` {..<n}) = local.sum_mat h {..<n}\n[PROOF STEP]\nusing \\<open>f ` {..<n} = {..<n}\\<close>\n[PROOF STATE]\nproof (prove)\nusing this:\nf ` {..<n} = {..<n}\n\ngoal (1 subgoal):\n 1. local.sum_mat h (f ` {..<n}) = local.sum_mat h {..<n}\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlocal.sum_mat h (f ` {..<n}) = local.sum_mat h {..<n}\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nlocal.sum_mat h (f ` {..<n}) = local.sum_mat h {..<n}\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nhave \"... =sum_mat (\\<lambda>i. diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i))\n    {..<n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. local.sum_mat h {..<n} = local.sum_mat (\\<lambda>i. diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i)) {..<n}\n[PROOF STEP]\nproof (rule sum_mat_cong, (auto simp add:h h_def))\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i) \\<in> fc\n 2. \\<And>i. i < n \\<Longrightarrow> diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i) \\<in> fc\n[PROOF STEP]\nshow \"\\<And>i. i < n \\<Longrightarrow> diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i) \\<in> fc\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i) \\<in> fc\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\<in> carrier_mat n n\nComplex_Matrix.unitary A\nD \\<in> carrier_mat n n\ndiagonal_mat D\n0 < n\nbij_betw f {..<n} {..<n}\n\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i) \\<in> fc\n[PROOF STEP]\nunfolding fc_def\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\<in> carrier_mat n n\nComplex_Matrix.unitary A\nD \\<in> carrier_mat n n\ndiagonal_mat D\n0 < n\nbij_betw f {..<n} {..<n}\n\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i) \\<in> carrier_mat n n\n[PROOF STEP]\nby (metis fc_mats_carrier h h_def)\n[PROOF STATE]\nproof (state)\nthis:\n?i1 < n \\<Longrightarrow> diag_mat D ! ?i1 \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A ?i1) \\<in> fc\n\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i) \\<in> fc\n[PROOF STEP]\nshow \"\\<And>i. i < n \\<Longrightarrow> diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i) \\<in> fc\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i) \\<in> fc\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\<in> carrier_mat n n\nComplex_Matrix.unitary A\nD \\<in> carrier_mat n n\ndiagonal_mat D\n0 < n\nbij_betw f {..<n} {..<n}\n\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i) \\<in> fc\n[PROOF STEP]\nunfolding fc_def\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\<in> carrier_mat n n\nComplex_Matrix.unitary A\nD \\<in> carrier_mat n n\ndiagonal_mat D\n0 < n\nbij_betw f {..<n} {..<n}\n\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i) \\<in> carrier_mat n n\n[PROOF STEP]\nby (metis fc_mats_carrier h h_def)\n[PROOF STATE]\nproof (state)\nthis:\n?i1 < n \\<Longrightarrow> diag_mat D ! ?i1 \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A ?i1) \\<in> fc\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nlocal.sum_mat h {..<n} = local.sum_mat (\\<lambda>i. diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i)) {..<n}\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nlocal.sum_mat h {..<n} = local.sum_mat (\\<lambda>i. diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i)) {..<n}\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nhave \"... = A * D * (Complex_Matrix.adjoint A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. local.sum_mat (\\<lambda>i. diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i)) {..<n} = A * D * Complex_Matrix.adjoint A\n[PROOF STEP]\nusing weighted_sum_rank_1_proj_unitary assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>?A \\<in> fc; ?B \\<in> fc; diagonal_mat ?B; Complex_Matrix.unitary ?A\\<rbrakk> \\<Longrightarrow> local.sum_mat (\\<lambda>i. diag_mat ?B ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col ?A i)) {..<n} = ?A * ?B * Complex_Matrix.adjoint ?A\nA \\<in> carrier_mat n n\nComplex_Matrix.unitary A\nD \\<in> carrier_mat n n\ndiagonal_mat D\n0 < n\nbij_betw f {..<n} {..<n}\n\ngoal (1 subgoal):\n 1. local.sum_mat (\\<lambda>i. diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i)) {..<n} = A * D * Complex_Matrix.adjoint A\n[PROOF STEP]\nunfolding fc_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>?A \\<in> carrier_mat n n; ?B \\<in> carrier_mat n n; diagonal_mat ?B; Complex_Matrix.unitary ?A\\<rbrakk> \\<Longrightarrow> local.sum_mat (\\<lambda>i. diag_mat ?B ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col ?A i)) {..<n} = ?A * ?B * Complex_Matrix.adjoint ?A\nA \\<in> carrier_mat n n\nComplex_Matrix.unitary A\nD \\<in> carrier_mat n n\ndiagonal_mat D\n0 < n\nbij_betw f {..<n} {..<n}\n\ngoal (1 subgoal):\n 1. local.sum_mat (\\<lambda>i. diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i)) {..<n} = A * D * Complex_Matrix.adjoint A\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlocal.sum_mat (\\<lambda>i. diag_mat D ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A i)) {..<n} = A * D * Complex_Matrix.adjoint A\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nlocal.sum_mat g {..<n} = A * D * Complex_Matrix.adjoint A\n[PROOF STEP]\nhave sg: \"sum_mat g {..<n} = A * D * (Complex_Matrix.adjoint A)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlocal.sum_mat g {..<n} = A * D * Complex_Matrix.adjoint A\n\ngoal (1 subgoal):\n 1. local.sum_mat g {..<n} = A * D * Complex_Matrix.adjoint A\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nlocal.sum_mat g {..<n} = A * D * Complex_Matrix.adjoint A\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nhave \"(per_col A f) * (per_diag D f) * \n    (Complex_Matrix.adjoint (per_col A f)) = \n    sum_mat (\\<lambda>i. diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m \n    rank_1_proj (Matrix.col (per_col A f) i)) {..<n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f) = local.sum_mat (\\<lambda>i. diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i)) {..<n}\n[PROOF STEP]\nproof (rule weighted_sum_rank_1_proj_unitary[symmetric])\n[PROOF STATE]\nproof (state)\ngoal (4 subgoals):\n 1. per_col A f \\<in> fc\n 2. per_diag D f \\<in> fc\n 3. diagonal_mat (per_diag D f)\n 4. Complex_Matrix.unitary (per_col A f)\n[PROOF STEP]\nshow \"per_col A f \\<in> fc\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. per_col A f \\<in> fc\n[PROOF STEP]\nusing per_col_carrier[of A] assms\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\<in> carrier_mat ?n ?m \\<Longrightarrow> per_col A ?f \\<in> carrier_mat ?n ?m\nA \\<in> carrier_mat n n\nComplex_Matrix.unitary A\nD \\<in> carrier_mat n n\ndiagonal_mat D\n0 < n\nbij_betw f {..<n} {..<n}\n\ngoal (1 subgoal):\n 1. per_col A f \\<in> fc\n[PROOF STEP]\nunfolding fc_def\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\<in> carrier_mat ?n ?m \\<Longrightarrow> per_col A ?f \\<in> carrier_mat ?n ?m\nA \\<in> carrier_mat n n\nComplex_Matrix.unitary A\nD \\<in> carrier_mat n n\ndiagonal_mat D\n0 < n\nbij_betw f {..<n} {..<n}\n\ngoal (1 subgoal):\n 1. per_col A f \\<in> carrier_mat n n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nper_col A f \\<in> fc\n\ngoal (3 subgoals):\n 1. per_diag D f \\<in> fc\n 2. diagonal_mat (per_diag D f)\n 3. Complex_Matrix.unitary (per_col A f)\n[PROOF STEP]\nshow \"per_diag D f \\<in> fc\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. per_diag D f \\<in> fc\n[PROOF STEP]\nusing per_diag_carrier[of D] assms\n[PROOF STATE]\nproof (prove)\nusing this:\nper_diag D ?f \\<in> carrier_mat (dim_row D) (dim_col D)\nA \\<in> carrier_mat n n\nComplex_Matrix.unitary A\nD \\<in> carrier_mat n n\ndiagonal_mat D\n0 < n\nbij_betw f {..<n} {..<n}\n\ngoal (1 subgoal):\n 1. per_diag D f \\<in> fc\n[PROOF STEP]\nunfolding fc_def\n[PROOF STATE]\nproof (prove)\nusing this:\nper_diag D ?f \\<in> carrier_mat (dim_row D) (dim_col D)\nA \\<in> carrier_mat n n\nComplex_Matrix.unitary A\nD \\<in> carrier_mat n n\ndiagonal_mat D\n0 < n\nbij_betw f {..<n} {..<n}\n\ngoal (1 subgoal):\n 1. per_diag D f \\<in> carrier_mat n n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nper_diag D f \\<in> fc\n\ngoal (2 subgoals):\n 1. diagonal_mat (per_diag D f)\n 2. Complex_Matrix.unitary (per_col A f)\n[PROOF STEP]\nshow \"diagonal_mat (per_diag D f)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. diagonal_mat (per_diag D f)\n[PROOF STEP]\nusing assms per_diag_diagonal[of D]\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\<in> carrier_mat n n\nComplex_Matrix.unitary A\nD \\<in> carrier_mat n n\ndiagonal_mat D\n0 < n\nbij_betw f {..<n} {..<n}\n\\<lbrakk>D \\<in> carrier_mat ?n ?n; diagonal_mat D; bij_betw ?f {..<?n} {..<?n}\\<rbrakk> \\<Longrightarrow> diagonal_mat (per_diag D ?f)\n\ngoal (1 subgoal):\n 1. diagonal_mat (per_diag D f)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndiagonal_mat (per_diag D f)\n\ngoal (1 subgoal):\n 1. Complex_Matrix.unitary (per_col A f)\n[PROOF STEP]\nshow \"unitary (per_col A f)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.unitary (per_col A f)\n[PROOF STEP]\nusing per_col_unitary[of A] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>A \\<in> carrier_mat ?n ?n; Complex_Matrix.unitary A; bij_betw ?f {..<?n} {..<?n}\\<rbrakk> \\<Longrightarrow> Complex_Matrix.unitary (per_col A ?f)\nA \\<in> carrier_mat n n\nComplex_Matrix.unitary A\nD \\<in> carrier_mat n n\ndiagonal_mat D\n0 < n\nbij_betw f {..<n} {..<n}\n\ngoal (1 subgoal):\n 1. Complex_Matrix.unitary (per_col A f)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.unitary (per_col A f)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nper_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f) = local.sum_mat (\\<lambda>i. diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i)) {..<n}\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nper_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f) = local.sum_mat (\\<lambda>i. diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i)) {..<n}\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nhave \"... = sum_mat g {..<n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. local.sum_mat (\\<lambda>i. diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i)) {..<n} = local.sum_mat g {..<n}\n[PROOF STEP]\nproof (rule sum_mat_cong, (auto simp add: g_def))\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) = diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i))\n 2. \\<And>i. i < n \\<Longrightarrow> diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) \\<in> fc\n 3. \\<And>i. i < n \\<Longrightarrow> diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i)) \\<in> fc\n[PROOF STEP]\nshow \"\\<And>i. i < n \\<Longrightarrow> diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m \n      rank_1_proj (Matrix.col (per_col A f) i) \\<in> fc\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) \\<in> fc\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) \\<in> fc\n[PROOF STEP]\nfix i\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) \\<in> fc\n[PROOF STEP]\nassume \"i < n\"\n[PROOF STATE]\nproof (state)\nthis:\ni < n\n\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) \\<in> fc\n[PROOF STEP]\nhave \"dim_vec (Matrix.col (per_col A f) i) = n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_vec (Matrix.col (per_col A f) i) = n\n[PROOF STEP]\nusing assms per_col_col\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\<in> carrier_mat n n\nComplex_Matrix.unitary A\nD \\<in> carrier_mat n n\ndiagonal_mat D\n0 < n\nbij_betw f {..<n} {..<n}\n\\<lbrakk>?A \\<in> carrier_mat ?n ?m; ?j < ?m\\<rbrakk> \\<Longrightarrow> Matrix.col (per_col ?A ?f) ?j = Matrix.col ?A (?f ?j)\n\ngoal (1 subgoal):\n 1. dim_vec (Matrix.col (per_col A f) i) = n\n[PROOF STEP]\nby (metis carrier_matD(1) dim_col)\n[PROOF STATE]\nproof (state)\nthis:\ndim_vec (Matrix.col (per_col A f) i) = n\n\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) \\<in> fc\n[PROOF STEP]\nhence \"rank_1_proj (Matrix.col (per_col A f) i) \\<in> fc\"\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec (Matrix.col (per_col A f) i) = n\n\ngoal (1 subgoal):\n 1. rank_1_proj (Matrix.col (per_col A f) i) \\<in> fc\n[PROOF STEP]\nunfolding fc_def\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec (Matrix.col (per_col A f) i) = n\n\ngoal (1 subgoal):\n 1. rank_1_proj (Matrix.col (per_col A f) i) \\<in> carrier_mat n n\n[PROOF STEP]\nusing rank_1_proj_carrier\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec (Matrix.col (per_col A f) i) = n\nrank_1_proj ?v \\<in> carrier_mat (dim_vec ?v) (dim_vec ?v)\n\ngoal (1 subgoal):\n 1. rank_1_proj (Matrix.col (per_col A f) i) \\<in> carrier_mat n n\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nrank_1_proj (Matrix.col (per_col A f) i) \\<in> fc\n\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) \\<in> fc\n[PROOF STEP]\nthus \"diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m \n        rank_1_proj (Matrix.col (per_col A f) i) \\<in> fc\"\n[PROOF STATE]\nproof (prove)\nusing this:\nrank_1_proj (Matrix.col (per_col A f) i) \\<in> fc\n\ngoal (1 subgoal):\n 1. diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) \\<in> fc\n[PROOF STEP]\nunfolding fc_def\n[PROOF STATE]\nproof (prove)\nusing this:\nrank_1_proj (Matrix.col (per_col A f) i) \\<in> carrier_mat n n\n\ngoal (1 subgoal):\n 1. diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) \\<in> carrier_mat n n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndiag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) \\<in> fc\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n?i1 < n \\<Longrightarrow> diag_mat (per_diag D f) ! ?i1 \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) ?i1) \\<in> fc\n\ngoal (2 subgoals):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) = diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i))\n 2. \\<And>i. i < n \\<Longrightarrow> diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i)) \\<in> fc\n[PROOF STEP]\nshow \"\\<And>i. i < n \\<Longrightarrow> diag_mat D ! f i \\<cdot>\\<^sub>m \n      rank_1_proj (Matrix.col A (f i)) \\<in> fc\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i)) \\<in> fc\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i)) \\<in> fc\n[PROOF STEP]\nfix i\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i)) \\<in> fc\n[PROOF STEP]\nassume \"i < n\"\n[PROOF STATE]\nproof (state)\nthis:\ni < n\n\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i)) \\<in> fc\n[PROOF STEP]\nhence \"f i < n\"\n[PROOF STATE]\nproof (prove)\nusing this:\ni < n\n\ngoal (1 subgoal):\n 1. f i < n\n[PROOF STEP]\nusing \\<open>f ` {..<n} = {..<n}\\<close>\n[PROOF STATE]\nproof (prove)\nusing this:\ni < n\nf ` {..<n} = {..<n}\n\ngoal (1 subgoal):\n 1. f i < n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nf i < n\n\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i)) \\<in> fc\n[PROOF STEP]\nhence \"dim_vec (Matrix.col A (f i)) = n\"\n[PROOF STATE]\nproof (prove)\nusing this:\nf i < n\n\ngoal (1 subgoal):\n 1. dim_vec (Matrix.col A (f i)) = n\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nf i < n\nA \\<in> carrier_mat n n\nComplex_Matrix.unitary A\nD \\<in> carrier_mat n n\ndiagonal_mat D\n0 < n\nbij_betw f {..<n} {..<n}\n\ngoal (1 subgoal):\n 1. dim_vec (Matrix.col A (f i)) = n\n[PROOF STEP]\nby (metis carrier_matD(1) dim_col)\n[PROOF STATE]\nproof (state)\nthis:\ndim_vec (Matrix.col A (f i)) = n\n\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i)) \\<in> fc\n[PROOF STEP]\nhence \"rank_1_proj (Matrix.col A (f i)) \\<in> fc\"\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec (Matrix.col A (f i)) = n\n\ngoal (1 subgoal):\n 1. rank_1_proj (Matrix.col A (f i)) \\<in> fc\n[PROOF STEP]\nunfolding fc_def\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec (Matrix.col A (f i)) = n\n\ngoal (1 subgoal):\n 1. rank_1_proj (Matrix.col A (f i)) \\<in> carrier_mat n n\n[PROOF STEP]\nusing rank_1_proj_carrier\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec (Matrix.col A (f i)) = n\nrank_1_proj ?v \\<in> carrier_mat (dim_vec ?v) (dim_vec ?v)\n\ngoal (1 subgoal):\n 1. rank_1_proj (Matrix.col A (f i)) \\<in> carrier_mat n n\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nrank_1_proj (Matrix.col A (f i)) \\<in> fc\n\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i)) \\<in> fc\n[PROOF STEP]\nthus \"diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i)) \\<in> fc\"\n[PROOF STATE]\nproof (prove)\nusing this:\nrank_1_proj (Matrix.col A (f i)) \\<in> fc\n\ngoal (1 subgoal):\n 1. diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i)) \\<in> fc\n[PROOF STEP]\nunfolding fc_def\n[PROOF STATE]\nproof (prove)\nusing this:\nrank_1_proj (Matrix.col A (f i)) \\<in> carrier_mat n n\n\ngoal (1 subgoal):\n 1. diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i)) \\<in> carrier_mat n n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndiag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i)) \\<in> fc\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n?i1 < n \\<Longrightarrow> diag_mat D ! f ?i1 \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f ?i1)) \\<in> fc\n\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) = diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i))\n[PROOF STEP]\nshow \"\\<And>i. i < n \\<Longrightarrow>\n         diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m \n           rank_1_proj (Matrix.col (per_col A f) i) =\n         diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) = diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i))\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) = diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i))\n[PROOF STEP]\nfix i\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) = diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i))\n[PROOF STEP]\nassume \"i < n\"\n[PROOF STATE]\nproof (state)\nthis:\ni < n\n\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) = diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i))\n[PROOF STEP]\nhence \"f i < n\"\n[PROOF STATE]\nproof (prove)\nusing this:\ni < n\n\ngoal (1 subgoal):\n 1. f i < n\n[PROOF STEP]\nusing \\<open>f ` {..<n} = {..<n}\\<close>\n[PROOF STATE]\nproof (prove)\nusing this:\ni < n\nf ` {..<n} = {..<n}\n\ngoal (1 subgoal):\n 1. f i < n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nf i < n\n\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) = diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i))\n[PROOF STEP]\nhave \"Matrix.col (per_col A f) i = Matrix.col A (f i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Matrix.col (per_col A f) i = Matrix.col A (f i)\n[PROOF STEP]\nusing per_col_col assms \\<open>i < n\\<close>\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>?A \\<in> carrier_mat ?n ?m; ?j < ?m\\<rbrakk> \\<Longrightarrow> Matrix.col (per_col ?A ?f) ?j = Matrix.col ?A (?f ?j)\nA \\<in> carrier_mat n n\nComplex_Matrix.unitary A\nD \\<in> carrier_mat n n\ndiagonal_mat D\n0 < n\nbij_betw f {..<n} {..<n}\ni < n\n\ngoal (1 subgoal):\n 1. Matrix.col (per_col A f) i = Matrix.col A (f i)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nMatrix.col (per_col A f) i = Matrix.col A (f i)\n\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) = diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i))\n[PROOF STEP]\nhence \"rank_1_proj (Matrix.col (per_col A f) i) = \n        rank_1_proj (Matrix.col A (f i))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nMatrix.col (per_col A f) i = Matrix.col A (f i)\n\ngoal (1 subgoal):\n 1. rank_1_proj (Matrix.col (per_col A f) i) = rank_1_proj (Matrix.col A (f i))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nrank_1_proj (Matrix.col (per_col A f) i) = rank_1_proj (Matrix.col A (f i))\n\ngoal (1 subgoal):\n 1. \\<And>i. i < n \\<Longrightarrow> diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) = diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i))\n[PROOF STEP]\nthus \"diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m \n        rank_1_proj (Matrix.col (per_col A f) i) =\n        diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nrank_1_proj (Matrix.col (per_col A f) i) = rank_1_proj (Matrix.col A (f i))\n\ngoal (1 subgoal):\n 1. diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) = diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i))\n[PROOF STEP]\nusing assms per_diag_diag_mat[of D] \\<open>i < n\\<close> \\<open>f i < n\\<close>\n[PROOF STATE]\nproof (prove)\nusing this:\nrank_1_proj (Matrix.col (per_col A f) i) = rank_1_proj (Matrix.col A (f i))\nA \\<in> carrier_mat n n\nComplex_Matrix.unitary A\nD \\<in> carrier_mat n n\ndiagonal_mat D\n0 < n\nbij_betw f {..<n} {..<n}\n\\<lbrakk>D \\<in> carrier_mat ?n ?n; ?i < ?n; ?f ?i < ?n\\<rbrakk> \\<Longrightarrow> diag_mat (per_diag D ?f) ! ?i = diag_mat D ! ?f ?i\ni < n\nf i < n\n\ngoal (1 subgoal):\n 1. diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) = diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndiag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i) = diag_mat D ! f i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f i))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n?i1 < n \\<Longrightarrow> diag_mat (per_diag D f) ! ?i1 \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) ?i1) = diag_mat D ! f ?i1 \\<cdot>\\<^sub>m rank_1_proj (Matrix.col A (f ?i1))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nlocal.sum_mat (\\<lambda>i. diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i)) {..<n} = local.sum_mat g {..<n}\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nlocal.sum_mat (\\<lambda>i. diag_mat (per_diag D f) ! i \\<cdot>\\<^sub>m rank_1_proj (Matrix.col (per_col A f) i)) {..<n} = local.sum_mat g {..<n}\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nhave \"... = A * D * (Complex_Matrix.adjoint A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. local.sum_mat g {..<n} = A * D * Complex_Matrix.adjoint A\n[PROOF STEP]\nusing sg\n[PROOF STATE]\nproof (prove)\nusing this:\nlocal.sum_mat g {..<n} = A * D * Complex_Matrix.adjoint A\n\ngoal (1 subgoal):\n 1. local.sum_mat g {..<n} = A * D * Complex_Matrix.adjoint A\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlocal.sum_mat g {..<n} = A * D * Complex_Matrix.adjoint A\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nper_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f) = A * D * Complex_Matrix.adjoint A\n[PROOF STEP]\nhave \"(per_col A f) * (per_diag D f) * \n    (Complex_Matrix.adjoint (per_col A f)) = \n    A * D * (Complex_Matrix.adjoint A)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nper_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f) = A * D * Complex_Matrix.adjoint A\n\ngoal (1 subgoal):\n 1. per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f) = A * D * Complex_Matrix.adjoint A\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nper_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f) = A * D * Complex_Matrix.adjoint A\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nper_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f) = A * D * Complex_Matrix.adjoint A\n\ngoal (1 subgoal):\n 1. A * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nA * D * Complex_Matrix.adjoint A = per_col A f * per_diag D f * Complex_Matrix.adjoint (per_col A f)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 17040, "file": "Commuting_Hermitian_Commuting_Hermitian", "length": 143, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8976952838963489, "lm_q2_score": 0.8596637577007394, "lm_q1q2_score": 0.7717161010245673}}
{"text": "[STATEMENT]\nlemma catalan_Suc_aux:\n  \"(n + 2) * catalan (Suc n) = 2 * (2 * n + 1) * catalan n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (n + 2) * catalan (Suc n) = 2 * (2 * n + 1) * catalan n\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (n + 2) * catalan (Suc n) = 2 * (2 * n + 1) * catalan n\n[PROOF STEP]\nhave \"real (catalan (Suc n)) * real (n + 2) = real (catalan n) * 2 * real (2 * n + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (catalan (Suc n)) * real (n + 2) = real (catalan n) * 2 * real (2 * n + 1)\n[PROOF STEP]\nproof (cases n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. n = 0 \\<Longrightarrow> real (catalan (Suc n)) * real (n + 2) = real (catalan n) * 2 * real (2 * n + 1)\n 2. \\<And>nat. n = Suc nat \\<Longrightarrow> real (catalan (Suc n)) * real (n + 2) = real (catalan n) * 2 * real (2 * n + 1)\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\nn = Suc n\n\ngoal (2 subgoals):\n 1. n = 0 \\<Longrightarrow> real (catalan (Suc n)) * real (n + 2) = real (catalan n) * 2 * real (2 * n + 1)\n 2. \\<And>nat. n = Suc nat \\<Longrightarrow> real (catalan (Suc n)) * real (n + 2) = real (catalan n) * 2 * real (2 * n + 1)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nn = Suc n\n\ngoal (1 subgoal):\n 1. real (catalan (Suc n)) * real (n + 2) = real (catalan n) * 2 * real (2 * n + 1)\n[PROOF STEP]\nby (subst (1 2) of_nat_catalan_closed_form, subst (1 2) binomial_fact)\n         (simp_all add: divide_simps)\n[PROOF STATE]\nproof (state)\nthis:\nreal (catalan (Suc n)) * real (n + 2) = real (catalan n) * 2 * real (2 * n + 1)\n\ngoal (1 subgoal):\n 1. n = 0 \\<Longrightarrow> real (catalan (Suc n)) * real (n + 2) = real (catalan n) * 2 * real (2 * n + 1)\n[PROOF STEP]\nqed simp_all\n[PROOF STATE]\nproof (state)\nthis:\nreal (catalan (Suc n)) * real (n + 2) = real (catalan n) * 2 * real (2 * n + 1)\n\ngoal (1 subgoal):\n 1. (n + 2) * catalan (Suc n) = 2 * (2 * n + 1) * catalan n\n[PROOF STEP]\nhence \"real ((n + 2) * catalan (Suc n)) = real (2 * (2 * n + 1) * catalan n)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal (catalan (Suc n)) * real (n + 2) = real (catalan n) * 2 * real (2 * n + 1)\n\ngoal (1 subgoal):\n 1. real ((n + 2) * catalan (Suc n)) = real (2 * (2 * n + 1) * catalan n)\n[PROOF STEP]\nby (simp only: mult_ac of_nat_mult)\n[PROOF STATE]\nproof (state)\nthis:\nreal ((n + 2) * catalan (Suc n)) = real (2 * (2 * n + 1) * catalan n)\n\ngoal (1 subgoal):\n 1. (n + 2) * catalan (Suc n) = 2 * (2 * n + 1) * catalan n\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nreal ((n + 2) * catalan (Suc n)) = real (2 * (2 * n + 1) * catalan n)\n\ngoal (1 subgoal):\n 1. (n + 2) * catalan (Suc n) = 2 * (2 * n + 1) * catalan n\n[PROOF STEP]\nby (simp only: of_nat_eq_iff)\n[PROOF STATE]\nproof (state)\nthis:\n(n + 2) * catalan (Suc n) = 2 * (2 * n + 1) * catalan n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1366, "file": "Catalan_Numbers_Catalan_Numbers", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9073122288794595, "lm_q2_score": 0.8499711718571774, "lm_q1q2_score": 0.7711892384210217}}
{"text": "[STATEMENT]\ntheorem (in group) l_lagrange:\n     \"\\<lbrakk>finite(carrier G); subgroup H G\\<rbrakk>\n      \\<Longrightarrow> card(lcosets H) * card(H) = order(G)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>finite (carrier G); subgroup H G\\<rbrakk> \\<Longrightarrow> card (lcosets H) * card H = order G\n[PROOF STEP]\napply (simp (no_asm_simp) add: order_def lcosets_part_G [symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>finite (carrier G); subgroup H G\\<rbrakk> \\<Longrightarrow> card (lcosets H) * card H = card (\\<Union> (lcosets H))\n[PROOF STEP]\napply (subst mult.commute)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>finite (carrier G); subgroup H G\\<rbrakk> \\<Longrightarrow> card H * card (lcosets H) = card (\\<Union> (lcosets H))\n[PROOF STEP]\napply (rule card_partition)\n[PROOF STATE]\nproof (prove)\ngoal (4 subgoals):\n 1. \\<lbrakk>finite (carrier G); subgroup H G\\<rbrakk> \\<Longrightarrow> finite (lcosets H)\n 2. \\<lbrakk>finite (carrier G); subgroup H G\\<rbrakk> \\<Longrightarrow> finite (\\<Union> (lcosets H))\n 3. \\<And>c. \\<lbrakk>finite (carrier G); subgroup H G; c \\<in> lcosets H\\<rbrakk> \\<Longrightarrow> card c = card H\n 4. \\<And>c1 c2. \\<lbrakk>finite (carrier G); subgroup H G; c1 \\<in> lcosets H; c2 \\<in> lcosets H; c1 \\<noteq> c2\\<rbrakk> \\<Longrightarrow> c1 \\<inter> c2 = {}\n[PROOF STEP]\napply (simp add: lcosets_subset_PowG [THEN finite_subset])\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. \\<lbrakk>finite (carrier G); subgroup H G\\<rbrakk> \\<Longrightarrow> finite (\\<Union> (lcosets H))\n 2. \\<And>c. \\<lbrakk>finite (carrier G); subgroup H G; c \\<in> lcosets H\\<rbrakk> \\<Longrightarrow> card c = card H\n 3. \\<And>c1 c2. \\<lbrakk>finite (carrier G); subgroup H G; c1 \\<in> lcosets H; c2 \\<in> lcosets H; c1 \\<noteq> c2\\<rbrakk> \\<Longrightarrow> c1 \\<inter> c2 = {}\n[PROOF STEP]\napply (simp add: lcosets_part_G)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\<And>c. \\<lbrakk>finite (carrier G); subgroup H G; c \\<in> lcosets H\\<rbrakk> \\<Longrightarrow> card c = card H\n 2. \\<And>c1 c2. \\<lbrakk>finite (carrier G); subgroup H G; c1 \\<in> lcosets H; c2 \\<in> lcosets H; c1 \\<noteq> c2\\<rbrakk> \\<Longrightarrow> c1 \\<inter> c2 = {}\n[PROOF STEP]\napply (simp add: l_card_cosets_equal subgroup.subset)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>c1 c2. \\<lbrakk>finite (carrier G); subgroup H G; c1 \\<in> lcosets H; c2 \\<in> lcosets H; c1 \\<noteq> c2\\<rbrakk> \\<Longrightarrow> c1 \\<inter> c2 = {}\n[PROOF STEP]\napply (simp add: lcos_disjoint)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1060, "file": "Orbit_Stabiliser_Left_Coset", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9136765163620469, "lm_q2_score": 0.8438951064805861, "lm_q1q2_score": 0.7710471410641605}}
{"text": "[STATEMENT]\nlemma liminf_subseq_mono:\n  fixes X :: \"nat \\<Rightarrow> 'a :: complete_linorder\"\n  assumes \"strict_mono r\"\n  shows \"liminf X \\<le> liminf (X \\<circ> r) \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. liminf X \\<le> liminf (X \\<circ> r)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. liminf X \\<le> liminf (X \\<circ> r)\n[PROOF STEP]\nhave \"\\<And>n. (INF m\\<in>{n..}. X m) \\<le> (INF m\\<in>{n..}. (X \\<circ> r) m)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>n. Inf (X ` {n..}) \\<le> Inf ((X \\<circ> r) ` {n..})\n[PROOF STEP]\nproof (safe intro!: INF_mono)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>n m. n \\<le> m \\<Longrightarrow> \\<exists>ma\\<in>{n..}. X ma \\<le> (X \\<circ> r) m\n[PROOF STEP]\nfix n m :: nat\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>n m. n \\<le> m \\<Longrightarrow> \\<exists>ma\\<in>{n..}. X ma \\<le> (X \\<circ> r) m\n[PROOF STEP]\nassume \"n \\<le> m\"\n[PROOF STATE]\nproof (state)\nthis:\nn \\<le> m\n\ngoal (1 subgoal):\n 1. \\<And>n m. n \\<le> m \\<Longrightarrow> \\<exists>ma\\<in>{n..}. X ma \\<le> (X \\<circ> r) m\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nn \\<le> m\n[PROOF STEP]\nshow \"\\<exists>ma\\<in>{n..}. X ma \\<le> (X \\<circ> r) m\"\n[PROOF STATE]\nproof (prove)\nusing this:\nn \\<le> m\n\ngoal (1 subgoal):\n 1. \\<exists>ma\\<in>{n..}. X ma \\<le> (X \\<circ> r) m\n[PROOF STEP]\nusing seq_suble[OF \\<open>strict_mono r\\<close>, of m]\n[PROOF STATE]\nproof (prove)\nusing this:\nn \\<le> m\nm \\<le> r m\n\ngoal (1 subgoal):\n 1. \\<exists>ma\\<in>{n..}. X ma \\<le> (X \\<circ> r) m\n[PROOF STEP]\nby (intro bexI[of _ \"r m\"]) auto\n[PROOF STATE]\nproof (state)\nthis:\n\\<exists>ma\\<in>{n..}. X ma \\<le> (X \\<circ> r) m\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nInf (X ` {?n..}) \\<le> Inf ((X \\<circ> r) ` {?n..})\n\ngoal (1 subgoal):\n 1. liminf X \\<le> liminf (X \\<circ> r)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nInf (X ` {?n..}) \\<le> Inf ((X \\<circ> r) ` {?n..})\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nInf (X ` {?n..}) \\<le> Inf ((X \\<circ> r) ` {?n..})\n\ngoal (1 subgoal):\n 1. liminf X \\<le> liminf (X \\<circ> r)\n[PROOF STEP]\nby (auto intro!: SUP_mono simp: liminf_SUP_INF comp_def)\n[PROOF STATE]\nproof (state)\nthis:\nliminf X \\<le> liminf (X \\<circ> r)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1091, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8872045937171068, "lm_q2_score": 0.8688267728417087, "lm_q1q2_score": 0.7708271040095732}}
{"text": "[STATEMENT]\nlemma sin_double: \"sin(2 * x) = 2 * sin x * cos x\"\n  for x :: \"'a::{real_normed_field,banach}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin ((2::'a) * x) = (2::'a) * sin x * cos x\n[PROOF STEP]\nusing sin_add [where x=x and y=x]\n[PROOF STATE]\nproof (prove)\nusing this:\nsin (x + x) = sin x * cos x + cos x * sin x\n\ngoal (1 subgoal):\n 1. sin ((2::'a) * x) = (2::'a) * sin x * cos x\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 198, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.900529786117893, "lm_q2_score": 0.8558511506439708, "lm_q1q2_score": 0.7707194536381676}}
{"text": "[STATEMENT]\nlemma exponential_order_add:\n  assumes \"exponential_order M a f\" \"exponential_order M a g\"\n  shows \"exponential_order (2 * M) a (\\<lambda>x. f x + g x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exponential_order (2 * M) a (\\<lambda>x. f x + g x)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nexponential_order M a f\nexponential_order M a g\n\ngoal (1 subgoal):\n 1. exponential_order (2 * M) a (\\<lambda>x. f x + g x)\n[PROOF STEP]\napply (auto simp: exponential_order_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>\\<forall>\\<^sub>F t in at_top. norm (g t) \\<le> M * exp (a * t); 0 < M; \\<forall>\\<^sub>F t in at_top. norm (f t) \\<le> M * exp (a * t)\\<rbrakk> \\<Longrightarrow> \\<forall>\\<^sub>F t in at_top. norm (f t + g t) \\<le> 2 * M * exp (a * t)\n[PROOF STEP]\nsubgoal premises prems\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<forall>\\<^sub>F t in at_top. norm (f t + g t) \\<le> 2 * M * exp (a * t)\n[PROOF STEP]\nusing prems(1,3)\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<forall>\\<^sub>F t in at_top. norm (g t) \\<le> M * exp (a * t)\n\\<forall>\\<^sub>F t in at_top. norm (f t) \\<le> M * exp (a * t)\n\ngoal (1 subgoal):\n 1. \\<forall>\\<^sub>F t in at_top. norm (f t + g t) \\<le> 2 * M * exp (a * t)\n[PROOF STEP]\napply (eventually_elim)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>t. \\<lbrakk>norm (g t) \\<le> M * exp (a * t); norm (f t) \\<le> M * exp (a * t)\\<rbrakk> \\<Longrightarrow> norm (f t + g t) \\<le> 2 * M * exp (a * t)\n[PROOF STEP]\napply (rule norm_triangle_le)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>t. \\<lbrakk>norm (g t) \\<le> M * exp (a * t); norm (f t) \\<le> M * exp (a * t)\\<rbrakk> \\<Longrightarrow> norm (f t) + norm (g t) \\<le> 2 * M * exp (a * t)\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 820, "file": "Laplace_Transform_Uniqueness", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9086179068309441, "lm_q2_score": 0.8479677622198946, "lm_q1q2_score": 0.7704786931683604}}
{"text": "[STATEMENT]\nlemma contour_integrable_linepath_Reals_iff:\n  fixes a b :: complex and f :: \"complex \\<Rightarrow> complex\"\n  assumes \"a \\<in> Reals\" \"b \\<in> Reals\" \"Re a < Re b\"\n  shows   \"(f contour_integrable_on linepath a b) \\<longleftrightarrow>\n             (\\<lambda>x. f (of_real x)) integrable_on {Re a..Re b}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f contour_integrable_on linepath a b) = ((\\<lambda>x. f (complex_of_real x)) integrable_on {Re a..Re b})\n[PROOF STEP]\nusing has_contour_integral_linepath_Reals_iff[OF assms, of f]\n[PROOF STATE]\nproof (prove)\nusing this:\n(f has_contour_integral ?I) (linepath a b) = ((\\<lambda>x. f (complex_of_real x)) has_integral ?I) {Re a..Re b}\n\ngoal (1 subgoal):\n 1. (f contour_integrable_on linepath a b) = ((\\<lambda>x. f (complex_of_real x)) integrable_on {Re a..Re b})\n[PROOF STEP]\nby (auto simp: contour_integrable_on_def integrable_on_def)", "meta": {"llama_tokens": 366, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9294403979493139, "lm_q2_score": 0.8289388104343892, "lm_q1q2_score": 0.7704492178457696}}
{"text": "[STATEMENT]\nlemma contour_integrable_linepath_Reals_iff:\n  fixes a b :: complex and f :: \"complex \\<Rightarrow> complex\"\n  assumes \"a \\<in> Reals\" \"b \\<in> Reals\" \"Re a < Re b\"\n  shows   \"(f contour_integrable_on linepath a b) \\<longleftrightarrow>\n             (\\<lambda>x. f (of_real x)) integrable_on {Re a..Re b}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (f contour_integrable_on linepath a b) = ((\\<lambda>x. f (complex_of_real x)) integrable_on {Re a..Re b})\n[PROOF STEP]\nusing has_contour_integral_linepath_Reals_iff[OF assms, of f]\n[PROOF STATE]\nproof (prove)\nusing this:\n(f has_contour_integral ?I) (linepath a b) = ((\\<lambda>x. f (complex_of_real x)) has_integral ?I) {Re a..Re b}\n\ngoal (1 subgoal):\n 1. (f contour_integrable_on linepath a b) = ((\\<lambda>x. f (complex_of_real x)) integrable_on {Re a..Re b})\n[PROOF STEP]\nby (auto simp: contour_integrable_on_def integrable_on_def)", "meta": {"llama_tokens": 366, "file": "Prime_Number_Theorem_Prime_Number_Theorem_Library", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9294403979493139, "lm_q2_score": 0.8289388019824946, "lm_q1q2_score": 0.7704492099902372}}
{"text": "[STATEMENT]\nlemma cos_plus_cos: \"cos w + cos z = 2 * cos ((w + z) / 2) * cos ((w - z) / 2)\"\n  for w :: \"'a::{real_normed_field,banach,field}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos w + cos z = (2::'a) * cos ((w + z) / (2::'a)) * cos ((w - z) / (2::'a))\n[PROOF STEP]\napply (simp add: mult.assoc cos_times_cos)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos w * (2::'a) + cos z * (2::'a) = (2::'a) * cos ((w + z) / (2::'a) - (w - z) / (2::'a)) + (2::'a) * cos ((w + z) / (2::'a) + (w - z) / (2::'a))\n[PROOF STEP]\napply (simp add: field_simps)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 313, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9196425267730008, "lm_q2_score": 0.8376199552262966, "lm_q1q2_score": 0.7703109320997993}}
{"text": "[STATEMENT]\nlemma card_multiset_only_sets':\n  assumes \"finite A\"\n  shows \"card {M. set_mset M \\<subseteq> A \\<and> size M = k \\<and> (\\<forall>x. count M x \\<le> 1)} = card A choose k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {M. set_mset M \\<subseteq> A \\<and> size M = k \\<and> (\\<forall>x. count M x \\<le> 1)} = card A choose k\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {M. set_mset M \\<subseteq> A \\<and> size M = k \\<and> (\\<forall>x. count M x \\<le> 1)} = card A choose k\n[PROOF STEP]\nfrom \\<open>finite A\\<close>\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\n[PROOF STEP]\nhave \"{M. set_mset M \\<subseteq> A \\<and> size M = k \\<and> (\\<forall>x. count M x \\<le> 1)} =\n    {M. M \\<subseteq># mset_set A \\<and> size M = k}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\n\ngoal (1 subgoal):\n 1. {M. set_mset M \\<subseteq> A \\<and> size M = k \\<and> (\\<forall>x. count M x \\<le> 1)} = {M. M \\<subseteq># mset_set A \\<and> size M = k}\n[PROOF STEP]\nusing msubset_mset_set_iff\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nfinite ?A \\<Longrightarrow> (set_mset ?M \\<subseteq> ?A \\<and> (\\<forall>x. count ?M x \\<le> 1)) = (?M \\<subseteq># mset_set ?A)\n\ngoal (1 subgoal):\n 1. {M. set_mset M \\<subseteq> A \\<and> size M = k \\<and> (\\<forall>x. count M x \\<le> 1)} = {M. M \\<subseteq># mset_set A \\<and> size M = k}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{M. set_mset M \\<subseteq> A \\<and> size M = k \\<and> (\\<forall>x. count M x \\<le> 1)} = {M. M \\<subseteq># mset_set A \\<and> size M = k}\n\ngoal (1 subgoal):\n 1. card {M. set_mset M \\<subseteq> A \\<and> size M = k \\<and> (\\<forall>x. count M x \\<le> 1)} = card A choose k\n[PROOF STEP]\nfrom this \\<open>finite A\\<close> card_multiset_only_sets\n[PROOF STATE]\nproof (chain)\npicking this:\n{M. set_mset M \\<subseteq> A \\<and> size M = k \\<and> (\\<forall>x. count M x \\<le> 1)} = {M. M \\<subseteq># mset_set A \\<and> size M = k}\nfinite A\nfinite ?A \\<Longrightarrow> card {M. M \\<subseteq># mset_set ?A \\<and> size M = ?k} = card ?A choose ?k\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n{M. set_mset M \\<subseteq> A \\<and> size M = k \\<and> (\\<forall>x. count M x \\<le> 1)} = {M. M \\<subseteq># mset_set A \\<and> size M = k}\nfinite A\nfinite ?A \\<Longrightarrow> card {M. M \\<subseteq># mset_set ?A \\<and> size M = ?k} = card ?A choose ?k\n\ngoal (1 subgoal):\n 1. card {M. set_mset M \\<subseteq> A \\<and> size M = k \\<and> (\\<forall>x. count M x \\<le> 1)} = card A choose k\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard {M. set_mset M \\<subseteq> A \\<and> size M = k \\<and> (\\<forall>x. count M x \\<le> 1)} = card A choose k\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1127, "file": "Twelvefold_Way_Twelvefold_Way_Entry5", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.89181104831338, "lm_q2_score": 0.8633916011860786, "lm_q1q2_score": 0.7699821689587244}}
{"text": "[STATEMENT]\nlemma hermitian_square_hermitian: \nfixes A::\"'a::conjugatable_field Matrix.mat\"\n  assumes \"hermitian A\"\n  shows \"hermitian (A * A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. hermitian (A * A)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. hermitian (A * A)\n[PROOF STEP]\nhave \"Complex_Matrix.adjoint (A * A) = Complex_Matrix.adjoint A * (Complex_Matrix.adjoint A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.adjoint (A * A) = Complex_Matrix.adjoint A * Complex_Matrix.adjoint A\n[PROOF STEP]\nusing adjoint_mult\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>?A \\<in> carrier_mat ?n ?m; ?B \\<in> carrier_mat ?m ?l\\<rbrakk> \\<Longrightarrow> Complex_Matrix.adjoint (?A * ?B) = Complex_Matrix.adjoint ?B * Complex_Matrix.adjoint ?A\n\ngoal (1 subgoal):\n 1. Complex_Matrix.adjoint (A * A) = Complex_Matrix.adjoint A * Complex_Matrix.adjoint A\n[PROOF STEP]\nby (metis assms hermitian_square)\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.adjoint (A * A) = Complex_Matrix.adjoint A * Complex_Matrix.adjoint A\n\ngoal (1 subgoal):\n 1. hermitian (A * A)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.adjoint (A * A) = Complex_Matrix.adjoint A * Complex_Matrix.adjoint A\n\ngoal (1 subgoal):\n 1. hermitian (A * A)\n[PROOF STEP]\nhave \"... = A * A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.adjoint A * Complex_Matrix.adjoint A = A * A\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nhermitian A\n\ngoal (1 subgoal):\n 1. Complex_Matrix.adjoint A * Complex_Matrix.adjoint A = A * A\n[PROOF STEP]\nunfolding hermitian_def\n[PROOF STATE]\nproof (prove)\nusing this:\nComplex_Matrix.adjoint A = A\n\ngoal (1 subgoal):\n 1. Complex_Matrix.adjoint A * Complex_Matrix.adjoint A = A * A\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.adjoint A * Complex_Matrix.adjoint A = A * A\n\ngoal (1 subgoal):\n 1. hermitian (A * A)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nComplex_Matrix.adjoint (A * A) = A * A\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nComplex_Matrix.adjoint (A * A) = A * A\n\ngoal (1 subgoal):\n 1. hermitian (A * A)\n[PROOF STEP]\nunfolding hermitian_def\n[PROOF STATE]\nproof (prove)\nusing this:\nComplex_Matrix.adjoint (A * A) = A * A\n\ngoal (1 subgoal):\n 1. Complex_Matrix.adjoint (A * A) = A * A\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nhermitian (A * A)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1040, "file": "Commuting_Hermitian_Spectral_Theory_Complements", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392786908831, "lm_q2_score": 0.8705972667296309, "lm_q1q2_score": 0.7696421797099172}}
{"text": "[STATEMENT]\nlemma independent_card_le_aff_dim:\n  fixes B :: \"'n::euclidean_space set\"\n  assumes \"B \\<subseteq> V\"\n  assumes \"\\<not> affine_dependent B\"\n  shows \"int (card B) \\<le> aff_dim V + 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. int (card B) \\<le> aff_dim V + 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. int (card B) \\<le> aff_dim V + 1\n[PROOF STEP]\nobtain T where T: \"\\<not> affine_dependent T \\<and> B \\<subseteq> T \\<and> T \\<subseteq> V \\<and> affine hull T = affine hull V\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<And>T. \\<not> affine_dependent T \\<and> B \\<subseteq> T \\<and> T \\<subseteq> V \\<and> affine hull T = affine hull V \\<Longrightarrow> thesis) \\<Longrightarrow> thesis\n[PROOF STEP]\nby (metis assms extend_to_affine_basis[of B V])\n[PROOF STATE]\nproof (state)\nthis:\n\\<not> affine_dependent T \\<and> B \\<subseteq> T \\<and> T \\<subseteq> V \\<and> affine hull T = affine hull V\n\ngoal (1 subgoal):\n 1. int (card B) \\<le> aff_dim V + 1\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\<not> affine_dependent T \\<and> B \\<subseteq> T \\<and> T \\<subseteq> V \\<and> affine hull T = affine hull V\n[PROOF STEP]\nhave \"of_nat (card T) = aff_dim V + 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<not> affine_dependent T \\<and> B \\<subseteq> T \\<and> T \\<subseteq> V \\<and> affine hull T = affine hull V\n\ngoal (1 subgoal):\n 1. int (card T) = aff_dim V + 1\n[PROOF STEP]\nusing aff_dim_unique\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<not> affine_dependent T \\<and> B \\<subseteq> T \\<and> T \\<subseteq> V \\<and> affine hull T = affine hull V\naffine hull ?B = affine hull ?V \\<and> \\<not> affine_dependent ?B \\<Longrightarrow> int (card ?B) = aff_dim ?V + 1\n\ngoal (1 subgoal):\n 1. int (card T) = aff_dim V + 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nint (card T) = aff_dim V + 1\n\ngoal (1 subgoal):\n 1. int (card B) \\<le> aff_dim V + 1\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nint (card T) = aff_dim V + 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nint (card T) = aff_dim V + 1\n\ngoal (1 subgoal):\n 1. int (card B) \\<le> aff_dim V + 1\n[PROOF STEP]\nusing T card_mono[of T B] aff_independent_finite[of T]\n[PROOF STATE]\nproof (prove)\nusing this:\nint (card T) = aff_dim V + 1\n\\<not> affine_dependent T \\<and> B \\<subseteq> T \\<and> T \\<subseteq> V \\<and> affine hull T = affine hull V\n\\<lbrakk>finite T; B \\<subseteq> T\\<rbrakk> \\<Longrightarrow> card B \\<le> card T\n\\<not> affine_dependent T \\<Longrightarrow> finite T\n\ngoal (1 subgoal):\n 1. int (card B) \\<le> aff_dim V + 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nint (card B) \\<le> aff_dim V + 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1127, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314828740729, "lm_q2_score": 0.8688267660487573, "lm_q1q2_score": 0.7694603371764461}}
{"text": "[STATEMENT]\nlemma card_Un:\n  \"\\<lbrakk> finite A; finite B \\<rbrakk> \\<Longrightarrow> card (A \\<union> B) = card (A) + card (B) - card(A \\<inter> B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>finite A; finite B\\<rbrakk> \\<Longrightarrow> card (A \\<union> B) = card A + card B - card (A \\<inter> B)\n[PROOF STEP]\nby(subst card_Un_Int) simp_all", "meta": {"llama_tokens": 143, "file": "Containers_Card_Datatype", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9207896802383028, "lm_q2_score": 0.8354835350552604, "lm_q1q2_score": 0.7693046170879001}}
{"text": "[STATEMENT]\nlemma poincare_distance_zero_x_axis:\n  assumes \"x \\<in> unit_disc\" and \"x \\<in> circline_set x_axis\"\n  shows \"poincare_distance 0\\<^sub>h x = (let x' = to_complex x in abs (ln (Re ((1 - x') / (1 + x')))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poincare_distance 0\\<^sub>h x = (let x' = to_complex x in \\<bar>ln (Re ((1 - x') / (1 + x')))\\<bar>)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\<in> unit_disc\nx \\<in> circline_set x_axis\n\ngoal (1 subgoal):\n 1. poincare_distance 0\\<^sub>h x = (let x' = to_complex x in \\<bar>ln (Re ((1 - x') / (1 + x')))\\<bar>)\n[PROOF STEP]\nusing poincare_distance_x_axis_x_axis[of \"0\\<^sub>h\" x]\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\<in> unit_disc\nx \\<in> circline_set x_axis\n\\<lbrakk>0\\<^sub>h \\<in> unit_disc; x \\<in> unit_disc; 0\\<^sub>h \\<in> circline_set x_axis; x \\<in> circline_set x_axis\\<rbrakk> \\<Longrightarrow> poincare_distance 0\\<^sub>h x = (let x' = to_complex 0\\<^sub>h; y' = to_complex x in \\<bar>ln (Re ((1 + x') * (1 - y') / ((1 - x') * (1 + y'))))\\<bar>)\n\ngoal (1 subgoal):\n 1. poincare_distance 0\\<^sub>h x = (let x' = to_complex x in \\<bar>ln (Re ((1 - x') / (1 + x')))\\<bar>)\n[PROOF STEP]\nby (simp add: Let_def)", "meta": {"llama_tokens": 548, "file": "Poincare_Disc_Poincare_Distance", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9111797124237605, "lm_q2_score": 0.8438951025545426, "lm_q1q2_score": 0.768940096861468}}
{"text": "[STATEMENT]\nlemma (in group) inter_subgroup_dvd_card:\n  assumes \"subgroup H G\" \"subgroup J G\"\n  shows \"card (H \\<inter> J) dvd card H\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (H \\<inter> J) dvd card H\n[PROOF STEP]\nusing subgroups_Inter_pair[of H J] assms sub_subgroup_dvd_card[of H \"H \\<inter> J\"]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>subgroup H G; subgroup J G\\<rbrakk> \\<Longrightarrow> subgroup (H \\<inter> J) G\nsubgroup H G\nsubgroup J G\n\\<lbrakk>subgroup H G; subgroup (H \\<inter> J) G; H \\<inter> J \\<subseteq> H\\<rbrakk> \\<Longrightarrow> card (H \\<inter> J) dvd card H\n\ngoal (1 subgoal):\n 1. card (H \\<inter> J) dvd card H\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 274, "file": "Finitely_Generated_Abelian_Groups_Miscellaneous_Groups", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505453836383, "lm_q2_score": 0.8499711832583695, "lm_q1q2_score": 0.7689268944950604}}
{"text": "[STATEMENT]\nlemma sum_dist:\n  assumes \"finite (A :: 'a::dioid_one_zero set)\"\n  and \"finite B\"\n  shows \"(\\<Sum>A) \\<cdot> (\\<Sum>B) = \\<Sum>{x \\<cdot> y |x y. x \\<in> A \\<and> y \\<in> B}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<Sum> A \\<cdot> \\<Sum> B = \\<Sum> {x \\<cdot> y |x y. x \\<in> A \\<and> y \\<in> B}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<Sum> A \\<cdot> \\<Sum> B = \\<Sum> {x \\<cdot> y |x y. x \\<in> A \\<and> y \\<in> B}\n[PROOF STEP]\nhave \"(\\<Sum>A) \\<cdot> (\\<Sum>B) = \\<Sum>{x \\<cdot> \\<Sum>B |x. x \\<in> A}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<Sum> A \\<cdot> \\<Sum> B = \\<Sum> {x \\<cdot> \\<Sum> B |x. x \\<in> A}\n[PROOF STEP]\nby (simp add: assms sum_distr)\n[PROOF STATE]\nproof (state)\nthis:\n\\<Sum> A \\<cdot> \\<Sum> B = \\<Sum> {x \\<cdot> \\<Sum> B |x. x \\<in> A}\n\ngoal (1 subgoal):\n 1. \\<Sum> A \\<cdot> \\<Sum> B = \\<Sum> {x \\<cdot> y |x y. x \\<in> A \\<and> y \\<in> B}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\<Sum> A \\<cdot> \\<Sum> B = \\<Sum> {x \\<cdot> \\<Sum> B |x. x \\<in> A}\n\ngoal (1 subgoal):\n 1. \\<Sum> A \\<cdot> \\<Sum> B = \\<Sum> {x \\<cdot> y |x y. x \\<in> A \\<and> y \\<in> B}\n[PROOF STEP]\nhave \"... = \\<Sum>{\\<Sum>{x \\<cdot> y |y. y \\<in> B} |x. x \\<in> A}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<Sum> {x \\<cdot> \\<Sum> B |x. x \\<in> A} = \\<Sum> {\\<Sum> {x \\<cdot> y |y. y \\<in> B} |x. x \\<in> A}\n[PROOF STEP]\nby (simp add: assms sum_distl)\n[PROOF STATE]\nproof (state)\nthis:\n\\<Sum> {x \\<cdot> \\<Sum> B |x. x \\<in> A} = \\<Sum> {\\<Sum> {x \\<cdot> y |y. y \\<in> B} |x. x \\<in> A}\n\ngoal (1 subgoal):\n 1. \\<Sum> A \\<cdot> \\<Sum> B = \\<Sum> {x \\<cdot> y |x y. x \\<in> A \\<and> y \\<in> B}\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\<Sum> A \\<cdot> \\<Sum> B = \\<Sum> {\\<Sum> {x \\<cdot> y |y. y \\<in> B} |x. x \\<in> A}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<Sum> A \\<cdot> \\<Sum> B = \\<Sum> {\\<Sum> {x \\<cdot> y |y. y \\<in> B} |x. x \\<in> A}\n\ngoal (1 subgoal):\n 1. \\<Sum> A \\<cdot> \\<Sum> B = \\<Sum> {x \\<cdot> y |x y. x \\<in> A \\<and> y \\<in> B}\n[PROOF STEP]\nby  (simp only: sum_flatten1 assms finite_cartesian_product)\n[PROOF STATE]\nproof (state)\nthis:\n\\<Sum> A \\<cdot> \\<Sum> B = \\<Sum> {x \\<cdot> y |x y. x \\<in> A \\<and> y \\<in> B}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1113, "file": "Kleene_Algebra_Finite_Suprema", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942377652497, "lm_q2_score": 0.8633916082162403, "lm_q1q2_score": 0.7686725737297908}}
{"text": "[STATEMENT]\nlemma primorial'_conv_primorial:\n  assumes \"n > 0\"\n  shows   \"primorial' n = primorial (nth_prime (n - 1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. primorial' n = primorial (real (nth_prime (n - 1)))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. primorial' n = primorial (real (nth_prime (n - 1)))\n[PROOF STEP]\nhave \"primorial (nth_prime (n - 1)) = (\\<Prod>k<nat (int (n - 1) + 1). nth_prime k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. primorial (real (nth_prime (n - 1))) = prod nth_prime {..<nat (int (n - 1) + 1)}\n[PROOF STEP]\nby (simp add: primorial_conv_primorial' primorial'_def)\n[PROOF STATE]\nproof (state)\nthis:\nprimorial (real (nth_prime (n - 1))) = prod nth_prime {..<nat (int (n - 1) + 1)}\n\ngoal (1 subgoal):\n 1. primorial' n = primorial (real (nth_prime (n - 1)))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nprimorial (real (nth_prime (n - 1))) = prod nth_prime {..<nat (int (n - 1) + 1)}\n\ngoal (1 subgoal):\n 1. primorial' n = primorial (real (nth_prime (n - 1)))\n[PROOF STEP]\nhave \"nat (int (n - 1) + 1) = n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. nat (int (n - 1) + 1) = n\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < n\n\ngoal (1 subgoal):\n 1. nat (int (n - 1) + 1) = n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nnat (int (n - 1) + 1) = n\n\ngoal (1 subgoal):\n 1. primorial' n = primorial (real (nth_prime (n - 1)))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nprimorial (real (nth_prime (n - 1))) = prod nth_prime {..<n}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nprimorial (real (nth_prime (n - 1))) = prod nth_prime {..<n}\n\ngoal (1 subgoal):\n 1. primorial' n = primorial (real (nth_prime (n - 1)))\n[PROOF STEP]\nby (simp add: primorial'_def)\n[PROOF STATE]\nproof (state)\nthis:\nprimorial' n = primorial (real (nth_prime (n - 1)))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 868, "file": "Prime_Distribution_Elementary_Primorial", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9059898153067649, "lm_q2_score": 0.8479677564567913, "lm_q1q2_score": 0.7682501510583802}}
{"text": "[STATEMENT]\nlemma matrix_mult_transpose_dot_row:\n  shows \"A ** transpose A = (\\<chi> i j. inner (row i A) (row j A))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A ** Finite_Cartesian_Product.transpose A = (\\<chi>i j. inner (row i A) (row j A))\n[PROOF STEP]\nby (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def row_def inner_vec_def)", "meta": {"llama_tokens": 139, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9124361700013356, "lm_q2_score": 0.8418256412990657, "lm_q1q2_score": 0.7681121639558376}}
{"text": "[STATEMENT]\nlemma four_block_diag_adjoint:\n  shows  \"(Complex_Matrix.adjoint (four_block_diag A1 A2)) = \n    (four_block_diag (Complex_Matrix.adjoint A1) \n    (Complex_Matrix.adjoint A2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.adjoint (four_block_diag A1 A2) = four_block_diag (Complex_Matrix.adjoint A1) (Complex_Matrix.adjoint A2)\n[PROOF STEP]\nby (rule eq_matI, \n        auto simp: four_block_mat_adjoint zero_adjoint adjoint_eval)", "meta": {"llama_tokens": 188, "file": "Commuting_Hermitian_Commuting_Hermitian", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9099070035949656, "lm_q2_score": 0.8438951104066293, "lm_q1q2_score": 0.7678660712585388}}
{"text": "[STATEMENT]\nlemma closed_compact_differences:\n  fixes S T :: \"'a::real_normed_vector set\"\n  assumes \"closed S\" \"compact T\"\n  shows \"closed (\\<Union>x\\<in> S. \\<Union>y \\<in> T. {x - y})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. closed (\\<Union>x\\<in>S. \\<Union>y\\<in>T. {x - y})\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. closed (\\<Union>x\\<in>S. \\<Union>y\\<in>T. {x - y})\n[PROOF STEP]\nhave \"(\\<Union>x\\<in> S. \\<Union>y \\<in> uminus ` T. {x + y}) = {x - y |x y. x \\<in> S \\<and> y \\<in> T}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Union>x\\<in>S. \\<Union>y\\<in>uminus ` T. {x + y}) = {x - y |x y. x \\<in> S \\<and> y \\<in> T}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Union>x\\<in>S. \\<Union>y\\<in>uminus ` T. {x + y}) = {x - y |x y. x \\<in> S \\<and> y \\<in> T}\n\ngoal (1 subgoal):\n 1. closed (\\<Union>x\\<in>S. \\<Union>y\\<in>T. {x - y})\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\<Union>x\\<in>S. \\<Union>y\\<in>uminus ` T. {x + y}) = {x - y |x y. x \\<in> S \\<and> y \\<in> T}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<Union>x\\<in>S. \\<Union>y\\<in>uminus ` T. {x + y}) = {x - y |x y. x \\<in> S \\<and> y \\<in> T}\n\ngoal (1 subgoal):\n 1. closed (\\<Union>x\\<in>S. \\<Union>y\\<in>T. {x - y})\n[PROOF STEP]\nusing closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<Union>x\\<in>S. \\<Union>y\\<in>uminus ` T. {x + y}) = {x - y |x y. x \\<in> S \\<and> y \\<in> T}\nclosed (\\<Union>x\\<in>S. \\<Union>y\\<in>uminus ` T. {x + y})\n\ngoal (1 subgoal):\n 1. closed (\\<Union>x\\<in>S. \\<Union>y\\<in>T. {x - y})\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nclosed (\\<Union>x\\<in>S. \\<Union>y\\<in>T. {x - y})\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 860, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.894789468908171, "lm_q2_score": 0.8577681068080748, "lm_q1q2_score": 0.7675218687371645}}
{"text": "[STATEMENT]\nlemma matrix'_id_eq_matrix_change_of_basis:\n  fixes X::\"'a::{field}^'n^'n\" and Y::\"'a^'n^'n\"\n  assumes basis_X: \"is_basis (set_of_vector X)\" and basis_Y: \"is_basis (set_of_vector Y)\"\n  shows \"matrix' X Y (id) = matrix_change_of_basis X Y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix' X Y id = matrix_change_of_basis X Y\n[PROOF STEP]\nunfolding matrix'_def matrix_change_of_basis_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<chi>i j. coord Y (id (X $ j)) $ i) = (\\<chi>i j. coord Y (X $ j) $ i)\n[PROOF STEP]\nunfolding id_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<chi>i j. coord Y (X $ j) $ i) = (\\<chi>i j. coord Y (X $ j) $ i)\n[PROOF STEP]\n..", "meta": {"llama_tokens": 319, "file": "Gauss_Jordan_Linear_Maps", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9136765163620469, "lm_q2_score": 0.8397339656668287, "lm_q1q2_score": 0.7672452044213548}}
{"text": "[STATEMENT]\nlemma ceiling_diff_floor_le_1: \"\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nhave \"of_int \\<lceil>x\\<rceil> - 1 < x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_int \\<lceil>x\\<rceil> - (1::'a) < x\n[PROOF STEP]\nusing ceiling_correct[of x]\n[PROOF STATE]\nproof (prove)\nusing this:\nof_int \\<lceil>x\\<rceil> - (1::'a) < x \\<and> x \\<le> of_int \\<lceil>x\\<rceil>\n\ngoal (1 subgoal):\n 1. of_int \\<lceil>x\\<rceil> - (1::'a) < x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nof_int \\<lceil>x\\<rceil> - (1::'a) < x\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nof_int \\<lceil>x\\<rceil> - (1::'a) < x\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nhave \"x < of_int \\<lfloor>x\\<rfloor> + 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x < of_int \\<lfloor>x\\<rfloor> + (1::'a)\n[PROOF STEP]\nusing floor_correct[of x]\n[PROOF STATE]\nproof (prove)\nusing this:\nof_int \\<lfloor>x\\<rfloor> \\<le> x \\<and> x < of_int (\\<lfloor>x\\<rfloor> + 1)\n\ngoal (1 subgoal):\n 1. x < of_int \\<lfloor>x\\<rfloor> + (1::'a)\n[PROOF STEP]\nby simp_all\n[PROOF STATE]\nproof (state)\nthis:\nx < of_int \\<lfloor>x\\<rfloor> + (1::'a)\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nof_int \\<lceil>x\\<rceil> - (1::'a) < of_int \\<lfloor>x\\<rfloor> + (1::'a)\n[PROOF STEP]\nhave \"of_int (\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor>) < (of_int 2::'a)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nof_int \\<lceil>x\\<rceil> - (1::'a) < of_int \\<lfloor>x\\<rfloor> + (1::'a)\n\ngoal (1 subgoal):\n 1. of_int (\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor>) < of_int 2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nof_int (\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor>) < of_int 2\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nof_int (\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor>) < of_int 2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nof_int (\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor>) < of_int 2\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nunfolding of_int_less_iff\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> < 2\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1380, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772253241803, "lm_q2_score": 0.8774767970940975, "lm_q1q2_score": 0.7669824840903575}}
{"text": "[STATEMENT]\nlemma ceiling_diff_floor_le_1: \"\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nhave \"of_int \\<lceil>x\\<rceil> - 1 < x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_int \\<lceil>x\\<rceil> - (1::'a) < x\n[PROOF STEP]\nusing ceiling_correct[of x]\n[PROOF STATE]\nproof (prove)\nusing this:\nof_int \\<lceil>x\\<rceil> - (1::'a) < x \\<and> x \\<le> of_int \\<lceil>x\\<rceil>\n\ngoal (1 subgoal):\n 1. of_int \\<lceil>x\\<rceil> - (1::'a) < x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nof_int \\<lceil>x\\<rceil> - (1::'a) < x\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nof_int \\<lceil>x\\<rceil> - (1::'a) < x\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nhave \"x < of_int \\<lfloor>x\\<rfloor> + 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x < of_int \\<lfloor>x\\<rfloor> + (1::'a)\n[PROOF STEP]\nusing floor_correct[of x]\n[PROOF STATE]\nproof (prove)\nusing this:\nof_int \\<lfloor>x\\<rfloor> \\<le> x \\<and> x < of_int (\\<lfloor>x\\<rfloor> + 1)\n\ngoal (1 subgoal):\n 1. x < of_int \\<lfloor>x\\<rfloor> + (1::'a)\n[PROOF STEP]\nby simp_all\n[PROOF STATE]\nproof (state)\nthis:\nx < of_int \\<lfloor>x\\<rfloor> + (1::'a)\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nof_int \\<lceil>x\\<rceil> - (1::'a) < of_int \\<lfloor>x\\<rfloor> + (1::'a)\n[PROOF STEP]\nhave \"of_int (\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor>) < (of_int 2::'a)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nof_int \\<lceil>x\\<rceil> - (1::'a) < of_int \\<lfloor>x\\<rfloor> + (1::'a)\n\ngoal (1 subgoal):\n 1. of_int (\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor>) < of_int 2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nof_int (\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor>) < of_int 2\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nof_int (\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor>) < of_int 2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nof_int (\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor>) < of_int 2\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nunfolding of_int_less_iff\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> < 2\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1380, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772253241803, "lm_q2_score": 0.8774767922879693, "lm_q1q2_score": 0.7669824798894302}}
{"text": "[STATEMENT]\nlemma ceiling_diff_floor_le_1: \"\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nhave \"of_int \\<lceil>x\\<rceil> - 1 < x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_int \\<lceil>x\\<rceil> - (1::'a) < x\n[PROOF STEP]\nusing ceiling_correct[of x]\n[PROOF STATE]\nproof (prove)\nusing this:\nof_int \\<lceil>x\\<rceil> - (1::'a) < x \\<and> x \\<le> of_int \\<lceil>x\\<rceil>\n\ngoal (1 subgoal):\n 1. of_int \\<lceil>x\\<rceil> - (1::'a) < x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nof_int \\<lceil>x\\<rceil> - (1::'a) < x\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nof_int \\<lceil>x\\<rceil> - (1::'a) < x\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nhave \"x < of_int \\<lfloor>x\\<rfloor> + 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x < of_int \\<lfloor>x\\<rfloor> + (1::'a)\n[PROOF STEP]\nusing floor_correct[of x]\n[PROOF STATE]\nproof (prove)\nusing this:\nof_int \\<lfloor>x\\<rfloor> \\<le> x \\<and> x < of_int (\\<lfloor>x\\<rfloor> + 1)\n\ngoal (1 subgoal):\n 1. x < of_int \\<lfloor>x\\<rfloor> + (1::'a)\n[PROOF STEP]\nby simp_all\n[PROOF STATE]\nproof (state)\nthis:\nx < of_int \\<lfloor>x\\<rfloor> + (1::'a)\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nof_int \\<lceil>x\\<rceil> - (1::'a) < of_int \\<lfloor>x\\<rfloor> + (1::'a)\n[PROOF STEP]\nhave \"of_int (\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor>) < (of_int 2::'a)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nof_int \\<lceil>x\\<rceil> - (1::'a) < of_int \\<lfloor>x\\<rfloor> + (1::'a)\n\ngoal (1 subgoal):\n 1. of_int (\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor>) < of_int 2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nof_int (\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor>) < of_int 2\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nof_int (\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor>) < of_int 2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nof_int (\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor>) < of_int 2\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nunfolding of_int_less_iff\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> < 2\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1380, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772253241803, "lm_q2_score": 0.8774767922879693, "lm_q1q2_score": 0.7669824798894302}}
{"text": "[STATEMENT]\nlemma ceiling_diff_floor_le_1: \"\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nhave \"of_int \\<lceil>x\\<rceil> - 1 < x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_int \\<lceil>x\\<rceil> - (1::'a) < x\n[PROOF STEP]\nusing ceiling_correct[of x]\n[PROOF STATE]\nproof (prove)\nusing this:\nof_int \\<lceil>x\\<rceil> - (1::'a) < x \\<and> x \\<le> of_int \\<lceil>x\\<rceil>\n\ngoal (1 subgoal):\n 1. of_int \\<lceil>x\\<rceil> - (1::'a) < x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nof_int \\<lceil>x\\<rceil> - (1::'a) < x\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nof_int \\<lceil>x\\<rceil> - (1::'a) < x\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nhave \"x < of_int \\<lfloor>x\\<rfloor> + 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x < of_int \\<lfloor>x\\<rfloor> + (1::'a)\n[PROOF STEP]\nusing floor_correct[of x]\n[PROOF STATE]\nproof (prove)\nusing this:\nof_int \\<lfloor>x\\<rfloor> \\<le> x \\<and> x < of_int (\\<lfloor>x\\<rfloor> + 1)\n\ngoal (1 subgoal):\n 1. x < of_int \\<lfloor>x\\<rfloor> + (1::'a)\n[PROOF STEP]\nby simp_all\n[PROOF STATE]\nproof (state)\nthis:\nx < of_int \\<lfloor>x\\<rfloor> + (1::'a)\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nof_int \\<lceil>x\\<rceil> - (1::'a) < of_int \\<lfloor>x\\<rfloor> + (1::'a)\n[PROOF STEP]\nhave \"of_int (\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor>) < (of_int 2::'a)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nof_int \\<lceil>x\\<rceil> - (1::'a) < of_int \\<lfloor>x\\<rfloor> + (1::'a)\n\ngoal (1 subgoal):\n 1. of_int (\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor>) < of_int 2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nof_int (\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor>) < of_int 2\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nof_int (\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor>) < of_int 2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nof_int (\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor>) < of_int 2\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nunfolding of_int_less_iff\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> < 2\n\ngoal (1 subgoal):\n 1. \\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\<lceil>x\\<rceil> - \\<lfloor>x\\<rfloor> \\<le> 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1380, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772253241803, "lm_q2_score": 0.8774767890838836, "lm_q1q2_score": 0.766982477088812}}
{"text": "[STATEMENT]\nlemma orthogonal_matrix_norm:\n  fixes A::\"real^'n^'n\"\n  assumes o: \"orthogonal_matrix A\" \n  shows \"norm (column i A) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (column i A) = 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. norm (column i A) = 1\n[PROOF STEP]\nhave \"1 = (transpose A ** A) $ i $ i\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 = (Finite_Cartesian_Product.transpose A ** A) $ i $ i\n[PROOF STEP]\nusing o\n[PROOF STATE]\nproof (prove)\nusing this:\northogonal_matrix A\n\ngoal (1 subgoal):\n 1. 1 = (Finite_Cartesian_Product.transpose A ** A) $ i $ i\n[PROOF STEP]\nunfolding orthogonal_matrix\n[PROOF STATE]\nproof (prove)\nusing this:\nFinite_Cartesian_Product.transpose A ** A = mat 1\n\ngoal (1 subgoal):\n 1. 1 = (Finite_Cartesian_Product.transpose A ** A) $ i $ i\n[PROOF STEP]\nby (simp add: mat_1_fun)\n[PROOF STATE]\nproof (state)\nthis:\n1 = (Finite_Cartesian_Product.transpose A ** A) $ i $ i\n\ngoal (1 subgoal):\n 1. norm (column i A) = 1\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n1 = (Finite_Cartesian_Product.transpose A ** A) $ i $ i\n\ngoal (1 subgoal):\n 1. norm (column i A) = 1\n[PROOF STEP]\nhave \"... = (column i A) \\<bullet> (column i A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (Finite_Cartesian_Product.transpose A ** A) $ i $ i = column i A \\<bullet> column i A\n[PROOF STEP]\nunfolding matrix_matrix_mult_inner_mult row_transpose\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. column i A \\<bullet> column i A = column i A \\<bullet> column i A\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\n(Finite_Cartesian_Product.transpose A ** A) $ i $ i = column i A \\<bullet> column i A\n\ngoal (1 subgoal):\n 1. norm (column i A) = 1\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n1 = column i A \\<bullet> column i A\n[PROOF STEP]\nshow \"norm (column i A) = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 = column i A \\<bullet> column i A\n\ngoal (1 subgoal):\n 1. norm (column i A) = 1\n[PROOF STEP]\nusing norm_eq_1\n[PROOF STATE]\nproof (prove)\nusing this:\n1 = column i A \\<bullet> column i A\n(norm ?x = 1) = (?x \\<bullet> ?x = 1)\n\ngoal (1 subgoal):\n 1. norm (column i A) = 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nnorm (column i A) = 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 974, "file": "QR_Decomposition_Miscellaneous_QR", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972717658209, "lm_q2_score": 0.880797081106935, "lm_q1q2_score": 0.766819535790996}}
{"text": "[STATEMENT]\nlemma sum_le_card_Max: \"\\<lbrakk> finite A; A \\<noteq> {} \\<rbrakk> \\<Longrightarrow> sum f A \\<le> card A * Max (f ` A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>finite A; A \\<noteq> {}\\<rbrakk> \\<Longrightarrow> sum f A \\<le> card A * Max (f ` A)\n[PROOF STEP]\nproof(induction A rule: finite_ne_induct)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\<And>x. sum f {x} \\<le> card {x} * Max (f ` {x})\n 2. \\<And>x F. \\<lbrakk>finite F; F \\<noteq> {}; x \\<notin> F; sum f F \\<le> card F * Max (f ` F)\\<rbrakk> \\<Longrightarrow> sum f (insert x F) \\<le> card (insert x F) * Max (f ` insert x F)\n[PROOF STEP]\ncase (singleton x)\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. \\<And>x. sum f {x} \\<le> card {x} * Max (f ` {x})\n 2. \\<And>x F. \\<lbrakk>finite F; F \\<noteq> {}; x \\<notin> F; sum f F \\<le> card F * Max (f ` F)\\<rbrakk> \\<Longrightarrow> sum f (insert x F) \\<le> card (insert x F) * Max (f ` insert x F)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f {x} \\<le> card {x} * Max (f ` {x})\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsum f {x} \\<le> card {x} * Max (f ` {x})\n\ngoal (1 subgoal):\n 1. \\<And>x F. \\<lbrakk>finite F; F \\<noteq> {}; x \\<notin> F; sum f F \\<le> card F * Max (f ` F)\\<rbrakk> \\<Longrightarrow> sum f (insert x F) \\<le> card (insert x F) * Max (f ` insert x F)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>x F. \\<lbrakk>finite F; F \\<noteq> {}; x \\<notin> F; sum f F \\<le> card F * Max (f ` F)\\<rbrakk> \\<Longrightarrow> sum f (insert x F) \\<le> card (insert x F) * Max (f ` insert x F)\n[PROOF STEP]\ncase (insert x F)\n[PROOF STATE]\nproof (state)\nthis:\nfinite F\nF \\<noteq> {}\nx \\<notin> F\nsum f F \\<le> card F * Max (f ` F)\n\ngoal (1 subgoal):\n 1. \\<And>x F. \\<lbrakk>finite F; F \\<noteq> {}; x \\<notin> F; sum f F \\<le> card F * Max (f ` F)\\<rbrakk> \\<Longrightarrow> sum f (insert x F) \\<le> card (insert x F) * Max (f ` insert x F)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite F\nF \\<noteq> {}\nx \\<notin> F\nsum f F \\<le> card F * Max (f ` F)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite F\nF \\<noteq> {}\nx \\<notin> F\nsum f F \\<le> card F * Max (f ` F)\n\ngoal (1 subgoal):\n 1. sum f (insert x F) \\<le> card (insert x F) * Max (f ` insert x F)\n[PROOF STEP]\nby (auto simp: max_def order.trans[of \"sum f F\" \"card F * Max (f ` F)\"])\n[PROOF STATE]\nproof (state)\nthis:\nsum f (insert x F) \\<le> card (insert x F) * Max (f ` insert x F)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1138, "file": "Approximation_Algorithms_Approx_LB_Hoare", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8824278695464501, "lm_q2_score": 0.8688267779364222, "lm_q1q2_score": 0.7666769626593437}}
{"text": "[STATEMENT]\ntheorem (in state_space)\n  shows \"\\<Gamma>\\<turnstile> \\<lbrace>True\\<rbrace>\n           \\<acute>S :== 0;; \\<acute>I :== 1;;\n           WHILE \\<acute>I \\<noteq> n\n           INV \\<lbrace>\\<acute>S = (SUMM j<\\<acute>I. j)\\<rbrace>\n           DO\n             \\<acute>S :== \\<acute>S + \\<acute>I;;\n             \\<acute>I :== \\<acute>I + 1\n           OD\n          \\<lbrace>\\<acute>S = (SUMM j<n. j)\\<rbrace>\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<Gamma>\\<turnstile> \\<lbrace>True\\<rbrace> \\<acute>S :== 0;; \\<acute>I :== 1;; WHILE \\<acute>I \\<noteq> n INV \\<lbrace>\\<acute>S = (SUMM j<\\<acute>I. j)\\<rbrace> DO \\<acute>S :== \\<acute>S + \\<acute>I;; \\<acute>I :== \\<acute>I + 1 OD \\<lbrace>\\<acute>S = (SUMM j<n. j)\\<rbrace>\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<Gamma>\\<turnstile> \\<lbrace>True\\<rbrace> \\<acute>S :== 0;; \\<acute>I :== 1;; WHILE \\<acute>I \\<noteq> n INV \\<lbrace>\\<acute>S = (SUMM j<\\<acute>I. j)\\<rbrace> DO \\<acute>S :== \\<acute>S + \\<acute>I;; \\<acute>I :== \\<acute>I + 1 OD \\<lbrace>\\<acute>S = (SUMM j<n. j)\\<rbrace>\n[PROOF STEP]\nlet ?sum = \"\\<lambda>k. SUMM j<k. j\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<Gamma>\\<turnstile> \\<lbrace>True\\<rbrace> \\<acute>S :== 0;; \\<acute>I :== 1;; WHILE \\<acute>I \\<noteq> n INV \\<lbrace>\\<acute>S = (SUMM j<\\<acute>I. j)\\<rbrace> DO \\<acute>S :== \\<acute>S + \\<acute>I;; \\<acute>I :== \\<acute>I + 1 OD \\<lbrace>\\<acute>S = (SUMM j<n. j)\\<rbrace>\n[PROOF STEP]\nlet ?inv = \"\\<lambda>s i. s = ?sum i\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<Gamma>\\<turnstile> \\<lbrace>True\\<rbrace> \\<acute>S :== 0;; \\<acute>I :== 1;; WHILE \\<acute>I \\<noteq> n INV \\<lbrace>\\<acute>S = (SUMM j<\\<acute>I. j)\\<rbrace> DO \\<acute>S :== \\<acute>S + \\<acute>I;; \\<acute>I :== \\<acute>I + 1 OD \\<lbrace>\\<acute>S = (SUMM j<n. j)\\<rbrace>\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<Gamma>\\<turnstile> \\<lbrace>True\\<rbrace> \\<acute>S :== 0;; \\<acute>I :== 1;; WHILE \\<acute>I \\<noteq> n INV \\<lbrace>\\<acute>S = (SUMM j<\\<acute>I. j)\\<rbrace> DO \\<acute>S :== \\<acute>S + \\<acute>I;; \\<acute>I :== \\<acute>I + 1 OD \\<lbrace>\\<acute>S = (SUMM j<n. j)\\<rbrace>\n[PROOF STEP]\nproof vcg\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. True \\<Longrightarrow> 0 = (SUMM j<1. j)\n 2. \\<And>I S. \\<lbrakk>S = (SUMM j<I. j); I \\<noteq> n\\<rbrakk> \\<Longrightarrow> S + I = (SUMM j<I + 1. j)\n 3. \\<And>I S. \\<lbrakk>S = (SUMM j<I. j); \\<not> I \\<noteq> n\\<rbrakk> \\<Longrightarrow> S = (SUMM j<n. j)\n[PROOF STEP]\nshow \"?inv 0 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 = (SUMM j<1. j)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n0 = (SUMM j<1. j)\n\ngoal (2 subgoals):\n 1. \\<And>I S. \\<lbrakk>S = (SUMM j<I. j); I \\<noteq> n\\<rbrakk> \\<Longrightarrow> S + I = (SUMM j<I + 1. j)\n 2. \\<And>I S. \\<lbrakk>S = (SUMM j<I. j); \\<not> I \\<noteq> n\\<rbrakk> \\<Longrightarrow> S = (SUMM j<n. j)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\<And>I S. \\<lbrakk>S = (SUMM j<I. j); I \\<noteq> n\\<rbrakk> \\<Longrightarrow> S + I = (SUMM j<I + 1. j)\n 2. \\<And>I S. \\<lbrakk>S = (SUMM j<I. j); \\<not> I \\<noteq> n\\<rbrakk> \\<Longrightarrow> S = (SUMM j<n. j)\n[PROOF STEP]\nfix i s\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\<And>I S. \\<lbrakk>S = (SUMM j<I. j); I \\<noteq> n\\<rbrakk> \\<Longrightarrow> S + I = (SUMM j<I + 1. j)\n 2. \\<And>I S. \\<lbrakk>S = (SUMM j<I. j); \\<not> I \\<noteq> n\\<rbrakk> \\<Longrightarrow> S = (SUMM j<n. j)\n[PROOF STEP]\nassume \"?inv s i\" \"i \\<noteq> n\"\n[PROOF STATE]\nproof (state)\nthis:\ns = (SUMM j<i. j)\ni \\<noteq> n\n\ngoal (2 subgoals):\n 1. \\<And>I S. \\<lbrakk>S = (SUMM j<I. j); I \\<noteq> n\\<rbrakk> \\<Longrightarrow> S + I = (SUMM j<I + 1. j)\n 2. \\<And>I S. \\<lbrakk>S = (SUMM j<I. j); \\<not> I \\<noteq> n\\<rbrakk> \\<Longrightarrow> S = (SUMM j<n. j)\n[PROOF STEP]\nthus \"?inv (s + i) (i + 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ns = (SUMM j<i. j)\ni \\<noteq> n\n\ngoal (1 subgoal):\n 1. s + i = (SUMM j<i + 1. j)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ns + i = (SUMM j<i + 1. j)\n\ngoal (1 subgoal):\n 1. \\<And>I S. \\<lbrakk>S = (SUMM j<I. j); \\<not> I \\<noteq> n\\<rbrakk> \\<Longrightarrow> S = (SUMM j<n. j)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>I S. \\<lbrakk>S = (SUMM j<I. j); \\<not> I \\<noteq> n\\<rbrakk> \\<Longrightarrow> S = (SUMM j<n. j)\n[PROOF STEP]\nfix i s\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>I S. \\<lbrakk>S = (SUMM j<I. j); \\<not> I \\<noteq> n\\<rbrakk> \\<Longrightarrow> S = (SUMM j<n. j)\n[PROOF STEP]\nassume x: \"?inv s i\" \"\\<not> i \\<noteq> n\"\n[PROOF STATE]\nproof (state)\nthis:\ns = (SUMM j<i. j)\n\\<not> i \\<noteq> n\n\ngoal (1 subgoal):\n 1. \\<And>I S. \\<lbrakk>S = (SUMM j<I. j); \\<not> I \\<noteq> n\\<rbrakk> \\<Longrightarrow> S = (SUMM j<n. j)\n[PROOF STEP]\nthus \"s = ?sum n\"\n[PROOF STATE]\nproof (prove)\nusing this:\ns = (SUMM j<i. j)\n\\<not> i \\<noteq> n\n\ngoal (1 subgoal):\n 1. s = (SUMM j<n. j)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ns = (SUMM j<n. j)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\<Gamma>\\<turnstile> \\<lbrace>True\\<rbrace> \\<acute>S :== 0;; \\<acute>I :== 1;; WHILE \\<acute>I \\<noteq> n INV \\<lbrace>\\<acute>S = (SUMM j<\\<acute>I. j)\\<rbrace> DO \\<acute>S :== \\<acute>S + \\<acute>I;; \\<acute>I :== \\<acute>I + 1 OD \\<lbrace>\\<acute>S = (SUMM j<n. j)\\<rbrace>\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2674, "file": "Simpl_ex_VcgExSP", "length": 19, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8976952948443461, "lm_q2_score": 0.8539127473751341, "lm_q1q2_score": 0.7665534555262666}}
{"text": "[STATEMENT]\nlemma ex_sphere_params:\n  assumes \"x*x + y*y + z*z = 1\"\n  shows \"\\<exists> \\<alpha> \\<beta>. x = cos \\<alpha> * cos \\<beta> \\<and> y = cos \\<alpha> * sin \\<beta> \\<and> z = sin \\<alpha> \\<and> -pi / 2 \\<le> \\<alpha> \\<and> \\<alpha> \\<le> pi / 2 \\<and> -pi \\<le> \\<beta> \\<and> \\<beta> < pi\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<exists>\\<alpha> \\<beta>. x = cos \\<alpha> * cos \\<beta> \\<and> y = cos \\<alpha> * sin \\<beta> \\<and> z = sin \\<alpha> \\<and> - pi / 2 \\<le> \\<alpha> \\<and> \\<alpha> \\<le> pi / 2 \\<and> - pi \\<le> \\<beta> \\<and> \\<beta> < pi\n[PROOF STEP]\nusing assms arcsin_bounded[of z] sphere_bounds[of x y z]\n[PROOF STATE]\nproof (prove)\nusing this:\nx * x + y * y + z * z = 1\n\\<lbrakk>- 1 \\<le> z; z \\<le> 1\\<rbrakk> \\<Longrightarrow> - (pi / 2) \\<le> arcsin z \\<and> arcsin z \\<le> pi / 2\nx * x + y * y + z * z = 1 \\<Longrightarrow> - 1 \\<le> x \\<and> x \\<le> 1\nx * x + y * y + z * z = 1 \\<Longrightarrow> - 1 \\<le> y \\<and> y \\<le> 1\nx * x + y * y + z * z = 1 \\<Longrightarrow> - 1 \\<le> z \\<and> z \\<le> 1\n\ngoal (1 subgoal):\n 1. \\<exists>\\<alpha> \\<beta>. x = cos \\<alpha> * cos \\<beta> \\<and> y = cos \\<alpha> * sin \\<beta> \\<and> z = sin \\<alpha> \\<and> - pi / 2 \\<le> \\<alpha> \\<and> \\<alpha> \\<le> pi / 2 \\<and> - pi \\<le> \\<beta> \\<and> \\<beta> < pi\n[PROOF STEP]\nby (rule_tac x=\"arcsin z\" in exI, rule_tac x=\"atan2 y x\" in exI) (simp add: sphere_params arcsin_bounded atan2_bounded)", "meta": {"llama_tokens": 614, "file": "Complex_Geometry_Riemann_Sphere", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.905989822921759, "lm_q2_score": 0.84594244507642, "lm_q1q2_score": 0.7664152460167856}}
{"text": "[STATEMENT]\nlemma card_extensional_funcset_inj_on:\n  assumes \"finite S\" \"finite T\" \"card S \\<le> card T\"\n  shows \"card {f \\<in> extensional_funcset S T. inj_on f S} = fact (card T) div (fact (card T - card S))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {f \\<in> S \\<rightarrow>\\<^sub>E T. inj_on f S} = fact (card T) div fact (card T - card S)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite S\nfinite T\ncard S \\<le> card T\n\ngoal (1 subgoal):\n 1. card {f \\<in> S \\<rightarrow>\\<^sub>E T. inj_on f S} = fact (card T) div fact (card T - card S)\n[PROOF STEP]\nproof (induct S arbitrary: T rule: finite_induct)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\<And>T. \\<lbrakk>finite T; card {} \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> {} \\<rightarrow>\\<^sub>E T. inj_on f {}} = fact (card T) div fact (card T - card {})\n 2. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\ncase empty\n[PROOF STATE]\nproof (state)\nthis:\nfinite T\ncard {} \\<le> card T\n\ngoal (2 subgoals):\n 1. \\<And>T. \\<lbrakk>finite T; card {} \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> {} \\<rightarrow>\\<^sub>E T. inj_on f {}} = fact (card T) div fact (card T - card {})\n 2. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite T\ncard {} \\<le> card T\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite T\ncard {} \\<le> card T\n\ngoal (1 subgoal):\n 1. card {f. f \\<in> {} \\<rightarrow>\\<^sub>E T \\<and> inj_on f {}} = fact (card T) div fact (card T - card {})\n[PROOF STEP]\nby (simp add: Collect_conv_if PiE_empty_domain)\n[PROOF STATE]\nproof (state)\nthis:\ncard {f. f \\<in> {} \\<rightarrow>\\<^sub>E T \\<and> inj_on f {}} = fact (card T) div fact (card T - card {})\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\ncase (insert x S)\n[PROOF STATE]\nproof (state)\nthis:\nfinite S\nx \\<notin> S\n\\<lbrakk>finite ?T; card S \\<le> card ?T\\<rbrakk> \\<Longrightarrow> card {f. f \\<in> S \\<rightarrow>\\<^sub>E ?T \\<and> inj_on f S} = fact (card ?T) div fact (card ?T - card S)\nfinite T\ncard (insert x S) \\<le> card T\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\n{\n[PROOF STATE]\nproof (state)\nthis:\nfinite S\nx \\<notin> S\n\\<lbrakk>finite ?T; card S \\<le> card ?T\\<rbrakk> \\<Longrightarrow> card {f. f \\<in> S \\<rightarrow>\\<^sub>E ?T \\<and> inj_on f S} = fact (card ?T) div fact (card ?T - card S)\nfinite T\ncard (insert x S) \\<le> card T\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nfix x\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nfrom \\<open>finite T\\<close>\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite T\n[PROOF STEP]\nhave \"finite (T - {x})\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite T\n\ngoal (1 subgoal):\n 1. finite (T - {x})\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nfinite (T - {x})\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nfrom \\<open>finite S\\<close> this\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite S\nfinite (T - {x})\n[PROOF STEP]\nhave \"finite (extensional_funcset S (T - {x}))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite S\nfinite (T - {x})\n\ngoal (1 subgoal):\n 1. finite (S \\<rightarrow>\\<^sub>E T - {x})\n[PROOF STEP]\nby (rule finite_PiE)\n[PROOF STATE]\nproof (state)\nthis:\nfinite (S \\<rightarrow>\\<^sub>E T - {x})\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nfinite (S \\<rightarrow>\\<^sub>E T - {x})\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nhave \"{f : extensional_funcset S (T - {x}). inj_on f S} \\<subseteq> (extensional_funcset S (T - {x}))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {f \\<in> S \\<rightarrow>\\<^sub>E T - {x}. inj_on f S} \\<subseteq> S \\<rightarrow>\\<^sub>E T - {x}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{f \\<in> S \\<rightarrow>\\<^sub>E T - {x}. inj_on f S} \\<subseteq> S \\<rightarrow>\\<^sub>E T - {x}\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite (S \\<rightarrow>\\<^sub>E T - {x})\n{f \\<in> S \\<rightarrow>\\<^sub>E T - {x}. inj_on f S} \\<subseteq> S \\<rightarrow>\\<^sub>E T - {x}\n[PROOF STEP]\nhave \"finite {f : extensional_funcset S (T - {x}). inj_on f S}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite (S \\<rightarrow>\\<^sub>E T - {x})\n{f \\<in> S \\<rightarrow>\\<^sub>E T - {x}. inj_on f S} \\<subseteq> S \\<rightarrow>\\<^sub>E T - {x}\n\ngoal (1 subgoal):\n 1. finite {f \\<in> S \\<rightarrow>\\<^sub>E T - {x}. inj_on f S}\n[PROOF STEP]\nby (auto intro: finite_subset)\n[PROOF STATE]\nproof (state)\nthis:\nfinite {f \\<in> S \\<rightarrow>\\<^sub>E T - {x}. inj_on f S}\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\n}\n[PROOF STATE]\nproof (state)\nthis:\nfinite {f \\<in> S \\<rightarrow>\\<^sub>E T - {?xa2}. inj_on f S}\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nnote finite_delete = this\n[PROOF STATE]\nproof (state)\nthis:\nfinite {f \\<in> S \\<rightarrow>\\<^sub>E T - {?xa2}. inj_on f S}\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nfrom insert\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite S\nx \\<notin> S\n\\<lbrakk>finite ?T; card S \\<le> card ?T\\<rbrakk> \\<Longrightarrow> card {f. f \\<in> S \\<rightarrow>\\<^sub>E ?T \\<and> inj_on f S} = fact (card ?T) div fact (card ?T - card S)\nfinite T\ncard (insert x S) \\<le> card T\n[PROOF STEP]\nhave hyps: \"\\<forall>y \\<in> T. card ({g. g \\<in> extensional_funcset S (T - {y}) \\<and> inj_on g S}) = fact (card T - 1) div fact ((card T - 1) - card S)\"(is \"\\<forall> _ \\<in> T. _ = ?k\")\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite S\nx \\<notin> S\n\\<lbrakk>finite ?T; card S \\<le> card ?T\\<rbrakk> \\<Longrightarrow> card {f. f \\<in> S \\<rightarrow>\\<^sub>E ?T \\<and> inj_on f S} = fact (card ?T) div fact (card ?T - card S)\nfinite T\ncard (insert x S) \\<le> card T\n\ngoal (1 subgoal):\n 1. \\<forall>y\\<in>T. card {g \\<in> S \\<rightarrow>\\<^sub>E T - {y}. inj_on g S} = fact (card T - 1) div fact (card T - 1 - card S)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\<forall>y\\<in>T. card {g \\<in> S \\<rightarrow>\\<^sub>E T - {y}. inj_on g S} = fact (card T - 1) div fact (card T - 1 - card S)\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nfrom extensional_funcset_extend_domain_inj_on_eq[OF \\<open>x \\<notin> S\\<close>]\n[PROOF STATE]\nproof (chain)\npicking this:\n{f \\<in> insert x S \\<rightarrow>\\<^sub>E ?T. inj_on f (insert x S)} = (\\<lambda>(y, g). g(x := y)) ` {(y, g). y \\<in> ?T \\<and> g \\<in> S \\<rightarrow>\\<^sub>E ?T - {y} \\<and> inj_on g S}\n[PROOF STEP]\nhave \"card {f. f \\<in> extensional_funcset (insert x S) T \\<and> inj_on f (insert x S)} =\n    card ((\\<lambda>(y, g). g(x := y)) ` {(y, g). y \\<in> T \\<and> g \\<in> extensional_funcset S (T - {y}) \\<and> inj_on g S})\"\n[PROOF STATE]\nproof (prove)\nusing this:\n{f \\<in> insert x S \\<rightarrow>\\<^sub>E ?T. inj_on f (insert x S)} = (\\<lambda>(y, g). g(x := y)) ` {(y, g). y \\<in> ?T \\<and> g \\<in> S \\<rightarrow>\\<^sub>E ?T - {y} \\<and> inj_on g S}\n\ngoal (1 subgoal):\n 1. card {f \\<in> insert x S \\<rightarrow>\\<^sub>E T. inj_on f (insert x S)} = card ((\\<lambda>(y, g). g(x := y)) ` {(y, g). y \\<in> T \\<and> g \\<in> S \\<rightarrow>\\<^sub>E T - {y} \\<and> inj_on g S})\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\ncard {f \\<in> insert x S \\<rightarrow>\\<^sub>E T. inj_on f (insert x S)} = card ((\\<lambda>(y, g). g(x := y)) ` {(y, g). y \\<in> T \\<and> g \\<in> S \\<rightarrow>\\<^sub>E T - {y} \\<and> inj_on g S})\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard {f \\<in> insert x S \\<rightarrow>\\<^sub>E T. inj_on f (insert x S)} = card ((\\<lambda>(y, g). g(x := y)) ` {(y, g). y \\<in> T \\<and> g \\<in> S \\<rightarrow>\\<^sub>E T - {y} \\<and> inj_on g S})\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nfrom extensional_funcset_extend_domain_inj_onI[OF \\<open>x \\<notin> S\\<close>, of T]\n[PROOF STATE]\nproof (chain)\npicking this:\ninj_on (\\<lambda>(y, g). g(x := y)) {(y, g). y \\<in> T \\<and> g \\<in> S \\<rightarrow>\\<^sub>E T - {y} \\<and> inj_on g S}\n[PROOF STEP]\nhave \"\\<dots> =  card {(y, g). y \\<in> T \\<and> g \\<in> extensional_funcset S (T - {y}) \\<and> inj_on g S}\"\n[PROOF STATE]\nproof (prove)\nusing this:\ninj_on (\\<lambda>(y, g). g(x := y)) {(y, g). y \\<in> T \\<and> g \\<in> S \\<rightarrow>\\<^sub>E T - {y} \\<and> inj_on g S}\n\ngoal (1 subgoal):\n 1. card ((\\<lambda>(y, g). g(x := y)) ` {(y, g). y \\<in> T \\<and> g \\<in> S \\<rightarrow>\\<^sub>E T - {y} \\<and> inj_on g S}) = card {(y, g). y \\<in> T \\<and> g \\<in> S \\<rightarrow>\\<^sub>E T - {y} \\<and> inj_on g S}\n[PROOF STEP]\nby (simp add: card_image)\n[PROOF STATE]\nproof (state)\nthis:\ncard ((\\<lambda>(y, g). g(x := y)) ` {(y, g). y \\<in> T \\<and> g \\<in> S \\<rightarrow>\\<^sub>E T - {y} \\<and> inj_on g S}) = card {(y, g). y \\<in> T \\<and> g \\<in> S \\<rightarrow>\\<^sub>E T - {y} \\<and> inj_on g S}\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ((\\<lambda>(y, g). g(x := y)) ` {(y, g). y \\<in> T \\<and> g \\<in> S \\<rightarrow>\\<^sub>E T - {y} \\<and> inj_on g S}) = card {(y, g). y \\<in> T \\<and> g \\<in> S \\<rightarrow>\\<^sub>E T - {y} \\<and> inj_on g S}\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nhave \"card {(y, g). y \\<in> T \\<and> g \\<in> extensional_funcset S (T - {y}) \\<and> inj_on g S} =\n    card {(y, g). y \\<in> T \\<and> g \\<in> {f \\<in> extensional_funcset S (T - {y}). inj_on f S}}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {(y, g). y \\<in> T \\<and> g \\<in> S \\<rightarrow>\\<^sub>E T - {y} \\<and> inj_on g S} = card {(y, g). y \\<in> T \\<and> g \\<in> {f \\<in> S \\<rightarrow>\\<^sub>E T - {y}. inj_on f S}}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncard {(y, g). y \\<in> T \\<and> g \\<in> S \\<rightarrow>\\<^sub>E T - {y} \\<and> inj_on g S} = card {(y, g). y \\<in> T \\<and> g \\<in> {f \\<in> S \\<rightarrow>\\<^sub>E T - {y}. inj_on f S}}\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard {(y, g). y \\<in> T \\<and> g \\<in> S \\<rightarrow>\\<^sub>E T - {y} \\<and> inj_on g S} = card {(y, g). y \\<in> T \\<and> g \\<in> {f \\<in> S \\<rightarrow>\\<^sub>E T - {y}. inj_on f S}}\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nfrom \\<open>finite T\\<close> finite_delete\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite T\nfinite {f \\<in> S \\<rightarrow>\\<^sub>E T - {?xa2}. inj_on f S}\n[PROOF STEP]\nhave \"... = (\\<Sum>y \\<in> T. card {g. g \\<in> extensional_funcset S (T - {y}) \\<and>  inj_on g S})\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite T\nfinite {f \\<in> S \\<rightarrow>\\<^sub>E T - {?xa2}. inj_on f S}\n\ngoal (1 subgoal):\n 1. card {(y, g). y \\<in> T \\<and> g \\<in> {f \\<in> S \\<rightarrow>\\<^sub>E T - {y}. inj_on f S}} = (\\<Sum>y\\<in>T. card {g \\<in> S \\<rightarrow>\\<^sub>E T - {y}. inj_on g S})\n[PROOF STEP]\nby (subst card_product_dependent) auto\n[PROOF STATE]\nproof (state)\nthis:\ncard {(y, g). y \\<in> T \\<and> g \\<in> {f \\<in> S \\<rightarrow>\\<^sub>E T - {y}. inj_on f S}} = (\\<Sum>y\\<in>T. card {g \\<in> S \\<rightarrow>\\<^sub>E T - {y}. inj_on g S})\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard {(y, g). y \\<in> T \\<and> g \\<in> {f \\<in> S \\<rightarrow>\\<^sub>E T - {y}. inj_on f S}} = (\\<Sum>y\\<in>T. card {g \\<in> S \\<rightarrow>\\<^sub>E T - {y}. inj_on g S})\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nfrom hyps\n[PROOF STATE]\nproof (chain)\npicking this:\n\\<forall>y\\<in>T. card {g \\<in> S \\<rightarrow>\\<^sub>E T - {y}. inj_on g S} = fact (card T - 1) div fact (card T - 1 - card S)\n[PROOF STEP]\nhave \"... = (card T) * ?k\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<forall>y\\<in>T. card {g \\<in> S \\<rightarrow>\\<^sub>E T - {y}. inj_on g S} = fact (card T - 1) div fact (card T - 1 - card S)\n\ngoal (1 subgoal):\n 1. (\\<Sum>y\\<in>T. card {g \\<in> S \\<rightarrow>\\<^sub>E T - {y}. inj_on g S}) = card T * (fact (card T - 1) div fact (card T - 1 - card S))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>y\\<in>T. card {g \\<in> S \\<rightarrow>\\<^sub>E T - {y}. inj_on g S}) = card T * (fact (card T - 1) div fact (card T - 1 - card S))\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>y\\<in>T. card {g \\<in> S \\<rightarrow>\\<^sub>E T - {y}. inj_on g S}) = card T * (fact (card T - 1) div fact (card T - 1 - card S))\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nhave \"... = card T * fact (card T - 1) div fact (card T - card (insert x S))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card T * (fact (card T - 1) div fact (card T - 1 - card S)) = card T * fact (card T - 1) div fact (card T - card (insert x S))\n[PROOF STEP]\nusing insert\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite S\nx \\<notin> S\n\\<lbrakk>finite ?T; card S \\<le> card ?T\\<rbrakk> \\<Longrightarrow> card {f. f \\<in> S \\<rightarrow>\\<^sub>E ?T \\<and> inj_on f S} = fact (card ?T) div fact (card ?T - card S)\nfinite T\ncard (insert x S) \\<le> card T\n\ngoal (1 subgoal):\n 1. card T * (fact (card T - 1) div fact (card T - 1 - card S)) = card T * fact (card T - 1) div fact (card T - card (insert x S))\n[PROOF STEP]\nunfolding div_mult1_eq[of \"card T\" \"fact (card T - 1)\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite S\nx \\<notin> S\n\\<lbrakk>finite ?T; card S \\<le> card ?T\\<rbrakk> \\<Longrightarrow> card {f. f \\<in> S \\<rightarrow>\\<^sub>E ?T \\<and> inj_on f S} = fact (card ?T) div fact (card ?T - card S)\nfinite T\ncard (insert x S) \\<le> card T\n\ngoal (1 subgoal):\n 1. card T * (fact (card T - 1) div fact (card T - 1 - card S)) = card T * (fact (card T - 1) div fact (card T - card (insert x S))) + card T * (fact (card T - 1) mod fact (card T - card (insert x S))) div fact (card T - card (insert x S))\n[PROOF STEP]\nby (simp add: fact_mod)\n[PROOF STATE]\nproof (state)\nthis:\ncard T * (fact (card T - 1) div fact (card T - 1 - card S)) = card T * fact (card T - 1) div fact (card T - card (insert x S))\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard T * (fact (card T - 1) div fact (card T - 1 - card S)) = card T * fact (card T - 1) div fact (card T - card (insert x S))\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nhave \"... = fact (card T) div fact (card T - card (insert x S))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card T * fact (card T - 1) div fact (card T - card (insert x S)) = fact (card T) div fact (card T - card (insert x S))\n[PROOF STEP]\nusing insert\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite S\nx \\<notin> S\n\\<lbrakk>finite ?T; card S \\<le> card ?T\\<rbrakk> \\<Longrightarrow> card {f. f \\<in> S \\<rightarrow>\\<^sub>E ?T \\<and> inj_on f S} = fact (card ?T) div fact (card ?T - card S)\nfinite T\ncard (insert x S) \\<le> card T\n\ngoal (1 subgoal):\n 1. card T * fact (card T - 1) div fact (card T - card (insert x S)) = fact (card T) div fact (card T - card (insert x S))\n[PROOF STEP]\nby (simp add: fact_reduce[of \"card T\"])\n[PROOF STATE]\nproof (state)\nthis:\ncard T * fact (card T - 1) div fact (card T - card (insert x S)) = fact (card T) div fact (card T - card (insert x S))\n\ngoal (1 subgoal):\n 1. \\<And>x F T. \\<lbrakk>finite F; x \\<notin> F; \\<And>T. \\<lbrakk>finite T; card F \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> F \\<rightarrow>\\<^sub>E T. inj_on f F} = fact (card T) div fact (card T - card F); finite T; card (insert x F) \\<le> card T\\<rbrakk> \\<Longrightarrow> card {f \\<in> insert x F \\<rightarrow>\\<^sub>E T. inj_on f (insert x F)} = fact (card T) div fact (card T - card (insert x F))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard {f \\<in> insert x S \\<rightarrow>\\<^sub>E T. inj_on f (insert x S)} = fact (card T) div fact (card T - card (insert x S))\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {f \\<in> insert x S \\<rightarrow>\\<^sub>E T. inj_on f (insert x S)} = fact (card T) div fact (card T - card (insert x S))\n\ngoal (1 subgoal):\n 1. card {f. f \\<in> insert x S \\<rightarrow>\\<^sub>E T \\<and> inj_on f (insert x S)} = fact (card T) div fact (card T - card (insert x S))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard {f. f \\<in> insert x S \\<rightarrow>\\<^sub>E T \\<and> inj_on f (insert x S)} = fact (card T) div fact (card T - card (insert x S))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 10788, "file": null, "length": 58, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213799730774, "lm_q2_score": 0.8519528038477825, "lm_q1q2_score": 0.7660089806675507}}
{"text": "[STATEMENT]\ntheorem orbit_stabiliser:\n  assumes finite: \"finite (carrier G)\"\n  shows \"order G = card (orbit x) * card (stabiliser x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. order G = card (orbit x) * card (stabiliser x)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. order G = card (orbit x) * card (stabiliser x)\n[PROOF STEP]\nhave \"card (carrier (G LMod (stabiliser x))) = card (orbit x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (carrier (G LMod stabiliser x)) = card (orbit x)\n[PROOF STEP]\nusing bij_betw_same_card orbit_stabiliser_bij\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw ?f ?A ?B \\<Longrightarrow> card ?A = card ?B\nbij_betw (\\<lambda>H. rep H \\<odot> ?x) (carrier (G LMod stabiliser ?x)) (orbit ?x)\n\ngoal (1 subgoal):\n 1. card (carrier (G LMod stabiliser x)) = card (orbit x)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncard (carrier (G LMod stabiliser x)) = card (orbit x)\n\ngoal (1 subgoal):\n 1. order G = card (orbit x) * card (stabiliser x)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ncard (carrier (G LMod stabiliser x)) = card (orbit x)\n\ngoal (1 subgoal):\n 1. order G = card (orbit x) * card (stabiliser x)\n[PROOF STEP]\nhave \"card (carrier (G LMod (stabiliser x))) * card (stabiliser x)  = order G\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (carrier (G LMod stabiliser x)) * card (stabiliser x) = order G\n[PROOF STEP]\nusing finite stabiliser_subgroup l_lagrange\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite (carrier G)\nsubgroup (stabiliser ?x) G\n\\<lbrakk>finite (carrier G); subgroup ?H G\\<rbrakk> \\<Longrightarrow> card (lcosets ?H) * card ?H = order G\n\ngoal (1 subgoal):\n 1. card (carrier (G LMod stabiliser x)) * card (stabiliser x) = order G\n[PROOF STEP]\nunfolding LFactGroup_def\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite (carrier G)\nsubgroup (stabiliser ?x) G\n\\<lbrakk>finite (carrier G); subgroup ?H G\\<rbrakk> \\<Longrightarrow> card (lcosets ?H) * card ?H = order G\n\ngoal (1 subgoal):\n 1. card (carrier \\<lparr>carrier = lcosets stabiliser x, mult = (<#>), one = stabiliser x\\<rparr>) * card (stabiliser x) = order G\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard (carrier (G LMod stabiliser x)) * card (stabiliser x) = order G\n\ngoal (1 subgoal):\n 1. order G = card (orbit x) * card (stabiliser x)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (carrier (G LMod stabiliser x)) = card (orbit x)\ncard (carrier (G LMod stabiliser x)) * card (stabiliser x) = order G\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (carrier (G LMod stabiliser x)) = card (orbit x)\ncard (carrier (G LMod stabiliser x)) * card (stabiliser x) = order G\n\ngoal (1 subgoal):\n 1. order G = card (orbit x) * card (stabiliser x)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\norder G = card (orbit x) * card (stabiliser x)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1174, "file": "Orbit_Stabiliser_Orbit_Stabiliser", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045907347108, "lm_q2_score": 0.863391599428538, "lm_q1q2_score": 0.7660049906147834}}
{"text": "[STATEMENT]\ntheorem card_partitions_with_k_parts:\n  \"card {N. number_partition n N \\<and> size N = k} = Partition n k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {N. number_partition n N \\<and> size N = k} = Partition n k\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card {N. number_partition n N \\<and> size N = k} = Partition n k\n[PROOF STEP]\nhave \"bij_betw count {N. number_partition n N \\<and> size N = k} {p. p partitions n \\<and> sum p {..n} = k}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bij_betw count {N. number_partition n N \\<and> size N = k} {p. p partitions n \\<and> sum p {..n} = k}\n[PROOF STEP]\nby (rule bij_betw_multiset_number_partition_with_size)\n[PROOF STATE]\nproof (state)\nthis:\nbij_betw count {N. number_partition n N \\<and> size N = k} {p. p partitions n \\<and> sum p {..n} = k}\n\ngoal (1 subgoal):\n 1. card {N. number_partition n N \\<and> size N = k} = Partition n k\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\nbij_betw count {N. number_partition n N \\<and> size N = k} {p. p partitions n \\<and> sum p {..n} = k}\n[PROOF STEP]\nhave \"card {N. number_partition n N \\<and> size N = k} = card {p. p partitions n \\<and> sum p {..n} = k}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw count {N. number_partition n N \\<and> size N = k} {p. p partitions n \\<and> sum p {..n} = k}\n\ngoal (1 subgoal):\n 1. card {N. number_partition n N \\<and> size N = k} = card {p. p partitions n \\<and> sum p {..n} = k}\n[PROOF STEP]\nby (rule bij_betw_same_card)\n[PROOF STATE]\nproof (state)\nthis:\ncard {N. number_partition n N \\<and> size N = k} = card {p. p partitions n \\<and> sum p {..n} = k}\n\ngoal (1 subgoal):\n 1. card {N. number_partition n N \\<and> size N = k} = Partition n k\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard {N. number_partition n N \\<and> size N = k} = card {p. p partitions n \\<and> sum p {..n} = k}\n\ngoal (1 subgoal):\n 1. card {N. number_partition n N \\<and> size N = k} = Partition n k\n[PROOF STEP]\nhave \"\\<dots> = Partition n k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {p. p partitions n \\<and> sum p {..n} = k} = Partition n k\n[PROOF STEP]\nby (rule card_partitions_k_parts)\n[PROOF STATE]\nproof (state)\nthis:\ncard {p. p partitions n \\<and> sum p {..n} = k} = Partition n k\n\ngoal (1 subgoal):\n 1. card {N. number_partition n N \\<and> size N = k} = Partition n k\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard {N. number_partition n N \\<and> size N = k} = Partition n k\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {N. number_partition n N \\<and> size N = k} = Partition n k\n\ngoal (1 subgoal):\n 1. card {N. number_partition n N \\<and> size N = k} = Partition n k\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard {N. number_partition n N \\<and> size N = k} = Partition n k\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1171, "file": "Card_Number_Partitions_Card_Number_Partitions", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.894789457685656, "lm_q2_score": 0.8558511488056151, "lm_q1q2_score": 0.765806585299422}}
{"text": "[STATEMENT]\nlemma norm_add_rule_thm:\n  fixes x1 x2 :: \"'a::real_normed_vector\"\n  shows \"norm x1 \\<le> b1 \\<Longrightarrow> norm x2 \\<le> b2 \\<Longrightarrow> norm (x1 + x2) \\<le> b1 + b2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>norm x1 \\<le> b1; norm x2 \\<le> b2\\<rbrakk> \\<Longrightarrow> norm (x1 + x2) \\<le> b1 + b2\n[PROOF STEP]\nby (rule order_trans [OF norm_triangle_ineq add_mono])", "meta": {"llama_tokens": 173, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.900529786117893, "lm_q2_score": 0.8499711737573762, "lm_q1q2_score": 0.7654243593101044}}
{"text": "[STATEMENT]\nlemma op_norm_triangle: \"\\<parallel>A + B\\<parallel>\\<^sub>o\\<^sub>p \\<le> (\\<parallel>A\\<parallel>\\<^sub>o\\<^sub>p) + (\\<parallel>B\\<parallel>\\<^sub>o\\<^sub>p)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<parallel>A + B\\<parallel>\\<^sub>o\\<^sub>p \\<le> (\\<parallel>A\\<parallel>\\<^sub>o\\<^sub>p) + (\\<parallel>B\\<parallel>\\<^sub>o\\<^sub>p)\n[PROOF STEP]\nusing onorm_triangle[OF blin_matrix_vector_mult[of A] blin_matrix_vector_mult[of B]]\n    matrix_vector_mult_add_rdistrib[symmetric, of A _ B]\n[PROOF STATE]\nproof (prove)\nusing this:\nonorm (\\<lambda>x. A *v x + B *v x) \\<le> (\\<parallel>A\\<parallel>\\<^sub>o\\<^sub>p) + (\\<parallel>B\\<parallel>\\<^sub>o\\<^sub>p)\nA *v ?x + B *v ?x = (A + B) *v ?x\n\ngoal (1 subgoal):\n 1. \\<parallel>A + B\\<parallel>\\<^sub>o\\<^sub>p \\<le> (\\<parallel>A\\<parallel>\\<^sub>o\\<^sub>p) + (\\<parallel>B\\<parallel>\\<^sub>o\\<^sub>p)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 362, "file": "Matrices_for_ODEs_MTX_Norms", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9161096181702032, "lm_q2_score": 0.8354835309589074, "lm_q1q2_score": 0.7653944985342578}}
{"text": "[STATEMENT]\nlemma matrix_inv_mult:\n  fixes A::\"'a::{semiring_1}^'n^'n\"\n  and B::\"'a::{semiring_1}^'n^'n\"\n  assumes \"invertible A\" and \"invertible B\"\n  shows \"matrix_inv (A ** B) = matrix_inv B ** matrix_inv A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix_inv (A ** B) = matrix_inv B ** matrix_inv A\n[PROOF STEP]\nproof (rule matrix_inv_unique[of \"A**B\"])\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. A ** B ** (matrix_inv B ** matrix_inv A) = mat (1::'a)\n 2. matrix_inv B ** matrix_inv A ** (A ** B) = mat (1::'a)\n[PROOF STEP]\nshow \"A ** B ** (matrix_inv B ** matrix_inv A) = mat 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. A ** B ** (matrix_inv B ** matrix_inv A) = mat (1::'a)\n[PROOF STEP]\nby (metis assms(1) assms(2) matrix_inv_right matrix_mul_assoc matrix_mul_lid)\n[PROOF STATE]\nproof (state)\nthis:\nA ** B ** (matrix_inv B ** matrix_inv A) = mat (1::'a)\n\ngoal (1 subgoal):\n 1. matrix_inv B ** matrix_inv A ** (A ** B) = mat (1::'a)\n[PROOF STEP]\nshow \" matrix_inv B ** matrix_inv A ** (A ** B) = mat 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix_inv B ** matrix_inv A ** (A ** B) = mat (1::'a)\n[PROOF STEP]\nby (metis assms(1) assms(2) matrix_inv_left matrix_mul_assoc matrix_mul_lid)\n[PROOF STATE]\nproof (state)\nthis:\nmatrix_inv B ** matrix_inv A ** (A ** B) = mat (1::'a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 603, "file": "QR_Decomposition_Miscellaneous_QR", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9161096158798117, "lm_q2_score": 0.8354835289107307, "lm_q1q2_score": 0.7653944947443191}}
{"text": "[STATEMENT]\nlemma Cauchy_Schwarz_complex_vec_norm:\nassumes \"dim_vec x = dim_vec y\"\nshows \"cmod (inner_prod x y) \\<le> vec_norm x * vec_norm y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. complex_of_real (cmod (Complex_Matrix.inner_prod x y)) \\<le> vec_norm x * vec_norm y\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. complex_of_real (cmod (Complex_Matrix.inner_prod x y)) \\<le> vec_norm x * vec_norm y\n[PROOF STEP]\nhave x: \"x \\<in> carrier_vec (dim_vec x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\<in> carrier_vec (dim_vec x)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nx \\<in> carrier_vec (dim_vec x)\n\ngoal (1 subgoal):\n 1. complex_of_real (cmod (Complex_Matrix.inner_prod x y)) \\<le> vec_norm x * vec_norm y\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nx \\<in> carrier_vec (dim_vec x)\n\ngoal (1 subgoal):\n 1. complex_of_real (cmod (Complex_Matrix.inner_prod x y)) \\<le> vec_norm x * vec_norm y\n[PROOF STEP]\nhave y: \"y \\<in> carrier_vec (dim_vec x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. y \\<in> carrier_vec (dim_vec x)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec x = dim_vec y\n\ngoal (1 subgoal):\n 1. y \\<in> carrier_vec (dim_vec x)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ny \\<in> carrier_vec (dim_vec x)\n\ngoal (1 subgoal):\n 1. complex_of_real (cmod (Complex_Matrix.inner_prod x y)) \\<le> vec_norm x * vec_norm y\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\<in> carrier_vec (dim_vec x)\ny \\<in> carrier_vec (dim_vec x)\n[PROOF STEP]\nhave \"(cmod (inner_prod x y))\\<^sup>2 = inner_prod x y * inner_prod y x\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\<in> carrier_vec (dim_vec x)\ny \\<in> carrier_vec (dim_vec x)\n\ngoal (1 subgoal):\n 1. complex_of_real ((cmod (Complex_Matrix.inner_prod x y))\\<^sup>2) = Complex_Matrix.inner_prod x y * Complex_Matrix.inner_prod y x\n[PROOF STEP]\nusing complex_norm_square\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\<in> carrier_vec (dim_vec x)\ny \\<in> carrier_vec (dim_vec x)\ncomplex_of_real ((cmod ?z)\\<^sup>2) = ?z * cnj ?z\n\ngoal (1 subgoal):\n 1. complex_of_real ((cmod (Complex_Matrix.inner_prod x y))\\<^sup>2) = Complex_Matrix.inner_prod x y * Complex_Matrix.inner_prod y x\n[PROOF STEP]\nby (metis inner_prod_swap mult_conj_cmod_square)\n[PROOF STATE]\nproof (state)\nthis:\ncomplex_of_real ((cmod (Complex_Matrix.inner_prod x y))\\<^sup>2) = Complex_Matrix.inner_prod x y * Complex_Matrix.inner_prod y x\n\ngoal (1 subgoal):\n 1. complex_of_real (cmod (Complex_Matrix.inner_prod x y)) \\<le> vec_norm x * vec_norm y\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncomplex_of_real ((cmod (Complex_Matrix.inner_prod x y))\\<^sup>2) = Complex_Matrix.inner_prod x y * Complex_Matrix.inner_prod y x\n\ngoal (1 subgoal):\n 1. complex_of_real (cmod (Complex_Matrix.inner_prod x y)) \\<le> vec_norm x * vec_norm y\n[PROOF STEP]\nhave \"... \\<le> inner_prod x x * inner_prod y y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod x y * Complex_Matrix.inner_prod y x \\<le> Complex_Matrix.inner_prod x x * Complex_Matrix.inner_prod y y\n[PROOF STEP]\nusing Cauchy_Schwarz_complex_vec x y\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>?x \\<in> carrier_vec ?n; ?y \\<in> carrier_vec ?n\\<rbrakk> \\<Longrightarrow> Complex_Matrix.inner_prod ?x ?y * Complex_Matrix.inner_prod ?y ?x \\<le> Complex_Matrix.inner_prod ?x ?x * Complex_Matrix.inner_prod ?y ?y\nx \\<in> carrier_vec (dim_vec x)\ny \\<in> carrier_vec (dim_vec x)\n\ngoal (1 subgoal):\n 1. Complex_Matrix.inner_prod x y * Complex_Matrix.inner_prod y x \\<le> Complex_Matrix.inner_prod x x * Complex_Matrix.inner_prod y y\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nComplex_Matrix.inner_prod x y * Complex_Matrix.inner_prod y x \\<le> Complex_Matrix.inner_prod x x * Complex_Matrix.inner_prod y y\n\ngoal (1 subgoal):\n 1. complex_of_real (cmod (Complex_Matrix.inner_prod x y)) \\<le> vec_norm x * vec_norm y\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncomplex_of_real ((cmod (Complex_Matrix.inner_prod x y))\\<^sup>2) \\<le> Complex_Matrix.inner_prod x x * Complex_Matrix.inner_prod y y\n[PROOF STEP]\nhave \"(cmod (inner_prod x y))\\<^sup>2 \\<le> inner_prod x x * inner_prod y y\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncomplex_of_real ((cmod (Complex_Matrix.inner_prod x y))\\<^sup>2) \\<le> Complex_Matrix.inner_prod x x * Complex_Matrix.inner_prod y y\n\ngoal (1 subgoal):\n 1. complex_of_real ((cmod (Complex_Matrix.inner_prod x y))\\<^sup>2) \\<le> Complex_Matrix.inner_prod x x * Complex_Matrix.inner_prod y y\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncomplex_of_real ((cmod (Complex_Matrix.inner_prod x y))\\<^sup>2) \\<le> Complex_Matrix.inner_prod x x * Complex_Matrix.inner_prod y y\n\ngoal (1 subgoal):\n 1. complex_of_real (cmod (Complex_Matrix.inner_prod x y)) \\<le> vec_norm x * vec_norm y\n[PROOF STEP]\nhence \"(cmod (inner_prod x y))\\<^sup>2 \\<le> Re (inner_prod x x) * Re (inner_prod y y)\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncomplex_of_real ((cmod (Complex_Matrix.inner_prod x y))\\<^sup>2) \\<le> Complex_Matrix.inner_prod x x * Complex_Matrix.inner_prod y y\n\ngoal (1 subgoal):\n 1. (cmod (Complex_Matrix.inner_prod x y))\\<^sup>2 \\<le> Re (Complex_Matrix.inner_prod x x) * Re (Complex_Matrix.inner_prod y y)\n[PROOF STEP]\nusing less_eq_complex_def\n[PROOF STATE]\nproof (prove)\nusing this:\ncomplex_of_real ((cmod (Complex_Matrix.inner_prod x y))\\<^sup>2) \\<le> Complex_Matrix.inner_prod x x * Complex_Matrix.inner_prod y y\n(?x \\<le> ?y) = (Re ?x \\<le> Re ?y \\<and> Im ?x = Im ?y)\n\ngoal (1 subgoal):\n 1. (cmod (Complex_Matrix.inner_prod x y))\\<^sup>2 \\<le> Re (Complex_Matrix.inner_prod x x) * Re (Complex_Matrix.inner_prod y y)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(cmod (Complex_Matrix.inner_prod x y))\\<^sup>2 \\<le> Re (Complex_Matrix.inner_prod x x) * Re (Complex_Matrix.inner_prod y y)\n\ngoal (1 subgoal):\n 1. complex_of_real (cmod (Complex_Matrix.inner_prod x y)) \\<le> vec_norm x * vec_norm y\n[PROOF STEP]\nhence \"sqrt ((cmod (inner_prod x y))\\<^sup>2) \\<le> \n    sqrt (Re (inner_prod x x) * Re (inner_prod y y))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(cmod (Complex_Matrix.inner_prod x y))\\<^sup>2 \\<le> Re (Complex_Matrix.inner_prod x x) * Re (Complex_Matrix.inner_prod y y)\n\ngoal (1 subgoal):\n 1. sqrt ((cmod (Complex_Matrix.inner_prod x y))\\<^sup>2) \\<le> sqrt (Re (Complex_Matrix.inner_prod x x) * Re (Complex_Matrix.inner_prod y y))\n[PROOF STEP]\nusing real_sqrt_le_iff\n[PROOF STATE]\nproof (prove)\nusing this:\n(cmod (Complex_Matrix.inner_prod x y))\\<^sup>2 \\<le> Re (Complex_Matrix.inner_prod x x) * Re (Complex_Matrix.inner_prod y y)\n(sqrt ?x \\<le> sqrt ?y) = (?x \\<le> ?y)\n\ngoal (1 subgoal):\n 1. sqrt ((cmod (Complex_Matrix.inner_prod x y))\\<^sup>2) \\<le> sqrt (Re (Complex_Matrix.inner_prod x x) * Re (Complex_Matrix.inner_prod y y))\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nsqrt ((cmod (Complex_Matrix.inner_prod x y))\\<^sup>2) \\<le> sqrt (Re (Complex_Matrix.inner_prod x x) * Re (Complex_Matrix.inner_prod y y))\n\ngoal (1 subgoal):\n 1. complex_of_real (cmod (Complex_Matrix.inner_prod x y)) \\<le> vec_norm x * vec_norm y\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsqrt ((cmod (Complex_Matrix.inner_prod x y))\\<^sup>2) \\<le> sqrt (Re (Complex_Matrix.inner_prod x x) * Re (Complex_Matrix.inner_prod y y))\n\ngoal (1 subgoal):\n 1. complex_of_real (cmod (Complex_Matrix.inner_prod x y)) \\<le> vec_norm x * vec_norm y\n[PROOF STEP]\nhave \"... =  sqrt (Re (inner_prod x x)) * sqrt (Re (inner_prod y y))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt (Re (Complex_Matrix.inner_prod x x) * Re (Complex_Matrix.inner_prod y y)) = sqrt (Re (Complex_Matrix.inner_prod x x)) * sqrt (Re (Complex_Matrix.inner_prod y y))\n[PROOF STEP]\nby (simp add: real_sqrt_mult)\n[PROOF STATE]\nproof (state)\nthis:\nsqrt (Re (Complex_Matrix.inner_prod x x) * Re (Complex_Matrix.inner_prod y y)) = sqrt (Re (Complex_Matrix.inner_prod x x)) * sqrt (Re (Complex_Matrix.inner_prod y y))\n\ngoal (1 subgoal):\n 1. complex_of_real (cmod (Complex_Matrix.inner_prod x y)) \\<le> vec_norm x * vec_norm y\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nsqrt ((cmod (Complex_Matrix.inner_prod x y))\\<^sup>2) \\<le> sqrt (Re (Complex_Matrix.inner_prod x x)) * sqrt (Re (Complex_Matrix.inner_prod y y))\n[PROOF STEP]\nhave \"sqrt ((cmod (inner_prod x y))\\<^sup>2) \\<le> \n    sqrt (Re (inner_prod x x)) * sqrt (Re (inner_prod y y))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsqrt ((cmod (Complex_Matrix.inner_prod x y))\\<^sup>2) \\<le> sqrt (Re (Complex_Matrix.inner_prod x x)) * sqrt (Re (Complex_Matrix.inner_prod y y))\n\ngoal (1 subgoal):\n 1. sqrt ((cmod (Complex_Matrix.inner_prod x y))\\<^sup>2) \\<le> sqrt (Re (Complex_Matrix.inner_prod x x)) * sqrt (Re (Complex_Matrix.inner_prod y y))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nsqrt ((cmod (Complex_Matrix.inner_prod x y))\\<^sup>2) \\<le> sqrt (Re (Complex_Matrix.inner_prod x x)) * sqrt (Re (Complex_Matrix.inner_prod y y))\n\ngoal (1 subgoal):\n 1. complex_of_real (cmod (Complex_Matrix.inner_prod x y)) \\<le> vec_norm x * vec_norm y\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsqrt ((cmod (Complex_Matrix.inner_prod x y))\\<^sup>2) \\<le> sqrt (Re (Complex_Matrix.inner_prod x x)) * sqrt (Re (Complex_Matrix.inner_prod y y))\n\ngoal (1 subgoal):\n 1. complex_of_real (cmod (Complex_Matrix.inner_prod x y)) \\<le> vec_norm x * vec_norm y\n[PROOF STEP]\nusing less_eq_complex_def\n[PROOF STATE]\nproof (prove)\nusing this:\nsqrt ((cmod (Complex_Matrix.inner_prod x y))\\<^sup>2) \\<le> sqrt (Re (Complex_Matrix.inner_prod x x)) * sqrt (Re (Complex_Matrix.inner_prod y y))\n(?x \\<le> ?y) = (Re ?x \\<le> Re ?y \\<and> Im ?x = Im ?y)\n\ngoal (1 subgoal):\n 1. complex_of_real (cmod (Complex_Matrix.inner_prod x y)) \\<le> vec_norm x * vec_norm y\n[PROOF STEP]\nby (simp add: vec_norm_def)\n[PROOF STATE]\nproof (state)\nthis:\ncomplex_of_real (cmod (Complex_Matrix.inner_prod x y)) \\<le> vec_norm x * vec_norm y\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 4143, "file": "Commuting_Hermitian_Spectral_Theory_Complements", "length": 34, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9111797027760039, "lm_q2_score": 0.8397339716830606, "lm_q1q2_score": 0.7651485507290845}}
{"text": "[STATEMENT]\ntheorem inverse_triangle_num_sums: \"(\\<lambda>n. 1 / triangle_num (Suc n)) sums 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<lambda>n. 1 / real (triangle_num (Suc n))) sums 2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\<lambda>n. 1 / real (triangle_num (Suc n))) sums 2\n[PROOF STEP]\nhave \"(\\<lambda>n. inverse (real (Suc n)) - inverse (real (Suc (Suc n)))) sums\n          (inverse (real (Suc 0)) - 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<lambda>n. inverse (real (Suc n)) - inverse (real (Suc (Suc n)))) sums (inverse (real (Suc 0)) - 0)\n[PROOF STEP]\nby (intro telescope_sums' LIMSEQ_inverse_real_of_nat)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>n. inverse (real (Suc n)) - inverse (real (Suc (Suc n)))) sums (inverse (real (Suc 0)) - 0)\n\ngoal (1 subgoal):\n 1. (\\<lambda>n. 1 / real (triangle_num (Suc n))) sums 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>n. inverse (real (Suc n)) - inverse (real (Suc (Suc n)))) sums (inverse (real (Suc 0)) - 0)\n\ngoal (1 subgoal):\n 1. (\\<lambda>n. 1 / real (triangle_num (Suc n))) sums 2\n[PROOF STEP]\nhave \"(\\<lambda>n. inverse (real (Suc n)) - inverse (real (Suc (Suc n)))) =\n               (\\<lambda>n. 1 / real (2 * triangle_num (Suc n)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<lambda>n. inverse (real (Suc n)) - inverse (real (Suc (Suc n)))) = (\\<lambda>n. 1 / real (2 * triangle_num (Suc n)))\n[PROOF STEP]\nby (auto simp: field_simps triangle_num_def)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>n. inverse (real (Suc n)) - inverse (real (Suc (Suc n)))) = (\\<lambda>n. 1 / real (2 * triangle_num (Suc n)))\n\ngoal (1 subgoal):\n 1. (\\<lambda>n. 1 / real (triangle_num (Suc n))) sums 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>n. inverse (real (Suc n)) - inverse (real (Suc (Suc n)))) = (\\<lambda>n. 1 / real (2 * triangle_num (Suc n)))\n\ngoal (1 subgoal):\n 1. (\\<lambda>n. 1 / real (triangle_num (Suc n))) sums 2\n[PROOF STEP]\nhave \"inverse (real (Suc 0)) - 0 = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse (real (Suc 0)) - 0 = 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ninverse (real (Suc 0)) - 0 = 1\n\ngoal (1 subgoal):\n 1. (\\<lambda>n. 1 / real (triangle_num (Suc n))) sums 2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\<lambda>n. 1 / real (2 * triangle_num (Suc n))) sums 1\n[PROOF STEP]\nhave \"(\\<lambda>n. 2 * (1 / real (2 * triangle_num (Suc n)))) sums (2 * 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<lambda>n. 1 / real (2 * triangle_num (Suc n))) sums 1\n\ngoal (1 subgoal):\n 1. (\\<lambda>n. 2 * (1 / real (2 * triangle_num (Suc n)))) sums (2 * 1)\n[PROOF STEP]\nby (intro sums_mult)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>n. 2 * (1 / real (2 * triangle_num (Suc n)))) sums (2 * 1)\n\ngoal (1 subgoal):\n 1. (\\<lambda>n. 1 / real (triangle_num (Suc n))) sums 2\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<lambda>n. 2 * (1 / real (2 * triangle_num (Suc n)))) sums (2 * 1)\n\ngoal (1 subgoal):\n 1. (\\<lambda>n. 1 / real (triangle_num (Suc n))) sums 2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>n. 1 / real (triangle_num (Suc n))) sums 2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1420, "file": null, "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8918110569397306, "lm_q2_score": 0.8577680995361899, "lm_q1q2_score": 0.7649670754565536}}
{"text": "[STATEMENT]\nlemma inner_prod_expand:\n  assumes \"dim_vec a = dim_vec b\" and \"dim_vec a = dim_vec c\" and \"dim_vec a = dim_vec d\"\n  shows \"\\<langle>a + b|c + d\\<rangle> = \\<langle>a|c\\<rangle> + \\<langle>a|d\\<rangle> + \\<langle>b|c\\<rangle> + \\<langle>b|d\\<rangle>\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<langle>a + b|c + d\\<rangle> = \\<langle>a|c\\<rangle> + \\<langle>a|d\\<rangle> + \\<langle>b|c\\<rangle> + \\<langle>b|d\\<rangle>\n[PROOF STEP]\napply (simp add: inner_prod_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<dim_vec d. cnj ((a + b) $ i) * (c $ i + d $ i)) = (\\<Sum>i = 0..<dim_vec c. cnj (a $ i) * c $ i) + (\\<Sum>i = 0..<dim_vec d. cnj (a $ i) * d $ i) + (\\<Sum>i = 0..<dim_vec c. cnj (b $ i) * c $ i) + (\\<Sum>i = 0..<dim_vec d. cnj (b $ i) * d $ i)\n[PROOF STEP]\nusing assms sum.cong\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec a = dim_vec b\ndim_vec a = dim_vec c\ndim_vec a = dim_vec d\n\\<lbrakk>?A = ?B; \\<And>x. x \\<in> ?B \\<Longrightarrow> ?g x = ?h x\\<rbrakk> \\<Longrightarrow> sum ?g ?A = sum ?h ?B\n\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<dim_vec d. cnj ((a + b) $ i) * (c $ i + d $ i)) = (\\<Sum>i = 0..<dim_vec c. cnj (a $ i) * c $ i) + (\\<Sum>i = 0..<dim_vec d. cnj (a $ i) * d $ i) + (\\<Sum>i = 0..<dim_vec c. cnj (b $ i) * c $ i) + (\\<Sum>i = 0..<dim_vec d. cnj (b $ i) * d $ i)\n[PROOF STEP]\nby (simp add: sum.distrib algebra_simps)", "meta": {"llama_tokens": 670, "file": "Isabelle_Marries_Dirac_No_Cloning", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314768368161, "lm_q2_score": 0.8633916152464016, "lm_q1q2_score": 0.7646467912991948}}
{"text": "[STATEMENT]\nlemma neg_prod_sum_lt:\n  fixes c :: \"'a::linordered_field\"\n  assumes \"c < 0\"\n  shows \"c * x + t < 0 \\<equiv> x > (- 1 / c) * t\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c * x + t < (0::'a) \\<equiv> - (1::'a) / c * t < x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. c * x + t < (0::'a) \\<equiv> - (1::'a) / c * t < x\n[PROOF STEP]\nhave \"c * x + t < 0 \\<longleftrightarrow> c * x < - t\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (c * x + t < (0::'a)) = (c * x < - t)\n[PROOF STEP]\nby (subst less_iff_diff_less_0 [of \"c * x\" \"- t\"]) simp\n[PROOF STATE]\nproof (state)\nthis:\n(c * x + t < (0::'a)) = (c * x < - t)\n\ngoal (1 subgoal):\n 1. c * x + t < (0::'a) \\<equiv> - (1::'a) / c * t < x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(c * x + t < (0::'a)) = (c * x < - t)\n\ngoal (1 subgoal):\n 1. c * x + t < (0::'a) \\<equiv> - (1::'a) / c * t < x\n[PROOF STEP]\nhave \"\\<dots> \\<longleftrightarrow> - t / c < x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (c * x < - t) = (- t / c < x)\n[PROOF STEP]\nby (simp only: neg_divide_less_eq[OF \\<open>c < 0\\<close>] algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(c * x < - t) = (- t / c < x)\n\ngoal (1 subgoal):\n 1. c * x + t < (0::'a) \\<equiv> - (1::'a) / c * t < x\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(c * x < - t) = (- t / c < x)\n\ngoal (1 subgoal):\n 1. c * x + t < (0::'a) \\<equiv> - (1::'a) / c * t < x\n[PROOF STEP]\nhave \"\\<dots> \\<longleftrightarrow> (- 1 / c) * t < x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (- t / c < x) = (- (1::'a) / c * t < x)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(- t / c < x) = (- (1::'a) / c * t < x)\n\ngoal (1 subgoal):\n 1. c * x + t < (0::'a) \\<equiv> - (1::'a) / c * t < x\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(c * x + t < (0::'a)) = (- (1::'a) / c * t < x)\n[PROOF STEP]\nshow \"PROP ?thesis\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(c * x + t < (0::'a)) = (- (1::'a) / c * t < x)\n\ngoal (1 subgoal):\n 1. c * x + t < (0::'a) \\<equiv> - (1::'a) / c * t < x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nc * x + t < (0::'a) \\<equiv> - (1::'a) / c * t < x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1109, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314677809304, "lm_q2_score": 0.863391595913457, "lm_q1q2_score": 0.7646467663585549}}
{"text": "[STATEMENT]\nlemma rank_transpose: \n  fixes A::\"'a::{field}^'n::{mod_type}^'m::{mod_type}\"\n  shows  \"rank (transpose A) = rank A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rank (Finite_Cartesian_Product.transpose A) = rank A\n[PROOF STEP]\nby (metis rank_def rank_eq_dim_col_space row_rank_def row_space_eq_col_space_transpose)", "meta": {"llama_tokens": 139, "file": "Gauss_Jordan_Gauss_Jordan", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9059898305367524, "lm_q2_score": 0.843895100591521, "lm_q1q2_score": 0.7645603791757077}}
{"text": "[STATEMENT]\nlemma setdist_eq_0_closed_compact:\n  assumes S: \"closed S\" and T: \"compact T\"\n    shows \"setdist S T = 0 \\<longleftrightarrow> S = {} \\<or> T = {} \\<or> S \\<inter> T \\<noteq> {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (setdist S T = 0) = (S = {} \\<or> T = {} \\<or> S \\<inter> T \\<noteq> {})\n[PROOF STEP]\nusing setdist_eq_0_compact_closed [OF T S]\n[PROOF STATE]\nproof (prove)\nusing this:\n(setdist T S = 0) = (T = {} \\<or> S = {} \\<or> T \\<inter> S \\<noteq> {})\n\ngoal (1 subgoal):\n 1. (setdist S T = 0) = (S = {} \\<or> T = {} \\<or> S \\<inter> T \\<noteq> {})\n[PROOF STEP]\nby (metis Int_commute setdist_sym)", "meta": {"llama_tokens": 278, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9059898203834278, "lm_q2_score": 0.8438951025545426, "lm_q1q2_score": 0.7645603723858444}}
{"text": "[STATEMENT]\nlemma pi_machin: \"pi = 16 * arctan (1 / 5) - 4 * arctan (1 / 239)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. pi = 16 * arctan (1 / 5) - 4 * arctan (1 / 239)\n[PROOF STEP]\nusing machin\n[PROOF STATE]\nproof (prove)\nusing this:\npi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)\n\ngoal (1 subgoal):\n 1. pi = 16 * arctan (1 / 5) - 4 * arctan (1 / 239)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 209, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9124361604769413, "lm_q2_score": 0.8376199694135332, "lm_q1q2_score": 0.7642747488304973}}
{"text": "[STATEMENT]\nlemma row_space_eq_col_space_transpose:\n  fixes A::\"'a::{field}^'columns^'rows\"\n  shows \"row_space A = col_space (transpose A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. row_space A = col_space (Finite_Cartesian_Product.transpose A)\n[PROOF STEP]\nunfolding col_space_def row_space_def columns_transpose[of A]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.span (rows A) = vec.span (rows A)\n[PROOF STEP]\n..", "meta": {"llama_tokens": 178, "file": "Rank_Nullity_Theorem_Fundamental_Subspaces", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8991213772699435, "lm_q2_score": 0.8499711699569787, "lm_q1q2_score": 0.7642272489714639}}
{"text": "[STATEMENT]\nlemma contour_integral_midpoint:\n  assumes \"continuous_on (closed_segment a b) f\"\n  shows \"contour_integral (linepath a b) f =\n         contour_integral (linepath a (midpoint a b)) f + contour_integral (linepath (midpoint a b) b) f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. contour_integral (linepath a b) f = contour_integral (linepath a (midpoint a b)) f + contour_integral (linepath (midpoint a b) b) f\n[PROOF STEP]\nproof (rule contour_integral_split)\n[PROOF STATE]\nproof (state)\ngoal (4 subgoals):\n 1. continuous_on (closed_segment a b) f\n 2. 0 \\<le> ?k\n 3. ?k \\<le> 1\n 4. midpoint a b - a = ?k *\\<^sub>R (b - a)\n[PROOF STEP]\nshow \"midpoint a b - a = (1/2) *\\<^sub>R (b - a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. midpoint a b - a = (1 / 2) *\\<^sub>R (b - a)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ncontinuous_on (closed_segment a b) f\n\ngoal (1 subgoal):\n 1. midpoint a b - a = (1 / 2) *\\<^sub>R (b - a)\n[PROOF STEP]\nby (auto simp: midpoint_def scaleR_conv_of_real)\n[PROOF STATE]\nproof (state)\nthis:\nmidpoint a b - a = (1 / 2) *\\<^sub>R (b - a)\n\ngoal (3 subgoals):\n 1. continuous_on (closed_segment a b) f\n 2. 0 \\<le> 1 / 2\n 3. 1 / 2 \\<le> 1\n[PROOF STEP]\nqed (use assms in auto)", "meta": {"llama_tokens": 550, "file": null, "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9099070133672954, "lm_q2_score": 0.8397339616560073, "lm_q1q2_score": 0.7640798210735046}}
{"text": "[STATEMENT]\nlemma trace_outer_prod:\n  fixes v w :: \"('a::conjugatable_field vec)\"\n  assumes v: \"v \\<in> carrier_vec n\" and w: \"w \\<in> carrier_vec n\"\n  shows \"trace (outer_prod v w) = inner_prod w v\" (is \"?lhs = ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trace (outer_prod v w) = inner_prod w v\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. trace (outer_prod v w) = inner_prod w v\n[PROOF STEP]\nhave \"(1\\<^sub>m n) * (outer_prod v w) = outer_prod v w\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1\\<^sub>m n * outer_prod v w = outer_prod v w\n[PROOF STEP]\napply (subst left_mult_one_mat)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. outer_prod v w \\<in> carrier_mat n ?nc\n 2. outer_prod v w = outer_prod v w\n[PROOF STEP]\nusing outer_prod_dim assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>?v \\<in> carrier_vec ?n; ?w \\<in> carrier_vec ?m\\<rbrakk> \\<Longrightarrow> outer_prod ?v ?w \\<in> carrier_mat ?n ?m\nv \\<in> carrier_vec n\nw \\<in> carrier_vec n\n\ngoal (2 subgoals):\n 1. outer_prod v w \\<in> carrier_mat n ?nc\n 2. outer_prod v w = outer_prod v w\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n1\\<^sub>m n * outer_prod v w = outer_prod v w\n\ngoal (1 subgoal):\n 1. trace (outer_prod v w) = inner_prod w v\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n1\\<^sub>m n * outer_prod v w = outer_prod v w\n\ngoal (1 subgoal):\n 1. trace (outer_prod v w) = inner_prod w v\n[PROOF STEP]\nhave \"1\\<^sub>m n *\\<^sub>v v = v\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1\\<^sub>m n *\\<^sub>v v = v\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nv \\<in> carrier_vec n\nw \\<in> carrier_vec n\n\ngoal (1 subgoal):\n 1. 1\\<^sub>m n *\\<^sub>v v = v\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n1\\<^sub>m n *\\<^sub>v v = v\n\ngoal (1 subgoal):\n 1. trace (outer_prod v w) = inner_prod w v\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n1\\<^sub>m n * outer_prod v w = outer_prod v w\n1\\<^sub>m n *\\<^sub>v v = v\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n1\\<^sub>m n * outer_prod v w = outer_prod v w\n1\\<^sub>m n *\\<^sub>v v = v\n\ngoal (1 subgoal):\n 1. trace (outer_prod v w) = inner_prod w v\n[PROOF STEP]\nusing trace_outer_prod_right[of \"1\\<^sub>m n\" n v w] assms\n[PROOF STATE]\nproof (prove)\nusing this:\n1\\<^sub>m n * outer_prod v w = outer_prod v w\n1\\<^sub>m n *\\<^sub>v v = v\n\\<lbrakk>1\\<^sub>m n \\<in> carrier_mat n n; v \\<in> carrier_vec n; w \\<in> carrier_vec n\\<rbrakk> \\<Longrightarrow> trace (1\\<^sub>m n * outer_prod v w) = inner_prod w (1\\<^sub>m n *\\<^sub>v v)\nv \\<in> carrier_vec n\nw \\<in> carrier_vec n\n\ngoal (1 subgoal):\n 1. trace (outer_prod v w) = inner_prod w v\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ntrace (outer_prod v w) = inner_prod w v\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1255, "file": "QHLProver_Complex_Matrix", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8774767970940975, "lm_q2_score": 0.8705972667296309, "lm_q1q2_score": 0.7639289011687922}}
{"text": "[STATEMENT]\nlemma mult_const_minus_self_real_le:\n  fixes x :: real\n  shows \"x * (c - x) \\<le> c\\<^sup>2 / 4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x * (c - x) \\<le> c\\<^sup>2 / 4\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x * (c - x) \\<le> c\\<^sup>2 / 4\n[PROOF STEP]\nhave \"x * (c - x) = -(x - c / 2)\\<^sup>2 + c\\<^sup>2 / 4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x * (c - x) = - (x - c / 2)\\<^sup>2 + c\\<^sup>2 / 4\n[PROOF STEP]\nby (simp add: field_simps power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\nx * (c - x) = - (x - c / 2)\\<^sup>2 + c\\<^sup>2 / 4\n\ngoal (1 subgoal):\n 1. x * (c - x) \\<le> c\\<^sup>2 / 4\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nx * (c - x) = - (x - c / 2)\\<^sup>2 + c\\<^sup>2 / 4\n\ngoal (1 subgoal):\n 1. x * (c - x) \\<le> c\\<^sup>2 / 4\n[PROOF STEP]\nhave \"\\<dots> \\<le> 0 + c\\<^sup>2 / 4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - (x - c / 2)\\<^sup>2 + c\\<^sup>2 / 4 \\<le> 0 + c\\<^sup>2 / 4\n[PROOF STEP]\nby (intro add_mono) auto\n[PROOF STATE]\nproof (state)\nthis:\n- (x - c / 2)\\<^sup>2 + c\\<^sup>2 / 4 \\<le> 0 + c\\<^sup>2 / 4\n\ngoal (1 subgoal):\n 1. x * (c - x) \\<le> c\\<^sup>2 / 4\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nx * (c - x) \\<le> 0 + c\\<^sup>2 / 4\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nx * (c - x) \\<le> 0 + c\\<^sup>2 / 4\n\ngoal (1 subgoal):\n 1. x * (c - x) \\<le> c\\<^sup>2 / 4\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nx * (c - x) \\<le> c\\<^sup>2 / 4\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 814, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767810736693, "lm_q2_score": 0.8705972801594706, "lm_q1q2_score": 0.7639288990058237}}
{"text": "[STATEMENT]\nlemma ack_3: \"ack (Suc (Suc (Suc 0))) j = 2 ^ (j+3) - 3\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ack (Suc (Suc (Suc 0))) j = 2 ^ (j + 3) - 3\n[PROOF STEP]\nproof (induct j)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. ack (Suc (Suc (Suc 0))) 0 = 2 ^ (0 + 3) - 3\n 2. \\<And>j. ack (Suc (Suc (Suc 0))) j = 2 ^ (j + 3) - 3 \\<Longrightarrow> ack (Suc (Suc (Suc 0))) (Suc j) = 2 ^ (Suc j + 3) - 3\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. ack (Suc (Suc (Suc 0))) 0 = 2 ^ (0 + 3) - 3\n 2. \\<And>j. ack (Suc (Suc (Suc 0))) j = 2 ^ (j + 3) - 3 \\<Longrightarrow> ack (Suc (Suc (Suc 0))) (Suc j) = 2 ^ (Suc j + 3) - 3\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ack (Suc (Suc (Suc 0))) 0 = 2 ^ (0 + 3) - 3\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nack (Suc (Suc (Suc 0))) 0 = 2 ^ (0 + 3) - 3\n\ngoal (1 subgoal):\n 1. \\<And>j. ack (Suc (Suc (Suc 0))) j = 2 ^ (j + 3) - 3 \\<Longrightarrow> ack (Suc (Suc (Suc 0))) (Suc j) = 2 ^ (Suc j + 3) - 3\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>j. ack (Suc (Suc (Suc 0))) j = 2 ^ (j + 3) - 3 \\<Longrightarrow> ack (Suc (Suc (Suc 0))) (Suc j) = 2 ^ (Suc j + 3) - 3\n[PROOF STEP]\ncase (Suc j)\n[PROOF STATE]\nproof (state)\nthis:\nack (Suc (Suc (Suc 0))) j = 2 ^ (j + 3) - 3\n\ngoal (1 subgoal):\n 1. \\<And>j. ack (Suc (Suc (Suc 0))) j = 2 ^ (j + 3) - 3 \\<Longrightarrow> ack (Suc (Suc (Suc 0))) (Suc j) = 2 ^ (Suc j + 3) - 3\n[PROOF STEP]\nwith less_le_trans\n[PROOF STATE]\nproof (chain)\npicking this:\n\\<lbrakk>?x < ?y; ?y \\<le> ?z\\<rbrakk> \\<Longrightarrow> ?x < ?z\nack (Suc (Suc (Suc 0))) j = 2 ^ (j + 3) - 3\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>?x < ?y; ?y \\<le> ?z\\<rbrakk> \\<Longrightarrow> ?x < ?z\nack (Suc (Suc (Suc 0))) j = 2 ^ (j + 3) - 3\n\ngoal (1 subgoal):\n 1. ack (Suc (Suc (Suc 0))) (Suc j) = 2 ^ (Suc j + 3) - 3\n[PROOF STEP]\nby (fastforce simp add: power_add algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nack (Suc (Suc (Suc 0))) (Suc j) = 2 ^ (Suc j + 3) - 3\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1170, "file": "Ackermanns_not_PR_Primrec", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467580102419, "lm_q2_score": 0.8688267626522814, "lm_q1q2_score": 0.7638262316582871}}
{"text": "[STATEMENT]\nlemma round_up_diff_round_down: \"round_up prec x - round_down prec x \\<le> 2 powr -prec\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. round_up prec x - round_down prec x \\<le> 2 powr - real_of_int prec\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. round_up prec x - round_down prec x \\<le> 2 powr - real_of_int prec\n[PROOF STEP]\nhave \"round_up prec x - round_down prec x = (\\<lceil>x * 2 powr prec\\<rceil> - \\<lfloor>x * 2 powr prec\\<rfloor>) * 2 powr -prec\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. round_up prec x - round_down prec x = real_of_int (\\<lceil>x * 2 powr real_of_int prec\\<rceil> - \\<lfloor>x * 2 powr real_of_int prec\\<rfloor>) * 2 powr real_of_int (- prec)\n[PROOF STEP]\nby (simp add: round_up_def round_down_def field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nround_up prec x - round_down prec x = real_of_int (\\<lceil>x * 2 powr real_of_int prec\\<rceil> - \\<lfloor>x * 2 powr real_of_int prec\\<rfloor>) * 2 powr real_of_int (- prec)\n\ngoal (1 subgoal):\n 1. round_up prec x - round_down prec x \\<le> 2 powr - real_of_int prec\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nround_up prec x - round_down prec x = real_of_int (\\<lceil>x * 2 powr real_of_int prec\\<rceil> - \\<lfloor>x * 2 powr real_of_int prec\\<rfloor>) * 2 powr real_of_int (- prec)\n\ngoal (1 subgoal):\n 1. round_up prec x - round_down prec x \\<le> 2 powr - real_of_int prec\n[PROOF STEP]\nhave \"\\<dots> \\<le> 1 * 2 powr -prec\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real_of_int (\\<lceil>x * 2 powr real_of_int prec\\<rceil> - \\<lfloor>x * 2 powr real_of_int prec\\<rfloor>) * 2 powr real_of_int (- prec) \\<le> 1 * 2 powr real_of_int (- prec)\n[PROOF STEP]\nby (rule mult_mono)\n      (auto simp flip: of_int_diff simp: ceiling_diff_floor_le_1)\n[PROOF STATE]\nproof (state)\nthis:\nreal_of_int (\\<lceil>x * 2 powr real_of_int prec\\<rceil> - \\<lfloor>x * 2 powr real_of_int prec\\<rfloor>) * 2 powr real_of_int (- prec) \\<le> 1 * 2 powr real_of_int (- prec)\n\ngoal (1 subgoal):\n 1. round_up prec x - round_down prec x \\<le> 2 powr - real_of_int prec\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nround_up prec x - round_down prec x \\<le> 1 * 2 powr real_of_int (- prec)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nround_up prec x - round_down prec x \\<le> 1 * 2 powr real_of_int (- prec)\n\ngoal (1 subgoal):\n 1. round_up prec x - round_down prec x \\<le> 2 powr - real_of_int prec\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nround_up prec x - round_down prec x \\<le> 2 powr - real_of_int prec\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1117, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970842359877, "lm_q2_score": 0.8670357477770336, "lm_q1q2_score": 0.7636825585703805}}
{"text": "[STATEMENT]\nlemma polyfun_Sum:\nassumes \"finite I\"\nassumes \"\\<And>i. i\\<in>I \\<Longrightarrow> polyfun N (f i)\"\nshows \"polyfun N (\\<lambda>x. \\<Sum>i\\<in>I. f i x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. polyfun N (\\<lambda>x. \\<Sum>i\\<in>I. f i x)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite I\n?i \\<in> I \\<Longrightarrow> polyfun N (f ?i)\n\ngoal (1 subgoal):\n 1. polyfun N (\\<lambda>x. \\<Sum>i\\<in>I. f i x)\n[PROOF STEP]\napply (induction I rule:finite_induct)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. (\\<And>i. i \\<in> {} \\<Longrightarrow> polyfun N (f i)) \\<Longrightarrow> polyfun N (\\<lambda>x. \\<Sum>i\\<in>{}. f i x)\n 2. \\<And>x F. \\<lbrakk>finite F; x \\<notin> F; (\\<And>i. i \\<in> F \\<Longrightarrow> polyfun N (f i)) \\<Longrightarrow> polyfun N (\\<lambda>x. \\<Sum>i\\<in>F. f i x); \\<And>i. i \\<in> insert x F \\<Longrightarrow> polyfun N (f i)\\<rbrakk> \\<Longrightarrow> polyfun N (\\<lambda>xa. \\<Sum>i\\<in>insert x F. f i xa)\n[PROOF STEP]\napply (simp add: polyfun_const)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>x F. \\<lbrakk>finite F; x \\<notin> F; (\\<And>i. i \\<in> F \\<Longrightarrow> polyfun N (f i)) \\<Longrightarrow> polyfun N (\\<lambda>x. \\<Sum>i\\<in>F. f i x); \\<And>i. i \\<in> insert x F \\<Longrightarrow> polyfun N (f i)\\<rbrakk> \\<Longrightarrow> polyfun N (\\<lambda>xa. \\<Sum>i\\<in>insert x F. f i xa)\n[PROOF STEP]\nusing comm_monoid_add_class.sum.insert polyfun_add\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>finite ?A; ?x \\<notin> ?A\\<rbrakk> \\<Longrightarrow> sum ?g (insert ?x ?A) = ?g ?x + sum ?g ?A\n\\<lbrakk>polyfun ?N ?f; polyfun ?N ?g\\<rbrakk> \\<Longrightarrow> polyfun ?N (\\<lambda>x. ?f x + ?g x)\n\ngoal (1 subgoal):\n 1. \\<And>x F. \\<lbrakk>finite F; x \\<notin> F; (\\<And>i. i \\<in> F \\<Longrightarrow> polyfun N (f i)) \\<Longrightarrow> polyfun N (\\<lambda>x. \\<Sum>i\\<in>F. f i x); \\<And>i. i \\<in> insert x F \\<Longrightarrow> polyfun N (f i)\\<rbrakk> \\<Longrightarrow> polyfun N (\\<lambda>xa. \\<Sum>i\\<in>insert x F. f i xa)\n[PROOF STEP]\nby fastforce", "meta": {"llama_tokens": 842, "file": "Polynomials_More_MPoly_Type", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942203004186, "lm_q2_score": 0.8577681104440172, "lm_q1q2_score": 0.7636659910863197}}
{"text": "[STATEMENT]\nlemma polyfun_eq_coeffs: \"(\\<forall>x. (\\<Sum>i\\<le>n. c i * x^i) = (\\<Sum>i\\<le>n. d i * x^i)) \\<longleftrightarrow> (\\<forall>i\\<le>n. c i = d i)\"\n  for c :: \"nat \\<Rightarrow> 'a::{idom,real_normed_div_algebra}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<forall>x. (\\<Sum>i\\<le>n. c i * x ^ i) = (\\<Sum>i\\<le>n. d i * x ^ i)) = (\\<forall>i\\<le>n. c i = d i)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\<forall>x. (\\<Sum>i\\<le>n. c i * x ^ i) = (\\<Sum>i\\<le>n. d i * x ^ i)) = (\\<forall>i\\<le>n. c i = d i)\n[PROOF STEP]\nhave \"(\\<forall>x. (\\<Sum>i\\<le>n. c i * x^i) = (\\<Sum>i\\<le>n. d i * x^i)) \\<longleftrightarrow> (\\<forall>x. (\\<Sum>i\\<le>n. (c i - d i) * x^i) = 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<forall>x. (\\<Sum>i\\<le>n. c i * x ^ i) = (\\<Sum>i\\<le>n. d i * x ^ i)) = (\\<forall>x. (\\<Sum>i\\<le>n. (c i - d i) * x ^ i) = (0::'a))\n[PROOF STEP]\nby (simp add: left_diff_distrib Groups_Big.sum_subtractf)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<forall>x. (\\<Sum>i\\<le>n. c i * x ^ i) = (\\<Sum>i\\<le>n. d i * x ^ i)) = (\\<forall>x. (\\<Sum>i\\<le>n. (c i - d i) * x ^ i) = (0::'a))\n\ngoal (1 subgoal):\n 1. (\\<forall>x. (\\<Sum>i\\<le>n. c i * x ^ i) = (\\<Sum>i\\<le>n. d i * x ^ i)) = (\\<forall>i\\<le>n. c i = d i)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\<forall>x. (\\<Sum>i\\<le>n. c i * x ^ i) = (\\<Sum>i\\<le>n. d i * x ^ i)) = (\\<forall>x. (\\<Sum>i\\<le>n. (c i - d i) * x ^ i) = (0::'a))\n\ngoal (1 subgoal):\n 1. (\\<forall>x. (\\<Sum>i\\<le>n. c i * x ^ i) = (\\<Sum>i\\<le>n. d i * x ^ i)) = (\\<forall>i\\<le>n. c i = d i)\n[PROOF STEP]\nhave \"\\<dots> \\<longleftrightarrow> (\\<forall>i\\<le>n. c i - d i = 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<forall>x. (\\<Sum>i\\<le>n. (c i - d i) * x ^ i) = (0::'a)) = (\\<forall>i\\<le>n. c i - d i = (0::'a))\n[PROOF STEP]\nby (rule polyfun_eq_0)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<forall>x. (\\<Sum>i\\<le>n. (c i - d i) * x ^ i) = (0::'a)) = (\\<forall>i\\<le>n. c i - d i = (0::'a))\n\ngoal (1 subgoal):\n 1. (\\<forall>x. (\\<Sum>i\\<le>n. c i * x ^ i) = (\\<Sum>i\\<le>n. d i * x ^ i)) = (\\<forall>i\\<le>n. c i = d i)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\<forall>x. (\\<Sum>i\\<le>n. c i * x ^ i) = (\\<Sum>i\\<le>n. d i * x ^ i)) = (\\<forall>i\\<le>n. c i - d i = (0::'a))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<forall>x. (\\<Sum>i\\<le>n. c i * x ^ i) = (\\<Sum>i\\<le>n. d i * x ^ i)) = (\\<forall>i\\<le>n. c i - d i = (0::'a))\n\ngoal (1 subgoal):\n 1. (\\<forall>x. (\\<Sum>i\\<le>n. c i * x ^ i) = (\\<Sum>i\\<le>n. d i * x ^ i)) = (\\<forall>i\\<le>n. c i = d i)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\<forall>x. (\\<Sum>i\\<le>n. c i * x ^ i) = (\\<Sum>i\\<le>n. d i * x ^ i)) = (\\<forall>i\\<le>n. c i = d i)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1422, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942144788077, "lm_q2_score": 0.8577681049901037, "lm_q1q2_score": 0.7636659812371399}}
{"text": "[STATEMENT]\nlemma homeomorphic_spheres:\n  fixes a b ::\"'a::real_normed_vector\"\n  assumes \"0 < d\"  \"0 < e\"\n  shows \"(sphere a d) homeomorphic (sphere b e)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sphere a d homeomorphic sphere b e\n[PROOF STEP]\nunfolding homeomorphic_minimal\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<exists>f g. (\\<forall>x\\<in>sphere a d. f x \\<in> sphere b e \\<and> g (f x) = x) \\<and> (\\<forall>y\\<in>sphere b e. g y \\<in> sphere a d \\<and> f (g y) = y) \\<and> continuous_on (sphere a d) f \\<and> continuous_on (sphere b e) g\n[PROOF STEP]\napply(rule_tac x=\"\\<lambda>x. b + (e/d) *\\<^sub>R (x - a)\" in exI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<exists>g. (\\<forall>x\\<in>sphere a d. b + (e / d) *\\<^sub>R (x - a) \\<in> sphere b e \\<and> g (b + (e / d) *\\<^sub>R (x - a)) = x) \\<and> (\\<forall>y\\<in>sphere b e. g y \\<in> sphere a d \\<and> b + (e / d) *\\<^sub>R (g y - a) = y) \\<and> continuous_on (sphere a d) (\\<lambda>x. b + (e / d) *\\<^sub>R (x - a)) \\<and> continuous_on (sphere b e) g\n[PROOF STEP]\napply(rule_tac x=\"\\<lambda>x. a + (d/e) *\\<^sub>R (x - b)\" in exI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<forall>x\\<in>sphere a d. b + (e / d) *\\<^sub>R (x - a) \\<in> sphere b e \\<and> a + (d / e) *\\<^sub>R (b + (e / d) *\\<^sub>R (x - a) - b) = x) \\<and> (\\<forall>y\\<in>sphere b e. a + (d / e) *\\<^sub>R (y - b) \\<in> sphere a d \\<and> b + (e / d) *\\<^sub>R (a + (d / e) *\\<^sub>R (y - b) - a) = y) \\<and> continuous_on (sphere a d) (\\<lambda>x. b + (e / d) *\\<^sub>R (x - a)) \\<and> continuous_on (sphere b e) (\\<lambda>x. a + (d / e) *\\<^sub>R (x - b))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < d\n0 < e\n\ngoal (1 subgoal):\n 1. (\\<forall>x\\<in>sphere a d. b + (e / d) *\\<^sub>R (x - a) \\<in> sphere b e \\<and> a + (d / e) *\\<^sub>R (b + (e / d) *\\<^sub>R (x - a) - b) = x) \\<and> (\\<forall>y\\<in>sphere b e. a + (d / e) *\\<^sub>R (y - b) \\<in> sphere a d \\<and> b + (e / d) *\\<^sub>R (a + (d / e) *\\<^sub>R (y - b) - a) = y) \\<and> continuous_on (sphere a d) (\\<lambda>x. b + (e / d) *\\<^sub>R (x - a)) \\<and> continuous_on (sphere b e) (\\<lambda>x. a + (d / e) *\\<^sub>R (x - b))\n[PROOF STEP]\nby (auto intro!: continuous_intros simp: dist_commute dist_norm pos_divide_less_eq)", "meta": {"llama_tokens": 1046, "file": null, "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513703624557, "lm_q2_score": 0.8519527944504227, "lm_q1q2_score": 0.7635638595103149}}
{"text": "[STATEMENT]\nlemma path_image_part_circlepath':\n  \"path_image (part_circlepath z r s t) = (\\<lambda>x. z + r * cis x) ` closed_segment s t\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. path_image (part_circlepath z r s t) = (\\<lambda>x. z + complex_of_real r * cis x) ` closed_segment s t\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. path_image (part_circlepath z r s t) = (\\<lambda>x. z + complex_of_real r * cis x) ` closed_segment s t\n[PROOF STEP]\nhave \"path_image (part_circlepath z r s t) = \n          (\\<lambda>x. z + r * exp(\\<i> * of_real x)) ` linepath s t ` {0..1}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. path_image (part_circlepath z r s t) = (\\<lambda>x. z + complex_of_real r * exp (\\<i> * complex_of_real x)) ` linepath s t ` {0..1}\n[PROOF STEP]\nby (simp add: image_image path_image_def part_circlepath_def)\n[PROOF STATE]\nproof (state)\nthis:\npath_image (part_circlepath z r s t) = (\\<lambda>x. z + complex_of_real r * exp (\\<i> * complex_of_real x)) ` linepath s t ` {0..1}\n\ngoal (1 subgoal):\n 1. path_image (part_circlepath z r s t) = (\\<lambda>x. z + complex_of_real r * cis x) ` closed_segment s t\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\npath_image (part_circlepath z r s t) = (\\<lambda>x. z + complex_of_real r * exp (\\<i> * complex_of_real x)) ` linepath s t ` {0..1}\n\ngoal (1 subgoal):\n 1. path_image (part_circlepath z r s t) = (\\<lambda>x. z + complex_of_real r * cis x) ` closed_segment s t\n[PROOF STEP]\nhave \"linepath s t ` {0..1} = closed_segment s t\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. linepath s t ` {0..1} = closed_segment s t\n[PROOF STEP]\nby (rule linepath_image_01)\n[PROOF STATE]\nproof (state)\nthis:\nlinepath s t ` {0..1} = closed_segment s t\n\ngoal (1 subgoal):\n 1. path_image (part_circlepath z r s t) = (\\<lambda>x. z + complex_of_real r * cis x) ` closed_segment s t\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\npath_image (part_circlepath z r s t) = (\\<lambda>x. z + complex_of_real r * exp (\\<i> * complex_of_real x)) ` closed_segment s t\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\npath_image (part_circlepath z r s t) = (\\<lambda>x. z + complex_of_real r * exp (\\<i> * complex_of_real x)) ` closed_segment s t\n\ngoal (1 subgoal):\n 1. path_image (part_circlepath z r s t) = (\\<lambda>x. z + complex_of_real r * cis x) ` closed_segment s t\n[PROOF STEP]\nby (simp add: cis_conv_exp)\n[PROOF STATE]\nproof (state)\nthis:\npath_image (part_circlepath z r s t) = (\\<lambda>x. z + complex_of_real r * cis x) ` closed_segment s t\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1060, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278726384089, "lm_q2_score": 0.865224084314688, "lm_q1q2_score": 0.7634978480773255}}
{"text": "[STATEMENT]\nlemma ring_iso_same_card: \"R \\<simeq> S \\<Longrightarrow> card (carrier R) = card (carrier S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. R \\<simeq> S \\<Longrightarrow> card (carrier R) = card (carrier S)\n[PROOF STEP]\nusing bij_betw_same_card\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw ?f ?A ?B \\<Longrightarrow> card ?A = card ?B\n\ngoal (1 subgoal):\n 1. R \\<simeq> S \\<Longrightarrow> card (carrier R) = card (carrier S)\n[PROOF STEP]\nunfolding is_ring_iso_def ring_iso_def\n[PROOF STATE]\nproof (prove)\nusing this:\nbij_betw ?f ?A ?B \\<Longrightarrow> card ?A = card ?B\n\ngoal (1 subgoal):\n 1. {h \\<in> ring_hom R S. bij_betw h (carrier R) (carrier S)} \\<noteq> {} \\<Longrightarrow> card (carrier R) = card (carrier S)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 302, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9161096204605946, "lm_q2_score": 0.8333246015211008, "lm_q1q2_score": 0.7634166844199719}}
{"text": "[STATEMENT]\nlemma mult_bounds_enclose_zero1:\n  \"min (la * lb) (min (la * ub) (min (lb * ua) (ua * ub))) \\<le> 0\"\n  \"0 \\<le> max (la * lb) (max (la * ub) (max (lb * ua) (ua * ub)))\"\n  if \"la \\<le> 0\" \"0 \\<le> ua\"\n  for la lb ua ub:: \"'a::linordered_idom\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. min (la * lb) (min (la * ub) (min (lb * ua) (ua * ub))) \\<le> (0::'a) &&& (0::'a) \\<le> max (la * lb) (max (la * ub) (max (lb * ua) (ua * ub)))\n[PROOF STEP]\nsubgoal\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. min (la * lb) (min (la * ub) (min (lb * ua) (ua * ub))) \\<le> (0::'a)\n[PROOF STEP]\nby (metis (no_types, opaque_lifting) that eq_iff min_le_iff_disj mult_zero_left mult_zero_right\n        zero_le_mult_iff)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (0::'a) \\<le> max (la * lb) (max (la * ub) (max (lb * ua) (ua * ub)))\n[PROOF STEP]\nsubgoal\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (0::'a) \\<le> max (la * lb) (max (la * ub) (max (lb * ua) (ua * ub)))\n[PROOF STEP]\nby (metis that le_max_iff_disj mult_zero_right order_refl zero_le_mult_iff)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 544, "file": null, "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9184802395624259, "lm_q2_score": 0.8311430394931456, "lm_q1q2_score": 0.7633884580243071}}
{"text": "[STATEMENT]\nlemma cosine_law_triangle'':\n  \"cos (angle b a c) = (dist a b ^ 2 + dist a c ^ 2 - dist b c ^ 2) / (2 * dist a b * dist a c)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. cos (angle b a c) = ((dist a b)\\<^sup>2 + (dist a c)\\<^sup>2 - (dist b c)\\<^sup>2) / (2 * dist a b * dist a c)\n[PROOF STEP]\nusing cosine_law_triangle[of b c a]\n[PROOF STATE]\nproof (prove)\nusing this:\n(dist b c)\\<^sup>2 = (dist a b)\\<^sup>2 + (dist a c)\\<^sup>2 - 2 * dist a b * dist a c * cos (angle b a c)\n\ngoal (1 subgoal):\n 1. cos (angle b a c) = ((dist a b)\\<^sup>2 + (dist a c)\\<^sup>2 - (dist b c)\\<^sup>2) / (2 * dist a b * dist a c)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 303, "file": "Triangle_Triangle", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.936285002192296, "lm_q2_score": 0.8152324938410784, "lm_q1q2_score": 0.7632899572832251}}
{"text": "[STATEMENT]\nlemma isCont_powreal_exponent_less_one: \n  assumes \"0 < a\" \n  and \"a < 1\" \n  shows \"isCont (\\<lambda>x. a pow\\<^sub>\\<real> x) x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. isCont ((pow\\<^sub>\\<real>) a) x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. isCont ((pow\\<^sub>\\<real>) a) x\n[PROOF STEP]\nhave \"1 < inverse a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 1 < inverse a\n[PROOF STEP]\nby (simp add: assms one_less_inverse)\n[PROOF STATE]\nproof (state)\nthis:\n1 < inverse a\n\ngoal (1 subgoal):\n 1. isCont ((pow\\<^sub>\\<real>) a) x\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n1 < inverse a\n[PROOF STEP]\nhave \"isCont ((pow\\<^sub>\\<real>) (inverse a)) x\"\n[PROOF STATE]\nproof (prove)\nusing this:\n1 < inverse a\n\ngoal (1 subgoal):\n 1. isCont ((pow\\<^sub>\\<real>) (inverse a)) x\n[PROOF STEP]\nby (simp add: isCont_powreal_exponent_gt_one)\n[PROOF STATE]\nproof (state)\nthis:\nisCont ((pow\\<^sub>\\<real>) (inverse a)) x\n\ngoal (1 subgoal):\n 1. isCont ((pow\\<^sub>\\<real>) a) x\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nisCont ((pow\\<^sub>\\<real>) (inverse a)) x\n[PROOF STEP]\nhave \"isCont (\\<lambda>x. inverse (inverse a pow\\<^sub>\\<real> x)) x\"\n[PROOF STATE]\nproof (prove)\nusing this:\nisCont ((pow\\<^sub>\\<real>) (inverse a)) x\n\ngoal (1 subgoal):\n 1. isCont (\\<lambda>x. inverse (inverse a pow\\<^sub>\\<real> x)) x\n[PROOF STEP]\nusing assms(1) continuous_at_within_inverse powreal_not_zero\n[PROOF STATE]\nproof (prove)\nusing this:\nisCont ((pow\\<^sub>\\<real>) (inverse a)) x\n0 < a\n\\<lbrakk>continuous (at ?a within ?s) ?f; ?f ?a \\<noteq> (0::?'b)\\<rbrakk> \\<Longrightarrow> continuous (at ?a within ?s) (\\<lambda>x. inverse (?f x))\n0 < ?a \\<Longrightarrow> ?a pow\\<^sub>\\<real> ?x \\<noteq> 0\n\ngoal (1 subgoal):\n 1. isCont (\\<lambda>x. inverse (inverse a pow\\<^sub>\\<real> x)) x\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\nisCont (\\<lambda>x. inverse (inverse a pow\\<^sub>\\<real> x)) x\n\ngoal (1 subgoal):\n 1. isCont ((pow\\<^sub>\\<real>) a) x\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nisCont (\\<lambda>x. inverse (inverse a pow\\<^sub>\\<real> x)) x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nisCont (\\<lambda>x. inverse (inverse a pow\\<^sub>\\<real> x)) x\n\ngoal (1 subgoal):\n 1. isCont ((pow\\<^sub>\\<real>) a) x\n[PROOF STEP]\nusing assms(1) powreal_inverse\n[PROOF STATE]\nproof (prove)\nusing this:\nisCont (\\<lambda>x. inverse (inverse a pow\\<^sub>\\<real> x)) x\n0 < a\ninverse (?a pow\\<^sub>\\<real> ?x) = inverse ?a pow\\<^sub>\\<real> ?x\n\ngoal (1 subgoal):\n 1. isCont ((pow\\<^sub>\\<real>) a) x\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nisCont ((pow\\<^sub>\\<real>) a) x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1141, "file": "Real_Power_RealPower", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314828740729, "lm_q2_score": 0.8615382147637196, "lm_q1q2_score": 0.7630053666938745}}
{"text": "[STATEMENT]\nlemma contour_integral_id [simp]: \"contour_integral (linepath a b) (\\<lambda>y. y) = (b^2 - a^2)/2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. contour_integral (linepath a b) (\\<lambda>y. y) = (b\\<^sup>2 - a\\<^sup>2) / 2\n[PROOF STEP]\nusing contour_integral_primitive [of UNIV \"\\<lambda>x. x^2/2\" \"\\<lambda>x. x\" \"linepath a b\"] contour_integral_unique\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>\\<And>x. x \\<in> UNIV \\<Longrightarrow> ((\\<lambda>x. x\\<^sup>2 / 2) has_field_derivative x) (at x); valid_path (linepath a b); path_image (linepath a b) \\<subseteq> UNIV\\<rbrakk> \\<Longrightarrow> ((\\<lambda>x. x) has_contour_integral (pathfinish (linepath a b))\\<^sup>2 / 2 - (pathstart (linepath a b))\\<^sup>2 / 2) (linepath a b)\n(?f has_contour_integral ?i) ?g \\<Longrightarrow> contour_integral ?g ?f = ?i\n\ngoal (1 subgoal):\n 1. contour_integral (linepath a b) (\\<lambda>y. y) = (b\\<^sup>2 - a\\<^sup>2) / 2\n[PROOF STEP]\nby (simp add: has_field_der_id)", "meta": {"llama_tokens": 411, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9086179043564153, "lm_q2_score": 0.8397339756938818, "lm_q1q2_score": 0.7629973252118558}}
{"text": "[STATEMENT]\nlemma norm_columns_fst_QR_decomposition:\n  fixes A::\"real^'n::{mod_type}^'m::{mod_type}\"\n  assumes \"rank A = ncols A\"\n  shows \"norm (column i (fst (QR_decomposition A))) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm (column i (fst (QR_decomposition A))) = 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. norm (column i (fst (QR_decomposition A))) = 1\n[PROOF STEP]\nhave \"vec.independent (columns (Gram_Schmidt_matrix A))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec.independent (columns (Gram_Schmidt_matrix A))\n[PROOF STEP]\nby (metis assms full_rank_imp_is_basis2 independent_columns_Gram_Schmidt_matrix)\n[PROOF STATE]\nproof (state)\nthis:\nvec.independent (columns (Gram_Schmidt_matrix A))\n\ngoal (1 subgoal):\n 1. norm (column i (fst (QR_decomposition A))) = 1\n[PROOF STEP]\nhence \"column i (Gram_Schmidt_matrix A) \\<noteq> 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nvec.independent (columns (Gram_Schmidt_matrix A))\n\ngoal (1 subgoal):\n 1. column i (Gram_Schmidt_matrix A) \\<noteq> 0\n[PROOF STEP]\nusing vec.dependent_zero[of \"columns (Gram_Schmidt_matrix A)\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nvec.independent (columns (Gram_Schmidt_matrix A))\n0 \\<in> columns (Gram_Schmidt_matrix A) \\<Longrightarrow> vec.dependent (columns (Gram_Schmidt_matrix A))\n\ngoal (1 subgoal):\n 1. column i (Gram_Schmidt_matrix A) \\<noteq> 0\n[PROOF STEP]\nunfolding columns_def\n[PROOF STATE]\nproof (prove)\nusing this:\nvec.independent {column i (Gram_Schmidt_matrix A) |i. i \\<in> UNIV}\n0 \\<in> {column i (Gram_Schmidt_matrix A) |i. i \\<in> UNIV} \\<Longrightarrow> vec.dependent {column i (Gram_Schmidt_matrix A) |i. i \\<in> UNIV}\n\ngoal (1 subgoal):\n 1. column i (Gram_Schmidt_matrix A) \\<noteq> 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncolumn i (Gram_Schmidt_matrix A) \\<noteq> 0\n\ngoal (1 subgoal):\n 1. norm (column i (fst (QR_decomposition A))) = 1\n[PROOF STEP]\nthus \"norm (column i (fst (QR_decomposition A))) = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncolumn i (Gram_Schmidt_matrix A) \\<noteq> 0\n\ngoal (1 subgoal):\n 1. norm (column i (fst (QR_decomposition A))) = 1\n[PROOF STEP]\nunfolding QR_decomposition_def Let_def fst_conv\n[PROOF STATE]\nproof (prove)\nusing this:\ncolumn i (Gram_Schmidt_matrix A) \\<noteq> 0\n\ngoal (1 subgoal):\n 1. norm (column i (divide_by_norm (Gram_Schmidt_matrix A))) = 1\n[PROOF STEP]\nby (rule norm_column_divide_by_norm)\n[PROOF STATE]\nproof (state)\nthis:\nnorm (column i (fst (QR_decomposition A))) = 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1067, "file": "QR_Decomposition_QR_Decomposition", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.901920681802153, "lm_q2_score": 0.8459424431344437, "lm_q1q2_score": 0.7629729850771965}}
{"text": "[STATEMENT]\nlemma sorted_list_of_multiset_image_commute:\n  assumes \"mono f\"\n  shows \"sorted_list_of_multiset (image_mset f M) = map f (sorted_list_of_multiset M)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sorted_list_of_multiset (image_mset f M) = map f (sorted_list_of_multiset M)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sorted_list_of_multiset (image_mset f M) = map f (sorted_list_of_multiset M)\n[PROOF STEP]\nhave \"sorted (sorted_list_of_multiset (image_mset f M))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sorted (sorted_list_of_multiset (image_mset f M))\n[PROOF STEP]\nby (simp add:sorted_sorted_list_of_multiset)\n[PROOF STATE]\nproof (state)\nthis:\nsorted (sorted_list_of_multiset (image_mset f M))\n\ngoal (1 subgoal):\n 1. sorted_list_of_multiset (image_mset f M) = map f (sorted_list_of_multiset M)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nsorted (sorted_list_of_multiset (image_mset f M))\n\ngoal (1 subgoal):\n 1. sorted_list_of_multiset (image_mset f M) = map f (sorted_list_of_multiset M)\n[PROOF STEP]\nhave \" sorted_wrt (\\<lambda>x y. f x \\<le> f y) (sorted_list_of_multiset M)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sorted_wrt (\\<lambda>x y. f x \\<le> f y) (sorted_list_of_multiset M)\n[PROOF STEP]\nby (rule sorted_wrt_mono_rel[where P=\"\\<lambda>x y. x \\<le> y\"]) \n      (auto intro: monoD[OF assms] sorted_sorted_list_of_multiset)\n[PROOF STATE]\nproof (state)\nthis:\nsorted_wrt (\\<lambda>x y. f x \\<le> f y) (sorted_list_of_multiset M)\n\ngoal (1 subgoal):\n 1. sorted_list_of_multiset (image_mset f M) = map f (sorted_list_of_multiset M)\n[PROOF STEP]\nhence \"sorted (map f (sorted_list_of_multiset M))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsorted_wrt (\\<lambda>x y. f x \\<le> f y) (sorted_list_of_multiset M)\n\ngoal (1 subgoal):\n 1. sorted (map f (sorted_list_of_multiset M))\n[PROOF STEP]\nby (subst sorted_wrt_map)\n[PROOF STATE]\nproof (state)\nthis:\nsorted (map f (sorted_list_of_multiset M))\n\ngoal (1 subgoal):\n 1. sorted_list_of_multiset (image_mset f M) = map f (sorted_list_of_multiset M)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nsorted (sorted_list_of_multiset (image_mset f M))\nsorted (map f (sorted_list_of_multiset M))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsorted (sorted_list_of_multiset (image_mset f M))\nsorted (map f (sorted_list_of_multiset M))\n\ngoal (1 subgoal):\n 1. sorted_list_of_multiset (image_mset f M) = map f (sorted_list_of_multiset M)\n[PROOF STEP]\nby (intro list_eq_iff, auto)\n[PROOF STATE]\nproof (state)\nthis:\nsorted_list_of_multiset (image_mset f M) = map f (sorted_list_of_multiset M)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1169, "file": "Frequency_Moments_Frequency_Moments_Preliminary_Results", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.90192067652954, "lm_q2_score": 0.8459424353665381, "lm_q1q2_score": 0.7629729736108347}}
{"text": "[STATEMENT]\nlemma index_matrix_prod [simp]:\n  assumes \"i < dim_row A\" and \"j < dim_col B\" and \"dim_col A = dim_row B\"\n  shows \"(A * B) $$ (i,j) = (\\<Sum>k<dim_row B. (A $$ (i,k)) * (B $$ (k,j)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (A * B) $$ (i, j) = (\\<Sum>k<dim_row B. A $$ (i, k) * B $$ (k, j))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ni < dim_row A\nj < dim_col B\ndim_col A = dim_row B\n\ngoal (1 subgoal):\n 1. (A * B) $$ (i, j) = (\\<Sum>k<dim_row B. A $$ (i, k) * B $$ (k, j))\n[PROOF STEP]\napply(simp add: scalar_prod_def atLeast0LessThan)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\n.", "meta": {"llama_tokens": 312, "file": "Isabelle_Marries_Dirac_Basics", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9304582632076909, "lm_q2_score": 0.8198933315126792, "lm_q1q2_score": 0.7628765252548549}}
{"text": "[STATEMENT]\nlemma real_sqrt_sum_squares_triangle_ineq:\n  \"sqrt ((a + c)\\<^sup>2 + (b + d)\\<^sup>2) \\<le> sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt ((a + c)\\<^sup>2 + (b + d)\\<^sup>2) \\<le> sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sqrt ((a + c)\\<^sup>2 + (b + d)\\<^sup>2) \\<le> sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2)\n[PROOF STEP]\nhave \"(a * c + b * d) \\<le> (sqrt (a\\<^sup>2 + b\\<^sup>2) * sqrt (c\\<^sup>2 + d\\<^sup>2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a * c + b * d \\<le> sqrt (a\\<^sup>2 + b\\<^sup>2) * sqrt (c\\<^sup>2 + d\\<^sup>2)\n[PROOF STEP]\nby (rule power2_le_imp_le) (simp_all add: power2_sum power_mult_distrib ring_distribs L2_set_mult_ineq_lemma add.commute)\n[PROOF STATE]\nproof (state)\nthis:\na * c + b * d \\<le> sqrt (a\\<^sup>2 + b\\<^sup>2) * sqrt (c\\<^sup>2 + d\\<^sup>2)\n\ngoal (1 subgoal):\n 1. sqrt ((a + c)\\<^sup>2 + (b + d)\\<^sup>2) \\<le> sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\na * c + b * d \\<le> sqrt (a\\<^sup>2 + b\\<^sup>2) * sqrt (c\\<^sup>2 + d\\<^sup>2)\n[PROOF STEP]\nhave \"(a + c)\\<^sup>2 + (b + d)\\<^sup>2 \\<le> (sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2))\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\nusing this:\na * c + b * d \\<le> sqrt (a\\<^sup>2 + b\\<^sup>2) * sqrt (c\\<^sup>2 + d\\<^sup>2)\n\ngoal (1 subgoal):\n 1. (a + c)\\<^sup>2 + (b + d)\\<^sup>2 \\<le> (sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2))\\<^sup>2\n[PROOF STEP]\nby (simp add: power2_sum)\n[PROOF STATE]\nproof (state)\nthis:\n(a + c)\\<^sup>2 + (b + d)\\<^sup>2 \\<le> (sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2))\\<^sup>2\n\ngoal (1 subgoal):\n 1. sqrt ((a + c)\\<^sup>2 + (b + d)\\<^sup>2) \\<le> sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(a + c)\\<^sup>2 + (b + d)\\<^sup>2 \\<le> (sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2))\\<^sup>2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(a + c)\\<^sup>2 + (b + d)\\<^sup>2 \\<le> (sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2))\\<^sup>2\n\ngoal (1 subgoal):\n 1. sqrt ((a + c)\\<^sup>2 + (b + d)\\<^sup>2) \\<le> sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2)\n[PROOF STEP]\nby (auto intro: power2_le_imp_le)\n[PROOF STATE]\nproof (state)\nthis:\nsqrt ((a + c)\\<^sup>2 + (b + d)\\<^sup>2) \\<le> sqrt (a\\<^sup>2 + b\\<^sup>2) + sqrt (c\\<^sup>2 + d\\<^sup>2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1354, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933094088947399, "lm_q2_score": 0.8539127585282744, "lm_q1q2_score": 0.7628083015685696}}
{"text": "[STATEMENT]\nlemma real_of_int_div_aux:\n  \"(real_of_int x) / (real_of_int d) =\n    real_of_int (x div d) + (real_of_int (x mod d)) / (real_of_int d)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real_of_int x / real_of_int d = real_of_int (x div d) + real_of_int (x mod d) / real_of_int d\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. real_of_int x / real_of_int d = real_of_int (x div d) + real_of_int (x mod d) / real_of_int d\n[PROOF STEP]\nhave \"x = (x div d) * d + x mod d\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x = x div d * d + x mod d\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nx = x div d * d + x mod d\n\ngoal (1 subgoal):\n 1. real_of_int x / real_of_int d = real_of_int (x div d) + real_of_int (x mod d) / real_of_int d\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx = x div d * d + x mod d\n[PROOF STEP]\nhave \"real_of_int x = real_of_int (x div d) * real_of_int d + real_of_int(x mod d)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx = x div d * d + x mod d\n\ngoal (1 subgoal):\n 1. real_of_int x = real_of_int (x div d) * real_of_int d + real_of_int (x mod d)\n[PROOF STEP]\nby (metis of_int_add of_int_mult)\n[PROOF STATE]\nproof (state)\nthis:\nreal_of_int x = real_of_int (x div d) * real_of_int d + real_of_int (x mod d)\n\ngoal (1 subgoal):\n 1. real_of_int x / real_of_int d = real_of_int (x div d) + real_of_int (x mod d) / real_of_int d\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nreal_of_int x = real_of_int (x div d) * real_of_int d + real_of_int (x mod d)\n[PROOF STEP]\nhave \"real_of_int x / real_of_int d = \\<dots> / real_of_int d\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal_of_int x = real_of_int (x div d) * real_of_int d + real_of_int (x mod d)\n\ngoal (1 subgoal):\n 1. real_of_int x / real_of_int d = (real_of_int (x div d) * real_of_int d + real_of_int (x mod d)) / real_of_int d\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nreal_of_int x / real_of_int d = (real_of_int (x div d) * real_of_int d + real_of_int (x mod d)) / real_of_int d\n\ngoal (1 subgoal):\n 1. real_of_int x / real_of_int d = real_of_int (x div d) + real_of_int (x mod d) / real_of_int d\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nreal_of_int x / real_of_int d = (real_of_int (x div d) * real_of_int d + real_of_int (x mod d)) / real_of_int d\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nreal_of_int x / real_of_int d = (real_of_int (x div d) * real_of_int d + real_of_int (x mod d)) / real_of_int d\n\ngoal (1 subgoal):\n 1. real_of_int x / real_of_int d = real_of_int (x div d) + real_of_int (x mod d) / real_of_int d\n[PROOF STEP]\nby (auto simp add: add_divide_distrib algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nreal_of_int x / real_of_int d = real_of_int (x div d) + real_of_int (x mod d) / real_of_int d\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1316, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045966995027, "lm_q2_score": 0.8596637559030338, "lm_q1q2_score": 0.7626976358531308}}
{"text": "[STATEMENT]\nlemma card_UNIV_option: \"CARD('a option) = (if CARD('a) = 0 then 0 else CARD('a) + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. CARD('a option) = (if CARD('a) = 0 then 0 else CARD('a) + 1)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. CARD('a option) = (if CARD('a) = 0 then 0 else CARD('a) + 1)\n[PROOF STEP]\nhave \"(None :: 'a option) \\<notin> range Some\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. None \\<notin> range Some\n[PROOF STEP]\nby clarsimp\n[PROOF STATE]\nproof (state)\nthis:\nNone \\<notin> range Some\n\ngoal (1 subgoal):\n 1. CARD('a option) = (if CARD('a) = 0 then 0 else CARD('a) + 1)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nNone \\<notin> range Some\n\ngoal (1 subgoal):\n 1. CARD('a option) = (if CARD('a) = 0 then 0 else CARD('a) + 1)\n[PROOF STEP]\nby (simp add: UNIV_option_conv card_eq_0_iff finite_range_Some card_image)\n[PROOF STATE]\nproof (state)\nthis:\nCARD('a option) = (if CARD('a) = 0 then 0 else CARD('a) + 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 474, "file": null, "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9059898102301019, "lm_q2_score": 0.8418256452674009, "lm_q1q2_score": 0.7626854566026456}}
{"text": "[STATEMENT]\nlemma cong_mersenne_number_int:\n  fixes k :: int\n  shows \"[k mod 2 ^ n + k div 2 ^ n = k] (mod (2 ^ n - 1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [k mod 2 ^ n + k div 2 ^ n = k] (mod 2 ^ n - 1)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. [k mod 2 ^ n + k div 2 ^ n = k] (mod 2 ^ n - 1)\n[PROOF STEP]\nhave \"k = (2 ^ n - 1 + 1) * (k div 2 ^ n) + (k mod 2 ^ n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. k = (2 ^ n - 1 + 1) * (k div 2 ^ n) + k mod 2 ^ n\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nk = (2 ^ n - 1 + 1) * (k div 2 ^ n) + k mod 2 ^ n\n\ngoal (1 subgoal):\n 1. [k mod 2 ^ n + k div 2 ^ n = k] (mod 2 ^ n - 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nk = (2 ^ n - 1 + 1) * (k div 2 ^ n) + k mod 2 ^ n\n\ngoal (1 subgoal):\n 1. [k mod 2 ^ n + k div 2 ^ n = k] (mod 2 ^ n - 1)\n[PROOF STEP]\nhave \"[\\<dots> = (0 + 1) * (k div 2 ^ n) + (k mod 2 ^ n)] (mod (2 ^ n - 1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. [(2 ^ n - 1 + 1) * (k div 2 ^ n) + k mod 2 ^ n = (0 + 1) * (k div 2 ^ n) + k mod 2 ^ n] (mod 2 ^ n - 1)\n[PROOF STEP]\nby (intro cong_add cong_mult cong_refl) (auto simp: cong_def)\n[PROOF STATE]\nproof (state)\nthis:\n[(2 ^ n - 1 + 1) * (k div 2 ^ n) + k mod 2 ^ n = (0 + 1) * (k div 2 ^ n) + k mod 2 ^ n] (mod 2 ^ n - 1)\n\ngoal (1 subgoal):\n 1. [k mod 2 ^ n + k div 2 ^ n = k] (mod 2 ^ n - 1)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n[k = (0 + 1) * (k div 2 ^ n) + k mod 2 ^ n] (mod 2 ^ n - 1)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n[k = (0 + 1) * (k div 2 ^ n) + k mod 2 ^ n] (mod 2 ^ n - 1)\n\ngoal (1 subgoal):\n 1. [k mod 2 ^ n + k div 2 ^ n = k] (mod 2 ^ n - 1)\n[PROOF STEP]\nby (simp add: cong_sym add_ac)\n[PROOF STATE]\nproof (state)\nthis:\n[k mod 2 ^ n + k div 2 ^ n = k] (mod 2 ^ n - 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 991, "file": "Mersenne_Primes_Lucas_Lehmer_Code", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213691605412, "lm_q2_score": 0.847967764140929, "lm_q1q2_score": 0.7624259370983949}}
{"text": "[STATEMENT]\nlemma diagonal_mat_smult:\n  fixes A::\"'a::{ring} Matrix.mat\"\n  assumes \"diagonal_mat A\"\n  shows \"diagonal_mat (x \\<cdot>\\<^sub>mA)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. diagonal_mat (x \\<cdot>\\<^sub>m A)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ndiagonal_mat A\n\ngoal (1 subgoal):\n 1. diagonal_mat (x \\<cdot>\\<^sub>m A)\n[PROOF STEP]\nunfolding diagonal_mat_def uminus_mat_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<forall>i<dim_row A. \\<forall>j<dim_col A. i \\<noteq> j \\<longrightarrow> A $$ (i, j) = (0::'a)\n\ngoal (1 subgoal):\n 1. \\<forall>i<dim_row (x \\<cdot>\\<^sub>m A). \\<forall>j<dim_col (x \\<cdot>\\<^sub>m A). i \\<noteq> j \\<longrightarrow> (x \\<cdot>\\<^sub>m A) $$ (i, j) = (0::'a)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 332, "file": "Commuting_Hermitian_Spectral_Theory_Complements", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8991213664574069, "lm_q2_score": 0.8479677526147223, "lm_q1q2_score": 0.7624259244427656}}
{"text": "[STATEMENT]\nlemma effective_scalar_product_distributivity:\n assumes \"length v1 = length v2\" and \"length w1 = length w2\"\n shows \"(scalar_product v1 v2)*(scalar_product w1 w2)\n      = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2) \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nusing assms scalar_product_distributivity\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v1 = length v2\nlength w1 = length w2\n\\<forall>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = ?n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n\ngoal (1 subgoal):\n 1. scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 363, "file": "Matrix_Tensor_Matrix_Tensor", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9173026528034426, "lm_q2_score": 0.8311430499496096, "lm_q1q2_score": 0.7624097245779211}}
{"text": "[STATEMENT]\nlemma norm_eq_on_real_2_vec:\n  fixes x :: \"real ^ 2\"\n  shows \"norm x = sqrt ((x $ 1) ^ 2 + (x $ 2) ^ 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. norm x = sqrt ((x $ 1)\\<^sup>2 + (x $ 2)\\<^sup>2)\n[PROOF STEP]\nby (simp add: norm_eq_sqrt_inner inner_vec_def UNIV_2 power2_eq_square)", "meta": {"llama_tokens": 148, "file": "Ptolemys_Theorem_Ptolemys_Theorem", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9149009526726545, "lm_q2_score": 0.8333245911726382, "lm_q1q2_score": 0.762409462349397}}
{"text": "[STATEMENT]\nlemma card_PiE: \"finite S \\<Longrightarrow> card (\\<Pi>\\<^sub>E i \\<in> S. T i) = (\\<Prod> i\\<in>S. card (T i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite S \\<Longrightarrow> card (Pi\\<^sub>E S T) = (\\<Prod>i\\<in>S. card (T i))\n[PROOF STEP]\nproof (induct rule: finite_induct)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. card (Pi\\<^sub>E {} T) = (\\<Prod>i\\<in>{}. card (T i))\n 2. \\<And>x F. \\<lbrakk>finite F; x \\<notin> F; card (Pi\\<^sub>E F T) = (\\<Prod>i\\<in>F. card (T i))\\<rbrakk> \\<Longrightarrow> card (Pi\\<^sub>E (insert x F) T) = (\\<Prod>i\\<in>insert x F. card (T i))\n[PROOF STEP]\ncase empty\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. card (Pi\\<^sub>E {} T) = (\\<Prod>i\\<in>{}. card (T i))\n 2. \\<And>x F. \\<lbrakk>finite F; x \\<notin> F; card (Pi\\<^sub>E F T) = (\\<Prod>i\\<in>F. card (T i))\\<rbrakk> \\<Longrightarrow> card (Pi\\<^sub>E (insert x F) T) = (\\<Prod>i\\<in>insert x F. card (T i))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (Pi\\<^sub>E {} T) = (\\<Prod>i\\<in>{}. card (T i))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncard (Pi\\<^sub>E {} T) = (\\<Prod>i\\<in>{}. card (T i))\n\ngoal (1 subgoal):\n 1. \\<And>x F. \\<lbrakk>finite F; x \\<notin> F; card (Pi\\<^sub>E F T) = (\\<Prod>i\\<in>F. card (T i))\\<rbrakk> \\<Longrightarrow> card (Pi\\<^sub>E (insert x F) T) = (\\<Prod>i\\<in>insert x F. card (T i))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>x F. \\<lbrakk>finite F; x \\<notin> F; card (Pi\\<^sub>E F T) = (\\<Prod>i\\<in>F. card (T i))\\<rbrakk> \\<Longrightarrow> card (Pi\\<^sub>E (insert x F) T) = (\\<Prod>i\\<in>insert x F. card (T i))\n[PROOF STEP]\ncase (insert x S)\n[PROOF STATE]\nproof (state)\nthis:\nfinite S\nx \\<notin> S\ncard (Pi\\<^sub>E S T) = (\\<Prod>i\\<in>S. card (T i))\n\ngoal (1 subgoal):\n 1. \\<And>x F. \\<lbrakk>finite F; x \\<notin> F; card (Pi\\<^sub>E F T) = (\\<Prod>i\\<in>F. card (T i))\\<rbrakk> \\<Longrightarrow> card (Pi\\<^sub>E (insert x F) T) = (\\<Prod>i\\<in>insert x F. card (T i))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite S\nx \\<notin> S\ncard (Pi\\<^sub>E S T) = (\\<Prod>i\\<in>S. card (T i))\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite S\nx \\<notin> S\ncard (Pi\\<^sub>E S T) = (\\<Prod>i\\<in>S. card (T i))\n\ngoal (1 subgoal):\n 1. card (Pi\\<^sub>E (insert x S) T) = (\\<Prod>i\\<in>insert x S. card (T i))\n[PROOF STEP]\nby (simp add: PiE_insert_eq inj_combinator card_image card_cartesian_product)\n[PROOF STATE]\nproof (state)\nthis:\ncard (Pi\\<^sub>E (insert x S) T) = (\\<Prod>i\\<in>insert x S. card (T i))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1216, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.888758793492457, "lm_q2_score": 0.8577681104440172, "lm_q1q2_score": 0.7623489509345294}}
{"text": "[STATEMENT]\nlemma n_leaves_append [simp]:\n     \"n_leaves (append t1 t2) = n_leaves t1 * n_leaves t2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n_leaves (Binary_Tree.append t1 t2) = n_leaves t1 * n_leaves t2\n[PROOF STEP]\napply (induct t1)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. n_leaves (Binary_Tree.append Lf t2) = n_leaves Lf * n_leaves t2\n 2. \\<And>x1 t11 t12. \\<lbrakk>n_leaves (Binary_Tree.append t11 t2) = n_leaves t11 * n_leaves t2; n_leaves (Binary_Tree.append t12 t2) = n_leaves t12 * n_leaves t2\\<rbrakk> \\<Longrightarrow> n_leaves (Binary_Tree.append (Br x1 t11 t12) t2) = n_leaves (Br x1 t11 t12) * n_leaves t2\n[PROOF STEP]\napply (metis append.simps(1) n_leaves.simps(1) nat_mult_1 plus_nat.simps(1)\n              Suc_eq_plus1)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>x1 t11 t12. \\<lbrakk>n_leaves (Binary_Tree.append t11 t2) = n_leaves t11 * n_leaves t2; n_leaves (Binary_Tree.append t12 t2) = n_leaves t12 * n_leaves t2\\<rbrakk> \\<Longrightarrow> n_leaves (Binary_Tree.append (Br x1 t11 t12) t2) = n_leaves (Br x1 t11 t12) * n_leaves t2\n[PROOF STEP]\nby (simp add: distrib_right)", "meta": {"llama_tokens": 537, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.919642533380189, "lm_q2_score": 0.8289388083214156, "lm_q1q2_score": 0.7623273857018615}}
{"text": "[STATEMENT]\nlemma norm_dot:\n \"\\<parallel>v\\<parallel> = sqrt (v\\<cdot>v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<parallel>v\\<parallel> = sqrt (v \\<cdot> v)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<parallel>v\\<parallel> = sqrt (v \\<cdot> v)\n[PROOF STEP]\nhave \"sqrt (v\\<cdot>v) = sqrt (\\<Sum>j\\<in>{1..(vlen v)}. v\\<^bsub>j\\<^esub>*v\\<^bsub>j\\<^esub>)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt (v \\<cdot> v) = sqrt (\\<Sum>j = 1..vlen v. v\\<^bsub>j\\<^esub> * v\\<^bsub>j\\<^esub>)\n[PROOF STEP]\nunfolding dot_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt (\\<Sum>j = 1..vlen v. v\\<^bsub>j\\<^esub> * v\\<^bsub>j\\<^esub>) = sqrt (\\<Sum>j = 1..vlen v. v\\<^bsub>j\\<^esub> * v\\<^bsub>j\\<^esub>)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsqrt (v \\<cdot> v) = sqrt (\\<Sum>j = 1..vlen v. v\\<^bsub>j\\<^esub> * v\\<^bsub>j\\<^esub>)\n\ngoal (1 subgoal):\n 1. \\<parallel>v\\<parallel> = sqrt (v \\<cdot> v)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsqrt (v \\<cdot> v) = sqrt (\\<Sum>j = 1..vlen v. v\\<^bsub>j\\<^esub> * v\\<^bsub>j\\<^esub>)\n\ngoal (1 subgoal):\n 1. \\<parallel>v\\<parallel> = sqrt (v \\<cdot> v)\n[PROOF STEP]\nwith real_sq\n[PROOF STATE]\nproof (chain)\npicking this:\n?a * ?a = ?a\\<^sup>2\nsqrt (v \\<cdot> v) = sqrt (\\<Sum>j = 1..vlen v. v\\<^bsub>j\\<^esub> * v\\<^bsub>j\\<^esub>)\n[PROOF STEP]\nhave \"\\<dots> = sqrt (\\<Sum>j\\<in>{1..(vlen v)}. v\\<^bsub>j\\<^esub>^2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n?a * ?a = ?a\\<^sup>2\nsqrt (v \\<cdot> v) = sqrt (\\<Sum>j = 1..vlen v. v\\<^bsub>j\\<^esub> * v\\<^bsub>j\\<^esub>)\n\ngoal (1 subgoal):\n 1. sqrt (\\<Sum>j = 1..vlen v. v\\<^bsub>j\\<^esub> * v\\<^bsub>j\\<^esub>) = sqrt (\\<Sum>j = 1..vlen v. (v\\<^bsub>j\\<^esub>)\\<^sup>2)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsqrt (\\<Sum>j = 1..vlen v. v\\<^bsub>j\\<^esub> * v\\<^bsub>j\\<^esub>) = sqrt (\\<Sum>j = 1..vlen v. (v\\<^bsub>j\\<^esub>)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. \\<parallel>v\\<parallel> = sqrt (v \\<cdot> v)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsqrt (\\<Sum>j = 1..vlen v. v\\<^bsub>j\\<^esub> * v\\<^bsub>j\\<^esub>) = sqrt (\\<Sum>j = 1..vlen v. (v\\<^bsub>j\\<^esub>)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. \\<parallel>v\\<parallel> = sqrt (v \\<cdot> v)\n[PROOF STEP]\nhave \"\\<dots> = \\<parallel>v\\<parallel>\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt (\\<Sum>j = 1..vlen v. (v\\<^bsub>j\\<^esub>)\\<^sup>2) = \\<parallel>v\\<parallel>\n[PROOF STEP]\nunfolding norm_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sqrt (\\<Sum>j = 1..vlen v. (v\\<^bsub>j\\<^esub>)\\<^sup>2) = sqrt (\\<Sum>j = 1..vlen v. (v\\<^bsub>j\\<^esub>)\\<^sup>2)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nsqrt (\\<Sum>j = 1..vlen v. (v\\<^bsub>j\\<^esub>)\\<^sup>2) = \\<parallel>v\\<parallel>\n\ngoal (1 subgoal):\n 1. \\<parallel>v\\<parallel> = sqrt (v \\<cdot> v)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nsqrt (v \\<cdot> v) = \\<parallel>v\\<parallel>\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsqrt (v \\<cdot> v) = \\<parallel>v\\<parallel>\n\ngoal (1 subgoal):\n 1. \\<parallel>v\\<parallel> = sqrt (v \\<cdot> v)\n[PROOF STEP]\n..\n[PROOF STATE]\nproof (state)\nthis:\n\\<parallel>v\\<parallel> = sqrt (v \\<cdot> v)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1555, "file": "Cauchy_CauchySchwarz", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9032942041005328, "lm_q2_score": 0.8438951005915208, "lm_q1q2_score": 0.7622855532331569}}
{"text": "[STATEMENT]\nlemma convex_hull_list_combination:\n  assumes Vs: \"set Vs \\<subseteq> carrier_vec n\"\n    and x: \"x \\<in> convex_hull_list Vs\"\n    and y: \"y \\<in> convex_hull_list Vs\"\n    and l0: \"0 \\<le> l\" and l1: \"l \\<le> 1\"\n  shows \"l \\<cdot>\\<^sub>v x + (1 - l) \\<cdot>\\<^sub>v y \\<in> convex_hull_list Vs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. l \\<cdot>\\<^sub>v x + ((1::'a) - l) \\<cdot>\\<^sub>v y \\<in> convex_hull_list Vs\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. l \\<cdot>\\<^sub>v x + ((1::'a) - l) \\<cdot>\\<^sub>v y \\<in> convex_hull_list Vs\n[PROOF STEP]\nfrom x\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\<in> convex_hull_list Vs\n[PROOF STEP]\nobtain cx where x: \"lincomb_list cx Vs = x\" and cx0: \"\\<forall> i < length Vs. cx i \\<ge> 0\"\n    and cx1: \"sum cx {0..<length Vs} = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\<in> convex_hull_list Vs\n\ngoal (1 subgoal):\n 1. (\\<And>cx. \\<lbrakk>lincomb_list cx Vs = x; \\<forall>i<length Vs. (0::'a) \\<le> cx i; sum cx {0..<length Vs} = (1::'a)\\<rbrakk> \\<Longrightarrow> thesis) \\<Longrightarrow> thesis\n[PROOF STEP]\nunfolding convex_hull_list_def convex_lincomb_list_def nonneg_lincomb_list_def\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\<in> {x. \\<exists>c. (lincomb_list c Vs = x \\<and> (\\<forall>i<length Vs. (0::'a) \\<le> c i)) \\<and> sum c {0..<length Vs} = (1::'a)}\n\ngoal (1 subgoal):\n 1. (\\<And>cx. \\<lbrakk>lincomb_list cx Vs = x; \\<forall>i<length Vs. (0::'a) \\<le> cx i; sum cx {0..<length Vs} = (1::'a)\\<rbrakk> \\<Longrightarrow> thesis) \\<Longrightarrow> thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlincomb_list cx Vs = x\n\\<forall>i<length Vs. (0::'a) \\<le> cx i\nsum cx {0..<length Vs} = (1::'a)\n\ngoal (1 subgoal):\n 1. l \\<cdot>\\<^sub>v x + ((1::'a) - l) \\<cdot>\\<^sub>v y \\<in> convex_hull_list Vs\n[PROOF STEP]\nfrom y\n[PROOF STATE]\nproof (chain)\npicking this:\ny \\<in> convex_hull_list Vs\n[PROOF STEP]\nobtain cy where y: \"lincomb_list cy Vs = y\" and cy0: \"\\<forall> i < length Vs. cy i \\<ge> 0\"\n    and cy1: \"sum cy {0..<length Vs} = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\ny \\<in> convex_hull_list Vs\n\ngoal (1 subgoal):\n 1. (\\<And>cy. \\<lbrakk>lincomb_list cy Vs = y; \\<forall>i<length Vs. (0::'a) \\<le> cy i; sum cy {0..<length Vs} = (1::'a)\\<rbrakk> \\<Longrightarrow> thesis) \\<Longrightarrow> thesis\n[PROOF STEP]\nunfolding convex_hull_list_def convex_lincomb_list_def nonneg_lincomb_list_def\n[PROOF STATE]\nproof (prove)\nusing this:\ny \\<in> {x. \\<exists>c. (lincomb_list c Vs = x \\<and> (\\<forall>i<length Vs. (0::'a) \\<le> c i)) \\<and> sum c {0..<length Vs} = (1::'a)}\n\ngoal (1 subgoal):\n 1. (\\<And>cy. \\<lbrakk>lincomb_list cy Vs = y; \\<forall>i<length Vs. (0::'a) \\<le> cy i; sum cy {0..<length Vs} = (1::'a)\\<rbrakk> \\<Longrightarrow> thesis) \\<Longrightarrow> thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlincomb_list cy Vs = y\n\\<forall>i<length Vs. (0::'a) \\<le> cy i\nsum cy {0..<length Vs} = (1::'a)\n\ngoal (1 subgoal):\n 1. l \\<cdot>\\<^sub>v x + ((1::'a) - l) \\<cdot>\\<^sub>v y \\<in> convex_hull_list Vs\n[PROOF STEP]\nlet ?c = \"\\<lambda> i. l * cx i + (1 - l) * cy i\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. l \\<cdot>\\<^sub>v x + ((1::'a) - l) \\<cdot>\\<^sub>v y \\<in> convex_hull_list Vs\n[PROOF STEP]\nhave \"set Vs \\<subseteq> carrier_vec n \\<Longrightarrow>\n        lincomb_list ?c Vs = l \\<cdot>\\<^sub>v lincomb_list cx Vs + (1 - l) \\<cdot>\\<^sub>v lincomb_list cy Vs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set Vs \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) Vs = l \\<cdot>\\<^sub>v lincomb_list cx Vs + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy Vs\n[PROOF STEP]\nproof (induction Vs rule: rev_induct)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. set [] \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) [] = l \\<cdot>\\<^sub>v lincomb_list cx [] + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy []\n 2. \\<And>x xs. \\<lbrakk>set xs \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) xs = l \\<cdot>\\<^sub>v lincomb_list cx xs + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy xs; set (xs @ [x]) \\<subseteq> carrier_vec n\\<rbrakk> \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) (xs @ [x]) = l \\<cdot>\\<^sub>v lincomb_list cx (xs @ [x]) + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy (xs @ [x])\n[PROOF STEP]\ncase (snoc v Vs)\n[PROOF STATE]\nproof (state)\nthis:\nset Vs \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) Vs = l \\<cdot>\\<^sub>v lincomb_list cx Vs + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy Vs\nset (Vs @ [v]) \\<subseteq> carrier_vec n\n\ngoal (2 subgoals):\n 1. set [] \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) [] = l \\<cdot>\\<^sub>v lincomb_list cx [] + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy []\n 2. \\<And>x xs. \\<lbrakk>set xs \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) xs = l \\<cdot>\\<^sub>v lincomb_list cx xs + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy xs; set (xs @ [x]) \\<subseteq> carrier_vec n\\<rbrakk> \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) (xs @ [x]) = l \\<cdot>\\<^sub>v lincomb_list cx (xs @ [x]) + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy (xs @ [x])\n[PROOF STEP]\nhave v: \"v \\<in> carrier_vec n\" and Vs: \"set Vs \\<subseteq> carrier_vec n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. v \\<in> carrier_vec n &&& set Vs \\<subseteq> carrier_vec n\n[PROOF STEP]\nusing snoc.prems\n[PROOF STATE]\nproof (prove)\nusing this:\nset (Vs @ [v]) \\<subseteq> carrier_vec n\n\ngoal (1 subgoal):\n 1. v \\<in> carrier_vec n &&& set Vs \\<subseteq> carrier_vec n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nv \\<in> carrier_vec n\nset Vs \\<subseteq> carrier_vec n\n\ngoal (2 subgoals):\n 1. set [] \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) [] = l \\<cdot>\\<^sub>v lincomb_list cx [] + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy []\n 2. \\<And>x xs. \\<lbrakk>set xs \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) xs = l \\<cdot>\\<^sub>v lincomb_list cx xs + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy xs; set (xs @ [x]) \\<subseteq> carrier_vec n\\<rbrakk> \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) (xs @ [x]) = l \\<cdot>\\<^sub>v lincomb_list cx (xs @ [x]) + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy (xs @ [x])\n[PROOF STEP]\nhave \"lincomb_list ?c (Vs @ [v]) = lincomb_list ?c Vs + ?c (length Vs) \\<cdot>\\<^sub>v v\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) (Vs @ [v]) = lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) Vs + (l * cx (length Vs) + ((1::'a) - l) * cy (length Vs)) \\<cdot>\\<^sub>v v\n[PROOF STEP]\nusing snoc.prems\n[PROOF STATE]\nproof (prove)\nusing this:\nset (Vs @ [v]) \\<subseteq> carrier_vec n\n\ngoal (1 subgoal):\n 1. lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) (Vs @ [v]) = lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) Vs + (l * cx (length Vs) + ((1::'a) - l) * cy (length Vs)) \\<cdot>\\<^sub>v v\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) (Vs @ [v]) = lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) Vs + (l * cx (length Vs) + ((1::'a) - l) * cy (length Vs)) \\<cdot>\\<^sub>v v\n\ngoal (2 subgoals):\n 1. set [] \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) [] = l \\<cdot>\\<^sub>v lincomb_list cx [] + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy []\n 2. \\<And>x xs. \\<lbrakk>set xs \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) xs = l \\<cdot>\\<^sub>v lincomb_list cx xs + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy xs; set (xs @ [x]) \\<subseteq> carrier_vec n\\<rbrakk> \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) (xs @ [x]) = l \\<cdot>\\<^sub>v lincomb_list cx (xs @ [x]) + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy (xs @ [x])\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nlincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) (Vs @ [v]) = lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) Vs + (l * cx (length Vs) + ((1::'a) - l) * cy (length Vs)) \\<cdot>\\<^sub>v v\n\ngoal (2 subgoals):\n 1. set [] \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) [] = l \\<cdot>\\<^sub>v lincomb_list cx [] + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy []\n 2. \\<And>x xs. \\<lbrakk>set xs \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) xs = l \\<cdot>\\<^sub>v lincomb_list cx xs + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy xs; set (xs @ [x]) \\<subseteq> carrier_vec n\\<rbrakk> \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) (xs @ [x]) = l \\<cdot>\\<^sub>v lincomb_list cx (xs @ [x]) + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy (xs @ [x])\n[PROOF STEP]\nhave \"lincomb_list ?c Vs =\n               l \\<cdot>\\<^sub>v lincomb_list cx Vs + (1 - l) \\<cdot>\\<^sub>v lincomb_list cy Vs\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) Vs = l \\<cdot>\\<^sub>v lincomb_list cx Vs + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy Vs\n[PROOF STEP]\nby (rule snoc.IH[OF Vs])\n[PROOF STATE]\nproof (state)\nthis:\nlincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) Vs = l \\<cdot>\\<^sub>v lincomb_list cx Vs + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy Vs\n\ngoal (2 subgoals):\n 1. set [] \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) [] = l \\<cdot>\\<^sub>v lincomb_list cx [] + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy []\n 2. \\<And>x xs. \\<lbrakk>set xs \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) xs = l \\<cdot>\\<^sub>v lincomb_list cx xs + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy xs; set (xs @ [x]) \\<subseteq> carrier_vec n\\<rbrakk> \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) (xs @ [x]) = l \\<cdot>\\<^sub>v lincomb_list cx (xs @ [x]) + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy (xs @ [x])\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nlincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) Vs = l \\<cdot>\\<^sub>v lincomb_list cx Vs + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy Vs\n\ngoal (2 subgoals):\n 1. set [] \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) [] = l \\<cdot>\\<^sub>v lincomb_list cx [] + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy []\n 2. \\<And>x xs. \\<lbrakk>set xs \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) xs = l \\<cdot>\\<^sub>v lincomb_list cx xs + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy xs; set (xs @ [x]) \\<subseteq> carrier_vec n\\<rbrakk> \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) (xs @ [x]) = l \\<cdot>\\<^sub>v lincomb_list cx (xs @ [x]) + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy (xs @ [x])\n[PROOF STEP]\nhave \"?c (length Vs) \\<cdot>\\<^sub>v v =\n               l \\<cdot>\\<^sub>v (cx (length Vs) \\<cdot>\\<^sub>v v) + (1 - l) \\<cdot>\\<^sub>v (cy (length Vs) \\<cdot>\\<^sub>v v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (l * cx (length Vs) + ((1::'a) - l) * cy (length Vs)) \\<cdot>\\<^sub>v v = l \\<cdot>\\<^sub>v (cx (length Vs) \\<cdot>\\<^sub>v v) + ((1::'a) - l) \\<cdot>\\<^sub>v (cy (length Vs) \\<cdot>\\<^sub>v v)\n[PROOF STEP]\nusing add_smult_distrib_vec smult_smult_assoc\n[PROOF STATE]\nproof (prove)\nusing this:\n(?a + ?b) \\<cdot>\\<^sub>v ?v = ?a \\<cdot>\\<^sub>v ?v + ?b \\<cdot>\\<^sub>v ?v\n?a \\<cdot>\\<^sub>v (?b \\<cdot>\\<^sub>v ?v) = ?a * ?b \\<cdot>\\<^sub>v ?v\n\ngoal (1 subgoal):\n 1. (l * cx (length Vs) + ((1::'a) - l) * cy (length Vs)) \\<cdot>\\<^sub>v v = l \\<cdot>\\<^sub>v (cx (length Vs) \\<cdot>\\<^sub>v v) + ((1::'a) - l) \\<cdot>\\<^sub>v (cy (length Vs) \\<cdot>\\<^sub>v v)\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\n(l * cx (length Vs) + ((1::'a) - l) * cy (length Vs)) \\<cdot>\\<^sub>v v = l \\<cdot>\\<^sub>v (cx (length Vs) \\<cdot>\\<^sub>v v) + ((1::'a) - l) \\<cdot>\\<^sub>v (cy (length Vs) \\<cdot>\\<^sub>v v)\n\ngoal (2 subgoals):\n 1. set [] \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) [] = l \\<cdot>\\<^sub>v lincomb_list cx [] + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy []\n 2. \\<And>x xs. \\<lbrakk>set xs \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) xs = l \\<cdot>\\<^sub>v lincomb_list cx xs + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy xs; set (xs @ [x]) \\<subseteq> carrier_vec n\\<rbrakk> \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) (xs @ [x]) = l \\<cdot>\\<^sub>v lincomb_list cx (xs @ [x]) + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy (xs @ [x])\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(l * cx (length Vs) + ((1::'a) - l) * cy (length Vs)) \\<cdot>\\<^sub>v v = l \\<cdot>\\<^sub>v (cx (length Vs) \\<cdot>\\<^sub>v v) + ((1::'a) - l) \\<cdot>\\<^sub>v (cy (length Vs) \\<cdot>\\<^sub>v v)\n\ngoal (2 subgoals):\n 1. set [] \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) [] = l \\<cdot>\\<^sub>v lincomb_list cx [] + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy []\n 2. \\<And>x xs. \\<lbrakk>set xs \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) xs = l \\<cdot>\\<^sub>v lincomb_list cx xs + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy xs; set (xs @ [x]) \\<subseteq> carrier_vec n\\<rbrakk> \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) (xs @ [x]) = l \\<cdot>\\<^sub>v lincomb_list cx (xs @ [x]) + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy (xs @ [x])\n[PROOF STEP]\nhave \"l \\<cdot>\\<^sub>v lincomb_list cx Vs + (1 - l) \\<cdot>\\<^sub>v lincomb_list cy Vs + \\<dots> =\n                  l \\<cdot>\\<^sub>v (lincomb_list cx Vs + cx (length Vs) \\<cdot>\\<^sub>v v) +\n                  (1 - l) \\<cdot>\\<^sub>v (lincomb_list cy Vs + cy (length Vs) \\<cdot>\\<^sub>v v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. l \\<cdot>\\<^sub>v lincomb_list cx Vs + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy Vs + (l \\<cdot>\\<^sub>v (cx (length Vs) \\<cdot>\\<^sub>v v) + ((1::'a) - l) \\<cdot>\\<^sub>v (cy (length Vs) \\<cdot>\\<^sub>v v)) = l \\<cdot>\\<^sub>v (lincomb_list cx Vs + cx (length Vs) \\<cdot>\\<^sub>v v) + ((1::'a) - l) \\<cdot>\\<^sub>v (lincomb_list cy Vs + cy (length Vs) \\<cdot>\\<^sub>v v)\n[PROOF STEP]\nusing lincomb_list_carrier[OF Vs] v\n[PROOF STATE]\nproof (prove)\nusing this:\nlincomb_list ?c Vs \\<in> carrier_vec n\nv \\<in> carrier_vec n\n\ngoal (1 subgoal):\n 1. l \\<cdot>\\<^sub>v lincomb_list cx Vs + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy Vs + (l \\<cdot>\\<^sub>v (cx (length Vs) \\<cdot>\\<^sub>v v) + ((1::'a) - l) \\<cdot>\\<^sub>v (cy (length Vs) \\<cdot>\\<^sub>v v)) = l \\<cdot>\\<^sub>v (lincomb_list cx Vs + cx (length Vs) \\<cdot>\\<^sub>v v) + ((1::'a) - l) \\<cdot>\\<^sub>v (lincomb_list cy Vs + cy (length Vs) \\<cdot>\\<^sub>v v)\n[PROOF STEP]\nby (simp add: M.add.m_assoc M.add.m_lcomm smult_r_distr)\n[PROOF STATE]\nproof (state)\nthis:\nl \\<cdot>\\<^sub>v lincomb_list cx Vs + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy Vs + (l \\<cdot>\\<^sub>v (cx (length Vs) \\<cdot>\\<^sub>v v) + ((1::'a) - l) \\<cdot>\\<^sub>v (cy (length Vs) \\<cdot>\\<^sub>v v)) = l \\<cdot>\\<^sub>v (lincomb_list cx Vs + cx (length Vs) \\<cdot>\\<^sub>v v) + ((1::'a) - l) \\<cdot>\\<^sub>v (lincomb_list cy Vs + cy (length Vs) \\<cdot>\\<^sub>v v)\n\ngoal (2 subgoals):\n 1. set [] \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) [] = l \\<cdot>\\<^sub>v lincomb_list cx [] + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy []\n 2. \\<And>x xs. \\<lbrakk>set xs \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) xs = l \\<cdot>\\<^sub>v lincomb_list cx xs + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy xs; set (xs @ [x]) \\<subseteq> carrier_vec n\\<rbrakk> \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) (xs @ [x]) = l \\<cdot>\\<^sub>v lincomb_list cx (xs @ [x]) + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy (xs @ [x])\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nlincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) (Vs @ [v]) = l \\<cdot>\\<^sub>v (lincomb_list cx Vs + cx (length Vs) \\<cdot>\\<^sub>v v) + ((1::'a) - l) \\<cdot>\\<^sub>v (lincomb_list cy Vs + cy (length Vs) \\<cdot>\\<^sub>v v)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nlincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) (Vs @ [v]) = l \\<cdot>\\<^sub>v (lincomb_list cx Vs + cx (length Vs) \\<cdot>\\<^sub>v v) + ((1::'a) - l) \\<cdot>\\<^sub>v (lincomb_list cy Vs + cy (length Vs) \\<cdot>\\<^sub>v v)\n\ngoal (1 subgoal):\n 1. lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) (Vs @ [v]) = l \\<cdot>\\<^sub>v lincomb_list cx (Vs @ [v]) + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy (Vs @ [v])\n[PROOF STEP]\nusing Vs v\n[PROOF STATE]\nproof (prove)\nusing this:\nlincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) (Vs @ [v]) = l \\<cdot>\\<^sub>v (lincomb_list cx Vs + cx (length Vs) \\<cdot>\\<^sub>v v) + ((1::'a) - l) \\<cdot>\\<^sub>v (lincomb_list cy Vs + cy (length Vs) \\<cdot>\\<^sub>v v)\nset Vs \\<subseteq> carrier_vec n\nv \\<in> carrier_vec n\n\ngoal (1 subgoal):\n 1. lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) (Vs @ [v]) = l \\<cdot>\\<^sub>v lincomb_list cx (Vs @ [v]) + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy (Vs @ [v])\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) (Vs @ [v]) = l \\<cdot>\\<^sub>v lincomb_list cx (Vs @ [v]) + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy (Vs @ [v])\n\ngoal (1 subgoal):\n 1. set [] \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) [] = l \\<cdot>\\<^sub>v lincomb_list cx [] + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy []\n[PROOF STEP]\nqed simp\n[PROOF STATE]\nproof (state)\nthis:\nset Vs \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) Vs = l \\<cdot>\\<^sub>v lincomb_list cx Vs + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy Vs\n\ngoal (1 subgoal):\n 1. l \\<cdot>\\<^sub>v x + ((1::'a) - l) \\<cdot>\\<^sub>v y \\<in> convex_hull_list Vs\n[PROOF STEP]\nhence \"lincomb_list ?c Vs = l \\<cdot>\\<^sub>v x + (1 - l) \\<cdot>\\<^sub>v y\"\n[PROOF STATE]\nproof (prove)\nusing this:\nset Vs \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) Vs = l \\<cdot>\\<^sub>v lincomb_list cx Vs + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy Vs\n\ngoal (1 subgoal):\n 1. lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) Vs = l \\<cdot>\\<^sub>v x + ((1::'a) - l) \\<cdot>\\<^sub>v y\n[PROOF STEP]\nusing Vs x y\n[PROOF STATE]\nproof (prove)\nusing this:\nset Vs \\<subseteq> carrier_vec n \\<Longrightarrow> lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) Vs = l \\<cdot>\\<^sub>v lincomb_list cx Vs + ((1::'a) - l) \\<cdot>\\<^sub>v lincomb_list cy Vs\nset Vs \\<subseteq> carrier_vec n\nlincomb_list cx Vs = x\nlincomb_list cy Vs = y\n\ngoal (1 subgoal):\n 1. lincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) Vs = l \\<cdot>\\<^sub>v x + ((1::'a) - l) \\<cdot>\\<^sub>v y\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) Vs = l \\<cdot>\\<^sub>v x + ((1::'a) - l) \\<cdot>\\<^sub>v y\n\ngoal (1 subgoal):\n 1. l \\<cdot>\\<^sub>v x + ((1::'a) - l) \\<cdot>\\<^sub>v y \\<in> convex_hull_list Vs\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nlincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) Vs = l \\<cdot>\\<^sub>v x + ((1::'a) - l) \\<cdot>\\<^sub>v y\n\ngoal (1 subgoal):\n 1. l \\<cdot>\\<^sub>v x + ((1::'a) - l) \\<cdot>\\<^sub>v y \\<in> convex_hull_list Vs\n[PROOF STEP]\nhave \"\\<forall> i < length Vs. ?c i \\<ge> 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<forall>i<length Vs. (0::'a) \\<le> l * cx i + ((1::'a) - l) * cy i\n[PROOF STEP]\nusing cx0 cy0 l0 l1\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<forall>i<length Vs. (0::'a) \\<le> cx i\n\\<forall>i<length Vs. (0::'a) \\<le> cy i\n(0::'a) \\<le> l\nl \\<le> (1::'a)\n\ngoal (1 subgoal):\n 1. \\<forall>i<length Vs. (0::'a) \\<le> l * cx i + ((1::'a) - l) * cy i\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\<forall>i<length Vs. (0::'a) \\<le> l * cx i + ((1::'a) - l) * cy i\n\ngoal (1 subgoal):\n 1. l \\<cdot>\\<^sub>v x + ((1::'a) - l) \\<cdot>\\<^sub>v y \\<in> convex_hull_list Vs\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\<forall>i<length Vs. (0::'a) \\<le> l * cx i + ((1::'a) - l) * cy i\n\ngoal (1 subgoal):\n 1. l \\<cdot>\\<^sub>v x + ((1::'a) - l) \\<cdot>\\<^sub>v y \\<in> convex_hull_list Vs\n[PROOF STEP]\nhave \"sum ?c {0..<length Vs} = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<length Vs. l * cx i + ((1::'a) - l) * cy i) = (1::'a)\n[PROOF STEP]\nproof(simp add: sum.distrib)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<length Vs. l * cx i) + (\\<Sum>i = 0..<length Vs. ((1::'a) - l) * cy i) = (1::'a)\n[PROOF STEP]\nhave \"(\\<Sum>i = 0..<length Vs. (1 - l) * cy i) = (1 - l) * sum cy {0..<length Vs}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<length Vs. ((1::'a) - l) * cy i) = ((1::'a) - l) * sum cy {0..<length Vs}\n[PROOF STEP]\nusing sum_distrib_left\n[PROOF STATE]\nproof (prove)\nusing this:\n?r * sum ?f ?A = (\\<Sum>n\\<in>?A. ?r * ?f n)\n\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<length Vs. ((1::'a) - l) * cy i) = ((1::'a) - l) * sum cy {0..<length Vs}\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>i = 0..<length Vs. ((1::'a) - l) * cy i) = ((1::'a) - l) * sum cy {0..<length Vs}\n\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<length Vs. l * cx i) + (\\<Sum>i = 0..<length Vs. ((1::'a) - l) * cy i) = (1::'a)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>i = 0..<length Vs. ((1::'a) - l) * cy i) = ((1::'a) - l) * sum cy {0..<length Vs}\n\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<length Vs. l * cx i) + (\\<Sum>i = 0..<length Vs. ((1::'a) - l) * cy i) = (1::'a)\n[PROOF STEP]\nhave \"(\\<Sum>i = 0..<length Vs. l * cx i) = l * sum cx {0..<length Vs}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<length Vs. l * cx i) = l * sum cx {0..<length Vs}\n[PROOF STEP]\nusing sum_distrib_left\n[PROOF STATE]\nproof (prove)\nusing this:\n?r * sum ?f ?A = (\\<Sum>n\\<in>?A. ?r * ?f n)\n\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<length Vs. l * cx i) = l * sum cx {0..<length Vs}\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>i = 0..<length Vs. l * cx i) = l * sum cx {0..<length Vs}\n\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<length Vs. l * cx i) + (\\<Sum>i = 0..<length Vs. ((1::'a) - l) * cy i) = (1::'a)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\<Sum>i = 0..<length Vs. ((1::'a) - l) * cy i) = ((1::'a) - l) * sum cy {0..<length Vs}\n(\\<Sum>i = 0..<length Vs. l * cx i) = l * sum cx {0..<length Vs}\n[PROOF STEP]\nshow \"(\\<Sum>i = 0..<length Vs. l * cx i) + (\\<Sum>i = 0..<length Vs. (1 - l) * cy i) = 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<Sum>i = 0..<length Vs. ((1::'a) - l) * cy i) = ((1::'a) - l) * sum cy {0..<length Vs}\n(\\<Sum>i = 0..<length Vs. l * cx i) = l * sum cx {0..<length Vs}\n\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<length Vs. l * cx i) + (\\<Sum>i = 0..<length Vs. ((1::'a) - l) * cy i) = (1::'a)\n[PROOF STEP]\nusing cx1 cy1\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<Sum>i = 0..<length Vs. ((1::'a) - l) * cy i) = ((1::'a) - l) * sum cy {0..<length Vs}\n(\\<Sum>i = 0..<length Vs. l * cx i) = l * sum cx {0..<length Vs}\nsum cx {0..<length Vs} = (1::'a)\nsum cy {0..<length Vs} = (1::'a)\n\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<length Vs. l * cx i) + (\\<Sum>i = 0..<length Vs. ((1::'a) - l) * cy i) = (1::'a)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>i = 0..<length Vs. l * cx i) + (\\<Sum>i = 0..<length Vs. ((1::'a) - l) * cy i) = (1::'a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>i = 0..<length Vs. l * cx i + ((1::'a) - l) * cy i) = (1::'a)\n\ngoal (1 subgoal):\n 1. l \\<cdot>\\<^sub>v x + ((1::'a) - l) \\<cdot>\\<^sub>v y \\<in> convex_hull_list Vs\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nlincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) Vs = l \\<cdot>\\<^sub>v x + ((1::'a) - l) \\<cdot>\\<^sub>v y\n\\<forall>i<length Vs. (0::'a) \\<le> l * cx i + ((1::'a) - l) * cy i\n(\\<Sum>i = 0..<length Vs. l * cx i + ((1::'a) - l) * cy i) = (1::'a)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nlincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) Vs = l \\<cdot>\\<^sub>v x + ((1::'a) - l) \\<cdot>\\<^sub>v y\n\\<forall>i<length Vs. (0::'a) \\<le> l * cx i + ((1::'a) - l) * cy i\n(\\<Sum>i = 0..<length Vs. l * cx i + ((1::'a) - l) * cy i) = (1::'a)\n\ngoal (1 subgoal):\n 1. l \\<cdot>\\<^sub>v x + ((1::'a) - l) \\<cdot>\\<^sub>v y \\<in> convex_hull_list Vs\n[PROOF STEP]\nunfolding convex_hull_list_def convex_lincomb_list_def nonneg_lincomb_list_def\n[PROOF STATE]\nproof (prove)\nusing this:\nlincomb_list (\\<lambda>i. l * cx i + ((1::'a) - l) * cy i) Vs = l \\<cdot>\\<^sub>v x + ((1::'a) - l) \\<cdot>\\<^sub>v y\n\\<forall>i<length Vs. (0::'a) \\<le> l * cx i + ((1::'a) - l) * cy i\n(\\<Sum>i = 0..<length Vs. l * cx i + ((1::'a) - l) * cy i) = (1::'a)\n\ngoal (1 subgoal):\n 1. l \\<cdot>\\<^sub>v x + ((1::'a) - l) \\<cdot>\\<^sub>v y \\<in> {x. \\<exists>c. (lincomb_list c Vs = x \\<and> (\\<forall>i<length Vs. (0::'a) \\<le> c i)) \\<and> sum c {0..<length Vs} = (1::'a)}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nl \\<cdot>\\<^sub>v x + ((1::'a) - l) \\<cdot>\\<^sub>v y \\<in> convex_hull_list Vs\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 11804, "file": "Linear_Inequalities_Convex_Hull", "length": 62, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467738423874, "lm_q2_score": 0.8670357512127872, "lm_q1q2_score": 0.7622516834847327}}
{"text": "[STATEMENT]\nlemma bijective_map_preimage:\n  assumes \"bijective_map f S T\"\n  shows \"bijective_map (inverse_map f S T) T S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bijective_map (inverse_map f S T) T S\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. inverse_map f S T \\<in> T \\<rightarrow>\\<^sub>E S\n 2. bij_betw (inverse_map f S T) T S\n[PROOF STEP]\nshow \"inverse_map f S T \\<in> T \\<rightarrow>\\<^sub>E S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inverse_map f S T \\<in> T \\<rightarrow>\\<^sub>E S\n[PROOF STEP]\nby (simp add: assms bij_betw_imp_funcset bij_betw_inv_into bijective.bijective bijective_map.axioms(2) inverse_map_def)\n[PROOF STATE]\nproof (state)\nthis:\ninverse_map f S T \\<in> T \\<rightarrow>\\<^sub>E S\n\ngoal (1 subgoal):\n 1. bij_betw (inverse_map f S T) T S\n[PROOF STEP]\nshow \"bij_betw (inverse_map f S T) T S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bij_betw (inverse_map f S T) T S\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nbijective_map f S T\n\ngoal (1 subgoal):\n 1. bij_betw (inverse_map f S T) T S\n[PROOF STEP]\nby (simp add: bij_betw_inv_into bijective_def bijective_map_def inverse_map_def)\n[PROOF STATE]\nproof (state)\nthis:\nbij_betw (inverse_map f S T) T S\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 561, "file": "Grothendieck_Schemes_Set_Extras", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942377652497, "lm_q2_score": 0.8558511396138365, "lm_q1q2_score": 0.761959337983021}}
{"text": "[STATEMENT]\nlemma inter_sorted_correct :\n  assumes l1_OK: \"sorted (rev l1)\"\n  assumes l2_OK: \"sorted (rev l2)\"\n    shows \"sorted (rev (inter_sorted_rev l1 l2)) \\<and> set (inter_sorted_rev l1 l2) = set l1 \\<inter> set l2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Sorted_Less.sorted (rev (inter_sorted_rev l1 l2)) \\<and> set (inter_sorted_rev l1 l2) = set l1 \\<inter> set l2\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nSorted_Less.sorted (rev l1)\nSorted_Less.sorted (rev l2)\n\ngoal (1 subgoal):\n 1. Sorted_Less.sorted (rev (inter_sorted_rev l1 l2)) \\<and> set (inter_sorted_rev l1 l2) = set l1 \\<inter> set l2\n[PROOF STEP]\nproof (induct l1 arbitrary: l2)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\<And>l2. \\<lbrakk>Sorted_Less.sorted (rev []); Sorted_Less.sorted (rev l2)\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev [] l2)) \\<and> set (inter_sorted_rev [] l2) = set [] \\<inter> set l2\n 2. \\<And>a l1 l2. \\<lbrakk>\\<And>l2. \\<lbrakk>Sorted_Less.sorted (rev l1); Sorted_Less.sorted (rev l2)\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev l1 l2)) \\<and> set (inter_sorted_rev l1 l2) = set l1 \\<inter> set l2; Sorted_Less.sorted (rev (a # l1)); Sorted_Less.sorted (rev l2)\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (a # l1) l2)) \\<and> set (inter_sorted_rev (a # l1) l2) = set (a # l1) \\<inter> set l2\n[PROOF STEP]\ncase Nil\n[PROOF STATE]\nproof (state)\nthis:\nSorted_Less.sorted (rev [])\nSorted_Less.sorted (rev l2)\n\ngoal (2 subgoals):\n 1. \\<And>l2. \\<lbrakk>Sorted_Less.sorted (rev []); Sorted_Less.sorted (rev l2)\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev [] l2)) \\<and> set (inter_sorted_rev [] l2) = set [] \\<inter> set l2\n 2. \\<And>a l1 l2. \\<lbrakk>\\<And>l2. \\<lbrakk>Sorted_Less.sorted (rev l1); Sorted_Less.sorted (rev l2)\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev l1 l2)) \\<and> set (inter_sorted_rev l1 l2) = set l1 \\<inter> set l2; Sorted_Less.sorted (rev (a # l1)); Sorted_Less.sorted (rev l2)\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (a # l1) l2)) \\<and> set (inter_sorted_rev (a # l1) l2) = set (a # l1) \\<inter> set l2\n[PROOF STEP]\nthus ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nSorted_Less.sorted (rev [])\nSorted_Less.sorted (rev l2)\n\ngoal (1 subgoal):\n 1. Sorted_Less.sorted (rev (inter_sorted_rev [] l2)) \\<and> set (inter_sorted_rev [] l2) = set [] \\<inter> set l2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nSorted_Less.sorted (rev (inter_sorted_rev [] l2)) \\<and> set (inter_sorted_rev [] l2) = set [] \\<inter> set l2\n\ngoal (1 subgoal):\n 1. \\<And>a l1 l2. \\<lbrakk>\\<And>l2. \\<lbrakk>Sorted_Less.sorted (rev l1); Sorted_Less.sorted (rev l2)\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev l1 l2)) \\<and> set (inter_sorted_rev l1 l2) = set l1 \\<inter> set l2; Sorted_Less.sorted (rev (a # l1)); Sorted_Less.sorted (rev l2)\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (a # l1) l2)) \\<and> set (inter_sorted_rev (a # l1) l2) = set (a # l1) \\<inter> set l2\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>a l1 l2. \\<lbrakk>\\<And>l2. \\<lbrakk>Sorted_Less.sorted (rev l1); Sorted_Less.sorted (rev l2)\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev l1 l2)) \\<and> set (inter_sorted_rev l1 l2) = set l1 \\<inter> set l2; Sorted_Less.sorted (rev (a # l1)); Sorted_Less.sorted (rev l2)\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (a # l1) l2)) \\<and> set (inter_sorted_rev (a # l1) l2) = set (a # l1) \\<inter> set l2\n[PROOF STEP]\ncase (Cons x1 l1 l2)\n[PROOF STATE]\nproof (state)\nthis:\n\\<lbrakk>Sorted_Less.sorted (rev l1); Sorted_Less.sorted (rev ?l2.0)\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev l1 ?l2.0)) \\<and> set (inter_sorted_rev l1 ?l2.0) = set l1 \\<inter> set ?l2.0\nSorted_Less.sorted (rev (x1 # l1))\nSorted_Less.sorted (rev l2)\n\ngoal (1 subgoal):\n 1. \\<And>a l1 l2. \\<lbrakk>\\<And>l2. \\<lbrakk>Sorted_Less.sorted (rev l1); Sorted_Less.sorted (rev l2)\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev l1 l2)) \\<and> set (inter_sorted_rev l1 l2) = set l1 \\<inter> set l2; Sorted_Less.sorted (rev (a # l1)); Sorted_Less.sorted (rev l2)\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (a # l1) l2)) \\<and> set (inter_sorted_rev (a # l1) l2) = set (a # l1) \\<inter> set l2\n[PROOF STEP]\nnote x1_l1_props = Cons(2)\n[PROOF STATE]\nproof (state)\nthis:\nSorted_Less.sorted (rev (x1 # l1))\n\ngoal (1 subgoal):\n 1. \\<And>a l1 l2. \\<lbrakk>\\<And>l2. \\<lbrakk>Sorted_Less.sorted (rev l1); Sorted_Less.sorted (rev l2)\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev l1 l2)) \\<and> set (inter_sorted_rev l1 l2) = set l1 \\<inter> set l2; Sorted_Less.sorted (rev (a # l1)); Sorted_Less.sorted (rev l2)\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (a # l1) l2)) \\<and> set (inter_sorted_rev (a # l1) l2) = set (a # l1) \\<inter> set l2\n[PROOF STEP]\nnote l2_props = Cons(3)\n[PROOF STATE]\nproof (state)\nthis:\nSorted_Less.sorted (rev l2)\n\ngoal (1 subgoal):\n 1. \\<And>a l1 l2. \\<lbrakk>\\<And>l2. \\<lbrakk>Sorted_Less.sorted (rev l1); Sorted_Less.sorted (rev l2)\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev l1 l2)) \\<and> set (inter_sorted_rev l1 l2) = set l1 \\<inter> set l2; Sorted_Less.sorted (rev (a # l1)); Sorted_Less.sorted (rev l2)\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (a # l1) l2)) \\<and> set (inter_sorted_rev (a # l1) l2) = set (a # l1) \\<inter> set l2\n[PROOF STEP]\nfrom x1_l1_props\n[PROOF STATE]\nproof (chain)\npicking this:\nSorted_Less.sorted (rev (x1 # l1))\n[PROOF STEP]\nhave l1_props: \"sorted (rev l1)\"\n                    and x1_nin_l1: \"x1 \\<notin> set l1\"\n                    and x1_gt: \"\\<And>x. x \\<in> set l1 \\<Longrightarrow> x1 > x\"\n[PROOF STATE]\nproof (prove)\nusing this:\nSorted_Less.sorted (rev (x1 # l1))\n\ngoal (1 subgoal):\n 1. Sorted_Less.sorted (rev l1) &&& x1 \\<notin> set l1 &&& (\\<And>x. x \\<in> set l1 \\<Longrightarrow> x < x1)\n[PROOF STEP]\nby (auto simp add: Ball_def sorted_wrt_append)\n[PROOF STATE]\nproof (state)\nthis:\nSorted_Less.sorted (rev l1)\nx1 \\<notin> set l1\n?x \\<in> set l1 \\<Longrightarrow> ?x < x1\n\ngoal (1 subgoal):\n 1. \\<And>a l1 l2. \\<lbrakk>\\<And>l2. \\<lbrakk>Sorted_Less.sorted (rev l1); Sorted_Less.sorted (rev l2)\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev l1 l2)) \\<and> set (inter_sorted_rev l1 l2) = set l1 \\<inter> set l2; Sorted_Less.sorted (rev (a # l1)); Sorted_Less.sorted (rev l2)\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (a # l1) l2)) \\<and> set (inter_sorted_rev (a # l1) l2) = set (a # l1) \\<inter> set l2\n[PROOF STEP]\nnote ind_hyp_l1 = Cons(1)[OF l1_props]\n[PROOF STATE]\nproof (state)\nthis:\nSorted_Less.sorted (rev ?l2.0) \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev l1 ?l2.0)) \\<and> set (inter_sorted_rev l1 ?l2.0) = set l1 \\<inter> set ?l2.0\n\ngoal (1 subgoal):\n 1. \\<And>a l1 l2. \\<lbrakk>\\<And>l2. \\<lbrakk>Sorted_Less.sorted (rev l1); Sorted_Less.sorted (rev l2)\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev l1 l2)) \\<and> set (inter_sorted_rev l1 l2) = set l1 \\<inter> set l2; Sorted_Less.sorted (rev (a # l1)); Sorted_Less.sorted (rev l2)\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (a # l1) l2)) \\<and> set (inter_sorted_rev (a # l1) l2) = set (a # l1) \\<inter> set l2\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) l2)) \\<and> set (inter_sorted_rev (x1 # l1) l2) = set (x1 # l1) \\<inter> set l2\n[PROOF STEP]\nusing l2_props\n[PROOF STATE]\nproof (prove)\nusing this:\nSorted_Less.sorted (rev l2)\n\ngoal (1 subgoal):\n 1. Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) l2)) \\<and> set (inter_sorted_rev (x1 # l1) l2) = set (x1 # l1) \\<inter> set l2\n[PROOF STEP]\nproof (induct l2)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. Sorted_Less.sorted (rev []) \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) [])) \\<and> set (inter_sorted_rev (x1 # l1) []) = set (x1 # l1) \\<inter> set []\n 2. \\<And>a l2. \\<lbrakk>Sorted_Less.sorted (rev l2) \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) l2)) \\<and> set (inter_sorted_rev (x1 # l1) l2) = set (x1 # l1) \\<inter> set l2; Sorted_Less.sorted (rev (a # l2))\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (a # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (a # l2)) = set (x1 # l1) \\<inter> set (a # l2)\n[PROOF STEP]\ncase Nil\n[PROOF STATE]\nproof (state)\nthis:\nSorted_Less.sorted (rev [])\n\ngoal (2 subgoals):\n 1. Sorted_Less.sorted (rev []) \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) [])) \\<and> set (inter_sorted_rev (x1 # l1) []) = set (x1 # l1) \\<inter> set []\n 2. \\<And>a l2. \\<lbrakk>Sorted_Less.sorted (rev l2) \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) l2)) \\<and> set (inter_sorted_rev (x1 # l1) l2) = set (x1 # l1) \\<inter> set l2; Sorted_Less.sorted (rev (a # l2))\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (a # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (a # l2)) = set (x1 # l1) \\<inter> set (a # l2)\n[PROOF STEP]\nwith x1_l1_props\n[PROOF STATE]\nproof (chain)\npicking this:\nSorted_Less.sorted (rev (x1 # l1))\nSorted_Less.sorted (rev [])\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nSorted_Less.sorted (rev (x1 # l1))\nSorted_Less.sorted (rev [])\n\ngoal (1 subgoal):\n 1. Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) [])) \\<and> set (inter_sorted_rev (x1 # l1) []) = set (x1 # l1) \\<inter> set []\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nSorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) [])) \\<and> set (inter_sorted_rev (x1 # l1) []) = set (x1 # l1) \\<inter> set []\n\ngoal (1 subgoal):\n 1. \\<And>a l2. \\<lbrakk>Sorted_Less.sorted (rev l2) \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) l2)) \\<and> set (inter_sorted_rev (x1 # l1) l2) = set (x1 # l1) \\<inter> set l2; Sorted_Less.sorted (rev (a # l2))\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (a # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (a # l2)) = set (x1 # l1) \\<inter> set (a # l2)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>a l2. \\<lbrakk>Sorted_Less.sorted (rev l2) \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) l2)) \\<and> set (inter_sorted_rev (x1 # l1) l2) = set (x1 # l1) \\<inter> set l2; Sorted_Less.sorted (rev (a # l2))\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (a # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (a # l2)) = set (x1 # l1) \\<inter> set (a # l2)\n[PROOF STEP]\ncase (Cons x2 l2)\n[PROOF STATE]\nproof (state)\nthis:\nSorted_Less.sorted (rev l2) \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) l2)) \\<and> set (inter_sorted_rev (x1 # l1) l2) = set (x1 # l1) \\<inter> set l2\nSorted_Less.sorted (rev (x2 # l2))\n\ngoal (1 subgoal):\n 1. \\<And>a l2. \\<lbrakk>Sorted_Less.sorted (rev l2) \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) l2)) \\<and> set (inter_sorted_rev (x1 # l1) l2) = set (x1 # l1) \\<inter> set l2; Sorted_Less.sorted (rev (a # l2))\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (a # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (a # l2)) = set (x1 # l1) \\<inter> set (a # l2)\n[PROOF STEP]\nnote x2_l2_props = Cons(2)\n[PROOF STATE]\nproof (state)\nthis:\nSorted_Less.sorted (rev (x2 # l2))\n\ngoal (1 subgoal):\n 1. \\<And>a l2. \\<lbrakk>Sorted_Less.sorted (rev l2) \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) l2)) \\<and> set (inter_sorted_rev (x1 # l1) l2) = set (x1 # l1) \\<inter> set l2; Sorted_Less.sorted (rev (a # l2))\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (a # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (a # l2)) = set (x1 # l1) \\<inter> set (a # l2)\n[PROOF STEP]\n(* sorted (rev (x2 # l2))*)\n[PROOF STATE]\nproof (state)\nthis:\nSorted_Less.sorted (rev (x2 # l2))\n\ngoal (1 subgoal):\n 1. \\<And>a l2. \\<lbrakk>Sorted_Less.sorted (rev l2) \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) l2)) \\<and> set (inter_sorted_rev (x1 # l1) l2) = set (x1 # l1) \\<inter> set l2; Sorted_Less.sorted (rev (a # l2))\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (a # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (a # l2)) = set (x1 # l1) \\<inter> set (a # l2)\n[PROOF STEP]\nfrom x2_l2_props\n[PROOF STATE]\nproof (chain)\npicking this:\nSorted_Less.sorted (rev (x2 # l2))\n[PROOF STEP]\nhave l2_props: \"sorted (rev l2)\"\n                    and x2_nin_l2: \"x2 \\<notin> set l2\"\n                    and x2_gt: \"\\<And>x. x \\<in> set l2 \\<Longrightarrow> x2 > x\"\n[PROOF STATE]\nproof (prove)\nusing this:\nSorted_Less.sorted (rev (x2 # l2))\n\ngoal (1 subgoal):\n 1. Sorted_Less.sorted (rev l2) &&& x2 \\<notin> set l2 &&& (\\<And>x. x \\<in> set l2 \\<Longrightarrow> x < x2)\n[PROOF STEP]\nby (auto simp  add: Ball_def sorted_wrt_append )\n[PROOF STATE]\nproof (state)\nthis:\nSorted_Less.sorted (rev l2)\nx2 \\<notin> set l2\n?x \\<in> set l2 \\<Longrightarrow> ?x < x2\n\ngoal (1 subgoal):\n 1. \\<And>a l2. \\<lbrakk>Sorted_Less.sorted (rev l2) \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) l2)) \\<and> set (inter_sorted_rev (x1 # l1) l2) = set (x1 # l1) \\<inter> set l2; Sorted_Less.sorted (rev (a # l2))\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (a # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (a # l2)) = set (x1 # l1) \\<inter> set (a # l2)\n[PROOF STEP]\nnote ind_hyp_l2 = Cons(1)[OF l2_props]\n[PROOF STATE]\nproof (state)\nthis:\nSorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) l2)) \\<and> set (inter_sorted_rev (x1 # l1) l2) = set (x1 # l1) \\<inter> set l2\n\ngoal (1 subgoal):\n 1. \\<And>a l2. \\<lbrakk>Sorted_Less.sorted (rev l2) \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) l2)) \\<and> set (inter_sorted_rev (x1 # l1) l2) = set (x1 # l1) \\<inter> set l2; Sorted_Less.sorted (rev (a # l2))\\<rbrakk> \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (a # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (a # l2)) = set (x1 # l1) \\<inter> set (a # l2)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nproof (cases \"x1 > x2\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. x2 < x1 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n 2. \\<not> x2 < x1 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nx2 < x1\n\ngoal (2 subgoals):\n 1. x2 < x1 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n 2. \\<not> x2 < x1 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nnote x1_gt_x2 = this\n[PROOF STATE]\nproof (state)\nthis:\nx2 < x1\n\ngoal (2 subgoals):\n 1. x2 < x1 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n 2. \\<not> x2 < x1 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nhave \"set l1 \\<inter> set (x2 # l2) = set (x1 # l1)\\<inter> set (x2 # l2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set l1 \\<inter> set (x2 # l2) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nusing x1_gt_x2 x1_nin_l1 x2_nin_l2 x1_gt x2_gt\n[PROOF STATE]\nproof (prove)\nusing this:\nx2 < x1\nx1 \\<notin> set l1\nx2 \\<notin> set l2\n?x \\<in> set l1 \\<Longrightarrow> ?x < x1\n?x \\<in> set l2 \\<Longrightarrow> ?x < x2\n\ngoal (1 subgoal):\n 1. set l1 \\<inter> set (x2 # l2) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\nset l1 \\<inter> set (x2 # l2) = set (x1 # l1) \\<inter> set (x2 # l2)\n\ngoal (2 subgoals):\n 1. x2 < x1 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n 2. \\<not> x2 < x1 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nset l1 \\<inter> set (x2 # l2) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nset l1 \\<inter> set (x2 # l2) = set (x1 # l1) \\<inter> set (x2 # l2)\n\ngoal (1 subgoal):\n 1. Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nusing ind_hyp_l1[OF x2_l2_props]\n[PROOF STATE]\nproof (prove)\nusing this:\nset l1 \\<inter> set (x2 # l2) = set (x1 # l1) \\<inter> set (x2 # l2)\nSorted_Less.sorted (rev (inter_sorted_rev l1 (x2 # l2))) \\<and> set (inter_sorted_rev l1 (x2 # l2)) = set l1 \\<inter> set (x2 # l2)\n\ngoal (1 subgoal):\n 1. Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nusing x1_gt_x2 x1_nin_l1 x2_nin_l2 x1_gt x2_gt\n[PROOF STATE]\nproof (prove)\nusing this:\nset l1 \\<inter> set (x2 # l2) = set (x1 # l1) \\<inter> set (x2 # l2)\nSorted_Less.sorted (rev (inter_sorted_rev l1 (x2 # l2))) \\<and> set (inter_sorted_rev l1 (x2 # l2)) = set l1 \\<inter> set (x2 # l2)\nx2 < x1\nx1 \\<notin> set l1\nx2 \\<notin> set l2\n?x \\<in> set l1 \\<Longrightarrow> ?x < x1\n?x \\<in> set l2 \\<Longrightarrow> ?x < x2\n\ngoal (1 subgoal):\n 1. Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nby (auto simp add:Ball_def sorted_wrt_append)\n[PROOF STATE]\nproof (state)\nthis:\nSorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n\ngoal (1 subgoal):\n 1. \\<not> x2 < x1 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<not> x2 < x1 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\n\\<not> x2 < x1\n\ngoal (1 subgoal):\n 1. \\<not> x2 < x1 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nnote x2_ge_x1 = this\n[PROOF STATE]\nproof (state)\nthis:\n\\<not> x2 < x1\n\ngoal (1 subgoal):\n 1. \\<not> x2 < x1 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nproof (cases \"x1 = x2\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. x1 = x2 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n 2. x1 \\<noteq> x2 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nx1 = x2\n\ngoal (2 subgoals):\n 1. x1 = x2 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n 2. x1 \\<noteq> x2 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nnote x1_eq_x2 = this\n[PROOF STATE]\nproof (state)\nthis:\nx1 = x2\n\ngoal (2 subgoals):\n 1. x1 = x2 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n 2. x1 \\<noteq> x2 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx1 = x2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nx1 = x2\n\ngoal (1 subgoal):\n 1. Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nusing ind_hyp_l1[OF l2_props]\n[PROOF STATE]\nproof (prove)\nusing this:\nx1 = x2\nSorted_Less.sorted (rev (inter_sorted_rev l1 l2)) \\<and> set (inter_sorted_rev l1 l2) = set l1 \\<inter> set l2\n\ngoal (1 subgoal):\n 1. Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nusing x1_eq_x2  x1_nin_l1 x2_nin_l2 x1_gt x2_gt\n[PROOF STATE]\nproof (prove)\nusing this:\nx1 = x2\nSorted_Less.sorted (rev (inter_sorted_rev l1 l2)) \\<and> set (inter_sorted_rev l1 l2) = set l1 \\<inter> set l2\nx1 = x2\nx1 \\<notin> set l1\nx2 \\<notin> set l2\n?x \\<in> set l1 \\<Longrightarrow> ?x < x1\n?x \\<in> set l2 \\<Longrightarrow> ?x < x2\n\ngoal (1 subgoal):\n 1. Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nby (auto simp add:Ball_def sorted_wrt_append)\n[PROOF STATE]\nproof (state)\nthis:\nSorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n\ngoal (1 subgoal):\n 1. x1 \\<noteq> x2 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x1 \\<noteq> x2 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nx1 \\<noteq> x2\n\ngoal (1 subgoal):\n 1. x1 \\<noteq> x2 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nnote x1_neq_x2 = this\n[PROOF STATE]\nproof (state)\nthis:\nx1 \\<noteq> x2\n\ngoal (1 subgoal):\n 1. x1 \\<noteq> x2 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nwith x2_ge_x1\n[PROOF STATE]\nproof (chain)\npicking this:\n\\<not> x2 < x1\nx1 \\<noteq> x2\n[PROOF STEP]\nhave x2_gt_x1 : \"x2 > x1\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<not> x2 < x1\nx1 \\<noteq> x2\n\ngoal (1 subgoal):\n 1. x1 < x2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nx1 < x2\n\ngoal (1 subgoal):\n 1. x1 \\<noteq> x2 \\<Longrightarrow> Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nfrom ind_hyp_l2 x2_ge_x1 x1_neq_x2 x2_gt x2_nin_l2 x1_gt\n[PROOF STATE]\nproof (chain)\npicking this:\nSorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) l2)) \\<and> set (inter_sorted_rev (x1 # l1) l2) = set (x1 # l1) \\<inter> set l2\n\\<not> x2 < x1\nx1 \\<noteq> x2\n?x \\<in> set l2 \\<Longrightarrow> ?x < x2\nx2 \\<notin> set l2\n?x \\<in> set l1 \\<Longrightarrow> ?x < x1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nSorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) l2)) \\<and> set (inter_sorted_rev (x1 # l1) l2) = set (x1 # l1) \\<inter> set l2\n\\<not> x2 < x1\nx1 \\<noteq> x2\n?x \\<in> set l2 \\<Longrightarrow> ?x < x2\nx2 \\<notin> set l2\n?x \\<in> set l1 \\<Longrightarrow> ?x < x1\n\ngoal (1 subgoal):\n 1. Sorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nSorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nSorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nSorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) (x2 # l2))) \\<and> set (inter_sorted_rev (x1 # l1) (x2 # l2)) = set (x1 # l1) \\<inter> set (x2 # l2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nSorted_Less.sorted (rev (inter_sorted_rev (x1 # l1) l2)) \\<and> set (inter_sorted_rev (x1 # l1) l2) = set (x1 # l1) \\<inter> set l2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 11992, "file": "Dominance_CHK_Sorted_List_Operations2", "length": 65, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513842182775, "lm_q2_score": 0.84997116805678, "lm_q1q2_score": 0.7617878359165152}}
{"text": "[STATEMENT]\nlemma fixes p1 :: \"'a::comm_ring\" shows\n  \"(sq p1 + sq q1 + sq r1 + sq s1 + sq t1 + sq u1 + sq v1 + sq w1) * \n   (sq p2 + sq q2 + sq r2 + sq s2 + sq t2 + sq u2 + sq v2 + sq w2) \n    = sq (p1*p2 - q1*q2 - r1*r2 - s1*s2 - t1*t2 - u1*u2 - v1*v2 - w1*w2) + \n      sq (p1*q2 + q1*p2 + r1*s2 - s1*r2 + t1*u2 - u1*t2 - v1*w2 + w1*v2) +\n      sq (p1*r2 - q1*s2 + r1*p2 + s1*q2 + t1*v2 + u1*w2 - v1*t2 - w1*u2) +\n      sq (p1*s2 + q1*r2 - r1*q2 + s1*p2 + t1*w2 - u1*v2 + v1*u2 - w1*t2) +\n      sq (p1*t2 - q1*u2 - r1*v2 - s1*w2 + t1*p2 + u1*q2 + v1*r2 + w1*s2) +\n      sq (p1*u2 + q1*t2 - r1*w2 + s1*v2 - t1*q2 + u1*p2 - v1*s2 + w1*r2) +\n      sq (p1*v2 + q1*w2 + r1*t2 - s1*u2 - t1*r2 + u1*s2 + v1*p2 - w1*q2) +\n      sq (p1*w2 - q1*v2 + r1*u2 + s1*t2 - t1*s2 - u1*r2 + v1*q2 + w1*p2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (sq p1 + sq q1 + sq r1 + sq s1 + sq t1 + sq u1 + sq v1 + sq w1) * (sq p2 + sq q2 + sq r2 + sq s2 + sq t2 + sq u2 + sq v2 + sq w2) = sq (p1 * p2 - q1 * q2 - r1 * r2 - s1 * s2 - t1 * t2 - u1 * u2 - v1 * v2 - w1 * w2) + sq (p1 * q2 + q1 * p2 + r1 * s2 - s1 * r2 + t1 * u2 - u1 * t2 - v1 * w2 + w1 * v2) + sq (p1 * r2 - q1 * s2 + r1 * p2 + s1 * q2 + t1 * v2 + u1 * w2 - v1 * t2 - w1 * u2) + sq (p1 * s2 + q1 * r2 - r1 * q2 + s1 * p2 + t1 * w2 - u1 * v2 + v1 * u2 - w1 * t2) + sq (p1 * t2 - q1 * u2 - r1 * v2 - s1 * w2 + t1 * p2 + u1 * q2 + v1 * r2 + w1 * s2) + sq (p1 * u2 + q1 * t2 - r1 * w2 + s1 * v2 - t1 * q2 + u1 * p2 - v1 * s2 + w1 * r2) + sq (p1 * v2 + q1 * w2 + r1 * t2 - s1 * u2 - t1 * r2 + u1 * s2 + v1 * p2 - w1 * q2) + sq (p1 * w2 - q1 * v2 + r1 * u2 + s1 * t2 - t1 * s2 - u1 * r2 + v1 * q2 + w1 * p2)\n[PROOF STEP]\nby (simp only: sq_def algebra_simps)", "meta": {"llama_tokens": 1037, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9263037323284109, "lm_q2_score": 0.8221891305219504, "lm_q1q2_score": 0.7615968602823336}}
{"text": "[STATEMENT]\nlemma prod_not_prime: \n  assumes \"prime (x::nat)\" \n    and \"prime y\" \n    and \"x > 2\" \n    and \"y > 2\" \n  shows \"\\<not> prime ((x-1)*(y-1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<not> prime ((x - 1) * (y - 1))\n[PROOF STEP]\nby (metis assms One_nat_def Suc_diff_1 nat_neq_iff numeral_2_eq_2 prime_gt_0_nat prime_product)", "meta": {"llama_tokens": 161, "file": "Sigma_Commit_Crypto_Number_Theory_Aux", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9207896758909756, "lm_q2_score": 0.8267117962054049, "lm_q1q2_score": 0.7612276868832211}}
{"text": "[STATEMENT]\nlemma prod_not_prime: \n  assumes \"prime (x::nat)\" \n    and \"prime y\" \n    and \"x > 2\" \n    and \"y > 2\" \n  shows \"\\<not> prime ((x-1)*(y-1))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<not> prime ((x - 1) * (y - 1))\n[PROOF STEP]\nby (metis assms One_nat_def Suc_diff_1 nat_neq_iff numeral_2_eq_2 prime_gt_0_nat prime_product)", "meta": {"llama_tokens": 161, "file": "Multi_Party_Computation_Number_Theory_Aux", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9207896758909756, "lm_q2_score": 0.8267117940706735, "lm_q1q2_score": 0.7612276849175824}}
{"text": "[STATEMENT]\nlemma norm_pos:\n  \"\\<parallel>v\\<parallel> \\<ge> 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\<le> \\<parallel>v\\<parallel>\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 0 \\<le> \\<parallel>v\\<parallel>\n[PROOF STEP]\nhave \"\\<forall>j. v\\<^bsub>j\\<^esub>^2 \\<ge> 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<forall>j. 0 \\<le> (v\\<^bsub>j\\<^esub>)\\<^sup>2\n[PROOF STEP]\nunfolding ith_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<forall>j. 0 \\<le> (fst v j)\\<^sup>2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\<forall>j. 0 \\<le> (v\\<^bsub>j\\<^esub>)\\<^sup>2\n\ngoal (1 subgoal):\n 1. 0 \\<le> \\<parallel>v\\<parallel>\n[PROOF STEP]\nhave \"(\\<Sum>j\\<in>{1..(vlen v)}. v\\<^bsub>j\\<^esub>^2) \\<ge> 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\<le> (\\<Sum>j = 1..vlen v. (v\\<^bsub>j\\<^esub>)\\<^sup>2)\n[PROOF STEP]\nby (simp add: sum_nonneg)\n[PROOF STATE]\nproof (state)\nthis:\n0 \\<le> (\\<Sum>j = 1..vlen v. (v\\<^bsub>j\\<^esub>)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. 0 \\<le> \\<parallel>v\\<parallel>\n[PROOF STEP]\nwith real_sqrt_ge_zero\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\<le> ?x \\<Longrightarrow> 0 \\<le> sqrt ?x\n0 \\<le> (\\<Sum>j = 1..vlen v. (v\\<^bsub>j\\<^esub>)\\<^sup>2)\n[PROOF STEP]\nhave \"sqrt (\\<Sum>j\\<in>{1..(vlen v)}. v\\<^bsub>j\\<^esub>^2) \\<ge> 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\<le> ?x \\<Longrightarrow> 0 \\<le> sqrt ?x\n0 \\<le> (\\<Sum>j = 1..vlen v. (v\\<^bsub>j\\<^esub>)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. 0 \\<le> sqrt (\\<Sum>j = 1..vlen v. (v\\<^bsub>j\\<^esub>)\\<^sup>2)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n0 \\<le> sqrt (\\<Sum>j = 1..vlen v. (v\\<^bsub>j\\<^esub>)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. 0 \\<le> \\<parallel>v\\<parallel>\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\<le> sqrt (\\<Sum>j = 1..vlen v. (v\\<^bsub>j\\<^esub>)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. 0 \\<le> \\<parallel>v\\<parallel>\n[PROOF STEP]\nunfolding norm_def\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\<le> sqrt (\\<Sum>j = 1..vlen v. (v\\<^bsub>j\\<^esub>)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. 0 \\<le> sqrt (\\<Sum>j = 1..vlen v. (v\\<^bsub>j\\<^esub>)\\<^sup>2)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n0 \\<le> \\<parallel>v\\<parallel>\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1134, "file": "Cauchy_CauchySchwarz", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933094032139576, "lm_q2_score": 0.8519528076067262, "lm_q1q2_score": 0.7610574541296202}}
{"text": "[STATEMENT]\nlemma inner_prod_of_unit_vec:\n  fixes n i:: nat\n  assumes \"i < n\"\n  shows \"\\<langle>unit_vec n i| unit_vec n i\\<rangle> = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<langle>unit_vec n i|unit_vec n i\\<rangle> = 1\n[PROOF STEP]\nby (auto simp add: inner_prod_def unit_vec_def)\n    (simp add: assms sum.cong[of \"{0..<n}\" \"{0..<n}\"\n        \"\\<lambda>j. cnj (if j = i then 1 else 0) * (if j = i then 1 else 0)\" \"\\<lambda>j. (if j = i then 1 else 0)\"])", "meta": {"llama_tokens": 203, "file": "Isabelle_Marries_Dirac_No_Cloning", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.89330940889474, "lm_q2_score": 0.8519527982093666, "lm_q1q2_score": 0.761057450574629}}
{"text": "[STATEMENT]\nlemma horner_sum_eq_sum:\n  \\<open>horner_sum f a xs = (\\<Sum>n = 0..<length xs. f (xs ! n) * a ^ n)\\<close>\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. horner_sum f a xs = (\\<Sum>n = 0..<length xs. f (xs ! n) * a ^ n)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. horner_sum f a xs = (\\<Sum>n = 0..<length xs. f (xs ! n) * a ^ n)\n[PROOF STEP]\nhave \\<open>(*) a ^^ n = (*) (a ^ n)\\<close> for n\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (*) a ^^ n = (*) (a ^ n)\n[PROOF STEP]\nby (induction n) (simp_all add: ac_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(*) a ^^ ?n1 = (*) (a ^ ?n1)\n\ngoal (1 subgoal):\n 1. horner_sum f a xs = (\\<Sum>n = 0..<length xs. f (xs ! n) * a ^ n)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(*) a ^^ ?n1 = (*) (a ^ ?n1)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(*) a ^^ ?n1 = (*) (a ^ ?n1)\n\ngoal (1 subgoal):\n 1. horner_sum f a xs = (\\<Sum>n = 0..<length xs. f (xs ! n) * a ^ n)\n[PROOF STEP]\nby (simp add: horner_sum_eq_sum_funpow ac_simps)\n[PROOF STATE]\nproof (state)\nthis:\nhorner_sum f a xs = (\\<Sum>n = 0..<length xs. f (xs ! n) * a ^ n)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 578, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045996818986, "lm_q2_score": 0.8577681122619883, "lm_q1q2_score": 0.7610158146592952}}
{"text": "[STATEMENT]\nlemma real_of_int_div_aux:\n  \"(real_of_int x) / (real_of_int d) =\n    real_of_int (x div d) + (real_of_int (x mod d)) / (real_of_int d)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real_of_int x / real_of_int d = real_of_int (x div d) + real_of_int (x mod d) / real_of_int d\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. real_of_int x / real_of_int d = real_of_int (x div d) + real_of_int (x mod d) / real_of_int d\n[PROOF STEP]\nhave \"x = (x div d) * d + x mod d\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x = x div d * d + x mod d\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nx = x div d * d + x mod d\n\ngoal (1 subgoal):\n 1. real_of_int x / real_of_int d = real_of_int (x div d) + real_of_int (x mod d) / real_of_int d\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nx = x div d * d + x mod d\n[PROOF STEP]\nhave \"real_of_int x = real_of_int (x div d) * real_of_int d + real_of_int(x mod d)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx = x div d * d + x mod d\n\ngoal (1 subgoal):\n 1. real_of_int x = real_of_int (x div d) * real_of_int d + real_of_int (x mod d)\n[PROOF STEP]\nby (metis of_int_add of_int_mult)\n[PROOF STATE]\nproof (state)\nthis:\nreal_of_int x = real_of_int (x div d) * real_of_int d + real_of_int (x mod d)\n\ngoal (1 subgoal):\n 1. real_of_int x / real_of_int d = real_of_int (x div d) + real_of_int (x mod d) / real_of_int d\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nreal_of_int x = real_of_int (x div d) * real_of_int d + real_of_int (x mod d)\n[PROOF STEP]\nhave \"real_of_int x / real_of_int d = \\<dots> / real_of_int d\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal_of_int x = real_of_int (x div d) * real_of_int d + real_of_int (x mod d)\n\ngoal (1 subgoal):\n 1. real_of_int x / real_of_int d = (real_of_int (x div d) * real_of_int d + real_of_int (x mod d)) / real_of_int d\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nreal_of_int x / real_of_int d = (real_of_int (x div d) * real_of_int d + real_of_int (x mod d)) / real_of_int d\n\ngoal (1 subgoal):\n 1. real_of_int x / real_of_int d = real_of_int (x div d) + real_of_int (x mod d) / real_of_int d\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nreal_of_int x / real_of_int d = (real_of_int (x div d) * real_of_int d + real_of_int (x mod d)) / real_of_int d\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nreal_of_int x / real_of_int d = (real_of_int (x div d) * real_of_int d + real_of_int (x mod d)) / real_of_int d\n\ngoal (1 subgoal):\n 1. real_of_int x / real_of_int d = real_of_int (x div d) + real_of_int (x mod d) / real_of_int d\n[PROOF STEP]\nby (auto simp add: add_divide_distrib algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\nreal_of_int x / real_of_int d = real_of_int (x div d) + real_of_int (x mod d) / real_of_int d\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1316, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045966995027, "lm_q2_score": 0.8577681031721325, "lm_q1q2_score": 0.7610158040365292}}
{"text": "[STATEMENT]\nlemma lin_indpt_cols_imp_det_not_0:\n  fixes A::\"'a mat\"\n  assumes A: \"A \\<in> carrier_mat n n\" and li: \"lin_indpt (set (cols A))\" and d: \"distinct (cols A)\" \n  shows \"det A \\<noteq> 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. det A \\<noteq> (0::'a)\n[PROOF STEP]\nusing A li d det_rank_iff lin_indpt_full_rank\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\<in> carrier_mat n n\nlin_indpt (set (cols A))\ndistinct (cols A)\n?A \\<in> carrier_mat n n \\<Longrightarrow> (det ?A \\<noteq> (0::'a)) = (local.rank ?A = n)\n\\<lbrakk>?A \\<in> carrier_mat n ?nc; distinct (cols ?A); lin_indpt (set (cols ?A))\\<rbrakk> \\<Longrightarrow> local.rank ?A = ?nc\n\ngoal (1 subgoal):\n 1. det A \\<noteq> (0::'a)\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 322, "file": "Modular_arithmetic_LLL_and_HNF_algorithms_HNF_Mod_Det_Soundness", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.933430805473952, "lm_q2_score": 0.8152324826183822, "lm_q1q2_score": 0.7609631128990061}}
{"text": "[STATEMENT]\nlemma set_to_nat_mono: \"\\<lbrakk> finite B; A \\<subseteq> B\\<rbrakk> \\<Longrightarrow> set_to_nat A \\<le> set_to_nat B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>finite B; A \\<subseteq> B\\<rbrakk> \\<Longrightarrow> set_to_nat A \\<le> set_to_nat B\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<lbrakk>finite B; A \\<subseteq> B\\<rbrakk> \\<Longrightarrow> set_to_nat A \\<le> set_to_nat B\n[PROOF STEP]\nassume b_finite: \"finite B\"\n[PROOF STATE]\nproof (state)\nthis:\nfinite B\n\ngoal (1 subgoal):\n 1. \\<lbrakk>finite B; A \\<subseteq> B\\<rbrakk> \\<Longrightarrow> set_to_nat A \\<le> set_to_nat B\n[PROOF STEP]\nassume a_le_b: \"A \\<subseteq> B\"\n[PROOF STATE]\nproof (state)\nthis:\nA \\<subseteq> B\n\ngoal (1 subgoal):\n 1. \\<lbrakk>finite B; A \\<subseteq> B\\<rbrakk> \\<Longrightarrow> set_to_nat A \\<le> set_to_nat B\n[PROOF STEP]\nlet ?f = \"\\<lambda> (x::nat). (2::nat) ^ x\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<lbrakk>finite B; A \\<subseteq> B\\<rbrakk> \\<Longrightarrow> set_to_nat A \\<le> set_to_nat B\n[PROOF STEP]\nhave S1: \"set_to_nat A = sum ?f A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set_to_nat A = sum ((^) 2) A\n[PROOF STEP]\nby (simp add: set_to_nat_def)\n[PROOF STATE]\nproof (state)\nthis:\nset_to_nat A = sum ((^) 2) A\n\ngoal (1 subgoal):\n 1. \\<lbrakk>finite B; A \\<subseteq> B\\<rbrakk> \\<Longrightarrow> set_to_nat A \\<le> set_to_nat B\n[PROOF STEP]\nhave S2: \"set_to_nat B = sum ?f B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set_to_nat B = sum ((^) 2) B\n[PROOF STEP]\nby (simp add: set_to_nat_def)\n[PROOF STATE]\nproof (state)\nthis:\nset_to_nat B = sum ((^) 2) B\n\ngoal (1 subgoal):\n 1. \\<lbrakk>finite B; A \\<subseteq> B\\<rbrakk> \\<Longrightarrow> set_to_nat A \\<le> set_to_nat B\n[PROOF STEP]\nhave S3: \"\\<And> x. x  \\<in> B - A \\<Longrightarrow> 0 \\<le> ?f x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>x. x \\<in> B - A \\<Longrightarrow> 0 \\<le> 2 ^ x\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n?x \\<in> B - A \\<Longrightarrow> 0 \\<le> 2 ^ ?x\n\ngoal (1 subgoal):\n 1. \\<lbrakk>finite B; A \\<subseteq> B\\<rbrakk> \\<Longrightarrow> set_to_nat A \\<le> set_to_nat B\n[PROOF STEP]\nfrom b_finite a_le_b S3\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite B\nA \\<subseteq> B\n?x \\<in> B - A \\<Longrightarrow> 0 \\<le> 2 ^ ?x\n[PROOF STEP]\nhave \"sum ?f A \\<le> sum ?f B\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite B\nA \\<subseteq> B\n?x \\<in> B - A \\<Longrightarrow> 0 \\<le> 2 ^ ?x\n\ngoal (1 subgoal):\n 1. sum ((^) 2) A \\<le> sum ((^) 2) B\n[PROOF STEP]\nby (rule sum_mono2)\n[PROOF STATE]\nproof (state)\nthis:\nsum ((^) 2) A \\<le> sum ((^) 2) B\n\ngoal (1 subgoal):\n 1. \\<lbrakk>finite B; A \\<subseteq> B\\<rbrakk> \\<Longrightarrow> set_to_nat A \\<le> set_to_nat B\n[PROOF STEP]\nwith S1 S2\n[PROOF STATE]\nproof (chain)\npicking this:\nset_to_nat A = sum ((^) 2) A\nset_to_nat B = sum ((^) 2) B\nsum ((^) 2) A \\<le> sum ((^) 2) B\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nset_to_nat A = sum ((^) 2) A\nset_to_nat B = sum ((^) 2) B\nsum ((^) 2) A \\<le> sum ((^) 2) B\n\ngoal (1 subgoal):\n 1. set_to_nat A \\<le> set_to_nat B\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nset_to_nat A \\<le> set_to_nat B\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1487, "file": "Recursion-Theory-I_PRecFinSet", "length": 17, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767746654976, "lm_q2_score": 0.8670357683915538, "lm_q1q2_score": 0.760803749567842}}
{"text": "[STATEMENT]\ntheorem simplify_sum_of_powers: \"(x - 1::nat) * (\\<Sum>i=0 .. n . x^i)  = x^(n + 1) - 1\" (is \"?l = ?r\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x - 1) * sum ((^) x) {0..n} = x ^ (n + 1) - 1\n[PROOF STEP]\nproof (cases)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. ?P \\<Longrightarrow> (x - 1) * sum ((^) x) {0..n} = x ^ (n + 1) - 1\n 2. \\<not> ?P \\<Longrightarrow> (x - 1) * sum ((^) x) {0..n} = x ^ (n + 1) - 1\n[PROOF STEP]\nassume \"n = 0\"\n[PROOF STATE]\nproof (state)\nthis:\nn = 0\n\ngoal (2 subgoals):\n 1. ?P \\<Longrightarrow> (x - 1) * sum ((^) x) {0..n} = x ^ (n + 1) - 1\n 2. \\<not> ?P \\<Longrightarrow> (x - 1) * sum ((^) x) {0..n} = x ^ (n + 1) - 1\n[PROOF STEP]\nthus \"?l = x^(n+1) - 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\nn = 0\n\ngoal (1 subgoal):\n 1. (x - 1) * sum ((^) x) {0..n} = x ^ (n + 1) - 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(x - 1) * sum ((^) x) {0..n} = x ^ (n + 1) - 1\n\ngoal (1 subgoal):\n 1. n \\<noteq> 0 \\<Longrightarrow> (x - 1) * sum ((^) x) {0..n} = x ^ (n + 1) - 1\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. n \\<noteq> 0 \\<Longrightarrow> (x - 1) * sum ((^) x) {0..n} = x ^ (n + 1) - 1\n[PROOF STEP]\nassume \"n\\<noteq>0\"\n[PROOF STATE]\nproof (state)\nthis:\nn \\<noteq> 0\n\ngoal (1 subgoal):\n 1. n \\<noteq> 0 \\<Longrightarrow> (x - 1) * sum ((^) x) {0..n} = x ^ (n + 1) - 1\n[PROOF STEP]\nhence n0: \"n>0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nn \\<noteq> 0\n\ngoal (1 subgoal):\n 1. 0 < n\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 < n\n\ngoal (1 subgoal):\n 1. n \\<noteq> 0 \\<Longrightarrow> (x - 1) * sum ((^) x) {0..n} = x ^ (n + 1) - 1\n[PROOF STEP]\nhave \"?l  = (x::nat)*(\\<Sum>i=0 .. n . x^i) - (\\<Sum>i=0 .. n . x^i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x - 1) * sum ((^) x) {0..n} = x * sum ((^) x) {0..n} - sum ((^) x) {0..n}\n[PROOF STEP]\nby (metis diff_mult_distrib nat_mult_1)\n[PROOF STATE]\nproof (state)\nthis:\n(x - 1) * sum ((^) x) {0..n} = x * sum ((^) x) {0..n} - sum ((^) x) {0..n}\n\ngoal (1 subgoal):\n 1. n \\<noteq> 0 \\<Longrightarrow> (x - 1) * sum ((^) x) {0..n} = x ^ (n + 1) - 1\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(x - 1) * sum ((^) x) {0..n} = x * sum ((^) x) {0..n} - sum ((^) x) {0..n}\n\ngoal (1 subgoal):\n 1. n \\<noteq> 0 \\<Longrightarrow> (x - 1) * sum ((^) x) {0..n} = x ^ (n + 1) - 1\n[PROOF STEP]\nhave \"... = (\\<Sum>i=0 .. n . x^(Suc i))    - (\\<Sum>i=0 .. n . x^i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x * sum ((^) x) {0..n} - sum ((^) x) {0..n} = (\\<Sum>i = 0..n. x ^ Suc i) - sum ((^) x) {0..n}\n[PROOF STEP]\nby (simp add: sum_distrib_left)\n[PROOF STATE]\nproof (state)\nthis:\nx * sum ((^) x) {0..n} - sum ((^) x) {0..n} = (\\<Sum>i = 0..n. x ^ Suc i) - sum ((^) x) {0..n}\n\ngoal (1 subgoal):\n 1. n \\<noteq> 0 \\<Longrightarrow> (x - 1) * sum ((^) x) {0..n} = x ^ (n + 1) - 1\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nx * sum ((^) x) {0..n} - sum ((^) x) {0..n} = (\\<Sum>i = 0..n. x ^ Suc i) - sum ((^) x) {0..n}\n\ngoal (1 subgoal):\n 1. n \\<noteq> 0 \\<Longrightarrow> (x - 1) * sum ((^) x) {0..n} = x ^ (n + 1) - 1\n[PROOF STEP]\nhave \"... = (\\<Sum>i=Suc 0 .. Suc n . x^i)  - (\\<Sum>i=0 .. n . x^i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..n. x ^ Suc i) - sum ((^) x) {0..n} = sum ((^) x) {Suc 0..Suc n} - sum ((^) x) {0..n}\n[PROOF STEP]\nby (metis sum.shift_bounds_cl_Suc_ivl)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>i = 0..n. x ^ Suc i) - sum ((^) x) {0..n} = sum ((^) x) {Suc 0..Suc n} - sum ((^) x) {0..n}\n\ngoal (1 subgoal):\n 1. n \\<noteq> 0 \\<Longrightarrow> (x - 1) * sum ((^) x) {0..n} = x ^ (n + 1) - 1\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>i = 0..n. x ^ Suc i) - sum ((^) x) {0..n} = sum ((^) x) {Suc 0..Suc n} - sum ((^) x) {0..n}\n\ngoal (1 subgoal):\n 1. n \\<noteq> 0 \\<Longrightarrow> (x - 1) * sum ((^) x) {0..n} = x ^ (n + 1) - 1\n[PROOF STEP]\nhave \"... = ((\\<Sum>i=Suc 0 .. n. x^i)+x^(Suc n)) - (x^0 + (\\<Sum>i=Suc 0 .. n. x^i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum ((^) x) {Suc 0..Suc n} - sum ((^) x) {0..n} = sum ((^) x) {Suc 0..n} + x ^ Suc n - (x ^ 0 + sum ((^) x) {Suc 0..n})\n[PROOF STEP]\nby (simp add: sum.union_disjoint diff_add_inverse sum.atLeast_Suc_atMost)\n[PROOF STATE]\nproof (state)\nthis:\nsum ((^) x) {Suc 0..Suc n} - sum ((^) x) {0..n} = sum ((^) x) {Suc 0..n} + x ^ Suc n - (x ^ 0 + sum ((^) x) {Suc 0..n})\n\ngoal (1 subgoal):\n 1. n \\<noteq> 0 \\<Longrightarrow> (x - 1) * sum ((^) x) {0..n} = x ^ (n + 1) - 1\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(x - 1) * sum ((^) x) {0..n} = sum ((^) x) {Suc 0..n} + x ^ Suc n - (x ^ 0 + sum ((^) x) {Suc 0..n})\n[PROOF STEP]\nshow \"?thesis\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(x - 1) * sum ((^) x) {0..n} = sum ((^) x) {Suc 0..n} + x ^ Suc n - (x ^ 0 + sum ((^) x) {Suc 0..n})\n\ngoal (1 subgoal):\n 1. (x - 1) * sum ((^) x) {0..n} = x ^ (n + 1) - 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n(x - 1) * sum ((^) x) {0..n} = x ^ (n + 1) - 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2677, "file": "Perfect-Number-Thm_PerfectBasics", "length": 23, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9059898203834278, "lm_q2_score": 0.8397339716830606, "lm_q1q2_score": 0.7607904301749986}}
{"text": "[STATEMENT]\nlemma integrable_I: \n  \"(\\<lambda>x. x powr (of_nat n - 1/2) * sqrt (4 - x)) integrable_on {0..4}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<lambda>x. x powr (real n - 1 / 2) * sqrt (4 - x)) integrable_on {0..4}\n[PROOF STEP]\nproof (cases \"n = 0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. n = 0 \\<Longrightarrow> (\\<lambda>x. x powr (real n - 1 / 2) * sqrt (4 - x)) integrable_on {0..4}\n 2. n \\<noteq> 0 \\<Longrightarrow> (\\<lambda>x. x powr (real n - 1 / 2) * sqrt (4 - x)) integrable_on {0..4}\n[PROOF STEP]\ncase True\n[PROOF STATE]\nproof (state)\nthis:\nn = 0\n\ngoal (2 subgoals):\n 1. n = 0 \\<Longrightarrow> (\\<lambda>x. x powr (real n - 1 / 2) * sqrt (4 - x)) integrable_on {0..4}\n 2. n \\<noteq> 0 \\<Longrightarrow> (\\<lambda>x. x powr (real n - 1 / 2) * sqrt (4 - x)) integrable_on {0..4}\n[PROOF STEP]\nwith has_integral_I0\n[PROOF STATE]\nproof (chain)\npicking this:\n((\\<lambda>x. x powr - (1 / 2) * sqrt (4 - x)) has_integral 2 * pi) {0..4}\nn = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n((\\<lambda>x. x powr - (1 / 2) * sqrt (4 - x)) has_integral 2 * pi) {0..4}\nn = 0\n\ngoal (1 subgoal):\n 1. (\\<lambda>x. x powr (real n - 1 / 2) * sqrt (4 - x)) integrable_on {0..4}\n[PROOF STEP]\nby (simp add: has_integral_integrable)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>x. x powr (real n - 1 / 2) * sqrt (4 - x)) integrable_on {0..4}\n\ngoal (1 subgoal):\n 1. n \\<noteq> 0 \\<Longrightarrow> (\\<lambda>x. x powr (real n - 1 / 2) * sqrt (4 - x)) integrable_on {0..4}\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. n \\<noteq> 0 \\<Longrightarrow> (\\<lambda>x. x powr (real n - 1 / 2) * sqrt (4 - x)) integrable_on {0..4}\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nn \\<noteq> 0\n\ngoal (1 subgoal):\n 1. n \\<noteq> 0 \\<Longrightarrow> (\\<lambda>x. x powr (real n - 1 / 2) * sqrt (4 - x)) integrable_on {0..4}\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nn \\<noteq> 0\n\ngoal (1 subgoal):\n 1. (\\<lambda>x. x powr (real n - 1 / 2) * sqrt (4 - x)) integrable_on {0..4}\n[PROOF STEP]\nby (intro integrable_continuous_real continuous_on_mult continuous_on_powr')\n                  (auto intro!: continuous_intros)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>x. x powr (real n - 1 / 2) * sqrt (4 - x)) integrable_on {0..4}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1082, "file": "Catalan_Numbers_Catalan_Numbers", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8947894745194281, "lm_q2_score": 0.8499711737573762, "lm_q1q2_score": 0.7605452599230242}}
{"text": "[STATEMENT]\nlemma frequently_eventually_at_top:\n  fixes P Q::\"'a::linorder \\<Rightarrow> bool\"\n  assumes \"frequently P at_top\" \"eventually Q at_top\"\n  shows \"frequently (\\<lambda>x. P x \\<and> (\\<forall>y\\<ge>x. Q y) ) at_top\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<exists>\\<^sub>F x in at_top. P x \\<and> (\\<forall>y\\<ge>x. Q y)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfrequently P at_top\neventually Q at_top\n\ngoal (1 subgoal):\n 1. \\<exists>\\<^sub>F x in at_top. P x \\<and> (\\<forall>y\\<ge>x. Q y)\n[PROOF STEP]\nunfolding frequently_def eventually_at_top_linorder\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<nexists>N. \\<forall>n\\<ge>N. \\<not> P n\n\\<exists>N. \\<forall>n\\<ge>N. Q n\n\ngoal (1 subgoal):\n 1. \\<nexists>N. \\<forall>n\\<ge>N. \\<not> (P n \\<and> (\\<forall>y\\<ge>n. Q y))\n[PROOF STEP]\nby (metis (mono_tags, opaque_lifting) le_cases order_trans)", "meta": {"llama_tokens": 382, "file": "Irrational_Series_Erdos_Straus_Irrational_Series_Erdos_Straus", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.91243616285804, "lm_q2_score": 0.8333245911726382, "lm_q1q2_score": 0.7603554923848069}}
{"text": "[STATEMENT]\nlemma infsetsum_diff:\n  assumes \"f abs_summable_on A\" and \"g abs_summable_on A\"\n  shows   \"infsetsum (\\<lambda>x. f x - g x) A = infsetsum f A - infsetsum g A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>\\<^sub>ax\\<in>A. f x - g x) = infsetsum f A - infsetsum g A\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nf abs_summable_on A\ng abs_summable_on A\n\ngoal (1 subgoal):\n 1. (\\<Sum>\\<^sub>ax\\<in>A. f x - g x) = infsetsum f A - infsetsum g A\n[PROOF STEP]\nunfolding infsetsum_def abs_summable_on_def\n[PROOF STATE]\nproof (prove)\nusing this:\nintegrable (count_space A) f\nintegrable (count_space A) g\n\ngoal (1 subgoal):\n 1. LINT x|count_space A. f x - g x = integral\\<^sup>L (count_space A) f - integral\\<^sup>L (count_space A) g\n[PROOF STEP]\nby (rule Bochner_Integration.integral_diff)", "meta": {"llama_tokens": 355, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9196425267730008, "lm_q2_score": 0.8267117898012104, "lm_q1q2_score": 0.760279319285815}}
{"text": "[STATEMENT]\nlemma convex_differences:\n  assumes \"convex S\" \"convex T\"\n  shows \"convex (\\<Union>x\\<in> S. \\<Union>y \\<in> T. {x - y})\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. convex (\\<Union>x\\<in>S. \\<Union>y\\<in>T. {x - y})\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. convex (\\<Union>x\\<in>S. \\<Union>y\\<in>T. {x - y})\n[PROOF STEP]\nhave \"{x - y| x y. x \\<in> S \\<and> y \\<in> T} = {x + y |x y. x \\<in> S \\<and> y \\<in> uminus ` T}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {x - y |x y. x \\<in> S \\<and> y \\<in> T} = {x + y |x y. x \\<in> S \\<and> y \\<in> uminus ` T}\n[PROOF STEP]\nby (auto simp: diff_conv_add_uminus simp del: add_uminus_conv_diff)\n[PROOF STATE]\nproof (state)\nthis:\n{x - y |x y. x \\<in> S \\<and> y \\<in> T} = {x + y |x y. x \\<in> S \\<and> y \\<in> uminus ` T}\n\ngoal (1 subgoal):\n 1. convex (\\<Union>x\\<in>S. \\<Union>y\\<in>T. {x - y})\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n{x - y |x y. x \\<in> S \\<and> y \\<in> T} = {x + y |x y. x \\<in> S \\<and> y \\<in> uminus ` T}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n{x - y |x y. x \\<in> S \\<and> y \\<in> T} = {x + y |x y. x \\<in> S \\<and> y \\<in> uminus ` T}\n\ngoal (1 subgoal):\n 1. convex (\\<Union>x\\<in>S. \\<Union>y\\<in>T. {x - y})\n[PROOF STEP]\nusing convex_sums[OF assms(1) convex_negations[OF assms(2)]]\n[PROOF STATE]\nproof (prove)\nusing this:\n{x - y |x y. x \\<in> S \\<and> y \\<in> T} = {x + y |x y. x \\<in> S \\<and> y \\<in> uminus ` T}\nconvex (\\<Union>x\\<in>S. \\<Union>y\\<in>uminus ` T. {x + y})\n\ngoal (1 subgoal):\n 1. convex (\\<Union>x\\<in>S. \\<Union>y\\<in>T. {x - y})\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nconvex (\\<Union>x\\<in>S. \\<Union>y\\<in>T. {x - y})\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 852, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278664544911, "lm_q2_score": 0.8615382129861583, "lm_q1q2_score": 0.7602453271543906}}
{"text": "[STATEMENT]\nlemma lagrange_basis_poly_0: assumes \"x' \\<in> set (map fst xs_ys)\" and \"x' \\<noteq> x\" \n  shows \"poly (lagrange_basis_poly (map fst xs_ys) x) x' = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poly (lagrange_basis_poly (map fst xs_ys) x) x' = (0::'a)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. poly (lagrange_basis_poly (map fst xs_ys) x) x' = (0::'a)\n[PROOF STEP]\nlet ?f = \"\\<lambda>xi. smult (inverse (x - xi)) [:- xi, 1:]\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. poly (lagrange_basis_poly (map fst xs_ys) x) x' = (0::'a)\n[PROOF STEP]\nlet ?xs = \"filter (\\<lambda>c. c\\<noteq>x) (map fst xs_ys)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. poly (lagrange_basis_poly (map fst xs_ys) x) x' = (0::'a)\n[PROOF STEP]\nhave mem: \"?f x' \\<in> set (map ?f ?xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. smult (inverse (x - x')) [:- x', 1::'a:] \\<in> set (map (\\<lambda>xi. smult (inverse (x - xi)) [:- xi, 1::'a:]) (filter (\\<lambda>c. c \\<noteq> x) (map fst xs_ys)))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nx' \\<in> set (map fst xs_ys)\nx' \\<noteq> x\n\ngoal (1 subgoal):\n 1. smult (inverse (x - x')) [:- x', 1::'a:] \\<in> set (map (\\<lambda>xi. smult (inverse (x - xi)) [:- xi, 1::'a:]) (filter (\\<lambda>c. c \\<noteq> x) (map fst xs_ys)))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nsmult (inverse (x - x')) [:- x', 1::'a:] \\<in> set (map (\\<lambda>xi. smult (inverse (x - xi)) [:- xi, 1::'a:]) (filter (\\<lambda>c. c \\<noteq> x) (map fst xs_ys)))\n\ngoal (1 subgoal):\n 1. poly (lagrange_basis_poly (map fst xs_ys) x) x' = (0::'a)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poly (lagrange_basis_poly (map fst xs_ys) x) x' = (0::'a)\n[PROOF STEP]\nunfolding lagrange_basis_poly_def Let_def poly_prod_list prod_list_map_remove1[OF mem]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poly (smult (inverse (x - x')) [:- x', 1::'a:]) x' * (\\<Prod>p\\<leftarrow>remove1 (smult (inverse (x - x')) [:- x', 1::'a:]) (map (\\<lambda>xi. smult (inverse (x - xi)) [:- xi, 1::'a:]) (filter (\\<lambda>xa. xa \\<noteq> x) (map fst xs_ys))). poly p x') = (0::'a)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\npoly (lagrange_basis_poly (map fst xs_ys) x) x' = (0::'a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1101, "file": "Polynomial_Interpolation_Lagrange_Interpolation", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8902942261220292, "lm_q2_score": 0.8539127566694178, "lm_q1q2_score": 0.760233596874728}}
{"text": "[STATEMENT]\nlemma sphere_params_on_sphere:\n  fixes \\<alpha> \\<beta> :: real\n  assumes \"x = cos \\<alpha> * cos \\<beta>\" and \"y = cos \\<alpha> * sin \\<beta>\" \"z = sin \\<alpha>\"\n  shows \"x*x + y*y + z*z = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x * x + y * y + z * z = 1\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x * x + y * y + z * z = 1\n[PROOF STEP]\nhave \"x*x + y*y = (cos \\<alpha> * cos \\<alpha>) * (cos \\<beta> * cos \\<beta>) + (cos \\<alpha> * cos \\<alpha>) * (sin \\<beta> * sin \\<beta>)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x * x + y * y = cos \\<alpha> * cos \\<alpha> * (cos \\<beta> * cos \\<beta>) + cos \\<alpha> * cos \\<alpha> * (sin \\<beta> * sin \\<beta>)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nx = cos \\<alpha> * cos \\<beta>\ny = cos \\<alpha> * sin \\<beta>\nz = sin \\<alpha>\n\ngoal (1 subgoal):\n 1. x * x + y * y = cos \\<alpha> * cos \\<alpha> * (cos \\<beta> * cos \\<beta>) + cos \\<alpha> * cos \\<alpha> * (sin \\<beta> * sin \\<beta>)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nx * x + y * y = cos \\<alpha> * cos \\<alpha> * (cos \\<beta> * cos \\<beta>) + cos \\<alpha> * cos \\<alpha> * (sin \\<beta> * sin \\<beta>)\n\ngoal (1 subgoal):\n 1. x * x + y * y + z * z = 1\n[PROOF STEP]\nhence \"x*x + y*y = cos \\<alpha> * cos \\<alpha>\"\n[PROOF STATE]\nproof (prove)\nusing this:\nx * x + y * y = cos \\<alpha> * cos \\<alpha> * (cos \\<beta> * cos \\<beta>) + cos \\<alpha> * cos \\<alpha> * (sin \\<beta> * sin \\<beta>)\n\ngoal (1 subgoal):\n 1. x * x + y * y = cos \\<alpha> * cos \\<alpha>\n[PROOF STEP]\nusing sin_cos_squared_add3[of \\<beta>]\n[PROOF STATE]\nproof (prove)\nusing this:\nx * x + y * y = cos \\<alpha> * cos \\<alpha> * (cos \\<beta> * cos \\<beta>) + cos \\<alpha> * cos \\<alpha> * (sin \\<beta> * sin \\<beta>)\ncos \\<beta> * cos \\<beta> + sin \\<beta> * sin \\<beta> = 1\n\ngoal (1 subgoal):\n 1. x * x + y * y = cos \\<alpha> * cos \\<alpha>\n[PROOF STEP]\nby (subst (asm) distrib_left[symmetric]) (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nx * x + y * y = cos \\<alpha> * cos \\<alpha>\n\ngoal (1 subgoal):\n 1. x * x + y * y + z * z = 1\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nx * x + y * y = cos \\<alpha> * cos \\<alpha>\n\ngoal (1 subgoal):\n 1. x * x + y * y + z * z = 1\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nx * x + y * y = cos \\<alpha> * cos \\<alpha>\nx = cos \\<alpha> * cos \\<beta>\ny = cos \\<alpha> * sin \\<beta>\nz = sin \\<alpha>\n\ngoal (1 subgoal):\n 1. x * x + y * y + z * z = 1\n[PROOF STEP]\nusing sin_cos_squared_add3[of \\<alpha>]\n[PROOF STATE]\nproof (prove)\nusing this:\nx * x + y * y = cos \\<alpha> * cos \\<alpha>\nx = cos \\<alpha> * cos \\<beta>\ny = cos \\<alpha> * sin \\<beta>\nz = sin \\<alpha>\ncos \\<alpha> * cos \\<alpha> + sin \\<alpha> * sin \\<alpha> = 1\n\ngoal (1 subgoal):\n 1. x * x + y * y + z * z = 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nx * x + y * y + z * z = 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1264, "file": "Complex_Geometry_Riemann_Sphere", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942319436397, "lm_q2_score": 0.853912747375134, "lm_q1q2_score": 0.7602335935712281}}
{"text": "[STATEMENT]\nlemma sym_group_card_carrier: \"card (carrier (sym_group n)) = fact n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (carrier (sym_group n)) = fact n\n[PROOF STEP]\nusing card_permutations[of \"{1..n}\" n]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>card {1..n} = n; finite {1..n}\\<rbrakk> \\<Longrightarrow> card {p. p permutes {1..n}} = fact n\n\ngoal (1 subgoal):\n 1. card (carrier (sym_group n)) = fact n\n[PROOF STEP]\nunfolding sym_group_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>card {1..n} = n; finite {1..n}\\<rbrakk> \\<Longrightarrow> card {p. p permutes {1..n}} = fact n\n\ngoal (1 subgoal):\n 1. card (carrier \\<lparr>carrier = {p. p permutes {1..n}}, monoid.mult = (\\<circ>), one = id\\<rparr>) = fact n\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 321, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9219218348550491, "lm_q2_score": 0.824461928533133, "lm_q1q2_score": 0.7600894539213984}}
{"text": "[STATEMENT]\nlemma complex_abs_le_norm: \"\\<bar>Re z\\<bar> + \\<bar>Im z\\<bar> \\<le> sqrt 2 * norm z\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<bar>Re z\\<bar> + \\<bar>Im z\\<bar> \\<le> sqrt 2 * cmod z\n[PROOF STEP]\nunfolding norm_complex_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<bar>Re z\\<bar> + \\<bar>Im z\\<bar> \\<le> sqrt 2 * sqrt ((Re z)\\<^sup>2 + (Im z)\\<^sup>2)\n[PROOF STEP]\napply (rule abs_sqrt_wlog [where x=\"Re z\"])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>x. 0 \\<le> x \\<Longrightarrow> x + \\<bar>Im z\\<bar> \\<le> sqrt 2 * sqrt (x\\<^sup>2 + (Im z)\\<^sup>2)\n[PROOF STEP]\napply (rule abs_sqrt_wlog [where x=\"Im z\"])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>x xa. \\<lbrakk>0 \\<le> x; 0 \\<le> xa\\<rbrakk> \\<Longrightarrow> x + xa \\<le> sqrt 2 * sqrt (x\\<^sup>2 + xa\\<^sup>2)\n[PROOF STEP]\napply (rule power2_le_imp_le)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\<And>x xa. \\<lbrakk>0 \\<le> x; 0 \\<le> xa\\<rbrakk> \\<Longrightarrow> (x + xa)\\<^sup>2 \\<le> (sqrt 2 * sqrt (x\\<^sup>2 + xa\\<^sup>2))\\<^sup>2\n 2. \\<And>x xa. \\<lbrakk>0 \\<le> x; 0 \\<le> xa\\<rbrakk> \\<Longrightarrow> 0 \\<le> sqrt 2 * sqrt (x\\<^sup>2 + xa\\<^sup>2)\n[PROOF STEP]\napply (simp_all add: power2_sum add.commute sum_squares_bound real_sqrt_mult [symmetric])\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 644, "file": null, "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392909114835, "lm_q2_score": 0.8596637469145054, "lm_q1q2_score": 0.7599765292446083}}
{"text": "[STATEMENT]\nlemma crb_lem_neg: \n  fixes x:: \"real\"\n  fixes p:: \"real poly\"\n  assumes x: \"poly p x = 0\" \n  assumes p: \"p \\<noteq> 0\" \n  shows \"x > -crb p\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real_of_int (- crb p) < x\n[PROOF STEP]\nusing cauchy_root_bound[of p x]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>poly p x = 0; p \\<noteq> 0\\<rbrakk> \\<Longrightarrow> norm x \\<le> 1 + max_list_non_empty (map (\\<lambda>i. norm (coeff p i)) [0..<degree p]) / norm (lead_coeff p)\n\ngoal (1 subgoal):\n 1. real_of_int (- crb p) < x\n[PROOF STEP]\napply (auto)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<lbrakk>poly p x = 0; p \\<noteq> 0\\<rbrakk> \\<Longrightarrow> \\<bar>x\\<bar> \\<le> 1 + max_list_non_empty (map (\\<lambda>i. \\<bar>coeff p i\\<bar>) [0..<degree p]) / \\<bar>lead_coeff p\\<bar>) \\<Longrightarrow> - real_of_int (crb p) < x\n[PROOF STEP]\nunfolding crb_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<lbrakk>poly p x = 0; p \\<noteq> 0\\<rbrakk> \\<Longrightarrow> \\<bar>x\\<bar> \\<le> 1 + max_list_non_empty (map (\\<lambda>i. \\<bar>coeff p i\\<bar>) [0..<degree p]) / \\<bar>lead_coeff p\\<bar>) \\<Longrightarrow> - real_of_int \\<lceil>2 + max_list_non_empty (map (\\<lambda>i. norm (coeff p i)) [0..<degree p]) / norm (lead_coeff p)\\<rceil> < x\n[PROOF STEP]\napply (auto)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<lbrakk>poly p x = 0; p \\<noteq> 0\\<rbrakk> \\<Longrightarrow> \\<bar>x\\<bar> \\<le> 1 + max_list_non_empty (map (\\<lambda>i. \\<bar>coeff p i\\<bar>) [0..<degree p]) / \\<bar>lead_coeff p\\<bar>) \\<Longrightarrow> - real_of_int \\<lceil>2 + max_list_non_empty (map (\\<lambda>i. \\<bar>coeff p i\\<bar>) [0..<degree p]) / \\<bar>lead_coeff p\\<bar>\\<rceil> < x\n[PROOF STEP]\nusing p x\n[PROOF STATE]\nproof (prove)\nusing this:\np \\<noteq> 0\npoly p x = 0\n\ngoal (1 subgoal):\n 1. (\\<lbrakk>poly p x = 0; p \\<noteq> 0\\<rbrakk> \\<Longrightarrow> \\<bar>x\\<bar> \\<le> 1 + max_list_non_empty (map (\\<lambda>i. \\<bar>coeff p i\\<bar>) [0..<degree p]) / \\<bar>lead_coeff p\\<bar>) \\<Longrightarrow> - real_of_int \\<lceil>2 + max_list_non_empty (map (\\<lambda>i. \\<bar>coeff p i\\<bar>) [0..<degree p]) / \\<bar>lead_coeff p\\<bar>\\<rceil> < x\n[PROOF STEP]\nby linarith", "meta": {"llama_tokens": 973, "file": "BenOr_Kozen_Reif_BKR_Decision", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9005297807787537, "lm_q2_score": 0.8438951005915208, "lm_q1q2_score": 0.7599526699359466}}
{"text": "[STATEMENT]\nlemma real_inverse_ge_one_lemma: \n      \"\\<lbrakk> 0 < (a::real); a < 1 \\<rbrakk> \\<Longrightarrow> inverse a \\<ge> 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>0 < a; a < 1\\<rbrakk> \\<Longrightarrow> 1 \\<le> inverse a\n[PROOF STEP]\nby (metis less_eq_real_def one_le_inverse_iff)", "meta": {"llama_tokens": 134, "file": "Real_Power_RealPower", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9241418158002492, "lm_q2_score": 0.8221891327004132, "lm_q1q2_score": 0.7598193580249919}}
{"text": "[STATEMENT]\nlemma gcd_eq_0_iff [simp]: \"gcd a b = 0 \\<longleftrightarrow> a = 0 \\<and> b = 0\"\n  (is \"?P \\<longleftrightarrow> ?Q\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (gcd a b = (0::'a)) = (a = (0::'a) \\<and> b = (0::'a))\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. gcd a b = (0::'a) \\<Longrightarrow> a = (0::'a) \\<and> b = (0::'a)\n 2. a = (0::'a) \\<and> b = (0::'a) \\<Longrightarrow> gcd a b = (0::'a)\n[PROOF STEP]\nassume ?P\n[PROOF STATE]\nproof (state)\nthis:\ngcd a b = (0::'a)\n\ngoal (2 subgoals):\n 1. gcd a b = (0::'a) \\<Longrightarrow> a = (0::'a) \\<and> b = (0::'a)\n 2. a = (0::'a) \\<and> b = (0::'a) \\<Longrightarrow> gcd a b = (0::'a)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ngcd a b = (0::'a)\n[PROOF STEP]\nhave \"0 dvd gcd a b\"\n[PROOF STATE]\nproof (prove)\nusing this:\ngcd a b = (0::'a)\n\ngoal (1 subgoal):\n 1. (0::'a) dvd gcd a b\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(0::'a) dvd gcd a b\n\ngoal (2 subgoals):\n 1. gcd a b = (0::'a) \\<Longrightarrow> a = (0::'a) \\<and> b = (0::'a)\n 2. a = (0::'a) \\<and> b = (0::'a) \\<Longrightarrow> gcd a b = (0::'a)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(0::'a) dvd gcd a b\n[PROOF STEP]\nhave \"0 dvd a\" and \"0 dvd b\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(0::'a) dvd gcd a b\n\ngoal (1 subgoal):\n 1. (0::'a) dvd a &&& (0::'a) dvd b\n[PROOF STEP]\nby (blast intro: dvd_trans)+\n[PROOF STATE]\nproof (state)\nthis:\n(0::'a) dvd a\n(0::'a) dvd b\n\ngoal (2 subgoals):\n 1. gcd a b = (0::'a) \\<Longrightarrow> a = (0::'a) \\<and> b = (0::'a)\n 2. a = (0::'a) \\<and> b = (0::'a) \\<Longrightarrow> gcd a b = (0::'a)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(0::'a) dvd a\n(0::'a) dvd b\n[PROOF STEP]\nshow ?Q\n[PROOF STATE]\nproof (prove)\nusing this:\n(0::'a) dvd a\n(0::'a) dvd b\n\ngoal (1 subgoal):\n 1. a = (0::'a) \\<and> b = (0::'a)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\na = (0::'a) \\<and> b = (0::'a)\n\ngoal (1 subgoal):\n 1. a = (0::'a) \\<and> b = (0::'a) \\<Longrightarrow> gcd a b = (0::'a)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. a = (0::'a) \\<and> b = (0::'a) \\<Longrightarrow> gcd a b = (0::'a)\n[PROOF STEP]\nassume ?Q\n[PROOF STATE]\nproof (state)\nthis:\na = (0::'a) \\<and> b = (0::'a)\n\ngoal (1 subgoal):\n 1. a = (0::'a) \\<and> b = (0::'a) \\<Longrightarrow> gcd a b = (0::'a)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\na = (0::'a) \\<and> b = (0::'a)\n[PROOF STEP]\nshow ?P\n[PROOF STATE]\nproof (prove)\nusing this:\na = (0::'a) \\<and> b = (0::'a)\n\ngoal (1 subgoal):\n 1. gcd a b = (0::'a)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ngcd a b = (0::'a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1391, "file": null, "length": 17, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314707995591, "lm_q2_score": 0.8577681068080749, "lm_q1q2_score": 0.7596664300373887}}
{"text": "[STATEMENT]\nlemma dirichlet_prod_commutes:\n  \"dirichlet_prod (f :: nat \\<Rightarrow> 'a :: comm_semiring_0) g = dirichlet_prod g f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dirichlet_prod f g = dirichlet_prod g f\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>x. dirichlet_prod f g x = dirichlet_prod g f x\n[PROOF STEP]\nfix n :: nat\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>x. dirichlet_prod f g x = dirichlet_prod g f x\n[PROOF STEP]\nshow \"dirichlet_prod f g n = dirichlet_prod g f n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dirichlet_prod f g n = dirichlet_prod g f n\n[PROOF STEP]\nproof (cases \"n = 0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. n = 0 \\<Longrightarrow> dirichlet_prod f g n = dirichlet_prod g f n\n 2. n \\<noteq> 0 \\<Longrightarrow> dirichlet_prod f g n = dirichlet_prod g f n\n[PROOF STEP]\ncase False\n[PROOF STATE]\nproof (state)\nthis:\nn \\<noteq> 0\n\ngoal (2 subgoals):\n 1. n = 0 \\<Longrightarrow> dirichlet_prod f g n = dirichlet_prod g f n\n 2. n \\<noteq> 0 \\<Longrightarrow> dirichlet_prod f g n = dirichlet_prod g f n\n[PROOF STEP]\nhave \"(\\<Sum>(r,d) | r * d = n. f r * g d) = (\\<Sum>(d,r) | r * d = n. f r * g d)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>(r, d)\\<in>{(r, d). r * d = n}. f r * g d) = (\\<Sum>(d, r)\\<in>{(d, r). r * d = n}. f r * g d)\n[PROOF STEP]\nby (rule sum.reindex_bij_witness [of _ \"\\<lambda>(x,y). (y,x)\" \"\\<lambda>(x,y). (y,x)\"]) auto\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>(r, d)\\<in>{(r, d). r * d = n}. f r * g d) = (\\<Sum>(d, r)\\<in>{(d, r). r * d = n}. f r * g d)\n\ngoal (2 subgoals):\n 1. n = 0 \\<Longrightarrow> dirichlet_prod f g n = dirichlet_prod g f n\n 2. n \\<noteq> 0 \\<Longrightarrow> dirichlet_prod f g n = dirichlet_prod g f n\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<Sum>(r, d)\\<in>{(r, d). r * d = n}. f r * g d) = (\\<Sum>(d, r)\\<in>{(d, r). r * d = n}. f r * g d)\n\ngoal (1 subgoal):\n 1. dirichlet_prod f g n = dirichlet_prod g f n\n[PROOF STEP]\nby (simp add: dirichlet_prod_altdef2 mult.commute)\n[PROOF STATE]\nproof (state)\nthis:\ndirichlet_prod f g n = dirichlet_prod g f n\n\ngoal (1 subgoal):\n 1. n = 0 \\<Longrightarrow> dirichlet_prod f g n = dirichlet_prod g f n\n[PROOF STEP]\nqed (simp add: dirichlet_prod_def)\n[PROOF STATE]\nproof (state)\nthis:\ndirichlet_prod f g n = dirichlet_prod g f n\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1084, "file": "Dirichlet_Series_Dirichlet_Product", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505376715775, "lm_q2_score": 0.8397339736884712, "lm_q1q2_score": 0.7596657907983658}}
{"text": "[STATEMENT]\nlemma sincos_total_pi_half:\n  assumes \"0 \\<le> x\" \"0 \\<le> y\" \"x\\<^sup>2 + y\\<^sup>2 = 1\"\n  shows \"\\<exists>t. 0 \\<le> t \\<and> t \\<le> pi/2 \\<and> x = cos t \\<and> y = sin t\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<exists>t\\<ge>0. t \\<le> pi / 2 \\<and> x = cos t \\<and> y = sin t\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<exists>t\\<ge>0. t \\<le> pi / 2 \\<and> x = cos t \\<and> y = sin t\n[PROOF STEP]\nhave x1: \"x \\<le> 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. x \\<le> 1\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\<le> x\n0 \\<le> y\nx\\<^sup>2 + y\\<^sup>2 = 1\n\ngoal (1 subgoal):\n 1. x \\<le> 1\n[PROOF STEP]\nby (metis le_add_same_cancel1 power2_le_imp_le power_one zero_le_power2)\n[PROOF STATE]\nproof (state)\nthis:\nx \\<le> 1\n\ngoal (1 subgoal):\n 1. \\<exists>t\\<ge>0. t \\<le> pi / 2 \\<and> x = cos t \\<and> y = sin t\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\<le> x\n0 \\<le> y\nx\\<^sup>2 + y\\<^sup>2 = 1\nx \\<le> 1\n[PROOF STEP]\nhave *: \"0 \\<le> arccos x\" \"cos (arccos x) = x\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\<le> x\n0 \\<le> y\nx\\<^sup>2 + y\\<^sup>2 = 1\nx \\<le> 1\n\ngoal (1 subgoal):\n 1. 0 \\<le> arccos x &&& cos (arccos x) = x\n[PROOF STEP]\nby (auto simp: arccos)\n[PROOF STATE]\nproof (state)\nthis:\n0 \\<le> arccos x\ncos (arccos x) = x\n\ngoal (1 subgoal):\n 1. \\<exists>t\\<ge>0. t \\<le> pi / 2 \\<and> x = cos t \\<and> y = sin t\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\n0 \\<le> x\n0 \\<le> y\nx\\<^sup>2 + y\\<^sup>2 = 1\n[PROOF STEP]\nhave \"y = sqrt (1 - x\\<^sup>2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\<le> x\n0 \\<le> y\nx\\<^sup>2 + y\\<^sup>2 = 1\n\ngoal (1 subgoal):\n 1. y = sqrt (1 - x\\<^sup>2)\n[PROOF STEP]\nby (metis abs_of_nonneg add.commute add_diff_cancel real_sqrt_abs)\n[PROOF STATE]\nproof (state)\nthis:\ny = sqrt (1 - x\\<^sup>2)\n\ngoal (1 subgoal):\n 1. \\<exists>t\\<ge>0. t \\<le> pi / 2 \\<and> x = cos t \\<and> y = sin t\n[PROOF STEP]\nwith x1 * assms arccos_le_pi2 [of x]\n[PROOF STATE]\nproof (chain)\npicking this:\nx \\<le> 1\n0 \\<le> arccos x\ncos (arccos x) = x\n0 \\<le> x\n0 \\<le> y\nx\\<^sup>2 + y\\<^sup>2 = 1\n\\<lbrakk>0 \\<le> x; x \\<le> 1\\<rbrakk> \\<Longrightarrow> arccos x \\<le> pi / 2\ny = sqrt (1 - x\\<^sup>2)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nx \\<le> 1\n0 \\<le> arccos x\ncos (arccos x) = x\n0 \\<le> x\n0 \\<le> y\nx\\<^sup>2 + y\\<^sup>2 = 1\n\\<lbrakk>0 \\<le> x; x \\<le> 1\\<rbrakk> \\<Longrightarrow> arccos x \\<le> pi / 2\ny = sqrt (1 - x\\<^sup>2)\n\ngoal (1 subgoal):\n 1. \\<exists>t\\<ge>0. t \\<le> pi / 2 \\<and> x = cos t \\<and> y = sin t\n[PROOF STEP]\nby (rule_tac x=\"arccos x\" in exI) (auto simp: sin_arccos)\n[PROOF STATE]\nproof (state)\nthis:\n\\<exists>t\\<ge>0. t \\<le> pi / 2 \\<and> x = cos t \\<and> y = sin t\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1394, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972751232809, "lm_q2_score": 0.8723473663814338, "lm_q1q2_score": 0.7594632401326467}}
{"text": "[STATEMENT]\nlemma pos_prod_sum_lt:\n  fixes c :: \"'a::linordered_field\"\n  assumes \"c > 0\"\n  shows \"c * x + t < 0 \\<equiv> x < (- 1 / c) * t\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c * x + t < (0::'a) \\<equiv> x < - (1::'a) / c * t\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. c * x + t < (0::'a) \\<equiv> x < - (1::'a) / c * t\n[PROOF STEP]\nhave \"c * x + t < 0 \\<longleftrightarrow> c * x < - t\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (c * x + t < (0::'a)) = (c * x < - t)\n[PROOF STEP]\nby (subst less_iff_diff_less_0 [of \"c * x\" \"- t\"]) simp\n[PROOF STATE]\nproof (state)\nthis:\n(c * x + t < (0::'a)) = (c * x < - t)\n\ngoal (1 subgoal):\n 1. c * x + t < (0::'a) \\<equiv> x < - (1::'a) / c * t\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(c * x + t < (0::'a)) = (c * x < - t)\n\ngoal (1 subgoal):\n 1. c * x + t < (0::'a) \\<equiv> x < - (1::'a) / c * t\n[PROOF STEP]\nhave \"\\<dots> \\<longleftrightarrow> - t / c > x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (c * x < - t) = (x < - t / c)\n[PROOF STEP]\nby (simp only: pos_less_divide_eq[OF \\<open>c > 0\\<close>] algebra_simps)\n[PROOF STATE]\nproof (state)\nthis:\n(c * x < - t) = (x < - t / c)\n\ngoal (1 subgoal):\n 1. c * x + t < (0::'a) \\<equiv> x < - (1::'a) / c * t\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(c * x < - t) = (x < - t / c)\n\ngoal (1 subgoal):\n 1. c * x + t < (0::'a) \\<equiv> x < - (1::'a) / c * t\n[PROOF STEP]\nhave \"\\<dots> \\<longleftrightarrow> (- 1 / c) * t > x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x < - t / c) = (x < - (1::'a) / c * t)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(x < - t / c) = (x < - (1::'a) / c * t)\n\ngoal (1 subgoal):\n 1. c * x + t < (0::'a) \\<equiv> x < - (1::'a) / c * t\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(c * x + t < (0::'a)) = (x < - (1::'a) / c * t)\n[PROOF STEP]\nshow \"PROP ?thesis\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(c * x + t < (0::'a)) = (x < - (1::'a) / c * t)\n\ngoal (1 subgoal):\n 1. c * x + t < (0::'a) \\<equiv> x < - (1::'a) / c * t\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nc * x + t < (0::'a) \\<equiv> x < - (1::'a) / c * t\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1118, "file": null, "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473746782093, "lm_q2_score": 0.8705972600147106, "lm_q1q2_score": 0.7594632341758751}}
{"text": "[STATEMENT]\nlemma ket_psim_norm:\n  shows \"\\<parallel>ket_psim\\<parallel> = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<parallel>ket_psim\\<parallel> = 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<parallel>ket_psim\\<parallel> = 1\n[PROOF STEP]\nhave \"dim_vec ket_psim = 2\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_vec ket_psim = 2\\<^sup>2\n[PROOF STEP]\nunfolding ket_psim_def ket_01_def ket_10_def ket_0_def ket_1_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_vec (complex_of_real (1 / sqrt 2) \\<cdot>\\<^sub>v ((unit_vec 2 0 \\<otimes> unit_vec 2 1) - (unit_vec 2 1 \\<otimes> unit_vec 2 0))) = 2\\<^sup>2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndim_vec ket_psim = 2\\<^sup>2\n\ngoal (1 subgoal):\n 1. \\<parallel>ket_psim\\<parallel> = 1\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ndim_vec ket_psim = 2\\<^sup>2\n\ngoal (1 subgoal):\n 1. \\<parallel>ket_psim\\<parallel> = 1\n[PROOF STEP]\nhave \"(\\<Sum>i<4. (cmod (vec_index ket_psim i))\\<^sup>2) = 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>i<4. (cmod (ket_psim $ i))\\<^sup>2) = 1\n[PROOF STEP]\napply (auto simp add: ket_psim_def ket_01_def ket_10_def ket_0_def ket_1_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>i<4. (cmod (((if i div 2 = 0 then 1 else 0) * (if i mod 2 = Suc 0 then 1 else 0) - (if i div 2 = Suc 0 then 1 else 0) * (if i mod 2 = 0 then 1 else 0)) / complex_of_real (sqrt 2)))\\<^sup>2) = 1\n[PROOF STEP]\napply (simp add: sum_4_elems)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>i<4. (cmod (ket_psim $ i))\\<^sup>2) = 1\n\ngoal (1 subgoal):\n 1. \\<parallel>ket_psim\\<parallel> = 1\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ndim_vec ket_psim = 2\\<^sup>2\n(\\<Sum>i<4. (cmod (ket_psim $ i))\\<^sup>2) = 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec ket_psim = 2\\<^sup>2\n(\\<Sum>i<4. (cmod (ket_psim $ i))\\<^sup>2) = 1\n\ngoal (1 subgoal):\n 1. \\<parallel>ket_psim\\<parallel> = 1\n[PROOF STEP]\nby (simp add: cpx_vec_length_def)\n[PROOF STATE]\nproof (state)\nthis:\n\\<parallel>ket_psim\\<parallel> = 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1085, "file": "Projective_Measurements_CHSH_Inequality", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8976952948443462, "lm_q2_score": 0.84594244507642, "lm_q1q2_score": 0.759398552654224}}
{"text": "[STATEMENT]\nlemma sum_squared: \"(\\<Sum>i=0..n. i)^2 = (\\<Sum>i=0..n. i^3)\" for n :: nat\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum> {0..n})\\<^sup>2 = (\\<Sum>i = 0..n. i ^ 3)\n[PROOF STEP]\nproof (induct n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. (\\<Sum> {0..0})\\<^sup>2 = (\\<Sum>i = 0..0. i ^ 3)\n 2. \\<And>n. (\\<Sum> {0..n})\\<^sup>2 = (\\<Sum>i = 0..n. i ^ 3) \\<Longrightarrow> (\\<Sum> {0..Suc n})\\<^sup>2 = (\\<Sum>i = 0..Suc n. i ^ 3)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. (\\<Sum> {0..0})\\<^sup>2 = (\\<Sum>i = 0..0. i ^ 3)\n 2. \\<And>n. (\\<Sum> {0..n})\\<^sup>2 = (\\<Sum>i = 0..n. i ^ 3) \\<Longrightarrow> (\\<Sum> {0..Suc n})\\<^sup>2 = (\\<Sum>i = 0..Suc n. i ^ 3)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum> {0..0})\\<^sup>2 = (\\<Sum>i = 0..0. i ^ 3)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum> {0..0})\\<^sup>2 = (\\<Sum>i = 0..0. i ^ 3)\n\ngoal (1 subgoal):\n 1. \\<And>n. (\\<Sum> {0..n})\\<^sup>2 = (\\<Sum>i = 0..n. i ^ 3) \\<Longrightarrow> (\\<Sum> {0..Suc n})\\<^sup>2 = (\\<Sum>i = 0..Suc n. i ^ 3)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>n. (\\<Sum> {0..n})\\<^sup>2 = (\\<Sum>i = 0..n. i ^ 3) \\<Longrightarrow> (\\<Sum> {0..Suc n})\\<^sup>2 = (\\<Sum>i = 0..Suc n. i ^ 3)\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum> {0..n})\\<^sup>2 = (\\<Sum>i = 0..n. i ^ 3)\n\ngoal (1 subgoal):\n 1. \\<And>n. (\\<Sum> {0..n})\\<^sup>2 = (\\<Sum>i = 0..n. i ^ 3) \\<Longrightarrow> (\\<Sum> {0..Suc n})\\<^sup>2 = (\\<Sum>i = 0..Suc n. i ^ 3)\n[PROOF STEP]\nhave \"(\\<Sum>i = 0..Suc n. i)^2 =\n        (\\<Sum>i = 0..n. i^3) + (2*(\\<Sum>i = 0..n. i)*(n+1) + (n+1)^2)\"\n    (is \"_ = ?A + ?B\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum> {0..Suc n})\\<^sup>2 = (\\<Sum>i = 0..n. i ^ 3) + (2 * \\<Sum> {0..n} * (n + 1) + (n + 1)\\<^sup>2)\n[PROOF STEP]\nusing Suc\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<Sum> {0..n})\\<^sup>2 = (\\<Sum>i = 0..n. i ^ 3)\n\ngoal (1 subgoal):\n 1. (\\<Sum> {0..Suc n})\\<^sup>2 = (\\<Sum>i = 0..n. i ^ 3) + (2 * \\<Sum> {0..n} * (n + 1) + (n + 1)\\<^sup>2)\n[PROOF STEP]\nby (simp add: eval_nat_numeral)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum> {0..Suc n})\\<^sup>2 = (\\<Sum>i = 0..n. i ^ 3) + (2 * \\<Sum> {0..n} * (n + 1) + (n + 1)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. \\<And>n. (\\<Sum> {0..n})\\<^sup>2 = (\\<Sum>i = 0..n. i ^ 3) \\<Longrightarrow> (\\<Sum> {0..Suc n})\\<^sup>2 = (\\<Sum>i = 0..Suc n. i ^ 3)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum> {0..Suc n})\\<^sup>2 = (\\<Sum>i = 0..n. i ^ 3) + (2 * \\<Sum> {0..n} * (n + 1) + (n + 1)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. \\<And>n. (\\<Sum> {0..n})\\<^sup>2 = (\\<Sum>i = 0..n. i ^ 3) \\<Longrightarrow> (\\<Sum> {0..Suc n})\\<^sup>2 = (\\<Sum>i = 0..Suc n. i ^ 3)\n[PROOF STEP]\nhave \"?B = (n+1)^3\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * \\<Sum> {0..n} * (n + 1) + (n + 1)\\<^sup>2 = (n + 1) ^ 3\n[PROOF STEP]\nusing sum_of_naturals\n[PROOF STATE]\nproof (prove)\nusing this:\n2 * \\<Sum> {0..?n} = ?n * Suc ?n\n\ngoal (1 subgoal):\n 1. 2 * \\<Sum> {0..n} * (n + 1) + (n + 1)\\<^sup>2 = (n + 1) ^ 3\n[PROOF STEP]\nby (simp add: eval_nat_numeral)\n[PROOF STATE]\nproof (state)\nthis:\n2 * \\<Sum> {0..n} * (n + 1) + (n + 1)\\<^sup>2 = (n + 1) ^ 3\n\ngoal (1 subgoal):\n 1. \\<And>n. (\\<Sum> {0..n})\\<^sup>2 = (\\<Sum>i = 0..n. i ^ 3) \\<Longrightarrow> (\\<Sum> {0..Suc n})\\<^sup>2 = (\\<Sum>i = 0..Suc n. i ^ 3)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n2 * \\<Sum> {0..n} * (n + 1) + (n + 1)\\<^sup>2 = (n + 1) ^ 3\n\ngoal (1 subgoal):\n 1. \\<And>n. (\\<Sum> {0..n})\\<^sup>2 = (\\<Sum>i = 0..n. i ^ 3) \\<Longrightarrow> (\\<Sum> {0..Suc n})\\<^sup>2 = (\\<Sum>i = 0..Suc n. i ^ 3)\n[PROOF STEP]\nhave \"?A + (n+1)^3 = (\\<Sum>i=0..Suc n. i^3)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..n. i ^ 3) + (n + 1) ^ 3 = (\\<Sum>i = 0..Suc n. i ^ 3)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>i = 0..n. i ^ 3) + (n + 1) ^ 3 = (\\<Sum>i = 0..Suc n. i ^ 3)\n\ngoal (1 subgoal):\n 1. \\<And>n. (\\<Sum> {0..n})\\<^sup>2 = (\\<Sum>i = 0..n. i ^ 3) \\<Longrightarrow> (\\<Sum> {0..Suc n})\\<^sup>2 = (\\<Sum>i = 0..Suc n. i ^ 3)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\<Sum> {0..Suc n})\\<^sup>2 = (\\<Sum>i = 0..Suc n. i ^ 3)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<Sum> {0..Suc n})\\<^sup>2 = (\\<Sum>i = 0..Suc n. i ^ 3)\n\ngoal (1 subgoal):\n 1. (\\<Sum> {0..Suc n})\\<^sup>2 = (\\<Sum>i = 0..Suc n. i ^ 3)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum> {0..Suc n})\\<^sup>2 = (\\<Sum>i = 0..Suc n. i ^ 3)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2465, "file": null, "length": 20, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8976952866333484, "lm_q2_score": 0.8459424392504911, "lm_q1q2_score": 0.7593985404782835}}
{"text": "[STATEMENT]\nlemma rank_mat_mult_lt_min_rank_factor: \n  fixes A :: \"'a::{conjugatable_ordered_field} mat\"\n  assumes \"A \\<in> carrier_mat n m\"\n  assumes \"B \\<in> carrier_mat m nc\" \n  shows \"rank n (A * B) \\<le> min (rank n A) (rank m B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rank n (A * B) \\<le> min (rank n A) (rank m B)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. rank n (A * B) \\<le> min (rank n A) (rank m B)\n[PROOF STEP]\nhave 1: \"rank n (A * B) \\<le> (rank n A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rank n (A * B) \\<le> rank n A\n[PROOF STEP]\nusing assms(1) assms(2) vec_space.rank_mat_mul_right\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\<in> carrier_mat n m\nB \\<in> carrier_mat m nc\n\\<lbrakk>?A \\<in> carrier_mat ?n ?d; ?B \\<in> carrier_mat ?d ?nc\\<rbrakk> \\<Longrightarrow> rank ?n (?A * ?B) \\<le> rank ?n ?A\n\ngoal (1 subgoal):\n 1. rank n (A * B) \\<le> rank n A\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nrank n (A * B) \\<le> rank n A\n\ngoal (1 subgoal):\n 1. rank n (A * B) \\<le> min (rank n A) (rank m B)\n[PROOF STEP]\nhave \"rank n (A* B) \\<le> rank m B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rank n (A * B) \\<le> rank m B\n[PROOF STEP]\nby (meson assms(1) assms(2) rank_mat_mul_left)\n[PROOF STATE]\nproof (state)\nthis:\nrank n (A * B) \\<le> rank m B\n\ngoal (1 subgoal):\n 1. rank n (A * B) \\<le> min (rank n A) (rank m B)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nrank n (A * B) \\<le> rank m B\n\ngoal (1 subgoal):\n 1. rank n (A * B) \\<le> min (rank n A) (rank m B)\n[PROOF STEP]\nusing 1\n[PROOF STATE]\nproof (prove)\nusing this:\nrank n (A * B) \\<le> rank m B\nrank n (A * B) \\<le> rank n A\n\ngoal (1 subgoal):\n 1. rank n (A * B) \\<le> min (rank n A) (rank m B)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nrank n (A * B) \\<le> min (rank n A) (rank m B)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 883, "file": "Fishers_Inequality_Rank_Argument_General", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357598021707, "lm_q2_score": 0.8757869948899665, "lm_q1q2_score": 0.7593386425392818}}
{"text": "[STATEMENT]\nlemma horner_schema':\n  fixes x :: real and a :: \"nat \\<Rightarrow> real\"\n  shows \"a 0 - x * (\\<Sum> i=0..<n. (-1)^i * a (Suc i) * x^i) = (\\<Sum> i=0..<Suc n. (-1)^i * a i * x^i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a 0 - x * (\\<Sum>i = 0..<n. (- 1) ^ i * a (Suc i) * x ^ i) = (\\<Sum>i = 0..<Suc n. (- 1) ^ i * a i * x ^ i)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. a 0 - x * (\\<Sum>i = 0..<n. (- 1) ^ i * a (Suc i) * x ^ i) = (\\<Sum>i = 0..<Suc n. (- 1) ^ i * a i * x ^ i)\n[PROOF STEP]\nhave shift_pow: \"\\<And>i. - (x * ((-1)^i * a (Suc i) * x ^ i)) = (-1)^(Suc i) * a (Suc i) * x ^ (Suc i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>i. - (x * ((- 1) ^ i * a (Suc i) * x ^ i)) = (- 1) ^ Suc i * a (Suc i) * x ^ Suc i\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n- (x * ((- 1) ^ ?i * a (Suc ?i) * x ^ ?i)) = (- 1) ^ Suc ?i * a (Suc ?i) * x ^ Suc ?i\n\ngoal (1 subgoal):\n 1. a 0 - x * (\\<Sum>i = 0..<n. (- 1) ^ i * a (Suc i) * x ^ i) = (\\<Sum>i = 0..<Suc n. (- 1) ^ i * a i * x ^ i)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a 0 - x * (\\<Sum>i = 0..<n. (- 1) ^ i * a (Suc i) * x ^ i) = (\\<Sum>i = 0..<Suc n. (- 1) ^ i * a i * x ^ i)\n[PROOF STEP]\nunfolding sum_distrib_left shift_pow uminus_add_conv_diff [symmetric] sum_negf[symmetric]\n    sum.atLeast_Suc_lessThan[OF zero_less_Suc]\n    sum.reindex[OF inj_Suc, unfolded comp_def, symmetric, of \"\\<lambda> n. (-1)^n  *a n * x^n\"]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>n\\<in>Suc ` {0..<n}. (- 1) ^ n * a n * x ^ n) + a 0 = (- 1) ^ 0 * a 0 * x ^ 0 + (\\<Sum>i = Suc 0..<Suc n. (- 1) ^ i * a i * x ^ i)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\na 0 - x * (\\<Sum>i = 0..<n. (- 1) ^ i * a (Suc i) * x ^ i) = (\\<Sum>i = 0..<Suc n. (- 1) ^ i * a i * x ^ i)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 982, "file": null, "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933094032139577, "lm_q2_score": 0.8499711699569787, "lm_q1q2_score": 0.7592872385833381}}
{"text": "[STATEMENT]\nlemma transpose_minus_1:\n  assumes \"dim_row Q = dim_col Q\"\n  shows \"transpose_mat (Q - (1\\<^sub>m (dim_row Q))) =  (transpose_mat Q - (1\\<^sub>m (dim_row Q)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (Q - 1\\<^sub>m (dim_row Q))\\<^sup>T = Q\\<^sup>T - 1\\<^sub>m (dim_row Q)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_row Q = dim_col Q\n\ngoal (1 subgoal):\n 1. (Q - 1\\<^sub>m (dim_row Q))\\<^sup>T = Q\\<^sup>T - 1\\<^sub>m (dim_row Q)\n[PROOF STEP]\nunfolding mat_eq_iff\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_row Q = dim_col Q\n\ngoal (1 subgoal):\n 1. dim_row (Q - 1\\<^sub>m (dim_row Q))\\<^sup>T = dim_row (Q\\<^sup>T - 1\\<^sub>m (dim_row Q)) \\<and> dim_col (Q - 1\\<^sub>m (dim_row Q))\\<^sup>T = dim_col (Q\\<^sup>T - 1\\<^sub>m (dim_row Q)) \\<and> (\\<forall>i j. i < dim_row (Q\\<^sup>T - 1\\<^sub>m (dim_row Q)) \\<longrightarrow> j < dim_col (Q\\<^sup>T - 1\\<^sub>m (dim_row Q)) \\<longrightarrow> (Q - 1\\<^sub>m (dim_row Q))\\<^sup>T $$ (i, j) = (Q\\<^sup>T - 1\\<^sub>m (dim_row Q)) $$ (i, j))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 497, "file": "Berlekamp_Zassenhaus_Berlekamp_Type_Based", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206659843131, "lm_q2_score": 0.8418256412990658, "lm_q1q2_score": 0.7592599430431248}}
{"text": "[STATEMENT]\nlemma arcsin_lt_bounded:\n  assumes \"- 1 < y\" \"y < 1\"\n  shows  \"- (pi/2) < arcsin y \\<and> arcsin y < pi/2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - (pi / 2) < arcsin y \\<and> arcsin y < pi / 2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. - (pi / 2) < arcsin y \\<and> arcsin y < pi / 2\n[PROOF STEP]\nhave \"arcsin y \\<noteq> pi/2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. arcsin y \\<noteq> pi / 2\n[PROOF STEP]\nby (metis arcsin assms not_less not_less_iff_gr_or_eq sin_pi_half)\n[PROOF STATE]\nproof (state)\nthis:\narcsin y \\<noteq> pi / 2\n\ngoal (1 subgoal):\n 1. - (pi / 2) < arcsin y \\<and> arcsin y < pi / 2\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\narcsin y \\<noteq> pi / 2\n\ngoal (1 subgoal):\n 1. - (pi / 2) < arcsin y \\<and> arcsin y < pi / 2\n[PROOF STEP]\nhave \"arcsin y \\<noteq> - pi/2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. arcsin y \\<noteq> - pi / 2\n[PROOF STEP]\nby (metis arcsin assms minus_divide_left not_less not_less_iff_gr_or_eq sin_minus sin_pi_half)\n[PROOF STATE]\nproof (state)\nthis:\narcsin y \\<noteq> - pi / 2\n\ngoal (1 subgoal):\n 1. - (pi / 2) < arcsin y \\<and> arcsin y < pi / 2\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\narcsin y \\<noteq> pi / 2\narcsin y \\<noteq> - pi / 2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\narcsin y \\<noteq> pi / 2\narcsin y \\<noteq> - pi / 2\n\ngoal (1 subgoal):\n 1. - (pi / 2) < arcsin y \\<and> arcsin y < pi / 2\n[PROOF STEP]\nusing arcsin_bounded [of y] assms\n[PROOF STATE]\nproof (prove)\nusing this:\narcsin y \\<noteq> pi / 2\narcsin y \\<noteq> - pi / 2\n\\<lbrakk>- 1 \\<le> y; y \\<le> 1\\<rbrakk> \\<Longrightarrow> - (pi / 2) \\<le> arcsin y \\<and> arcsin y \\<le> pi / 2\n- 1 < y\ny < 1\n\ngoal (1 subgoal):\n 1. - (pi / 2) < arcsin y \\<and> arcsin y < pi / 2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n- (pi / 2) < arcsin y \\<and> arcsin y < pi / 2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 958, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8652240895276223, "lm_q2_score": 0.8774767842777551, "lm_q1q2_score": 0.7592140517583466}}
{"text": "[STATEMENT]\nlemma of_nat_binomial_Suc:\n  assumes \"k \\<le> n\"\n  shows   \"(of_nat (Suc n choose k) :: 'a :: field_char_0) = \n             of_nat (Suc n) / of_nat (Suc n - k) * of_nat (n choose k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_nat (Suc n choose k) = of_nat (Suc n) / of_nat (Suc n - k) * of_nat (n choose k)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nk \\<le> n\n\ngoal (1 subgoal):\n 1. of_nat (Suc n choose k) = of_nat (Suc n) / of_nat (Suc n - k) * of_nat (n choose k)\n[PROOF STEP]\nby (simp add: binomial_fact divide_simps fact_diff_Suc of_nat_diff del: of_nat_Suc)", "meta": {"llama_tokens": 274, "file": "Bernoulli_Bernoulli", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9207896802383028, "lm_q2_score": 0.8244619220634456, "lm_q1q2_score": 0.7591560295854566}}
{"text": "[STATEMENT]\nlemma poincare_distance_ge0:\n  assumes \"u \\<in> unit_disc\" and \"v \\<in> unit_disc\"\n  shows \"poincare_distance u v \\<ge> 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\<le> poincare_distance u v\n[PROOF STEP]\nusing poincare_distance_formula'_ge_1\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>?u \\<in> unit_disc; ?v \\<in> unit_disc\\<rbrakk> \\<Longrightarrow> 1 \\<le> poincare_distance_formula' (to_complex ?u) (to_complex ?v)\n\ngoal (1 subgoal):\n 1. 0 \\<le> poincare_distance u v\n[PROOF STEP]\nunfolding poincare_distance_formula[OF assms(1) assms(2)]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>?u \\<in> unit_disc; ?v \\<in> unit_disc\\<rbrakk> \\<Longrightarrow> 1 \\<le> poincare_distance_formula' (to_complex ?u) (to_complex ?v)\n\ngoal (1 subgoal):\n 1. 0 \\<le> poincare_distance_formula (to_complex u) (to_complex v)\n[PROOF STEP]\nunfolding poincare_distance_formula_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>?u \\<in> unit_disc; ?v \\<in> unit_disc\\<rbrakk> \\<Longrightarrow> 1 \\<le> poincare_distance_formula' (to_complex ?u) (to_complex ?v)\n\ngoal (1 subgoal):\n 1. 0 \\<le> arcosh (poincare_distance_formula' (to_complex u) (to_complex v))\n[PROOF STEP]\nunfolding poincare_distance_formula'_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>?u \\<in> unit_disc; ?v \\<in> unit_disc\\<rbrakk> \\<Longrightarrow> 1 \\<le> 1 + 2 * ((cmod (to_complex ?u - to_complex ?v))\\<^sup>2 / ((1 - (cmod (to_complex ?u))\\<^sup>2) * (1 - (cmod (to_complex ?v))\\<^sup>2)))\n\ngoal (1 subgoal):\n 1. 0 \\<le> arcosh (1 + 2 * ((cmod (to_complex u - to_complex v))\\<^sup>2 / ((1 - (cmod (to_complex u))\\<^sup>2) * (1 - (cmod (to_complex v))\\<^sup>2))))\n[PROOF STEP]\nby (rule arcosh_ge_0, simp_all add: assms)", "meta": {"llama_tokens": 754, "file": "Poincare_Disc_Poincare_Distance", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.888758793492457, "lm_q2_score": 0.8539127455162773, "lm_q1q2_score": 0.7589224614528781}}
{"text": "[STATEMENT]\nlemma effective_matrix_match_condn_2: \n assumes \"(matrix_match A1 A2 B1 B2) \"\n shows \"\\<forall>i j.((i<(row_length A1)*(row_length B1))\n         \\<and>(j<(length A2)*(length B2))\n            \\<longrightarrow> ((A1 \\<otimes> B1)\\<circ>(A2 \\<otimes> B2))!j!i\n           =  scalar_product \n                  (vec_vec_Tensor \n                           (row A1 (i div row_length B1)) \n                           (row B1 (i mod row_length B1))) \n                  (vec_vec_Tensor \n                           (col A2 (j div length B2)) \n                           (col B2 (j mod length B2))))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<forall>i j. i < row_length A1 * row_length B1 \\<and> j < length A2 * length B2 \\<longrightarrow> ((A1 \\<otimes> B1) \\<circ> (A2 \\<otimes> B2)) ! j ! i = scalar_product (vec_vec_Tensor (row A1 (i div row_length B1)) (row B1 (i mod row_length B1))) (vec_vec_Tensor (col A2 (j div length B2)) (col B2 (j mod length B2)))\n[PROOF STEP]\nusing assms matrix_match_condn_2\n[PROOF STATE]\nproof (prove)\nusing this:\nmatrix_match A1 A2 B1 B2\nmatrix_match ?A1.0 ?A2.0 ?B1.0 ?B2.0 \\<and> ?i < row_length ?A1.0 * row_length ?B1.0 \\<and> ?j < length ?A2.0 * length ?B2.0 \\<Longrightarrow> ((?A1.0 \\<otimes> ?B1.0) \\<circ> (?A2.0 \\<otimes> ?B2.0)) ! ?j ! ?i = scalar_product (vec_vec_Tensor (row ?A1.0 (?i div row_length ?B1.0)) (row ?B1.0 (?i mod row_length ?B1.0))) (vec_vec_Tensor (col ?A2.0 (?j div length ?B2.0)) (col ?B2.0 (?j mod length ?B2.0)))\n\ngoal (1 subgoal):\n 1. \\<forall>i j. i < row_length A1 * row_length B1 \\<and> j < length A2 * length B2 \\<longrightarrow> ((A1 \\<otimes> B1) \\<circ> (A2 \\<otimes> B2)) ! j ! i = scalar_product (vec_vec_Tensor (row A1 (i div row_length B1)) (row B1 (i mod row_length B1))) (vec_vec_Tensor (col A2 (j div length B2)) (col B2 (j mod length B2)))\n[PROOF STEP]\nunfolding matrix_match_def\n[PROOF STATE]\nproof (prove)\nusing this:\nmat (row_length A1) (length A1) A1 \\<and> mat (row_length A2) (length A2) A2 \\<and> mat (row_length B1) (length B1) B1 \\<and> mat (row_length B2) (length B2) B2 \\<and> length A1 = row_length A2 \\<and> length B1 = row_length B2 \\<and> A1 \\<noteq> [] \\<and> A2 \\<noteq> [] \\<and> B1 \\<noteq> [] \\<and> B2 \\<noteq> []\n(mat (row_length ?A1.0) (length ?A1.0) ?A1.0 \\<and> mat (row_length ?A2.0) (length ?A2.0) ?A2.0 \\<and> mat (row_length ?B1.0) (length ?B1.0) ?B1.0 \\<and> mat (row_length ?B2.0) (length ?B2.0) ?B2.0 \\<and> length ?A1.0 = row_length ?A2.0 \\<and> length ?B1.0 = row_length ?B2.0 \\<and> ?A1.0 \\<noteq> [] \\<and> ?A2.0 \\<noteq> [] \\<and> ?B1.0 \\<noteq> [] \\<and> ?B2.0 \\<noteq> []) \\<and> ?i < row_length ?A1.0 * row_length ?B1.0 \\<and> ?j < length ?A2.0 * length ?B2.0 \\<Longrightarrow> ((?A1.0 \\<otimes> ?B1.0) \\<circ> (?A2.0 \\<otimes> ?B2.0)) ! ?j ! ?i = scalar_product (vec_vec_Tensor (row ?A1.0 (?i div row_length ?B1.0)) (row ?B1.0 (?i mod row_length ?B1.0))) (vec_vec_Tensor (col ?A2.0 (?j div length ?B2.0)) (col ?B2.0 (?j mod length ?B2.0)))\n\ngoal (1 subgoal):\n 1. \\<forall>i j. i < row_length A1 * row_length B1 \\<and> j < length A2 * length B2 \\<longrightarrow> ((A1 \\<otimes> B1) \\<circ> (A2 \\<otimes> B2)) ! j ! i = scalar_product (vec_vec_Tensor (row A1 (i div row_length B1)) (row B1 (i mod row_length B1))) (vec_vec_Tensor (col A2 (j div length B2)) (col B2 (j mod length B2)))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 1416, "file": "Matrix_Tensor_Matrix_Tensor", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9230391706552536, "lm_q2_score": 0.8221891283434876, "lm_q1q2_score": 0.7589127711479386}}
{"text": "[STATEMENT]\nlemma odd_round_up:\nassumes \"odd x\"\nshows \"round (real_of_int x / 2) = (x+1) div 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. round (real_of_int x / 2) = (x + 1) div 2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. round (real_of_int x / 2) = (x + 1) div 2\n[PROOF STEP]\nhave \"round (real_of_int x / 2) = round (real_of_int (x+1) /2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. round (real_of_int x / 2) = round (real_of_int (x + 1) / 2)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nodd x\n\ngoal (1 subgoal):\n 1. round (real_of_int x / 2) = round (real_of_int (x + 1) / 2)\n[PROOF STEP]\nunfolding round_def\n[PROOF STATE]\nproof (prove)\nusing this:\nodd x\n\ngoal (1 subgoal):\n 1. \\<lfloor>real_of_int x / 2 + 1 / 2\\<rfloor> = \\<lfloor>real_of_int (x + 1) / 2 + 1 / 2\\<rfloor>\n[PROOF STEP]\nby (metis (no_types, opaque_lifting) add.commute \n      add_divide_distrib even_add even_succ_div_2 \n      floor_divide_of_int_eq odd_one of_int_add \n      of_int_hom.hom_one of_int_numeral)\n[PROOF STATE]\nproof (state)\nthis:\nround (real_of_int x / 2) = round (real_of_int (x + 1) / 2)\n\ngoal (1 subgoal):\n 1. round (real_of_int x / 2) = (x + 1) div 2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nround (real_of_int x / 2) = round (real_of_int (x + 1) / 2)\n\ngoal (1 subgoal):\n 1. round (real_of_int x / 2) = (x + 1) div 2\n[PROOF STEP]\nhave \"\\<dots> = (x+1) div 2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. round (real_of_int (x + 1) / 2) = (x + 1) div 2\n[PROOF STEP]\nby (metis add_divide_distrib calculation \n    floor_divide_of_int_eq of_int_add of_int_hom.hom_one \n    of_int_numeral round_def)\n[PROOF STATE]\nproof (state)\nthis:\nround (real_of_int (x + 1) / 2) = (x + 1) div 2\n\ngoal (1 subgoal):\n 1. round (real_of_int x / 2) = (x + 1) div 2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nround (real_of_int x / 2) = (x + 1) div 2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nround (real_of_int x / 2) = (x + 1) div 2\n\ngoal (1 subgoal):\n 1. round (real_of_int x / 2) = (x + 1) div 2\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nround (real_of_int x / 2) = (x + 1) div 2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1092, "file": "CRYSTALS-Kyber_Abs_Qr", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9059898203834277, "lm_q2_score": 0.8376199613065411, "lm_q1q2_score": 0.7588751582936869}}
{"text": "[STATEMENT]\nlemma poly_mult_degree_eq:\n  assumes \"subring K R\" \"polynomial K p1\" \"polynomial K p2\"\n  shows \"degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else (degree p1) + (degree p2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nproof (cases p1)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. p1 = [] \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n 2. \\<And>a list. p1 = a # list \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\ncase Nil\n[PROOF STATE]\nproof (state)\nthis:\np1 = []\n\ngoal (2 subgoals):\n 1. p1 = [] \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n 2. \\<And>a list. p1 = a # list \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\np1 = []\n\ngoal (1 subgoal):\n 1. degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndegree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n\ngoal (1 subgoal):\n 1. \\<And>a list. p1 = a # list \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>a list. p1 = a # list \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\ncase (Cons a p1')\n[PROOF STATE]\nproof (state)\nthis:\np1 = a # p1'\n\ngoal (1 subgoal):\n 1. \\<And>a list. p1 = a # list \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nnote p1 = Cons\n[PROOF STATE]\nproof (state)\nthis:\np1 = a # p1'\n\ngoal (1 subgoal):\n 1. \\<And>a list. p1 = a # list \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nproof (cases p2)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. p2 = [] \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n 2. \\<And>a list. p2 = a # list \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\ncase Nil\n[PROOF STATE]\nproof (state)\nthis:\np2 = []\n\ngoal (2 subgoals):\n 1. p2 = [] \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n 2. \\<And>a list. p2 = a # list \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\np2 = []\n\ngoal (1 subgoal):\n 1. degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nusing poly_mult_zero(2)[OF polynomial_in_carrier[OF assms(1-2)]]\n[PROOF STATE]\nproof (prove)\nusing this:\np2 = []\npoly_mult p1 [] = []\n\ngoal (1 subgoal):\n 1. degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndegree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n\ngoal (1 subgoal):\n 1. \\<And>a list. p2 = a # list \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>a list. p2 = a # list \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\ncase (Cons b p2')\n[PROOF STATE]\nproof (state)\nthis:\np2 = b # p2'\n\ngoal (1 subgoal):\n 1. \\<And>a list. p2 = a # list \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nnote p2 = Cons\n[PROOF STATE]\nproof (state)\nthis:\np2 = b # p2'\n\ngoal (1 subgoal):\n 1. \\<And>a list. p2 = a # list \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nhave a: \"a \\<in> carrier R\" and b: \"b \\<in> carrier R\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a \\<in> carrier R &&& b \\<in> carrier R\n[PROOF STEP]\nusing p1 p2 polynomial_in_carrier[OF assms(1-2)] polynomial_in_carrier[OF assms(1,3)]\n[PROOF STATE]\nproof (prove)\nusing this:\np1 = a # p1'\np2 = b # p2'\nset p1 \\<subseteq> carrier R\nset p2 \\<subseteq> carrier R\n\ngoal (1 subgoal):\n 1. a \\<in> carrier R &&& b \\<in> carrier R\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\na \\<in> carrier R\nb \\<in> carrier R\n\ngoal (1 subgoal):\n 1. \\<And>a list. p2 = a # list \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nhave \"(coeff (poly_mult p1 p2)) ((degree p1) + (degree p2)) = a \\<otimes> b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. local.coeff (poly_mult p1 p2) (degree p1 + degree p2) = a \\<otimes> b\n[PROOF STEP]\nusing poly_mult_lead_coeff_aux[OF assms] p1 p2\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>p1 \\<noteq> []; p2 \\<noteq> []\\<rbrakk> \\<Longrightarrow> local.coeff (poly_mult p1 p2) (degree p1 + degree p2) = lead_coeff p1 \\<otimes> lead_coeff p2\np1 = a # p1'\np2 = b # p2'\n\ngoal (1 subgoal):\n 1. local.coeff (poly_mult p1 p2) (degree p1 + degree p2) = a \\<otimes> b\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlocal.coeff (poly_mult p1 p2) (degree p1 + degree p2) = a \\<otimes> b\n\ngoal (1 subgoal):\n 1. \\<And>a list. p2 = a # list \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nhence neq0: \"(coeff (poly_mult p1 p2)) ((degree p1) + (degree p2)) \\<noteq> \\<zero>\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlocal.coeff (poly_mult p1 p2) (degree p1 + degree p2) = a \\<otimes> b\n\ngoal (1 subgoal):\n 1. local.coeff (poly_mult p1 p2) (degree p1 + degree p2) \\<noteq> \\<zero>\n[PROOF STEP]\nusing assms(2-3) integral[of a b] lead_coeff_in_carrier[OF assms(1)] p1 p2\n[PROOF STATE]\nproof (prove)\nusing this:\nlocal.coeff (poly_mult p1 p2) (degree p1 + degree p2) = a \\<otimes> b\npolynomial K p1\npolynomial K p2\n\\<lbrakk>a \\<otimes> b = \\<zero>; a \\<in> carrier R; b \\<in> carrier R\\<rbrakk> \\<Longrightarrow> a = \\<zero> \\<or> b = \\<zero>\npolynomial K (?a # ?p) \\<Longrightarrow> ?a \\<in> carrier R - {\\<zero>}\np1 = a # p1'\np2 = b # p2'\n\ngoal (1 subgoal):\n 1. local.coeff (poly_mult p1 p2) (degree p1 + degree p2) \\<noteq> \\<zero>\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlocal.coeff (poly_mult p1 p2) (degree p1 + degree p2) \\<noteq> \\<zero>\n\ngoal (1 subgoal):\n 1. \\<And>a list. p2 = a # list \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nlocal.coeff (poly_mult p1 p2) (degree p1 + degree p2) \\<noteq> \\<zero>\n\ngoal (1 subgoal):\n 1. \\<And>a list. p2 = a # list \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nhave eq0: \"\\<And>i. i > (degree p1) + (degree p2) \\<Longrightarrow> (coeff (poly_mult p1 p2)) i = \\<zero>\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>i. degree p1 + degree p2 < i \\<Longrightarrow> local.coeff (poly_mult p1 p2) i = \\<zero>\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>i. degree p1 + degree p2 < i \\<Longrightarrow> local.coeff (poly_mult p1 p2) i = \\<zero>\n[PROOF STEP]\nhave aux_lemma: \"degree (poly_mult p1 p2) \\<le> (degree p1) + (degree p2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. degree (poly_mult p1 p2) \\<le> degree p1 + degree p2\n[PROOF STEP]\nproof (induct p1)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. degree (poly_mult [] p2) \\<le> degree [] + degree p2\n 2. \\<And>a p1. degree (poly_mult p1 p2) \\<le> degree p1 + degree p2 \\<Longrightarrow> degree (poly_mult (a # p1) p2) \\<le> degree (a # p1) + degree p2\n[PROOF STEP]\ncase Nil\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. degree (poly_mult [] p2) \\<le> degree [] + degree p2\n 2. \\<And>a p1. degree (poly_mult p1 p2) \\<le> degree p1 + degree p2 \\<Longrightarrow> degree (poly_mult (a # p1) p2) \\<le> degree (a # p1) + degree p2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. degree (poly_mult [] p2) \\<le> degree [] + degree p2\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndegree (poly_mult [] p2) \\<le> degree [] + degree p2\n\ngoal (1 subgoal):\n 1. \\<And>a p1. degree (poly_mult p1 p2) \\<le> degree p1 + degree p2 \\<Longrightarrow> degree (poly_mult (a # p1) p2) \\<le> degree (a # p1) + degree p2\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>a p1. degree (poly_mult p1 p2) \\<le> degree p1 + degree p2 \\<Longrightarrow> degree (poly_mult (a # p1) p2) \\<le> degree (a # p1) + degree p2\n[PROOF STEP]\ncase (Cons a p1)\n[PROOF STATE]\nproof (state)\nthis:\ndegree (poly_mult p1 p2) \\<le> degree p1 + degree p2\n\ngoal (1 subgoal):\n 1. \\<And>a p1. degree (poly_mult p1 p2) \\<le> degree p1 + degree p2 \\<Longrightarrow> degree (poly_mult (a # p1) p2) \\<le> degree (a # p1) + degree p2\n[PROOF STEP]\nlet ?a_p2 = \"(map (\\<lambda>b. a \\<otimes> b) p2) @ (replicate (degree (a # p1)) \\<zero>)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>a p1. degree (poly_mult p1 p2) \\<le> degree p1 + degree p2 \\<Longrightarrow> degree (poly_mult (a # p1) p2) \\<le> degree (a # p1) + degree p2\n[PROOF STEP]\nhave \"poly_mult (a # p1) p2 = poly_add ?a_p2 (poly_mult p1 p2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poly_mult (a # p1) p2 = poly_add (map ((\\<otimes>) a) p2 @ replicate (degree (a # p1)) \\<zero>) (poly_mult p1 p2)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\npoly_mult (a # p1) p2 = poly_add (map ((\\<otimes>) a) p2 @ replicate (degree (a # p1)) \\<zero>) (poly_mult p1 p2)\n\ngoal (1 subgoal):\n 1. \\<And>a p1. degree (poly_mult p1 p2) \\<le> degree p1 + degree p2 \\<Longrightarrow> degree (poly_mult (a # p1) p2) \\<le> degree (a # p1) + degree p2\n[PROOF STEP]\nhence \"degree (poly_mult (a # p1) p2) \\<le> max (degree ?a_p2) (degree (poly_mult p1 p2))\"\n[PROOF STATE]\nproof (prove)\nusing this:\npoly_mult (a # p1) p2 = poly_add (map ((\\<otimes>) a) p2 @ replicate (degree (a # p1)) \\<zero>) (poly_mult p1 p2)\n\ngoal (1 subgoal):\n 1. degree (poly_mult (a # p1) p2) \\<le> max (degree (map ((\\<otimes>) a) p2 @ replicate (degree (a # p1)) \\<zero>)) (degree (poly_mult p1 p2))\n[PROOF STEP]\nusing poly_add_degree[of ?a_p2 \"poly_mult p1 p2\"]\n[PROOF STATE]\nproof (prove)\nusing this:\npoly_mult (a # p1) p2 = poly_add (map ((\\<otimes>) a) p2 @ replicate (degree (a # p1)) \\<zero>) (poly_mult p1 p2)\ndegree (poly_add (map ((\\<otimes>) a) p2 @ replicate (degree (a # p1)) \\<zero>) (poly_mult p1 p2)) \\<le> max (degree (map ((\\<otimes>) a) p2 @ replicate (degree (a # p1)) \\<zero>)) (degree (poly_mult p1 p2))\n\ngoal (1 subgoal):\n 1. degree (poly_mult (a # p1) p2) \\<le> max (degree (map ((\\<otimes>) a) p2 @ replicate (degree (a # p1)) \\<zero>)) (degree (poly_mult p1 p2))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndegree (poly_mult (a # p1) p2) \\<le> max (degree (map ((\\<otimes>) a) p2 @ replicate (degree (a # p1)) \\<zero>)) (degree (poly_mult p1 p2))\n\ngoal (1 subgoal):\n 1. \\<And>a p1. degree (poly_mult p1 p2) \\<le> degree p1 + degree p2 \\<Longrightarrow> degree (poly_mult (a # p1) p2) \\<le> degree (a # p1) + degree p2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ndegree (poly_mult (a # p1) p2) \\<le> max (degree (map ((\\<otimes>) a) p2 @ replicate (degree (a # p1)) \\<zero>)) (degree (poly_mult p1 p2))\n\ngoal (1 subgoal):\n 1. \\<And>a p1. degree (poly_mult p1 p2) \\<le> degree p1 + degree p2 \\<Longrightarrow> degree (poly_mult (a # p1) p2) \\<le> degree (a # p1) + degree p2\n[PROOF STEP]\nhave \" ... \\<le> max ((degree (a # p1)) + (degree p2)) (degree (poly_mult p1 p2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. max (degree (map ((\\<otimes>) a) p2 @ replicate (degree (a # p1)) \\<zero>)) (degree (poly_mult p1 p2)) \\<le> max (degree (a # p1) + degree p2) (degree (poly_mult p1 p2))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nmax (degree (map ((\\<otimes>) a) p2 @ replicate (degree (a # p1)) \\<zero>)) (degree (poly_mult p1 p2)) \\<le> max (degree (a # p1) + degree p2) (degree (poly_mult p1 p2))\n\ngoal (1 subgoal):\n 1. \\<And>a p1. degree (poly_mult p1 p2) \\<le> degree p1 + degree p2 \\<Longrightarrow> degree (poly_mult (a # p1) p2) \\<le> degree (a # p1) + degree p2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmax (degree (map ((\\<otimes>) a) p2 @ replicate (degree (a # p1)) \\<zero>)) (degree (poly_mult p1 p2)) \\<le> max (degree (a # p1) + degree p2) (degree (poly_mult p1 p2))\n\ngoal (1 subgoal):\n 1. \\<And>a p1. degree (poly_mult p1 p2) \\<le> degree p1 + degree p2 \\<Longrightarrow> degree (poly_mult (a # p1) p2) \\<le> degree (a # p1) + degree p2\n[PROOF STEP]\nhave \" ... \\<le> max ((degree (a # p1)) + (degree p2)) ((degree p1) + (degree p2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. max (degree (a # p1) + degree p2) (degree (poly_mult p1 p2)) \\<le> max (degree (a # p1) + degree p2) (degree p1 + degree p2)\n[PROOF STEP]\nusing Cons\n[PROOF STATE]\nproof (prove)\nusing this:\ndegree (poly_mult p1 p2) \\<le> degree p1 + degree p2\n\ngoal (1 subgoal):\n 1. max (degree (a # p1) + degree p2) (degree (poly_mult p1 p2)) \\<le> max (degree (a # p1) + degree p2) (degree p1 + degree p2)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nmax (degree (a # p1) + degree p2) (degree (poly_mult p1 p2)) \\<le> max (degree (a # p1) + degree p2) (degree p1 + degree p2)\n\ngoal (1 subgoal):\n 1. \\<And>a p1. degree (poly_mult p1 p2) \\<le> degree p1 + degree p2 \\<Longrightarrow> degree (poly_mult (a # p1) p2) \\<le> degree (a # p1) + degree p2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmax (degree (a # p1) + degree p2) (degree (poly_mult p1 p2)) \\<le> max (degree (a # p1) + degree p2) (degree p1 + degree p2)\n\ngoal (1 subgoal):\n 1. \\<And>a p1. degree (poly_mult p1 p2) \\<le> degree p1 + degree p2 \\<Longrightarrow> degree (poly_mult (a # p1) p2) \\<le> degree (a # p1) + degree p2\n[PROOF STEP]\nhave \" ... \\<le> (degree (a # p1)) + (degree p2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. max (degree (a # p1) + degree p2) (degree p1 + degree p2) \\<le> degree (a # p1) + degree p2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nmax (degree (a # p1) + degree p2) (degree p1 + degree p2) \\<le> degree (a # p1) + degree p2\n\ngoal (1 subgoal):\n 1. \\<And>a p1. degree (poly_mult p1 p2) \\<le> degree p1 + degree p2 \\<Longrightarrow> degree (poly_mult (a # p1) p2) \\<le> degree (a # p1) + degree p2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ndegree (poly_mult (a # p1) p2) \\<le> degree (a # p1) + degree p2\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\ndegree (poly_mult (a # p1) p2) \\<le> degree (a # p1) + degree p2\n\ngoal (1 subgoal):\n 1. degree (poly_mult (a # p1) p2) \\<le> degree (a # p1) + degree p2\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ndegree (poly_mult (a # p1) p2) \\<le> degree (a # p1) + degree p2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ndegree (poly_mult p1 p2) \\<le> degree p1 + degree p2\n\ngoal (1 subgoal):\n 1. \\<And>i. degree p1 + degree p2 < i \\<Longrightarrow> local.coeff (poly_mult p1 p2) i = \\<zero>\n[PROOF STEP]\nfix i\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>i. degree p1 + degree p2 < i \\<Longrightarrow> local.coeff (poly_mult p1 p2) i = \\<zero>\n[PROOF STEP]\nshow \"i > (degree p1) + (degree p2) \\<Longrightarrow> (coeff (poly_mult p1 p2)) i = \\<zero>\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. degree p1 + degree p2 < i \\<Longrightarrow> local.coeff (poly_mult p1 p2) i = \\<zero>\n[PROOF STEP]\nusing coeff_degree aux_lemma\n[PROOF STATE]\nproof (prove)\nusing this:\ndegree ?p < ?i \\<Longrightarrow> local.coeff ?p ?i = \\<zero>\ndegree (poly_mult p1 p2) \\<le> degree p1 + degree p2\n\ngoal (1 subgoal):\n 1. degree p1 + degree p2 < i \\<Longrightarrow> local.coeff (poly_mult p1 p2) i = \\<zero>\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndegree p1 + degree p2 < i \\<Longrightarrow> local.coeff (poly_mult p1 p2) i = \\<zero>\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ndegree p1 + degree p2 < ?i \\<Longrightarrow> local.coeff (poly_mult p1 p2) ?i = \\<zero>\n\ngoal (1 subgoal):\n 1. \\<And>a list. p2 = a # list \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ndegree p1 + degree p2 < ?i \\<Longrightarrow> local.coeff (poly_mult p1 p2) ?i = \\<zero>\n\ngoal (1 subgoal):\n 1. \\<And>a list. p2 = a # list \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nhave \"polynomial K (poly_mult p1 p2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. polynomial K (poly_mult p1 p2)\n[PROOF STEP]\nby (simp add: assms poly_mult_closed)\n[PROOF STATE]\nproof (state)\nthis:\npolynomial K (poly_mult p1 p2)\n\ngoal (1 subgoal):\n 1. \\<And>a list. p2 = a # list \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nlocal.coeff (poly_mult p1 p2) (degree p1 + degree p2) \\<noteq> \\<zero>\ndegree p1 + degree p2 < ?i \\<Longrightarrow> local.coeff (poly_mult p1 p2) ?i = \\<zero>\npolynomial K (poly_mult p1 p2)\n[PROOF STEP]\nhave \"degree (poly_mult p1 p2) = degree p1 + degree p2\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlocal.coeff (poly_mult p1 p2) (degree p1 + degree p2) \\<noteq> \\<zero>\ndegree p1 + degree p2 < ?i \\<Longrightarrow> local.coeff (poly_mult p1 p2) ?i = \\<zero>\npolynomial K (poly_mult p1 p2)\n\ngoal (1 subgoal):\n 1. degree (poly_mult p1 p2) = degree p1 + degree p2\n[PROOF STEP]\nby (metis (no_types) assms(1) coeff.simps(1) coeff_degree domain.poly_mult_one(1) domain_axioms eq0 lead_coeff_simp length_greater_0_conv neq0 normalize_length_lt not_less_iff_gr_or_eq poly_mult_one'(1) polynomial_in_carrier)\n[PROOF STATE]\nproof (state)\nthis:\ndegree (poly_mult p1 p2) = degree p1 + degree p2\n\ngoal (1 subgoal):\n 1. \\<And>a list. p2 = a # list \\<Longrightarrow> degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndegree (poly_mult p1 p2) = degree p1 + degree p2\n\ngoal (1 subgoal):\n 1. degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nusing p1 p2\n[PROOF STATE]\nproof (prove)\nusing this:\ndegree (poly_mult p1 p2) = degree p1 + degree p2\np1 = a # p1'\np2 = b # p2'\n\ngoal (1 subgoal):\n 1. degree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ndegree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ndegree (poly_mult p1 p2) = (if p1 = [] \\<or> p2 = [] then 0 else degree p1 + degree p2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 8454, "file": null, "length": 72, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278726384089, "lm_q2_score": 0.8596637559030338, "lm_q1q2_score": 0.7585912593058586}}
{"text": "[STATEMENT]\nlemma scalar_product_distributivity:\n\"\\<forall>v1 v2 w1 w2.((length v1 = length v2)\\<and>(length v1 = n)\\<and> (length w1 = length w2)\n           \\<longrightarrow>  (scalar_product v1 v2)*(scalar_product w1 w2)\n      = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)) \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<forall>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\napply (rule allI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>v1. \\<forall>v2 w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\napply (rule allI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>v1 v2. \\<forall>w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\napply (rule allI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>v1 v2 w1. \\<forall>w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\napply (rule allI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nproof(induct \"n\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n 2. \\<And>n v1 v2 w1 w2. (\\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)) \\<Longrightarrow> length v1 = length v2 \\<and> length v1 = Suc n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. \\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n 2. \\<And>n v1 v2 w1 w2. (\\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)) \\<Longrightarrow> length v1 = length v2 \\<and> length v1 = Suc n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nhave \"((length v1 = length v2)\\<and>(length v1 = 0)\\<and> (length w1 = length w2))\n           \\<longrightarrow>length v1 = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> length v1 = 0\n[PROOF STEP]\nusing 0\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> length v1 = 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> length v1 = 0\n\ngoal (2 subgoals):\n 1. \\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n 2. \\<And>n v1 v2 w1 w2. (\\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)) \\<Longrightarrow> length v1 = length v2 \\<and> length v1 = Suc n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> length v1 = 0\n[PROOF STEP]\nhave 1:\"((length v1 = length v2)\n                 \\<and>(length v1 = 0)\n                 \\<and>(length w1 = length w2))\n                        \\<longrightarrow>v1 = []\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> length v1 = 0\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> v1 = []\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> v1 = []\n\ngoal (2 subgoals):\n 1. \\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n 2. \\<And>n v1 v2 w1 w2. (\\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)) \\<Longrightarrow> length v1 = length v2 \\<and> length v1 = Suc n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> v1 = []\n\ngoal (2 subgoals):\n 1. \\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n 2. \\<And>n v1 v2 w1 w2. (\\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)) \\<Longrightarrow> length v1 = length v2 \\<and> length v1 = Suc n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nhave \"((length v1 = length v2)\n                   \\<and>(length v1 = 0)\n                   \\<and>(length w1 = length w2))\n           \\<longrightarrow>length v2 = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> length v2 = 0\n[PROOF STEP]\nusing 0\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> length v2 = 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> length v2 = 0\n\ngoal (2 subgoals):\n 1. \\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n 2. \\<And>n v1 v2 w1 w2. (\\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)) \\<Longrightarrow> length v1 = length v2 \\<and> length v1 = Suc n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> length v2 = 0\n\ngoal (2 subgoals):\n 1. \\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n 2. \\<And>n v1 v2 w1 w2. (\\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)) \\<Longrightarrow> length v1 = length v2 \\<and> length v1 = Suc n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> length v2 = 0\n[PROOF STEP]\nhave 2:\"((length v1 = length v2)\n                          \\<and>(length v1 = 0)\n                          \\<and>(length w1 = length w2))\n                                    \\<longrightarrow>v2 = []\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> length v2 = 0\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> v2 = []\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> v2 = []\n\ngoal (2 subgoals):\n 1. \\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n 2. \\<And>n v1 v2 w1 w2. (\\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)) \\<Longrightarrow> length v1 = length v2 \\<and> length v1 = Suc n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> v1 = []\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> length v2 = 0\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> v2 = []\n[PROOF STEP]\nhave 3:\n         \"((length v1 = length v2)\\<and>(length v1 = 0)\\<and> (length w1 = length w2))\n           \\<longrightarrow>scalar_product v1 v2 = zer\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> v1 = []\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> length v2 = 0\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> v2 = []\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 = zer\n[PROOF STEP]\nunfolding scalar_product_def scalar_prodI_def\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> v1 = []\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> length v2 = 0\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> v2 = []\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> foldr (\\<lambda>(x, y). (+) (x * y)) (zip v1 v2) zer = zer\n[PROOF STEP]\nusing zip_Nil\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> v1 = []\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> length v2 = 0\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> v2 = []\nzip [] [] = []\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> foldr (\\<lambda>(x, y). (+) (x * y)) (zip v1 v2) zer = zer\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 = zer\n\ngoal (2 subgoals):\n 1. \\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n 2. \\<And>n v1 v2 w1 w2. (\\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)) \\<Longrightarrow> length v1 = length v2 \\<and> length v1 = Suc n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 = zer\n[PROOF STEP]\nhave 4:\"f zer (scalar_product w1 w2) = zer\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 = zer\n\ngoal (1 subgoal):\n 1. zer * scalar_product w1 w2 = zer\n[PROOF STEP]\nusing zer_left_mult\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 = zer\nzer * ?x = zer\n\ngoal (1 subgoal):\n 1. zer * scalar_product w1 w2 = zer\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nzer * scalar_product w1 w2 = zer\n\ngoal (2 subgoals):\n 1. \\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n 2. \\<And>n v1 v2 w1 w2. (\\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)) \\<Longrightarrow> length v1 = length v2 \\<and> length v1 = Suc n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nhave \"((length v1 = length v2)\\<and>(length v1 = 0)\\<and> (length w1 = length w2))\n           \\<longrightarrow>vec_vec_Tensor v1 w1 = []\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> vec_vec_Tensor v1 w1 = []\n[PROOF STEP]\nusing 1\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> v1 = []\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> vec_vec_Tensor v1 w1 = []\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> vec_vec_Tensor v1 w1 = []\n\ngoal (2 subgoals):\n 1. \\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n 2. \\<And>n v1 v2 w1 w2. (\\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)) \\<Longrightarrow> length v1 = length v2 \\<and> length v1 = Suc n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> vec_vec_Tensor v1 w1 = []\n\ngoal (2 subgoals):\n 1. \\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n 2. \\<And>n v1 v2 w1 w2. (\\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)) \\<Longrightarrow> length v1 = length v2 \\<and> length v1 = Suc n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nhave \"((length v1 = length v2)\n                   \\<and>(length v1 = 0)\n                   \\<and>(length w1 = length w2))\n                    \\<longrightarrow>vec_vec_Tensor v2 w2 = []\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> vec_vec_Tensor v2 w2 = []\n[PROOF STEP]\nusing 2\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> v2 = []\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> vec_vec_Tensor v2 w2 = []\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> vec_vec_Tensor v2 w2 = []\n\ngoal (2 subgoals):\n 1. \\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n 2. \\<And>n v1 v2 w1 w2. (\\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)) \\<Longrightarrow> length v1 = length v2 \\<and> length v1 = Suc n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> vec_vec_Tensor v1 w1 = []\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> vec_vec_Tensor v2 w2 = []\n[PROOF STEP]\nhave \"((length v1 = length v2)\n                      \\<and>(length v1 = 0)\n                      \\<and>(length w1 = length w2))\n                         \\<longrightarrow> scalar_product \n                                 (vec_vec_Tensor v1 w1) \n                                 (vec_vec_Tensor v2 w2)  = zer\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> vec_vec_Tensor v1 w1 = []\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> vec_vec_Tensor v2 w2 = []\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2) = zer\n[PROOF STEP]\nunfolding scalar_product_def scalar_prodI_def\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> vec_vec_Tensor v1 w1 = []\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> vec_vec_Tensor v2 w2 = []\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> foldr (\\<lambda>(x, y). (+) (x * y)) (zip (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)) zer = zer\n[PROOF STEP]\nusing zip_Nil\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> vec_vec_Tensor v1 w1 = []\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> vec_vec_Tensor v2 w2 = []\nzip [] [] = []\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> foldr (\\<lambda>(x, y). (+) (x * y)) (zip (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)) zer = zer\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2) = zer\n\ngoal (2 subgoals):\n 1. \\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n 2. \\<And>n v1 v2 w1 w2. (\\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)) \\<Longrightarrow> length v1 = length v2 \\<and> length v1 = Suc n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nwith 3 4\n[PROOF STATE]\nproof (chain)\npicking this:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 = zer\nzer * scalar_product w1 w2 = zer\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2) = zer\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 = zer\nzer * scalar_product w1 w2 = zer\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2) = zer\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlength v1 = length v2 \\<and> length v1 = 0 \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n\ngoal (1 subgoal):\n 1. \\<And>n v1 v2 w1 w2. (\\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)) \\<Longrightarrow> length v1 = length v2 \\<and> length v1 = Suc n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>n v1 v2 w1 w2. (\\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)) \\<Longrightarrow> length v1 = length v2 \\<and> length v1 = Suc n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\ncase (Suc k)\n[PROOF STATE]\nproof (state)\nthis:\nlength ?v1.0 = length ?v2.0 \\<and> length ?v1.0 = k \\<and> length ?w1.0 = length ?w2.0 \\<longrightarrow> scalar_product ?v1.0 ?v2.0 * scalar_product ?w1.0 ?w2.0 = scalar_product (vec_vec_Tensor ?v1.0 ?w1.0) (vec_vec_Tensor ?v2.0 ?w2.0)\n\ngoal (1 subgoal):\n 1. \\<And>n v1 v2 w1 w2. (\\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)) \\<Longrightarrow> length v1 = length v2 \\<and> length v1 = Suc n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nhave \"((length v1 = length v2)\\<and>(length v1 = Suc k)\n                    \\<and> (length w1 = length w2))\n           \\<Longrightarrow>  f (scalar_product v1 v2) (scalar_product w1 w2)\n      = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nassume assms:\"((length v1 = length v2)\\<and>(length v1 = Suc k)\n                    \\<and> (length w1 = length w2))\"\n[PROOF STATE]\nproof (state)\nthis:\nlength v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nhave \"length v1 = Suc k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length v1 = Suc k\n[PROOF STEP]\nusing Suc assms\n[PROOF STATE]\nproof (prove)\nusing this:\nlength ?v1.0 = length ?v2.0 \\<and> length ?v1.0 = k \\<and> length ?w1.0 = length ?w2.0 \\<longrightarrow> scalar_product ?v1.0 ?v2.0 * scalar_product ?w1.0 ?w2.0 = scalar_product (vec_vec_Tensor ?v1.0 ?w1.0) (vec_vec_Tensor ?v2.0 ?w2.0)\nlength v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2\n\ngoal (1 subgoal):\n 1. length v1 = Suc k\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlength v1 = Suc k\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nlength v1 = Suc k\n[PROOF STEP]\nhave \"(\\<exists>a1 u1.(v1 = a1#u1)\\<and>(length u1 = k))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v1 = Suc k\n\ngoal (1 subgoal):\n 1. \\<exists>a1 u1. v1 = a1 # u1 \\<and> length u1 = k\n[PROOF STEP]\nusing assms Suc_length_conv\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v1 = Suc k\nlength v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2\n(Suc ?n = length ?xs) = (\\<exists>y ys. ?xs = y # ys \\<and> length ys = ?n)\n\ngoal (1 subgoal):\n 1. \\<exists>a1 u1. v1 = a1 # u1 \\<and> length u1 = k\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\n\\<exists>a1 u1. v1 = a1 # u1 \\<and> length u1 = k\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\<exists>a1 u1. v1 = a1 # u1 \\<and> length u1 = k\n[PROOF STEP]\nobtain a1 u1 where \"(v1 = a1#u1)\\<and>(length u1 = k)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<exists>a1 u1. v1 = a1 # u1 \\<and> length u1 = k\n\ngoal (1 subgoal):\n 1. (\\<And>a1 u1. v1 = a1 # u1 \\<and> length u1 = k \\<Longrightarrow> thesis) \\<Longrightarrow> thesis\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<exists>a1 u1. v1 = a1 # u1 \\<and> length u1 = k\nlength v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2\n\ngoal (1 subgoal):\n 1. (\\<And>a1 u1. v1 = a1 # u1 \\<and> length u1 = k \\<Longrightarrow> thesis) \\<Longrightarrow> thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nv1 = a1 # u1 \\<and> length u1 = k\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nv1 = a1 # u1 \\<and> length u1 = k\n[PROOF STEP]\nhave Cons_1:\"(v1 = a1#u1)\\<and>(length u1 = k)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nv1 = a1 # u1 \\<and> length u1 = k\n\ngoal (1 subgoal):\n 1. v1 = a1 # u1 \\<and> length u1 = k\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nv1 = a1 # u1 \\<and> length u1 = k\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nv1 = a1 # u1 \\<and> length u1 = k\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nhave \"length v2 = Suc k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length v2 = Suc k\n[PROOF STEP]\nusing assms Suc\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2\nlength ?v1.0 = length ?v2.0 \\<and> length ?v1.0 = k \\<and> length ?w1.0 = length ?w2.0 \\<longrightarrow> scalar_product ?v1.0 ?v2.0 * scalar_product ?w1.0 ?w2.0 = scalar_product (vec_vec_Tensor ?v1.0 ?w1.0) (vec_vec_Tensor ?v2.0 ?w2.0)\n\ngoal (1 subgoal):\n 1. length v2 = Suc k\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlength v2 = Suc k\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nlength v2 = Suc k\n[PROOF STEP]\nhave \"(\\<exists>a2 u2.(v2 = a2#u2)\\<and>(length u2 = k))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v2 = Suc k\n\ngoal (1 subgoal):\n 1. \\<exists>a2 u2. v2 = a2 # u2 \\<and> length u2 = k\n[PROOF STEP]\nusing  Suc_length_conv\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v2 = Suc k\n(Suc ?n = length ?xs) = (\\<exists>y ys. ?xs = y # ys \\<and> length ys = ?n)\n\ngoal (1 subgoal):\n 1. \\<exists>a2 u2. v2 = a2 # u2 \\<and> length u2 = k\n[PROOF STEP]\nby metis\n[PROOF STATE]\nproof (state)\nthis:\n\\<exists>a2 u2. v2 = a2 # u2 \\<and> length u2 = k\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\<exists>a2 u2. v2 = a2 # u2 \\<and> length u2 = k\n[PROOF STEP]\nobtain a2 u2 where \"(v2 = a2#u2)\\<and>(length u2 = k)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<exists>a2 u2. v2 = a2 # u2 \\<and> length u2 = k\n\ngoal (1 subgoal):\n 1. (\\<And>a2 u2. v2 = a2 # u2 \\<and> length u2 = k \\<Longrightarrow> thesis) \\<Longrightarrow> thesis\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nv2 = a2 # u2 \\<and> length u2 = k\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nv2 = a2 # u2 \\<and> length u2 = k\n[PROOF STEP]\nhave Cons_2: \"(v2 = a2#u2)\\<and>(length u2 = k)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nv2 = a2 # u2 \\<and> length u2 = k\n\ngoal (1 subgoal):\n 1. v2 = a2 # u2 \\<and> length u2 = k\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nv2 = a2 # u2 \\<and> length u2 = k\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nv2 = a2 # u2 \\<and> length u2 = k\n[PROOF STEP]\nhave \"length u1 = length u2\"\n[PROOF STATE]\nproof (prove)\nusing this:\nv2 = a2 # u2 \\<and> length u2 = k\n\ngoal (1 subgoal):\n 1. length u1 = length u2\n[PROOF STEP]\nusing Cons_1\n[PROOF STATE]\nproof (prove)\nusing this:\nv2 = a2 # u2 \\<and> length u2 = k\nv1 = a1 # u1 \\<and> length u1 = k\n\ngoal (1 subgoal):\n 1. length u1 = length u2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlength u1 = length u2\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nlength u1 = length u2\n[PROOF STEP]\nhave Cons_3:\"(scalar_product u1 u2) * scalar_product w1 w2 =\n         scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlength u1 = length u2\n\ngoal (1 subgoal):\n 1. scalar_product u1 u2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\n[PROOF STEP]\nusing Suc Cons_1 Cons_2 assms\n[PROOF STATE]\nproof (prove)\nusing this:\nlength u1 = length u2\nlength ?v1.0 = length ?v2.0 \\<and> length ?v1.0 = k \\<and> length ?w1.0 = length ?w2.0 \\<longrightarrow> scalar_product ?v1.0 ?v2.0 * scalar_product ?w1.0 ?w2.0 = scalar_product (vec_vec_Tensor ?v1.0 ?w1.0) (vec_vec_Tensor ?v2.0 ?w2.0)\nv1 = a1 # u1 \\<and> length u1 = k\nv2 = a2 # u2 \\<and> length u2 = k\nlength v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2\n\ngoal (1 subgoal):\n 1. scalar_product u1 u2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product u1 u2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nscalar_product u1 u2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\n[PROOF STEP]\nhave \"zip v1 v2 = (a1,a2)#(zip u1 u2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nscalar_product u1 u2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\n\ngoal (1 subgoal):\n 1. zip v1 v2 = (a1, a2) # zip u1 u2\n[PROOF STEP]\nusing  zip_Cons Cons_1 Cons_2\n[PROOF STATE]\nproof (prove)\nusing this:\nscalar_product u1 u2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\nlength ?v = length ?w \\<Longrightarrow> zip (?a # ?v) (?b # ?w) = (?a, ?b) # zip ?v ?w\nv1 = a1 # u1 \\<and> length u1 = k\nv2 = a2 # u2 \\<and> length u2 = k\n\ngoal (1 subgoal):\n 1. zip v1 v2 = (a1, a2) # zip u1 u2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nzip v1 v2 = (a1, a2) # zip u1 u2\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nzip v1 v2 = (a1, a2) # zip u1 u2\n[PROOF STEP]\nhave Cons_4:\"scalar_product v1 v2 =  (a1*a2)+ (scalar_product u1 u2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nzip v1 v2 = (a1, a2) # zip u1 u2\n\ngoal (1 subgoal):\n 1. scalar_product v1 v2 = a1 * a2 + scalar_product u1 u2\n[PROOF STEP]\nunfolding scalar_product_def scalar_prodI_def\n[PROOF STATE]\nproof (prove)\nusing this:\nzip v1 v2 = (a1, a2) # zip u1 u2\n\ngoal (1 subgoal):\n 1. foldr (\\<lambda>(x, y). (+) (x * y)) (zip v1 v2) zer = a1 * a2 + foldr (\\<lambda>(x, y). (+) (x * y)) (zip u1 u2) zer\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product v1 v2 = a1 * a2 + scalar_product u1 u2\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nscalar_product v1 v2 = a1 * a2 + scalar_product u1 u2\n[PROOF STEP]\nhave \"f (scalar_product v1 v2) (scalar_product w1 w2)\n                      = ((a1*a2)+ (scalar_product u1 u2))*(scalar_product w1 w2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nscalar_product v1 v2 = a1 * a2 + scalar_product u1 u2\n\ngoal (1 subgoal):\n 1. scalar_product v1 v2 * scalar_product w1 w2 = a1 * a2 + scalar_product u1 u2 * scalar_product w1 w2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product v1 v2 * scalar_product w1 w2 = a1 * a2 + scalar_product u1 u2 * scalar_product w1 w2\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nscalar_product v1 v2 * scalar_product w1 w2 = a1 * a2 + scalar_product u1 u2 * scalar_product w1 w2\n[PROOF STEP]\nhave \"... = ((a1*a2)*(scalar_product w1 w2)) \n                                     + ((scalar_product u1 u2)*(scalar_product w1 w2))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nscalar_product v1 v2 * scalar_product w1 w2 = a1 * a2 + scalar_product u1 u2 * scalar_product w1 w2\n\ngoal (1 subgoal):\n 1. a1 * a2 + scalar_product u1 u2 * scalar_product w1 w2 = a1 * a2 * scalar_product w1 w2 + (scalar_product u1 u2 * scalar_product w1 w2)\n[PROOF STEP]\nusing plus_right_distributivity\n[PROOF STATE]\nproof (prove)\nusing this:\nscalar_product v1 v2 * scalar_product w1 w2 = a1 * a2 + scalar_product u1 u2 * scalar_product w1 w2\n?a + ?b * ?c = ?a * ?c + (?b * ?c)\n\ngoal (1 subgoal):\n 1. a1 * a2 + scalar_product u1 u2 * scalar_product w1 w2 = a1 * a2 * scalar_product w1 w2 + (scalar_product u1 u2 * scalar_product w1 w2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\na1 * a2 + scalar_product u1 u2 * scalar_product w1 w2 = a1 * a2 * scalar_product w1 w2 + (scalar_product u1 u2 * scalar_product w1 w2)\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\na1 * a2 + scalar_product u1 u2 * scalar_product w1 w2 = a1 * a2 * scalar_product w1 w2 + (scalar_product u1 u2 * scalar_product w1 w2)\n[PROOF STEP]\nhave Cons_5:\"... = ((a1*a2)*(scalar_product w1 w2))\n                       + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\na1 * a2 + scalar_product u1 u2 * scalar_product w1 w2 = a1 * a2 * scalar_product w1 w2 + (scalar_product u1 u2 * scalar_product w1 w2)\n\ngoal (1 subgoal):\n 1. a1 * a2 * scalar_product w1 w2 + (scalar_product u1 u2 * scalar_product w1 w2) = a1 * a2 * scalar_product w1 w2 + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\n[PROOF STEP]\nusing Cons_3\n[PROOF STATE]\nproof (prove)\nusing this:\na1 * a2 + scalar_product u1 u2 * scalar_product w1 w2 = a1 * a2 * scalar_product w1 w2 + (scalar_product u1 u2 * scalar_product w1 w2)\nscalar_product u1 u2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\n\ngoal (1 subgoal):\n 1. a1 * a2 * scalar_product w1 w2 + (scalar_product u1 u2 * scalar_product w1 w2) = a1 * a2 * scalar_product w1 w2 + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\na1 * a2 * scalar_product w1 w2 + (scalar_product u1 u2 * scalar_product w1 w2) = a1 * a2 * scalar_product w1 w2 + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\na1 * a2 * scalar_product w1 w2 + (scalar_product u1 u2 * scalar_product w1 w2) = a1 * a2 * scalar_product w1 w2 + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\n[PROOF STEP]\nhave Cons_6:\"... = (scalar_product (times a1 w1) (times a2 w2))\n                    +  scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\na1 * a2 * scalar_product w1 w2 + (scalar_product u1 u2 * scalar_product w1 w2) = a1 * a2 * scalar_product w1 w2 + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\n\ngoal (1 subgoal):\n 1. a1 * a2 * scalar_product w1 w2 + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2) = scalar_product (local.times a1 w1) (local.times a2 w2) + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\n[PROOF STEP]\nusing assms effective_scalar_product_times\n[PROOF STATE]\nproof (prove)\nusing this:\na1 * a2 * scalar_product w1 w2 + (scalar_product u1 u2 * scalar_product w1 w2) = a1 * a2 * scalar_product w1 w2 + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\nlength v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2\nlength ?w1.0 = length ?w2.0 \\<Longrightarrow> ?x * ?y * scalar_product ?w1.0 ?w2.0 = scalar_product (local.times ?x ?w1.0) (local.times ?y ?w2.0)\n\ngoal (1 subgoal):\n 1. a1 * a2 * scalar_product w1 w2 + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2) = scalar_product (local.times a1 w1) (local.times a2 w2) + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\na1 * a2 * scalar_product w1 w2 + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2) = scalar_product (local.times a1 w1) (local.times a2 w2) + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\na1 * a2 * scalar_product w1 w2 + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2) = scalar_product (local.times a1 w1) (local.times a2 w2) + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\n[PROOF STEP]\nhave \"scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n                        = scalar_product (vec_vec_Tensor (a1#u1) w1) (vec_vec_Tensor (a2#u2) w2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\na1 * a2 * scalar_product w1 w2 + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2) = scalar_product (local.times a1 w1) (local.times a2 w2) + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\n\ngoal (1 subgoal):\n 1. scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2) = scalar_product (vec_vec_Tensor (a1 # u1) w1) (vec_vec_Tensor (a2 # u2) w2)\n[PROOF STEP]\nusing Cons_1 Cons_2\n[PROOF STATE]\nproof (prove)\nusing this:\na1 * a2 * scalar_product w1 w2 + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2) = scalar_product (local.times a1 w1) (local.times a2 w2) + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\nv1 = a1 # u1 \\<and> length u1 = k\nv2 = a2 # u2 \\<and> length u2 = k\n\ngoal (1 subgoal):\n 1. scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2) = scalar_product (vec_vec_Tensor (a1 # u1) w1) (vec_vec_Tensor (a2 # u2) w2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2) = scalar_product (vec_vec_Tensor (a1 # u1) w1) (vec_vec_Tensor (a2 # u2) w2)\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2) = scalar_product (vec_vec_Tensor (a1 # u1) w1) (vec_vec_Tensor (a2 # u2) w2)\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nhave \"(vec_vec_Tensor (a1#u1) w1) = (times a1 w1)@(vec_vec_Tensor u1 w1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec_vec_Tensor (a1 # u1) w1 = local.times a1 w1 @ vec_vec_Tensor u1 w1\n[PROOF STEP]\nusing vec_vec_Tensor.simps\n[PROOF STATE]\nproof (prove)\nusing this:\nvec_vec_Tensor [] ?ys = []\nvec_vec_Tensor (?x # ?xs) ?ys = local.times ?x ?ys @ vec_vec_Tensor ?xs ?ys\n\ngoal (1 subgoal):\n 1. vec_vec_Tensor (a1 # u1) w1 = local.times a1 w1 @ vec_vec_Tensor u1 w1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nvec_vec_Tensor (a1 # u1) w1 = local.times a1 w1 @ vec_vec_Tensor u1 w1\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nvec_vec_Tensor (a1 # u1) w1 = local.times a1 w1 @ vec_vec_Tensor u1 w1\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nhave \"(vec_vec_Tensor (a2#u2) w2) = (times a2 w2)@(vec_vec_Tensor u2 w2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec_vec_Tensor (a2 # u2) w2 = local.times a2 w2 @ vec_vec_Tensor u2 w2\n[PROOF STEP]\nusing vec_vec_Tensor.simps\n[PROOF STATE]\nproof (prove)\nusing this:\nvec_vec_Tensor [] ?ys = []\nvec_vec_Tensor (?x # ?xs) ?ys = local.times ?x ?ys @ vec_vec_Tensor ?xs ?ys\n\ngoal (1 subgoal):\n 1. vec_vec_Tensor (a2 # u2) w2 = local.times a2 w2 @ vec_vec_Tensor u2 w2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nvec_vec_Tensor (a2 # u2) w2 = local.times a2 w2 @ vec_vec_Tensor u2 w2\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nv1 = a1 # u1 \\<and> length u1 = k\nscalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2) = scalar_product (vec_vec_Tensor (a1 # u1) w1) (vec_vec_Tensor (a2 # u2) w2)\nvec_vec_Tensor (a1 # u1) w1 = local.times a1 w1 @ vec_vec_Tensor u1 w1\nvec_vec_Tensor (a2 # u2) w2 = local.times a2 w2 @ vec_vec_Tensor u2 w2\n[PROOF STEP]\nhave Cons_7:\"scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n                      = scalar_product ((times a1 w1)@(vec_vec_Tensor u1 w1)) \n                                ((times a2 w2)@(vec_vec_Tensor u2 w2))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nv1 = a1 # u1 \\<and> length u1 = k\nscalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2) = scalar_product (vec_vec_Tensor (a1 # u1) w1) (vec_vec_Tensor (a2 # u2) w2)\nvec_vec_Tensor (a1 # u1) w1 = local.times a1 w1 @ vec_vec_Tensor u1 w1\nvec_vec_Tensor (a2 # u2) w2 = local.times a2 w2 @ vec_vec_Tensor u2 w2\n\ngoal (1 subgoal):\n 1. scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2) = scalar_product (local.times a1 w1 @ vec_vec_Tensor u1 w1) (local.times a2 w2 @ vec_vec_Tensor u2 w2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2) = scalar_product (local.times a1 w1 @ vec_vec_Tensor u1 w1) (local.times a2 w2 @ vec_vec_Tensor u2 w2)\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2) = scalar_product (local.times a1 w1 @ vec_vec_Tensor u1 w1) (local.times a2 w2 @ vec_vec_Tensor u2 w2)\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nhave \"length (vec_vec_Tensor u2 w2) = length (vec_vec_Tensor u1 w1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (vec_vec_Tensor u2 w2) = length (vec_vec_Tensor u1 w1)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2\n\ngoal (1 subgoal):\n 1. length (vec_vec_Tensor u2 w2) = length (vec_vec_Tensor u1 w1)\n[PROOF STEP]\nby (metis Cons_1 Cons_2 vec_vec_Tensor_length)\n[PROOF STATE]\nproof (state)\nthis:\nlength (vec_vec_Tensor u2 w2) = length (vec_vec_Tensor u1 w1)\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nlength (vec_vec_Tensor u2 w2) = length (vec_vec_Tensor u1 w1)\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nhave \"length (times a1 w1) = (length (times a2 w2))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. length (local.times a1 w1) = length (local.times a2 w2)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2\n\ngoal (1 subgoal):\n 1. length (local.times a1 w1) = length (local.times a2 w2)\n[PROOF STEP]\nby (metis preserving_length)\n[PROOF STATE]\nproof (state)\nthis:\nlength (local.times a1 w1) = length (local.times a2 w2)\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nscalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2) = scalar_product (local.times a1 w1 @ vec_vec_Tensor u1 w1) (local.times a2 w2 @ vec_vec_Tensor u2 w2)\nlength (vec_vec_Tensor u2 w2) = length (vec_vec_Tensor u1 w1)\nlength (local.times a1 w1) = length (local.times a2 w2)\n[PROOF STEP]\nhave \"scalar_product ((times a1 w1)@(vec_vec_Tensor u1 w1)) \n                                ((times a2 w2)@(vec_vec_Tensor u2 w2)) = \n                    (scalar_product (times a1 w1) (times a2 w2))\n                    +  scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nscalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2) = scalar_product (local.times a1 w1 @ vec_vec_Tensor u1 w1) (local.times a2 w2 @ vec_vec_Tensor u2 w2)\nlength (vec_vec_Tensor u2 w2) = length (vec_vec_Tensor u1 w1)\nlength (local.times a1 w1) = length (local.times a2 w2)\n\ngoal (1 subgoal):\n 1. scalar_product (local.times a1 w1 @ vec_vec_Tensor u1 w1) (local.times a2 w2 @ vec_vec_Tensor u2 w2) = scalar_product (local.times a1 w1) (local.times a2 w2) + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\n[PROOF STEP]\nusing effective_scalar_product_append\n[PROOF STATE]\nproof (prove)\nusing this:\nscalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2) = scalar_product (local.times a1 w1 @ vec_vec_Tensor u1 w1) (local.times a2 w2 @ vec_vec_Tensor u2 w2)\nlength (vec_vec_Tensor u2 w2) = length (vec_vec_Tensor u1 w1)\nlength (local.times a1 w1) = length (local.times a2 w2)\n\\<lbrakk>length ?zs = length ?ws; length ?xs = length ?ys\\<rbrakk> \\<Longrightarrow> scalar_product (?xs @ ?zs) (?ys @ ?ws) = scalar_product ?xs ?ys + scalar_product ?zs ?ws\n\ngoal (1 subgoal):\n 1. scalar_product (local.times a1 w1 @ vec_vec_Tensor u1 w1) (local.times a2 w2 @ vec_vec_Tensor u2 w2) = scalar_product (local.times a1 w1) (local.times a2 w2) + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product (local.times a1 w1 @ vec_vec_Tensor u1 w1) (local.times a2 w2 @ vec_vec_Tensor u2 w2) = scalar_product (local.times a1 w1) (local.times a2 w2) + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nscalar_product (local.times a1 w1 @ vec_vec_Tensor u1 w1) (local.times a2 w2 @ vec_vec_Tensor u2 w2) = scalar_product (local.times a1 w1) (local.times a2 w2) + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nscalar_product (local.times a1 w1 @ vec_vec_Tensor u1 w1) (local.times a2 w2 @ vec_vec_Tensor u2 w2) = scalar_product (local.times a1 w1) (local.times a2 w2) + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\n\ngoal (1 subgoal):\n 1. scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nusing Cons_6 Cons_7 \\<open>a1 * a2 + scalar_product u1 u2 * scalar_product w1 w2 \n                 = a1 * a2 * scalar_product w1 w2 \n                  + (scalar_product u1 u2 * scalar_product w1 w2)\\<close>\n[PROOF STATE]\nproof (prove)\nusing this:\nscalar_product (local.times a1 w1 @ vec_vec_Tensor u1 w1) (local.times a2 w2 @ vec_vec_Tensor u2 w2) = scalar_product (local.times a1 w1) (local.times a2 w2) + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\na1 * a2 * scalar_product w1 w2 + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2) = scalar_product (local.times a1 w1) (local.times a2 w2) + scalar_product (vec_vec_Tensor u1 w1) (vec_vec_Tensor u2 w2)\nscalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2) = scalar_product (local.times a1 w1 @ vec_vec_Tensor u1 w1) (local.times a2 w2 @ vec_vec_Tensor u2 w2)\na1 * a2 + scalar_product u1 u2 * scalar_product w1 w2 = a1 * a2 * scalar_product w1 w2 + (scalar_product u1 u2 * scalar_product w1 w2)\n\ngoal (1 subgoal):\n 1. scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nby (metis Cons_3 Cons_4 )\n[PROOF STATE]\nproof (state)\nthis:\nscalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nlength v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n\ngoal (1 subgoal):\n 1. \\<And>n v1 v2 w1 w2. (\\<And>v1 v2 w1 w2. length v1 = length v2 \\<and> length v1 = n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)) \\<Longrightarrow> length v1 = length v2 \\<and> length v1 = Suc n \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nlength v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nlength v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<Longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n\ngoal (1 subgoal):\n 1. length v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlength v1 = length v2 \\<and> length v1 = Suc k \\<and> length w1 = length w2 \\<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 23641, "file": "Matrix_Tensor_Matrix_Tensor", "length": 144, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.92522995296862, "lm_q2_score": 0.8198933381139645, "lm_q1q2_score": 0.7585898746624682}}
{"text": "[STATEMENT]\nlemma trace_adjoint_positive:\n  fixes A :: \"complex mat\"\n  shows \"trace (A * adjoint A) \\<ge> 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\<le> trace (A * adjoint A)\n[PROOF STEP]\napply (auto simp add: trace_def adjoint_col)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\<le> (\\<Sum>i = 0..<dim_row A. inner_prod (row A i) (row A i))\n[PROOF STEP]\napply (rule sum_nonneg)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>x. x \\<in> {0..<dim_row A} \\<Longrightarrow> 0 \\<le> inner_prod (row A x) (row A x)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 249, "file": "QHLProver_Complex_Matrix", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9032942171172603, "lm_q2_score": 0.8397339656668287, "lm_q1q2_score": 0.7585268351037904}}
{"text": "[STATEMENT]\ntheorem sum_of_odds:\n  \"(\\<Sum>i::nat=0..<n. 2 * i + 1) = n^Suc (Suc 0)\"\n  (is \"?P n\" is \"?S n = _\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<n. 2 * i + 1) = n ^ Suc (Suc 0)\n[PROOF STEP]\nproof (induct n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. (\\<Sum>i = 0..<0. 2 * i + 1) = 0 ^ Suc (Suc 0)\n 2. \\<And>n. (\\<Sum>i = 0..<n. 2 * i + 1) = n ^ Suc (Suc 0) \\<Longrightarrow> (\\<Sum>i = 0..<Suc n. 2 * i + 1) = Suc n ^ Suc (Suc 0)\n[PROOF STEP]\nshow \"?P 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<0. 2 * i + 1) = 0 ^ Suc (Suc 0)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>i = 0..<0. 2 * i + 1) = 0 ^ Suc (Suc 0)\n\ngoal (1 subgoal):\n 1. \\<And>n. (\\<Sum>i = 0..<n. 2 * i + 1) = n ^ Suc (Suc 0) \\<Longrightarrow> (\\<Sum>i = 0..<Suc n. 2 * i + 1) = Suc n ^ Suc (Suc 0)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>n. (\\<Sum>i = 0..<n. 2 * i + 1) = n ^ Suc (Suc 0) \\<Longrightarrow> (\\<Sum>i = 0..<Suc n. 2 * i + 1) = Suc n ^ Suc (Suc 0)\n[PROOF STEP]\nfix n\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>n. (\\<Sum>i = 0..<n. 2 * i + 1) = n ^ Suc (Suc 0) \\<Longrightarrow> (\\<Sum>i = 0..<Suc n. 2 * i + 1) = Suc n ^ Suc (Suc 0)\n[PROOF STEP]\nhave \"?S (n + 1) = ?S n + 2 * n + 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<n + 1. 2 * i + 1) = (\\<Sum>i = 0..<n. 2 * i + 1) + 2 * n + 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>i = 0..<n + 1. 2 * i + 1) = (\\<Sum>i = 0..<n. 2 * i + 1) + 2 * n + 1\n\ngoal (1 subgoal):\n 1. \\<And>n. (\\<Sum>i = 0..<n. 2 * i + 1) = n ^ Suc (Suc 0) \\<Longrightarrow> (\\<Sum>i = 0..<Suc n. 2 * i + 1) = Suc n ^ Suc (Suc 0)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>i = 0..<n + 1. 2 * i + 1) = (\\<Sum>i = 0..<n. 2 * i + 1) + 2 * n + 1\n\ngoal (1 subgoal):\n 1. \\<And>n. (\\<Sum>i = 0..<n. 2 * i + 1) = n ^ Suc (Suc 0) \\<Longrightarrow> (\\<Sum>i = 0..<Suc n. 2 * i + 1) = Suc n ^ Suc (Suc 0)\n[PROOF STEP]\nassume \"?S n = n^Suc (Suc 0)\"\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>i = 0..<n. 2 * i + 1) = n ^ Suc (Suc 0)\n\ngoal (1 subgoal):\n 1. \\<And>n. (\\<Sum>i = 0..<n. 2 * i + 1) = n ^ Suc (Suc 0) \\<Longrightarrow> (\\<Sum>i = 0..<Suc n. 2 * i + 1) = Suc n ^ Suc (Suc 0)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>i = 0..<n. 2 * i + 1) = n ^ Suc (Suc 0)\n\ngoal (1 subgoal):\n 1. \\<And>n. (\\<Sum>i = 0..<n. 2 * i + 1) = n ^ Suc (Suc 0) \\<Longrightarrow> (\\<Sum>i = 0..<Suc n. 2 * i + 1) = Suc n ^ Suc (Suc 0)\n[PROOF STEP]\nhave \"\\<dots> + 2 * n + 1 = (n + 1)^Suc (Suc 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n ^ Suc (Suc 0) + 2 * n + 1 = (n + 1) ^ Suc (Suc 0)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nn ^ Suc (Suc 0) + 2 * n + 1 = (n + 1) ^ Suc (Suc 0)\n\ngoal (1 subgoal):\n 1. \\<And>n. (\\<Sum>i = 0..<n. 2 * i + 1) = n ^ Suc (Suc 0) \\<Longrightarrow> (\\<Sum>i = 0..<Suc n. 2 * i + 1) = Suc n ^ Suc (Suc 0)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\<Sum>i = 0..<n + 1. 2 * i + 1) = (n + 1) ^ Suc (Suc 0)\n[PROOF STEP]\nshow \"?P (Suc n)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<Sum>i = 0..<n + 1. 2 * i + 1) = (n + 1) ^ Suc (Suc 0)\n\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<Suc n. 2 * i + 1) = Suc n ^ Suc (Suc 0)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>i = 0..<Suc n. 2 * i + 1) = Suc n ^ Suc (Suc 0)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1928, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942261220292, "lm_q2_score": 0.8519528019683105, "lm_q1q2_score": 0.7584886605208714}}
{"text": "[STATEMENT]\nlemma matrix_works:\n  assumes lf: \"Vector_Spaces.linear (*s) (*s) f\"\n  shows \"matrix f *v x = f (x::'a::field ^ 'n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. matrix f *v x = f x\n[PROOF STEP]\napply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult.commute)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<forall>i. (\\<Sum>j\\<in>UNIV. x $ j * f (axis j (1::'a)) $ i) = f x $ i\n[PROOF STEP]\napply clarify\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>i. (\\<Sum>j\\<in>UNIV. x $ j * f (axis j (1::'a)) $ i) = f x $ i\n[PROOF STEP]\napply (rule linear_componentwise[OF lf, symmetric])\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 307, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9149009549929797, "lm_q2_score": 0.8289388083214156, "lm_q1q2_score": 0.7583969073640058}}
{"text": "[STATEMENT]\nlemma cauchy_schwarz_ineq:\n  assumes \"dim_vec v = dim_vec w\"\n  shows \"(cmod(\\<langle>v|w\\<rangle>))\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nproof (cases \"\\<langle>v|v\\<rangle> = 0\")\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\<langle>v|v\\<rangle> = 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n 2. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\ncase c0:True\n[PROOF STATE]\nproof (state)\nthis:\n\\<langle>v|v\\<rangle> = 0\n\ngoal (2 subgoals):\n 1. \\<langle>v|v\\<rangle> = 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n 2. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\<langle>v|v\\<rangle> = 0\n[PROOF STEP]\nhave \"\\<And>i. i < dim_vec v \\<Longrightarrow> v $ i = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<langle>v|v\\<rangle> = 0\n\ngoal (1 subgoal):\n 1. \\<And>i. i < dim_vec v \\<Longrightarrow> v $ i = 0\n[PROOF STEP]\nby(metis index_zero_vec(1) inner_prod_with_itself_nonneg_reals_non0)\n[PROOF STATE]\nproof (state)\nthis:\n?i < dim_vec v \\<Longrightarrow> v $ ?i = 0\n\ngoal (2 subgoals):\n 1. \\<langle>v|v\\<rangle> = 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n 2. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n?i < dim_vec v \\<Longrightarrow> v $ ?i = 0\n[PROOF STEP]\nhave \"(cmod(\\<langle>v|w\\<rangle>))\\<^sup>2 = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\n?i < dim_vec v \\<Longrightarrow> v $ ?i = 0\n\ngoal (1 subgoal):\n 1. (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 = 0\n[PROOF STEP]\nby (simp add: assms inner_prod_def)\n[PROOF STATE]\nproof (state)\nthis:\n(cmod \\<langle>v|w\\<rangle>)\\<^sup>2 = 0\n\ngoal (2 subgoals):\n 1. \\<langle>v|v\\<rangle> = 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n 2. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n(cmod \\<langle>v|w\\<rangle>)\\<^sup>2 = 0\n\ngoal (2 subgoals):\n 1. \\<langle>v|v\\<rangle> = 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n 2. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nhave \"Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>) = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>) = 0\n[PROOF STEP]\nby (simp add: c0)\n[PROOF STATE]\nproof (state)\nthis:\nRe (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>) = 0\n\ngoal (2 subgoals):\n 1. \\<langle>v|v\\<rangle> = 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n 2. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n(cmod \\<langle>v|w\\<rangle>)\\<^sup>2 = 0\nRe (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>) = 0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(cmod \\<langle>v|w\\<rangle>)\\<^sup>2 = 0\nRe (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>) = 0\n\ngoal (1 subgoal):\n 1. (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n\ngoal (1 subgoal):\n 1. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\ncase c1:False\n[PROOF STATE]\nproof (state)\nthis:\n\\<langle>v|v\\<rangle> \\<noteq> 0\n\ngoal (1 subgoal):\n 1. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nhave \"dim_vec w = dim_vec (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. dim_vec w = dim_vec (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v)\n[PROOF STEP]\nby (simp add: assms)\n[PROOF STATE]\nproof (state)\nthis:\ndim_vec w = dim_vec (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v)\n\ngoal (1 subgoal):\n 1. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ndim_vec w = dim_vec (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v)\n[PROOF STEP]\nhave \"\\<langle>w + -\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + -\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = \\<langle>w|w\\<rangle> + \\<langle>w|-\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> +\n\\<langle>-\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w\\<rangle> + \\<langle>-\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|-\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle>\"\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec w = dim_vec (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v)\n\ngoal (1 subgoal):\n 1. \\<langle>w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = \\<langle>w|w\\<rangle> + \\<langle>w|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> + \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w\\<rangle> + \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle>\n[PROOF STEP]\nusing inner_prod_expand[of \"w\" \"-\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\" \"w\" \"-\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\"]\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec w = dim_vec (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v)\n\\<lbrakk>dim_vec w = dim_vec (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v); dim_vec w = dim_vec w; dim_vec w = dim_vec (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v)\\<rbrakk> \\<Longrightarrow> \\<langle>w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = \\<langle>w|w\\<rangle> + \\<langle>w|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> + \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w\\<rangle> + \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle>\n\ngoal (1 subgoal):\n 1. \\<langle>w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = \\<langle>w|w\\<rangle> + \\<langle>w|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> + \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w\\<rangle> + \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle>\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\<langle>w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = \\<langle>w|w\\<rangle> + \\<langle>w|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> + \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w\\<rangle> + \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle>\n\ngoal (1 subgoal):\n 1. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\<langle>w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = \\<langle>w|w\\<rangle> + \\<langle>w|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> + \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w\\<rangle> + \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle>\n\ngoal (1 subgoal):\n 1. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nhave \"\\<langle>w|-\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = -\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle> * \\<langle>w|v\\<rangle>\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<langle>w|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> * \\<langle>w|v\\<rangle>\n[PROOF STEP]\nusing assms inner_prod_distrib_right[of \"w\" \"v\" \"-\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle>\"]\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec v = dim_vec w\ndim_vec w = dim_vec v \\<Longrightarrow> \\<langle>w|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> * \\<langle>w|v\\<rangle>\n\ngoal (1 subgoal):\n 1. \\<langle>w|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> * \\<langle>w|v\\<rangle>\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\<langle>w|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> * \\<langle>w|v\\<rangle>\n\ngoal (1 subgoal):\n 1. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\<langle>w|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> * \\<langle>w|v\\<rangle>\n\ngoal (1 subgoal):\n 1. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nhave \"\\<langle>-\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w\\<rangle> = cnj(-\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle>) * \\<langle>v|w\\<rangle>\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w\\<rangle> = cnj (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle>) * \\<langle>v|w\\<rangle>\n[PROOF STEP]\nusing assms inner_prod_distrib_left[of \"v\" \"w\" \"-\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle>\"]\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec v = dim_vec w\ndim_vec v = dim_vec w \\<Longrightarrow> \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w\\<rangle> = cnj (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle>) * \\<langle>v|w\\<rangle>\n\ngoal (1 subgoal):\n 1. \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w\\<rangle> = cnj (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle>) * \\<langle>v|w\\<rangle>\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w\\<rangle> = cnj (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle>) * \\<langle>v|w\\<rangle>\n\ngoal (1 subgoal):\n 1. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w\\<rangle> = cnj (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle>) * \\<langle>v|w\\<rangle>\n\ngoal (1 subgoal):\n 1. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nhave \"\\<langle>-\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|-\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = cnj(-\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle>) * (-\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle>) * \\<langle>v|v\\<rangle>\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = cnj (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle>) * (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle>) * \\<langle>v|v\\<rangle>\n[PROOF STEP]\nusing inner_prod_distrib_left[of \"v\" \"-\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\" \"-\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle>\"]\ninner_prod_distrib_right[of \"v\" \"v\" \"-\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle>\"]\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_vec v = dim_vec (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v) \\<Longrightarrow> \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = cnj (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle>) * \\<langle>v|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle>\ndim_vec v = dim_vec v \\<Longrightarrow> \\<langle>v|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> * \\<langle>v|v\\<rangle>\n\ngoal (1 subgoal):\n 1. \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = cnj (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle>) * (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle>) * \\<langle>v|v\\<rangle>\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = cnj (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle>) * (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle>) * \\<langle>v|v\\<rangle>\n\ngoal (1 subgoal):\n 1. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n\\<langle>w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = \\<langle>w|w\\<rangle> + \\<langle>w|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> + \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w\\<rangle> + \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle>\n\\<langle>w|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> * \\<langle>w|v\\<rangle>\n\\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w\\<rangle> = cnj (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle>) * \\<langle>v|w\\<rangle>\n\\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = cnj (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle>) * (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle>) * \\<langle>v|v\\<rangle>\n[PROOF STEP]\nhave \"\\<langle>w + -\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + -\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = \\<langle>w|w\\<rangle> -  cmod(\\<langle>v|w\\<rangle>)^2 / \\<langle>v|v\\<rangle>\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<langle>w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = \\<langle>w|w\\<rangle> + \\<langle>w|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> + \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w\\<rangle> + \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle>\n\\<langle>w|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> * \\<langle>w|v\\<rangle>\n\\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w\\<rangle> = cnj (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle>) * \\<langle>v|w\\<rangle>\n\\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = cnj (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle>) * (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle>) * \\<langle>v|v\\<rangle>\n\ngoal (1 subgoal):\n 1. \\<langle>w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = \\<langle>w|w\\<rangle> - complex_of_real ((cmod \\<langle>v|w\\<rangle>)\\<^sup>2) / \\<langle>v|v\\<rangle>\n[PROOF STEP]\nusing assms inner_prod_cnj[of \"w\" \"v\"] inner_prod_cnj[of \"v\" \"v\"] complex_norm_square\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<langle>w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = \\<langle>w|w\\<rangle> + \\<langle>w|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> + \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w\\<rangle> + \\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle>\n\\<langle>w|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> * \\<langle>w|v\\<rangle>\n\\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w\\<rangle> = cnj (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle>) * \\<langle>v|w\\<rangle>\n\\<langle>- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = cnj (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle>) * (- \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle>) * \\<langle>v|v\\<rangle>\ndim_vec v = dim_vec w\ndim_vec w = dim_vec v \\<Longrightarrow> \\<langle>v|w\\<rangle> = cnj \\<langle>w|v\\<rangle>\ndim_vec v = dim_vec v \\<Longrightarrow> \\<langle>v|v\\<rangle> = cnj \\<langle>v|v\\<rangle>\ncomplex_of_real ((cmod ?z)\\<^sup>2) = ?z * cnj ?z\n\ngoal (1 subgoal):\n 1. \\<langle>w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = \\<langle>w|w\\<rangle> - complex_of_real ((cmod \\<langle>v|w\\<rangle>)\\<^sup>2) / \\<langle>v|v\\<rangle>\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\<langle>w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = \\<langle>w|w\\<rangle> - complex_of_real ((cmod \\<langle>v|w\\<rangle>)\\<^sup>2) / \\<langle>v|v\\<rangle>\n\ngoal (1 subgoal):\n 1. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\<langle>w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = \\<langle>w|w\\<rangle> - complex_of_real ((cmod \\<langle>v|w\\<rangle>)\\<^sup>2) / \\<langle>v|v\\<rangle>\n\ngoal (1 subgoal):\n 1. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nhave \"Re(\\<langle>w + -\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + -\\<langle>v|w\\<rangle>/\\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle>) \\<ge> 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 \\<le> Re \\<langle>w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle>\n[PROOF STEP]\nusing inner_prod_with_itself_Re\n[PROOF STATE]\nproof (prove)\nusing this:\n0 \\<le> Re \\<langle>?u|?u\\<rangle>\n\ngoal (1 subgoal):\n 1. 0 \\<le> Re \\<langle>w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle>\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n0 \\<le> Re \\<langle>w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle>\n\ngoal (1 subgoal):\n 1. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n\\<langle>w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = \\<langle>w|w\\<rangle> - complex_of_real ((cmod \\<langle>v|w\\<rangle>)\\<^sup>2) / \\<langle>v|v\\<rangle>\n0 \\<le> Re \\<langle>w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle>\n[PROOF STEP]\nhave \"Re(\\<langle>w|w\\<rangle>) \\<ge> cmod(\\<langle>v|w\\<rangle>)^2/Re(\\<langle>v|v\\<rangle>)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<langle>w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = \\<langle>w|w\\<rangle> - complex_of_real ((cmod \\<langle>v|w\\<rangle>)\\<^sup>2) / \\<langle>v|v\\<rangle>\n0 \\<le> Re \\<langle>w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle>\n\ngoal (1 subgoal):\n 1. (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 / Re \\<langle>v|v\\<rangle> \\<le> Re \\<langle>w|w\\<rangle>\n[PROOF STEP]\nusing inner_prod_with_itself_real\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<langle>w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle> = \\<langle>w|w\\<rangle> - complex_of_real ((cmod \\<langle>v|w\\<rangle>)\\<^sup>2) / \\<langle>v|v\\<rangle>\n0 \\<le> Re \\<langle>w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v|w + - \\<langle>v|w\\<rangle> / \\<langle>v|v\\<rangle> \\<cdot>\\<^sub>v v\\<rangle>\n\\<langle>?u|?u\\<rangle> \\<in> \\<real>\n\ngoal (1 subgoal):\n 1. (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 / Re \\<langle>v|v\\<rangle> \\<le> Re \\<langle>w|w\\<rangle>\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(cmod \\<langle>v|w\\<rangle>)\\<^sup>2 / Re \\<langle>v|v\\<rangle> \\<le> Re \\<langle>w|w\\<rangle>\n\ngoal (1 subgoal):\n 1. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n(cmod \\<langle>v|w\\<rangle>)\\<^sup>2 / Re \\<langle>v|v\\<rangle> \\<le> Re \\<langle>w|w\\<rangle>\n\ngoal (1 subgoal):\n 1. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nhave c2:\"Re(\\<langle>v|v\\<rangle>) > 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < Re \\<langle>v|v\\<rangle>\n[PROOF STEP]\nusing inner_prod_with_itself_Re_non0 inner_prod_with_itself_eq0 c1\n[PROOF STATE]\nproof (prove)\nusing this:\n?u \\<noteq> 0\\<^sub>v (dim_vec ?u) \\<Longrightarrow> 0 < Re \\<langle>?u|?u\\<rangle>\n?u = 0\\<^sub>v (dim_vec ?u) \\<Longrightarrow> \\<langle>?u|?u\\<rangle> = 0\n\\<langle>v|v\\<rangle> \\<noteq> 0\n\ngoal (1 subgoal):\n 1. 0 < Re \\<langle>v|v\\<rangle>\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n0 < Re \\<langle>v|v\\<rangle>\n\ngoal (1 subgoal):\n 1. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n(cmod \\<langle>v|w\\<rangle>)\\<^sup>2 / Re \\<langle>v|v\\<rangle> \\<le> Re \\<langle>w|w\\<rangle>\n0 < Re \\<langle>v|v\\<rangle>\n[PROOF STEP]\nhave \"Re(\\<langle>w|w\\<rangle>) * Re(\\<langle>v|v\\<rangle>) \\<ge> cmod(\\<langle>v|w\\<rangle>)^2/Re(\\<langle>v|v\\<rangle>) * Re(\\<langle>v|v\\<rangle>)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(cmod \\<langle>v|w\\<rangle>)\\<^sup>2 / Re \\<langle>v|v\\<rangle> \\<le> Re \\<langle>w|w\\<rangle>\n0 < Re \\<langle>v|v\\<rangle>\n\ngoal (1 subgoal):\n 1. (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 / Re \\<langle>v|v\\<rangle> * Re \\<langle>v|v\\<rangle> \\<le> Re \\<langle>w|w\\<rangle> * Re \\<langle>v|v\\<rangle>\n[PROOF STEP]\nby (metis less_numeral_extra(3) nonzero_divide_eq_eq pos_divide_le_eq)\n[PROOF STATE]\nproof (state)\nthis:\n(cmod \\<langle>v|w\\<rangle>)\\<^sup>2 / Re \\<langle>v|v\\<rangle> * Re \\<langle>v|v\\<rangle> \\<le> Re \\<langle>w|w\\<rangle> * Re \\<langle>v|v\\<rangle>\n\ngoal (1 subgoal):\n 1. \\<langle>v|v\\<rangle> \\<noteq> 0 \\<Longrightarrow> (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(cmod \\<langle>v|w\\<rangle>)\\<^sup>2 / Re \\<langle>v|v\\<rangle> * Re \\<langle>v|v\\<rangle> \\<le> Re \\<langle>w|w\\<rangle> * Re \\<langle>v|v\\<rangle>\n\ngoal (1 subgoal):\n 1. (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nusing inner_prod_with_itself_Im c2\n[PROOF STATE]\nproof (prove)\nusing this:\n(cmod \\<langle>v|w\\<rangle>)\\<^sup>2 / Re \\<langle>v|v\\<rangle> * Re \\<langle>v|v\\<rangle> \\<le> Re \\<langle>w|w\\<rangle> * Re \\<langle>v|v\\<rangle>\nIm \\<langle>?u|?u\\<rangle> = 0\n0 < Re \\<langle>v|v\\<rangle>\n\ngoal (1 subgoal):\n 1. (cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n[PROOF STEP]\nby (simp add: mult.commute)\n[PROOF STATE]\nproof (state)\nthis:\n(cmod \\<langle>v|w\\<rangle>)\\<^sup>2 \\<le> Re (\\<langle>v|v\\<rangle> * \\<langle>w|w\\<rangle>)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 11887, "file": "Isabelle_Marries_Dirac_No_Cloning", "length": 57, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.912436167620237, "lm_q2_score": 0.8311430394931456, "lm_q1q2_score": 0.7583649696993611}}
{"text": "[STATEMENT]\nlemma powreal_mult_base_lemma3: \n  assumes \"0 < a\" \n  and \"a < 1\" \n  and \"0 < b\" \n  and \"b < 1\" \nshows \"(a * b) pow\\<^sub>\\<real> x = (a pow\\<^sub>\\<real> x) * (b pow\\<^sub>\\<real> x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a * b) pow\\<^sub>\\<real> x = a pow\\<^sub>\\<real> x * b pow\\<^sub>\\<real> x\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (a * b) pow\\<^sub>\\<real> x = a pow\\<^sub>\\<real> x * b pow\\<^sub>\\<real> x\n[PROOF STEP]\nhave \"a * b < 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a * b < 1\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < a\na < 1\n0 < b\nb < 1\n\ngoal (1 subgoal):\n 1. a * b < 1\n[PROOF STEP]\nby (metis less_trans mult.left_neutral mult_less_cancel_right_disj)\n[PROOF STATE]\nproof (state)\nthis:\na * b < 1\n\ngoal (1 subgoal):\n 1. (a * b) pow\\<^sub>\\<real> x = a pow\\<^sub>\\<real> x * b pow\\<^sub>\\<real> x\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\na * b < 1\n\ngoal (1 subgoal):\n 1. (a * b) pow\\<^sub>\\<real> x = a pow\\<^sub>\\<real> x * b pow\\<^sub>\\<real> x\n[PROOF STEP]\nhave \"(inverse a * inverse b) powa - x = \n                   inverse a powa - x * inverse b powa - x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (inverse a * inverse b) powa - x = inverse a powa - x * inverse b powa - x\n[PROOF STEP]\nby (simp add: assms powa_mult_base real_inverse_ge_one_lemma)\n[PROOF STATE]\nproof (state)\nthis:\n(inverse a * inverse b) powa - x = inverse a powa - x * inverse b powa - x\n\ngoal (1 subgoal):\n 1. (a * b) pow\\<^sub>\\<real> x = a pow\\<^sub>\\<real> x * b pow\\<^sub>\\<real> x\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\na * b < 1\n(inverse a * inverse b) powa - x = inverse a powa - x * inverse b powa - x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\na * b < 1\n(inverse a * inverse b) powa - x = inverse a powa - x * inverse b powa - x\n\ngoal (1 subgoal):\n 1. (a * b) pow\\<^sub>\\<real> x = a pow\\<^sub>\\<real> x * b pow\\<^sub>\\<real> x\n[PROOF STEP]\nusing powreal_def assms\n[PROOF STATE]\nproof (prove)\nusing this:\na * b < 1\n(inverse a * inverse b) powa - x = inverse a powa - x * inverse b powa - x\n?a pow\\<^sub>\\<real> ?x = (if 0 < ?a \\<and> ?a < 1 then inverse ?a powa - ?x else if 1 \\<le> ?a then ?a powa ?x else 0)\n0 < a\na < 1\n0 < b\nb < 1\n\ngoal (1 subgoal):\n 1. (a * b) pow\\<^sub>\\<real> x = a pow\\<^sub>\\<real> x * b pow\\<^sub>\\<real> x\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(a * b) pow\\<^sub>\\<real> x = a pow\\<^sub>\\<real> x * b pow\\<^sub>\\<real> x\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1131, "file": "Real_Power_RealPower", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.884039278690883, "lm_q2_score": 0.8577681031721325, "lm_q1q2_score": 0.7583006952123389}}
{"text": "[STATEMENT]\nlemma sum_Suc_reindex:\n  fixes f :: \"nat \\<Rightarrow> 'a::ab_group_add\"\n    shows  \"sum f {0..n} = f 0 - f (Suc n) + sum (\\<lambda>i. f (Suc i)) {0..n}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sum f {0..n} = f 0 - f (Suc n) + (\\<Sum>i = 0..n. f (Suc i))\n[PROOF STEP]\nby (induct n) auto", "meta": {"llama_tokens": 149, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9196425377849806, "lm_q2_score": 0.8244619220634456, "lm_q1q2_score": 0.75821025431351}}
{"text": "[STATEMENT]\nlemma col_space_eq_range:\n  fixes f::\"('a::field^'n::{finite, wellorder}) \\<Rightarrow> ('a^'m)\"\n  assumes lf: \"Vector_Spaces.linear (*s) (*s) f\"\n  shows \"col_space (matrix f) = range f\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. col_space (matrix f) = range f\n[PROOF STEP]\nunfolding col_space_eq\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {y. \\<exists>x. matrix f *v x = y} = range f\n[PROOF STEP]\nunfolding matrix_works[OF lf]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {y. \\<exists>x. f x = y} = range f\n[PROOF STEP]\nby blast", "meta": {"llama_tokens": 245, "file": "Rank_Nullity_Theorem_Fundamental_Subspaces", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9196425267730008, "lm_q2_score": 0.8244619220634457, "lm_q1q2_score": 0.7582102452345522}}
{"text": "[STATEMENT]\nlemma complex_vector_affinity_eq:\n  fixes x :: \"'a :: complex_vector\"\n  assumes m0: \"m \\<noteq> 0\"\n  shows \"m *\\<^sub>C x + c = y \\<longleftrightarrow> x = inverse m *\\<^sub>C y - (inverse m *\\<^sub>C c)\"\n    (is \"?lhs \\<longleftrightarrow> ?rhs\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (m *\\<^sub>C x + c = y) = (x = y /\\<^sub>C m - c /\\<^sub>C m)\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. m *\\<^sub>C x + c = y \\<Longrightarrow> x = y /\\<^sub>C m - c /\\<^sub>C m\n 2. x = y /\\<^sub>C m - c /\\<^sub>C m \\<Longrightarrow> m *\\<^sub>C x + c = y\n[PROOF STEP]\nassume ?lhs\n[PROOF STATE]\nproof (state)\nthis:\nm *\\<^sub>C x + c = y\n\ngoal (2 subgoals):\n 1. m *\\<^sub>C x + c = y \\<Longrightarrow> x = y /\\<^sub>C m - c /\\<^sub>C m\n 2. x = y /\\<^sub>C m - c /\\<^sub>C m \\<Longrightarrow> m *\\<^sub>C x + c = y\n[PROOF STEP]\nhence \"m *\\<^sub>C x = y - c\"\n[PROOF STATE]\nproof (prove)\nusing this:\nm *\\<^sub>C x + c = y\n\ngoal (1 subgoal):\n 1. m *\\<^sub>C x = y - c\n[PROOF STEP]\nby (simp add: field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nm *\\<^sub>C x = y - c\n\ngoal (2 subgoals):\n 1. m *\\<^sub>C x + c = y \\<Longrightarrow> x = y /\\<^sub>C m - c /\\<^sub>C m\n 2. x = y /\\<^sub>C m - c /\\<^sub>C m \\<Longrightarrow> m *\\<^sub>C x + c = y\n[PROOF STEP]\nhence \"inverse m *\\<^sub>C (m *\\<^sub>C x) = inverse m *\\<^sub>C (y - c)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nm *\\<^sub>C x = y - c\n\ngoal (1 subgoal):\n 1. m *\\<^sub>C x /\\<^sub>C m = (y - c) /\\<^sub>C m\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nm *\\<^sub>C x /\\<^sub>C m = (y - c) /\\<^sub>C m\n\ngoal (2 subgoals):\n 1. m *\\<^sub>C x + c = y \\<Longrightarrow> x = y /\\<^sub>C m - c /\\<^sub>C m\n 2. x = y /\\<^sub>C m - c /\\<^sub>C m \\<Longrightarrow> m *\\<^sub>C x + c = y\n[PROOF STEP]\nthus \"x = inverse m *\\<^sub>C y - (inverse m *\\<^sub>C c)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nm *\\<^sub>C x /\\<^sub>C m = (y - c) /\\<^sub>C m\n\ngoal (1 subgoal):\n 1. x = y /\\<^sub>C m - c /\\<^sub>C m\n[PROOF STEP]\nusing m0\n[PROOF STATE]\nproof (prove)\nusing this:\nm *\\<^sub>C x /\\<^sub>C m = (y - c) /\\<^sub>C m\nm \\<noteq> 0\n\ngoal (1 subgoal):\n 1. x = y /\\<^sub>C m - c /\\<^sub>C m\n[PROOF STEP]\nby (simp add: complex_vector.scale_right_diff_distrib)\n[PROOF STATE]\nproof (state)\nthis:\nx = y /\\<^sub>C m - c /\\<^sub>C m\n\ngoal (1 subgoal):\n 1. x = y /\\<^sub>C m - c /\\<^sub>C m \\<Longrightarrow> m *\\<^sub>C x + c = y\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. x = y /\\<^sub>C m - c /\\<^sub>C m \\<Longrightarrow> m *\\<^sub>C x + c = y\n[PROOF STEP]\nassume ?rhs\n[PROOF STATE]\nproof (state)\nthis:\nx = y /\\<^sub>C m - c /\\<^sub>C m\n\ngoal (1 subgoal):\n 1. x = y /\\<^sub>C m - c /\\<^sub>C m \\<Longrightarrow> m *\\<^sub>C x + c = y\n[PROOF STEP]\nwith m0\n[PROOF STATE]\nproof (chain)\npicking this:\nm \\<noteq> 0\nx = y /\\<^sub>C m - c /\\<^sub>C m\n[PROOF STEP]\nshow \"m *\\<^sub>C x + c = y\"\n[PROOF STATE]\nproof (prove)\nusing this:\nm \\<noteq> 0\nx = y /\\<^sub>C m - c /\\<^sub>C m\n\ngoal (1 subgoal):\n 1. m *\\<^sub>C x + c = y\n[PROOF STEP]\nby (simp add: complex_vector.scale_right_diff_distrib)\n[PROOF STATE]\nproof (state)\nthis:\nm *\\<^sub>C x + c = y\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1492, "file": "Complex_Bounded_Operators_Complex_Vector_Spaces0", "length": 15, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9073122288794594, "lm_q2_score": 0.8354835309589073, "lm_q1q2_score": 0.758044424666407}}
{"text": "[STATEMENT]\nlemma null_space_eq_ker:\n  fixes f::\"('a::field^'n) => ('a^'m)\"\n  assumes lf: \"Vector_Spaces.linear (*s) (*s) f\"\n  shows \"null_space (matrix f) = {x. f x = 0}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. null_space (matrix f) = {x. f x = 0}\n[PROOF STEP]\nunfolding null_space_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {x. matrix f *v x = 0} = {x. f x = 0}\n[PROOF STEP]\nusing matrix_works [OF lf]\n[PROOF STATE]\nproof (prove)\nusing this:\nmatrix f *v ?x = f ?x\n\ngoal (1 subgoal):\n 1. {x. matrix f *v x = 0} = {x. f x = 0}\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 272, "file": "Rank_Nullity_Theorem_Fundamental_Subspaces", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9073122163480668, "lm_q2_score": 0.8354835411997897, "lm_q1q2_score": 0.7580444234883126}}
{"text": "[STATEMENT]\nlemma ListSum_disj_union: \n  \"distinct A \\<Longrightarrow> distinct B \\<Longrightarrow> distinct C \\<Longrightarrow> \n  set C = set A \\<union> set B  \\<Longrightarrow> \n  set A \\<inter> set B = {} \\<Longrightarrow>\n  (\\<Sum>\\<^bsub>a \\<in> C\\<^esub> (f a)) = (\\<Sum>\\<^bsub>a \\<in> A\\<^esub> f a) + (\\<Sum>\\<^bsub>a \\<in> B\\<^esub> (f a::nat))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>distinct A; distinct B; distinct C; set C = set A \\<union> set B; set A \\<inter> set B = {}\\<rbrakk> \\<Longrightarrow> ListSum C f = ListSum A f + ListSum B f\n[PROOF STEP]\nby (simp add: ListSum_conv_sum sum.union_disjoint)", "meta": {"llama_tokens": 238, "file": "Flyspeck-Tame_ListSum", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9046505402422645, "lm_q2_score": 0.8376199572530448, "lm_q1q2_score": 0.7577533468466695}}
{"text": "[STATEMENT]\ntheorem Digamma_of_nat:\n  \"Digamma (of_nat (Suc n) :: 'a :: {real_normed_field,banach}) = harm n - euler_mascheroni\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Digamma (of_nat (Suc n)) = harm n - euler_mascheroni\n[PROOF STEP]\nproof (induction n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. Digamma (of_nat (Suc 0)) = harm 0 - euler_mascheroni\n 2. \\<And>n. Digamma (of_nat (Suc n)) = harm n - euler_mascheroni \\<Longrightarrow> Digamma (of_nat (Suc (Suc n))) = harm (Suc n) - euler_mascheroni\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\nDigamma (of_nat (Suc n)) = harm n - euler_mascheroni\n\ngoal (2 subgoals):\n 1. Digamma (of_nat (Suc 0)) = harm 0 - euler_mascheroni\n 2. \\<And>n. Digamma (of_nat (Suc n)) = harm n - euler_mascheroni \\<Longrightarrow> Digamma (of_nat (Suc (Suc n))) = harm (Suc n) - euler_mascheroni\n[PROOF STEP]\nhave \"Digamma (of_nat (Suc (Suc n)) :: 'a) = Digamma (of_nat (Suc n) + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Digamma (of_nat (Suc (Suc n))) = Digamma (of_nat (Suc n) + (1::'a))\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nDigamma (of_nat (Suc (Suc n))) = Digamma (of_nat (Suc n) + (1::'a))\n\ngoal (2 subgoals):\n 1. Digamma (of_nat (Suc 0)) = harm 0 - euler_mascheroni\n 2. \\<And>n. Digamma (of_nat (Suc n)) = harm n - euler_mascheroni \\<Longrightarrow> Digamma (of_nat (Suc (Suc n))) = harm (Suc n) - euler_mascheroni\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nDigamma (of_nat (Suc (Suc n))) = Digamma (of_nat (Suc n) + (1::'a))\n\ngoal (2 subgoals):\n 1. Digamma (of_nat (Suc 0)) = harm 0 - euler_mascheroni\n 2. \\<And>n. Digamma (of_nat (Suc n)) = harm n - euler_mascheroni \\<Longrightarrow> Digamma (of_nat (Suc (Suc n))) = harm (Suc n) - euler_mascheroni\n[PROOF STEP]\nhave \"\\<dots> = Digamma (of_nat (Suc n)) + inverse (of_nat (Suc n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Digamma (of_nat (Suc n) + (1::'a)) = Digamma (of_nat (Suc n)) + inverse (of_nat (Suc n))\n[PROOF STEP]\nby (subst Digamma_plus1) (simp_all add: inverse_eq_divide del: of_nat_Suc)\n[PROOF STATE]\nproof (state)\nthis:\nDigamma (of_nat (Suc n) + (1::'a)) = Digamma (of_nat (Suc n)) + inverse (of_nat (Suc n))\n\ngoal (2 subgoals):\n 1. Digamma (of_nat (Suc 0)) = harm 0 - euler_mascheroni\n 2. \\<And>n. Digamma (of_nat (Suc n)) = harm n - euler_mascheroni \\<Longrightarrow> Digamma (of_nat (Suc (Suc n))) = harm (Suc n) - euler_mascheroni\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nDigamma (of_nat (Suc n) + (1::'a)) = Digamma (of_nat (Suc n)) + inverse (of_nat (Suc n))\n\ngoal (2 subgoals):\n 1. Digamma (of_nat (Suc 0)) = harm 0 - euler_mascheroni\n 2. \\<And>n. Digamma (of_nat (Suc n)) = harm n - euler_mascheroni \\<Longrightarrow> Digamma (of_nat (Suc (Suc n))) = harm (Suc n) - euler_mascheroni\n[PROOF STEP]\nhave \"Digamma (of_nat (Suc n) :: 'a) = harm n - euler_mascheroni \"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Digamma (of_nat (Suc n)) = harm n - euler_mascheroni\n[PROOF STEP]\nby (rule Suc)\n[PROOF STATE]\nproof (state)\nthis:\nDigamma (of_nat (Suc n)) = harm n - euler_mascheroni\n\ngoal (2 subgoals):\n 1. Digamma (of_nat (Suc 0)) = harm 0 - euler_mascheroni\n 2. \\<And>n. Digamma (of_nat (Suc n)) = harm n - euler_mascheroni \\<Longrightarrow> Digamma (of_nat (Suc (Suc n))) = harm (Suc n) - euler_mascheroni\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nDigamma (of_nat (Suc n)) = harm n - euler_mascheroni\n\ngoal (2 subgoals):\n 1. Digamma (of_nat (Suc 0)) = harm 0 - euler_mascheroni\n 2. \\<And>n. Digamma (of_nat (Suc n)) = harm n - euler_mascheroni \\<Longrightarrow> Digamma (of_nat (Suc (Suc n))) = harm (Suc n) - euler_mascheroni\n[PROOF STEP]\nhave \"\\<dots> + inverse (of_nat (Suc n)) = harm (Suc n) - euler_mascheroni\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. harm n - euler_mascheroni + inverse (of_nat (Suc n)) = harm (Suc n) - euler_mascheroni\n[PROOF STEP]\nby (simp add: harm_Suc)\n[PROOF STATE]\nproof (state)\nthis:\nharm n - euler_mascheroni + inverse (of_nat (Suc n)) = harm (Suc n) - euler_mascheroni\n\ngoal (2 subgoals):\n 1. Digamma (of_nat (Suc 0)) = harm 0 - euler_mascheroni\n 2. \\<And>n. Digamma (of_nat (Suc n)) = harm n - euler_mascheroni \\<Longrightarrow> Digamma (of_nat (Suc (Suc n))) = harm (Suc n) - euler_mascheroni\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nDigamma (of_nat (Suc (Suc n))) = harm (Suc n) - euler_mascheroni\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nDigamma (of_nat (Suc (Suc n))) = harm (Suc n) - euler_mascheroni\n\ngoal (1 subgoal):\n 1. Digamma (of_nat (Suc (Suc n))) = harm (Suc n) - euler_mascheroni\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nDigamma (of_nat (Suc (Suc n))) = harm (Suc n) - euler_mascheroni\n\ngoal (1 subgoal):\n 1. Digamma (of_nat (Suc 0)) = harm 0 - euler_mascheroni\n[PROOF STEP]\nqed (simp add: harm_def)", "meta": {"llama_tokens": 2145, "file": null, "length": 17, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045877523148, "lm_q2_score": 0.8539127510928476, "lm_q1q2_score": 0.7575953103097749}}
{"text": "[STATEMENT]\nlemma mset_set_Diff:\n  assumes \"finite A\" \"B \\<subseteq> A\"\n  shows  \"mset_set (A - B) = mset_set A - mset_set B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mset_set (A - B) = mset_set A - mset_set B\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. mset_set (A - B) = mset_set A - mset_set B\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\nB \\<subseteq> A\n[PROOF STEP]\nhave \"mset_set ((A - B) \\<union> B) = mset_set (A - B) + mset_set B\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nB \\<subseteq> A\n\ngoal (1 subgoal):\n 1. mset_set (A - B \\<union> B) = mset_set (A - B) + mset_set B\n[PROOF STEP]\nby (intro mset_set_Union) (auto dest: finite_subset)\n[PROOF STATE]\nproof (state)\nthis:\nmset_set (A - B \\<union> B) = mset_set (A - B) + mset_set B\n\ngoal (1 subgoal):\n 1. mset_set (A - B) = mset_set A - mset_set B\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nmset_set (A - B \\<union> B) = mset_set (A - B) + mset_set B\n\ngoal (1 subgoal):\n 1. mset_set (A - B) = mset_set A - mset_set B\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\nB \\<subseteq> A\n[PROOF STEP]\nhave \"A - B \\<union> B = A\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nB \\<subseteq> A\n\ngoal (1 subgoal):\n 1. A - B \\<union> B = A\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nA - B \\<union> B = A\n\ngoal (1 subgoal):\n 1. mset_set (A - B) = mset_set A - mset_set B\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nmset_set A = mset_set (A - B) + mset_set B\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nmset_set A = mset_set (A - B) + mset_set B\n\ngoal (1 subgoal):\n 1. mset_set (A - B) = mset_set A - mset_set B\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nmset_set (A - B) = mset_set A - mset_set B\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 873, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933094145755219, "lm_q2_score": 0.8479677526147223, "lm_q1q2_score": 0.7574975766671787}}
{"text": "[STATEMENT]\nlemma prime_dvd_choose:\n  assumes \"0 < k\" \"k < p\" \"prime p\" \n  shows \"p dvd (p choose k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p dvd p choose k\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. p dvd p choose k\n[PROOF STEP]\nhave \"k \\<le> p\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. k \\<le> p\n[PROOF STEP]\nusing \\<open>k < p\\<close>\n[PROOF STATE]\nproof (prove)\nusing this:\nk < p\n\ngoal (1 subgoal):\n 1. k \\<le> p\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nk \\<le> p\n\ngoal (1 subgoal):\n 1. p dvd p choose k\n[PROOF STEP]\nhave \"p dvd fact p\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p dvd fact p\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < k\nk < p\nprime p\n\ngoal (1 subgoal):\n 1. p dvd fact p\n[PROOF STEP]\nby (simp add: prime_dvd_fact_iff)\n[PROOF STATE]\nproof (state)\nthis:\np dvd fact p\n\ngoal (1 subgoal):\n 1. p dvd p choose k\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\np dvd fact p\n\ngoal (1 subgoal):\n 1. p dvd p choose k\n[PROOF STEP]\nhave \"\\<not> p dvd fact k * fact (p - k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<not> p dvd fact k * fact (p - k)\n[PROOF STEP]\nunfolding prime_dvd_mult_iff[OF assms(3)] prime_dvd_fact_iff[OF assms(3)]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<not> (p \\<le> k \\<or> p \\<le> p - k)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < k\nk < p\nprime p\n\ngoal (1 subgoal):\n 1. \\<not> (p \\<le> k \\<or> p \\<le> p - k)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\<not> p dvd fact k * fact (p - k)\n\ngoal (1 subgoal):\n 1. p dvd p choose k\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\np dvd fact p\n\\<not> p dvd fact k * fact (p - k)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\np dvd fact p\n\\<not> p dvd fact k * fact (p - k)\n\ngoal (1 subgoal):\n 1. p dvd p choose k\n[PROOF STEP]\nunfolding binomial_fact_lemma[OF \\<open>k \\<le> p\\<close>, symmetric]\n[PROOF STATE]\nproof (prove)\nusing this:\np dvd fact k * fact (p - k) * (p choose k)\n\\<not> p dvd fact k * fact (p - k)\n\ngoal (1 subgoal):\n 1. p dvd p choose k\n[PROOF STEP]\nusing assms prime_dvd_multD\n[PROOF STATE]\nproof (prove)\nusing this:\np dvd fact k * fact (p - k) * (p choose k)\n\\<not> p dvd fact k * fact (p - k)\n0 < k\nk < p\nprime p\n\\<lbrakk>prime ?p; ?p dvd ?a * ?b\\<rbrakk> \\<Longrightarrow> ?p dvd ?a \\<or> ?p dvd ?b\n\ngoal (1 subgoal):\n 1. p dvd p choose k\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\np dvd p choose k\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1182, "file": "Mersenne_Primes_Lucas_Lehmer_Auxiliary", "length": 18, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8791467801752451, "lm_q2_score": 0.8615382147637196, "lm_q1q2_score": 0.7574185475074529}}
{"text": "[STATEMENT]\nlemma set_integral_Un:\n  fixes f :: \"_ \\<Rightarrow> _ :: {banach, second_countable_topology}\"\n  assumes \"A \\<inter> B = {}\"\n  and \"set_integrable M A f\"\n  and \"set_integrable M B f\"\nshows \"LINT x:A\\<union>B|M. f x = (LINT x:A|M. f x) + (LINT x:B|M. f x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set_lebesgue_integral M (A \\<union> B) f = set_lebesgue_integral M A f + set_lebesgue_integral M B f\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\<inter> B = {}\nset_integrable M A f\nset_integrable M B f\n\ngoal (1 subgoal):\n 1. set_lebesgue_integral M (A \\<union> B) f = set_lebesgue_integral M A f + set_lebesgue_integral M B f\n[PROOF STEP]\nunfolding set_integrable_def set_lebesgue_integral_def\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\<inter> B = {}\nintegrable M (\\<lambda>x. indicat_real A x *\\<^sub>R f x)\nintegrable M (\\<lambda>x. indicat_real B x *\\<^sub>R f x)\n\ngoal (1 subgoal):\n 1. LINT x|M. indicat_real (A \\<union> B) x *\\<^sub>R f x = (LINT x|M. indicat_real A x *\\<^sub>R f x) + (LINT x|M. indicat_real B x *\\<^sub>R f x)\n[PROOF STEP]\nby (auto simp add: indicator_union_arith indicator_inter_arith[symmetric] scaleR_add_left)", "meta": {"llama_tokens": 510, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9136765234137297, "lm_q2_score": 0.828938806208442, "lm_q1q2_score": 0.7573819265792567}}
{"text": "[STATEMENT]\ntheorem s1_1_th: \"(\\<lambda> y. (nat_to_pr n) (c_pair x y)) = nat_to_pr (s1_1 n x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<lambda>y. nat_to_pr n (c_pair x y)) = nat_to_pr (s1_1 n x)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\<lambda>y. nat_to_pr n (c_pair x y)) = nat_to_pr (s1_1 n x)\n[PROOF STEP]\nhave \"nat_to_pr (s1_1 n x) = nat_to_pr (comp_by_index n (index_of_c_pair_n x))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. nat_to_pr (s1_1 n x) = nat_to_pr (comp_by_index n (index_of_c_pair_n x))\n[PROOF STEP]\nby (simp add: s1_1_def)\n[PROOF STATE]\nproof (state)\nthis:\nnat_to_pr (s1_1 n x) = nat_to_pr (comp_by_index n (index_of_c_pair_n x))\n\ngoal (1 subgoal):\n 1. (\\<lambda>y. nat_to_pr n (c_pair x y)) = nat_to_pr (s1_1 n x)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nnat_to_pr (s1_1 n x) = nat_to_pr (comp_by_index n (index_of_c_pair_n x))\n\ngoal (1 subgoal):\n 1. (\\<lambda>y. nat_to_pr n (c_pair x y)) = nat_to_pr (s1_1 n x)\n[PROOF STEP]\nhave \"\\<dots> = (\\<lambda> z. (nat_to_pr n) ((nat_to_pr (index_of_c_pair_n x)) z))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. nat_to_pr (comp_by_index n (index_of_c_pair_n x)) = (\\<lambda>z. nat_to_pr n (nat_to_pr (index_of_c_pair_n x) z))\n[PROOF STEP]\nby (simp add: comp_by_index_main)\n[PROOF STATE]\nproof (state)\nthis:\nnat_to_pr (comp_by_index n (index_of_c_pair_n x)) = (\\<lambda>z. nat_to_pr n (nat_to_pr (index_of_c_pair_n x) z))\n\ngoal (1 subgoal):\n 1. (\\<lambda>y. nat_to_pr n (c_pair x y)) = nat_to_pr (s1_1 n x)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nnat_to_pr (comp_by_index n (index_of_c_pair_n x)) = (\\<lambda>z. nat_to_pr n (nat_to_pr (index_of_c_pair_n x) z))\n\ngoal (1 subgoal):\n 1. (\\<lambda>y. nat_to_pr n (c_pair x y)) = nat_to_pr (s1_1 n x)\n[PROOF STEP]\nhave \"\\<dots> = (\\<lambda> z. (nat_to_pr n) ((\\<lambda> u. c_pair x u) z))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<lambda>z. nat_to_pr n (nat_to_pr (index_of_c_pair_n x) z)) = (\\<lambda>z. nat_to_pr n (c_pair x z))\n[PROOF STEP]\nby (simp add: index_of_c_pair_n_main)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>z. nat_to_pr n (nat_to_pr (index_of_c_pair_n x) z)) = (\\<lambda>z. nat_to_pr n (c_pair x z))\n\ngoal (1 subgoal):\n 1. (\\<lambda>y. nat_to_pr n (c_pair x y)) = nat_to_pr (s1_1 n x)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nnat_to_pr (s1_1 n x) = (\\<lambda>z. nat_to_pr n (c_pair x z))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nnat_to_pr (s1_1 n x) = (\\<lambda>z. nat_to_pr n (c_pair x z))\n\ngoal (1 subgoal):\n 1. (\\<lambda>y. nat_to_pr n (c_pair x y)) = nat_to_pr (s1_1 n x)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>y. nat_to_pr n (c_pair x y)) = nat_to_pr (s1_1 n x)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1404, "file": "Recursion-Theory-I_PRecFun2", "length": 13, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.91610961358942, "lm_q2_score": 0.8267117983401363, "lm_q1q2_score": 0.7573586261271967}}
{"text": "[STATEMENT]\nlemma ex_NleastTr:\"\\<lbrakk>a \\<in> (A::nat set); \\<not> (\\<exists>m. m\\<in>A \\<and> (\\<forall>x\\<in>A. m \\<le> x))\\<rbrakk> \\<Longrightarrow>\n                        (ndec_seq A a n) \\<le> (a - n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>a \\<in> A; \\<nexists>m. m \\<in> A \\<and> (\\<forall>x\\<in>A. m \\<le> x)\\<rbrakk> \\<Longrightarrow> ndec_seq A a n \\<le> a - n\n[PROOF STEP]\napply (induct_tac n)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\<lbrakk>a \\<in> A; \\<nexists>m. m \\<in> A \\<and> (\\<forall>x\\<in>A. m \\<le> x)\\<rbrakk> \\<Longrightarrow> ndec_seq A a 0 \\<le> a - 0\n 2. \\<And>n. \\<lbrakk>a \\<in> A; \\<nexists>m. m \\<in> A \\<and> (\\<forall>x\\<in>A. m \\<le> x); ndec_seq A a n \\<le> a - n\\<rbrakk> \\<Longrightarrow> ndec_seq A a (Suc n) \\<le> a - Suc n\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>n. \\<lbrakk>a \\<in> A; \\<nexists>m. m \\<in> A \\<and> (\\<forall>x\\<in>A. m \\<le> x); ndec_seq A a n \\<le> a - n\\<rbrakk> \\<Longrightarrow> ndec_seq A a (Suc n) \\<le> a - Suc n\n[PROOF STEP]\napply (frule_tac n = n in ndec_seqn1[of \"a\" \"A\"], assumption+)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>n. \\<lbrakk>a \\<in> A; \\<nexists>m. m \\<in> A \\<and> (\\<forall>x\\<in>A. m \\<le> x); ndec_seq A a n \\<le> a - n; ndec_seq A a (Suc n) \\<le> ndec_seq A a n - 1\\<rbrakk> \\<Longrightarrow> ndec_seq A a (Suc n) \\<le> a - Suc n\n[PROOF STEP]\napply (subgoal_tac \"ndec_seq A a n - 1 \\<le> (a - n) - 1\")\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\<And>n. \\<lbrakk>a \\<in> A; \\<nexists>m. m \\<in> A \\<and> (\\<forall>x\\<in>A. m \\<le> x); ndec_seq A a n \\<le> a - n; ndec_seq A a (Suc n) \\<le> ndec_seq A a n - 1; ndec_seq A a n - 1 \\<le> a - n - 1\\<rbrakk> \\<Longrightarrow> ndec_seq A a (Suc n) \\<le> a - Suc n\n 2. \\<And>n. \\<lbrakk>a \\<in> A; \\<nexists>m. m \\<in> A \\<and> (\\<forall>x\\<in>A. m \\<le> x); ndec_seq A a n \\<le> a - n; ndec_seq A a (Suc n) \\<le> ndec_seq A a n - 1\\<rbrakk> \\<Longrightarrow> ndec_seq A a n - 1 \\<le> a - n - 1\n[PROOF STEP]\nprefer 2\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\<And>n. \\<lbrakk>a \\<in> A; \\<nexists>m. m \\<in> A \\<and> (\\<forall>x\\<in>A. m \\<le> x); ndec_seq A a n \\<le> a - n; ndec_seq A a (Suc n) \\<le> ndec_seq A a n - 1\\<rbrakk> \\<Longrightarrow> ndec_seq A a n - 1 \\<le> a - n - 1\n 2. \\<And>n. \\<lbrakk>a \\<in> A; \\<nexists>m. m \\<in> A \\<and> (\\<forall>x\\<in>A. m \\<le> x); ndec_seq A a n \\<le> a - n; ndec_seq A a (Suc n) \\<le> ndec_seq A a n - 1; ndec_seq A a n - 1 \\<le> a - n - 1\\<rbrakk> \\<Longrightarrow> ndec_seq A a (Suc n) \\<le> a - Suc n\n[PROOF STEP]\napply arith\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>n. \\<lbrakk>a \\<in> A; \\<nexists>m. m \\<in> A \\<and> (\\<forall>x\\<in>A. m \\<le> x); ndec_seq A a n \\<le> a - n; ndec_seq A a (Suc n) \\<le> ndec_seq A a n - 1; ndec_seq A a n - 1 \\<le> a - n - 1\\<rbrakk> \\<Longrightarrow> ndec_seq A a (Suc n) \\<le> a - Suc n\n[PROOF STEP]\napply arith\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1488, "file": "Group-Ring-Module_Algebra1", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9263037262250327, "lm_q2_score": 0.8175744784160989, "lm_q1q2_score": 0.75732228582332}}
{"text": "[STATEMENT]\nlemma inner_axis_axis:\n  \"inner (axis i x) (axis j y) = (if i = j then inner x y else 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inner (axis i x) (axis j y) = (if i = j then inner x y else 0)\n[PROOF STEP]\nby (simp add: inner_vec_def axis_def sum.neutral sum.remove [of _ j])", "meta": {"llama_tokens": 123, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.905989810230102, "lm_q2_score": 0.8354835411997897, "lm_q1q2_score": 0.7569395749419712}}
{"text": "[STATEMENT]\nlemma orthogonal_matrix2:\n  fixes A::\"real^'n^'n\"\n  shows \"orthogonal_matrix A = ((pairwise orthogonal (columns A)) \\<and> (\\<forall>i. norm (column i A) = 1) \\<and>\n  (card (columns A) = ncols A))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. orthogonal_matrix A = (pairwise orthogonal (columns A) \\<and> (\\<forall>i. norm (column i A) = 1) \\<and> card (columns A) = ncols A)\n[PROOF STEP]\nusing orthogonal_matrix_intro[of A] \n    orthogonal_matrix_is_orthogonal[of A]\n    orthogonal_matrix_norm[of A]\n    orthogonal_matrix_card[of A]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>pairwise orthogonal (columns A); \\<forall>i. norm (column i A) = 1; card (columns A) = ncols A\\<rbrakk> \\<Longrightarrow> orthogonal_matrix A\northogonal_matrix A \\<Longrightarrow> pairwise orthogonal (columns A)\northogonal_matrix A \\<Longrightarrow> norm (column ?i A) = 1\northogonal_matrix A \\<Longrightarrow> card (columns A) = ncols A\n\ngoal (1 subgoal):\n 1. orthogonal_matrix A = (pairwise orthogonal (columns A) \\<and> (\\<forall>i. norm (column i A) = 1) \\<and> card (columns A) = ncols A)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 388, "file": "QR_Decomposition_Miscellaneous_QR", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.899121388082479, "lm_q2_score": 0.8418256492357358, "lm_q1q2_score": 0.7569034462642689}}
{"text": "[STATEMENT]\nlemma zmod_eq_diff_mod_0_conv: \"\n  ((a::int) mod m = b mod m) = ((b - a) mod m = 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (a mod m = b mod m) = ((b - a) mod m = 0)\n[PROOF STEP]\napply (rule iffI)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. a mod m = b mod m \\<Longrightarrow> (b - a) mod m = 0\n 2. (b - a) mod m = 0 \\<Longrightarrow> a mod m = b mod m\n[PROOF STEP]\napply (rule zmod_eq_imp_diff_mod_0, assumption)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (b - a) mod m = 0 \\<Longrightarrow> a mod m = b mod m\n[PROOF STEP]\napply (rule zdiff_mod_0_imp_mod_eq, assumption)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 306, "file": "List-Infinite_CommonArith_Util_Div", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.899121388082479, "lm_q2_score": 0.8418256412990657, "lm_q1q2_score": 0.756903439128239}}
{"text": "[STATEMENT]\nlemma autonomous_linear_sol_is_exp:\n  assumes \"D X = (\\<lambda>t. A *\\<^sub>V X t) on {t\\<^sub>0--t}\" and \"X t\\<^sub>0 = s\" \n  shows \"X t = exp ((t - t\\<^sub>0) *\\<^sub>R A) *\\<^sub>V s\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. X t = exp ((t - t\\<^sub>0) *\\<^sub>R A) *\\<^sub>V s\n[PROOF STEP]\napply(rule sq_mtx_unique_sol_autonomous_affine[of \"\\<lambda>s. {t\\<^sub>0--t}\" _ t X A 0])\n[PROOF STATE]\nproof (prove)\ngoal (4 subgoals):\n 1. is_interval {t\\<^sub>0--t}\n 2. t \\<in> {t\\<^sub>0--t}\n 3. X \\<in> Sols (\\<lambda>t s. A *\\<^sub>V s + 0) (\\<lambda>s. {t\\<^sub>0--t}) UNIV ?t\\<^sub>0 ?s\n 4. (\\<lambda>a. exp ((a - t\\<^sub>0) *\\<^sub>R A) *\\<^sub>V s) \\<in> Sols (\\<lambda>t s. A *\\<^sub>V s + 0) (\\<lambda>s. {t\\<^sub>0--t}) UNIV ?t\\<^sub>0 ?s\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n D X = (\\<lambda>t. A *\\<^sub>V X t) on {t\\<^sub>0--t}\nX t\\<^sub>0 = s\n\ngoal (4 subgoals):\n 1. is_interval {t\\<^sub>0--t}\n 2. t \\<in> {t\\<^sub>0--t}\n 3. X \\<in> Sols (\\<lambda>t s. A *\\<^sub>V s + 0) (\\<lambda>s. {t\\<^sub>0--t}) UNIV ?t\\<^sub>0 ?s\n 4. (\\<lambda>a. exp ((a - t\\<^sub>0) *\\<^sub>R A) *\\<^sub>V s) \\<in> Sols (\\<lambda>t s. A *\\<^sub>V s + 0) (\\<lambda>s. {t\\<^sub>0--t}) UNIV ?t\\<^sub>0 ?s\n[PROOF STEP]\napply(simp_all add: ivp_sols_def)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\<lbrakk>\\<forall>i.  D (\\<lambda>x. X x $ i) = (\\<lambda>x. A *\\<^sub>V X x $ i) on {t\\<^sub>0--t}; X t\\<^sub>0 = s\\<rbrakk> \\<Longrightarrow> X ?t\\<^sub>0 = ?s \\<and> ?t\\<^sub>0 \\<in> {t\\<^sub>0--t}\n 2. \\<lbrakk>\\<forall>i.  D (\\<lambda>x. X x $ i) = (\\<lambda>x. A *\\<^sub>V X x $ i) on {t\\<^sub>0--t}; X t\\<^sub>0 = s\\<rbrakk> \\<Longrightarrow> (\\<forall>i.  D (\\<lambda>x. exp ((x - t\\<^sub>0) *\\<^sub>R A) *\\<^sub>V s $ i) = (\\<lambda>x. A *\\<^sub>V (exp ((x - t\\<^sub>0) *\\<^sub>R A) *\\<^sub>V s) $ i) on {t\\<^sub>0--t}) \\<and> exp ((?t\\<^sub>0 - t\\<^sub>0) *\\<^sub>R A) *\\<^sub>V s = ?s \\<and> ?t\\<^sub>0 \\<in> {t\\<^sub>0--t}\n[PROOF STEP]\nusing has_vderiv_on_sq_mtx_linear\n[PROOF STATE]\nproof (prove)\nusing this:\n D (\\<lambda>t. exp ((t - ?t\\<^sub>0) *\\<^sub>R ?A) *\\<^sub>V ?s) = (\\<lambda>t. ?A *\\<^sub>V (exp ((t - ?t\\<^sub>0) *\\<^sub>R ?A) *\\<^sub>V ?s)) on {?t\\<^sub>0--?t}\n\ngoal (2 subgoals):\n 1. \\<lbrakk>\\<forall>i.  D (\\<lambda>x. X x $ i) = (\\<lambda>x. A *\\<^sub>V X x $ i) on {t\\<^sub>0--t}; X t\\<^sub>0 = s\\<rbrakk> \\<Longrightarrow> X ?t\\<^sub>0 = ?s \\<and> ?t\\<^sub>0 \\<in> {t\\<^sub>0--t}\n 2. \\<lbrakk>\\<forall>i.  D (\\<lambda>x. X x $ i) = (\\<lambda>x. A *\\<^sub>V X x $ i) on {t\\<^sub>0--t}; X t\\<^sub>0 = s\\<rbrakk> \\<Longrightarrow> (\\<forall>i.  D (\\<lambda>x. exp ((x - t\\<^sub>0) *\\<^sub>R A) *\\<^sub>V s $ i) = (\\<lambda>x. A *\\<^sub>V (exp ((x - t\\<^sub>0) *\\<^sub>R A) *\\<^sub>V s) $ i) on {t\\<^sub>0--t}) \\<and> exp ((?t\\<^sub>0 - t\\<^sub>0) *\\<^sub>R A) *\\<^sub>V s = ?s \\<and> ?t\\<^sub>0 \\<in> {t\\<^sub>0--t}\n[PROOF STEP]\nby force+", "meta": {"llama_tokens": 1427, "file": "Matrices_for_ODEs_MTX_Flows", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.925229959153748, "lm_q2_score": 0.8175744761936437, "lm_q1q2_score": 0.7564443992137918}}
{"text": "[STATEMENT]\nlemma span_not_univ_subset_hyperplane:\n  fixes S :: \"'a::euclidean_space set\"\n  assumes SU: \"span S \\<noteq> UNIV\"\n  shows \"\\<exists> a. a \\<noteq>0 \\<and> span S \\<subseteq> {x. a \\<bullet> x = 0}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<exists>a. a \\<noteq> (0::'a) \\<and> span S \\<subseteq> {x. a \\<bullet> x = 0}\n[PROOF STEP]\nusing span_not_univ_orthogonal[OF SU]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<exists>a. a \\<noteq> (0::'a) \\<and> (\\<forall>x\\<in>span S. a \\<bullet> x = 0)\n\ngoal (1 subgoal):\n 1. \\<exists>a. a \\<noteq> (0::'a) \\<and> span S \\<subseteq> {x. a \\<bullet> x = 0}\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 279, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9124361533336451, "lm_q2_score": 0.8289388125473629, "lm_q1q2_score": 0.7563537414696754}}
{"text": "[STATEMENT]\nlemma card_UNIV_set: \"CARD('a set) = (if CARD('a) = 0 then 0 else 2 ^ CARD('a))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. CARD('a set) = (if CARD('a) = 0 then 0 else 2 ^ CARD('a))\n[PROOF STEP]\nby(simp add: card_eq_0_iff card_Pow flip: Pow_UNIV)", "meta": {"llama_tokens": 130, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9099070060380482, "lm_q2_score": 0.8311430415844385, "lm_q1q2_score": 0.7562628765574535}}
{"text": "[STATEMENT]\nlemma prime_dvd_choose:\n  assumes \"0 < k\" \"k < p\" \"prime p\" \n  shows \"p dvd (p choose k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p dvd p choose k\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. p dvd p choose k\n[PROOF STEP]\nhave \"k \\<le> p\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. k \\<le> p\n[PROOF STEP]\nusing \\<open>k < p\\<close>\n[PROOF STATE]\nproof (prove)\nusing this:\nk < p\n\ngoal (1 subgoal):\n 1. k \\<le> p\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nk \\<le> p\n\ngoal (1 subgoal):\n 1. p dvd p choose k\n[PROOF STEP]\nhave \"p dvd fact p\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. p dvd fact p\n[PROOF STEP]\nusing \\<open>prime p\\<close>\n[PROOF STATE]\nproof (prove)\nusing this:\nprime p\n\ngoal (1 subgoal):\n 1. p dvd fact p\n[PROOF STEP]\nby (simp add: prime_dvd_fact_iff)\n[PROOF STATE]\nproof (state)\nthis:\np dvd fact p\n\ngoal (1 subgoal):\n 1. p dvd p choose k\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\np dvd fact p\n\ngoal (1 subgoal):\n 1. p dvd p choose k\n[PROOF STEP]\nhave \"\\<not> p dvd fact k * fact (p - k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<not> p dvd fact k * fact (p - k)\n[PROOF STEP]\nunfolding prime_dvd_mult_iff[OF \\<open>prime p\\<close>] prime_dvd_fact_iff[OF \\<open>prime p\\<close>]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<not> (p \\<le> k \\<or> p \\<le> p - k)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < k\nk < p\nprime p\n\ngoal (1 subgoal):\n 1. \\<not> (p \\<le> k \\<or> p \\<le> p - k)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\<not> p dvd fact k * fact (p - k)\n\ngoal (1 subgoal):\n 1. p dvd p choose k\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\np dvd fact p\n\\<not> p dvd fact k * fact (p - k)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\np dvd fact p\n\\<not> p dvd fact k * fact (p - k)\n\ngoal (1 subgoal):\n 1. p dvd p choose k\n[PROOF STEP]\nunfolding binomial_fact_lemma[OF \\<open>k \\<le> p\\<close>, symmetric]\n[PROOF STATE]\nproof (prove)\nusing this:\np dvd fact k * fact (p - k) * (p choose k)\n\\<not> p dvd fact k * fact (p - k)\n\ngoal (1 subgoal):\n 1. p dvd p choose k\n[PROOF STEP]\nusing assms prime_dvd_multD\n[PROOF STATE]\nproof (prove)\nusing this:\np dvd fact k * fact (p - k) * (p choose k)\n\\<not> p dvd fact k * fact (p - k)\n0 < k\nk < p\nprime p\n\\<lbrakk>prime ?p; ?p dvd ?a * ?b\\<rbrakk> \\<Longrightarrow> ?p dvd ?a \\<or> ?p dvd ?b\n\ngoal (1 subgoal):\n 1. p dvd p choose k\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\np dvd p choose k\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1188, "file": "Probabilistic_Prime_Tests_Algebraic_Auxiliaries", "length": 18, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8774767874818408, "lm_q2_score": 0.8615382076534742, "lm_q1q2_score": 0.7559797787446336}}
{"text": "[STATEMENT]\ntheorem sum_of_naturals:\n  \"2 * (\\<Sum>i::nat=0..n. i) = n * (n + 1)\"\n  (is \"?P n\" is \"?S n = _\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * \\<Sum> {0..n} = n * (n + 1)\n[PROOF STEP]\nproof (induct n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. 2 * \\<Sum> {0..0} = 0 * (0 + 1)\n 2. \\<And>n. 2 * \\<Sum> {0..n} = n * (n + 1) \\<Longrightarrow> 2 * \\<Sum> {0..Suc n} = Suc n * (Suc n + 1)\n[PROOF STEP]\nshow \"?P 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * \\<Sum> {0..0} = 0 * (0 + 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n2 * \\<Sum> {0..0} = 0 * (0 + 1)\n\ngoal (1 subgoal):\n 1. \\<And>n. 2 * \\<Sum> {0..n} = n * (n + 1) \\<Longrightarrow> 2 * \\<Sum> {0..Suc n} = Suc n * (Suc n + 1)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>n. 2 * \\<Sum> {0..n} = n * (n + 1) \\<Longrightarrow> 2 * \\<Sum> {0..Suc n} = Suc n * (Suc n + 1)\n[PROOF STEP]\nfix n\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>n. 2 * \\<Sum> {0..n} = n * (n + 1) \\<Longrightarrow> 2 * \\<Sum> {0..Suc n} = Suc n * (Suc n + 1)\n[PROOF STEP]\nhave \"?S (n + 1) = ?S n + 2 * (n + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * \\<Sum> {0..n + 1} = 2 * \\<Sum> {0..n} + 2 * (n + 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n2 * \\<Sum> {0..n + 1} = 2 * \\<Sum> {0..n} + 2 * (n + 1)\n\ngoal (1 subgoal):\n 1. \\<And>n. 2 * \\<Sum> {0..n} = n * (n + 1) \\<Longrightarrow> 2 * \\<Sum> {0..Suc n} = Suc n * (Suc n + 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n2 * \\<Sum> {0..n + 1} = 2 * \\<Sum> {0..n} + 2 * (n + 1)\n\ngoal (1 subgoal):\n 1. \\<And>n. 2 * \\<Sum> {0..n} = n * (n + 1) \\<Longrightarrow> 2 * \\<Sum> {0..Suc n} = Suc n * (Suc n + 1)\n[PROOF STEP]\nassume \"?S n = n * (n + 1)\"\n[PROOF STATE]\nproof (state)\nthis:\n2 * \\<Sum> {0..n} = n * (n + 1)\n\ngoal (1 subgoal):\n 1. \\<And>n. 2 * \\<Sum> {0..n} = n * (n + 1) \\<Longrightarrow> 2 * \\<Sum> {0..Suc n} = Suc n * (Suc n + 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n2 * \\<Sum> {0..n} = n * (n + 1)\n\ngoal (1 subgoal):\n 1. \\<And>n. 2 * \\<Sum> {0..n} = n * (n + 1) \\<Longrightarrow> 2 * \\<Sum> {0..Suc n} = Suc n * (Suc n + 1)\n[PROOF STEP]\nhave \"\\<dots> + 2 * (n + 1) = (n + 1) * (n + 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n * (n + 1) + 2 * (n + 1) = (n + 1) * (n + 2)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nn * (n + 1) + 2 * (n + 1) = (n + 1) * (n + 2)\n\ngoal (1 subgoal):\n 1. \\<And>n. 2 * \\<Sum> {0..n} = n * (n + 1) \\<Longrightarrow> 2 * \\<Sum> {0..Suc n} = Suc n * (Suc n + 1)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n2 * \\<Sum> {0..n + 1} = (n + 1) * (n + 2)\n[PROOF STEP]\nshow \"?P (Suc n)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n2 * \\<Sum> {0..n + 1} = (n + 1) * (n + 2)\n\ngoal (1 subgoal):\n 1. 2 * \\<Sum> {0..Suc n} = Suc n * (Suc n + 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n2 * \\<Sum> {0..Suc n} = Suc n * (Suc n + 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1601, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045966995027, "lm_q2_score": 0.8519527963298947, "lm_q1q2_score": 0.7558564370748778}}
{"text": "[STATEMENT]\nlemma index_mat_four_block[simp]:\n  \"i < dim_row A + dim_row D \\<Longrightarrow> j < dim_col A + dim_col D \\<Longrightarrow> four_block_mat A B C D $$ (i,j)\n  = (if i < dim_row A then\n      if j < dim_col A then A $$ (i,j) else B $$ (i,j - dim_col A)\n      else if j < dim_col A then C $$ (i - dim_row A, j) else D $$ (i - dim_row A, j - dim_col A))\"\n  \"dim_row (four_block_mat A B C D) = dim_row A + dim_row D\"\n  \"dim_col (four_block_mat A B C D) = dim_col A + dim_col D\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<lbrakk>i < dim_row A + dim_row D; j < dim_col A + dim_col D\\<rbrakk> \\<Longrightarrow> four_block_mat A B C D $$ (i, j) = (if i < dim_row A then if j < dim_col A then A $$ (i, j) else B $$ (i, j - dim_col A) else if j < dim_col A then C $$ (i - dim_row A, j) else D $$ (i - dim_row A, j - dim_col A))) &&& dim_row (four_block_mat A B C D) = dim_row A + dim_row D &&& dim_col (four_block_mat A B C D) = dim_col A + dim_col D\n[PROOF STEP]\nunfolding four_block_mat_def Let_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<lbrakk>i < dim_row A + dim_row D; j < dim_col A + dim_col D\\<rbrakk> \\<Longrightarrow> mat (dim_row A + dim_row D) (dim_col A + dim_col D) (\\<lambda>(i, j). if i < dim_row A then if j < dim_col A then A $$ (i, j) else B $$ (i, j - dim_col A) else if j < dim_col A then C $$ (i - dim_row A, j) else D $$ (i - dim_row A, j - dim_col A)) $$ (i, j) = (if i < dim_row A then if j < dim_col A then A $$ (i, j) else B $$ (i, j - dim_col A) else if j < dim_col A then C $$ (i - dim_row A, j) else D $$ (i - dim_row A, j - dim_col A))) &&& dim_row (mat (dim_row A + dim_row D) (dim_col A + dim_col D) (\\<lambda>(i, j). if i < dim_row A then if j < dim_col A then A $$ (i, j) else B $$ (i, j - dim_col A) else if j < dim_col A then C $$ (i - dim_row A, j) else D $$ (i - dim_row A, j - dim_col A))) = dim_row A + dim_row D &&& dim_col (mat (dim_row A + dim_row D) (dim_col A + dim_col D) (\\<lambda>(i, j). if i < dim_row A then if j < dim_col A then A $$ (i, j) else B $$ (i, j - dim_col A) else if j < dim_col A then C $$ (i - dim_row A, j) else D $$ (i - dim_row A, j - dim_col A))) = dim_col A + dim_col D\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 929, "file": "Jordan_Normal_Form_Matrix", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9046505402422644, "lm_q2_score": 0.8354835330070839, "lm_q1q2_score": 0.7558206294983743}}
{"text": "[STATEMENT]\nlemma card_seqs [simp]: \"card (seqs n) = 2 ^ n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (seqs n) = 2 ^ n\n[PROOF STEP]\nproof (induction n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. card (seqs 0) = 2 ^ 0\n 2. \\<And>n. card (seqs n) = 2 ^ n \\<Longrightarrow> card (seqs (Suc n)) = 2 ^ Suc n\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\ncard (seqs n) = 2 ^ n\n\ngoal (2 subgoals):\n 1. card (seqs 0) = 2 ^ 0\n 2. \\<And>n. card (seqs n) = 2 ^ n \\<Longrightarrow> card (seqs (Suc n)) = 2 ^ Suc n\n[PROOF STEP]\nhave \"card (seqs (Suc n)) = card ((#) True ` seqs n \\<union> (#) False ` seqs n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (seqs (Suc n)) = card ((#) True ` seqs n \\<union> (#) False ` seqs n)\n[PROOF STEP]\nby (auto simp: seqs_Suc)\n[PROOF STATE]\nproof (state)\nthis:\ncard (seqs (Suc n)) = card ((#) True ` seqs n \\<union> (#) False ` seqs n)\n\ngoal (2 subgoals):\n 1. card (seqs 0) = 2 ^ 0\n 2. \\<And>n. card (seqs n) = 2 ^ n \\<Longrightarrow> card (seqs (Suc n)) = 2 ^ Suc n\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (seqs (Suc n)) = card ((#) True ` seqs n \\<union> (#) False ` seqs n)\n\ngoal (2 subgoals):\n 1. card (seqs 0) = 2 ^ 0\n 2. \\<And>n. card (seqs n) = 2 ^ n \\<Longrightarrow> card (seqs (Suc n)) = 2 ^ Suc n\n[PROOF STEP]\nfrom Suc.IH\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (seqs n) = 2 ^ n\n[PROOF STEP]\nhave \"\\<dots> = 2 ^ Suc n\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (seqs n) = 2 ^ n\n\ngoal (1 subgoal):\n 1. card ((#) True ` seqs n \\<union> (#) False ` seqs n) = 2 ^ Suc n\n[PROOF STEP]\nby (subst card_Un_disjoint) (auto simp: card_image)\n[PROOF STATE]\nproof (state)\nthis:\ncard ((#) True ` seqs n \\<union> (#) False ` seqs n) = 2 ^ Suc n\n\ngoal (2 subgoals):\n 1. card (seqs 0) = 2 ^ 0\n 2. \\<And>n. card (seqs n) = 2 ^ n \\<Longrightarrow> card (seqs (Suc n)) = 2 ^ Suc n\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard (seqs (Suc n)) = 2 ^ Suc n\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (seqs (Suc n)) = 2 ^ Suc n\n\ngoal (1 subgoal):\n 1. card (seqs (Suc n)) = 2 ^ Suc n\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard (seqs (Suc n)) = 2 ^ Suc n\n\ngoal (1 subgoal):\n 1. card (seqs 0) = 2 ^ 0\n[PROOF STEP]\nqed auto", "meta": {"llama_tokens": 1078, "file": "IMO2019_IMO2019_Q5", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8933094088947399, "lm_q2_score": 0.8459424295406088, "lm_q1q2_score": 0.7556883316919014}}
{"text": "[STATEMENT]\nlemma lower_asymptotic_density_eq_upper:\n  assumes \"lower_asymptotic_density A = l\" \"upper_asymptotic_density A = l\"\n  shows \"(\\<lambda>n. card(A \\<inter> {..<n})/n) \\<longlonglongrightarrow> l\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<lambda>n. real (card (A \\<inter> {..<n})) / real n) \\<longlonglongrightarrow> l\n[PROOF STEP]\napply (rule limsup_le_liminf_real)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. limsup (\\<lambda>x. ereal (real (card (A \\<inter> {..<x})) / real x)) \\<le> ereal l\n 2. ereal l \\<le> liminf (\\<lambda>x. ereal (real (card (A \\<inter> {..<x})) / real x))\n[PROOF STEP]\nusing upper_asymptotic_density_in_01(1)[of A] lower_asymptotic_density_in_01(1)[of A] assms\n[PROOF STATE]\nproof (prove)\nusing this:\nereal (upper_asymptotic_density A) = limsup (\\<lambda>x. ereal (real (card (A \\<inter> {..<x})) / real x))\nereal (lower_asymptotic_density A) = liminf (\\<lambda>x. ereal (real (card (A \\<inter> {..<x})) / real x))\nlower_asymptotic_density A = l\nupper_asymptotic_density A = l\n\ngoal (2 subgoals):\n 1. limsup (\\<lambda>x. ereal (real (card (A \\<inter> {..<x})) / real x)) \\<le> ereal l\n 2. ereal l \\<le> liminf (\\<lambda>x. ereal (real (card (A \\<inter> {..<x})) / real x))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 526, "file": "Ergodic_Theory_Asymptotic_Density", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9241418158002492, "lm_q2_score": 0.8175744828610095, "lm_q1q2_score": 0.7555547671431231}}
{"text": "[STATEMENT]\nlemma real_inverse_gt_one_lemma: \n      \"\\<lbrakk> 0 < (a::real); a < 1 \\<rbrakk> \\<Longrightarrow> inverse a > 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>0 < a; a < 1\\<rbrakk> \\<Longrightarrow> 1 < inverse a\n[PROOF STEP]\nby (metis one_less_inverse_iff)", "meta": {"llama_tokens": 123, "file": "Real_Power_RealPower", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9241418241572634, "lm_q2_score": 0.8175744673038222, "lm_q1q2_score": 0.7555547595985572}}
{"text": "[STATEMENT]\ntheorem sum_of_squares:\n  \"6 * (\\<Sum>i::nat=0..n. i^Suc (Suc 0)) = n * (n + 1) * (2 * n + 1)\"\n  (is \"?P n\" is \"?S n = _\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 6 * (\\<Sum>i = 0..n. i ^ Suc (Suc 0)) = n * (n + 1) * (2 * n + 1)\n[PROOF STEP]\nproof (induct n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. 6 * (\\<Sum>i = 0..0. i ^ Suc (Suc 0)) = 0 * (0 + 1) * (2 * 0 + 1)\n 2. \\<And>n. 6 * (\\<Sum>i = 0..n. i ^ Suc (Suc 0)) = n * (n + 1) * (2 * n + 1) \\<Longrightarrow> 6 * (\\<Sum>i = 0..Suc n. i ^ Suc (Suc 0)) = Suc n * (Suc n + 1) * (2 * Suc n + 1)\n[PROOF STEP]\nshow \"?P 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 6 * (\\<Sum>i = 0..0. i ^ Suc (Suc 0)) = 0 * (0 + 1) * (2 * 0 + 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n6 * (\\<Sum>i = 0..0. i ^ Suc (Suc 0)) = 0 * (0 + 1) * (2 * 0 + 1)\n\ngoal (1 subgoal):\n 1. \\<And>n. 6 * (\\<Sum>i = 0..n. i ^ Suc (Suc 0)) = n * (n + 1) * (2 * n + 1) \\<Longrightarrow> 6 * (\\<Sum>i = 0..Suc n. i ^ Suc (Suc 0)) = Suc n * (Suc n + 1) * (2 * Suc n + 1)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>n. 6 * (\\<Sum>i = 0..n. i ^ Suc (Suc 0)) = n * (n + 1) * (2 * n + 1) \\<Longrightarrow> 6 * (\\<Sum>i = 0..Suc n. i ^ Suc (Suc 0)) = Suc n * (Suc n + 1) * (2 * Suc n + 1)\n[PROOF STEP]\nfix n\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>n. 6 * (\\<Sum>i = 0..n. i ^ Suc (Suc 0)) = n * (n + 1) * (2 * n + 1) \\<Longrightarrow> 6 * (\\<Sum>i = 0..Suc n. i ^ Suc (Suc 0)) = Suc n * (Suc n + 1) * (2 * Suc n + 1)\n[PROOF STEP]\nhave \"?S (n + 1) = ?S n + 6 * (n + 1)^Suc (Suc 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 6 * (\\<Sum>i = 0..n + 1. i ^ Suc (Suc 0)) = 6 * (\\<Sum>i = 0..n. i ^ Suc (Suc 0)) + 6 * (n + 1) ^ Suc (Suc 0)\n[PROOF STEP]\nby (simp add: distrib)\n[PROOF STATE]\nproof (state)\nthis:\n6 * (\\<Sum>i = 0..n + 1. i ^ Suc (Suc 0)) = 6 * (\\<Sum>i = 0..n. i ^ Suc (Suc 0)) + 6 * (n + 1) ^ Suc (Suc 0)\n\ngoal (1 subgoal):\n 1. \\<And>n. 6 * (\\<Sum>i = 0..n. i ^ Suc (Suc 0)) = n * (n + 1) * (2 * n + 1) \\<Longrightarrow> 6 * (\\<Sum>i = 0..Suc n. i ^ Suc (Suc 0)) = Suc n * (Suc n + 1) * (2 * Suc n + 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n6 * (\\<Sum>i = 0..n + 1. i ^ Suc (Suc 0)) = 6 * (\\<Sum>i = 0..n. i ^ Suc (Suc 0)) + 6 * (n + 1) ^ Suc (Suc 0)\n\ngoal (1 subgoal):\n 1. \\<And>n. 6 * (\\<Sum>i = 0..n. i ^ Suc (Suc 0)) = n * (n + 1) * (2 * n + 1) \\<Longrightarrow> 6 * (\\<Sum>i = 0..Suc n. i ^ Suc (Suc 0)) = Suc n * (Suc n + 1) * (2 * Suc n + 1)\n[PROOF STEP]\nassume \"?S n = n * (n + 1) * (2 * n + 1)\"\n[PROOF STATE]\nproof (state)\nthis:\n6 * (\\<Sum>i = 0..n. i ^ Suc (Suc 0)) = n * (n + 1) * (2 * n + 1)\n\ngoal (1 subgoal):\n 1. \\<And>n. 6 * (\\<Sum>i = 0..n. i ^ Suc (Suc 0)) = n * (n + 1) * (2 * n + 1) \\<Longrightarrow> 6 * (\\<Sum>i = 0..Suc n. i ^ Suc (Suc 0)) = Suc n * (Suc n + 1) * (2 * Suc n + 1)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n6 * (\\<Sum>i = 0..n. i ^ Suc (Suc 0)) = n * (n + 1) * (2 * n + 1)\n\ngoal (1 subgoal):\n 1. \\<And>n. 6 * (\\<Sum>i = 0..n. i ^ Suc (Suc 0)) = n * (n + 1) * (2 * n + 1) \\<Longrightarrow> 6 * (\\<Sum>i = 0..Suc n. i ^ Suc (Suc 0)) = Suc n * (Suc n + 1) * (2 * Suc n + 1)\n[PROOF STEP]\nhave \"\\<dots> + 6 * (n + 1)^Suc (Suc 0) =\n      (n + 1) * (n + 2) * (2 * (n + 1) + 1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. n * (n + 1) * (2 * n + 1) + 6 * (n + 1) ^ Suc (Suc 0) = (n + 1) * (n + 2) * (2 * (n + 1) + 1)\n[PROOF STEP]\nby (simp add: distrib)\n[PROOF STATE]\nproof (state)\nthis:\nn * (n + 1) * (2 * n + 1) + 6 * (n + 1) ^ Suc (Suc 0) = (n + 1) * (n + 2) * (2 * (n + 1) + 1)\n\ngoal (1 subgoal):\n 1. \\<And>n. 6 * (\\<Sum>i = 0..n. i ^ Suc (Suc 0)) = n * (n + 1) * (2 * n + 1) \\<Longrightarrow> 6 * (\\<Sum>i = 0..Suc n. i ^ Suc (Suc 0)) = Suc n * (Suc n + 1) * (2 * Suc n + 1)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n6 * (\\<Sum>i = 0..n + 1. i ^ Suc (Suc 0)) = (n + 1) * (n + 2) * (2 * (n + 1) + 1)\n[PROOF STEP]\nshow \"?P (Suc n)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n6 * (\\<Sum>i = 0..n + 1. i ^ Suc (Suc 0)) = (n + 1) * (n + 2) * (2 * (n + 1) + 1)\n\ngoal (1 subgoal):\n 1. 6 * (\\<Sum>i = 0..Suc n. i ^ Suc (Suc 0)) = Suc n * (Suc n + 1) * (2 * Suc n + 1)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n6 * (\\<Sum>i = 0..Suc n. i ^ Suc (Suc 0)) = Suc n * (Suc n + 1) * (2 * Suc n + 1)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2454, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9019206791658465, "lm_q2_score": 0.8376199714402812, "lm_q1q2_score": 0.7554667735242954}}
{"text": "[STATEMENT]\ntheorem heap_minimum: \n  assumes \n    \"l < r\" \n    \"is_heap (\\<le>) heap l r\"\n  shows \"heap l = Min_mset (arr_mset heap l r)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. heap l = Min_mset (arr_mset heap l r)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. heap l = Min_mset (arr_mset heap l r)\n[PROOF STEP]\nhave \"(\\<forall>x \\<in># (arr_mset heap l r). (heap l) \\<le> x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<forall>x\\<in>#arr_mset heap l r. heap l \\<le> x\n[PROOF STEP]\nusing assms(2) heap_first_el_alt transp_le\n[PROOF STATE]\nproof (prove)\nusing this:\nis_heap (\\<le>) heap l r\n\\<lbrakk>transp ?cmp; is_heap ?cmp ?heap ?l ?r; ?x \\<in># arr_mset ?heap ?l ?r; ?heap ?l \\<noteq> ?x\\<rbrakk> \\<Longrightarrow> ?cmp (?heap ?l) ?x\ntransp (\\<le>)\n\ngoal (1 subgoal):\n 1. \\<forall>x\\<in>#arr_mset heap l r. heap l \\<le> x\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\<forall>x\\<in>#arr_mset heap l r. heap l \\<le> x\n\ngoal (1 subgoal):\n 1. heap l = Min_mset (arr_mset heap l r)\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<forall>x\\<in>#arr_mset heap l r. heap l \\<le> x\n\ngoal (1 subgoal):\n 1. heap l = Min_mset (arr_mset heap l r)\n[PROOF STEP]\nby (simp add: assms(1) dual_order.antisym)\n[PROOF STATE]\nproof (state)\nthis:\nheap l = Min_mset (arr_mset heap l r)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 640, "file": "IMP2_Binary_Heap_IMP2_Binary_Heap", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887588023318195, "lm_q2_score": 0.8499711699569786, "lm_q1q2_score": 0.7554193590275398}}
{"text": "[STATEMENT]\ntheorem powerset_enum_correct: \"set (map set (powerset_enum xs)) = Pow (set xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set (map set (powerset_enum xs)) = Pow (set xs)\n[PROOF STEP]\nproof(standard)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. set (map set (powerset_enum xs)) \\<subseteq> Pow (set xs)\n 2. Pow (set xs) \\<subseteq> set (map set (powerset_enum xs))\n[PROOF STEP]\nshow \"set (map set (powerset_enum xs)) \\<subseteq> Pow (set xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set (map set (powerset_enum xs)) \\<subseteq> Pow (set xs)\n[PROOF STEP]\nusing filter_bool_list_not_elem\n[PROOF STATE]\nproof (prove)\nusing this:\n?x \\<notin> set ?xs \\<Longrightarrow> ?x \\<notin> set (filter_bool_list ?bs ?xs)\n\ngoal (1 subgoal):\n 1. set (map set (powerset_enum xs)) \\<subseteq> Pow (set xs)\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\nset (map set (powerset_enum xs)) \\<subseteq> Pow (set xs)\n\ngoal (1 subgoal):\n 1. Pow (set xs) \\<subseteq> set (map set (powerset_enum xs))\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. Pow (set xs) \\<subseteq> set (map set (powerset_enum xs))\n[PROOF STEP]\nhave \"\\<And>x. x \\<subseteq> set xs \\<Longrightarrow> x \\<in> (\\<lambda>x. set (filter_bool_list x xs)) ` {zs. length zs = length xs}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>x. x \\<subseteq> set xs \\<Longrightarrow> x \\<in> (\\<lambda>x. set (filter_bool_list x xs)) ` {zs. length zs = length xs}\n[PROOF STEP]\nunfolding image_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>x. x \\<subseteq> set xs \\<Longrightarrow> x \\<in> {y. \\<exists>x\\<in>{zs. length zs = length xs}. y = set (filter_bool_list x xs)}\n[PROOF STEP]\nusing filter_bool_list_exist_length image_def\n[PROOF STATE]\nproof (prove)\nusing this:\n?A \\<subseteq> set ?xs \\<Longrightarrow> \\<exists>bs. length bs = length ?xs \\<and> ?A = set (filter_bool_list bs ?xs)\n?f ` ?A = {y. \\<exists>x\\<in>?A. y = ?f x}\n\ngoal (1 subgoal):\n 1. \\<And>x. x \\<subseteq> set xs \\<Longrightarrow> x \\<in> {y. \\<exists>x\\<in>{zs. length zs = length xs}. y = set (filter_bool_list x xs)}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n?x \\<subseteq> set xs \\<Longrightarrow> ?x \\<in> (\\<lambda>x. set (filter_bool_list x xs)) ` {zs. length zs = length xs}\n\ngoal (1 subgoal):\n 1. Pow (set xs) \\<subseteq> set (map set (powerset_enum xs))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n?x \\<subseteq> set xs \\<Longrightarrow> ?x \\<in> (\\<lambda>x. set (filter_bool_list x xs)) ` {zs. length zs = length xs}\n[PROOF STEP]\nshow \"Pow (set xs) \\<subseteq> set (map set (powerset_enum xs))\"\n[PROOF STATE]\nproof (prove)\nusing this:\n?x \\<subseteq> set xs \\<Longrightarrow> ?x \\<in> (\\<lambda>x. set (filter_bool_list x xs)) ` {zs. length zs = length xs}\n\ngoal (1 subgoal):\n 1. Pow (set xs) \\<subseteq> set (map set (powerset_enum xs))\n[PROOF STEP]\nusing all_bool_lists_correct\n[PROOF STATE]\nproof (prove)\nusing this:\n?x \\<subseteq> set xs \\<Longrightarrow> ?x \\<in> (\\<lambda>x. set (filter_bool_list x xs)) ` {zs. length zs = length xs}\nset (all_bool_lists ?x) = {xs. length xs = ?x}\n\ngoal (1 subgoal):\n 1. Pow (set xs) \\<subseteq> set (map set (powerset_enum xs))\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nPow (set xs) \\<subseteq> set (map set (powerset_enum xs))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1312, "file": "Combinatorial_Enumeration_Algorithms_Powerset", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.888758793492457, "lm_q2_score": 0.8499711699569787, "lm_q1q2_score": 0.7554193515143366}}
{"text": "[STATEMENT]\nlemma (in prime_number_theorem) ln_divisor_count_primorial'_asymp_equiv:\n  \"(\\<lambda>k. ln (divisor_count (primorial' k))) \\<sim>[at_top]\n     (\\<lambda>k. ln 2 * ln (primorial' k) / ln (ln (primorial' k)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<lambda>k. ln (real (divisor_count (primorial' k)))) \\<sim>[sequentially] (\\<lambda>k. ln 2 * ln (real (primorial' k)) / ln (ln (real (primorial' k))))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\<lambda>k. ln (real (divisor_count (primorial' k)))) \\<sim>[sequentially] (\\<lambda>k. ln 2 * ln (real (primorial' k)) / ln (ln (real (primorial' k))))\n[PROOF STEP]\nhave \"(\\<lambda>k. ln 2 * (ln (primorial' k) / ln (ln (primorial' k)))) \\<sim>[at_top] (\\<lambda>k. ln 2 * k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<lambda>k. ln 2 * (ln (real (primorial' k)) / ln (ln (real (primorial' k))))) \\<sim>[sequentially] (\\<lambda>x. ln 2 * real x)\n[PROOF STEP]\nby (intro asymp_equiv_intros ln_over_ln_ln_primorial'_asymp_equiv)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>k. ln 2 * (ln (real (primorial' k)) / ln (ln (real (primorial' k))))) \\<sim>[sequentially] (\\<lambda>x. ln 2 * real x)\n\ngoal (1 subgoal):\n 1. (\\<lambda>k. ln (real (divisor_count (primorial' k)))) \\<sim>[sequentially] (\\<lambda>k. ln 2 * ln (real (primorial' k)) / ln (ln (real (primorial' k))))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>k. ln 2 * (ln (real (primorial' k)) / ln (ln (real (primorial' k))))) \\<sim>[sequentially] (\\<lambda>x. ln 2 * real x)\n\ngoal (1 subgoal):\n 1. (\\<lambda>k. ln (real (divisor_count (primorial' k)))) \\<sim>[sequentially] (\\<lambda>k. ln 2 * ln (real (primorial' k)) / ln (ln (real (primorial' k))))\n[PROOF STEP]\nhave \"\\<dots> \\<sim>[at_top] (\\<lambda>k. ln (divisor_count (primorial' k)))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<lambda>x. ln 2 * real x) \\<sim>[sequentially] (\\<lambda>k. ln (real (divisor_count (primorial' k))))\n[PROOF STEP]\nby (simp add: ln_realpow mult_ac)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>x. ln 2 * real x) \\<sim>[sequentially] (\\<lambda>k. ln (real (divisor_count (primorial' k))))\n\ngoal (1 subgoal):\n 1. (\\<lambda>k. ln (real (divisor_count (primorial' k)))) \\<sim>[sequentially] (\\<lambda>k. ln 2 * ln (real (primorial' k)) / ln (ln (real (primorial' k))))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\<And>c d. c \\<sim>[sequentially] d \\<Longrightarrow> c \\<sim>[sequentially] d) \\<Longrightarrow> (\\<lambda>k. ln 2 * (ln (real (primorial' k)) / ln (ln (real (primorial' k))))) \\<sim>[sequentially] (\\<lambda>a. ln (real (divisor_count (primorial' a))))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<And>c d. c \\<sim>[sequentially] d \\<Longrightarrow> c \\<sim>[sequentially] d) \\<Longrightarrow> (\\<lambda>k. ln 2 * (ln (real (primorial' k)) / ln (ln (real (primorial' k))))) \\<sim>[sequentially] (\\<lambda>a. ln (real (divisor_count (primorial' a))))\n\ngoal (1 subgoal):\n 1. (\\<lambda>k. ln (real (divisor_count (primorial' k)))) \\<sim>[sequentially] (\\<lambda>k. ln 2 * ln (real (primorial' k)) / ln (ln (real (primorial' k))))\n[PROOF STEP]\nby (simp add: asymp_equiv_sym mult_ac)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>k. ln (real (divisor_count (primorial' k)))) \\<sim>[sequentially] (\\<lambda>k. ln 2 * ln (real (primorial' k)) / ln (ln (real (primorial' k))))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1431, "file": "Prime_Distribution_Elementary_PNT_Consequences", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9111797148356995, "lm_q2_score": 0.8289388083214156, "lm_q1q2_score": 0.7553122269825521}}
{"text": "[STATEMENT]\nlemma edge_density_Un:\n  assumes \"disjnt X1 X2\" \"finite X1\" \"finite X2\" \"finite Y\"\n  shows \"edge_density (X1 \\<union> X2) Y = (edge_density X1 Y * card X1 + edge_density X2 Y * card X2) / (card X1 + card X2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. edge_density (X1 \\<union> X2) Y = (edge_density X1 Y * real (card X1) + edge_density X2 Y * real (card X2)) / real (card X1 + card X2)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ndisjnt X1 X2\nfinite X1\nfinite X2\nfinite Y\n\ngoal (1 subgoal):\n 1. edge_density (X1 \\<union> X2) Y = (edge_density X1 Y * real (card X1) + edge_density X2 Y * real (card X2)) / real (card X1 + card X2)\n[PROOF STEP]\nunfolding edge_density_def\n[PROOF STATE]\nproof (prove)\nusing this:\ndisjnt X1 X2\nfinite X1\nfinite X2\nfinite Y\n\ngoal (1 subgoal):\n 1. real (card (all_edges_between (X1 \\<union> X2) Y)) / real (card (X1 \\<union> X2) * card Y) = (real (card (all_edges_between X1 Y)) / real (card X1 * card Y) * real (card X1) + real (card (all_edges_between X2 Y)) / real (card X2 * card Y) * real (card X2)) / real (card X1 + card X2)\n[PROOF STEP]\nby (simp add: all_edges_between_disjnt1 all_edges_between_Un1 finite_all_edges_between card_Un_disjnt divide_simps)", "meta": {"llama_tokens": 523, "file": "Undirected_Graph_Theory_Undirected_Graph_Basics", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9111797148356995, "lm_q2_score": 0.8289388019824946, "lm_q1q2_score": 0.7553122212066558}}
{"text": "[STATEMENT]\nlemma sigma_algebra_intersection:\n  assumes \"sigma_algebra \\<Omega> A\"\n          \"sigma_algebra \\<Omega> B\"\n  shows \"sigma_algebra \\<Omega> (A \\<inter> B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sigma_algebra \\<Omega> (A \\<inter> B)\n[PROOF STEP]\napply (subst sigma_algebra_iff)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. algebra \\<Omega> (A \\<inter> B) \\<and> (\\<forall>Aa. range Aa \\<subseteq> A \\<inter> B \\<longrightarrow> \\<Union> (range Aa) \\<in> A \\<inter> B)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nsigma_algebra \\<Omega> A\nsigma_algebra \\<Omega> B\n\ngoal (1 subgoal):\n 1. algebra \\<Omega> (A \\<inter> B) \\<and> (\\<forall>Aa. range Aa \\<subseteq> A \\<inter> B \\<longrightarrow> \\<Union> (range Aa) \\<in> A \\<inter> B)\n[PROOF STEP]\nby (auto simp add: sigma_algebra_iff algebra_intersection)", "meta": {"llama_tokens": 320, "file": "Ergodic_Theory_SG_Library_Complement", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9161096090086367, "lm_q2_score": 0.8244619263765706, "lm_q1q2_score": 0.7552974930153474}}
{"text": "[STATEMENT]\nlemma card_3_eq':\n    \"\\<lbrakk>card X = 3; card {a,b,c} = 3; {a,b,c} \\<subseteq>X\\<rbrakk> \\<Longrightarrow> X = {a,b,c}\"\n    \"\\<lbrakk>card X = 3; a \\<in> X; b \\<in> X; c \\<in> X; a \\<noteq> b; a \\<noteq> c; b \\<noteq> c\\<rbrakk> \\<Longrightarrow> X = {a,b,c}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<lbrakk>card X = 3; card {a, b, c} = 3; {a, b, c} \\<subseteq> X\\<rbrakk> \\<Longrightarrow> X = {a, b, c}) &&& (\\<lbrakk>card X = 3; a \\<in> X; b \\<in> X; c \\<in> X; a \\<noteq> b; a \\<noteq> c; b \\<noteq> c\\<rbrakk> \\<Longrightarrow> X = {a, b, c})\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\<lbrakk>card X = 3; card {a, b, c} = 3; {a, b, c} \\<subseteq> X\\<rbrakk> \\<Longrightarrow> X = {a, b, c}\n 2. \\<lbrakk>card X = 3; a \\<in> X; b \\<in> X; c \\<in> X; a \\<noteq> b; a \\<noteq> c; b \\<noteq> c\\<rbrakk> \\<Longrightarrow> X = {a, b, c}\n[PROOF STEP]\nshow \"\\<lbrakk>card X = 3; card {a,b,c} = 3; {a,b,c} \\<subseteq>X\\<rbrakk> \\<Longrightarrow> X = {a,b,c}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>card X = 3; card {a, b, c} = 3; {a, b, c} \\<subseteq> X\\<rbrakk> \\<Longrightarrow> X = {a, b, c}\n[PROOF STEP]\nby (metis card.infinite card_subset_eq zero_neq_numeral)\n[PROOF STATE]\nproof (state)\nthis:\n\\<lbrakk>card X = 3; card {a, b, c} = 3; {a, b, c} \\<subseteq> X\\<rbrakk> \\<Longrightarrow> X = {a, b, c}\n\ngoal (1 subgoal):\n 1. \\<lbrakk>card X = 3; a \\<in> X; b \\<in> X; c \\<in> X; a \\<noteq> b; a \\<noteq> c; b \\<noteq> c\\<rbrakk> \\<Longrightarrow> X = {a, b, c}\n[PROOF STEP]\nthus \"\\<lbrakk>card X = 3; a \\<in> X; b \\<in> X; c \\<in> X; a \\<noteq> b; a \\<noteq> c; b \\<noteq> c\\<rbrakk> \\<Longrightarrow> X = {a,b,c}\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>card X = 3; card {a, b, c} = 3; {a, b, c} \\<subseteq> X\\<rbrakk> \\<Longrightarrow> X = {a, b, c}\n\ngoal (1 subgoal):\n 1. \\<lbrakk>card X = 3; a \\<in> X; b \\<in> X; c \\<in> X; a \\<noteq> b; a \\<noteq> c; b \\<noteq> c\\<rbrakk> \\<Longrightarrow> X = {a, b, c}\n[PROOF STEP]\nby (meson card_3_dist empty_subsetI insert_subset)\n[PROOF STATE]\nproof (state)\nthis:\n\\<lbrakk>card X = 3; a \\<in> X; b \\<in> X; c \\<in> X; a \\<noteq> b; a \\<noteq> c; b \\<noteq> c\\<rbrakk> \\<Longrightarrow> X = {a, b, c}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1095, "file": "Schutz_Spacetime_Util", "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9086178994073576, "lm_q2_score": 0.8311430541321951, "lm_q1q2_score": 0.7551914559526108}}
{"text": "[STATEMENT]\nlemma fib_closed_form:\n  fixes \\<phi> \\<psi> :: real\n  defines \"\\<phi> \\<equiv> (1 + sqrt 5) / 2\"\n    and \"\\<psi> \\<equiv> (1 - sqrt 5) / 2\"\n  shows \"of_nat (fib n) = (\\<phi> ^ n - \\<psi> ^ n) / sqrt 5\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (fib n) = (\\<phi> ^ n - \\<psi> ^ n) / sqrt 5\n[PROOF STEP]\nproof (induct n rule: fib.induct)\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. real (fib 0) = (\\<phi> ^ 0 - \\<psi> ^ 0) / sqrt 5\n 2. real (fib (Suc 0)) = (\\<phi> ^ Suc 0 - \\<psi> ^ Suc 0) / sqrt 5\n 3. \\<And>n. \\<lbrakk>real (fib (Suc n)) = (\\<phi> ^ Suc n - \\<psi> ^ Suc n) / sqrt 5; real (fib n) = (\\<phi> ^ n - \\<psi> ^ n) / sqrt 5\\<rbrakk> \\<Longrightarrow> real (fib (Suc (Suc n))) = (\\<phi> ^ Suc (Suc n) - \\<psi> ^ Suc (Suc n)) / sqrt 5\n[PROOF STEP]\nfix n :: nat\n[PROOF STATE]\nproof (state)\ngoal (3 subgoals):\n 1. real (fib 0) = (\\<phi> ^ 0 - \\<psi> ^ 0) / sqrt 5\n 2. real (fib (Suc 0)) = (\\<phi> ^ Suc 0 - \\<psi> ^ Suc 0) / sqrt 5\n 3. \\<And>n. \\<lbrakk>real (fib (Suc n)) = (\\<phi> ^ Suc n - \\<psi> ^ Suc n) / sqrt 5; real (fib n) = (\\<phi> ^ n - \\<psi> ^ n) / sqrt 5\\<rbrakk> \\<Longrightarrow> real (fib (Suc (Suc n))) = (\\<phi> ^ Suc (Suc n) - \\<psi> ^ Suc (Suc n)) / sqrt 5\n[PROOF STEP]\nassume IH1: \"of_nat (fib n) = (\\<phi> ^ n - \\<psi> ^ n) / sqrt 5\"\n[PROOF STATE]\nproof (state)\nthis:\nreal (fib n) = (\\<phi> ^ n - \\<psi> ^ n) / sqrt 5\n\ngoal (3 subgoals):\n 1. real (fib 0) = (\\<phi> ^ 0 - \\<psi> ^ 0) / sqrt 5\n 2. real (fib (Suc 0)) = (\\<phi> ^ Suc 0 - \\<psi> ^ Suc 0) / sqrt 5\n 3. \\<And>n. \\<lbrakk>real (fib (Suc n)) = (\\<phi> ^ Suc n - \\<psi> ^ Suc n) / sqrt 5; real (fib n) = (\\<phi> ^ n - \\<psi> ^ n) / sqrt 5\\<rbrakk> \\<Longrightarrow> real (fib (Suc (Suc n))) = (\\<phi> ^ Suc (Suc n) - \\<psi> ^ Suc (Suc n)) / sqrt 5\n[PROOF STEP]\nassume IH2: \"of_nat (fib (Suc n)) = (\\<phi> ^ Suc n - \\<psi> ^ Suc n) / sqrt 5\"\n[PROOF STATE]\nproof (state)\nthis:\nreal (fib (Suc n)) = (\\<phi> ^ Suc n - \\<psi> ^ Suc n) / sqrt 5\n\ngoal (3 subgoals):\n 1. real (fib 0) = (\\<phi> ^ 0 - \\<psi> ^ 0) / sqrt 5\n 2. real (fib (Suc 0)) = (\\<phi> ^ Suc 0 - \\<psi> ^ Suc 0) / sqrt 5\n 3. \\<And>n. \\<lbrakk>real (fib (Suc n)) = (\\<phi> ^ Suc n - \\<psi> ^ Suc n) / sqrt 5; real (fib n) = (\\<phi> ^ n - \\<psi> ^ n) / sqrt 5\\<rbrakk> \\<Longrightarrow> real (fib (Suc (Suc n))) = (\\<phi> ^ Suc (Suc n) - \\<psi> ^ Suc (Suc n)) / sqrt 5\n[PROOF STEP]\nhave \"of_nat (fib (Suc (Suc n))) = of_nat (fib (Suc n)) + of_nat (fib n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. of_nat (fib (Suc (Suc n))) = of_nat (fib (Suc n)) + of_nat (fib n)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nof_nat (fib (Suc (Suc n))) = of_nat (fib (Suc n)) + of_nat (fib n)\n\ngoal (3 subgoals):\n 1. real (fib 0) = (\\<phi> ^ 0 - \\<psi> ^ 0) / sqrt 5\n 2. real (fib (Suc 0)) = (\\<phi> ^ Suc 0 - \\<psi> ^ Suc 0) / sqrt 5\n 3. \\<And>n. \\<lbrakk>real (fib (Suc n)) = (\\<phi> ^ Suc n - \\<psi> ^ Suc n) / sqrt 5; real (fib n) = (\\<phi> ^ n - \\<psi> ^ n) / sqrt 5\\<rbrakk> \\<Longrightarrow> real (fib (Suc (Suc n))) = (\\<phi> ^ Suc (Suc n) - \\<psi> ^ Suc (Suc n)) / sqrt 5\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nof_nat (fib (Suc (Suc n))) = of_nat (fib (Suc n)) + of_nat (fib n)\n\ngoal (3 subgoals):\n 1. real (fib 0) = (\\<phi> ^ 0 - \\<psi> ^ 0) / sqrt 5\n 2. real (fib (Suc 0)) = (\\<phi> ^ Suc 0 - \\<psi> ^ Suc 0) / sqrt 5\n 3. \\<And>n. \\<lbrakk>real (fib (Suc n)) = (\\<phi> ^ Suc n - \\<psi> ^ Suc n) / sqrt 5; real (fib n) = (\\<phi> ^ n - \\<psi> ^ n) / sqrt 5\\<rbrakk> \\<Longrightarrow> real (fib (Suc (Suc n))) = (\\<phi> ^ Suc (Suc n) - \\<psi> ^ Suc (Suc n)) / sqrt 5\n[PROOF STEP]\nhave \"\\<dots> = (\\<phi>^n * (\\<phi> + 1) - \\<psi>^n * (\\<psi> + 1)) / sqrt 5\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (fib (Suc n)) + real (fib n) = (\\<phi> ^ n * (\\<phi> + 1) - \\<psi> ^ n * (\\<psi> + 1)) / sqrt 5\n[PROOF STEP]\nby (simp add: IH1 IH2 field_simps)\n[PROOF STATE]\nproof (state)\nthis:\nreal (fib (Suc n)) + real (fib n) = (\\<phi> ^ n * (\\<phi> + 1) - \\<psi> ^ n * (\\<psi> + 1)) / sqrt 5\n\ngoal (3 subgoals):\n 1. real (fib 0) = (\\<phi> ^ 0 - \\<psi> ^ 0) / sqrt 5\n 2. real (fib (Suc 0)) = (\\<phi> ^ Suc 0 - \\<psi> ^ Suc 0) / sqrt 5\n 3. \\<And>n. \\<lbrakk>real (fib (Suc n)) = (\\<phi> ^ Suc n - \\<psi> ^ Suc n) / sqrt 5; real (fib n) = (\\<phi> ^ n - \\<psi> ^ n) / sqrt 5\\<rbrakk> \\<Longrightarrow> real (fib (Suc (Suc n))) = (\\<phi> ^ Suc (Suc n) - \\<psi> ^ Suc (Suc n)) / sqrt 5\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreal (fib (Suc n)) + real (fib n) = (\\<phi> ^ n * (\\<phi> + 1) - \\<psi> ^ n * (\\<psi> + 1)) / sqrt 5\n\ngoal (3 subgoals):\n 1. real (fib 0) = (\\<phi> ^ 0 - \\<psi> ^ 0) / sqrt 5\n 2. real (fib (Suc 0)) = (\\<phi> ^ Suc 0 - \\<psi> ^ Suc 0) / sqrt 5\n 3. \\<And>n. \\<lbrakk>real (fib (Suc n)) = (\\<phi> ^ Suc n - \\<psi> ^ Suc n) / sqrt 5; real (fib n) = (\\<phi> ^ n - \\<psi> ^ n) / sqrt 5\\<rbrakk> \\<Longrightarrow> real (fib (Suc (Suc n))) = (\\<phi> ^ Suc (Suc n) - \\<psi> ^ Suc (Suc n)) / sqrt 5\n[PROOF STEP]\nhave \"\\<phi> + 1 = \\<phi>\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<phi> + 1 = \\<phi>\\<^sup>2\n[PROOF STEP]\nby (simp add: \\<phi>_def field_simps power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\n\\<phi> + 1 = \\<phi>\\<^sup>2\n\ngoal (3 subgoals):\n 1. real (fib 0) = (\\<phi> ^ 0 - \\<psi> ^ 0) / sqrt 5\n 2. real (fib (Suc 0)) = (\\<phi> ^ Suc 0 - \\<psi> ^ Suc 0) / sqrt 5\n 3. \\<And>n. \\<lbrakk>real (fib (Suc n)) = (\\<phi> ^ Suc n - \\<psi> ^ Suc n) / sqrt 5; real (fib n) = (\\<phi> ^ n - \\<psi> ^ n) / sqrt 5\\<rbrakk> \\<Longrightarrow> real (fib (Suc (Suc n))) = (\\<phi> ^ Suc (Suc n) - \\<psi> ^ Suc (Suc n)) / sqrt 5\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\<phi> + 1 = \\<phi>\\<^sup>2\n\ngoal (3 subgoals):\n 1. real (fib 0) = (\\<phi> ^ 0 - \\<psi> ^ 0) / sqrt 5\n 2. real (fib (Suc 0)) = (\\<phi> ^ Suc 0 - \\<psi> ^ Suc 0) / sqrt 5\n 3. \\<And>n. \\<lbrakk>real (fib (Suc n)) = (\\<phi> ^ Suc n - \\<psi> ^ Suc n) / sqrt 5; real (fib n) = (\\<phi> ^ n - \\<psi> ^ n) / sqrt 5\\<rbrakk> \\<Longrightarrow> real (fib (Suc (Suc n))) = (\\<phi> ^ Suc (Suc n) - \\<psi> ^ Suc (Suc n)) / sqrt 5\n[PROOF STEP]\nhave \"\\<psi> + 1 = \\<psi>\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<psi> + 1 = \\<psi>\\<^sup>2\n[PROOF STEP]\nby (simp add: \\<psi>_def field_simps power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\n\\<psi> + 1 = \\<psi>\\<^sup>2\n\ngoal (3 subgoals):\n 1. real (fib 0) = (\\<phi> ^ 0 - \\<psi> ^ 0) / sqrt 5\n 2. real (fib (Suc 0)) = (\\<phi> ^ Suc 0 - \\<psi> ^ Suc 0) / sqrt 5\n 3. \\<And>n. \\<lbrakk>real (fib (Suc n)) = (\\<phi> ^ Suc n - \\<psi> ^ Suc n) / sqrt 5; real (fib n) = (\\<phi> ^ n - \\<psi> ^ n) / sqrt 5\\<rbrakk> \\<Longrightarrow> real (fib (Suc (Suc n))) = (\\<phi> ^ Suc (Suc n) - \\<psi> ^ Suc (Suc n)) / sqrt 5\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n\\<psi> + 1 = \\<psi>\\<^sup>2\n\ngoal (3 subgoals):\n 1. real (fib 0) = (\\<phi> ^ 0 - \\<psi> ^ 0) / sqrt 5\n 2. real (fib (Suc 0)) = (\\<phi> ^ Suc 0 - \\<psi> ^ Suc 0) / sqrt 5\n 3. \\<And>n. \\<lbrakk>real (fib (Suc n)) = (\\<phi> ^ Suc n - \\<psi> ^ Suc n) / sqrt 5; real (fib n) = (\\<phi> ^ n - \\<psi> ^ n) / sqrt 5\\<rbrakk> \\<Longrightarrow> real (fib (Suc (Suc n))) = (\\<phi> ^ Suc (Suc n) - \\<psi> ^ Suc (Suc n)) / sqrt 5\n[PROOF STEP]\nhave \"\\<phi>^n * \\<phi>\\<^sup>2 - \\<psi>^n * \\<psi>\\<^sup>2 = \\<phi> ^ Suc (Suc n) - \\<psi> ^ Suc (Suc n)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<phi> ^ n * \\<phi>\\<^sup>2 - \\<psi> ^ n * \\<psi>\\<^sup>2 = \\<phi> ^ Suc (Suc n) - \\<psi> ^ Suc (Suc n)\n[PROOF STEP]\nby (simp add: power2_eq_square)\n[PROOF STATE]\nproof (state)\nthis:\n\\<phi> ^ n * \\<phi>\\<^sup>2 - \\<psi> ^ n * \\<psi>\\<^sup>2 = \\<phi> ^ Suc (Suc n) - \\<psi> ^ Suc (Suc n)\n\ngoal (3 subgoals):\n 1. real (fib 0) = (\\<phi> ^ 0 - \\<psi> ^ 0) / sqrt 5\n 2. real (fib (Suc 0)) = (\\<phi> ^ Suc 0 - \\<psi> ^ Suc 0) / sqrt 5\n 3. \\<And>n. \\<lbrakk>real (fib (Suc n)) = (\\<phi> ^ Suc n - \\<psi> ^ Suc n) / sqrt 5; real (fib n) = (\\<phi> ^ n - \\<psi> ^ n) / sqrt 5\\<rbrakk> \\<Longrightarrow> real (fib (Suc (Suc n))) = (\\<phi> ^ Suc (Suc n) - \\<psi> ^ Suc (Suc n)) / sqrt 5\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nreal (fib (Suc (Suc n))) = (\\<phi> ^ Suc (Suc n) - \\<psi> ^ Suc (Suc n)) / sqrt 5\n[PROOF STEP]\nshow \"of_nat (fib (Suc (Suc n))) = (\\<phi> ^ Suc (Suc n) - \\<psi> ^ Suc (Suc n)) / sqrt 5\"\n[PROOF STATE]\nproof (prove)\nusing this:\nreal (fib (Suc (Suc n))) = (\\<phi> ^ Suc (Suc n) - \\<psi> ^ Suc (Suc n)) / sqrt 5\n\ngoal (1 subgoal):\n 1. real (fib (Suc (Suc n))) = (\\<phi> ^ Suc (Suc n) - \\<psi> ^ Suc (Suc n)) / sqrt 5\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nreal (fib (Suc (Suc n))) = (\\<phi> ^ Suc (Suc n) - \\<psi> ^ Suc (Suc n)) / sqrt 5\n\ngoal (2 subgoals):\n 1. real (fib 0) = (\\<phi> ^ 0 - \\<psi> ^ 0) / sqrt 5\n 2. real (fib (Suc 0)) = (\\<phi> ^ Suc 0 - \\<psi> ^ Suc 0) / sqrt 5\n[PROOF STEP]\nqed (simp_all add: \\<phi>_def \\<psi>_def field_simps)", "meta": {"llama_tokens": 4404, "file": null, "length": 22, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.905989815306765, "lm_q2_score": 0.8333245994514084, "lm_q1q2_score": 0.7549835999475654}}
{"text": "[STATEMENT]\nlemma (*bt_map_append:*)\n     \"bt_map f (append t1 t2) = append (bt_map f t1) (bt_map f t2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. bt_map f (Binary_Tree.append t1 t2) = Binary_Tree.append (bt_map f t1) (bt_map f t2)\n[PROOF STEP]\napply (induct t1)\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. bt_map f (Binary_Tree.append Lf t2) = Binary_Tree.append (bt_map f Lf) (bt_map f t2)\n 2. \\<And>x1 t11 t12. \\<lbrakk>bt_map f (Binary_Tree.append t11 t2) = Binary_Tree.append (bt_map f t11) (bt_map f t2); bt_map f (Binary_Tree.append t12 t2) = Binary_Tree.append (bt_map f t12) (bt_map f t2)\\<rbrakk> \\<Longrightarrow> bt_map f (Binary_Tree.append (Br x1 t11 t12) t2) = Binary_Tree.append (bt_map f (Br x1 t11 t12)) (bt_map f t2)\n[PROOF STEP]\napply (metis append.simps(1) bt_map.simps(1))\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>x1 t11 t12. \\<lbrakk>bt_map f (Binary_Tree.append t11 t2) = Binary_Tree.append (bt_map f t11) (bt_map f t2); bt_map f (Binary_Tree.append t12 t2) = Binary_Tree.append (bt_map f t12) (bt_map f t2)\\<rbrakk> \\<Longrightarrow> bt_map f (Binary_Tree.append (Br x1 t11 t12) t2) = Binary_Tree.append (bt_map f (Br x1 t11 t12)) (bt_map f t2)\n[PROOF STEP]\nby (metis bt_map_append)", "meta": {"llama_tokens": 583, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942203004186, "lm_q2_score": 0.8479677526147223, "lm_q1q2_score": 0.7549407891540225}}
{"text": "[STATEMENT]\nlemma set_to_nat_of_interval: \"set_to_nat {i. (i::nat)<m} = 2 ^ m - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set_to_nat {i. i < m} = 2 ^ m - 1\n[PROOF STEP]\nproof (induct m)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. set_to_nat {i. i < 0} = 2 ^ 0 - 1\n 2. \\<And>m. set_to_nat {i. i < m} = 2 ^ m - 1 \\<Longrightarrow> set_to_nat {i. i < Suc m} = 2 ^ Suc m - 1\n[PROOF STEP]\nshow \"set_to_nat {i. i < 0} = 2 ^ 0 - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set_to_nat {i. i < 0} = 2 ^ 0 - 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. set_to_nat {i. i < 0} = 2 ^ 0 - 1\n[PROOF STEP]\nhave S1: \"{i. (i::nat) < 0} = {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {i. i < 0} = {}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{i. i < 0} = {}\n\ngoal (1 subgoal):\n 1. set_to_nat {i. i < 0} = 2 ^ 0 - 1\n[PROOF STEP]\nwith set_to_nat_at_empty\n[PROOF STATE]\nproof (chain)\npicking this:\nset_to_nat {} = 0\n{i. i < 0} = {}\n[PROOF STEP]\nhave \"set_to_nat {i. i<0} = 0\"\n[PROOF STATE]\nproof (prove)\nusing this:\nset_to_nat {} = 0\n{i. i < 0} = {}\n\ngoal (1 subgoal):\n 1. set_to_nat {i. i < 0} = 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nset_to_nat {i. i < 0} = 0\n\ngoal (1 subgoal):\n 1. set_to_nat {i. i < 0} = 2 ^ 0 - 1\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nset_to_nat {i. i < 0} = 0\n\ngoal (1 subgoal):\n 1. set_to_nat {i. i < 0} = 2 ^ 0 - 1\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nset_to_nat {i. i < 0} = 2 ^ 0 - 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nset_to_nat {i. i < 0} = 2 ^ 0 - 1\n\ngoal (1 subgoal):\n 1. \\<And>m. set_to_nat {i. i < m} = 2 ^ m - 1 \\<Longrightarrow> set_to_nat {i. i < Suc m} = 2 ^ Suc m - 1\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>m. set_to_nat {i. i < m} = 2 ^ m - 1 \\<Longrightarrow> set_to_nat {i. i < Suc m} = 2 ^ Suc m - 1\n[PROOF STEP]\nfix n\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>m. set_to_nat {i. i < m} = 2 ^ m - 1 \\<Longrightarrow> set_to_nat {i. i < Suc m} = 2 ^ Suc m - 1\n[PROOF STEP]\nshow \"set_to_nat {i. i < Suc n} = 2 ^ Suc n - 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set_to_nat {i. i < Suc n} = 2 ^ Suc n - 1\n[PROOF STEP]\nby (unfold set_to_nat_def, rule two_power_sum)\n[PROOF STATE]\nproof (state)\nthis:\nset_to_nat {i. i < Suc n} = 2 ^ Suc n - 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1296, "file": "Recursion-Theory-I_PRecFinSet", "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392848011833, "lm_q2_score": 0.8539127529517043, "lm_q1q2_score": 0.7548924194020342}}
{"text": "[STATEMENT]\nlemma poly_roots_set_same:\n  fixes a b c:: \"real\"\n  shows \"{(x::real). a * x\\<^sup>2 + b * x + c = 0} = {x. poly [:c, b, a:] x = 0}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {x. a * x\\<^sup>2 + b * x + c = 0} = {x. poly [:c, b, a:] x = 0}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. {x. a * x\\<^sup>2 + b * x + c = 0} = {x. poly [:c, b, a:] x = 0}\n[PROOF STEP]\nhave \"\\<forall>x. a*x^2 + b*x + c = poly [:c, b, a:] x\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<forall>x. a * x\\<^sup>2 + b * x + c = poly [:c, b, a:] x\n[PROOF STEP]\nproof clarsimp\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>x. a * x\\<^sup>2 + b * x = x * (b + x * a)\n[PROOF STEP]\nfix x\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>x. a * x\\<^sup>2 + b * x = x * (b + x * a)\n[PROOF STEP]\nshow \"a * x\\<^sup>2 + b * x = x * (b + x * a)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. a * x\\<^sup>2 + b * x = x * (b + x * a)\n[PROOF STEP]\nusing quadratic_poly_eval[of c b a x]\n[PROOF STATE]\nproof (prove)\nusing this:\npoly [:c, b, a:] x = a * x\\<^sup>2 + b * x + c\n\ngoal (1 subgoal):\n 1. a * x\\<^sup>2 + b * x = x * (b + x * a)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\na * x\\<^sup>2 + b * x = x * (b + x * a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n\\<forall>x. a * x\\<^sup>2 + b * x + c = poly [:c, b, a:] x\n\ngoal (1 subgoal):\n 1. {x. a * x\\<^sup>2 + b * x + c = 0} = {x. poly [:c, b, a:] x = 0}\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n\\<forall>x. a * x\\<^sup>2 + b * x + c = poly [:c, b, a:] x\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<forall>x. a * x\\<^sup>2 + b * x + c = poly [:c, b, a:] x\n\ngoal (1 subgoal):\n 1. {x. a * x\\<^sup>2 + b * x + c = 0} = {x. poly [:c, b, a:] x = 0}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{x. a * x\\<^sup>2 + b * x + c = 0} = {x. poly [:c, b, a:] x = 0}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 994, "file": "Virtual_Substitution_QE", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8840392756357326, "lm_q2_score": 0.8539127548105611, "lm_q1q2_score": 0.7548924132188414}}
{"text": "[STATEMENT]\nlemma divisor_count_upper_bound':\n  fixes n :: nat\n  shows \"real (divisor_count n) \\<le> 2 * sqrt n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (divisor_count n) \\<le> 2 * sqrt (real n)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. real (divisor_count n) \\<le> 2 * sqrt (real n)\n[PROOF STEP]\nhave \"real (divisor_count n) \\<le> 2 * real (nat \\<lfloor>sqrt n\\<rfloor>)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (divisor_count n) \\<le> 2 * real (nat \\<lfloor>sqrt (real n)\\<rfloor>)\n[PROOF STEP]\nusing divisor_count_upper_bound[of n]\n[PROOF STATE]\nproof (prove)\nusing this:\ndivisor_count n \\<le> 2 * nat \\<lfloor>sqrt (real n)\\<rfloor>\n\ngoal (1 subgoal):\n 1. real (divisor_count n) \\<le> 2 * real (nat \\<lfloor>sqrt (real n)\\<rfloor>)\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\nreal (divisor_count n) \\<le> 2 * real (nat \\<lfloor>sqrt (real n)\\<rfloor>)\n\ngoal (1 subgoal):\n 1. real (divisor_count n) \\<le> 2 * sqrt (real n)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreal (divisor_count n) \\<le> 2 * real (nat \\<lfloor>sqrt (real n)\\<rfloor>)\n\ngoal (1 subgoal):\n 1. real (divisor_count n) \\<le> 2 * sqrt (real n)\n[PROOF STEP]\nhave \"\\<dots> \\<le> 2 * sqrt n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * real (nat \\<lfloor>sqrt (real n)\\<rfloor>) \\<le> 2 * sqrt (real n)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n2 * real (nat \\<lfloor>sqrt (real n)\\<rfloor>) \\<le> 2 * sqrt (real n)\n\ngoal (1 subgoal):\n 1. real (divisor_count n) \\<le> 2 * sqrt (real n)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nreal (divisor_count n) \\<le> 2 * sqrt (real n)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nreal (divisor_count n) \\<le> 2 * sqrt (real n)\n\ngoal (1 subgoal):\n 1. real (divisor_count n) \\<le> 2 * sqrt (real n)\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nreal (divisor_count n) \\<le> 2 * sqrt (real n)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 900, "file": "Gauss_Sums_Polya_Vinogradov", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8652240825770432, "lm_q2_score": 0.872347368040789, "lm_q1q2_score": 0.7547759512015899}}
{"text": "[STATEMENT]\nlemma EVEN_ODD_infinite:\n  shows \"infinite EVEN\"\n  and   \"infinite ODD\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. infinite EVEN &&& infinite ODD\n[PROOF STEP]\nunfolding infinite_iff_countable_subset\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<exists>f. inj f \\<and> range f \\<subseteq> EVEN &&& \\<exists>f. inj f \\<and> range f \\<subseteq> ODD\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\<exists>f. inj f \\<and> range f \\<subseteq> EVEN\n 2. \\<exists>f. inj f \\<and> range f \\<subseteq> ODD\n[PROOF STEP]\nlet ?f = \"\\<lambda>n. atom (2*n)\"\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\<exists>f. inj f \\<and> range f \\<subseteq> EVEN\n 2. \\<exists>f. inj f \\<and> range f \\<subseteq> ODD\n[PROOF STEP]\nhave \"inj ?f \\<and> range ?f \\<subseteq> EVEN\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inj (\\<lambda>n. atom (2 * n)) \\<and> range (\\<lambda>n. atom (2 * n)) \\<subseteq> EVEN\n[PROOF STEP]\nby (auto simp add: inj_on_def)\n[PROOF STATE]\nproof (state)\nthis:\ninj (\\<lambda>n. atom (2 * n)) \\<and> range (\\<lambda>n. atom (2 * n)) \\<subseteq> EVEN\n\ngoal (2 subgoals):\n 1. \\<exists>f. inj f \\<and> range f \\<subseteq> EVEN\n 2. \\<exists>f. inj f \\<and> range f \\<subseteq> ODD\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ninj (\\<lambda>n. atom (2 * n)) \\<and> range (\\<lambda>n. atom (2 * n)) \\<subseteq> EVEN\n[PROOF STEP]\nshow \"\\<exists>f::nat\\<Rightarrow>atom. inj f \\<and> range f \\<subseteq> EVEN\"\n[PROOF STATE]\nproof (prove)\nusing this:\ninj (\\<lambda>n. atom (2 * n)) \\<and> range (\\<lambda>n. atom (2 * n)) \\<subseteq> EVEN\n\ngoal (1 subgoal):\n 1. \\<exists>f. inj f \\<and> range f \\<subseteq> EVEN\n[PROOF STEP]\nby (rule_tac exI)\n[PROOF STATE]\nproof (state)\nthis:\n\\<exists>f. inj f \\<and> range f \\<subseteq> EVEN\n\ngoal (1 subgoal):\n 1. \\<exists>f. inj f \\<and> range f \\<subseteq> ODD\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<exists>f. inj f \\<and> range f \\<subseteq> ODD\n[PROOF STEP]\nlet ?f = \"\\<lambda>n. atom (2*n+1)\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<exists>f. inj f \\<and> range f \\<subseteq> ODD\n[PROOF STEP]\nhave \"inj ?f \\<and> range ?f \\<subseteq> ODD\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inj (\\<lambda>n. atom (2 * n + 1)) \\<and> range (\\<lambda>n. atom (2 * n + 1)) \\<subseteq> ODD\n[PROOF STEP]\nby (auto simp add: inj_on_def)\n[PROOF STATE]\nproof (state)\nthis:\ninj (\\<lambda>n. atom (2 * n + 1)) \\<and> range (\\<lambda>n. atom (2 * n + 1)) \\<subseteq> ODD\n\ngoal (1 subgoal):\n 1. \\<exists>f. inj f \\<and> range f \\<subseteq> ODD\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ninj (\\<lambda>n. atom (2 * n + 1)) \\<and> range (\\<lambda>n. atom (2 * n + 1)) \\<subseteq> ODD\n[PROOF STEP]\nshow \"\\<exists>f::nat\\<Rightarrow>atom. inj f \\<and> range f \\<subseteq> ODD\"\n[PROOF STATE]\nproof (prove)\nusing this:\ninj (\\<lambda>n. atom (2 * n + 1)) \\<and> range (\\<lambda>n. atom (2 * n + 1)) \\<subseteq> ODD\n\ngoal (1 subgoal):\n 1. \\<exists>f. inj f \\<and> range f \\<subseteq> ODD\n[PROOF STEP]\nby (rule_tac exI)\n[PROOF STATE]\nproof (state)\nthis:\n\\<exists>f. inj f \\<and> range f \\<subseteq> ODD\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1346, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.863391602943619, "lm_q2_score": 0.8740772466456689, "lm_q1q2_score": 0.7546709550779491}}
{"text": "[STATEMENT]\nlemma lcm_eq_0_iff: \"lcm a b = 0 \\<longleftrightarrow> a = 0 \\<or> b = 0\"\n  (is \"?P \\<longleftrightarrow> ?Q\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (lcm a b = (0::'a)) = (a = (0::'a) \\<or> b = (0::'a))\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. lcm a b = (0::'a) \\<Longrightarrow> a = (0::'a) \\<or> b = (0::'a)\n 2. a = (0::'a) \\<or> b = (0::'a) \\<Longrightarrow> lcm a b = (0::'a)\n[PROOF STEP]\nassume ?P\n[PROOF STATE]\nproof (state)\nthis:\nlcm a b = (0::'a)\n\ngoal (2 subgoals):\n 1. lcm a b = (0::'a) \\<Longrightarrow> a = (0::'a) \\<or> b = (0::'a)\n 2. a = (0::'a) \\<or> b = (0::'a) \\<Longrightarrow> lcm a b = (0::'a)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nlcm a b = (0::'a)\n[PROOF STEP]\nhave \"0 dvd lcm a b\"\n[PROOF STATE]\nproof (prove)\nusing this:\nlcm a b = (0::'a)\n\ngoal (1 subgoal):\n 1. (0::'a) dvd lcm a b\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(0::'a) dvd lcm a b\n\ngoal (2 subgoals):\n 1. lcm a b = (0::'a) \\<Longrightarrow> a = (0::'a) \\<or> b = (0::'a)\n 2. a = (0::'a) \\<or> b = (0::'a) \\<Longrightarrow> lcm a b = (0::'a)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(0::'a) dvd lcm a b\n\ngoal (2 subgoals):\n 1. lcm a b = (0::'a) \\<Longrightarrow> a = (0::'a) \\<or> b = (0::'a)\n 2. a = (0::'a) \\<or> b = (0::'a) \\<Longrightarrow> lcm a b = (0::'a)\n[PROOF STEP]\nhave \"lcm a b dvd (a * b)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. lcm a b dvd a * b\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlcm a b dvd a * b\n\ngoal (2 subgoals):\n 1. lcm a b = (0::'a) \\<Longrightarrow> a = (0::'a) \\<or> b = (0::'a)\n 2. a = (0::'a) \\<or> b = (0::'a) \\<Longrightarrow> lcm a b = (0::'a)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(0::'a) dvd a * b\n[PROOF STEP]\nshow ?Q\n[PROOF STATE]\nproof (prove)\nusing this:\n(0::'a) dvd a * b\n\ngoal (1 subgoal):\n 1. a = (0::'a) \\<or> b = (0::'a)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\na = (0::'a) \\<or> b = (0::'a)\n\ngoal (1 subgoal):\n 1. a = (0::'a) \\<or> b = (0::'a) \\<Longrightarrow> lcm a b = (0::'a)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. a = (0::'a) \\<or> b = (0::'a) \\<Longrightarrow> lcm a b = (0::'a)\n[PROOF STEP]\nassume ?Q\n[PROOF STATE]\nproof (state)\nthis:\na = (0::'a) \\<or> b = (0::'a)\n\ngoal (1 subgoal):\n 1. a = (0::'a) \\<or> b = (0::'a) \\<Longrightarrow> lcm a b = (0::'a)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\na = (0::'a) \\<or> b = (0::'a)\n[PROOF STEP]\nshow ?P\n[PROOF STATE]\nproof (prove)\nusing this:\na = (0::'a) \\<or> b = (0::'a)\n\ngoal (1 subgoal):\n 1. lcm a b = (0::'a)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nlcm a b = (0::'a)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1407, "file": null, "length": 17, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757869884059266, "lm_q2_score": 0.8615382147637196, "lm_q1q2_score": 0.7545239585045365}}
{"text": "[STATEMENT]\nlemma infnorm_triangle:\n  fixes x :: \"'a::euclidean_space\"\n  shows \"infnorm (x + y) \\<le> infnorm x + infnorm y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. infnorm (x + y) \\<le> infnorm x + infnorm y\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. infnorm (x + y) \\<le> infnorm x + infnorm y\n[PROOF STEP]\nhave *: \"\\<And>a b c d :: real. \\<bar>a\\<bar> \\<le> c \\<Longrightarrow> \\<bar>b\\<bar> \\<le> d \\<Longrightarrow> \\<bar>a + b\\<bar> \\<le> c + d\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>a b c d. \\<lbrakk>\\<bar>a\\<bar> \\<le> c; \\<bar>b\\<bar> \\<le> d\\<rbrakk> \\<Longrightarrow> \\<bar>a + b\\<bar> \\<le> c + d\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n\\<lbrakk>\\<bar>?a\\<bar> \\<le> ?c; \\<bar>?b\\<bar> \\<le> ?d\\<rbrakk> \\<Longrightarrow> \\<bar>?a + ?b\\<bar> \\<le> ?c + ?d\n\ngoal (1 subgoal):\n 1. infnorm (x + y) \\<le> infnorm x + infnorm y\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. infnorm (x + y) \\<le> infnorm x + infnorm y\n[PROOF STEP]\nby (auto simp: infnorm_Max inner_add_left intro!: *)\n[PROOF STATE]\nproof (state)\nthis:\ninfnorm (x + y) \\<le> infnorm x + infnorm y\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 534, "file": null, "length": 6, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314798554444, "lm_q2_score": 0.8519528076067262, "lm_q1q2_score": 0.7545162257677457}}
{"text": "[STATEMENT]\nlemma permutation_mat_id_1: assumes p: \"p permutes {..<n}\" \n  shows \"permutation_mat n p * permutation_mat n (inv p) = 1\\<^sub>m n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. permutation_mat n p * permutation_mat n (inv p) = 1\\<^sub>m n\n[PROOF STEP]\nby (subst permutation_mat_left[OF _ p, of _ n], force, unfold permutation_mat_def, rule eq_matI, \n   auto simp: permutes_lt[OF permutes_inv[OF p]] permutes_iff[OF permutes_inv[OF p]])", "meta": {"llama_tokens": 183, "file": "Perron_Frobenius_Perron_Frobenius_General", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513842182777, "lm_q2_score": 0.8418256432832333, "lm_q1q2_score": 0.7544873980630399}}
{"text": "[STATEMENT]\nlemma append_rows_le: assumes A: \"A \\<in> carrier_mat nr1 nc\" \n  and B: \"B \\<in> carrier_mat nr2 nc\" \n  and a: \"a \\<in> carrier_vec nr1\" \n  and v: \"v \\<in> carrier_vec nc\"\nshows \"(A @\\<^sub>r B) *\\<^sub>v v \\<le> (a @\\<^sub>v b) \\<longleftrightarrow> A *\\<^sub>v v \\<le> a \\<and> B *\\<^sub>v v \\<le> b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((A @\\<^sub>r B) *\\<^sub>v v \\<le> a @\\<^sub>v b) = (A *\\<^sub>v v \\<le> a \\<and> B *\\<^sub>v v \\<le> b)\n[PROOF STEP]\nunfolding mat_mult_append[OF A B v]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (A *\\<^sub>v v @\\<^sub>v B *\\<^sub>v v \\<le> a @\\<^sub>v b) = (A *\\<^sub>v v \\<le> a \\<and> B *\\<^sub>v v \\<le> b)\n[PROOF STEP]\nby (rule append_vec_le[OF _ a], insert A v, auto)", "meta": {"llama_tokens": 358, "file": "Jordan_Normal_Form_Matrix", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9124361652391386, "lm_q2_score": 0.8267117919359419, "lm_q1q2_score": 0.7543217371920075}}
{"text": "[STATEMENT]\nlemma  trace_mul_sym: \"trace ((A::'a::comm_semiring_1^'n^'m) ** B) = trace (B**A)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. trace (A ** B) = trace (B ** A)\n[PROOF STEP]\napply (simp add: trace_def matrix_matrix_mult_def)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>i\\<in>UNIV. \\<Sum>k\\<in>UNIV. A $ i $ k * B $ k $ i) = (\\<Sum>i\\<in>UNIV. \\<Sum>k\\<in>UNIV. B $ i $ k * A $ k $ i)\n[PROOF STEP]\napply (subst sum.swap)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>j\\<in>UNIV. \\<Sum>i\\<in>UNIV. A $ i $ j * B $ j $ i) = (\\<Sum>i\\<in>UNIV. \\<Sum>k\\<in>UNIV. B $ i $ k * A $ k $ i)\n[PROOF STEP]\napply (simp add: mult.commute)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 351, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9099070084811306, "lm_q2_score": 0.8289388104343892, "lm_q1q2_score": 0.7542572332162621}}
{"text": "[STATEMENT]\nlemma measure_UNION_AE:\n  assumes I: \"finite I\"\n  shows \"(\\<And>i. i \\<in> I \\<Longrightarrow> F i \\<in> fmeasurable M) \\<Longrightarrow> pairwise (\\<lambda>i j. AE x in M. x \\<notin> F i \\<or> x \\<notin> F j) I \\<Longrightarrow>\n    measure M (\\<Union>i\\<in>I. F i) = (\\<Sum>i\\<in>I. measure M (F i))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>\\<And>i. i \\<in> I \\<Longrightarrow> F i \\<in> fmeasurable M; pairwise (\\<lambda>i j. AE x in M. x \\<notin> F i \\<or> x \\<notin> F j) I\\<rbrakk> \\<Longrightarrow> Sigma_Algebra.measure M (\\<Union> (F ` I)) = (\\<Sum>i\\<in>I. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\nunfolding AE_pairwise[OF countable_finite, OF I]\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>\\<And>i. i \\<in> I \\<Longrightarrow> F i \\<in> fmeasurable M; AE x in M. pairwise (\\<lambda>i j. x \\<notin> F i \\<or> x \\<notin> F j) I\\<rbrakk> \\<Longrightarrow> Sigma_Algebra.measure M (\\<Union> (F ` I)) = (\\<Sum>i\\<in>I. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\nusing I\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite I\n\ngoal (1 subgoal):\n 1. \\<lbrakk>\\<And>i. i \\<in> I \\<Longrightarrow> F i \\<in> fmeasurable M; AE x in M. pairwise (\\<lambda>i j. x \\<notin> F i \\<or> x \\<notin> F j) I\\<rbrakk> \\<Longrightarrow> Sigma_Algebra.measure M (\\<Union> (F ` I)) = (\\<Sum>i\\<in>I. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\nproof (induction I rule: finite_induct)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\<lbrakk>\\<And>i. i \\<in> {} \\<Longrightarrow> F i \\<in> fmeasurable M; AE x in M. pairwise (\\<lambda>i j. x \\<notin> F i \\<or> x \\<notin> F j) {}\\<rbrakk> \\<Longrightarrow> Sigma_Algebra.measure M (\\<Union> (F ` {})) = (\\<Sum>i\\<in>{}. Sigma_Algebra.measure M (F i))\n 2. \\<And>x Fa. \\<lbrakk>finite Fa; x \\<notin> Fa; \\<lbrakk>\\<And>i. i \\<in> Fa \\<Longrightarrow> F i \\<in> fmeasurable M; AE x in M. pairwise (\\<lambda>i j. x \\<notin> F i \\<or> x \\<notin> F j) Fa\\<rbrakk> \\<Longrightarrow> Sigma_Algebra.measure M (\\<Union> (F ` Fa)) = (\\<Sum>i\\<in>Fa. Sigma_Algebra.measure M (F i)); \\<And>i. i \\<in> insert x Fa \\<Longrightarrow> F i \\<in> fmeasurable M; AE xa in M. pairwise (\\<lambda>i j. xa \\<notin> F i \\<or> xa \\<notin> F j) (insert x Fa)\\<rbrakk> \\<Longrightarrow> Sigma_Algebra.measure M (\\<Union> (F ` insert x Fa)) = (\\<Sum>i\\<in>insert x Fa. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\ncase (insert x I)\n[PROOF STATE]\nproof (state)\nthis:\nfinite I\nx \\<notin> I\n\\<lbrakk>\\<And>i. i \\<in> I \\<Longrightarrow> F i \\<in> fmeasurable M; AE x in M. pairwise (\\<lambda>i j. x \\<notin> F i \\<or> x \\<notin> F j) I\\<rbrakk> \\<Longrightarrow> Sigma_Algebra.measure M (\\<Union> (F ` I)) = (\\<Sum>i\\<in>I. Sigma_Algebra.measure M (F i))\n?i \\<in> insert x I \\<Longrightarrow> F ?i \\<in> fmeasurable M\nAE xa in M. pairwise (\\<lambda>i j. xa \\<notin> F i \\<or> xa \\<notin> F j) (insert x I)\n\ngoal (2 subgoals):\n 1. \\<lbrakk>\\<And>i. i \\<in> {} \\<Longrightarrow> F i \\<in> fmeasurable M; AE x in M. pairwise (\\<lambda>i j. x \\<notin> F i \\<or> x \\<notin> F j) {}\\<rbrakk> \\<Longrightarrow> Sigma_Algebra.measure M (\\<Union> (F ` {})) = (\\<Sum>i\\<in>{}. Sigma_Algebra.measure M (F i))\n 2. \\<And>x Fa. \\<lbrakk>finite Fa; x \\<notin> Fa; \\<lbrakk>\\<And>i. i \\<in> Fa \\<Longrightarrow> F i \\<in> fmeasurable M; AE x in M. pairwise (\\<lambda>i j. x \\<notin> F i \\<or> x \\<notin> F j) Fa\\<rbrakk> \\<Longrightarrow> Sigma_Algebra.measure M (\\<Union> (F ` Fa)) = (\\<Sum>i\\<in>Fa. Sigma_Algebra.measure M (F i)); \\<And>i. i \\<in> insert x Fa \\<Longrightarrow> F i \\<in> fmeasurable M; AE xa in M. pairwise (\\<lambda>i j. xa \\<notin> F i \\<or> xa \\<notin> F j) (insert x Fa)\\<rbrakk> \\<Longrightarrow> Sigma_Algebra.measure M (\\<Union> (F ` insert x Fa)) = (\\<Sum>i\\<in>insert x Fa. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\nhave \"measure M (F x \\<union> \\<Union>(F ` I)) = measure M (F x) + measure M (\\<Union>(F ` I))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M (F x \\<union> \\<Union> (F ` I)) = Sigma_Algebra.measure M (F x) + Sigma_Algebra.measure M (\\<Union> (F ` I))\n[PROOF STEP]\nby (rule measure_Un_AE) (use insert in \\<open>auto simp: pairwise_insert\\<close>)\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M (F x \\<union> \\<Union> (F ` I)) = Sigma_Algebra.measure M (F x) + Sigma_Algebra.measure M (\\<Union> (F ` I))\n\ngoal (2 subgoals):\n 1. \\<lbrakk>\\<And>i. i \\<in> {} \\<Longrightarrow> F i \\<in> fmeasurable M; AE x in M. pairwise (\\<lambda>i j. x \\<notin> F i \\<or> x \\<notin> F j) {}\\<rbrakk> \\<Longrightarrow> Sigma_Algebra.measure M (\\<Union> (F ` {})) = (\\<Sum>i\\<in>{}. Sigma_Algebra.measure M (F i))\n 2. \\<And>x Fa. \\<lbrakk>finite Fa; x \\<notin> Fa; \\<lbrakk>\\<And>i. i \\<in> Fa \\<Longrightarrow> F i \\<in> fmeasurable M; AE x in M. pairwise (\\<lambda>i j. x \\<notin> F i \\<or> x \\<notin> F j) Fa\\<rbrakk> \\<Longrightarrow> Sigma_Algebra.measure M (\\<Union> (F ` Fa)) = (\\<Sum>i\\<in>Fa. Sigma_Algebra.measure M (F i)); \\<And>i. i \\<in> insert x Fa \\<Longrightarrow> F i \\<in> fmeasurable M; AE xa in M. pairwise (\\<lambda>i j. xa \\<notin> F i \\<or> xa \\<notin> F j) (insert x Fa)\\<rbrakk> \\<Longrightarrow> Sigma_Algebra.measure M (\\<Union> (F ` insert x Fa)) = (\\<Sum>i\\<in>insert x Fa. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\nwith insert\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite I\nx \\<notin> I\n\\<lbrakk>\\<And>i. i \\<in> I \\<Longrightarrow> F i \\<in> fmeasurable M; AE x in M. pairwise (\\<lambda>i j. x \\<notin> F i \\<or> x \\<notin> F j) I\\<rbrakk> \\<Longrightarrow> Sigma_Algebra.measure M (\\<Union> (F ` I)) = (\\<Sum>i\\<in>I. Sigma_Algebra.measure M (F i))\n?i \\<in> insert x I \\<Longrightarrow> F ?i \\<in> fmeasurable M\nAE xa in M. pairwise (\\<lambda>i j. xa \\<notin> F i \\<or> xa \\<notin> F j) (insert x I)\nSigma_Algebra.measure M (F x \\<union> \\<Union> (F ` I)) = Sigma_Algebra.measure M (F x) + Sigma_Algebra.measure M (\\<Union> (F ` I))\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite I\nx \\<notin> I\n\\<lbrakk>\\<And>i. i \\<in> I \\<Longrightarrow> F i \\<in> fmeasurable M; AE x in M. pairwise (\\<lambda>i j. x \\<notin> F i \\<or> x \\<notin> F j) I\\<rbrakk> \\<Longrightarrow> Sigma_Algebra.measure M (\\<Union> (F ` I)) = (\\<Sum>i\\<in>I. Sigma_Algebra.measure M (F i))\n?i \\<in> insert x I \\<Longrightarrow> F ?i \\<in> fmeasurable M\nAE xa in M. pairwise (\\<lambda>i j. xa \\<notin> F i \\<or> xa \\<notin> F j) (insert x I)\nSigma_Algebra.measure M (F x \\<union> \\<Union> (F ` I)) = Sigma_Algebra.measure M (F x) + Sigma_Algebra.measure M (\\<Union> (F ` I))\n\ngoal (1 subgoal):\n 1. Sigma_Algebra.measure M (\\<Union> (F ` insert x I)) = (\\<Sum>i\\<in>insert x I. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\nby (simp add: pairwise_insert )\n[PROOF STATE]\nproof (state)\nthis:\nSigma_Algebra.measure M (\\<Union> (F ` insert x I)) = (\\<Sum>i\\<in>insert x I. Sigma_Algebra.measure M (F i))\n\ngoal (1 subgoal):\n 1. \\<lbrakk>\\<And>i. i \\<in> {} \\<Longrightarrow> F i \\<in> fmeasurable M; AE x in M. pairwise (\\<lambda>i j. x \\<notin> F i \\<or> x \\<notin> F j) {}\\<rbrakk> \\<Longrightarrow> Sigma_Algebra.measure M (\\<Union> (F ` {})) = (\\<Sum>i\\<in>{}. Sigma_Algebra.measure M (F i))\n[PROOF STEP]\nqed simp", "meta": {"llama_tokens": 2869, "file": null, "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9173026641072387, "lm_q2_score": 0.8221891370573388, "lm_q1q2_score": 0.7541962858227285}}
{"text": "[STATEMENT]\nlemma tan_half: \"tan x = sin (2 * x) / (cos (2 * x) + 1)\"\n  for x :: \"'a::{real_normed_field,banach,field}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. tan x = sin ((2::'a) * x) / (cos ((2::'a) * x) + (1::'a))\n[PROOF STEP]\nunfolding tan_def sin_double cos_double sin_squared_eq\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin x / cos x = (2::'a) * sin x * cos x / ((cos x)\\<^sup>2 - ((1::'a) - (cos x)\\<^sup>2) + (1::'a))\n[PROOF STEP]\nby (simp add: power2_eq_square)", "meta": {"llama_tokens": 234, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9173026573249612, "lm_q2_score": 0.8221891370573388, "lm_q1q2_score": 0.7541962802464136}}
{"text": "[STATEMENT]\nlemma real_polynomial_function_sum_of_powers:\n  \"\\<exists>p. real_polynomial_function p \\<and> (\\<forall>n. (\\<Sum>i\\<le>n. real i ^ j) = p (real n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<exists>p. real_polynomial_function p \\<and> (\\<forall>n. (\\<Sum>i\\<le>n. real i ^ j) = p (real n))\n[PROOF STEP]\nproof (intro exI conjI strip)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. real_polynomial_function ?p\n 2. \\<And>n. (\\<Sum>i\\<le>n. real i ^ j) = ?p (real n)\n[PROOF STEP]\nlet ?p = \"\\<lambda>n. (bernpoly (Suc j) (1 + n) - bernpoly (Suc j) 0) / (Suc j)\"\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. real_polynomial_function ?p\n 2. \\<And>n. (\\<Sum>i\\<le>n. real i ^ j) = ?p (real n)\n[PROOF STEP]\nshow \"real_polynomial_function ?p\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real_polynomial_function (\\<lambda>n. (bernpoly (Suc j) (1 + n) - bernpoly (Suc j) 0) / real (Suc j))\n[PROOF STEP]\nby (force simp add: bernpoly_def)\n[PROOF STATE]\nproof (state)\nthis:\nreal_polynomial_function (\\<lambda>n. (bernpoly (Suc j) (1 + n) - bernpoly (Suc j) 0) / real (Suc j))\n\ngoal (1 subgoal):\n 1. \\<And>n. (\\<Sum>i\\<le>n. real i ^ j) = (bernpoly (Suc j) (1 + real n) - bernpoly (Suc j) 0) / real (Suc j)\n[PROOF STEP]\nqed (simp add: add.commute sum_of_powers)", "meta": {"llama_tokens": 576, "file": "Khovanskii_Theorem_Khovanskii", "length": 5, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9173026550642019, "lm_q2_score": 0.8221891283434876, "lm_q1q2_score": 0.7541962703944031}}
{"text": "[STATEMENT]\nlemma card_union_disjoint_fset: \n  shows \"xs |\\<inter>| ys = {||} \\<Longrightarrow> card_fset (xs |\\<union>| ys) = card_fset xs + card_fset ys\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. xs |\\<inter>| ys = {||} \\<Longrightarrow> card_fset (xs |\\<union>| ys) = card_fset xs + card_fset ys\n[PROOF STEP]\nunfolding card_fset union_fset\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. xs |\\<inter>| ys = {||} \\<Longrightarrow> card (fset xs \\<union> fset ys) = card (fset xs) + card (fset ys)\n[PROOF STEP]\napply (rule card_Un_disjoint[OF finite_fset finite_fset])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. xs |\\<inter>| ys = {||} \\<Longrightarrow> fset xs \\<inter> fset ys = {}\n[PROOF STEP]\nby (metis inter_fset fset_simps(1))", "meta": {"llama_tokens": 322, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9196425267730008, "lm_q2_score": 0.8198933381139645, "lm_q1q2_score": 0.7540087811474766}}
{"text": "[STATEMENT]\nlemma (in real_inner) parallelogram_law: \"(norm (x + y))\\<^sup>2 + (norm (x - y))\\<^sup>2 = 2 * (norm x)\\<^sup>2 + 2 * (norm y)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (norm (x + y))\\<^sup>2 + (norm (x - y))\\<^sup>2 = 2 * (norm x)\\<^sup>2 + 2 * (norm y)\\<^sup>2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (norm (x + y))\\<^sup>2 + (norm (x - y))\\<^sup>2 = 2 * (norm x)\\<^sup>2 + 2 * (norm y)\\<^sup>2\n[PROOF STEP]\nhave \"(norm (x + y))\\<^sup>2 + (norm (x - y))\\<^sup>2 = inner (x + y) (x + y) + inner (x - y) (x - y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (norm (x + y))\\<^sup>2 + (norm (x - y))\\<^sup>2 = (x + y) \\<bullet> (x + y) + (x - y) \\<bullet> (x - y)\n[PROOF STEP]\nby (simp add: norm_eq_sqrt_inner)\n[PROOF STATE]\nproof (state)\nthis:\n(norm (x + y))\\<^sup>2 + (norm (x - y))\\<^sup>2 = (x + y) \\<bullet> (x + y) + (x - y) \\<bullet> (x - y)\n\ngoal (1 subgoal):\n 1. (norm (x + y))\\<^sup>2 + (norm (x - y))\\<^sup>2 = 2 * (norm x)\\<^sup>2 + 2 * (norm y)\\<^sup>2\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(norm (x + y))\\<^sup>2 + (norm (x - y))\\<^sup>2 = (x + y) \\<bullet> (x + y) + (x - y) \\<bullet> (x - y)\n\ngoal (1 subgoal):\n 1. (norm (x + y))\\<^sup>2 + (norm (x - y))\\<^sup>2 = 2 * (norm x)\\<^sup>2 + 2 * (norm y)\\<^sup>2\n[PROOF STEP]\nhave \"\\<dots> = 2 * (norm x)\\<^sup>2 + 2 * (norm y)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x + y) \\<bullet> (x + y) + (x - y) \\<bullet> (x - y) = 2 * (norm x)\\<^sup>2 + 2 * (norm y)\\<^sup>2\n[PROOF STEP]\nby (simp add: algebra_simps norm_eq_sqrt_inner)\n[PROOF STATE]\nproof (state)\nthis:\n(x + y) \\<bullet> (x + y) + (x - y) \\<bullet> (x - y) = 2 * (norm x)\\<^sup>2 + 2 * (norm y)\\<^sup>2\n\ngoal (1 subgoal):\n 1. (norm (x + y))\\<^sup>2 + (norm (x - y))\\<^sup>2 = 2 * (norm x)\\<^sup>2 + 2 * (norm y)\\<^sup>2\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(norm (x + y))\\<^sup>2 + (norm (x - y))\\<^sup>2 = 2 * (norm x)\\<^sup>2 + 2 * (norm y)\\<^sup>2\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(norm (x + y))\\<^sup>2 + (norm (x - y))\\<^sup>2 = 2 * (norm x)\\<^sup>2 + 2 * (norm y)\\<^sup>2\n\ngoal (1 subgoal):\n 1. (norm (x + y))\\<^sup>2 + (norm (x - y))\\<^sup>2 = 2 * (norm x)\\<^sup>2 + 2 * (norm y)\\<^sup>2\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(norm (x + y))\\<^sup>2 + (norm (x - y))\\<^sup>2 = 2 * (norm x)\\<^sup>2 + 2 * (norm y)\\<^sup>2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1232, "file": "Ordinary_Differential_Equations_ODE_Auxiliarities", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970748488297, "lm_q2_score": 0.8558511469672594, "lm_q1q2_score": 0.753831186754778}}
{"text": "[STATEMENT]\nlemma exp1': \"exp 3 (A) < 3 * ((exp 3 A) * (exp 3 B)) + C\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exp 3 A < 3 * (exp 3 A * exp 3 B) + C\n[PROOF STEP]\napply(subgoal_tac \"exp 3 (A) < 3 * ((exp 3 A) * (exp 3 B))\")\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. exp 3 A < 3 * (exp 3 A * exp 3 B) \\<Longrightarrow> exp 3 A < 3 * (exp 3 A * exp 3 B) + C\n 2. exp 3 A < 3 * (exp 3 A * exp 3 B)\n[PROOF STEP]\napply arith\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. exp 3 A < 3 * (exp 3 A * exp 3 B)\n[PROOF STEP]\napply(case_tac \"exp 3 A\")\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. exp 3 A = 0 \\<Longrightarrow> exp 3 A < 3 * (exp 3 A * exp 3 B)\n 2. \\<And>nat. exp 3 A = Suc nat \\<Longrightarrow> exp 3 A < 3 * (exp 3 A * exp 3 B)\n[PROOF STEP]\nusing exp3Min[of A]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < exp 3 A\n\ngoal (2 subgoals):\n 1. exp 3 A = 0 \\<Longrightarrow> exp 3 A < 3 * (exp 3 A * exp 3 B)\n 2. \\<And>nat. exp 3 A = Suc nat \\<Longrightarrow> exp 3 A < 3 * (exp 3 A * exp 3 B)\n[PROOF STEP]\napply arith\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>nat. exp 3 A = Suc nat \\<Longrightarrow> exp 3 A < 3 * (exp 3 A * exp 3 B)\n[PROOF STEP]\napply(case_tac \"exp 3 B\")\n[PROOF STATE]\nproof (prove)\ngoal (2 subgoals):\n 1. \\<And>nat. \\<lbrakk>exp 3 A = Suc nat; exp 3 B = 0\\<rbrakk> \\<Longrightarrow> exp 3 A < 3 * (exp 3 A * exp 3 B)\n 2. \\<And>nat nata. \\<lbrakk>exp 3 A = Suc nat; exp 3 B = Suc nata\\<rbrakk> \\<Longrightarrow> exp 3 A < 3 * (exp 3 A * exp 3 B)\n[PROOF STEP]\nusing exp3Min[of B]\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < exp 3 B\n\ngoal (2 subgoals):\n 1. \\<And>nat. \\<lbrakk>exp 3 A = Suc nat; exp 3 B = 0\\<rbrakk> \\<Longrightarrow> exp 3 A < 3 * (exp 3 A * exp 3 B)\n 2. \\<And>nat nata. \\<lbrakk>exp 3 A = Suc nat; exp 3 B = Suc nata\\<rbrakk> \\<Longrightarrow> exp 3 A < 3 * (exp 3 A * exp 3 B)\n[PROOF STEP]\napply arith\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>nat nata. \\<lbrakk>exp 3 A = Suc nat; exp 3 B = Suc nata\\<rbrakk> \\<Longrightarrow> exp 3 A < 3 * (exp 3 A * exp 3 B)\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 1036, "file": "Completeness_Completeness", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.897695292107347, "lm_q2_score": 0.8397339596505965, "lm_q1q2_score": 0.7538252222010013}}
{"text": "[STATEMENT]\nlemma has_derivative_divide'[derivative_intros]:\n  fixes f :: \"_ \\<Rightarrow> 'a::real_normed_field\"\n  assumes f: \"(f has_derivative f') (at x within S)\"\n    and g: \"(g has_derivative g') (at x within S)\"\n    and x: \"g x \\<noteq> 0\"\n  shows \"((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)\n[PROOF STEP]\nhave \"f' h / g x - f x * (inverse (g x) * g' h * inverse (g x)) =\n      (f' h * g x - f x * g' h) / (g x * g x)\" for h\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f' h / g x - f x * (inverse (g x) * g' h * inverse (g x)) = (f' h * g x - f x * g' h) / (g x * g x)\n[PROOF STEP]\nby (simp add: field_simps x)\n[PROOF STATE]\nproof (state)\nthis:\nf' ?h / g x - f x * (inverse (g x) * g' ?h * inverse (g x)) = (f' ?h * g x - f x * g' ?h) / (g x * g x)\n\ngoal (1 subgoal):\n 1. ((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nf' ?h / g x - f x * (inverse (g x) * g' ?h * inverse (g x)) = (f' ?h * g x - f x * g' ?h) / (g x * g x)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nf' ?h / g x - f x * (inverse (g x) * g' ?h * inverse (g x)) = (f' ?h * g x - f x * g' ?h) / (g x * g x)\n\ngoal (1 subgoal):\n 1. ((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)\n[PROOF STEP]\nusing has_derivative_divide [OF f g] x\n[PROOF STATE]\nproof (prove)\nusing this:\nf' ?h / g x - f x * (inverse (g x) * g' ?h * inverse (g x)) = (f' ?h * g x - f x * g' ?h) / (g x * g x)\ng x \\<noteq> (0::'a) \\<Longrightarrow> ((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. - f x * (inverse (g x) * g' h * inverse (g x)) + f' h / g x)) (at x within S)\ng x \\<noteq> (0::'a)\n\ngoal (1 subgoal):\n 1. ((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1190, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206686206199, "lm_q2_score": 0.8354835432479661, "lm_q1q2_score": 0.7535398759477302}}
{"text": "[STATEMENT]\nlemma has_derivative_divide'[derivative_intros]:\n  fixes f :: \"_ \\<Rightarrow> 'a::real_normed_field\"\n  assumes f: \"(f has_derivative f') (at x within S)\"\n    and g: \"(g has_derivative g') (at x within S)\"\n    and x: \"g x \\<noteq> 0\"\n  shows \"((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)\n[PROOF STEP]\nhave \"f' h / g x - f x * (inverse (g x) * g' h * inverse (g x)) =\n      (f' h * g x - f x * g' h) / (g x * g x)\" for h\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f' h / g x - f x * (inverse (g x) * g' h * inverse (g x)) = (f' h * g x - f x * g' h) / (g x * g x)\n[PROOF STEP]\nby (simp add: field_simps x)\n[PROOF STATE]\nproof (state)\nthis:\nf' ?h / g x - f x * (inverse (g x) * g' ?h * inverse (g x)) = (f' ?h * g x - f x * g' ?h) / (g x * g x)\n\ngoal (1 subgoal):\n 1. ((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nf' ?h / g x - f x * (inverse (g x) * g' ?h * inverse (g x)) = (f' ?h * g x - f x * g' ?h) / (g x * g x)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nf' ?h / g x - f x * (inverse (g x) * g' ?h * inverse (g x)) = (f' ?h * g x - f x * g' ?h) / (g x * g x)\n\ngoal (1 subgoal):\n 1. ((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)\n[PROOF STEP]\nusing has_derivative_divide [OF f g] x\n[PROOF STATE]\nproof (prove)\nusing this:\nf' ?h / g x - f x * (inverse (g x) * g' ?h * inverse (g x)) = (f' ?h * g x - f x * g' ?h) / (g x * g x)\ng x \\<noteq> (0::'a) \\<Longrightarrow> ((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. - f x * (inverse (g x) * g' h * inverse (g x)) + f' h / g x)) (at x within S)\ng x \\<noteq> (0::'a)\n\ngoal (1 subgoal):\n 1. ((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1190, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206686206199, "lm_q2_score": 0.8354835330070839, "lm_q1q2_score": 0.7535398667112668}}
{"text": "[STATEMENT]\nlemma has_derivative_divide'[derivative_intros]:\n  fixes f :: \"_ \\<Rightarrow> 'a::real_normed_field\"\n  assumes f: \"(f has_derivative f') (at x within S)\"\n    and g: \"(g has_derivative g') (at x within S)\"\n    and x: \"g x \\<noteq> 0\"\n  shows \"((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)\n[PROOF STEP]\nhave \"f' h / g x - f x * (inverse (g x) * g' h * inverse (g x)) =\n      (f' h * g x - f x * g' h) / (g x * g x)\" for h\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. f' h / g x - f x * (inverse (g x) * g' h * inverse (g x)) = (f' h * g x - f x * g' h) / (g x * g x)\n[PROOF STEP]\nby (simp add: field_simps x)\n[PROOF STATE]\nproof (state)\nthis:\nf' ?h / g x - f x * (inverse (g x) * g' ?h * inverse (g x)) = (f' ?h * g x - f x * g' ?h) / (g x * g x)\n\ngoal (1 subgoal):\n 1. ((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nf' ?h / g x - f x * (inverse (g x) * g' ?h * inverse (g x)) = (f' ?h * g x - f x * g' ?h) / (g x * g x)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nf' ?h / g x - f x * (inverse (g x) * g' ?h * inverse (g x)) = (f' ?h * g x - f x * g' ?h) / (g x * g x)\n\ngoal (1 subgoal):\n 1. ((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)\n[PROOF STEP]\nusing has_derivative_divide [OF f g] x\n[PROOF STATE]\nproof (prove)\nusing this:\nf' ?h / g x - f x * (inverse (g x) * g' ?h * inverse (g x)) = (f' ?h * g x - f x * g' ?h) / (g x * g x)\ng x \\<noteq> (0::'a) \\<Longrightarrow> ((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. - f x * (inverse (g x) * g' h * inverse (g x)) + f' h / g x)) (at x within S)\ng x \\<noteq> (0::'a)\n\ngoal (1 subgoal):\n 1. ((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n((\\<lambda>x. f x / g x) has_derivative (\\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1190, "file": null, "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9019206686206199, "lm_q2_score": 0.8354835330070839, "lm_q1q2_score": 0.7535398667112668}}
{"text": "[STATEMENT]\nlemma winding_number_exp_2pi:\n    \"\\<lbrakk>path p; z \\<notin> path_image p\\<rbrakk>\n     \\<Longrightarrow> pathfinish p - z = exp (2 * pi * \\<i> * winding_number p z) * (pathstart p - z)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>path p; z \\<notin> path_image p\\<rbrakk> \\<Longrightarrow> pathfinish p - z = exp (complex_of_real (2 * pi) * \\<i> * winding_number p z) * (pathstart p - z)\n[PROOF STEP]\nusing winding_number [of p z 1]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>path p; z \\<notin> path_image p; 0 < 1\\<rbrakk> \\<Longrightarrow> \\<exists>pa. winding_number_prop p z 1 pa (winding_number p z)\n\ngoal (1 subgoal):\n 1. \\<lbrakk>path p; z \\<notin> path_image p\\<rbrakk> \\<Longrightarrow> pathfinish p - z = exp (complex_of_real (2 * pi) * \\<i> * winding_number p z) * (pathstart p - z)\n[PROOF STEP]\nunfolding valid_path_def path_image_def pathstart_def pathfinish_def winding_number_prop_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>path p; z \\<notin> p ` {0..1}; 0 < 1\\<rbrakk> \\<Longrightarrow> \\<exists>pa. pa piecewise_C1_differentiable_on {0..1} \\<and> z \\<notin> pa ` {0..1} \\<and> pa 0 = p 0 \\<and> pa 1 = p 1 \\<and> (\\<forall>t\\<in>{0..1}. cmod (p t - pa t) < 1) \\<and> contour_integral pa (\\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \\<i> * winding_number p z\n\ngoal (1 subgoal):\n 1. \\<lbrakk>path p; z \\<notin> p ` {0..1}\\<rbrakk> \\<Longrightarrow> p 1 - z = exp (complex_of_real (2 * pi) * \\<i> * winding_number p z) * (p 0 - z)\n[PROOF STEP]\nby (force dest: winding_number_exp_integral(2) [of _ 0 1 z] simp: field_simps contour_integral_integral exp_minus)", "meta": {"llama_tokens": 678, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972717658209, "lm_q2_score": 0.8652240860523328, "lm_q1q2_score": 0.7532617287832368}}
{"text": "[STATEMENT]\nlemma coprime_eq_empty_prime_inter:\n  assumes \"(n::nat) \\<noteq> 0\" \"m \\<noteq> 0\"\n  shows \"coprime n m \\<longleftrightarrow> (prime_factors n) \\<inter> (prime_factors m) = {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. coprime n m = (prime_factors n \\<inter> prime_factors m = {})\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. coprime n m \\<Longrightarrow> prime_factors n \\<inter> prime_factors m = {}\n 2. prime_factors n \\<inter> prime_factors m = {} \\<Longrightarrow> coprime n m\n[PROOF STEP]\nshow \"coprime n m \\<Longrightarrow> prime_factors n \\<inter> prime_factors m = {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. coprime n m \\<Longrightarrow> prime_factors n \\<inter> prime_factors m = {}\n[PROOF STEP]\nproof (rule ccontr)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<lbrakk>coprime n m; prime_factors n \\<inter> prime_factors m \\<noteq> {}\\<rbrakk> \\<Longrightarrow> False\n[PROOF STEP]\nassume cp: \"coprime n m\"\n[PROOF STATE]\nproof (state)\nthis:\ncoprime n m\n\ngoal (1 subgoal):\n 1. \\<lbrakk>coprime n m; prime_factors n \\<inter> prime_factors m \\<noteq> {}\\<rbrakk> \\<Longrightarrow> False\n[PROOF STEP]\nassume pf: \"prime_factors n \\<inter> prime_factors m \\<noteq> {}\"\n[PROOF STATE]\nproof (state)\nthis:\nprime_factors n \\<inter> prime_factors m \\<noteq> {}\n\ngoal (1 subgoal):\n 1. \\<lbrakk>coprime n m; prime_factors n \\<inter> prime_factors m \\<noteq> {}\\<rbrakk> \\<Longrightarrow> False\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nprime_factors n \\<inter> prime_factors m \\<noteq> {}\n[PROOF STEP]\nobtain p where p: \"p \\<in> prime_factors n\" \"p \\<in> prime_factors m\"\n[PROOF STATE]\nproof (prove)\nusing this:\nprime_factors n \\<inter> prime_factors m \\<noteq> {}\n\ngoal (1 subgoal):\n 1. (\\<And>p. \\<lbrakk>p \\<in># prime_factorization n; p \\<in># prime_factorization m\\<rbrakk> \\<Longrightarrow> thesis) \\<Longrightarrow> thesis\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\np \\<in># prime_factorization n\np \\<in># prime_factorization m\n\ngoal (1 subgoal):\n 1. \\<lbrakk>coprime n m; prime_factors n \\<inter> prime_factors m \\<noteq> {}\\<rbrakk> \\<Longrightarrow> False\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\np \\<in># prime_factorization n\np \\<in># prime_factorization m\n[PROOF STEP]\nhave p_dvd: \"p dvd n\" \"p dvd m\"\n[PROOF STATE]\nproof (prove)\nusing this:\np \\<in># prime_factorization n\np \\<in># prime_factorization m\n\ngoal (1 subgoal):\n 1. p dvd n &&& p dvd m\n[PROOF STEP]\nby blast+\n[PROOF STATE]\nproof (state)\nthis:\np dvd n\np dvd m\n\ngoal (1 subgoal):\n 1. \\<lbrakk>coprime n m; prime_factors n \\<inter> prime_factors m \\<noteq> {}\\<rbrakk> \\<Longrightarrow> False\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\np dvd n\np dvd m\n\ngoal (1 subgoal):\n 1. \\<lbrakk>coprime n m; prime_factors n \\<inter> prime_factors m \\<noteq> {}\\<rbrakk> \\<Longrightarrow> False\n[PROOF STEP]\nhave \"\\<not>is_unit p\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<not> is_unit p\n[PROOF STEP]\nusing p\n[PROOF STATE]\nproof (prove)\nusing this:\np \\<in># prime_factorization n\np \\<in># prime_factorization m\n\ngoal (1 subgoal):\n 1. \\<not> is_unit p\n[PROOF STEP]\nusing not_prime_unit\n[PROOF STATE]\nproof (prove)\nusing this:\np \\<in># prime_factorization n\np \\<in># prime_factorization m\nis_unit ?x \\<Longrightarrow> \\<not> prime ?x\n\ngoal (1 subgoal):\n 1. \\<not> is_unit p\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n\\<not> is_unit p\n\ngoal (1 subgoal):\n 1. \\<lbrakk>coprime n m; prime_factors n \\<inter> prime_factors m \\<noteq> {}\\<rbrakk> \\<Longrightarrow> False\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\np dvd n\np dvd m\n\\<not> is_unit p\n[PROOF STEP]\nshow \"False\"\n[PROOF STATE]\nproof (prove)\nusing this:\np dvd n\np dvd m\n\\<not> is_unit p\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nusing cp\n[PROOF STATE]\nproof (prove)\nusing this:\np dvd n\np dvd m\n\\<not> is_unit p\ncoprime n m\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nunfolding coprime_def\n[PROOF STATE]\nproof (prove)\nusing this:\np dvd n\np dvd m\n\\<not> is_unit p\n\\<forall>c. c dvd n \\<longrightarrow> c dvd m \\<longrightarrow> is_unit c\n\ngoal (1 subgoal):\n 1. False\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nFalse\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ncoprime n m \\<Longrightarrow> prime_factors n \\<inter> prime_factors m = {}\n\ngoal (1 subgoal):\n 1. prime_factors n \\<inter> prime_factors m = {} \\<Longrightarrow> coprime n m\n[PROOF STEP]\nassume assm: \"prime_factors n \\<inter> prime_factors m = {}\"\n[PROOF STATE]\nproof (state)\nthis:\nprime_factors n \\<inter> prime_factors m = {}\n\ngoal (1 subgoal):\n 1. prime_factors n \\<inter> prime_factors m = {} \\<Longrightarrow> coprime n m\n[PROOF STEP]\nshow \"coprime n m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. coprime n m\n[PROOF STEP]\nunfolding coprime_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<forall>c. c dvd n \\<longrightarrow> c dvd m \\<longrightarrow> is_unit c\n[PROOF STEP]\nproof\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>c. c dvd n \\<longrightarrow> c dvd m \\<longrightarrow> is_unit c\n[PROOF STEP]\nfix c\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>c. c dvd n \\<longrightarrow> c dvd m \\<longrightarrow> is_unit c\n[PROOF STEP]\nshow \"c dvd n \\<longrightarrow> c dvd m \\<longrightarrow> is_unit c\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c dvd n \\<longrightarrow> c dvd m \\<longrightarrow> is_unit c\n[PROOF STEP]\nproof(rule; rule)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<lbrakk>c dvd n; c dvd m\\<rbrakk> \\<Longrightarrow> is_unit c\n[PROOF STEP]\nassume c: \"c dvd n\" \"c dvd m\"\n[PROOF STATE]\nproof (state)\nthis:\nc dvd n\nc dvd m\n\ngoal (1 subgoal):\n 1. \\<lbrakk>c dvd n; c dvd m\\<rbrakk> \\<Longrightarrow> is_unit c\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nc dvd n\nc dvd m\n[PROOF STEP]\nhave \"prime_factors c \\<subseteq> prime_factors n\" \"prime_factors c \\<subseteq> prime_factors m\"\n[PROOF STATE]\nproof (prove)\nusing this:\nc dvd n\nc dvd m\n\ngoal (1 subgoal):\n 1. prime_factors c \\<subseteq> prime_factors n &&& prime_factors c \\<subseteq> prime_factors m\n[PROOF STEP]\nusing assms dvd_prime_factors\n[PROOF STATE]\nproof (prove)\nusing this:\nc dvd n\nc dvd m\nn \\<noteq> 0\nm \\<noteq> 0\n\\<lbrakk>?y \\<noteq> (0::?'a); ?x dvd ?y\\<rbrakk> \\<Longrightarrow> prime_factors ?x \\<subseteq> prime_factors ?y\n\ngoal (1 subgoal):\n 1. prime_factors c \\<subseteq> prime_factors n &&& prime_factors c \\<subseteq> prime_factors m\n[PROOF STEP]\nby blast+\n[PROOF STATE]\nproof (state)\nthis:\nprime_factors c \\<subseteq> prime_factors n\nprime_factors c \\<subseteq> prime_factors m\n\ngoal (1 subgoal):\n 1. \\<lbrakk>c dvd n; c dvd m\\<rbrakk> \\<Longrightarrow> is_unit c\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nprime_factors c \\<subseteq> prime_factors n\nprime_factors c \\<subseteq> prime_factors m\n[PROOF STEP]\nhave \"prime_factors c = {}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nprime_factors c \\<subseteq> prime_factors n\nprime_factors c \\<subseteq> prime_factors m\n\ngoal (1 subgoal):\n 1. prime_factors c = {}\n[PROOF STEP]\nusing assm\n[PROOF STATE]\nproof (prove)\nusing this:\nprime_factors c \\<subseteq> prime_factors n\nprime_factors c \\<subseteq> prime_factors m\nprime_factors n \\<inter> prime_factors m = {}\n\ngoal (1 subgoal):\n 1. prime_factors c = {}\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nprime_factors c = {}\n\ngoal (1 subgoal):\n 1. \\<lbrakk>c dvd n; c dvd m\\<rbrakk> \\<Longrightarrow> is_unit c\n[PROOF STEP]\nthus \"is_unit c\"\n[PROOF STATE]\nproof (prove)\nusing this:\nprime_factors c = {}\n\ngoal (1 subgoal):\n 1. is_unit c\n[PROOF STEP]\nusing assms c\n[PROOF STATE]\nproof (prove)\nusing this:\nprime_factors c = {}\nn \\<noteq> 0\nm \\<noteq> 0\nc dvd n\nc dvd m\n\ngoal (1 subgoal):\n 1. is_unit c\n[PROOF STEP]\nby (metis dvd_0_left_iff prime_factorization_empty_iff set_mset_eq_empty_iff)\n[PROOF STATE]\nproof (state)\nthis:\nis_unit c\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nc dvd n \\<longrightarrow> c dvd m \\<longrightarrow> is_unit c\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ncoprime n m\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 3394, "file": "Finitely_Generated_Abelian_Groups_General_Auxiliary", "length": 44, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972784807408, "lm_q2_score": 0.8652240773641087, "lm_q1q2_score": 0.7532617270292029}}
{"text": "[STATEMENT]\nlemma geometric_sums:\n  assumes \"norm c < 1\"\n  shows \"(\\<lambda>n. c^n) sums (1 / (1 - c))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nhave neq_0: \"c - 1 \\<noteq> 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c - (1::'a) \\<noteq> (0::'a)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm c < 1\n\ngoal (1 subgoal):\n 1. c - (1::'a) \\<noteq> (0::'a)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nc - (1::'a) \\<noteq> (0::'a)\n\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nc - (1::'a) \\<noteq> (0::'a)\n[PROOF STEP]\nhave \"(\\<lambda>n. c ^ n / (c - 1) - 1 / (c - 1)) \\<longlonglongrightarrow> 0 / (c - 1) - 1 / (c - 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nc - (1::'a) \\<noteq> (0::'a)\n\ngoal (1 subgoal):\n 1. (\\<lambda>n. c ^ n / (c - (1::'a)) - (1::'a) / (c - (1::'a))) \\<longlonglongrightarrow> (0::'a) / (c - (1::'a)) - (1::'a) / (c - (1::'a))\n[PROOF STEP]\nby (intro tendsto_intros assms)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>n. c ^ n / (c - (1::'a)) - (1::'a) / (c - (1::'a))) \\<longlonglongrightarrow> (0::'a) / (c - (1::'a)) - (1::'a) / (c - (1::'a))\n\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\<lambda>n. c ^ n / (c - (1::'a)) - (1::'a) / (c - (1::'a))) \\<longlonglongrightarrow> (0::'a) / (c - (1::'a)) - (1::'a) / (c - (1::'a))\n[PROOF STEP]\nhave \"(\\<lambda>n. (c ^ n - 1) / (c - 1)) \\<longlonglongrightarrow> 1 / (1 - c)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<lambda>n. c ^ n / (c - (1::'a)) - (1::'a) / (c - (1::'a))) \\<longlonglongrightarrow> (0::'a) / (c - (1::'a)) - (1::'a) / (c - (1::'a))\n\ngoal (1 subgoal):\n 1. (\\<lambda>n. (c ^ n - (1::'a)) / (c - (1::'a))) \\<longlonglongrightarrow> (1::'a) / ((1::'a) - c)\n[PROOF STEP]\nby (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>n. (c ^ n - (1::'a)) / (c - (1::'a))) \\<longlonglongrightarrow> (1::'a) / ((1::'a) - c)\n\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nwith neq_0\n[PROOF STATE]\nproof (chain)\npicking this:\nc - (1::'a) \\<noteq> (0::'a)\n(\\<lambda>n. (c ^ n - (1::'a)) / (c - (1::'a))) \\<longlonglongrightarrow> (1::'a) / ((1::'a) - c)\n[PROOF STEP]\nshow \"(\\<lambda>n. c ^ n) sums (1 / (1 - c))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nc - (1::'a) \\<noteq> (0::'a)\n(\\<lambda>n. (c ^ n - (1::'a)) / (c - (1::'a))) \\<longlonglongrightarrow> (1::'a) / ((1::'a) - c)\n\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nby (simp add: sums_def geometric_sum)\n[PROOF STATE]\nproof (state)\nthis:\n(^) c sums ((1::'a) / ((1::'a) - c))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1488, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894576856559, "lm_q2_score": 0.8418256512199033, "lm_q1q2_score": 0.7532567179209313}}
{"text": "[STATEMENT]\nlemma geometric_sums:\n  assumes \"norm c < 1\"\n  shows \"(\\<lambda>n. c^n) sums (1 / (1 - c))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nhave neq_0: \"c - 1 \\<noteq> 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c - (1::'a) \\<noteq> (0::'a)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm c < 1\n\ngoal (1 subgoal):\n 1. c - (1::'a) \\<noteq> (0::'a)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nc - (1::'a) \\<noteq> (0::'a)\n\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nc - (1::'a) \\<noteq> (0::'a)\n[PROOF STEP]\nhave \"(\\<lambda>n. c ^ n / (c - 1) - 1 / (c - 1)) \\<longlonglongrightarrow> 0 / (c - 1) - 1 / (c - 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nc - (1::'a) \\<noteq> (0::'a)\n\ngoal (1 subgoal):\n 1. (\\<lambda>n. c ^ n / (c - (1::'a)) - (1::'a) / (c - (1::'a))) \\<longlonglongrightarrow> (0::'a) / (c - (1::'a)) - (1::'a) / (c - (1::'a))\n[PROOF STEP]\nby (intro tendsto_intros assms)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>n. c ^ n / (c - (1::'a)) - (1::'a) / (c - (1::'a))) \\<longlonglongrightarrow> (0::'a) / (c - (1::'a)) - (1::'a) / (c - (1::'a))\n\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\<lambda>n. c ^ n / (c - (1::'a)) - (1::'a) / (c - (1::'a))) \\<longlonglongrightarrow> (0::'a) / (c - (1::'a)) - (1::'a) / (c - (1::'a))\n[PROOF STEP]\nhave \"(\\<lambda>n. (c ^ n - 1) / (c - 1)) \\<longlonglongrightarrow> 1 / (1 - c)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<lambda>n. c ^ n / (c - (1::'a)) - (1::'a) / (c - (1::'a))) \\<longlonglongrightarrow> (0::'a) / (c - (1::'a)) - (1::'a) / (c - (1::'a))\n\ngoal (1 subgoal):\n 1. (\\<lambda>n. (c ^ n - (1::'a)) / (c - (1::'a))) \\<longlonglongrightarrow> (1::'a) / ((1::'a) - c)\n[PROOF STEP]\nby (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>n. (c ^ n - (1::'a)) / (c - (1::'a))) \\<longlonglongrightarrow> (1::'a) / ((1::'a) - c)\n\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nwith neq_0\n[PROOF STATE]\nproof (chain)\npicking this:\nc - (1::'a) \\<noteq> (0::'a)\n(\\<lambda>n. (c ^ n - (1::'a)) / (c - (1::'a))) \\<longlonglongrightarrow> (1::'a) / ((1::'a) - c)\n[PROOF STEP]\nshow \"(\\<lambda>n. c ^ n) sums (1 / (1 - c))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nc - (1::'a) \\<noteq> (0::'a)\n(\\<lambda>n. (c ^ n - (1::'a)) / (c - (1::'a))) \\<longlonglongrightarrow> (1::'a) / ((1::'a) - c)\n\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nby (simp add: sums_def geometric_sum)\n[PROOF STATE]\nproof (state)\nthis:\n(^) c sums ((1::'a) / ((1::'a) - c))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1488, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894576856559, "lm_q2_score": 0.8418256512199032, "lm_q1q2_score": 0.7532567179209312}}
{"text": "[STATEMENT]\ntheorem sum_of_cubes:\n  \"4 * (\\<Sum>i::nat=0..n. i^3) = (n * (n + 1))^Suc (Suc 0)\"\n  (is \"?P n\" is \"?S n = _\")\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 4 * (\\<Sum>i = 0..n. i ^ 3) = (n * (n + 1)) ^ Suc (Suc 0)\n[PROOF STEP]\nproof (induct n)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. 4 * (\\<Sum>i = 0..0. i ^ 3) = (0 * (0 + 1)) ^ Suc (Suc 0)\n 2. \\<And>n. 4 * (\\<Sum>i = 0..n. i ^ 3) = (n * (n + 1)) ^ Suc (Suc 0) \\<Longrightarrow> 4 * (\\<Sum>i = 0..Suc n. i ^ 3) = (Suc n * (Suc n + 1)) ^ Suc (Suc 0)\n[PROOF STEP]\nshow \"?P 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 4 * (\\<Sum>i = 0..0. i ^ 3) = (0 * (0 + 1)) ^ Suc (Suc 0)\n[PROOF STEP]\nby (simp add: power_eq_if)\n[PROOF STATE]\nproof (state)\nthis:\n4 * (\\<Sum>i = 0..0. i ^ 3) = (0 * (0 + 1)) ^ Suc (Suc 0)\n\ngoal (1 subgoal):\n 1. \\<And>n. 4 * (\\<Sum>i = 0..n. i ^ 3) = (n * (n + 1)) ^ Suc (Suc 0) \\<Longrightarrow> 4 * (\\<Sum>i = 0..Suc n. i ^ 3) = (Suc n * (Suc n + 1)) ^ Suc (Suc 0)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>n. 4 * (\\<Sum>i = 0..n. i ^ 3) = (n * (n + 1)) ^ Suc (Suc 0) \\<Longrightarrow> 4 * (\\<Sum>i = 0..Suc n. i ^ 3) = (Suc n * (Suc n + 1)) ^ Suc (Suc 0)\n[PROOF STEP]\nfix n\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>n. 4 * (\\<Sum>i = 0..n. i ^ 3) = (n * (n + 1)) ^ Suc (Suc 0) \\<Longrightarrow> 4 * (\\<Sum>i = 0..Suc n. i ^ 3) = (Suc n * (Suc n + 1)) ^ Suc (Suc 0)\n[PROOF STEP]\nhave \"?S (n + 1) = ?S n + 4 * (n + 1)^3\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 4 * (\\<Sum>i = 0..n + 1. i ^ 3) = 4 * (\\<Sum>i = 0..n. i ^ 3) + 4 * (n + 1) ^ 3\n[PROOF STEP]\nby (simp add: power_eq_if distrib)\n[PROOF STATE]\nproof (state)\nthis:\n4 * (\\<Sum>i = 0..n + 1. i ^ 3) = 4 * (\\<Sum>i = 0..n. i ^ 3) + 4 * (n + 1) ^ 3\n\ngoal (1 subgoal):\n 1. \\<And>n. 4 * (\\<Sum>i = 0..n. i ^ 3) = (n * (n + 1)) ^ Suc (Suc 0) \\<Longrightarrow> 4 * (\\<Sum>i = 0..Suc n. i ^ 3) = (Suc n * (Suc n + 1)) ^ Suc (Suc 0)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n4 * (\\<Sum>i = 0..n + 1. i ^ 3) = 4 * (\\<Sum>i = 0..n. i ^ 3) + 4 * (n + 1) ^ 3\n\ngoal (1 subgoal):\n 1. \\<And>n. 4 * (\\<Sum>i = 0..n. i ^ 3) = (n * (n + 1)) ^ Suc (Suc 0) \\<Longrightarrow> 4 * (\\<Sum>i = 0..Suc n. i ^ 3) = (Suc n * (Suc n + 1)) ^ Suc (Suc 0)\n[PROOF STEP]\nassume \"?S n = (n * (n + 1))^Suc (Suc 0)\"\n[PROOF STATE]\nproof (state)\nthis:\n4 * (\\<Sum>i = 0..n. i ^ 3) = (n * (n + 1)) ^ Suc (Suc 0)\n\ngoal (1 subgoal):\n 1. \\<And>n. 4 * (\\<Sum>i = 0..n. i ^ 3) = (n * (n + 1)) ^ Suc (Suc 0) \\<Longrightarrow> 4 * (\\<Sum>i = 0..Suc n. i ^ 3) = (Suc n * (Suc n + 1)) ^ Suc (Suc 0)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n4 * (\\<Sum>i = 0..n. i ^ 3) = (n * (n + 1)) ^ Suc (Suc 0)\n\ngoal (1 subgoal):\n 1. \\<And>n. 4 * (\\<Sum>i = 0..n. i ^ 3) = (n * (n + 1)) ^ Suc (Suc 0) \\<Longrightarrow> 4 * (\\<Sum>i = 0..Suc n. i ^ 3) = (Suc n * (Suc n + 1)) ^ Suc (Suc 0)\n[PROOF STEP]\nhave \"\\<dots> + 4 * (n + 1)^3 = ((n + 1) * ((n + 1) + 1))^Suc (Suc 0)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (n * (n + 1)) ^ Suc (Suc 0) + 4 * (n + 1) ^ 3 = ((n + 1) * (n + 1 + 1)) ^ Suc (Suc 0)\n[PROOF STEP]\nby (simp add: power_eq_if distrib)\n[PROOF STATE]\nproof (state)\nthis:\n(n * (n + 1)) ^ Suc (Suc 0) + 4 * (n + 1) ^ 3 = ((n + 1) * (n + 1 + 1)) ^ Suc (Suc 0)\n\ngoal (1 subgoal):\n 1. \\<And>n. 4 * (\\<Sum>i = 0..n. i ^ 3) = (n * (n + 1)) ^ Suc (Suc 0) \\<Longrightarrow> 4 * (\\<Sum>i = 0..Suc n. i ^ 3) = (Suc n * (Suc n + 1)) ^ Suc (Suc 0)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n4 * (\\<Sum>i = 0..n + 1. i ^ 3) = ((n + 1) * (n + 1 + 1)) ^ Suc (Suc 0)\n[PROOF STEP]\nshow \"?P (Suc n)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n4 * (\\<Sum>i = 0..n + 1. i ^ 3) = ((n + 1) * (n + 1 + 1)) ^ Suc (Suc 0)\n\ngoal (1 subgoal):\n 1. 4 * (\\<Sum>i = 0..Suc n. i ^ 3) = (Suc n * (Suc n + 1)) ^ Suc (Suc 0)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n4 * (\\<Sum>i = 0..Suc n. i ^ 3) = (Suc n * (Suc n + 1)) ^ Suc (Suc 0)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2246, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.894789457685656, "lm_q2_score": 0.8418256472515683, "lm_q1q2_score": 0.7532567143701071}}
{"text": "[STATEMENT]\nlemma geometric_sums:\n  assumes \"norm c < 1\"\n  shows \"(\\<lambda>n. c^n) sums (1 / (1 - c))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nhave neq_0: \"c - 1 \\<noteq> 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c - (1::'a) \\<noteq> (0::'a)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm c < 1\n\ngoal (1 subgoal):\n 1. c - (1::'a) \\<noteq> (0::'a)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nc - (1::'a) \\<noteq> (0::'a)\n\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nc - (1::'a) \\<noteq> (0::'a)\n[PROOF STEP]\nhave \"(\\<lambda>n. c ^ n / (c - 1) - 1 / (c - 1)) \\<longlonglongrightarrow> 0 / (c - 1) - 1 / (c - 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nc - (1::'a) \\<noteq> (0::'a)\n\ngoal (1 subgoal):\n 1. (\\<lambda>n. c ^ n / (c - (1::'a)) - (1::'a) / (c - (1::'a))) \\<longlonglongrightarrow> (0::'a) / (c - (1::'a)) - (1::'a) / (c - (1::'a))\n[PROOF STEP]\nby (intro tendsto_intros assms)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>n. c ^ n / (c - (1::'a)) - (1::'a) / (c - (1::'a))) \\<longlonglongrightarrow> (0::'a) / (c - (1::'a)) - (1::'a) / (c - (1::'a))\n\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\<lambda>n. c ^ n / (c - (1::'a)) - (1::'a) / (c - (1::'a))) \\<longlonglongrightarrow> (0::'a) / (c - (1::'a)) - (1::'a) / (c - (1::'a))\n[PROOF STEP]\nhave \"(\\<lambda>n. (c ^ n - 1) / (c - 1)) \\<longlonglongrightarrow> 1 / (1 - c)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<lambda>n. c ^ n / (c - (1::'a)) - (1::'a) / (c - (1::'a))) \\<longlonglongrightarrow> (0::'a) / (c - (1::'a)) - (1::'a) / (c - (1::'a))\n\ngoal (1 subgoal):\n 1. (\\<lambda>n. (c ^ n - (1::'a)) / (c - (1::'a))) \\<longlonglongrightarrow> (1::'a) / ((1::'a) - c)\n[PROOF STEP]\nby (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>n. (c ^ n - (1::'a)) / (c - (1::'a))) \\<longlonglongrightarrow> (1::'a) / ((1::'a) - c)\n\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nwith neq_0\n[PROOF STATE]\nproof (chain)\npicking this:\nc - (1::'a) \\<noteq> (0::'a)\n(\\<lambda>n. (c ^ n - (1::'a)) / (c - (1::'a))) \\<longlonglongrightarrow> (1::'a) / ((1::'a) - c)\n[PROOF STEP]\nshow \"(\\<lambda>n. c ^ n) sums (1 / (1 - c))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nc - (1::'a) \\<noteq> (0::'a)\n(\\<lambda>n. (c ^ n - (1::'a)) / (c - (1::'a))) \\<longlonglongrightarrow> (1::'a) / ((1::'a) - c)\n\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nby (simp add: sums_def geometric_sum)\n[PROOF STATE]\nproof (state)\nthis:\n(^) c sums ((1::'a) / ((1::'a) - c))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1488, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894576856559, "lm_q2_score": 0.8418256432832332, "lm_q1q2_score": 0.7532567108192826}}
{"text": "[STATEMENT]\nlemma geometric_sums:\n  assumes \"norm c < 1\"\n  shows \"(\\<lambda>n. c^n) sums (1 / (1 - c))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nhave neq_0: \"c - 1 \\<noteq> 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. c - (1::'a) \\<noteq> (0::'a)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nnorm c < 1\n\ngoal (1 subgoal):\n 1. c - (1::'a) \\<noteq> (0::'a)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nc - (1::'a) \\<noteq> (0::'a)\n\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nc - (1::'a) \\<noteq> (0::'a)\n[PROOF STEP]\nhave \"(\\<lambda>n. c ^ n / (c - 1) - 1 / (c - 1)) \\<longlonglongrightarrow> 0 / (c - 1) - 1 / (c - 1)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nc - (1::'a) \\<noteq> (0::'a)\n\ngoal (1 subgoal):\n 1. (\\<lambda>n. c ^ n / (c - (1::'a)) - (1::'a) / (c - (1::'a))) \\<longlonglongrightarrow> (0::'a) / (c - (1::'a)) - (1::'a) / (c - (1::'a))\n[PROOF STEP]\nby (intro tendsto_intros assms)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>n. c ^ n / (c - (1::'a)) - (1::'a) / (c - (1::'a))) \\<longlonglongrightarrow> (0::'a) / (c - (1::'a)) - (1::'a) / (c - (1::'a))\n\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\<lambda>n. c ^ n / (c - (1::'a)) - (1::'a) / (c - (1::'a))) \\<longlonglongrightarrow> (0::'a) / (c - (1::'a)) - (1::'a) / (c - (1::'a))\n[PROOF STEP]\nhave \"(\\<lambda>n. (c ^ n - 1) / (c - 1)) \\<longlonglongrightarrow> 1 / (1 - c)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<lambda>n. c ^ n / (c - (1::'a)) - (1::'a) / (c - (1::'a))) \\<longlonglongrightarrow> (0::'a) / (c - (1::'a)) - (1::'a) / (c - (1::'a))\n\ngoal (1 subgoal):\n 1. (\\<lambda>n. (c ^ n - (1::'a)) / (c - (1::'a))) \\<longlonglongrightarrow> (1::'a) / ((1::'a) - c)\n[PROOF STEP]\nby (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<lambda>n. (c ^ n - (1::'a)) / (c - (1::'a))) \\<longlonglongrightarrow> (1::'a) / ((1::'a) - c)\n\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nwith neq_0\n[PROOF STATE]\nproof (chain)\npicking this:\nc - (1::'a) \\<noteq> (0::'a)\n(\\<lambda>n. (c ^ n - (1::'a)) / (c - (1::'a))) \\<longlonglongrightarrow> (1::'a) / ((1::'a) - c)\n[PROOF STEP]\nshow \"(\\<lambda>n. c ^ n) sums (1 / (1 - c))\"\n[PROOF STATE]\nproof (prove)\nusing this:\nc - (1::'a) \\<noteq> (0::'a)\n(\\<lambda>n. (c ^ n - (1::'a)) / (c - (1::'a))) \\<longlonglongrightarrow> (1::'a) / ((1::'a) - c)\n\ngoal (1 subgoal):\n 1. (^) c sums ((1::'a) / ((1::'a) - c))\n[PROOF STEP]\nby (simp add: sums_def geometric_sum)\n[PROOF STATE]\nproof (state)\nthis:\n(^) c sums ((1::'a) / ((1::'a) - c))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1488, "file": null, "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8947894576856559, "lm_q2_score": 0.8418256432832332, "lm_q1q2_score": 0.7532567108192826}}
{"text": "[STATEMENT]\nlemma stieltjes_gamma_aux6: \"(\\<Sum>t<k. (k choose t) * Ln 2 ^ (k - t) * stieltjes_gamma t) -\n                  Ln 2 ^ (k + 1) / of_nat (k + 1) =\n                (-1)^k * fact k * (\\<Sum>i=1..k+1. (-Ln 2) ^ i * A (k-(i-1)) / fact i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>t<k. of_nat (k choose t) * Ln 2 ^ (k - t) * stieltjes_gamma t) - Ln 2 ^ (k + 1) / of_nat (k + 1) = (- 1) ^ k * fact k * (\\<Sum>i = 1..k + 1. (- Ln 2) ^ i * A (k - (i - 1)) / fact i)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\<Sum>t<k. of_nat (k choose t) * Ln 2 ^ (k - t) * stieltjes_gamma t) - Ln 2 ^ (k + 1) / of_nat (k + 1) = (- 1) ^ k * fact k * (\\<Sum>i = 1..k + 1. (- Ln 2) ^ i * A (k - (i - 1)) / fact i)\n[PROOF STEP]\nhave \"(\\<Sum>t<k. (k choose t) * Ln 2 ^ (k - t) * stieltjes_gamma t) -\n          Ln 2 ^ (k + 1) / of_nat (k + 1) = (- 1) ^ k * (deriv ^^ k) g 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>t<k. of_nat (k choose t) * Ln 2 ^ (k - t) * stieltjes_gamma t) - Ln 2 ^ (k + 1) / of_nat (k + 1) = (- 1) ^ k * (deriv ^^ k) g 1\n[PROOF STEP]\nusing stieltjes_gamma_aux5[of k]\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<Sum>t<k. of_nat (k choose t) * Ln 2 ^ (k - t) * stieltjes_gamma t) - Ln 2 ^ (k + 1) / of_nat (k + 1) = (- 1) ^ k * (deriv ^^ k) g 1\n\ngoal (1 subgoal):\n 1. (\\<Sum>t<k. of_nat (k choose t) * Ln 2 ^ (k - t) * stieltjes_gamma t) - Ln 2 ^ (k + 1) / of_nat (k + 1) = (- 1) ^ k * (deriv ^^ k) g 1\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>t<k. of_nat (k choose t) * Ln 2 ^ (k - t) * stieltjes_gamma t) - Ln 2 ^ (k + 1) / of_nat (k + 1) = (- 1) ^ k * (deriv ^^ k) g 1\n\ngoal (1 subgoal):\n 1. (\\<Sum>t<k. of_nat (k choose t) * Ln 2 ^ (k - t) * stieltjes_gamma t) - Ln 2 ^ (k + 1) / of_nat (k + 1) = (- 1) ^ k * fact k * (\\<Sum>i = 1..k + 1. (- Ln 2) ^ i * A (k - (i - 1)) / fact i)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>t<k. of_nat (k choose t) * Ln 2 ^ (k - t) * stieltjes_gamma t) - Ln 2 ^ (k + 1) / of_nat (k + 1) = (- 1) ^ k * (deriv ^^ k) g 1\n\ngoal (1 subgoal):\n 1. (\\<Sum>t<k. of_nat (k choose t) * Ln 2 ^ (k - t) * stieltjes_gamma t) - Ln 2 ^ (k + 1) / of_nat (k + 1) = (- 1) ^ k * fact k * (\\<Sum>i = 1..k + 1. (- Ln 2) ^ i * A (k - (i - 1)) / fact i)\n[PROOF STEP]\nhave \"(deriv ^^ k) g 1 = fact k * fps_nth G k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (deriv ^^ k) g 1 = fact k * G $ k\n[PROOF STEP]\nby (rule stieltjes_gamma_aux2)\n[PROOF STATE]\nproof (state)\nthis:\n(deriv ^^ k) g 1 = fact k * G $ k\n\ngoal (1 subgoal):\n 1. (\\<Sum>t<k. of_nat (k choose t) * Ln 2 ^ (k - t) * stieltjes_gamma t) - Ln 2 ^ (k + 1) / of_nat (k + 1) = (- 1) ^ k * fact k * (\\<Sum>i = 1..k + 1. (- Ln 2) ^ i * A (k - (i - 1)) / fact i)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n(deriv ^^ k) g 1 = fact k * G $ k\n\ngoal (1 subgoal):\n 1. (\\<Sum>t<k. of_nat (k choose t) * Ln 2 ^ (k - t) * stieltjes_gamma t) - Ln 2 ^ (k + 1) / of_nat (k + 1) = (- 1) ^ k * fact k * (\\<Sum>i = 1..k + 1. (- Ln 2) ^ i * A (k - (i - 1)) / fact i)\n[PROOF STEP]\nhave \"fps_nth G k = (\\<Sum>i=1..k + 1. (-Ln 2) ^ i * A (k - (i - 1)) / fact i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. G $ k = (\\<Sum>i = 1..k + 1. (- Ln 2) ^ i * A (k - (i - 1)) / fact i)\n[PROOF STEP]\nby (rule stieltjes_gamma_aux4)\n[PROOF STATE]\nproof (state)\nthis:\nG $ k = (\\<Sum>i = 1..k + 1. (- Ln 2) ^ i * A (k - (i - 1)) / fact i)\n\ngoal (1 subgoal):\n 1. (\\<Sum>t<k. of_nat (k choose t) * Ln 2 ^ (k - t) * stieltjes_gamma t) - Ln 2 ^ (k + 1) / of_nat (k + 1) = (- 1) ^ k * fact k * (\\<Sum>i = 1..k + 1. (- Ln 2) ^ i * A (k - (i - 1)) / fact i)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\<Sum>t<k. of_nat (k choose t) * Ln 2 ^ (k - t) * stieltjes_gamma t) - Ln 2 ^ (k + 1) / of_nat (k + 1) = (- 1) ^ k * (fact k * (\\<Sum>i = 1..k + 1. (- Ln 2) ^ i * A (k - (i - 1)) / fact i))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<Sum>t<k. of_nat (k choose t) * Ln 2 ^ (k - t) * stieltjes_gamma t) - Ln 2 ^ (k + 1) / of_nat (k + 1) = (- 1) ^ k * (fact k * (\\<Sum>i = 1..k + 1. (- Ln 2) ^ i * A (k - (i - 1)) / fact i))\n\ngoal (1 subgoal):\n 1. (\\<Sum>t<k. of_nat (k choose t) * Ln 2 ^ (k - t) * stieltjes_gamma t) - Ln 2 ^ (k + 1) / of_nat (k + 1) = (- 1) ^ k * fact k * (\\<Sum>i = 1..k + 1. (- Ln 2) ^ i * A (k - (i - 1)) / fact i)\n[PROOF STEP]\nby (simp add: mult_ac)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Sum>t<k. of_nat (k choose t) * Ln 2 ^ (k - t) * stieltjes_gamma t) - Ln 2 ^ (k + 1) / of_nat (k + 1) = (- 1) ^ k * fact k * (\\<Sum>i = 1..k + 1. (- Ln 2) ^ i * A (k - (i - 1)) / fact i)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 2448, "file": "Zeta_Function_Zeta_Laurent_Expansion", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9161096112990283, "lm_q2_score": 0.822189134878876, "lm_q1q2_score": 0.7532153687681715}}
{"text": "[STATEMENT]\nlemma kronecker_inverse_index:\n  assumes \"r < dim_row A\" \"s < dim_col A\"\n  assumes \"v < dim_row B\" \"w < dim_col B\"\n  shows \"kronecker_product A B $$ (dim_row B*r+v, dim_col B*s+w) = A $$ (r,s) * B $$ (v,w)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. kronecker_product A B $$ (dim_row B * r + v, dim_col B * s + w) = A $$ (r, s) * B $$ (v, w)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. kronecker_product A B $$ (dim_row B * r + v, dim_col B * s + w) = A $$ (r, s) * B $$ (v, w)\n[PROOF STEP]\nfrom arith[OF assms(1) assms(3)]\n[PROOF STATE]\nproof (chain)\npicking this:\ndim_row B * r + v < dim_row A * dim_row B\n[PROOF STEP]\nhave \"dim_row B*r+v < dim_row A * dim_row B\"\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_row B * r + v < dim_row A * dim_row B\n\ngoal (1 subgoal):\n 1. dim_row B * r + v < dim_row A * dim_row B\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ndim_row B * r + v < dim_row A * dim_row B\n\ngoal (1 subgoal):\n 1. kronecker_product A B $$ (dim_row B * r + v, dim_col B * s + w) = A $$ (r, s) * B $$ (v, w)\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ndim_row B * r + v < dim_row A * dim_row B\n\ngoal (1 subgoal):\n 1. kronecker_product A B $$ (dim_row B * r + v, dim_col B * s + w) = A $$ (r, s) * B $$ (v, w)\n[PROOF STEP]\nfrom arith[OF assms(2) assms(4)]\n[PROOF STATE]\nproof (chain)\npicking this:\ndim_col B * s + w < dim_col A * dim_col B\n[PROOF STEP]\nhave \"dim_col B * s + w < dim_col A * dim_col B\"\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_col B * s + w < dim_col A * dim_col B\n\ngoal (1 subgoal):\n 1. dim_col B * s + w < dim_col A * dim_col B\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ndim_col B * s + w < dim_col A * dim_col B\n\ngoal (1 subgoal):\n 1. kronecker_product A B $$ (dim_row B * r + v, dim_col B * s + w) = A $$ (r, s) * B $$ (v, w)\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ndim_row B * r + v < dim_row A * dim_row B\ndim_col B * s + w < dim_col A * dim_col B\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_row B * r + v < dim_row A * dim_row B\ndim_col B * s + w < dim_col A * dim_col B\n\ngoal (1 subgoal):\n 1. kronecker_product A B $$ (dim_row B * r + v, dim_col B * s + w) = A $$ (r, s) * B $$ (v, w)\n[PROOF STEP]\nunfolding kronecker_product_def Let_def\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_row B * r + v < dim_row A * dim_row B\ndim_col B * s + w < dim_col A * dim_col B\n\ngoal (1 subgoal):\n 1. mat (dim_row A * dim_row B) (dim_col A * dim_col B) (\\<lambda>(i, j). A $$ (i div dim_row B, j div dim_col B) * B $$ (i mod dim_row B, j mod dim_col B)) $$ (dim_row B * r + v, dim_col B * s + w) = A $$ (r, s) * B $$ (v, w)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ndim_row B * r + v < dim_row A * dim_row B\ndim_col B * s + w < dim_col A * dim_col B\nr < dim_row A\ns < dim_col A\nv < dim_row B\nw < dim_col B\n\ngoal (1 subgoal):\n 1. mat (dim_row A * dim_row B) (dim_col A * dim_col B) (\\<lambda>(i, j). A $$ (i div dim_row B, j div dim_col B) * B $$ (i mod dim_row B, j mod dim_col B)) $$ (dim_row B * r + v, dim_col B * s + w) = A $$ (r, s) * B $$ (v, w)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nkronecker_product A B $$ (dim_row B * r + v, dim_col B * s + w) = A $$ (r, s) * B $$ (v, w)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1531, "file": "BenOr_Kozen_Reif_More_Matrix", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473779969193, "lm_q2_score": 0.8633916047011594, "lm_q1q2_score": 0.753177402545609}}
{"text": "[STATEMENT]\nlemma sin_cube_plus_cos_cube:\n  \"sin x ^ 3 + cos x ^ 3 = 1/2 * (sin x + cos x) * (2 - sin (2 * x))\"\n  for x :: \"'a::{real_normed_field,banach}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin x ^ 3 + cos x ^ 3 = (1::'a) / (2::'a) * (sin x + cos x) * ((2::'a) - sin ((2::'a) * x))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. sin x ^ 3 + cos x ^ 3 = (1::'a) / (2::'a) * (sin x + cos x) * ((2::'a) - sin ((2::'a) * x))\n[PROOF STEP]\nhave \"sin x ^ 3 + cos x ^ 3 = (sin x + cos x) * (cos x ^ 2 - cos x * sin x + sin x ^ 2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin x ^ 3 + cos x ^ 3 = (sin x + cos x) * ((cos x)\\<^sup>2 - cos x * sin x + (sin x)\\<^sup>2)\n[PROOF STEP]\nby (smt (z3) add.commute combine_common_factor diff_add_cancel distrib_left\n          mult.commute mult.left_commute power2_eq_square power3_eq_cube)\n[PROOF STATE]\nproof (state)\nthis:\nsin x ^ 3 + cos x ^ 3 = (sin x + cos x) * ((cos x)\\<^sup>2 - cos x * sin x + (sin x)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. sin x ^ 3 + cos x ^ 3 = (1::'a) / (2::'a) * (sin x + cos x) * ((2::'a) - sin ((2::'a) * x))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nsin x ^ 3 + cos x ^ 3 = (sin x + cos x) * ((cos x)\\<^sup>2 - cos x * sin x + (sin x)\\<^sup>2)\n\ngoal (1 subgoal):\n 1. sin x ^ 3 + cos x ^ 3 = (1::'a) / (2::'a) * (sin x + cos x) * ((2::'a) - sin ((2::'a) * x))\n[PROOF STEP]\nhave \"\\<dots> = (sin x + cos x) * (1 - cos x * sin x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (sin x + cos x) * ((cos x)\\<^sup>2 - cos x * sin x + (sin x)\\<^sup>2) = (sin x + cos x) * ((1::'a) - cos x * sin x)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n(sin x + cos x) * ((cos x)\\<^sup>2 - cos x * sin x + (sin x)\\<^sup>2) = (sin x + cos x) * ((1::'a) - cos x * sin x)\n\ngoal (1 subgoal):\n 1. sin x ^ 3 + cos x ^ 3 = (1::'a) / (2::'a) * (sin x + cos x) * ((2::'a) - sin ((2::'a) * x))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nsin x ^ 3 + cos x ^ 3 = (sin x + cos x) * ((1::'a) - cos x * sin x)\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nsin x ^ 3 + cos x ^ 3 = (sin x + cos x) * ((1::'a) - cos x * sin x)\n\ngoal (1 subgoal):\n 1. sin x ^ 3 + cos x ^ 3 = (1::'a) / (2::'a) * (sin x + cos x) * ((2::'a) - sin ((2::'a) * x))\n[PROOF STEP]\nby (smt (z3) mult.commute mult.left_neutral mult_2 nonzero_mult_div_cancel_left\n          right_diff_distrib' sin_add times_divide_eq_left zero_neq_numeral)\n[PROOF STATE]\nproof (state)\nthis:\nsin x ^ 3 + cos x ^ 3 = (1::'a) / (2::'a) * (sin x + cos x) * ((2::'a) - sin ((2::'a) * x))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1275, "file": "Hyperdual_AnalyticTestFunction", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473713594992, "lm_q2_score": 0.8633916047011595, "lm_q1q2_score": 0.7531773968149164}}
{"text": "[STATEMENT]\nlemma ld_ld_1_less:\n  assumes \"x > 0\" \"y > 0\" shows \"log 2 x + log 2 y + 1 < 2 * log 2 (x+y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. log 2 x + log 2 y + 1 < 2 * log 2 (x + y)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. log 2 x + log 2 y + 1 < 2 * log 2 (x + y)\n[PROOF STEP]\nhave \"2 powr (log 2 x + log 2 y + 1) = 2*x*y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 powr (log 2 x + log 2 y + 1) = 2 * x * y\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x\n0 < y\n\ngoal (1 subgoal):\n 1. 2 powr (log 2 x + log 2 y + 1) = 2 * x * y\n[PROOF STEP]\nby(simp add: powr_add)\n[PROOF STATE]\nproof (state)\nthis:\n2 powr (log 2 x + log 2 y + 1) = 2 * x * y\n\ngoal (1 subgoal):\n 1. log 2 x + log 2 y + 1 < 2 * log 2 (x + y)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n2 powr (log 2 x + log 2 y + 1) = 2 * x * y\n\ngoal (1 subgoal):\n 1. log 2 x + log 2 y + 1 < 2 * log 2 (x + y)\n[PROOF STEP]\nhave \"\\<dots> < (x+y)^2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * x * y < (x + y)\\<^sup>2\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x\n0 < y\n\ngoal (1 subgoal):\n 1. 2 * x * y < (x + y)\\<^sup>2\n[PROOF STEP]\nby(simp add: numeral_eq_Suc algebra_simps add_pos_pos)\n[PROOF STATE]\nproof (state)\nthis:\n2 * x * y < (x + y)\\<^sup>2\n\ngoal (1 subgoal):\n 1. log 2 x + log 2 y + 1 < 2 * log 2 (x + y)\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n2 * x * y < (x + y)\\<^sup>2\n\ngoal (1 subgoal):\n 1. log 2 x + log 2 y + 1 < 2 * log 2 (x + y)\n[PROOF STEP]\nhave \"\\<dots> = 2 powr (2 * log 2 (x+y))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (x + y)\\<^sup>2 = 2 powr (2 * log 2 (x + y))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < x\n0 < y\n\ngoal (1 subgoal):\n 1. (x + y)\\<^sup>2 = 2 powr (2 * log 2 (x + y))\n[PROOF STEP]\nby(simp add: powr_add log_powr[symmetric])\n[PROOF STATE]\nproof (state)\nthis:\n(x + y)\\<^sup>2 = 2 powr (2 * log 2 (x + y))\n\ngoal (1 subgoal):\n 1. log 2 x + log 2 y + 1 < 2 * log 2 (x + y)\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n2 powr (log 2 x + log 2 y + 1) < 2 powr (2 * log 2 (x + y))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n2 powr (log 2 x + log 2 y + 1) < 2 powr (2 * log 2 (x + y))\n\ngoal (1 subgoal):\n 1. log 2 x + log 2 y + 1 < 2 * log 2 (x + y)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nlog 2 x + log 2 y + 1 < 2 * log 2 (x + y)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1299, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8902942261220292, "lm_q2_score": 0.8459424314825853, "lm_q1q2_score": 0.753137662380576}}
{"text": "[STATEMENT]\nlemma mem_convex_alt:\n  assumes \"convex S\" \"x \\<in> S\" \"y \\<in> S\" \"u \\<ge> 0\" \"v \\<ge> 0\" \"u + v > 0\"\n  shows \"((u/(u+v)) *\\<^sub>R x + (v/(u+v)) *\\<^sub>R y) \\<in> S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (u / (u + v)) *\\<^sub>R x + (v / (u + v)) *\\<^sub>R y \\<in> S\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nconvex S\nx \\<in> S\ny \\<in> S\n0 \\<le> u\n0 \\<le> v\n0 < u + v\n\ngoal (1 subgoal):\n 1. (u / (u + v)) *\\<^sub>R x + (v / (u + v)) *\\<^sub>R y \\<in> S\n[PROOF STEP]\nby (simp add: convex_def zero_le_divide_iff add_divide_distrib [symmetric])", "meta": {"llama_tokens": 297, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9184802350995703, "lm_q2_score": 0.8198933337131076, "lm_q1q2_score": 0.7530558219053856}}
{"text": "[STATEMENT]\nlemma vec_of_poly_n_add: \"vec_of_poly_n (a + b) n = vec_of_poly_n a n + vec_of_poly_n b n\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec_of_poly_n (a + b) n = vec_of_poly_n a n + vec_of_poly_n b n\n[PROOF STEP]\nproof (induct n arbitrary: a b)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\<And>a b. vec_of_poly_n (a + b) 0 = vec_of_poly_n a 0 + vec_of_poly_n b 0\n 2. \\<And>n a b. (\\<And>a b. vec_of_poly_n (a + b) n = vec_of_poly_n a n + vec_of_poly_n b n) \\<Longrightarrow> vec_of_poly_n (a + b) (Suc n) = vec_of_poly_n a (Suc n) + vec_of_poly_n b (Suc n)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. \\<And>a b. vec_of_poly_n (a + b) 0 = vec_of_poly_n a 0 + vec_of_poly_n b 0\n 2. \\<And>n a b. (\\<And>a b. vec_of_poly_n (a + b) n = vec_of_poly_n a n + vec_of_poly_n b n) \\<Longrightarrow> vec_of_poly_n (a + b) (Suc n) = vec_of_poly_n a (Suc n) + vec_of_poly_n b (Suc n)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec_of_poly_n (a + b) 0 = vec_of_poly_n a 0 + vec_of_poly_n b 0\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nvec_of_poly_n (a + b) 0 = vec_of_poly_n a 0 + vec_of_poly_n b 0\n\ngoal (1 subgoal):\n 1. \\<And>n a b. (\\<And>a b. vec_of_poly_n (a + b) n = vec_of_poly_n a n + vec_of_poly_n b n) \\<Longrightarrow> vec_of_poly_n (a + b) (Suc n) = vec_of_poly_n a (Suc n) + vec_of_poly_n b (Suc n)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>n a b. (\\<And>a b. vec_of_poly_n (a + b) n = vec_of_poly_n a n + vec_of_poly_n b n) \\<Longrightarrow> vec_of_poly_n (a + b) (Suc n) = vec_of_poly_n a (Suc n) + vec_of_poly_n b (Suc n)\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\nvec_of_poly_n (?a + ?b) n = vec_of_poly_n ?a n + vec_of_poly_n ?b n\n\ngoal (1 subgoal):\n 1. \\<And>n a b. (\\<And>a b. vec_of_poly_n (a + b) n = vec_of_poly_n a n + vec_of_poly_n b n) \\<Longrightarrow> vec_of_poly_n (a + b) (Suc n) = vec_of_poly_n a (Suc n) + vec_of_poly_n b (Suc n)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nvec_of_poly_n (?a + ?b) n = vec_of_poly_n ?a n + vec_of_poly_n ?b n\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nvec_of_poly_n (?a + ?b) n = vec_of_poly_n ?a n + vec_of_poly_n ?b n\n\ngoal (1 subgoal):\n 1. vec_of_poly_n (a + b) (Suc n) = vec_of_poly_n a (Suc n) + vec_of_poly_n b (Suc n)\n[PROOF STEP]\nby (cases a, cases b, auto)\n[PROOF STATE]\nproof (state)\nthis:\nvec_of_poly_n (a + b) (Suc n) = vec_of_poly_n a (Suc n) + vec_of_poly_n b (Suc n)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1333, "file": "LLL_Basis_Reduction_Missing_Lemmas", "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772253241802, "lm_q2_score": 0.8615382165412808, "lm_q1q2_score": 0.7530509338251454}}
{"text": "[STATEMENT]\nlemma csegment_midpoint_subset: \"closed_segment (midpoint a b) b \\<subseteq> closed_segment a b\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. closed_segment (midpoint a b) b \\<subseteq> closed_segment a b\n[PROOF STEP]\napply (clarsimp simp: midpoint_def in_segment)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>u. \\<lbrakk>0 \\<le> u; u \\<le> 1\\<rbrakk> \\<Longrightarrow> \\<exists>ua\\<ge>0. ua \\<le> 1 \\<and> ((1 - u) / 2) *\\<^sub>R (a + b) + u *\\<^sub>R b = (1 - ua) *\\<^sub>R a + ua *\\<^sub>R b\n[PROOF STEP]\napply (rule_tac x=\"(1 + u) / 2\" in exI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>u. \\<lbrakk>0 \\<le> u; u \\<le> 1\\<rbrakk> \\<Longrightarrow> 0 \\<le> (1 + u) / 2 \\<and> (1 + u) / 2 \\<le> 1 \\<and> ((1 - u) / 2) *\\<^sub>R (a + b) + u *\\<^sub>R b = (1 - (1 + u) / 2) *\\<^sub>R a + ((1 + u) / 2) *\\<^sub>R b\n[PROOF STEP]\napply (auto simp: algebra_simps add_divide_distrib diff_divide_distrib)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>u. \\<lbrakk>0 \\<le> u; u \\<le> 1\\<rbrakk> \\<Longrightarrow> u *\\<^sub>R b = (u / 2) *\\<^sub>R b + (u / 2) *\\<^sub>R b\n[PROOF STEP]\nby (metis field_sum_of_halves scaleR_left.add)", "meta": {"llama_tokens": 547, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513786759491, "lm_q2_score": 0.839733963661418, "lm_q1q2_score": 0.7526127226525652}}
{"text": "[STATEMENT]\nlemma poly_of_list_split_at: assumes \"split_at n f = (f0,f1)\" \n  shows \"poly_of_list f = monom_mult n (poly_of_list f1) + poly_of_list f0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poly_of_list f = monom_mult n (poly_of_list f1) + poly_of_list f0\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. poly_of_list f = monom_mult n (poly_of_list f1) + poly_of_list f0\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\nsplit_at n f = (f0, f1)\n[PROOF STEP]\nhave id: \"f1 = drop n f\" \"f0 = take n f\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsplit_at n f = (f0, f1)\n\ngoal (1 subgoal):\n 1. f1 = drop n f &&& f0 = take n f\n[PROOF STEP]\nunfolding split_at_def\n[PROOF STATE]\nproof (prove)\nusing this:\n(take n f, drop n f) = (f0, f1)\n\ngoal (1 subgoal):\n 1. f1 = drop n f &&& f0 = take n f\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nf1 = drop n f\nf0 = take n f\n\ngoal (1 subgoal):\n 1. poly_of_list f = monom_mult n (poly_of_list f1) + poly_of_list f0\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poly_of_list f = monom_mult n (poly_of_list f1) + poly_of_list f0\n[PROOF STEP]\nunfolding id\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. poly_of_list f = monom_mult n (poly_of_list (drop n f)) + poly_of_list (take n f)\n[PROOF STEP]\nproof (rule poly_eqI)\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>na. coeff (poly_of_list f) na = coeff (monom_mult n (poly_of_list (drop n f)) + poly_of_list (take n f)) na\n[PROOF STEP]\nfix i\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>na. coeff (poly_of_list f) na = coeff (monom_mult n (poly_of_list (drop n f)) + poly_of_list (take n f)) na\n[PROOF STEP]\nshow \"coeff (poly_of_list f) i = \n      coeff (monom_mult n (poly_of_list (drop n f)) + poly_of_list (take n f)) i\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. coeff (poly_of_list f) i = coeff (monom_mult n (poly_of_list (drop n f)) + poly_of_list (take n f)) i\n[PROOF STEP]\nunfolding monom_mult_def coeff_monom_mult coeff_add poly_of_list_def coeff_Poly\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. nth_default (0::'a) f i = (if n \\<le> i then (1::'a) * nth_default (0::'a) (drop n f) (i - n) else (0::'a)) + nth_default (0::'a) (take n f) i\n[PROOF STEP]\nby (cases \"n \\<le> i\"; cases \"i \\<ge> length f\", auto simp: nth_default_nth nth_default_beyond)\n[PROOF STATE]\nproof (state)\nthis:\ncoeff (poly_of_list f) i = coeff (monom_mult n (poly_of_list (drop n f)) + poly_of_list (take n f)) i\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\npoly_of_list f = monom_mult n (poly_of_list f1) + poly_of_list f0\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1228, "file": "Berlekamp_Zassenhaus_Karatsuba_Multiplication", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513675912913, "lm_q2_score": 0.8397339656668287, "lm_q1q2_score": 0.7526127151417537}}
{"text": "[STATEMENT]\nlemma prod_lessThan_split:\nfixes g :: \"nat \\<Rightarrow> real\" shows \"prod g {..<n+m} = prod g {..<n} * prod (\\<lambda>x. g (x+n)) {..<m}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. prod g {..<n + m} = prod g {..<n} * (\\<Prod>x<m. g (x + n))\n[PROOF STEP]\nusing Groups_Big.comm_monoid_mult_class.prod.union_inter_neutral[of \"{..<n}\" \"{n..<n+m}\" g, unfolded ivl_disj_un_one(2)[OF le_add1], OF finite_lessThan finite_atLeastLessThan]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<forall>x\\<in>{..<n} \\<inter> {n..<n + m}. g x = 1 \\<Longrightarrow> prod g {..<n + m} = prod g {..<n} * prod g {n..<n + m}\n\ngoal (1 subgoal):\n 1. prod g {..<n + m} = prod g {..<n} * (\\<Prod>x<m. g (x + n))\n[PROOF STEP]\nby (metis (no_types) add.commute add.left_neutral atLeast0LessThan empty_iff ivl_disj_int_one(2) prod.shift_bounds_nat_ivl)", "meta": {"llama_tokens": 387, "file": "Deep_Learning_DL_Network", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110569397307, "lm_q2_score": 0.843895106480586, "lm_q1q2_score": 0.752594986856718}}
{"text": "[STATEMENT]\nlemma injf_max_order_preserving2: \n  assumes \"m < n\" and E: \"\\<forall>x. \\<exists>y \\<in> E. x < y\"\n  shows  \"injf_max m E < injf_max n E\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. injf_max m E < injf_max n E\n[PROOF STEP]\nusing \\<open>m < n\\<close>\n[PROOF STATE]\nproof (prove)\nusing this:\nm < n\n\ngoal (1 subgoal):\n 1. injf_max m E < injf_max n E\n[PROOF STEP]\nproof (induction n arbitrary: m)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\<And>m. m < 0 \\<Longrightarrow> injf_max m E < injf_max 0 E\n 2. \\<And>n m. \\<lbrakk>\\<And>m. m < n \\<Longrightarrow> injf_max m E < injf_max n E; m < Suc n\\<rbrakk> \\<Longrightarrow> injf_max m E < injf_max (Suc n) E\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\nm < 0\n\ngoal (2 subgoals):\n 1. \\<And>m. m < 0 \\<Longrightarrow> injf_max m E < injf_max 0 E\n 2. \\<And>n m. \\<lbrakk>\\<And>m. m < n \\<Longrightarrow> injf_max m E < injf_max n E; m < Suc n\\<rbrakk> \\<Longrightarrow> injf_max m E < injf_max (Suc n) E\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nm < 0\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nm < 0\n\ngoal (1 subgoal):\n 1. injf_max m E < injf_max 0 E\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ninjf_max m E < injf_max 0 E\n\ngoal (1 subgoal):\n 1. \\<And>n m. \\<lbrakk>\\<And>m. m < n \\<Longrightarrow> injf_max m E < injf_max n E; m < Suc n\\<rbrakk> \\<Longrightarrow> injf_max m E < injf_max (Suc n) E\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>n m. \\<lbrakk>\\<And>m. m < n \\<Longrightarrow> injf_max m E < injf_max n E; m < Suc n\\<rbrakk> \\<Longrightarrow> injf_max m E < injf_max (Suc n) E\n[PROOF STEP]\ncase (Suc n)\n[PROOF STATE]\nproof (state)\nthis:\n?m < n \\<Longrightarrow> injf_max ?m E < injf_max n E\nm < Suc n\n\ngoal (1 subgoal):\n 1. \\<And>n m. \\<lbrakk>\\<And>m. m < n \\<Longrightarrow> injf_max m E < injf_max n E; m < Suc n\\<rbrakk> \\<Longrightarrow> injf_max m E < injf_max (Suc n) E\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n?m < n \\<Longrightarrow> injf_max ?m E < injf_max n E\nm < Suc n\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\n?m < n \\<Longrightarrow> injf_max ?m E < injf_max n E\nm < Suc n\n\ngoal (1 subgoal):\n 1. injf_max m E < injf_max (Suc n) E\n[PROOF STEP]\nby (metis E injf_max_order_preserving less_Suc_eq order_less_trans)\n[PROOF STATE]\nproof (state)\nthis:\ninjf_max m E < injf_max (Suc n) E\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1121, "file": null, "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.900529791457032, "lm_q2_score": 0.8354835452961425, "lm_q1q2_score": 0.752377822811317}}
{"text": "[STATEMENT]\nlemma card_bijections_domain_and_range_permutation:\n  assumes \"finite A\" \"finite B\"\n  shows \"card ({f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B) = iverson (card A = card B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card ({f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B) = iverson (card A = card B)\n[PROOF STEP]\nusing assms card_bijections_domain_and_range_permutation_eq_0 card_bijections_domain_and_range_permutation_eq_1\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nfinite B\ncard ?A \\<noteq> card ?B \\<Longrightarrow> card ({f \\<in> ?A \\<rightarrow>\\<^sub>E ?B. bij_betw f ?A ?B} // domain_and_range_permutation ?A ?B) = 0\n\\<lbrakk>finite ?A; finite ?B; card ?A = card ?B\\<rbrakk> \\<Longrightarrow> card ({f \\<in> ?A \\<rightarrow>\\<^sub>E ?B. bij_betw f ?A ?B} // domain_and_range_permutation ?A ?B) = 1\n\ngoal (1 subgoal):\n 1. card ({f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B) = iverson (card A = card B)\n[PROOF STEP]\nunfolding iverson_def\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\nfinite B\ncard ?A \\<noteq> card ?B \\<Longrightarrow> card ({f \\<in> ?A \\<rightarrow>\\<^sub>E ?B. bij_betw f ?A ?B} // domain_and_range_permutation ?A ?B) = 0\n\\<lbrakk>finite ?A; finite ?B; card ?A = card ?B\\<rbrakk> \\<Longrightarrow> card ({f \\<in> ?A \\<rightarrow>\\<^sub>E ?B. bij_betw f ?A ?B} // domain_and_range_permutation ?A ?B) = 1\n\ngoal (1 subgoal):\n 1. card ({f \\<in> A \\<rightarrow>\\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B) = (if card A = card B then 1 else 0)\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 667, "file": "Twelvefold_Way_Card_Bijections", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9005297807787537, "lm_q2_score": 0.8354835371034368, "lm_q1q2_score": 0.7523778065120157}}
{"text": "[STATEMENT]\nlemma not_icard_Diff_subset: \"\\<exists>(A::nat set) B. B \\<subseteq> A \\<and> \\<not> icard (A - B) = icard A - icard B\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<exists>A B. B \\<subseteq> A \\<and> icard (A - B) \\<noteq> icard A - icard B\n[PROOF STEP]\napply (rule_tac x=\"{0..}\" in exI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<exists>B\\<subseteq>{0..}. icard ({0..} - B) \\<noteq> icard {0..} - icard B\n[PROOF STEP]\napply (rule_tac x=\"{1..}\" in exI)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {1..} \\<subseteq> {0..} \\<and> icard ({0..} - {1..}) \\<noteq> icard {0..} - icard {1..}\n[PROOF STEP]\napply (simp add: set_diff_eq linorder_not_le icard_UNIV_nat eSuc_enat)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 348, "file": "List-Infinite_CommonSet_InfiniteSet2", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9124361604769413, "lm_q2_score": 0.8244619220634456, "lm_q1q2_score": 0.7522688706270095}}
{"text": "[STATEMENT]\nlemma sin_plus_sin: \"sin w + sin z = 2 * sin ((w + z) / 2) * cos ((w - z) / 2)\"\n  for w :: \"'a::{real_normed_field,banach}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin w + sin z = (2::'a) * sin ((w + z) / (2::'a)) * cos ((w - z) / (2::'a))\n[PROOF STEP]\napply (simp add: mult.assoc sin_times_cos)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. sin w * (2::'a) + sin z * (2::'a) = (2::'a) * sin ((w + z) / (2::'a) + (w - z) / (2::'a)) + (2::'a) * sin ((w + z) / (2::'a) - (w - z) / (2::'a))\n[PROOF STEP]\napply (simp add: field_simps)\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 311, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9149009642742806, "lm_q2_score": 0.8221891305219504, "lm_q1q2_score": 0.7522216283303648}}
{"text": "[STATEMENT]\nlemma precedesTransitivity: \n  assumes \n  \"precedes a b l\" and \"precedes b c l\" \n  shows \n  \"precedes a c l\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. precedes a c l\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nprecedes a b l\nprecedes b c l\n\ngoal (1 subgoal):\n 1. precedes a c l\n[PROOF STEP]\nunfolding precedes_def\n[PROOF STATE]\nproof (prove)\nusing this:\na \\<in> set l \\<and> b \\<in> set l \\<and> firstPos a l \\<le> firstPos b l\nb \\<in> set l \\<and> c \\<in> set l \\<and> firstPos b l \\<le> firstPos c l\n\ngoal (1 subgoal):\n 1. a \\<in> set l \\<and> c \\<in> set l \\<and> firstPos a l \\<le> firstPos c l\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 279, "file": "SATSolverVerification_MoreList", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.880797071719777, "lm_q2_score": 0.8539127473751341, "lm_q1q2_score": 0.7521238473922078}}
{"text": "[STATEMENT]\nlemma card_differenceset_commute: \"card (differenceset B A) = card (differenceset A B)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (differenceset B A) = card (differenceset A B)\n[PROOF STEP]\nby (metis card_minusset' differenceset_commute sumset_subset_carrier)", "meta": {"llama_tokens": 110, "file": "Pluennecke_Ruzsa_Inequality_Pluennecke_Ruzsa_Inequality", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9173026618464796, "lm_q2_score": 0.8198933293122506, "lm_q1q2_score": 0.7520903334082997}}
{"text": "[STATEMENT]\nlemma exproj_psi_minus_1_tensor:\n  \"(2 \\<cdot>\\<^sub>m tensor_P proj_psi (1\\<^sub>m K)) - 1\\<^sub>m d = tensor_P (2 \\<cdot>\\<^sub>m proj_psi - (1\\<^sub>m N)) (1\\<^sub>m K)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 \\<cdot>\\<^sub>m tensor_P proj_psi (1\\<^sub>m K) - 1\\<^sub>m d = tensor_P (2 \\<cdot>\\<^sub>m proj_psi - 1\\<^sub>m N) (1\\<^sub>m K)\n[PROOF STEP]\nunfolding ps2_P.ptensor_mat_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 \\<cdot>\\<^sub>m ps_P.tensor_mat proj_psi (1\\<^sub>m K) - 1\\<^sub>m d = ps_P.tensor_mat (2 \\<cdot>\\<^sub>m proj_psi - 1\\<^sub>m N) (1\\<^sub>m K)\n[PROOF STEP]\napply (subst ps_P.tensor_mat_id[symmetric, simplified ps_P_d])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 \\<cdot>\\<^sub>m ps_P.tensor_mat proj_psi (1\\<^sub>m K) - ps_P.tensor_mat (1\\<^sub>m ps_P.d1) (1\\<^sub>m ps_P.d2) = ps_P.tensor_mat (2 \\<cdot>\\<^sub>m proj_psi - 1\\<^sub>m N) (1\\<^sub>m K)\n[PROOF STEP]\napply (auto simp add: ps_P_d1 ps_P_d2)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 \\<cdot>\\<^sub>m ps_P.tensor_mat proj_psi (1\\<^sub>m K) - ps_P.tensor_mat (1\\<^sub>m N) (1\\<^sub>m K) = ps_P.tensor_mat (2 \\<cdot>\\<^sub>m proj_psi - 1\\<^sub>m N) (1\\<^sub>m K)\n[PROOF STEP]\napply (subst ps_P.tensor_mat_scale1[symmetric])\n[PROOF STATE]\nproof (prove)\ngoal (3 subgoals):\n 1. proj_psi \\<in> carrier_mat ps_P.d1 ps_P.d1\n 2. 1\\<^sub>m K \\<in> carrier_mat ps_P.d2 ps_P.d2\n 3. ps_P.tensor_mat (2 \\<cdot>\\<^sub>m proj_psi) (1\\<^sub>m K) - ps_P.tensor_mat (1\\<^sub>m N) (1\\<^sub>m K) = ps_P.tensor_mat (2 \\<cdot>\\<^sub>m proj_psi - 1\\<^sub>m N) (1\\<^sub>m K)\n[PROOF STEP]\napply (auto simp add: ps_P_d1 ps_P_d2 proj_psi_dim)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ps_P.tensor_mat (2 \\<cdot>\\<^sub>m proj_psi) (1\\<^sub>m K) - ps_P.tensor_mat (1\\<^sub>m N) (1\\<^sub>m K) = ps_P.tensor_mat (2 \\<cdot>\\<^sub>m proj_psi - 1\\<^sub>m N) (1\\<^sub>m K)\n[PROOF STEP]\napply (subst ps_P.tensor_mat_minus1)\n[PROOF STATE]\nproof (prove)\ngoal (4 subgoals):\n 1. 2 \\<cdot>\\<^sub>m proj_psi \\<in> carrier_mat ps_P.d1 ps_P.d1\n 2. 1\\<^sub>m N \\<in> carrier_mat ps_P.d1 ps_P.d1\n 3. 1\\<^sub>m K \\<in> carrier_mat ps_P.d2 ps_P.d2\n 4. ps_P.tensor_mat (2 \\<cdot>\\<^sub>m proj_psi) (1\\<^sub>m K) - ps_P.tensor_mat (1\\<^sub>m N) (1\\<^sub>m K) = ps_P.tensor_mat (2 \\<cdot>\\<^sub>m proj_psi) (1\\<^sub>m K) - ps_P.tensor_mat (1\\<^sub>m N) (1\\<^sub>m K)\n[PROOF STEP]\nby (auto simp add: ps_P_d1 ps_P_d2 proj_psi_dim)", "meta": {"llama_tokens": 1210, "file": "QHLProver_Grover", "length": 7, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9173026573249611, "lm_q2_score": 0.8198933315126792, "lm_q1q2_score": 0.7520903317195958}}
{"text": "[STATEMENT]\ntheorem card_partition_on:\n  assumes \"finite A\"\n  shows \"card {P. partition_on A P \\<and> card P = k} = Stirling (card A) k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {P. partition_on A P \\<and> card P = k} = Stirling (card A) k\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\n\ngoal (1 subgoal):\n 1. card {P. partition_on A P \\<and> card P = k} = Stirling (card A) k\n[PROOF STEP]\nproof (induct A arbitrary: k)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\<And>k. card {P. partition_on {} P \\<and> card P = k} = Stirling (card {}) k\n 2. \\<And>x F k. \\<lbrakk>finite F; x \\<notin> F; \\<And>k. card {P. partition_on F P \\<and> card P = k} = Stirling (card F) k\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert x F) P \\<and> card P = k} = Stirling (card (insert x F)) k\n[PROOF STEP]\ncase empty\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. \\<And>k. card {P. partition_on {} P \\<and> card P = k} = Stirling (card {}) k\n 2. \\<And>x F k. \\<lbrakk>finite F; x \\<notin> F; \\<And>k. card {P. partition_on F P \\<and> card P = k} = Stirling (card F) k\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert x F) P \\<and> card P = k} = Stirling (card (insert x F)) k\n[PROOF STEP]\nhave eq: \"{P. P = {} \\<and> card P = 0} = {{}}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {P. P = {} \\<and> card P = 0} = {{}}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n{P. P = {} \\<and> card P = 0} = {{}}\n\ngoal (2 subgoals):\n 1. \\<And>k. card {P. partition_on {} P \\<and> card P = k} = Stirling (card {}) k\n 2. \\<And>x F k. \\<lbrakk>finite F; x \\<notin> F; \\<And>k. card {P. partition_on F P \\<and> card P = k} = Stirling (card F) k\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert x F) P \\<and> card P = k} = Stirling (card (insert x F)) k\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {P. partition_on {} P \\<and> card P = k} = Stirling (card {}) k\n[PROOF STEP]\nby (cases k) (auto simp add: partition_on_empty eq)\n[PROOF STATE]\nproof (state)\nthis:\ncard {P. partition_on {} P \\<and> card P = k} = Stirling (card {}) k\n\ngoal (1 subgoal):\n 1. \\<And>x F k. \\<lbrakk>finite F; x \\<notin> F; \\<And>k. card {P. partition_on F P \\<and> card P = k} = Stirling (card F) k\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert x F) P \\<and> card P = k} = Stirling (card (insert x F)) k\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>x F k. \\<lbrakk>finite F; x \\<notin> F; \\<And>k. card {P. partition_on F P \\<and> card P = k} = Stirling (card F) k\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert x F) P \\<and> card P = k} = Stirling (card (insert x F)) k\n[PROOF STEP]\ncase (insert a A)\n[PROOF STATE]\nproof (state)\nthis:\nfinite A\na \\<notin> A\ncard {P. partition_on A P \\<and> card P = ?k} = Stirling (card A) ?k\n\ngoal (1 subgoal):\n 1. \\<And>x F k. \\<lbrakk>finite F; x \\<notin> F; \\<And>k. card {P. partition_on F P \\<and> card P = k} = Stirling (card F) k\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert x F) P \\<and> card P = k} = Stirling (card (insert x F)) k\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\na \\<notin> A\ncard {P. partition_on A P \\<and> card P = ?k} = Stirling (card A) ?k\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\na \\<notin> A\ncard {P. partition_on A P \\<and> card P = ?k} = Stirling (card A) ?k\n\ngoal (1 subgoal):\n 1. card {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n[PROOF STEP]\nproof (cases k)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\<lbrakk>finite A; a \\<notin> A; \\<And>k. card {P. partition_on A P \\<and> card P = k} = Stirling (card A) k; k = 0\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n 2. \\<And>nat. \\<lbrakk>finite A; a \\<notin> A; \\<And>k. card {P. partition_on A P \\<and> card P = k} = Stirling (card A) k; k = Suc nat\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\nk = 0\n\ngoal (2 subgoals):\n 1. \\<lbrakk>finite A; a \\<notin> A; \\<And>k. card {P. partition_on A P \\<and> card P = k} = Stirling (card A) k; k = 0\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n 2. \\<And>nat. \\<lbrakk>finite A; a \\<notin> A; \\<And>k. card {P. partition_on A P \\<and> card P = k} = Stirling (card A) k; k = Suc nat\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n[PROOF STEP]\nfrom insert(1)\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\n[PROOF STEP]\nhave empty: \"{P. partition_on (insert a A) P \\<and> card P = 0} = {}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\n\ngoal (1 subgoal):\n 1. {P. partition_on (insert a A) P \\<and> card P = 0} = {}\n[PROOF STEP]\nunfolding partition_on_def\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\n\ngoal (1 subgoal):\n 1. {P. (\\<Union> P = insert a A \\<and> disjoint P \\<and> {} \\<notin> P) \\<and> card P = 0} = {}\n[PROOF STEP]\nby (auto simp add: card_eq_0_iff finite_UnionD)\n[PROOF STATE]\nproof (state)\nthis:\n{P. partition_on (insert a A) P \\<and> card P = 0} = {}\n\ngoal (2 subgoals):\n 1. \\<lbrakk>finite A; a \\<notin> A; \\<And>k. card {P. partition_on A P \\<and> card P = k} = Stirling (card A) k; k = 0\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n 2. \\<And>nat. \\<lbrakk>finite A; a \\<notin> A; \\<And>k. card {P. partition_on A P \\<and> card P = k} = Stirling (card A) k; k = Suc nat\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n[PROOF STEP]\nfrom 0 insert\n[PROOF STATE]\nproof (chain)\npicking this:\nk = 0\nfinite A\na \\<notin> A\ncard {P. partition_on A P \\<and> card P = ?k} = Stirling (card A) ?k\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nk = 0\nfinite A\na \\<notin> A\ncard {P. partition_on A P \\<and> card P = ?k} = Stirling (card A) ?k\n\ngoal (1 subgoal):\n 1. card {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n[PROOF STEP]\nby (auto simp add: empty)\n[PROOF STATE]\nproof (state)\nthis:\ncard {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n\ngoal (1 subgoal):\n 1. \\<And>nat. \\<lbrakk>finite A; a \\<notin> A; \\<And>k. card {P. partition_on A P \\<and> card P = k} = Stirling (card A) k; k = Suc nat\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>nat. \\<lbrakk>finite A; a \\<notin> A; \\<And>k. card {P. partition_on A P \\<and> card P = k} = Stirling (card A) k; k = Suc nat\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n[PROOF STEP]\ncase (Suc k')\n[PROOF STATE]\nproof (state)\nthis:\nk = Suc k'\n\ngoal (1 subgoal):\n 1. \\<And>nat. \\<lbrakk>finite A; a \\<notin> A; \\<And>k. card {P. partition_on A P \\<and> card P = k} = Stirling (card A) k; k = Suc nat\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n[PROOF STEP]\nlet ?subexpr1 = \"do {\n      P <- {P. partition_on A P \\<and> card P = Suc k'};\n      p <- P;\n      {insert (insert a p) (P - {p})}\n    }\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>nat. \\<lbrakk>finite A; a \\<notin> A; \\<And>k. card {P. partition_on A P \\<and> card P = k} = Stirling (card A) k; k = Suc nat\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n[PROOF STEP]\nlet ?subexpr2 = \"do {\n      P <- {P. partition_on A P \\<and> card P = k'};\n      {insert {a} P}\n    }\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>nat. \\<lbrakk>finite A; a \\<notin> A; \\<And>k. card {P. partition_on A P \\<and> card P = k} = Stirling (card A) k; k = Suc nat\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n[PROOF STEP]\nlet ?expr = \"?subexpr1 \\<union> ?subexpr2\"\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>nat. \\<lbrakk>finite A; a \\<notin> A; \\<And>k. card {P. partition_on A P \\<and> card P = k} = Stirling (card A) k; k = Suc nat\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n[PROOF STEP]\nhave \"card {P. partition_on (insert a A) P \\<and> card P = k} = card ?expr\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {P. partition_on (insert a A) P \\<and> card P = k} = card (({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<union> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})))\n[PROOF STEP]\nusing \\<open>finite A\\<close> \\<open>a \\<notin> A\\<close> \\<open>k = Suc k'\\<close>\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\na \\<notin> A\nk = Suc k'\n\ngoal (1 subgoal):\n 1. card {P. partition_on (insert a A) P \\<and> card P = k} = card (({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<union> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})))\n[PROOF STEP]\nby (simp add: set_partition_on_insert_with_fixed_card_eq)\n[PROOF STATE]\nproof (state)\nthis:\ncard {P. partition_on (insert a A) P \\<and> card P = k} = card (({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<union> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})))\n\ngoal (1 subgoal):\n 1. \\<And>nat. \\<lbrakk>finite A; a \\<notin> A; \\<And>k. card {P. partition_on A P \\<and> card P = k} = Stirling (card A) k; k = Suc nat\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard {P. partition_on (insert a A) P \\<and> card P = k} = card (({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<union> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})))\n\ngoal (1 subgoal):\n 1. \\<And>nat. \\<lbrakk>finite A; a \\<notin> A; \\<And>k. card {P. partition_on A P \\<and> card P = k} = Stirling (card A) k; k = Suc nat\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n[PROOF STEP]\nhave \"card ?expr = Stirling (card A) k' + Stirling (card A) (Suc k') * Suc k'\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<union> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P}))) = Stirling (card A) k' + Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<union> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P}))) = Stirling (card A) k' + Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nhave \"finite ?subexpr1 \\<and> card ?subexpr1 = Stirling (card A) (Suc k') * Suc k'\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<and> card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<and> card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nfrom \\<open>finite A\\<close>\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\n[PROOF STEP]\nhave \"finite {P. partition_on A P \\<and> card P = Suc k'}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\n\ngoal (1 subgoal):\n 1. finite {P. partition_on A P \\<and> card P = Suc k'}\n[PROOF STEP]\nby (simp add: finitely_many_partition_on)\n[PROOF STATE]\nproof (state)\nthis:\nfinite {P. partition_on A P \\<and> card P = Suc k'}\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<and> card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nfinite {P. partition_on A P \\<and> card P = Suc k'}\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<and> card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nhave \"\\<forall>X\\<in>{P. partition_on A P \\<and> card P = Suc k'}. finite (X \\<bind> (\\<lambda>p. {insert (insert a p) (X - {p})}))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<forall>X\\<in>{P. partition_on A P \\<and> card P = Suc k'}. finite (X \\<bind> (\\<lambda>p. {insert (insert a p) (X - {p})}))\n[PROOF STEP]\nusing finite_elements \\<open>finite A\\<close> finite_bind\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>finite ?A; partition_on ?A ?P\\<rbrakk> \\<Longrightarrow> finite ?P\nfinite A\n\\<lbrakk>finite ?S; \\<forall>x\\<in>?S. finite (?f x)\\<rbrakk> \\<Longrightarrow> finite (?S \\<bind> ?f)\n\ngoal (1 subgoal):\n 1. \\<forall>X\\<in>{P. partition_on A P \\<and> card P = Suc k'}. finite (X \\<bind> (\\<lambda>p. {insert (insert a p) (X - {p})}))\n[PROOF STEP]\nby (metis (no_types, lifting) finite.emptyI finite_insert mem_Collect_eq)\n[PROOF STATE]\nproof (state)\nthis:\n\\<forall>X\\<in>{P. partition_on A P \\<and> card P = Suc k'}. finite (X \\<bind> (\\<lambda>p. {insert (insert a p) (X - {p})}))\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<and> card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\<forall>X\\<in>{P. partition_on A P \\<and> card P = Suc k'}. finite (X \\<bind> (\\<lambda>p. {insert (insert a p) (X - {p})}))\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<and> card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nhave \"disjoint_family_on (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})) {P. partition_on A P \\<and> card P = Suc k'}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. disjoint_family_on (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})) {P. partition_on A P \\<and> card P = Suc k'}\n[PROOF STEP]\nby (injectivity_solver rule: injectivity_subexpr1(1)[OF \\<open>a \\<notin> A\\<close>])\n[PROOF STATE]\nproof (state)\nthis:\ndisjoint_family_on (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})) {P. partition_on A P \\<and> card P = Suc k'}\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<and> card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ndisjoint_family_on (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})) {P. partition_on A P \\<and> card P = Suc k'}\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<and> card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nhave \"card (P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})) = Suc k'\"\n          if \"P \\<in> {P. partition_on A P \\<and> card P = Suc k'}\" for P\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})) = Suc k'\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})) = Suc k'\n[PROOF STEP]\nfrom that \\<open>finite A\\<close>\n[PROOF STATE]\nproof (chain)\npicking this:\nP \\<in> {P. partition_on A P \\<and> card P = Suc k'}\nfinite A\n[PROOF STEP]\nhave \"finite P\"\n[PROOF STATE]\nproof (prove)\nusing this:\nP \\<in> {P. partition_on A P \\<and> card P = Suc k'}\nfinite A\n\ngoal (1 subgoal):\n 1. finite P\n[PROOF STEP]\nusing finite_elements\n[PROOF STATE]\nproof (prove)\nusing this:\nP \\<in> {P. partition_on A P \\<and> card P = Suc k'}\nfinite A\n\\<lbrakk>finite ?A; partition_on ?A ?P\\<rbrakk> \\<Longrightarrow> finite ?P\n\ngoal (1 subgoal):\n 1. finite P\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nfinite P\n\ngoal (1 subgoal):\n 1. card (P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})) = Suc k'\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nfinite P\n\ngoal (1 subgoal):\n 1. card (P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})) = Suc k'\n[PROOF STEP]\nhave \"inj_on (\\<lambda>p. insert (insert a p) (P - {p})) P\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inj_on (\\<lambda>p. insert (insert a p) (P - {p})) P\n[PROOF STEP]\nusing that injectivity_subexpr1(2)[OF \\<open>a \\<notin> A\\<close>]\n[PROOF STATE]\nproof (prove)\nusing this:\nP \\<in> {P. partition_on A P \\<and> card P = Suc k'}\n\\<lbrakk>?X \\<in> ?P \\<and> ?X' \\<in> ?P'; insert (insert a ?X) (?P - {?X}) = insert (insert a ?X') (?P' - {?X'}); (partition_on A ?P \\<and> card ?P = Suc ?k') \\<and> partition_on A ?P' \\<and> card ?P' = Suc ?k'\\<rbrakk> \\<Longrightarrow> ?X = ?X'\n\ngoal (1 subgoal):\n 1. inj_on (\\<lambda>p. insert (insert a p) (P - {p})) P\n[PROOF STEP]\nby (simp add: inj_onI)\n[PROOF STATE]\nproof (state)\nthis:\ninj_on (\\<lambda>p. insert (insert a p) (P - {p})) P\n\ngoal (1 subgoal):\n 1. card (P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})) = Suc k'\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ninj_on (\\<lambda>p. insert (insert a p) (P - {p})) P\n\ngoal (1 subgoal):\n 1. card (P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})) = Suc k'\n[PROOF STEP]\nfrom that\n[PROOF STATE]\nproof (chain)\npicking this:\nP \\<in> {P. partition_on A P \\<and> card P = Suc k'}\n[PROOF STEP]\nhave \"card P = Suc k'\"\n[PROOF STATE]\nproof (prove)\nusing this:\nP \\<in> {P. partition_on A P \\<and> card P = Suc k'}\n\ngoal (1 subgoal):\n 1. card P = Suc k'\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard P = Suc k'\n\ngoal (1 subgoal):\n 1. card (P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})) = Suc k'\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite P\ninj_on (\\<lambda>p. insert (insert a p) (P - {p})) P\ncard P = Suc k'\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite P\ninj_on (\\<lambda>p. insert (insert a p) (P - {p})) P\ncard P = Suc k'\n\ngoal (1 subgoal):\n 1. card (P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})) = Suc k'\n[PROOF STEP]\nby (simp add: card_bind_singleton)\n[PROOF STATE]\nproof (state)\nthis:\ncard (P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})) = Suc k'\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n?P \\<in> {P. partition_on A P \\<and> card P = Suc k'} \\<Longrightarrow> card (?P \\<bind> (\\<lambda>p. {insert (insert a p) (?P - {p})})) = Suc k'\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<and> card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite {P. partition_on A P \\<and> card P = Suc k'}\n\\<forall>X\\<in>{P. partition_on A P \\<and> card P = Suc k'}. finite (X \\<bind> (\\<lambda>p. {insert (insert a p) (X - {p})}))\ndisjoint_family_on (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})) {P. partition_on A P \\<and> card P = Suc k'}\n?P \\<in> {P. partition_on A P \\<and> card P = Suc k'} \\<Longrightarrow> card (?P \\<bind> (\\<lambda>p. {insert (insert a p) (?P - {p})})) = Suc k'\n[PROOF STEP]\nhave \"card ?subexpr1 = card {P. partition_on A P \\<and> card P = Suc k'} * Suc k'\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite {P. partition_on A P \\<and> card P = Suc k'}\n\\<forall>X\\<in>{P. partition_on A P \\<and> card P = Suc k'}. finite (X \\<bind> (\\<lambda>p. {insert (insert a p) (X - {p})}))\ndisjoint_family_on (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})) {P. partition_on A P \\<and> card P = Suc k'}\n?P \\<in> {P. partition_on A P \\<and> card P = Suc k'} \\<Longrightarrow> card (?P \\<bind> (\\<lambda>p. {insert (insert a p) (?P - {p})})) = Suc k'\n\ngoal (1 subgoal):\n 1. card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = card {P. partition_on A P \\<and> card P = Suc k'} * Suc k'\n[PROOF STEP]\nby (subst card_bind_constant) simp+\n[PROOF STATE]\nproof (state)\nthis:\ncard ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = card {P. partition_on A P \\<and> card P = Suc k'} * Suc k'\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<and> card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nfrom this\n[PROOF STATE]\nproof (chain)\npicking this:\ncard ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = card {P. partition_on A P \\<and> card P = Suc k'} * Suc k'\n[PROOF STEP]\nhave \"card ?subexpr1 = Stirling (card A) (Suc k') * Suc k'\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = card {P. partition_on A P \\<and> card P = Suc k'} * Suc k'\n\ngoal (1 subgoal):\n 1. card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nusing insert.hyps(3)\n[PROOF STATE]\nproof (prove)\nusing this:\ncard ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = card {P. partition_on A P \\<and> card P = Suc k'} * Suc k'\ncard {P. partition_on A P \\<and> card P = ?k} = Stirling (card A) ?k\n\ngoal (1 subgoal):\n 1. card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncard ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<and> card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ncard ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<and> card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nhave \"finite ?subexpr1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})))\n[PROOF STEP]\nusing \\<open>finite {P. partition_on A P \\<and> card P = Suc k'}\\<close>\n          \\<open>\\<forall>X\\<in>{P. partition_on A P \\<and> card P = Suc k'}. finite (X \\<bind> (\\<lambda>p. {insert (insert a p) (X - {p})}))\\<close>\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite {P. partition_on A P \\<and> card P = Suc k'}\n\\<forall>X\\<in>{P. partition_on A P \\<and> card P = Suc k'}. finite (X \\<bind> (\\<lambda>p. {insert (insert a p) (X - {p})}))\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})))\n[PROOF STEP]\nby (auto intro: finite_bind)\n[PROOF STATE]\nproof (state)\nthis:\nfinite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})))\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<and> card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ncard ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\nfinite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\nfinite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})))\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<and> card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nfinite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<and> card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nfinite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<and> card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\n\ngoal (1 subgoal):\n 1. card (({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<union> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P}))) = Stirling (card A) k' + Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nfinite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<and> card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\n\ngoal (1 subgoal):\n 1. card (({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<union> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P}))) = Stirling (card A) k' + Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nhave \"finite ?subexpr2 \\<and> card ?subexpr2 = Stirling (card A) k'\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) \\<and> card ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) \\<and> card ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n[PROOF STEP]\nfrom \\<open>finite A\\<close>\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite A\n[PROOF STEP]\nhave \"finite {P. partition_on A P \\<and> card P = k'}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\n\ngoal (1 subgoal):\n 1. finite {P. partition_on A P \\<and> card P = k'}\n[PROOF STEP]\nby (simp add: finitely_many_partition_on)\n[PROOF STATE]\nproof (state)\nthis:\nfinite {P. partition_on A P \\<and> card P = k'}\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) \\<and> card ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nfinite {P. partition_on A P \\<and> card P = k'}\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) \\<and> card ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n[PROOF STEP]\nhave \" inj_on (insert {a}) {P. partition_on A P \\<and> card P = k'}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. inj_on (insert {a}) {P. partition_on A P \\<and> card P = k'}\n[PROOF STEP]\nusing injectivity_subexpr2[OF \\<open>a \\<notin> A\\<close>]\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>insert {a} ?P = insert {a} ?P'; (partition_on A ?P \\<and> card ?P = ?k') \\<and> partition_on A ?P' \\<and> card ?P' = ?k'\\<rbrakk> \\<Longrightarrow> ?P = ?P'\n\ngoal (1 subgoal):\n 1. inj_on (insert {a}) {P. partition_on A P \\<and> card P = k'}\n[PROOF STEP]\nby (simp add: inj_on_def)\n[PROOF STATE]\nproof (state)\nthis:\ninj_on (insert {a}) {P. partition_on A P \\<and> card P = k'}\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) \\<and> card ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite {P. partition_on A P \\<and> card P = k'}\ninj_on (insert {a}) {P. partition_on A P \\<and> card P = k'}\n[PROOF STEP]\nhave \"card ?subexpr2 = card {P. partition_on A P \\<and> card P = k'}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite {P. partition_on A P \\<and> card P = k'}\ninj_on (insert {a}) {P. partition_on A P \\<and> card P = k'}\n\ngoal (1 subgoal):\n 1. card ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = card {P. partition_on A P \\<and> card P = k'}\n[PROOF STEP]\nby (simp add: card_bind_singleton)\n[PROOF STATE]\nproof (state)\nthis:\ncard ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = card {P. partition_on A P \\<and> card P = k'}\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) \\<and> card ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = card {P. partition_on A P \\<and> card P = k'}\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) \\<and> card ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n[PROOF STEP]\nhave \"\\<dots> = Stirling (card A) k'\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card {P. partition_on A P \\<and> card P = k'} = Stirling (card A) k'\n[PROOF STEP]\nusing insert.hyps(3)\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {P. partition_on A P \\<and> card P = ?k} = Stirling (card A) ?k\n\ngoal (1 subgoal):\n 1. card {P. partition_on A P \\<and> card P = k'} = Stirling (card A) k'\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard {P. partition_on A P \\<and> card P = k'} = Stirling (card A) k'\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) \\<and> card ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n[PROOF STEP]\nhave \"card ?subexpr2 = Stirling (card A) k'\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n\ngoal (1 subgoal):\n 1. card ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) \\<and> card ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\ncard ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) \\<and> card ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n[PROOF STEP]\nhave \"finite ?subexpr2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P}))\n[PROOF STEP]\nby (simp add: \\<open>finite {P. partition_on A P \\<and> card P = k'}\\<close> finite_bind)\n[PROOF STATE]\nproof (state)\nthis:\nfinite ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P}))\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) \\<and> card ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\ncard ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\nfinite ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P}))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\nfinite ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P}))\n\ngoal (1 subgoal):\n 1. finite ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) \\<and> card ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\nfinite ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) \\<and> card ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\nfinite ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) \\<and> card ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n\ngoal (1 subgoal):\n 1. card (({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<union> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P}))) = Stirling (card A) k' + Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\nfinite ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) \\<and> card ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n\ngoal (1 subgoal):\n 1. card (({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<union> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P}))) = Stirling (card A) k' + Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nhave \"?subexpr1 \\<inter> ?subexpr2 = {}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<inter> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = {}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<inter> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = {}\n[PROOF STEP]\nhave \"\\<forall>P\\<in>?subexpr1. {a} \\<notin> P\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<forall>P\\<in>{P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})). {a} \\<notin> P\n[PROOF STEP]\nusing insert.hyps(2)\n[PROOF STATE]\nproof (prove)\nusing this:\na \\<notin> A\n\ngoal (1 subgoal):\n 1. \\<forall>P\\<in>{P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})). {a} \\<notin> P\n[PROOF STEP]\nby (force elim!: partition_onE)\n[PROOF STATE]\nproof (state)\nthis:\n\\<forall>P\\<in>{P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})). {a} \\<notin> P\n\ngoal (1 subgoal):\n 1. ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<inter> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = {}\n[PROOF STEP]\nmoreover\n[PROOF STATE]\nproof (state)\nthis:\n\\<forall>P\\<in>{P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})). {a} \\<notin> P\n\ngoal (1 subgoal):\n 1. ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<inter> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = {}\n[PROOF STEP]\nhave \"\\<forall>P\\<in>?subexpr2. {a} \\<in> P\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<forall>P\\<in>{P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P}). {a} \\<in> P\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\<forall>P\\<in>{P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P}). {a} \\<in> P\n\ngoal (1 subgoal):\n 1. ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<inter> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = {}\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\n\\<forall>P\\<in>{P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})). {a} \\<notin> P\n\\<forall>P\\<in>{P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P}). {a} \\<in> P\n[PROOF STEP]\nshow \"?subexpr1 \\<inter> ?subexpr2 = {}\"\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<forall>P\\<in>{P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})})). {a} \\<notin> P\n\\<forall>P\\<in>{P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P}). {a} \\<in> P\n\ngoal (1 subgoal):\n 1. ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<inter> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = {}\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\n({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<inter> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = {}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\n({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<inter> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = {}\n\ngoal (1 subgoal):\n 1. card (({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<union> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P}))) = Stirling (card A) k' + Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nultimately\n[PROOF STATE]\nproof (chain)\npicking this:\nfinite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<and> card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\nfinite ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) \\<and> card ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<inter> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = {}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<and> card ({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) = Stirling (card A) (Suc k') * Suc k'\nfinite ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) \\<and> card ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = Stirling (card A) k'\n({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<inter> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P})) = {}\n\ngoal (1 subgoal):\n 1. card (({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<union> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P}))) = Stirling (card A) k' + Stirling (card A) (Suc k') * Suc k'\n[PROOF STEP]\nby (simp add: card_Un_disjoint)\n[PROOF STATE]\nproof (state)\nthis:\ncard (({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<union> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P}))) = Stirling (card A) k' + Stirling (card A) (Suc k') * Suc k'\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ncard (({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<union> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P}))) = Stirling (card A) k' + Stirling (card A) (Suc k') * Suc k'\n\ngoal (1 subgoal):\n 1. \\<And>nat. \\<lbrakk>finite A; a \\<notin> A; \\<And>k. card {P. partition_on A P \\<and> card P = k} = Stirling (card A) k; k = Suc nat\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\ncard (({P. partition_on A P \\<and> card P = Suc k'} \\<bind> (\\<lambda>P. P \\<bind> (\\<lambda>p. {insert (insert a p) (P - {p})}))) \\<union> ({P. partition_on A P \\<and> card P = k'} \\<bind> (\\<lambda>P. {insert {a} P}))) = Stirling (card A) k' + Stirling (card A) (Suc k') * Suc k'\n\ngoal (1 subgoal):\n 1. \\<And>nat. \\<lbrakk>finite A; a \\<notin> A; \\<And>k. card {P. partition_on A P \\<and> card P = k} = Stirling (card A) k; k = Suc nat\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n[PROOF STEP]\nhave \"\\<dots> = Stirling (card (insert a A)) k\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. Stirling (card A) k' + Stirling (card A) (Suc k') * Suc k' = Stirling (card (insert a A)) k\n[PROOF STEP]\nusing insert(1, 2) \\<open>k = Suc k'\\<close>\n[PROOF STATE]\nproof (prove)\nusing this:\nfinite A\na \\<notin> A\nk = Suc k'\n\ngoal (1 subgoal):\n 1. Stirling (card A) k' + Stirling (card A) (Suc k') * Suc k' = Stirling (card (insert a A)) k\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nStirling (card A) k' + Stirling (card A) (Suc k') * Suc k' = Stirling (card (insert a A)) k\n\ngoal (1 subgoal):\n 1. \\<And>nat. \\<lbrakk>finite A; a \\<notin> A; \\<And>k. card {P. partition_on A P \\<and> card P = k} = Stirling (card A) k; k = Suc nat\\<rbrakk> \\<Longrightarrow> card {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\ncard {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n\ngoal (1 subgoal):\n 1. card {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\ncard {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed\n[PROOF STATE]\nproof (state)\nthis:\ncard {P. partition_on (insert a A) P \\<and> card P = k} = Stirling (card (insert a A)) k\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 19756, "file": "Card_Partitions_Card_Partitions", "length": 130, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.893309411735131, "lm_q2_score": 0.8418256492357358, "lm_q1q2_score": 0.7520107755023199}}
{"text": "[STATEMENT]\nlemma scalar_product_linear_right:\n  \"scalar_product a (b+c) = \n   scalar_product a b + scalar_product a (c :: ('a qr, 'k) vec)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. scalar_product a (b + c) = scalar_product a b + scalar_product a c\n[PROOF STEP]\nunfolding scalar_product_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>i\\<in>UNIV. a $ i * (b + c) $ i) = (\\<Sum>i\\<in>UNIV. a $ i * b $ i) + (\\<Sum>i\\<in>UNIV. a $ i * c $ i)\n[PROOF STEP]\nby auto (metis (no_types, lifting) distrib_left sum.cong sum.distrib)", "meta": {"llama_tokens": 233, "file": "CRYSTALS-Kyber_Crypto_Scheme", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933093946927837, "lm_q2_score": 0.8418256452674009, "lm_q1q2_score": 0.7520107576106839}}
{"text": "[STATEMENT]\nlemma drop_map: \"drop n (map f xs) = map f (drop n xs)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. drop n (map f xs) = map f (drop n xs)\n[PROOF STEP]\nproof (induct n arbitrary: xs)\n[PROOF STATE]\nproof (state)\ngoal (2 subgoals):\n 1. \\<And>xs. drop 0 (map f xs) = map f (drop 0 xs)\n 2. \\<And>n xs. (\\<And>xs. drop n (map f xs) = map f (drop n xs)) \\<Longrightarrow> drop (Suc n) (map f xs) = map f (drop (Suc n) xs)\n[PROOF STEP]\ncase 0\n[PROOF STATE]\nproof (state)\nthis:\n\n\ngoal (2 subgoals):\n 1. \\<And>xs. drop 0 (map f xs) = map f (drop 0 xs)\n 2. \\<And>n xs. (\\<And>xs. drop n (map f xs) = map f (drop n xs)) \\<Longrightarrow> drop (Suc n) (map f xs) = map f (drop (Suc n) xs)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. drop 0 (map f xs) = map f (drop 0 xs)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ndrop 0 (map f xs) = map f (drop 0 xs)\n\ngoal (1 subgoal):\n 1. \\<And>n xs. (\\<And>xs. drop n (map f xs) = map f (drop n xs)) \\<Longrightarrow> drop (Suc n) (map f xs) = map f (drop (Suc n) xs)\n[PROOF STEP]\nnext\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<And>n xs. (\\<And>xs. drop n (map f xs) = map f (drop n xs)) \\<Longrightarrow> drop (Suc n) (map f xs) = map f (drop (Suc n) xs)\n[PROOF STEP]\ncase Suc\n[PROOF STATE]\nproof (state)\nthis:\ndrop n_ (map f ?xs) = map f (drop n_ ?xs)\n\ngoal (1 subgoal):\n 1. \\<And>n xs. (\\<And>xs. drop n (map f xs) = map f (drop n xs)) \\<Longrightarrow> drop (Suc n) (map f xs) = map f (drop (Suc n) xs)\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ndrop n_ (map f ?xs) = map f (drop n_ ?xs)\n[PROOF STEP]\nshow ?case\n[PROOF STATE]\nproof (prove)\nusing this:\ndrop n_ (map f ?xs) = map f (drop n_ ?xs)\n\ngoal (1 subgoal):\n 1. drop (Suc n_) (map f xs) = map f (drop (Suc n_) xs)\n[PROOF STEP]\nby (cases xs) simp_all\n[PROOF STATE]\nproof (state)\nthis:\ndrop (Suc n_) (map f xs) = map f (drop (Suc n_) xs)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 896, "file": null, "length": 11, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.867035763237924, "lm_q2_score": 0.8670357546485407, "lm_q1q2_score": 0.7517510072862669}}
{"text": "[STATEMENT]\nlemma double_ccos_square:\n  \"2 * ccos (a::real) * ccos a = ccos (2 * a) + 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 2 * complex_of_real (cos a) * complex_of_real (cos a) = complex_of_real (cos (2 * a)) + 1\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 2 * complex_of_real (cos a) * complex_of_real (cos a) = complex_of_real (cos (2 * a)) + 1\n[PROOF STEP]\nhave eq: \"ccos (2 * a) = ccos a * ccos a - csin a * csin a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. complex_of_real (cos (2 * a)) = complex_of_real (cos a) * complex_of_real (cos a) - complex_of_real (sin a) * complex_of_real (sin a)\n[PROOF STEP]\nusing cos_add[of a a]\n[PROOF STATE]\nproof (prove)\nusing this:\ncos (a + a) = cos a * cos a - sin a * sin a\n\ngoal (1 subgoal):\n 1. complex_of_real (cos (2 * a)) = complex_of_real (cos a) * complex_of_real (cos a) - complex_of_real (sin a) * complex_of_real (sin a)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncomplex_of_real (cos (2 * a)) = complex_of_real (cos a) * complex_of_real (cos a) - complex_of_real (sin a) * complex_of_real (sin a)\n\ngoal (1 subgoal):\n 1. 2 * complex_of_real (cos a) * complex_of_real (cos a) = complex_of_real (cos (2 * a)) + 1\n[PROOF STEP]\nhave \"csin a * csin a = 1 - ccos a * ccos a\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. complex_of_real (sin a) * complex_of_real (sin a) = 1 - complex_of_real (cos a) * complex_of_real (cos a)\n[PROOF STEP]\nusing csin_ccos_squared_add[of a]\n[PROOF STATE]\nproof (prove)\nusing this:\ncomplex_of_real (cos a) * complex_of_real (cos a) + complex_of_real (sin a) * complex_of_real (sin a) = 1\n\ngoal (1 subgoal):\n 1. complex_of_real (sin a) * complex_of_real (sin a) = 1 - complex_of_real (cos a) * complex_of_real (cos a)\n[PROOF STEP]\nby (metis add_diff_cancel_left')\n[PROOF STATE]\nproof (state)\nthis:\ncomplex_of_real (sin a) * complex_of_real (sin a) = 1 - complex_of_real (cos a) * complex_of_real (cos a)\n\ngoal (1 subgoal):\n 1. 2 * complex_of_real (cos a) * complex_of_real (cos a) = complex_of_real (cos (2 * a)) + 1\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\ncomplex_of_real (sin a) * complex_of_real (sin a) = 1 - complex_of_real (cos a) * complex_of_real (cos a)\n[PROOF STEP]\nhave \"ccos a * ccos a - csin a * csin a = 2 * ccos a * ccos a - 1\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncomplex_of_real (sin a) * complex_of_real (sin a) = 1 - complex_of_real (cos a) * complex_of_real (cos a)\n\ngoal (1 subgoal):\n 1. complex_of_real (cos a) * complex_of_real (cos a) - complex_of_real (sin a) * complex_of_real (sin a) = 2 * complex_of_real (cos a) * complex_of_real (cos a) - 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\ncomplex_of_real (cos a) * complex_of_real (cos a) - complex_of_real (sin a) * complex_of_real (sin a) = 2 * complex_of_real (cos a) * complex_of_real (cos a) - 1\n\ngoal (1 subgoal):\n 1. 2 * complex_of_real (cos a) * complex_of_real (cos a) = complex_of_real (cos (2 * a)) + 1\n[PROOF STEP]\nwith eq\n[PROOF STATE]\nproof (chain)\npicking this:\ncomplex_of_real (cos (2 * a)) = complex_of_real (cos a) * complex_of_real (cos a) - complex_of_real (sin a) * complex_of_real (sin a)\ncomplex_of_real (cos a) * complex_of_real (cos a) - complex_of_real (sin a) * complex_of_real (sin a) = 2 * complex_of_real (cos a) * complex_of_real (cos a) - 1\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncomplex_of_real (cos (2 * a)) = complex_of_real (cos a) * complex_of_real (cos a) - complex_of_real (sin a) * complex_of_real (sin a)\ncomplex_of_real (cos a) * complex_of_real (cos a) - complex_of_real (sin a) * complex_of_real (sin a) = 2 * complex_of_real (cos a) * complex_of_real (cos a) - 1\n\ngoal (1 subgoal):\n 1. 2 * complex_of_real (cos a) * complex_of_real (cos a) = complex_of_real (cos (2 * a)) + 1\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n2 * complex_of_real (cos a) * complex_of_real (cos a) = complex_of_real (cos (2 * a)) + 1\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1718, "file": "QHLProver_Grover", "length": 14, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972616934406, "lm_q2_score": 0.8633916187614823, "lm_q1q2_score": 0.7516663790628135}}
{"text": "[STATEMENT]\nlemma choose_reduce_nat:\n  \"0 < n \\<Longrightarrow> 0 < k \\<Longrightarrow>\n    n choose k = ((n - 1) choose (k - 1)) + ((n - 1) choose k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>0 < n; 0 < k\\<rbrakk> \\<Longrightarrow> n choose k = n - 1 choose (k - 1) + (n - 1 choose k)\n[PROOF STEP]\nusing binomial_Suc_Suc [of \"n - 1\" \"k - 1\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nSuc (n - 1) choose Suc (k - 1) = n - 1 choose (k - 1) + (n - 1 choose Suc (k - 1))\n\ngoal (1 subgoal):\n 1. \\<lbrakk>0 < n; 0 < k\\<rbrakk> \\<Longrightarrow> n choose k = n - 1 choose (k - 1) + (n - 1 choose k)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 288, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9219218284193595, "lm_q2_score": 0.8152324983301568, "lm_q1q2_score": 0.7515806354474206}}
{"text": "[STATEMENT]\nlemma choose_reduce_nat:\n  \"0 < n \\<Longrightarrow> 0 < k \\<Longrightarrow>\n    n choose k = ((n - 1) choose (k - 1)) + ((n - 1) choose k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>0 < n; 0 < k\\<rbrakk> \\<Longrightarrow> n choose k = n - 1 choose (k - 1) + (n - 1 choose k)\n[PROOF STEP]\nusing binomial_Suc_Suc [of \"n - 1\" \"k - 1\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nSuc (n - 1) choose Suc (k - 1) = n - 1 choose (k - 1) + (n - 1 choose Suc (k - 1))\n\ngoal (1 subgoal):\n 1. \\<lbrakk>0 < n; 0 < k\\<rbrakk> \\<Longrightarrow> n choose k = n - 1 choose (k - 1) + (n - 1 choose k)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 288, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9219218284193595, "lm_q2_score": 0.8152324983301568, "lm_q1q2_score": 0.7515806354474206}}
{"text": "[STATEMENT]\nlemma choose_reduce_nat:\n  \"0 < n \\<Longrightarrow> 0 < k \\<Longrightarrow>\n    n choose k = ((n - 1) choose (k - 1)) + ((n - 1) choose k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>0 < n; 0 < k\\<rbrakk> \\<Longrightarrow> n choose k = n - 1 choose (k - 1) + (n - 1 choose k)\n[PROOF STEP]\nusing binomial_Suc_Suc [of \"n - 1\" \"k - 1\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nSuc (n - 1) choose Suc (k - 1) = n - 1 choose (k - 1) + (n - 1 choose Suc (k - 1))\n\ngoal (1 subgoal):\n 1. \\<lbrakk>0 < n; 0 < k\\<rbrakk> \\<Longrightarrow> n choose k = n - 1 choose (k - 1) + (n - 1 choose k)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 288, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9219218284193595, "lm_q2_score": 0.8152324938410784, "lm_q1q2_score": 0.7515806313088413}}
{"text": "[STATEMENT]\nlemma choose_reduce_nat:\n  \"0 < n \\<Longrightarrow> 0 < k \\<Longrightarrow>\n    n choose k = ((n - 1) choose (k - 1)) + ((n - 1) choose k)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>0 < n; 0 < k\\<rbrakk> \\<Longrightarrow> n choose k = n - 1 choose (k - 1) + (n - 1 choose k)\n[PROOF STEP]\nusing binomial_Suc_Suc [of \"n - 1\" \"k - 1\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nSuc (n - 1) choose Suc (k - 1) = n - 1 choose (k - 1) + (n - 1 choose Suc (k - 1))\n\ngoal (1 subgoal):\n 1. \\<lbrakk>0 < n; 0 < k\\<rbrakk> \\<Longrightarrow> n choose k = n - 1 choose (k - 1) + (n - 1 choose k)\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 288, "file": null, "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9219218284193595, "lm_q2_score": 0.8152324915965392, "lm_q1q2_score": 0.7515806292395516}}
{"text": "[STATEMENT]\nlemma desargues_config_3D_non_coplanar :\n  assumes \"desargues_config_3D A B C A' B' C' P \\<alpha> \\<beta> \\<gamma>\"\n  shows \"rk {A, B, C, A', B', C', \\<alpha>, \\<beta>, \\<gamma>} \\<ge> 4\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 4 \\<le> rk {A, B, C, A', B', C', \\<alpha>, \\<beta>, \\<gamma>}\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. 4 \\<le> rk {A, B, C, A', B', C', \\<alpha>, \\<beta>, \\<gamma>}\n[PROOF STEP]\nhave \"rk {A, B, C, A', B', C'} \\<le> rk {A, B, C, A', B', C', \\<alpha>, \\<beta>, \\<gamma>}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. rk {A, B, C, A', B', C'} \\<le> rk {A, B, C, A', B', C', \\<alpha>, \\<beta>, \\<gamma>}\n[PROOF STEP]\nusing matroid_ax_2\n[PROOF STATE]\nproof (prove)\nusing this:\n?X \\<subseteq> ?Y \\<longrightarrow> rk ?X \\<le> rk ?Y\n\ngoal (1 subgoal):\n 1. rk {A, B, C, A', B', C'} \\<le> rk {A, B, C, A', B', C', \\<alpha>, \\<beta>, \\<gamma>}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nrk {A, B, C, A', B', C'} \\<le> rk {A, B, C, A', B', C', \\<alpha>, \\<beta>, \\<gamma>}\n\ngoal (1 subgoal):\n 1. 4 \\<le> rk {A, B, C, A', B', C', \\<alpha>, \\<beta>, \\<gamma>}\n[PROOF STEP]\nthus \"4 \\<le> rk {A, B, C, A', B', C', \\<alpha>, \\<beta>, \\<gamma>}\"\n[PROOF STATE]\nproof (prove)\nusing this:\nrk {A, B, C, A', B', C'} \\<le> rk {A, B, C, A', B', C', \\<alpha>, \\<beta>, \\<gamma>}\n\ngoal (1 subgoal):\n 1. 4 \\<le> rk {A, B, C, A', B', C', \\<alpha>, \\<beta>, \\<gamma>}\n[PROOF STEP]\nusing matroid_ax_2 assms desargues_config_3D_def[of A B C A' B' C' P \\<alpha> \\<beta> \\<gamma>]\n[PROOF STATE]\nproof (prove)\nusing this:\nrk {A, B, C, A', B', C'} \\<le> rk {A, B, C, A', B', C', \\<alpha>, \\<beta>, \\<gamma>}\n?X \\<subseteq> ?Y \\<longrightarrow> rk ?X \\<le> rk ?Y\ndesargues_config_3D A B C A' B' C' P \\<alpha> \\<beta> \\<gamma>\ndesargues_config_3D A B C A' B' C' P \\<alpha> \\<beta> \\<gamma> \\<equiv> rk {A, B, C} = 3 \\<and> rk {A', B', C'} = 3 \\<and> rk {A, A', P} = 2 \\<and> rk {B, B', P} = 2 \\<and> rk {C, C', P} = 2 \\<and> 4 \\<le> rk {A, B, C, A', B', C'} \\<and> rk {B, C, \\<alpha>} = 2 \\<and> rk {B', C', \\<alpha>} = 2 \\<and> rk {A, C, \\<beta>} = 2 \\<and> rk {A', C', \\<beta>} = 2 \\<and> rk {A, B, \\<gamma>} = 2 \\<and> rk {A', B', \\<gamma>} = 2\n\ngoal (1 subgoal):\n 1. 4 \\<le> rk {A, B, C, A', B', C', \\<alpha>, \\<beta>, \\<gamma>}\n[PROOF STEP]\nby linarith\n[PROOF STATE]\nproof (state)\nthis:\n4 \\<le> rk {A, B, C, A', B', C', \\<alpha>, \\<beta>, \\<gamma>}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1201, "file": "Projective_Geometry_Desargues_3D", "length": 8, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8740772351648677, "lm_q2_score": 0.8596637541053281, "lm_q1q2_score": 0.7514125173598359}}
{"text": "[STATEMENT]\nlemma sq_norm_poly_le_linf_norm:\n  fixes p :: \"'a :: {conjugatable_ring_1_abs_real_line} poly\"\n  shows \"\\<parallel>p\\<parallel>\\<^sup>2 \\<le> of_nat (degree p + 1) * \\<parallel>p\\<parallel>\\<^sub>\\<infinity>\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<parallel>p\\<parallel>\\<^sup>2 \\<le> of_nat (degree p + 1) * \\<parallel>p\\<parallel>\\<^sub>\\<infinity>\\<^sup>2\n[PROOF STEP]\nusing sq_norm_vec_le_linf_norm[of \"vec_of_poly p\" \"degree p + 1\"]\n[PROOF STATE]\nproof (prove)\nusing this:\nvec_of_poly p \\<in> carrier_vec (degree p + 1) \\<Longrightarrow> \\<parallel>vec_of_poly p\\<parallel>\\<^sup>2 \\<le> of_nat (degree p + 1) * \\<parallel>vec_of_poly p\\<parallel>\\<^sub>\\<infinity>\\<^sup>2\n\ngoal (1 subgoal):\n 1. \\<parallel>p\\<parallel>\\<^sup>2 \\<le> of_nat (degree p + 1) * \\<parallel>p\\<parallel>\\<^sub>\\<infinity>\\<^sup>2\n[PROOF STEP]\nby (auto simp: carrier_dim_vec)", "meta": {"llama_tokens": 362, "file": "LLL_Basis_Reduction_Norms", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9111797027760039, "lm_q2_score": 0.8244619263765707, "lm_q1q2_score": 0.7512329730259353}}
{"text": "[STATEMENT]\ntheorem sums_eval_fps:\n  fixes f :: \"'a :: {banach, real_normed_div_algebra} fps\"\n  assumes \"norm z < fps_conv_radius f\"\n  shows   \"(\\<lambda>n. fps_nth f n * z ^ n) sums eval_fps f z\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<lambda>n. fps_nth f n * z ^ n) sums eval_fps f z\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nereal (norm z) < fps_conv_radius f\n\ngoal (1 subgoal):\n 1. (\\<lambda>n. fps_nth f n * z ^ n) sums eval_fps f z\n[PROOF STEP]\nunfolding eval_fps_def fps_conv_radius_def\n[PROOF STATE]\nproof (prove)\nusing this:\nereal (norm z) < conv_radius (fps_nth f)\n\ngoal (1 subgoal):\n 1. (\\<lambda>n. fps_nth f n * z ^ n) sums (\\<Sum>n. fps_nth f n * z ^ n)\n[PROOF STEP]\nby (intro summable_sums summable_in_conv_radius) simp_all", "meta": {"llama_tokens": 343, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9136765281148512, "lm_q2_score": 0.8221891239865619, "lm_q1q2_score": 0.7512149042578328}}
{"text": "[STATEMENT]\nlemma greatest_plus_one_eq_0:\n  fixes A::\"'a::{field}^'columns::{mod_type}^'rows::{mod_type}\" and k::nat\n  assumes \"Suc (to_nat (GREATEST n. \\<not> is_zero_row_upt_k n k A)) = nrows A\"\n  shows \"(GREATEST n. \\<not> is_zero_row_upt_k n k A) + 1 = 0\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (GREATEST n. \\<not> is_zero_row_upt_k n k A) + (1::'rows) = (0::'rows)\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (GREATEST n. \\<not> is_zero_row_upt_k n k A) + (1::'rows) = (0::'rows)\n[PROOF STEP]\nhave \"to_nat (GREATEST R. \\<not> is_zero_row_upt_k R k A) + 1 = card (UNIV::'rows set)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. mod_type_class.to_nat (GREATEST R. \\<not> is_zero_row_upt_k R k A) + 1 = CARD('rows)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nSuc (mod_type_class.to_nat (GREATEST n. \\<not> is_zero_row_upt_k n k A)) = nrows A\n\ngoal (1 subgoal):\n 1. mod_type_class.to_nat (GREATEST R. \\<not> is_zero_row_upt_k R k A) + 1 = CARD('rows)\n[PROOF STEP]\nunfolding nrows_def\n[PROOF STATE]\nproof (prove)\nusing this:\nSuc (mod_type_class.to_nat (GREATEST n. \\<not> is_zero_row_upt_k n k A)) = CARD('rows)\n\ngoal (1 subgoal):\n 1. mod_type_class.to_nat (GREATEST R. \\<not> is_zero_row_upt_k R k A) + 1 = CARD('rows)\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\nmod_type_class.to_nat (GREATEST R. \\<not> is_zero_row_upt_k R k A) + 1 = CARD('rows)\n\ngoal (1 subgoal):\n 1. (GREATEST n. \\<not> is_zero_row_upt_k n k A) + (1::'rows) = (0::'rows)\n[PROOF STEP]\nthus \"(GREATEST n. \\<not> is_zero_row_upt_k n k A) + (1::'rows) = (0::'rows)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nmod_type_class.to_nat (GREATEST R. \\<not> is_zero_row_upt_k R k A) + 1 = CARD('rows)\n\ngoal (1 subgoal):\n 1. (GREATEST n. \\<not> is_zero_row_upt_k n k A) + (1::'rows) = (0::'rows)\n[PROOF STEP]\nusing to_nat_plus_one_less_card\n[PROOF STATE]\nproof (prove)\nusing this:\nmod_type_class.to_nat (GREATEST R. \\<not> is_zero_row_upt_k R k A) + 1 = CARD('rows)\n\\<forall>a. a + (1::?'a) \\<noteq> (0::?'a) \\<longrightarrow> mod_type_class.to_nat a + 1 < CARD(?'a)\n\ngoal (1 subgoal):\n 1. (GREATEST n. \\<not> is_zero_row_upt_k n k A) + (1::'rows) = (0::'rows)\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\n(GREATEST n. \\<not> is_zero_row_upt_k n k A) + (1::'rows) = (0::'rows)\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1134, "file": "Gauss_Jordan_Gauss_Jordan", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9136765187126079, "lm_q2_score": 0.822189123986562, "lm_q1q2_score": 0.7512148965274107}}
{"text": "[STATEMENT]\ntheorem sigma_algebra_Inter:\n  \"(\\<And>i::nat. a i \\<in> \\<sigma>_algebra A) \\<Longrightarrow> (\\<Inter>i. a i) \\<in> \\<sigma>_algebra A\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<And>i. a i \\<in> \\<sigma>_algebra A) \\<Longrightarrow> \\<Inter> (range a) \\<in> \\<sigma>_algebra A\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. (\\<And>i. a i \\<in> \\<sigma>_algebra A) \\<Longrightarrow> \\<Inter> (range a) \\<in> \\<sigma>_algebra A\n[PROOF STEP]\nassume \"\\<And>i::nat. a i \\<in> \\<sigma>_algebra A\"\n[PROOF STATE]\nproof (state)\nthis:\na ?i \\<in> \\<sigma>_algebra A\n\ngoal (1 subgoal):\n 1. (\\<And>i. a i \\<in> \\<sigma>_algebra A) \\<Longrightarrow> \\<Inter> (range a) \\<in> \\<sigma>_algebra A\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\na ?i \\<in> \\<sigma>_algebra A\n[PROOF STEP]\nhave \"\\<And>i::nat. -(a i) \\<in> \\<sigma>_algebra A\"\n[PROOF STATE]\nproof (prove)\nusing this:\na ?i \\<in> \\<sigma>_algebra A\n\ngoal (1 subgoal):\n 1. \\<And>i. - a i \\<in> \\<sigma>_algebra A\n[PROOF STEP]\nby (rule \\<sigma>_algebra.complement)\n[PROOF STATE]\nproof (state)\nthis:\n- a ?i \\<in> \\<sigma>_algebra A\n\ngoal (1 subgoal):\n 1. (\\<And>i. a i \\<in> \\<sigma>_algebra A) \\<Longrightarrow> \\<Inter> (range a) \\<in> \\<sigma>_algebra A\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n- a ?i \\<in> \\<sigma>_algebra A\n[PROOF STEP]\nhave \"(\\<Union>i. -(a i)) \\<in> \\<sigma>_algebra A\"\n[PROOF STATE]\nproof (prove)\nusing this:\n- a ?i \\<in> \\<sigma>_algebra A\n\ngoal (1 subgoal):\n 1. (\\<Union>i. - a i) \\<in> \\<sigma>_algebra A\n[PROOF STEP]\nby (rule \\<sigma>_algebra.Union)\n[PROOF STATE]\nproof (state)\nthis:\n(\\<Union>i. - a i) \\<in> \\<sigma>_algebra A\n\ngoal (1 subgoal):\n 1. (\\<And>i. a i \\<in> \\<sigma>_algebra A) \\<Longrightarrow> \\<Inter> (range a) \\<in> \\<sigma>_algebra A\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\n(\\<Union>i. - a i) \\<in> \\<sigma>_algebra A\n[PROOF STEP]\nhave \"-(\\<Union>i. -(a i)) \\<in> \\<sigma>_algebra A\"\n[PROOF STATE]\nproof (prove)\nusing this:\n(\\<Union>i. - a i) \\<in> \\<sigma>_algebra A\n\ngoal (1 subgoal):\n 1. - (\\<Union>i. - a i) \\<in> \\<sigma>_algebra A\n[PROOF STEP]\nby (rule \\<sigma>_algebra.complement)\n[PROOF STATE]\nproof (state)\nthis:\n- (\\<Union>i. - a i) \\<in> \\<sigma>_algebra A\n\ngoal (1 subgoal):\n 1. (\\<And>i. a i \\<in> \\<sigma>_algebra A) \\<Longrightarrow> \\<Inter> (range a) \\<in> \\<sigma>_algebra A\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n- (\\<Union>i. - a i) \\<in> \\<sigma>_algebra A\n\ngoal (1 subgoal):\n 1. (\\<And>i. a i \\<in> \\<sigma>_algebra A) \\<Longrightarrow> \\<Inter> (range a) \\<in> \\<sigma>_algebra A\n[PROOF STEP]\nhave \"-(\\<Union>i. -(a i)) = (\\<Inter>i. a i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. - (\\<Union>i. - a i) = \\<Inter> (range a)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\n- (\\<Union>i. - a i) = \\<Inter> (range a)\n\ngoal (1 subgoal):\n 1. (\\<And>i. a i \\<in> \\<sigma>_algebra A) \\<Longrightarrow> \\<Inter> (range a) \\<in> \\<sigma>_algebra A\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n\\<Inter> (range a) \\<in> \\<sigma>_algebra A\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<Inter> (range a) \\<in> \\<sigma>_algebra A\n\ngoal (1 subgoal):\n 1. \\<Inter> (range a) \\<in> \\<sigma>_algebra A\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n\\<Inter> (range a) \\<in> \\<sigma>_algebra A\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1447, "file": null, "length": 18, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213718636754, "lm_q2_score": 0.8354835411997897, "lm_q1q2_score": 0.7512011077330765}}
{"text": "[STATEMENT]\nlemma set_integral_diff [simp, intro]:\n  assumes \"set_integrable M A f\" \"set_integrable M A g\"\n  shows \"set_integrable M A (\\<lambda>x. f x - g x)\" and \"LINT x:A|M. f x - g x =\n    (LINT x:A|M. f x) - (LINT x:A|M. g x)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set_integrable M A (\\<lambda>x. f x - g x) &&& LINT x:A|M. f x - g x = set_lebesgue_integral M A f - set_lebesgue_integral M A g\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nset_integrable M A f\nset_integrable M A g\n\ngoal (1 subgoal):\n 1. set_integrable M A (\\<lambda>x. f x - g x) &&& LINT x:A|M. f x - g x = set_lebesgue_integral M A f - set_lebesgue_integral M A g\n[PROOF STEP]\nunfolding set_integrable_def set_lebesgue_integral_def\n[PROOF STATE]\nproof (prove)\nusing this:\nintegrable M (\\<lambda>x. indicat_real A x *\\<^sub>R f x)\nintegrable M (\\<lambda>x. indicat_real A x *\\<^sub>R g x)\n\ngoal (1 subgoal):\n 1. integrable M (\\<lambda>x. indicat_real A x *\\<^sub>R (f x - g x)) &&& LINT x|M. indicat_real A x *\\<^sub>R (f x - g x) = (LINT x|M. indicat_real A x *\\<^sub>R f x) - (LINT x|M. indicat_real A x *\\<^sub>R g x)\n[PROOF STEP]\nby (simp_all add: scaleR_diff_right)", "meta": {"llama_tokens": 527, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8991213691605412, "lm_q2_score": 0.8354835432479661, "lm_q1q2_score": 0.7512011073162116}}
{"text": "[STATEMENT]\ntheorem integer_compositions_card:\n  \"card (integer_compositions n) = 2^(n-1)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (integer_compositions n) = 2 ^ (n - 1)\n[PROOF STEP]\nusing integer_composition_enum_correct integer_composition_enum_length\n    integer_composition_enum_distinct distinct_card\n[PROOF STATE]\nproof (prove)\nusing this:\nset (integer_composition_enum ?n) = integer_compositions ?n\nlength (integer_composition_enum ?n) = 2 ^ (?n - 1)\ndistinct (integer_composition_enum ?n)\ndistinct ?xs \\<Longrightarrow> card (set ?xs) = length ?xs\n\ngoal (1 subgoal):\n 1. card (integer_compositions n) = 2 ^ (n - 1)\n[PROOF STEP]\nby metis", "meta": {"llama_tokens": 245, "file": "Combinatorial_Enumeration_Algorithms_Integer_Compositions", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9086178944582997, "lm_q2_score": 0.8267117898012104, "lm_q1q2_score": 0.7511651257730282}}
{"text": "[STATEMENT]\nlemma div_less_imp_less_mult: \"\\<lbrakk> 0 < (m::nat); n div m < k \\<rbrakk> \\<Longrightarrow> n < k * m\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>0 < m; n div m < k\\<rbrakk> \\<Longrightarrow> n < k * m\n[PROOF STEP]\napply (rule ccontr, simp only: linorder_not_less)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>0 < m; n div m < k; k * m \\<le> n\\<rbrakk> \\<Longrightarrow> False\n[PROOF STEP]\napply (drule div_le_mono[of _ _ m])\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>0 < m; n div m < k; k * m div m \\<le> n div m\\<rbrakk> \\<Longrightarrow> False\n[PROOF STEP]\napply simp\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 306, "file": "List-Infinite_CommonArith_Util_Div", "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8962513842182777, "lm_q2_score": 0.837619961306541, "lm_q1q2_score": 0.7507180497698476}}
{"text": "[STATEMENT]\ntheorem time_vebt_succ: \n  fixes t defines \"u \\<equiv> 2^n\"\n  shows \"invar_vebt t n \\<Longrightarrow>   T\\<^sub>s\\<^sub>u\\<^sub>c\\<^sub>c t x \\<le> 54 + 27 * lb (lb u)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. invar_vebt t n \\<Longrightarrow> real (T\\<^sub>s\\<^sub>u\\<^sub>c\\<^sub>c t x) \\<le> 54 + 27 * log 2 (log 2 u)\n[PROOF STEP]\nusing succ_bound_size_univ\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>invar_vebt ?t ?n; ?u = 2 ^ ?n\\<rbrakk> \\<Longrightarrow> real (T\\<^sub>s\\<^sub>u\\<^sub>c\\<^sub>c ?t ?x) \\<le> 54 + 27 * log 2 (log 2 ?u)\n\ngoal (1 subgoal):\n 1. invar_vebt t n \\<Longrightarrow> real (T\\<^sub>s\\<^sub>u\\<^sub>c\\<^sub>c t x) \\<le> 54 + 27 * log 2 (log 2 u)\n[PROOF STEP]\nunfolding u_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<lbrakk>invar_vebt ?t ?n; ?u = 2 ^ ?n\\<rbrakk> \\<Longrightarrow> real (T\\<^sub>s\\<^sub>u\\<^sub>c\\<^sub>c ?t ?x) \\<le> 54 + 27 * log 2 (log 2 ?u)\n\ngoal (1 subgoal):\n 1. invar_vebt t n \\<Longrightarrow> real (T\\<^sub>s\\<^sub>u\\<^sub>c\\<^sub>c t x) \\<le> 54 + 27 * log 2 (log 2 (2 ^ n))\n[PROOF STEP]\nby presburger", "meta": {"llama_tokens": 520, "file": "Van_Emde_Boas_Trees_VEBT_Intf_Functional", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9207896715436482, "lm_q2_score": 0.8152324915965392, "lm_q1q2_score": 0.7506576581688873}}
{"text": "[STATEMENT]\nlemma convergent_prod_inverse:\n  assumes \"convergent_prod f\" \n  shows \"convergent_prod (\\<lambda>n. inverse (f n))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. convergent_prod (\\<lambda>n. inverse (f n))\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nconvergent_prod f\n\ngoal (1 subgoal):\n 1. convergent_prod (\\<lambda>n. inverse (f n))\n[PROOF STEP]\nunfolding convergent_prod_def\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<exists>M p. raw_has_prod f M p\n\ngoal (1 subgoal):\n 1. \\<exists>M p. raw_has_prod (\\<lambda>n. inverse (f n)) M p\n[PROOF STEP]\nby (blast intro: raw_has_prod_inverse elim: )", "meta": {"llama_tokens": 256, "file": null, "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.88720460564669, "lm_q2_score": 0.8459424295406088, "lm_q1q2_score": 0.7505240196003787}}
{"text": "[STATEMENT]\nlemma Ivl4:\n  \"{0..<4::nat} = {0, 1, 2, 3}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {0..<4} = {0, 1, 2, 3}\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. {0..<4} = {0, 1, 2, 3}\n[PROOF STEP]\nhave \"{0..<4::nat} = {0..<Suc (Suc (Suc (Suc 0)))}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {0..<4} = {0..<Suc (Suc (Suc (Suc 0)))}\n[PROOF STEP]\nby (simp add: eval_nat_numeral)\n[PROOF STATE]\nproof (state)\nthis:\n{0..<4} = {0..<Suc (Suc (Suc (Suc 0)))}\n\ngoal (1 subgoal):\n 1. {0..<4} = {0, 1, 2, 3}\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\n{0..<4} = {0..<Suc (Suc (Suc (Suc 0)))}\n\ngoal (1 subgoal):\n 1. {0..<4} = {0, 1, 2, 3}\n[PROOF STEP]\nhave \"... = {0, 1, 2, 3}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. {0..<Suc (Suc (Suc (Suc 0)))} = {0, 1, 2, 3}\n[PROOF STEP]\nby (simp add: atLeastLessThanSuc eval_nat_numeral insert_commute)\n[PROOF STATE]\nproof (state)\nthis:\n{0..<Suc (Suc (Suc (Suc 0)))} = {0, 1, 2, 3}\n\ngoal (1 subgoal):\n 1. {0..<4} = {0, 1, 2, 3}\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\n{0..<4} = {0, 1, 2, 3}\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\n{0..<4} = {0, 1, 2, 3}\n\ngoal (1 subgoal):\n 1. {0..<4} = {0, 1, 2, 3}\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\n{0..<4} = {0, 1, 2, 3}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 782, "file": "FFT_FFT", "length": 10, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8872045817875224, "lm_q2_score": 0.8459424295406088, "lm_q1q2_score": 0.7505239994168965}}
{"text": "[STATEMENT]\nlemma card1:\n  assumes \"card A = 1\"\n  shows \"\\<exists>a. A = {a}\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<exists>a. A = {a}\n[PROOF STEP]\nproof-\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. \\<exists>a. A = {a}\n[PROOF STEP]\nfrom assms\n[PROOF STATE]\nproof (chain)\npicking this:\ncard A = 1\n[PROOF STEP]\nobtain a where a: \"a \\<in> A\"\n[PROOF STATE]\nproof (prove)\nusing this:\ncard A = 1\n\ngoal (1 subgoal):\n 1. (\\<And>a. a \\<in> A \\<Longrightarrow> thesis) \\<Longrightarrow> thesis\n[PROOF STEP]\nby fastforce\n[PROOF STATE]\nproof (state)\nthis:\na \\<in> A\n\ngoal (1 subgoal):\n 1. \\<exists>a. A = {a}\n[PROOF STEP]\nwith assms\n[PROOF STATE]\nproof (chain)\npicking this:\ncard A = 1\na \\<in> A\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard A = 1\na \\<in> A\n\ngoal (1 subgoal):\n 1. \\<exists>a. A = {a}\n[PROOF STEP]\nusing card_ge_0_finite[of A] card_subset_eq[of A \"{a}\"]\n[PROOF STATE]\nproof (prove)\nusing this:\ncard A = 1\na \\<in> A\n0 < card A \\<Longrightarrow> finite A\n\\<lbrakk>finite A; {a} \\<subseteq> A; card {a} = card A\\<rbrakk> \\<Longrightarrow> {a} = A\n\ngoal (1 subgoal):\n 1. \\<exists>a. A = {a}\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\n\\<exists>a. A = {a}\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 571, "file": "Buildings_Prelim", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357460591569, "lm_q2_score": 0.8652240877899776, "lm_q1q2_score": 0.7501802124653367}}
{"text": "[STATEMENT]\nlemma insert_into_member_list_equivalence:\n  fixes new_el::'a\n    and Sets::\"'a set list\"\n    and S::\"'a set\"\n  assumes \"distinct Sets\"\n  shows \"set (insert_into_member_list new_el Sets S) = insert_into_member new_el (set Sets) S\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set (insert_into_member_list new_el Sets S) = insert_into_member new_el (set Sets) S\n[PROOF STEP]\nunfolding insert_into_member_list_def insert_into_member_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. set ((S \\<union> {new_el}) # remove1 S Sets) = insert (S \\<union> {new_el}) (set Sets - {S})\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\ndistinct Sets\n\ngoal (1 subgoal):\n 1. set ((S \\<union> {new_el}) # remove1 S Sets) = insert (S \\<union> {new_el}) (set Sets - {S})\n[PROOF STEP]\nby simp", "meta": {"llama_tokens": 334, "file": "Vickrey_Clarke_Groves_Partitions", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933094088947399, "lm_q2_score": 0.8397339716830605, "lm_q1q2_score": 0.750142257873027}}
{"text": "[STATEMENT]\nlemma  avl_height_upperbound:\n  defines \"\\<phi> \\<equiv> (1 + sqrt 5) / 2\"\n  assumes \"avl t\"\n  shows   \"height t \\<le> (1/log 2 \\<phi>) * log 2 (size1 t)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. real (height t) \\<le> 1 / log 2 \\<phi> * log 2 (real (size1 t))\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. real (height t) \\<le> 1 / log 2 \\<phi> * log 2 (real (size1 t))\n[PROOF STEP]\nhave \"\\<phi> > 0\" \"\\<phi> > 1\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. 0 < \\<phi> &&& 1 < \\<phi>\n[PROOF STEP]\nby(auto simp: \\<phi>_def pos_add_strict)\n[PROOF STATE]\nproof (state)\nthis:\n0 < \\<phi>\n1 < \\<phi>\n\ngoal (1 subgoal):\n 1. real (height t) \\<le> 1 / log 2 \\<phi> * log 2 (real (size1 t))\n[PROOF STEP]\nhence \"height t = log \\<phi> (\\<phi> ^ height t)\"\n[PROOF STATE]\nproof (prove)\nusing this:\n0 < \\<phi>\n1 < \\<phi>\n\ngoal (1 subgoal):\n 1. real (height t) = log \\<phi> (\\<phi> ^ height t)\n[PROOF STEP]\nby(simp add: log_nat_power)\n[PROOF STATE]\nproof (state)\nthis:\nreal (height t) = log \\<phi> (\\<phi> ^ height t)\n\ngoal (1 subgoal):\n 1. real (height t) \\<le> 1 / log 2 \\<phi> * log 2 (real (size1 t))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nreal (height t) = log \\<phi> (\\<phi> ^ height t)\n\ngoal (1 subgoal):\n 1. real (height t) \\<le> 1 / log 2 \\<phi> * log 2 (real (size1 t))\n[PROOF STEP]\nhave \"\\<dots> \\<le> log \\<phi> (size1 t)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. log \\<phi> (\\<phi> ^ height t) \\<le> log \\<phi> (real (size1 t))\n[PROOF STEP]\nusing avl_size_lowerbound[OF assms(2), folded \\<phi>_def] \\<open>1 < \\<phi>\\<close>\n[PROOF STATE]\nproof (prove)\nusing this:\n\\<phi> ^ height t \\<le> real (size1 t)\n1 < \\<phi>\n\ngoal (1 subgoal):\n 1. log \\<phi> (\\<phi> ^ height t) \\<le> log \\<phi> (real (size1 t))\n[PROOF STEP]\nby (simp add: le_log_of_power)\n[PROOF STATE]\nproof (state)\nthis:\nlog \\<phi> (\\<phi> ^ height t) \\<le> log \\<phi> (real (size1 t))\n\ngoal (1 subgoal):\n 1. real (height t) \\<le> 1 / log 2 \\<phi> * log 2 (real (size1 t))\n[PROOF STEP]\nalso\n[PROOF STATE]\nproof (state)\nthis:\nlog \\<phi> (\\<phi> ^ height t) \\<le> log \\<phi> (real (size1 t))\n\ngoal (1 subgoal):\n 1. real (height t) \\<le> 1 / log 2 \\<phi> * log 2 (real (size1 t))\n[PROOF STEP]\nhave \"\\<dots> = (1/log 2 \\<phi>) * log 2 (size1 t)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. log \\<phi> (real (size1 t)) = 1 / log 2 \\<phi> * log 2 (real (size1 t))\n[PROOF STEP]\nby(simp add: log_base_change[of 2 \\<phi>])\n[PROOF STATE]\nproof (state)\nthis:\nlog \\<phi> (real (size1 t)) = 1 / log 2 \\<phi> * log 2 (real (size1 t))\n\ngoal (1 subgoal):\n 1. real (height t) \\<le> 1 / log 2 \\<phi> * log 2 (real (size1 t))\n[PROOF STEP]\nfinally\n[PROOF STATE]\nproof (chain)\npicking this:\nreal (height t) \\<le> 1 / log 2 \\<phi> * log 2 (real (size1 t))\n[PROOF STEP]\nshow ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nreal (height t) \\<le> 1 / log 2 \\<phi> * log 2 (real (size1 t))\n\ngoal (1 subgoal):\n 1. real (height t) \\<le> 1 / log 2 \\<phi> * log 2 (real (size1 t))\n[PROOF STEP]\n.\n[PROOF STATE]\nproof (state)\nthis:\nreal (height t) \\<le> 1 / log 2 \\<phi> * log 2 (real (size1 t))\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 1420, "file": null, "length": 16, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8933094060543488, "lm_q2_score": 0.8397339716830606, "lm_q1q2_score": 0.7501422554878543}}
{"text": "[STATEMENT]\nlemma card_all_bi_edges: \n  assumes \"finite X\" \"finite Y\"\n  assumes \"X \\<inter> Y = {}\"\n  shows \"card (all_bi_edges X Y) = card X * card Y\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (all_bi_edges X Y) = card X * card Y\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. card (all_bi_edges X Y) = card X * card Y\n[PROOF STEP]\nhave \"card (all_bi_edges X Y) = card (X \\<times> Y)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (all_bi_edges X Y) = card (X \\<times> Y)\n[PROOF STEP]\nunfolding all_bi_edges_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. card (mk_edge ` (X \\<times> Y)) = card (X \\<times> Y)\n[PROOF STEP]\nusing inj_on_mk_edge assms card_image\n[PROOF STATE]\nproof (prove)\nusing this:\n?X \\<inter> ?Y = {} \\<Longrightarrow> inj_on mk_edge (?X \\<times> ?Y)\nfinite X\nfinite Y\nX \\<inter> Y = {}\ninj_on ?f ?A \\<Longrightarrow> card (?f ` ?A) = card ?A\n\ngoal (1 subgoal):\n 1. card (mk_edge ` (X \\<times> Y)) = card (X \\<times> Y)\n[PROOF STEP]\nby blast\n[PROOF STATE]\nproof (state)\nthis:\ncard (all_bi_edges X Y) = card (X \\<times> Y)\n\ngoal (1 subgoal):\n 1. card (all_bi_edges X Y) = card X * card Y\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (all_bi_edges X Y) = card (X \\<times> Y)\n\ngoal (1 subgoal):\n 1. card (all_bi_edges X Y) = card X * card Y\n[PROOF STEP]\nusing card_cartesian_product\n[PROOF STATE]\nproof (prove)\nusing this:\ncard (all_bi_edges X Y) = card (X \\<times> Y)\ncard (?A \\<times> ?B) = card ?A * card ?B\n\ngoal (1 subgoal):\n 1. card (all_bi_edges X Y) = card X * card Y\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\ncard (all_bi_edges X Y) = card X * card Y\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 754, "file": "Undirected_Graph_Theory_Bipartite_Graphs", "length": 9, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267762381844, "lm_q2_score": 0.8633916099737806, "lm_q1q2_score": 0.7501377491246156}}
{"text": "[STATEMENT]\nlemma ennreal_suminf_SUP_eq:\n  fixes f :: \"nat \\<Rightarrow> nat \\<Rightarrow> ennreal\"\n  shows \"(\\<And>i. incseq (\\<lambda>n. f n i)) \\<Longrightarrow> (\\<Sum>i. SUP n. f n i) = (SUP n. \\<Sum>i. f n i)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<And>i. incseq (\\<lambda>n. f n i)) \\<Longrightarrow> (\\<Sum>i. SUP n. f n i) = (SUP n. \\<Sum>i. f n i)\n[PROOF STEP]\napply (rule ennreal_suminf_SUP_eq_directed)\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<And>N n j. \\<lbrakk>\\<And>i. incseq (\\<lambda>n. f n i); n \\<in> UNIV; j \\<in> UNIV; finite N\\<rbrakk> \\<Longrightarrow> \\<exists>k\\<in>UNIV. \\<forall>i\\<in>N. f n i \\<le> f k i \\<and> f j i \\<le> f k i\n[PROOF STEP]\nsubgoal for N n j\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>\\<And>i. incseq (\\<lambda>n. f n i); n \\<in> UNIV; j \\<in> UNIV; finite N\\<rbrakk> \\<Longrightarrow> \\<exists>k\\<in>UNIV. \\<forall>i\\<in>N. f n i \\<le> f k i \\<and> f j i \\<le> f k i\n[PROOF STEP]\nby (auto simp: incseq_def intro!:exI[of _ \"max n j\"])\n[PROOF STATE]\nproof (prove)\ngoal:\nNo subgoals!\n[PROOF STEP]\ndone", "meta": {"llama_tokens": 487, "file": null, "length": 4, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9149009549929799, "lm_q2_score": 0.819893335913536, "lm_q1q2_score": 0.7501211960196741}}
{"text": "[STATEMENT]\nlemma sum_list_take_eq:\n  fixes xs :: \"nat list\"\n  shows \"k < i \\<Longrightarrow> i < length xs \\<Longrightarrow> sum_list (take i xs) =\n    sum_list (take k xs) + xs ! k + sum_list (take (i - Suc k) (drop (Suc k) xs))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. \\<lbrakk>k < i; i < length xs\\<rbrakk> \\<Longrightarrow> sum_list (take i xs) = sum_list (take k xs) + xs ! k + sum_list (take (i - Suc k) (drop (Suc k) xs))\n[PROOF STEP]\nby (subst id_take_nth_drop [of k]) (auto simp: min_def drop_take)", "meta": {"llama_tokens": 208, "file": "FO_Theory_Rewriting_Util_Multihole_Context", "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887588023318195, "lm_q2_score": 0.8438950966654774, "lm_q1q2_score": 0.7500191954061046}}
{"text": "[STATEMENT]\nlemma size_multiset_int_count:\n  assumes \"of_nat (card (set_mset A)) = (ca :: int)\"\n  assumes \"\\<And>p. p \\<in># A \\<Longrightarrow> of_nat (count A p) = (ca2 :: int)\"\n  shows \"of_nat (size A) =  ((ca :: int) * ca2)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. int (size A) = ca * ca2\n[PROOF STEP]\nproof -\n[PROOF STATE]\nproof (state)\ngoal (1 subgoal):\n 1. int (size A) = ca * ca2\n[PROOF STEP]\nhave \"size A = (\\<Sum> p \\<in> (set_mset A) . count A p)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. size A = sum (count A) (set_mset A)\n[PROOF STEP]\nusing size_multiset_overloaded_eq\n[PROOF STATE]\nproof (prove)\nusing this:\nsize ?x = sum (count ?x) (set_mset ?x)\n\ngoal (1 subgoal):\n 1. size A = sum (count A) (set_mset A)\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nsize A = sum (count A) (set_mset A)\n\ngoal (1 subgoal):\n 1. int (size A) = ca * ca2\n[PROOF STEP]\nthen\n[PROOF STATE]\nproof (chain)\npicking this:\nsize A = sum (count A) (set_mset A)\n[PROOF STEP]\nhave \"of_nat (size A) = (\\<Sum> p \\<in> (set_mset A) . ca2)\"\n[PROOF STATE]\nproof (prove)\nusing this:\nsize A = sum (count A) (set_mset A)\n\ngoal (1 subgoal):\n 1. int (size A) = (\\<Sum>p\\<in>set_mset A. ca2)\n[PROOF STEP]\nusing assms\n[PROOF STATE]\nproof (prove)\nusing this:\nsize A = sum (count A) (set_mset A)\nint (card (set_mset A)) = ca\n?p \\<in># A \\<Longrightarrow> int (count A ?p) = ca2\n\ngoal (1 subgoal):\n 1. int (size A) = (\\<Sum>p\\<in>set_mset A. ca2)\n[PROOF STEP]\nby simp\n[PROOF STATE]\nproof (state)\nthis:\nint (size A) = (\\<Sum>p\\<in>set_mset A. ca2)\n\ngoal (1 subgoal):\n 1. int (size A) = ca * ca2\n[PROOF STEP]\nthus ?thesis\n[PROOF STATE]\nproof (prove)\nusing this:\nint (size A) = (\\<Sum>p\\<in>set_mset A. ca2)\n\ngoal (1 subgoal):\n 1. int (size A) = ca * ca2\n[PROOF STEP]\nusing assms(1)\n[PROOF STATE]\nproof (prove)\nusing this:\nint (size A) = (\\<Sum>p\\<in>set_mset A. ca2)\nint (card (set_mset A)) = ca\n\ngoal (1 subgoal):\n 1. int (size A) = ca * ca2\n[PROOF STEP]\nby auto\n[PROOF STATE]\nproof (state)\nthis:\nint (size A) = ca * ca2\n\ngoal:\nNo subgoals!\n[PROOF STEP]\nqed", "meta": {"llama_tokens": 943, "file": "Design_Theory_Multisets_Extras", "length": 12, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8887587934924569, "lm_q2_score": 0.8438951025545426, "lm_q1q2_score": 0.7500191931805684}}
{"text": "[STATEMENT]\nlemma conjugate_transpose_rank:\n  fixes A::\"'a::{conjugatable_ordered_field} mat\"\n  shows \"vec_space.rank (dim_row A) A = vec_space.rank (dim_col A) (A\\<^sup>H)\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. vec_space.rank (dim_row A) A = vec_space.rank (dim_col A) A\\<^sup>H\n[PROOF STEP]\nusing  conjugatable_vec_space.conjugate_transpose_rank_le\n[PROOF STATE]\nproof (prove)\nusing this:\n?A \\<in> carrier_mat ?n ?nc \\<Longrightarrow> vec_space.rank ?nc ?A\\<^sup>H \\<le> vec_space.rank ?n ?A\n\ngoal (1 subgoal):\n 1. vec_space.rank (dim_row A) A = vec_space.rank (dim_col A) A\\<^sup>H\n[PROOF STEP]\nby (metis (no_types, lifting) Matrix.transpose_transpose carrier_matI conjugate_id dim_col_conjugate dual_order.antisym index_transpose_mat(2) transpose_conjugate)", "meta": {"llama_tokens": 321, "file": "BenOr_Kozen_Reif_More_Matrix", "length": 2, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8887587875995483, "lm_q2_score": 0.8438951025545426, "lm_q1q2_score": 0.7500191882075717}}
{"text": "[STATEMENT]\nlemma scalar_prod_split_head: assumes \n  \"A \\<in> carrier_mat n n\" \"B \\<in> carrier_mat n n\" \"n > 0\" \n  shows \"row A 0 \\<bullet> col B 0 = A $$ (0,0) * B $$ (0,0) + (\\<Sum>i = 1..<n. A $$ (0, i) * B $$ (i, 0))\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. row A 0 \\<bullet> col B 0 = A $$ (0, 0) * B $$ (0, 0) + (\\<Sum>i = 1..<n. A $$ (0, i) * B $$ (i, 0))\n[PROOF STEP]\nunfolding scalar_prod_def\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<dim_vec (col B 0). row A 0 $ i * col B 0 $ i) = A $$ (0, 0) * B $$ (0, 0) + (\\<Sum>i = 1..<n. A $$ (0, i) * B $$ (i, 0))\n[PROOF STEP]\nusing assms sum.atLeast_Suc_lessThan\n[PROOF STATE]\nproof (prove)\nusing this:\nA \\<in> carrier_mat n n\nB \\<in> carrier_mat n n\n0 < n\n?m < ?n \\<Longrightarrow> sum ?g {?m..<?n} = ?g ?m + sum ?g {Suc ?m..<?n}\n\ngoal (1 subgoal):\n 1. (\\<Sum>i = 0..<dim_vec (col B 0). row A 0 $ i * col B 0 $ i) = A $$ (0, 0) * B $$ (0, 0) + (\\<Sum>i = 1..<n. A $$ (0, i) * B $$ (i, 0))\n[PROOF STEP]\nby auto", "meta": {"llama_tokens": 514, "file": "Jordan_Normal_Form_Matrix_Comparison", "length": 3, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8976952975813454, "lm_q2_score": 0.8354835452961425, "lm_q1q2_score": 0.7500096498189381}}
{"text": "[STATEMENT]\nlemma (in prob_space) variance_eq:\n  fixes X :: \"'a \\<Rightarrow> real\"\n  assumes [simp]: \"integrable M X\"\n  assumes [simp]: \"integrable M (\\<lambda>x. (X x)\\<^sup>2)\"\n  shows \"variance X = expectation (\\<lambda>x. (X x)\\<^sup>2) - (expectation X)\\<^sup>2\"\n[PROOF STATE]\nproof (prove)\ngoal (1 subgoal):\n 1. expectation (\\<lambda>x. (X x - expectation X)\\<^sup>2) = expectation (\\<lambda>x. (X x)\\<^sup>2) - (expectation X)\\<^sup>2\n[PROOF STEP]\nby (simp add: field_simps prob_space power2_diff power2_eq_square[symmetric])", "meta": {"llama_tokens": 209, "file": null, "length": 1, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8976952838963489, "lm_q2_score": 0.8354835432479661, "lm_q1q2_score": 0.7500096365467105}}