(*
    Author:      René Thiemann
                 Sebastiaan Joosten
                 Akihisa Yamada
    License:     BSD
*)
section \<open>Algebraic Numbers -- Excluding Addition and Multiplication\<close>

text \<open>This theory contains basic definition and results on algebraic numbers, namely that
  algebraic numbers are closed under negation, inversion, $n$-th roots, and
  that every rational number is algebraic. For all of these closure properties, corresponding
  polynomial witnesses are available.

  Moreover, this theory contains the uniqueness result,
  that for every algebraic number there is exactly one content-free irreducible polynomial with
  positive leading coefficient for it.
  This result is stronger than similar ones which you find in many textbooks.
  The reason is that here we do not require a least degree construction.

  This is essential, since given some content-free irreducible polynomial for x,
  how should we check whether the degree is optimal. In the formalized result, this is
  not required. The result is proven via GCDs, and that the GCD does not change
  when executed on the rational numbers or on the reals or complex numbers, and that
  the GCD of a rational polynomial can be expressed via the GCD of integer polynomials.\<close>

text \<open>Many results are taken from the textbook \<^cite>\<open>\<open>pages 317ff\<close> in "AlgNumbers"\<close>.\<close>

theory Algebraic_Numbers_Prelim
imports
  "HOL-Computational_Algebra.Fundamental_Theorem_Algebra"
  Polynomial_Interpolation.Newton_Interpolation (* for lemma smult_1 *)
  Polynomial_Factorization.Gauss_Lemma
  Berlekamp_Zassenhaus.Unique_Factorization_Poly
  Polynomial_Factorization.Square_Free_Factorization
begin

lemma primitive_imp_unit_iff:
  fixes p :: "'a :: {comm_semiring_1,semiring_no_zero_divisors} poly"
  assumes pr: "primitive p"
  shows "p dvd 1 \<longleftrightarrow> degree p = 0"
proof
  assume "degree p = 0"
  from degree0_coeffs[OF this] obtain p0 where p: "p = [:p0:]" by auto
  then have "\<forall>c \<in> set (coeffs p). p0 dvd c" by (simp add: cCons_def)
  with pr have "p0 dvd 1" by (auto dest: primitiveD)
  with p show "p dvd 1" by auto
next
  assume "p dvd 1"
  then show "degree p = 0" by (auto simp: poly_dvd_1)
qed

lemma dvd_all_coeffs_imp_dvd:
  assumes "\<forall>a \<in> set (coeffs p). c dvd a" shows "[:c:] dvd p"
proof (insert assms, induct p)
  case 0
  then show ?case by simp
next
  case (pCons a p)
  have "pCons a p = [:a:] + pCons 0 p" by simp
  also have "[:c:] dvd ..."
  proof (rule dvd_add)
    from pCons show "[:c:] dvd [:a:]" by (auto simp: cCons_def)
    from pCons have "[:c:] dvd p" by auto
    from Rings.dvd_mult[OF this]
    show "[:c:] dvd pCons 0 p" by (subst pCons_0_as_mult)
  qed
  finally show ?case.
qed

lemma irreducible_content:
  fixes p :: "'a::{comm_semiring_1,semiring_no_zero_divisors} poly"
  assumes "irreducible p" shows "degree p = 0 \<or> primitive p"
proof(rule ccontr)
  assume not: "\<not>?thesis"
  then obtain c where c1: "\<not>c dvd 1" and "\<forall>a \<in> set (coeffs p). c dvd a" by (auto elim: not_primitiveE)
  from dvd_all_coeffs_imp_dvd[OF this(2)]
  obtain r where p: "p = r * [:c:]" by (elim dvdE, auto)
  from irreducibleD[OF assms this] have "r dvd 1 \<or> [:c:] dvd 1" by auto
  with c1 have "r dvd 1" unfolding const_poly_dvd_1 by auto
  then have "degree r = 0" unfolding poly_dvd_1 by auto
  with p have "degree p = 0" by auto
  with not show False by auto
qed

(* TODO: move *)
lemma linear_irreducible_field:
  fixes p :: "'a :: field poly"
  assumes deg: "degree p = 1" shows "irreducible p"
proof (intro irreducibleI)
  from deg show p0: "p \<noteq> 0" by auto
  from deg show "\<not> p dvd 1" by (auto simp: poly_dvd_1)
  fix a b assume p: "p = a * b"
  with p0 have a0: "a \<noteq> 0" and b0: "b \<noteq> 0" by auto
  from degree_mult_eq[OF this, folded p] assms
  consider "degree a = 1" "degree b = 0" | "degree a = 0" "degree b = 1" by force
  then show "a dvd 1 \<or> b dvd 1"
    by (cases; insert a0 b0, auto simp: primitive_imp_unit_iff)
qed

(* TODO: move *)
lemma linear_irreducible_int:
  fixes p :: "int poly"
  assumes deg: "degree p = 1" and cp: "content p dvd 1"
  shows "irreducible p"
proof (intro irreducibleI)
  from deg show p0: "p \<noteq> 0" by auto
  from deg show "\<not> p dvd 1" by (auto simp: poly_dvd_1)
  fix a b assume p: "p = a * b"
  note * = cp[unfolded p is_unit_content_iff, unfolded content_mult]
  have a1: "content a dvd 1" and b1: "content b dvd 1"
    using content_ge_0_int[of a] pos_zmult_eq_1_iff_lemma[OF *] * by (auto simp: abs_mult)
  with p0 have a0: "a \<noteq> 0" and b0: "b \<noteq> 0" by auto
  from degree_mult_eq[OF this, folded p] assms
  consider "degree a = 1" "degree b = 0" | "degree a = 0" "degree b = 1" by force
  then show "a dvd 1 \<or> b dvd 1"
    by (cases; insert a1 b1, auto simp: primitive_imp_unit_iff)
qed

lemma irreducible_connect_rev:
  fixes p :: "'a :: {comm_semiring_1,semiring_no_zero_divisors} poly"
  assumes irr: "irreducible p" and deg: "degree p > 0"
  shows "irreducible\<^sub>d p"
proof(intro irreducible\<^sub>dI deg)
  fix q r
  assume degq: "degree q > 0" and diff: "degree q < degree p" and p: "p = q * r"
  from degq have nu: "\<not> q dvd 1" by (auto simp: poly_dvd_1)
  from irreducibleD[OF irr p] nu have "r dvd 1" by auto
  then have "degree r = 0" by (auto simp: poly_dvd_1)
  with degq diff show False unfolding p using degree_mult_le[of q r] by auto
qed

subsection \<open>Polynomial Evaluation of Integer and Rational Polynomials in Fields.\<close>

abbreviation ipoly where "ipoly f x \<equiv> poly (of_int_poly f) x"

lemma poly_map_poly_code[code_unfold]: "poly (map_poly h p) x = fold_coeffs (\<lambda> a b. h a + x * b) p 0"
  by (induct p, auto)

abbreviation real_of_int_poly :: "int poly \<Rightarrow> real poly" where
  "real_of_int_poly \<equiv> of_int_poly"

abbreviation real_of_rat_poly :: "rat poly \<Rightarrow> real poly" where
  "real_of_rat_poly \<equiv> map_poly of_rat"

lemma of_rat_of_int[simp]: "of_rat \<circ> of_int = of_int" by auto

lemma ipoly_of_rat[simp]: "ipoly p (of_rat y) = of_rat (ipoly p y)"
proof-
  have id: "of_int = of_rat o of_int" unfolding comp_def by auto
  show ?thesis by (subst id, subst map_poly_map_poly[symmetric], auto)
qed

lemma ipoly_of_real[simp]:
  "ipoly p (of_real x :: 'a :: {field,real_algebra_1}) = of_real (ipoly p x)"
proof -
  have id: "of_int = of_real o of_int" unfolding comp_def by auto
  show ?thesis by (subst id, subst map_poly_map_poly[symmetric], auto)
qed

lemma finite_ipoly_roots: assumes "p \<noteq> 0"
  shows "finite {x :: real. ipoly p x = 0}"
proof -
  let ?p = "real_of_int_poly p"
  from assms have "?p \<noteq> 0" by auto
  thus ?thesis by (rule poly_roots_finite)
qed

subsection \<open>Algebraic Numbers -- Definition, Inverse, and Roots\<close>

text \<open>A number @{term "x :: 'a :: field"} is algebraic iff it is the root of an integer polynomial.
  Whereas the Isabelle distribution this is defined via the embedding
  of integers in an field via @{const Ints}, we work with integer polynomials
  of type @{type int} and then use @{const ipoly} for evaluating the polynomial at
  a real or complex point.\<close>

lemma algebraic_altdef_ipoly:
  shows "algebraic x \<longleftrightarrow> (\<exists>p. ipoly p x = 0 \<and> p \<noteq> 0)"
unfolding algebraic_def
proof (safe, goal_cases)
  case (1 p)
  define the_int where "the_int = (\<lambda>x::'a. THE r. x = of_int r)"
  define p' where "p' = map_poly the_int p"
  have of_int_the_int: "of_int (the_int x) = x" if "x \<in> \<int>" for x
    unfolding the_int_def by (rule sym, rule theI') (insert that, auto simp: Ints_def)
  have the_int_0_iff: "the_int x = 0 \<longleftrightarrow> x = 0" if "x \<in> \<int>"
    using of_int_the_int[OF that] by auto
  have "map_poly of_int p' = map_poly (of_int \<circ> the_int) p"
      by (simp add: p'_def map_poly_map_poly)
  also from 1 of_int_the_int have "\<dots> = p"
    by (subst poly_eq_iff) (auto simp: coeff_map_poly)
  finally have p_p': "map_poly of_int p' = p" .
  show ?case
  proof (intro exI conjI notI)
    from 1 show "ipoly p' x = 0" by (simp add: p_p')
  next
    assume "p' = 0"
    hence "p = 0" by (simp add: p_p' [symmetric])
    with \<open>p \<noteq> 0\<close> show False by contradiction
  qed
next
  case (2 p)
  thus ?case by (intro exI[of _ "map_poly of_int p"], auto)
qed

text \<open>Definition of being algebraic with explicit witness polynomial.\<close>

definition represents :: "int poly \<Rightarrow> 'a :: field_char_0 \<Rightarrow> bool" (infix "represents" 51)
  where "p represents x = (ipoly p x = 0 \<and> p \<noteq> 0)"

lemma representsI[intro]: "ipoly p x = 0 \<Longrightarrow> p \<noteq> 0 \<Longrightarrow> p represents x"
  unfolding represents_def by auto

lemma representsD:
  assumes "p represents x" shows "p \<noteq> 0" and "ipoly p x = 0" using assms unfolding represents_def by auto

lemma representsE:
  assumes "p represents x" and "p \<noteq> 0 \<Longrightarrow> ipoly p x = 0 \<Longrightarrow> thesis"
  shows thesis using assms unfolding represents_def by auto

lemma represents_imp_degree:
  fixes x :: "'a :: field_char_0"
  assumes "p represents x" shows "degree p \<noteq> 0"
proof-
  from assms have "p \<noteq> 0" and px: "ipoly p x = 0" by (auto dest:representsD)
  then have "(of_int_poly p :: 'a poly) \<noteq> 0" by auto
  then have "degree (of_int_poly p :: 'a poly) \<noteq> 0" by (fold poly_zero[OF px])
  then show ?thesis by auto
qed

lemma representsE_full[elim]:
  assumes rep: "p represents x"
    and main: "p \<noteq> 0 \<Longrightarrow> ipoly p x = 0 \<Longrightarrow> degree p \<noteq> 0 \<Longrightarrow> thesis"
  shows thesis
  by (rule main, insert represents_imp_degree[OF rep] rep, auto elim: representsE)

lemma represents_of_rat[simp]: "p represents (of_rat x) = p represents x" by (auto elim!:representsE)
lemma represents_of_real[simp]: "p represents (of_real x) = p represents x" by (auto elim!:representsE)

lemma algebraic_iff_represents: "algebraic x \<longleftrightarrow> (\<exists> p. p represents x)"
  unfolding algebraic_altdef_ipoly represents_def ..

lemma represents_irr_non_0:
  assumes irr: "irreducible p" and ap: "p represents x" and x0: "x \<noteq> 0"
  shows "poly p 0 \<noteq> 0"
proof
  have nu: "\<not> [:0,1::int:] dvd 1" by (auto simp: poly_dvd_1)
  assume "poly p 0 = 0"
  hence dvd: "[: 0, 1 :] dvd p" by (unfold dvd_iff_poly_eq_0, simp)
  then obtain q where pq: "p = [:0,1:] * q" by (elim dvdE)
  from irreducibleD[OF irr this] nu have "q dvd 1" by auto
  from this obtain r where "q = [:r:]" "r dvd 1" by (auto simp add: poly_dvd_1 dest: degree0_coeffs)
  with pq have "p = [:0,r:]" by auto
  with ap have "x = 0" by (auto simp: of_int_hom.map_poly_pCons_hom)
  with x0 show False by auto
qed

text \<open>The polynomial encoding a rational number.\<close>

definition poly_rat :: "rat \<Rightarrow> int poly" where
  "poly_rat x = (case quotient_of x of (n,d) \<Rightarrow> [:-n,d:])"

definition abs_int_poly:: "int poly \<Rightarrow> int poly" where
  "abs_int_poly p \<equiv> if lead_coeff p < 0 then -p else p"

lemma pos_poly_abs_poly[simp]:
  shows "lead_coeff (abs_int_poly p) > 0 \<longleftrightarrow> p \<noteq> 0"
proof-
  have "p \<noteq> 0 \<longleftrightarrow> lead_coeff p * sgn (lead_coeff p) > 0" by (fold abs_sgn, auto)
  then show ?thesis by (auto simp: abs_int_poly_def mult.commute)
qed

lemma abs_int_poly_0[simp]: "abs_int_poly 0 = 0"
  by (auto simp: abs_int_poly_def)

lemma abs_int_poly_eq_0_iff[simp]: "abs_int_poly p = 0 \<longleftrightarrow> p = 0"
  by (auto simp: abs_int_poly_def sgn_eq_0_iff)

lemma degree_abs_int_poly[simp]: "degree (abs_int_poly p) = degree p"
  by (auto simp: abs_int_poly_def sgn_eq_0_iff)

lemma abs_int_poly_dvd[simp]: "abs_int_poly p dvd q \<longleftrightarrow> p dvd q"
  by (unfold abs_int_poly_def, auto)

(*TODO: move & generalize *)
lemma (in idom) irreducible_uminus[simp]: "irreducible (-x) \<longleftrightarrow> irreducible x"
proof-
  have "-x = -1 * x" by simp
  also have "irreducible ... \<longleftrightarrow> irreducible x" by (rule irreducible_mult_unit_left, auto)
  finally show ?thesis.
qed

lemma irreducible_abs_int_poly[simp]:
  "irreducible (abs_int_poly p) \<longleftrightarrow> irreducible p"
  by (unfold abs_int_poly_def, auto)

lemma coeff_abs_int_poly[simp]:
  "coeff (abs_int_poly p) n = (if lead_coeff p < 0 then - coeff p n else coeff p n)"
  by (simp add: abs_int_poly_def)

lemma lead_coeff_abs_int_poly[simp]:
  "lead_coeff (abs_int_poly p) = abs (lead_coeff p)"
  by auto

lemma ipoly_abs_int_poly_eq_zero_iff[simp]:
  "ipoly (abs_int_poly p) (x :: 'a :: comm_ring_1) = 0 \<longleftrightarrow> ipoly p x = 0"
  by (auto simp: abs_int_poly_def sgn_eq_0_iff of_int_poly_hom.hom_uminus)

lemma abs_int_poly_represents[simp]:
  "abs_int_poly p represents x \<longleftrightarrow> p represents x" by (auto elim!:representsE)

(* TODO: Move *)
lemma content_pCons[simp]: "content (pCons a p) = gcd a (content p)"
  by (unfold content_def coeffs_pCons_eq_cCons cCons_def, auto)

lemma content_uminus[simp]:
  fixes p :: "'a :: ring_gcd poly" shows "content (-p) = content p"
  by (induct p, auto)

lemma primitive_abs_int_poly[simp]:
  "primitive (abs_int_poly p) \<longleftrightarrow> primitive p"
  by (auto simp: abs_int_poly_def)

lemma abs_int_poly_inv[simp]: "smult (sgn (lead_coeff p)) (abs_int_poly p) = p"
  by (cases "lead_coeff p > 0", auto simp: abs_int_poly_def)

definition cf_pos :: "int poly \<Rightarrow> bool" where
  "cf_pos p = (content p = 1 \<and> lead_coeff p > 0)"

definition cf_pos_poly :: "int poly \<Rightarrow> int poly" where
  "cf_pos_poly f = (let
      c = content f;
      d = (sgn (lead_coeff f) * c)
    in sdiv_poly f d)"

lemma sgn_is_unit[intro!]:
  fixes x :: "'a :: linordered_idom" (* find/make better class *)
  assumes "x \<noteq> 0"
  shows "sgn x dvd 1" using assms by(cases x "0::'a" rule:linorder_cases, auto)

lemma cf_pos_poly_0[simp]: "cf_pos_poly 0 = 0" by (unfold cf_pos_poly_def sdiv_poly_def, auto)

lemma cf_pos_poly_eq_0[simp]: "cf_pos_poly f = 0 \<longleftrightarrow> f = 0"
proof(cases "f = 0")
  case True
  thus ?thesis unfolding cf_pos_poly_def Let_def by (simp add: sdiv_poly_def)
next
  case False
  then have lc0: "lead_coeff f \<noteq> 0" by auto
  then have s0: "sgn (lead_coeff f) \<noteq> 0" (is "?s \<noteq> 0") and "content f \<noteq> 0" (is "?c \<noteq> 0") by (auto simp: sgn_0_0)
  then have sc0: "?s * ?c \<noteq> 0" by auto
  { fix i
    from content_dvd_coeff sgn_is_unit[OF lc0]
    have "?s * ?c dvd coeff f i" by (auto simp: unit_dvd_iff)
    then have "coeff f i div (?s * ?c) = 0 \<longleftrightarrow> coeff f i = 0" by (auto simp:dvd_div_eq_0_iff)
  } note * = this
  show ?thesis unfolding cf_pos_poly_def Let_def sdiv_poly_def poly_eq_iff by (auto simp: coeff_map_poly *)
qed

lemma
  shows cf_pos_poly_main: "smult (sgn (lead_coeff f) * content f) (cf_pos_poly f) = f" (is ?g1)
    and content_cf_pos_poly[simp]: "content (cf_pos_poly f) = (if f = 0 then 0 else 1)" (is ?g2)
    and lead_coeff_cf_pos_poly[simp]: "lead_coeff (cf_pos_poly f) > 0 \<longleftrightarrow> f \<noteq> 0" (is ?g3)
    and cf_pos_poly_dvd[simp]: "cf_pos_poly f dvd f" (is ?g4)
proof(atomize(full), (cases "f = 0"; intro conjI))
  case True
  then show ?g1 ?g2 ?g3 ?g4 by simp_all
next
  case f0: False
  let ?s = "sgn (lead_coeff f)"
  have s: "?s \<in> {-1,1}" using f0 unfolding sgn_if by auto
  define g where "g \<equiv> smult ?s f"
  define d where "d \<equiv> ?s * content f"
  have "content g = content ([:?s:] * f)" unfolding g_def by simp
  also have "\<dots> = content [:?s:] * content f" unfolding content_mult by simp
  also have "content [:?s:] = 1" using s by (auto simp: content_def)
  finally have cg: "content g = content f" by simp
  from f0
  have d: "cf_pos_poly f = sdiv_poly f d"  by (auto simp: cf_pos_poly_def Let_def d_def)
  let ?g = "primitive_part g"
  define ng where "ng = primitive_part g"
  note d
  also have "sdiv_poly f d = sdiv_poly g (content g)" unfolding cg unfolding g_def d_def
    by (rule poly_eqI, unfold coeff_sdiv_poly coeff_smult, insert s, auto simp: div_minus_right)
  finally have fg: "cf_pos_poly f = primitive_part g" unfolding primitive_part_alt_def .
  have "lead_coeff f \<noteq> 0" using f0 by auto
  hence lg: "lead_coeff g > 0" unfolding g_def lead_coeff_smult
    by (meson linorder_neqE_linordered_idom sgn_greater sgn_less zero_less_mult_iff)
  hence g0: "g \<noteq> 0" by auto
  from f0 content_primitive_part[OF this]
  show ?g2 unfolding fg by auto
  from g0 have "content g \<noteq> 0" by simp
  with arg_cong[OF content_times_primitive_part[of g], of lead_coeff, unfolded lead_coeff_smult]
    lg content_ge_0_int[of g] have lg': "lead_coeff ng > 0" unfolding ng_def
    by (metis dual_order.antisym dual_order.strict_implies_order zero_less_mult_iff)
  with f0 show ?g3 unfolding fg ng_def by auto

  have d0: "d \<noteq> 0" using s f0 by (force simp add: d_def)
  have "smult d (cf_pos_poly f) = smult ?s (smult (content f) (sdiv_poly (smult ?s f) (content f)))"
    unfolding fg primitive_part_alt_def cg by (simp add: g_def d_def)
  also have "sdiv_poly (smult ?s f) (content f) = smult ?s (sdiv_poly f (content f))"
    using s by (metis cg g_def primitive_part_alt_def primitive_part_smult_int sgn_sgn)
  finally have "smult d (cf_pos_poly f) = smult (content f) (primitive_part f)"
    unfolding primitive_part_alt_def using s by auto
  also have "\<dots> = f" by (rule content_times_primitive_part)
  finally have df: "smult d (cf_pos_poly f) = f" .
  with d0 show ?g1 by (auto simp: d_def)
  from df have *: "f = cf_pos_poly f * [:d:]" by simp
  from dvdI[OF this] show ?g4.
qed

(* TODO: remove *)
lemma irreducible_connect_int:
  fixes p :: "int poly"
  assumes ir: "irreducible\<^sub>d p" and c: "content p = 1"
  shows "irreducible p"
  using c primitive_iff_content_eq_1 ir irreducible_primitive_connect by blast

lemma
  fixes x :: "'a :: {idom,ring_char_0}"
  shows ipoly_cf_pos_poly_eq_0[simp]: "ipoly (cf_pos_poly p) x = 0 \<longleftrightarrow> ipoly p x = 0"
    and degree_cf_pos_poly[simp]: "degree (cf_pos_poly p) = degree p"
    and cf_pos_cf_pos_poly[intro]: "p \<noteq> 0 \<Longrightarrow> cf_pos (cf_pos_poly p)"
proof-
  show "degree (cf_pos_poly p) = degree p"
    by (subst(3) cf_pos_poly_main[symmetric], auto simp:sgn_eq_0_iff)
  {
    assume p: "p \<noteq> 0"
    show "cf_pos (cf_pos_poly p)" using cf_pos_poly_main p by (auto simp: cf_pos_def)
    have "(ipoly (cf_pos_poly p) x = 0) = (ipoly p x = 0)"
      apply (subst(3) cf_pos_poly_main[symmetric]) by (auto simp: sgn_eq_0_iff hom_distribs)
  }
  then show "(ipoly (cf_pos_poly p) x = 0) = (ipoly p x = 0)" by (cases "p = 0", auto)
qed

lemma cf_pos_poly_eq_1: "cf_pos_poly f = 1 \<longleftrightarrow> degree f = 0 \<and> f \<noteq> 0" (is "?l \<longleftrightarrow> ?r")
proof(intro iffI conjI)
  assume ?r
  then have df0: "degree f = 0" and f0: "f \<noteq> 0" by auto
  from  degree0_coeffs[OF df0] obtain f0 where f: "f = [:f0:]" by auto
  show "cf_pos_poly f = 1" using f0 unfolding f cf_pos_poly_def Let_def sdiv_poly_def
    by (auto simp: content_def mult_sgn_abs)
next
  assume l: ?l
  then have "degree (cf_pos_poly f) = 0" by auto
  then show "degree f = 0" by simp
  from l have "cf_pos_poly f \<noteq> 0" by auto
  then show "f \<noteq> 0" by simp
qed

lemma irr_cf_poly_rat[simp]: "irreducible (poly_rat x)"
  "lead_coeff (poly_rat x) > 0" "primitive (poly_rat x)"
proof -
  obtain n d where x: "quotient_of x = (n,d)" by force
  hence id: "poly_rat x = [:-n,d:]" by (auto simp: poly_rat_def)
  from quotient_of_denom_pos[OF x] have d: "d > 0" by auto
  show "lead_coeff (poly_rat x) > 0" "primitive (poly_rat x)"
    unfolding id cf_pos_def using d quotient_of_coprime[OF x] by (auto simp: content_def)
  from this[unfolded cf_pos_def]
  show irr: "irreducible (poly_rat x)" unfolding id using d by (auto intro!: linear_irreducible_int)
qed

lemma poly_rat[simp]: "ipoly (poly_rat x) (of_rat x :: 'a :: field_char_0) = 0" "ipoly (poly_rat x) x = 0"
  "poly_rat x \<noteq> 0" "ipoly (poly_rat x) y = 0 \<longleftrightarrow> y = (of_rat x :: 'a)"
proof -
  from irr_cf_poly_rat(1)[of x] show "poly_rat x \<noteq> 0"
    unfolding Factorial_Ring.irreducible_def by auto
  obtain n d where x: "quotient_of x = (n,d)" by force
  hence id: "poly_rat x = [:-n,d:]" by (auto simp: poly_rat_def)
  from quotient_of_denom_pos[OF x] have d: "d \<noteq> 0" by auto
  have "y * of_int d = of_int n \<Longrightarrow> y = of_int n / of_int d" using d
    by (simp add: eq_divide_imp)
  with d id show "ipoly (poly_rat x) (of_rat x) = 0" "ipoly (poly_rat x) x = 0"
    "ipoly (poly_rat x) y = 0 \<longleftrightarrow> y = (of_rat x :: 'a)"
    by (auto simp: of_rat_minus of_rat_divide simp: quotient_of_div[OF x])
qed

lemma poly_rat_represents_of_rat: "(poly_rat x) represents (of_rat x)" by auto

lemma ipoly_smult_0_iff: assumes c: "c \<noteq> 0"
  shows "(ipoly (smult c p) x = (0 :: real)) = (ipoly p x = 0)"
  using c by (simp add: hom_distribs)

(* TODO *)
lemma not_irreducibleD:
  assumes "\<not> irreducible x" and "x \<noteq> 0" and "\<not> x dvd 1"
  shows "\<exists>y z. x = y * z \<and> \<not> y dvd 1 \<and> \<not> z dvd 1" using assms
  apply (unfold Factorial_Ring.irreducible_def) by auto

lemma cf_pos_poly_represents[simp]: "(cf_pos_poly p) represents x \<longleftrightarrow> p represents x"
  unfolding represents_def by auto

lemma coprime_prod: (* TODO: move *)
  "a \<noteq> 0 \<Longrightarrow> c \<noteq> 0 \<Longrightarrow> coprime (a * b) (c * d) \<Longrightarrow> coprime b (d::'a::{semiring_gcd})"
  by auto

lemma smult_prod: (* TODO: move or find corresponding lemma *)
  "smult a b = monom a 0 * b"
  by (simp add: monom_0)

lemma degree_map_poly_2:
  assumes "f (lead_coeff p) \<noteq> 0"
  shows   "degree (map_poly f p) = degree p"
proof (cases "p=0")
  case False thus ?thesis
    unfolding degree_eq_length_coeffs Polynomial.coeffs_map_poly
    using assms by (simp add:coeffs_def)
qed auto

lemma irreducible_cf_pos_poly:
  assumes irr: "irreducible p" and deg: "degree p \<noteq> 0"
  shows "irreducible (cf_pos_poly p)" (is "irreducible ?p")
proof (unfold irreducible_altdef, intro conjI allI impI)
  from irr show "?p \<noteq> 0" by auto
  from deg have "degree ?p \<noteq> 0" by simp
  then show "\<not> ?p dvd 1" unfolding poly_dvd_1 by auto
  fix b assume "b dvd cf_pos_poly p"
  also note cf_pos_poly_dvd
  finally have "b dvd p".
  with irr[unfolded irreducible_altdef] have "p dvd b \<or> b dvd 1" by auto
  then show "?p dvd b \<or> b dvd 1" by (auto dest: dvd_trans[OF cf_pos_poly_dvd])
qed

locale dvd_preserving_hom = comm_semiring_1_hom +
  assumes hom_eq_mult_hom_imp: "hom x = hom y * hz \<Longrightarrow> \<exists>z. hz = hom z \<and> x = y * z"
begin

  lemma hom_dvd_hom_iff[simp]: "hom x dvd hom y \<longleftrightarrow> x dvd y"
  proof
    assume "hom x dvd hom y"
    then obtain hz where "hom y = hom x * hz" by (elim dvdE)
    from hom_eq_mult_hom_imp[OF this] obtain z
    where "hz = hom z" and mult: "y = x * z" by auto
    then show "x dvd y" by auto
  qed auto

  sublocale unit_preserving_hom
  proof unfold_locales
    fix x assume "hom x dvd 1" then have "hom x dvd hom 1" by simp
    then show "x dvd 1" by (unfold hom_dvd_hom_iff)
  qed

  sublocale zero_hom_0
  proof (unfold_locales)
    fix a :: 'a
    assume "hom a = 0"
    then have "hom 0 dvd hom a" by auto
    then have "0 dvd a" by (unfold hom_dvd_hom_iff)
    then show "a = 0" by auto
  qed

end

lemma smult_inverse_monom:"p \<noteq> 0 \<Longrightarrow> smult (inverse c) (p::rat poly) = 1 \<longleftrightarrow> p = [: c :]"
  proof (cases "c=0")
    case True thus "p \<noteq> 0 \<Longrightarrow> ?thesis" by auto
  next
    case False thus ?thesis by (metis left_inverse right_inverse smult_1 smult_1_left smult_smult)
  qed

lemma of_int_monom:"of_int_poly p = [:rat_of_int c:] \<longleftrightarrow> p = [: c :]" by (induct p, auto)

lemma degree_0_content:
  fixes p :: "int poly"
  assumes deg: "degree p = 0" shows "content p = abs (coeff p 0)"
proof-
  from deg obtain a where p: "p = [:a:]" by (auto dest: degree0_coeffs)
  show ?thesis by (auto simp: p)
qed

lemma prime_elem_imp_gcd_eq:
  fixes x::"'a:: ring_gcd"
  shows "prime_elem x \<Longrightarrow> gcd x y = normalize x \<or> gcd x y = 1"
  using prime_elem_imp_coprime [of x y]
  by (auto simp add: gcd_proj1_iff intro: coprime_imp_gcd_eq_1)

lemma irreducible_pos_gcd:
  fixes p :: "int poly"
  assumes ir: "irreducible p" and pos: "lead_coeff p > 0" shows "gcd p q \<in> {1,p}"
proof-
  from pos have "[:sgn (lead_coeff p):] = 1" by auto
  with prime_elem_imp_gcd_eq[of p, unfolded prime_elem_iff_irreducible, OF ir, of q]
  show ?thesis by (auto simp: normalize_poly_def)
qed

lemma irreducible_pos_gcd_twice:
  fixes p q :: "int poly"
  assumes p: "irreducible p" "lead_coeff p > 0"
  and q: "irreducible q" "lead_coeff q > 0"
  shows "gcd p q = 1 \<or> p = q"
proof (cases "gcd p q = 1")
  case False note pq = this
  have "p = gcd p q" using irreducible_pos_gcd [OF p, of q] pq
    by auto
  also have "\<dots> = q" using irreducible_pos_gcd [OF q, of p] pq
    by (auto simp add: ac_simps)
  finally show ?thesis by auto
qed simp

interpretation of_rat_hom: field_hom_0' of_rat..

lemma poly_zero_imp_not_unit:
  assumes "poly p x = 0" shows "\<not> p dvd 1"
proof (rule notI)
  assume "p dvd 1"
  from poly_hom.hom_dvd_1[OF this] have "poly p x dvd 1" by auto
  with assms show False by auto
qed

lemma poly_prod_mset_zero_iff:
  fixes x :: "'a :: idom"
  shows "poly (prod_mset F) x = 0 \<longleftrightarrow> (\<exists>f \<in># F. poly f x = 0)"
  by (induct F, auto simp: poly_mult_zero_iff)

lemma algebraic_imp_represents_irreducible:
  fixes x :: "'a :: field_char_0"
  assumes "algebraic x"
  shows "\<exists>p. p represents x \<and> irreducible p"
proof -
  from assms obtain p
  where px0: "ipoly p x = 0" and p0: "p \<noteq> 0" unfolding algebraic_altdef_ipoly by auto
  from poly_zero_imp_not_unit[OF px0]
  have "\<not> p dvd 1" by (auto dest: of_int_poly_hom.hom_dvd_1[where 'a = 'a])
  from mset_factors_exist[OF p0 this]
  obtain F where F: "mset_factors F p" by auto
  then have "p = prod_mset F" by auto
  also have "(of_int_poly ... :: 'a poly) = prod_mset (image_mset of_int_poly F)" by simp
  finally have "poly ... x = 0" using px0 by auto
  from this[unfolded poly_prod_mset_zero_iff]
  obtain f where "f \<in># F" and fx0: "ipoly f x = 0" by auto
  with F have "irreducible f" by auto
  with fx0 show ?thesis by auto
qed

lemma algebraic_imp_represents_irreducible_cf_pos:
  assumes "algebraic (x::'a::field_char_0)"
  shows "\<exists>p. p represents x \<and> irreducible p \<and> lead_coeff p > 0 \<and> primitive p"
proof -
  from algebraic_imp_represents_irreducible[OF assms(1)]
  obtain p where px: "p represents x" and irr: "irreducible p" by auto
  let ?p = "cf_pos_poly p"
  from px irr represents_imp_degree
  have 1: "?p represents x" and 2: "irreducible ?p" and 3: "cf_pos ?p"
    by (auto intro: irreducible_cf_pos_poly)
  then show ?thesis by (auto intro: exI[of _ ?p] simp: cf_pos_def)
qed

lemma gcd_of_int_poly: "gcd (of_int_poly f) (of_int_poly g :: 'a :: {field_char_0,field_gcd} poly) =
  smult (inverse (of_int (lead_coeff (gcd f g)))) (of_int_poly (gcd f g))"
proof -
  let ?ia = "of_int_poly :: _ \<Rightarrow> 'a poly"
  let ?ir = "of_int_poly :: _ \<Rightarrow> rat poly"
  let ?ra = "map_poly of_rat :: _ \<Rightarrow> 'a poly"
  have id: "?ia x = ?ra (?ir x)" for x by (subst map_poly_map_poly, auto)
  show ?thesis
    unfolding id
    unfolding of_rat_hom.map_poly_gcd[symmetric]
    unfolding gcd_rat_to_gcd_int by (auto simp: hom_distribs)
qed

lemma algebraic_imp_represents_unique:
  fixes x :: "'a :: {field_char_0,field_gcd}"
  assumes "algebraic x"
  shows "\<exists>! p. p represents x \<and> irreducible p \<and> lead_coeff p > 0" (is "Ex1 ?p")
proof -
  from assms obtain p
  where p: "?p p" and cfp: "cf_pos p"
    by (auto simp: cf_pos_def dest: algebraic_imp_represents_irreducible_cf_pos)
  show ?thesis
  proof (rule ex1I)
    show "?p p" by fact
    fix q
    assume q: "?p q"
    then have "q represents x" by auto
    from represents_imp_degree[OF this] q irreducible_content[of q]
    have cfq: "cf_pos q" by (auto simp: cf_pos_def)
    show "q = p"
    proof (rule ccontr)
      let ?ia = "map_poly of_int :: int poly \<Rightarrow> 'a poly"
      assume "q \<noteq> p"
      with irreducible_pos_gcd_twice[of p q] p q cfp cfq have gcd: "gcd p q = 1" by auto
      from p q have rt: "ipoly p x = 0" "ipoly q x = 0" unfolding represents_def by auto
      define c :: 'a where "c = inverse (of_int (lead_coeff (gcd p q)))"
      have rt: "poly (?ia p) x = 0" "poly (?ia q) x = 0" using rt by auto
      hence "[:-x,1:] dvd ?ia p" "[:-x,1:] dvd ?ia q"
        unfolding poly_eq_0_iff_dvd by auto
      hence "[:-x,1:] dvd gcd (?ia p) (?ia q)" by (rule gcd_greatest)
      also have "\<dots> = smult c (?ia (gcd p q))" unfolding gcd_of_int_poly c_def ..
      also have "?ia (gcd p q) = 1" by (simp add: gcd)
      also have "smult c 1 = [: c :]" by simp
      finally show False using c_def gcd by (simp add: dvd_iff_poly_eq_0)
    qed
  qed
qed

lemma ipoly_poly_compose:
  fixes x :: "'a :: idom"
  shows "ipoly (p \<circ>\<^sub>p q) x = ipoly p (ipoly q x)"
proof (induct p)
  case (pCons a p)
  have "ipoly ((pCons a p) \<circ>\<^sub>p q) x = of_int a + ipoly (q * p \<circ>\<^sub>p q) x" by (simp add: hom_distribs)
  also have "ipoly (q * p \<circ>\<^sub>p q) x = ipoly q x * ipoly (p \<circ>\<^sub>p q) x" by (simp add: hom_distribs)
  also have "ipoly (p \<circ>\<^sub>p q) x = ipoly p (ipoly q x)" unfolding pCons(2) ..
  also have "of_int a + ipoly q x * \<dots> = ipoly (pCons a p) (ipoly q x)"
    unfolding map_poly_pCons[OF pCons(1)] by simp
  finally show ?case .
qed simp

lemma algebraic_0[simp]: "algebraic 0"
  unfolding algebraic_altdef_ipoly
  by (intro exI[of _ "[:0,1:]"], auto)

lemma algebraic_1[simp]: "algebraic 1"
  unfolding algebraic_altdef_ipoly
  by (intro exI[of _ "[:-1,1:]"], auto)

text \<open>Polynomial for unary minus.\<close>

definition poly_uminus :: "'a :: ring_1 poly \<Rightarrow> 'a poly" where [code del]:
  "poly_uminus p \<equiv> \<Sum>i\<le>degree p. monom ((-1)^i * coeff p i) i"

lemma poly_uminus_pCons_pCons[simp]:
  "poly_uminus (pCons a (pCons b p)) = pCons a (pCons (-b) (poly_uminus p))" (is "?l = ?r")
proof(cases "p = 0")
  case False
  then have deg: "degree (pCons a (pCons b p)) = Suc (Suc (degree p))" by simp
  show ?thesis
  by (unfold poly_uminus_def deg sum.atMost_Suc_shift monom_Suc monom_0 sum_pCons_0_commute, simp)
next
  case True
  then show ?thesis by (auto simp add: poly_uminus_def monom_0 monom_Suc)
qed

fun poly_uminus_inner :: "'a :: ring_1 list \<Rightarrow> 'a poly"
where "poly_uminus_inner [] = 0"
  |   "poly_uminus_inner [a] = [:a:]"
  |   "poly_uminus_inner (a#b#cs) = pCons a (pCons (-b) (poly_uminus_inner cs))"

lemma poly_uminus_code[code,simp]: "poly_uminus p = poly_uminus_inner (coeffs p)"
proof-
  have "poly_uminus (Poly as) = poly_uminus_inner as" for as :: "'a list"
  proof (induct "length as" arbitrary:as rule: less_induct)
    case less
    show ?case
    proof(cases as)
      case Nil
      then show ?thesis by (simp add: poly_uminus_def)
    next
      case [simp]: (Cons a bs)
      show ?thesis
      proof (cases bs)
        case Nil
        then show ?thesis by (simp add: poly_uminus_def monom_0)
      next
        case [simp]: (Cons b cs)
        show ?thesis by (simp add: less)
      qed
    qed
  qed
  from this[of "coeffs p"]
  show ?thesis by simp
qed

lemma poly_uminus_inner_0[simp]: "poly_uminus_inner as = 0 \<longleftrightarrow> Poly as = 0"
  by (induct as rule: poly_uminus_inner.induct, auto)

lemma degree_poly_uminus_inner[simp]: "degree (poly_uminus_inner as) = degree (Poly as)"
  by (induct as rule: poly_uminus_inner.induct, auto)

lemma ipoly_uminus_inner[simp]:
  "ipoly (poly_uminus_inner as) (x::'a::comm_ring_1) = ipoly (Poly as) (-x)"
  by (induct as rule: poly_uminus_inner.induct, auto simp: hom_distribs ring_distribs)

lemma represents_uminus: assumes alg: "p represents x"
  shows "(poly_uminus p) represents (-x)"
proof -
  from representsD[OF alg] have "p \<noteq> 0" and rp: "ipoly p x = 0" by auto
  hence 0: "poly_uminus p \<noteq> 0" by simp
  show ?thesis
    by (rule representsI[OF _ 0], insert rp, auto)
qed

lemma content_poly_uminus_inner[simp]:
  fixes as :: "'a :: ring_gcd list"
  shows "content (poly_uminus_inner as) = content (Poly as)"
  by (induct as rule: poly_uminus_inner.induct, auto)

text \<open>Multiplicative inverse is represented by @{const reflect_poly}.\<close>

lemma inverse_pow_minus: assumes "x \<noteq> (0 :: 'a :: field)"
  and "i \<le> n"
  shows "inverse x ^ n * x ^ i = inverse x ^ (n - i)"
  using assms by (simp add: field_class.field_divide_inverse power_diff power_inverse)

lemma (in inj_idom_hom) reflect_poly_hom:
  "reflect_poly (map_poly hom p) = map_poly hom (reflect_poly p)"
proof -
  obtain xs where xs: "rev (coeffs p) = xs" by auto
  show ?thesis unfolding reflect_poly_def coeffs_map_poly_hom rev_map
    xs by (induct xs, auto simp: hom_distribs)
qed

lemma ipoly_reflect_poly: assumes x: "(x :: 'a :: field_char_0) \<noteq> 0"
  shows "ipoly (reflect_poly p) x = x ^ (degree p) * ipoly p (inverse x)" (is "?l = ?r")
proof -
  let ?or = "of_int :: int \<Rightarrow> 'a"
  have hom: "inj_idom_hom ?or" ..
  show ?thesis
    using poly_reflect_poly_nz[OF x, of "map_poly ?or p"] by (simp add: inj_idom_hom.reflect_poly_hom[OF hom])
qed

lemma represents_inverse: assumes x: "x \<noteq> 0"
  and alg: "p represents x"
  shows "(reflect_poly p) represents (inverse x)"
proof (intro representsI)
  from representsD[OF alg] have "p \<noteq> 0" and rp: "ipoly p x = 0" by auto
  then show "reflect_poly p \<noteq> 0" by (metis reflect_poly_0 reflect_poly_at_0_eq_0_iff)
  show "ipoly (reflect_poly p) (inverse x) = 0" by (subst ipoly_reflect_poly, insert x, auto simp:rp)
qed

lemma inverse_roots: assumes x: "(x :: 'a :: field_char_0) \<noteq> 0"
  shows "ipoly (reflect_poly p) x = 0 \<longleftrightarrow> ipoly p (inverse x) = 0"
  using x by (auto simp: ipoly_reflect_poly)

context
  fixes n :: nat
begin
text \<open>Polynomial for n-th root.\<close>

definition poly_nth_root :: "'a :: idom poly \<Rightarrow> 'a poly" where
  "poly_nth_root p = p \<circ>\<^sub>p monom 1 n"

lemma ipoly_nth_root:
  fixes x :: "'a :: idom"
  shows "ipoly (poly_nth_root p) x = ipoly p (x ^ n)"
  unfolding poly_nth_root_def ipoly_poly_compose by (simp add: map_poly_monom poly_monom)

context
  assumes n: "n \<noteq> 0"
begin
lemma poly_nth_root_0[simp]: "poly_nth_root p = 0 \<longleftrightarrow> p = 0"
  unfolding poly_nth_root_def
  by (rule pcompose_eq_0, insert n, auto simp: degree_monom_eq)

lemma represents_nth_root:
  assumes y: "y^n = x" and alg: "p represents x"
  shows "(poly_nth_root p) represents y"
proof -
  from representsD[OF alg] have "p \<noteq> 0" and rp: "ipoly p x = 0" by auto
  hence 0: "poly_nth_root p \<noteq> 0" by simp
  show ?thesis
    by (rule representsI[OF _ 0], unfold ipoly_nth_root y rp, simp)
qed

lemma represents_nth_root_odd_real:
  assumes alg: "p represents x" and odd: "odd n"
  shows "(poly_nth_root p) represents (root n x)"
  by (rule represents_nth_root[OF odd_real_root_pow[OF odd] alg])

lemma represents_nth_root_pos_real:
  assumes alg: "p represents x" and pos: "x > 0"
  shows "(poly_nth_root p) represents (root n x)"
proof -
  from n have id: "Suc (n - 1) = n" by auto
  show ?thesis
  proof (rule represents_nth_root[OF _ alg])
    show "root n x ^ n = x" using id pos by auto
  qed
qed

lemma represents_nth_root_neg_real:
  assumes alg: "p represents x" and neg: "x < 0"
  shows "(poly_uminus (poly_nth_root (poly_uminus p))) represents (root n x)"
proof -
  have rt: "root n x = - root n (-x)" unfolding real_root_minus by simp
  show ?thesis unfolding rt
    by (rule represents_uminus[OF represents_nth_root_pos_real[OF represents_uminus[OF alg]]], insert neg, auto)
qed
end
end

lemma represents_csqrt:
  assumes alg: "p represents x" shows "(poly_nth_root 2 p) represents (csqrt x)"
  by (rule represents_nth_root[OF _ _ alg], auto)

lemma represents_sqrt:
  assumes alg: "p represents x" and pos: "x \<ge> 0"
  shows "(poly_nth_root 2 p) represents (sqrt x)"
  by (rule represents_nth_root[OF _ _ alg], insert pos, auto)

lemma represents_degree:
  assumes "p represents x" shows "degree p \<noteq> 0"
proof
  assume "degree p = 0"
  from degree0_coeffs[OF this] obtain c where p: "p = [:c:]" by auto
  from assms[unfolded represents_def p]
  show False by auto
qed

text \<open>Polynomial for multiplying a rational number with an algebraic number.\<close>

definition poly_mult_rat_main where
  "poly_mult_rat_main n d (f :: 'a :: idom poly) = (let fs = coeffs f; k = length fs in
    poly_of_list (map (\<lambda> (fi, i). fi * d ^ i * n ^ (k - Suc i)) (zip fs [0 ..< k])))"

definition poly_mult_rat :: "rat \<Rightarrow> int poly \<Rightarrow> int poly" where
  "poly_mult_rat r p \<equiv> case quotient_of r of (n,d) \<Rightarrow> poly_mult_rat_main n d p"

lemma coeff_poly_mult_rat_main: "coeff (poly_mult_rat_main n d f) i = coeff f i * n ^ (degree f - i) * d ^ i"
proof -
  have id: "coeff (poly_mult_rat_main n d f) i = (coeff f i * d ^ i) * n ^ (length (coeffs f) - Suc i)"
    unfolding poly_mult_rat_main_def Let_def poly_of_list_def coeff_Poly
    unfolding nth_default_coeffs_eq[symmetric]
    unfolding nth_default_def by auto
  show ?thesis unfolding id by (simp add: degree_eq_length_coeffs)
qed

lemma degree_poly_mult_rat_main: "n \<noteq> 0 \<Longrightarrow> degree (poly_mult_rat_main n d f) = (if d = 0 then 0 else degree f)"
proof (cases "d = 0")
  case True
  thus ?thesis unfolding degree_def unfolding coeff_poly_mult_rat_main by simp
next
  case False
  hence id: "(d = 0) = False" by simp
  show "n \<noteq> 0 \<Longrightarrow> ?thesis" unfolding degree_def coeff_poly_mult_rat_main id
    by (simp add: id)
qed

lemma ipoly_mult_rat_main:
  fixes x :: "'a :: {field,ring_char_0}"
  assumes "d \<noteq> 0" and "n \<noteq> 0"
  shows "ipoly (poly_mult_rat_main n d p) x = of_int n ^ degree p * ipoly p (x * of_int d / of_int n)"
proof -
  from assms have d: "(if d = 0 then t else f) = f" for t f :: 'b by simp
  show ?thesis
    unfolding poly_altdef of_int_hom.coeff_map_poly_hom mult.assoc[symmetric] of_int_mult[symmetric]
      sum_distrib_left
    unfolding of_int_hom.degree_map_poly_hom degree_poly_mult_rat_main[OF assms(2)] d
  proof (rule sum.cong[OF refl])
    fix i
    assume "i \<in> {..degree p}"
    hence i: "i \<le> degree p" by auto
    hence id: "of_int n ^ (degree p - i) = (of_int n ^ degree p / of_int n ^ i :: 'a)"
      by (simp add: assms(2) power_diff)
    thus "of_int (coeff (poly_mult_rat_main n d p) i) * x ^ i = of_int n ^ degree p * of_int (coeff p i) * (x * of_int d / of_int n) ^ i"
      unfolding coeff_poly_mult_rat_main
      by (simp add: field_simps)
  qed
qed

lemma degree_poly_mult_rat[simp]: assumes "r \<noteq> 0" shows "degree (poly_mult_rat r p) = degree p"
proof -
  obtain n d where quot: "quotient_of r = (n,d)" by force
  from quotient_of_div[OF quot] have r: "r = of_int n / of_int d" by auto
  from quotient_of_denom_pos[OF quot] have d: "d \<noteq> 0" by auto
  with assms r have n0: "n \<noteq> 0" by simp
  from quot have id: "poly_mult_rat r p = poly_mult_rat_main n d p"  unfolding poly_mult_rat_def by simp
  show ?thesis unfolding id degree_poly_mult_rat_main[OF n0] using d by simp
qed

lemma poly_mult_rat_main_0[simp]:
  assumes "n \<noteq> 0" "d \<noteq> 0" shows "poly_mult_rat_main n d p = 0 \<longleftrightarrow> p = 0"
proof
  assume "p = 0" thus "poly_mult_rat_main n d p = 0"
    by (simp add: poly_mult_rat_main_def)
next
  assume 0: "poly_mult_rat_main n d p = 0"
  {
    fix i
    from 0 have "coeff (poly_mult_rat_main n d p) i = 0" by simp
    hence "coeff p i = 0" unfolding coeff_poly_mult_rat_main using assms by simp
  }
  thus "p = 0" by (intro poly_eqI, auto)
qed

lemma poly_mult_rat_0[simp]: assumes r0: "r \<noteq> 0" shows "poly_mult_rat r p = 0 \<longleftrightarrow> p = 0"
proof -
  obtain n d where quot: "quotient_of r = (n,d)" by force
  from quotient_of_div[OF quot] have r: "r = of_int n / of_int d" by auto
  from quotient_of_denom_pos[OF quot] have d: "d \<noteq> 0" by auto
  from r r0 have n: "n \<noteq> 0" by simp
  from quot have id: "poly_mult_rat r p = poly_mult_rat_main n d p"  unfolding poly_mult_rat_def by simp
  show ?thesis unfolding id using n d by simp
qed

lemma represents_mult_rat:
  assumes r: "r \<noteq> 0" and "p represents x" shows "(poly_mult_rat r p) represents (of_rat r * x)"
  using assms
  unfolding represents_def ipoly_mult_rat[OF r] by (simp add: field_simps)

text \<open>Polynomial for adding a rational number on an algebraic number.
  Again, we do not have to factor afterwards.\<close>

definition poly_add_rat :: "rat \<Rightarrow> int poly \<Rightarrow> int poly" where
  "poly_add_rat r p \<equiv> case quotient_of r of (n,d) \<Rightarrow>
     (poly_mult_rat_main d 1 p \<circ>\<^sub>p [:-n,d:])"

lemma poly_add_rat_code[code]: "poly_add_rat r p \<equiv> case quotient_of r of (n,d) \<Rightarrow>
     let p' = (let fs = coeffs p; k = length fs in poly_of_list (map (\<lambda>(fi, i). fi * d ^ (k - Suc i)) (zip fs [0..<k])));
         p'' = p' \<circ>\<^sub>p [:-n,d:]
      in p''"
  unfolding poly_add_rat_def poly_mult_rat_main_def Let_def by simp

lemma degree_poly_add_rat[simp]: "degree (poly_add_rat r p) = degree p"
proof -
  obtain n d where quot: "quotient_of r = (n,d)" by force
  from quotient_of_div[OF quot] have r: "r = of_int n / of_int d" by auto
  from quotient_of_denom_pos[OF quot] have d: "d \<noteq> 0" "d > 0" by auto
  show ?thesis unfolding poly_add_rat_def quot split
    by (simp add: degree_poly_mult_rat_main d)
qed

lemma ipoly_add_rat: "ipoly (poly_add_rat r p) x = (of_int (snd (quotient_of r)) ^ degree p) * ipoly p (x - of_rat r)"
proof -
  obtain n d where quot: "quotient_of r = (n,d)" by force
  from quotient_of_div[OF quot] have r: "r = of_int n / of_int d" by auto
  from quotient_of_denom_pos[OF quot] have d: "d \<noteq> 0" "d > 0" by auto
  have id: "ipoly [:- n, 1:] (x / of_int d :: 'a) = - of_int n + x / of_int d" by simp
  show ?thesis unfolding poly_add_rat_def quot split
    by (simp add: ipoly_mult_rat_main ipoly_poly_compose d r degree_poly_mult_rat_main field_simps id of_rat_divide)
qed

lemma poly_add_rat_0[simp]: "poly_add_rat r p = 0 \<longleftrightarrow> p = 0"
proof -
  obtain n d where quot: "quotient_of r = (n,d)" by force
  from quotient_of_div[OF quot] have r: "r = of_int n / of_int d" by auto
  from quotient_of_denom_pos[OF quot] have d: "d \<noteq> 0" "d > 0" by auto
  show ?thesis unfolding poly_add_rat_def quot split
    by (simp add: d pcompose_eq_0)
qed

lemma add_rat_roots: "ipoly (poly_add_rat r p) x = 0 \<longleftrightarrow> ipoly p (x - of_rat r) = 0"
  unfolding ipoly_add_rat using quotient_of_nonzero by auto

lemma represents_add_rat:
  assumes "p represents x" shows "(poly_add_rat r p) represents (of_rat r + x)"
  using assms unfolding represents_def ipoly_add_rat by simp

(* TODO: move? *)
lemmas pos_mult[simplified,simp] = mult_less_cancel_left_pos[of _ 0] mult_less_cancel_left_pos[of _ _ 0]

lemma ipoly_add_rat_pos_neg:
  "ipoly (poly_add_rat r p) (x::'a::linordered_field) < 0 \<longleftrightarrow> ipoly p (x - of_rat r) < 0"
  "ipoly (poly_add_rat r p) (x::'a::linordered_field) > 0 \<longleftrightarrow> ipoly p (x - of_rat r) > 0"
  using quotient_of_nonzero unfolding ipoly_add_rat by auto

lemma sgn_ipoly_add_rat[simp]:
  "sgn (ipoly (poly_add_rat r p) (x::'a::linordered_field)) = sgn (ipoly p (x - of_rat r))" (is "sgn ?l = sgn ?r")
  using ipoly_add_rat_pos_neg[of r p x]
  by (cases ?r "0::'a" rule: linorder_cases,auto simp:  sgn_1_pos sgn_1_neg sgn_eq_0_iff)

lemma deg_nonzero_represents:
  assumes deg: "degree p \<noteq> 0" shows "\<exists> x :: complex. p represents x"
proof -
  let ?p = "of_int_poly p :: complex poly"
  from fundamental_theorem_algebra_factorized[of ?p]
  obtain as c where id: "smult c (\<Prod>a\<leftarrow>as. [:- a, 1:]) = ?p"
    and len: "length as = degree ?p" by blast
  have "degree ?p = degree p" by simp
  with deg len obtain b bs where as: "as = b # bs" by (cases as, auto)
  have "p represents b" unfolding represents_def id[symmetric] as using deg by auto
  thus ?thesis by blast
qed

end


theory Chapter5
imports Main
begin

inductive star :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"  for r where
refl:  "star r x x" |
step:  "r x y \<Longrightarrow> star r y z \<Longrightarrow> star r x z"

inductive iter :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" for r where
iter_0: "iter r 0 x x" |
iter_Suc: "r x y \<Longrightarrow> iter r n y z \<Longrightarrow> iter r (Suc n) x z"

text \<open>
\section*{Chapter 5}

\exercise
Give a readable, structured proof of the following lemma:
\<close>
lemma assumes T: "\<forall>x y. T x y \<or> T y x"
  and A: "\<forall>x y. A x y \<and> A y x \<longrightarrow> x = y"
  and TA: "\<forall>x y. T x y \<longrightarrow> A x y" and "A x y"
  shows "T x y"
proof-
  have "T x y \<or> T y x"
    using T by blast
  hence "T x y \<or> A y x"
    using TA by blast
  hence "T x y \<or> A x y \<and> A y x"
    using assms(4) by blast
  hence "T x y \<or> x = y"
    using A by blast
  hence "T x y"
    using T by blast
  then show ?thesis .
qed

(* your definition/proof here *)

text \<open>
Each step should use at most one of the assumptions @{text T}, @{text A}
or @{text TA}.
\endexercise

\exercise
Give a readable, structured proof of the following lemma:
\<close>
lemma "(\<exists>ys zs. xs = ys @ zs \<and> length ys = length zs)
      \<or> (\<exists>ys zs. xs = ys @ zs \<and> length ys = length zs + 1)"
(* your definition/proof here *)

text{*
Hint: There are predefined functions @{const take} and {const drop} of type
@{typ "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"} such that @{text"take k [x\<^sub>1,\<dots>] = [x\<^sub>1,\<dots>,x\<^sub>k]"}
and @{text"drop k [x\<^sub>1,\<dots>] = [x\<^bsub>k+1\<^esub>,\<dots>]"}. Let sledgehammer find and apply
the relevant @{const take} and @{const drop} lemmas for you.
\endexercise

\exercise
Give a structured proof by rule inversion:
*}
inductive ev :: "nat \<Rightarrow> bool" where
ev0: "ev 0" |
evSS: "ev n \<Longrightarrow> ev(Suc(Suc n))"
lemma assumes a: "ev(Suc(Suc n))" shows "ev n"
(* your definition/proof here *)

text{*
\exercise
Give a structured proof by rule inversions:
*}

lemma "\<not> ev(Suc(Suc(Suc 0)))"
(* your definition/proof here *)

text{*
If there are no cases to be proved you can close
a proof immediateley with \isacom{qed}.
\endexercise

\exercise
Recall predicate @{const star} from Section 4.5 and @{const iter}
from Exercise~\ref{exe:iter}.
*}

lemma "iter r n x y \<Longrightarrow> star r x y"
(* your definition/proof here *)

text{*
Prove this lemma in a structured style, do not just sledgehammer each case of the
required induction.
\endexercise

\exercise
Define a recursive function
*}

fun elems :: "'a list \<Rightarrow> 'a set" where
(* your definition/proof here *)

text{* that collects all elements of a list into a set. Prove *}

lemma "x \<in> elems xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> elems ys"
(* your definition/proof here *)

text{*
\endexercise

\exercise
Extend Exercise~\ref{exe:cfg} with a function that checks if some
\mbox{@{text "alpha list"}} is a balanced
string of parentheses. More precisely, define a recursive function *}
(* your definition/proof here *)
fun balanced :: "nat \<Rightarrow> alpha list \<Rightarrow> bool" where
(* your definition/proof here *)

text{* such that @{term"balanced n w"}
is true iff (informally) @{text"a\<^sup>n @ w \<in> S"}. Formally, prove *}

corollary "balanced n w \<longleftrightarrow> S (replicate n a @ w)"

text{* where @{const replicate} @{text"::"} @{typ"nat \<Rightarrow> 'a \<Rightarrow> 'a list"} is predefined
and @{term"replicate n x"} yields the list @{text"[x, \<dots>, x]"} of length @{text n}.
*}

end


(*  Title:      Prefix_Match.thy
    Authors:    Julius Michaelis, Cornelius Diekmann
*)
theory Prefix_Match
imports IP_Address
begin

section\<open>Prefix Match\<close>
text\<open>
  The main difference between the prefix match defined here and CIDR notation is a validity
  constraint imposed on prefix matches.

  For example, 192.168.42.42/16 is valid CIDR notation whereas for a prefix match,
  it must be 192.168.0.0/16.

  I.e. the last bits of the prefix must be set to zero.
\<close>

context
  notes [[typedef_overloaded]]
begin
  datatype 'a prefix_match = PrefixMatch (pfxm_prefix: "'a::len word") (pfxm_length: nat)
end

definition pfxm_mask :: "'a prefix_match \<Rightarrow> 'a::len word" where
  "pfxm_mask x \<equiv> mask (len_of (TYPE('a)) - pfxm_length x)"

definition valid_prefix :: "('a::len) prefix_match \<Rightarrow> bool" where
  "valid_prefix pf = ((pfxm_mask pf) AND pfxm_prefix pf = 0)"
text\<open>Note that @{const valid_prefix} looks very elegant as a definition. However, it hides something nasty:\<close>
lemma "valid_prefix (PrefixMatch (0::32 word) 42)" by eval

text\<open>When zeroing all least significant bits which exceed the @{const pfxm_length},
     you get a @{const valid_prefix}\<close>
lemma mk_valid_prefix:
  fixes base::"'a::len word"
  shows "valid_prefix (PrefixMatch (base AND NOT mask (len_of TYPE ('a) - len)) len)"
proof -
  have "mask (len - m) AND base AND NOT mask (len - m) = 0"
    for m len and base::"'a::len word" by(simp add: word_bw_lcs)
  thus ?thesis
  by(simp add: valid_prefix_def pfxm_mask_def pfxm_length_def pfxm_prefix_def)
qed

text\<open>The type @{typ "'a prefix_match"} usually requires @{const valid_prefix}.
      When we allow working on arbitrary IPs in CIDR notation,
      we will use the type @{typ "('i::len word \<times> nat)"} directly.\<close>

lemma valid_prefix_00: "valid_prefix (PrefixMatch 0 0)" by (simp add: valid_prefix_def)

definition prefix_match_to_CIDR :: "('i::len) prefix_match \<Rightarrow> ('i word \<times> nat)" where
  "prefix_match_to_CIDR pfx \<equiv> (pfxm_prefix pfx, pfxm_length pfx)"

lemma prefix_match_to_CIDR_def2: "prefix_match_to_CIDR = (\<lambda>pfx. (pfxm_prefix pfx, pfxm_length pfx))"
  unfolding prefix_match_to_CIDR_def fun_eq_iff by simp

definition "prefix_match_dtor m \<equiv> (case m of PrefixMatch p l \<Rightarrow> (p,l))"

text\<open>Some more or less random linear order on prefixes.
     Only used for serialization at the time of this writing.\<close>
instantiation prefix_match :: (len) linorder
begin
	definition "a \<le> b \<longleftrightarrow> (if pfxm_length a = pfxm_length b
                         then pfxm_prefix a \<le> pfxm_prefix b
                         else pfxm_length a > pfxm_length b)"
	definition "a < b \<longleftrightarrow> (a \<noteq> b \<and>
	                       (if pfxm_length a = pfxm_length b
	                        then pfxm_prefix a \<le> pfxm_prefix b
	                        else pfxm_length a > pfxm_length b))"
instance
by standard (auto simp: less_eq_prefix_match_def less_prefix_match_def prefix_match.expand
                  split: if_splits)
end

lemma "sorted_list_of_set
 {PrefixMatch 0 32 :: 32 prefix_match,
  PrefixMatch 42 32,
  PrefixMatch 0 0,
  PrefixMatch 0 1,
  PrefixMatch 12 31} =
    [PrefixMatch 0 32, PrefixMatch 0x2A 32, PrefixMatch 0xC 31, PrefixMatch 0 1, PrefixMatch 0 0]"
  by eval

context
begin

private lemma valid_prefix_E: "valid_prefix pf \<Longrightarrow> ((pfxm_mask pf) AND pfxm_prefix pf = 0)"
  unfolding valid_prefix_def .
private lemma valid_prefix_alt: fixes p::"'a::len prefix_match"
  shows "valid_prefix p = (pfxm_prefix p AND (2 ^ ((len_of TYPE ('a)) - pfxm_length p) - 1) = 0)"
  unfolding valid_prefix_def
  unfolding mask_def
  using word_bw_comms(1)
   arg_cong[where f = "\<lambda>x. (pfxm_prefix p AND x - 1 = 0)"]
   shiftl_1
  unfolding pfxm_prefix_def pfxm_mask_def mask_def
  by metis

subsection\<open>Address Semantics\<close>
  text\<open>Matching on a @{typ "'a::len prefix_match"}. Think of routing tables.\<close>
  definition prefix_match_semantics where
    "prefix_match_semantics m a \<equiv> pfxm_prefix m = (NOT pfxm_mask m) AND a"

lemma same_length_prefixes_distinct: "valid_prefix pfx1 \<Longrightarrow> valid_prefix pfx2 \<Longrightarrow> pfx1 \<noteq> pfx2 \<Longrightarrow> pfxm_length pfx1 = pfxm_length pfx2 \<Longrightarrow> prefix_match_semantics pfx1 w \<Longrightarrow> prefix_match_semantics pfx2 w \<Longrightarrow> False"
  by (simp add: pfxm_mask_def prefix_match.expand prefix_match_semantics_def)

subsection\<open>Relation between prefix and set\<close>

  definition prefix_to_wordset :: "'a::len prefix_match \<Rightarrow> 'a word set" where
    "prefix_to_wordset pfx = {pfxm_prefix pfx .. pfxm_prefix pfx OR pfxm_mask pfx}"

  private lemma pfx_not_empty: "valid_prefix pfx \<Longrightarrow> prefix_to_wordset pfx \<noteq> {}"
    unfolding valid_prefix_def prefix_to_wordset_def by(simp add: le_word_or2)

  lemma zero_prefix_match_all:
    "valid_prefix m \<Longrightarrow> pfxm_length m = 0 \<Longrightarrow> prefix_match_semantics m ip"
    by(simp add: pfxm_mask_def mask_2pm1 valid_prefix_alt prefix_match_semantics_def)

  lemma prefix_to_wordset_subset_ipset_from_cidr:
      "prefix_to_wordset pfx \<subseteq> ipset_from_cidr (pfxm_prefix pfx) (pfxm_length pfx)"
    apply(rule subsetI)
    apply(simp add: prefix_to_wordset_def addr_in_ipset_from_cidr_code)
    apply(intro impI conjI)
     apply (metis (erased, hide_lams) order_trans word_and_le2)
    apply(simp add: pfxm_mask_def)
    done

subsection\<open>Equivalence Proofs\<close>
  theorem prefix_match_semantics_wordset:
    assumes "valid_prefix pfx"
    shows "prefix_match_semantics pfx a \<longleftrightarrow> a \<in> prefix_to_wordset pfx"
    using assms
    unfolding valid_prefix_def pfxm_mask_def prefix_match_semantics_def prefix_to_wordset_def
    apply(cases pfx, rename_tac base len)
    apply(simp)
    apply(drule_tac base=base and len=len and a=a in zero_base_lsb_imp_set_eq_as_bit_operation)
    by (simp)

  private lemma valid_prefix_ipset_from_netmask_ipset_from_cidr:
    shows "ipset_from_netmask (pfxm_prefix pfx) (NOT pfxm_mask pfx) =
            ipset_from_cidr (pfxm_prefix pfx) (pfxm_length pfx)"
    apply(cases pfx)
    apply(simp add: ipset_from_cidr_alt2 pfxm_mask_def)
   done

  lemma prefix_match_semantics_ipset_from_netmask:
    assumes "valid_prefix pfx"
    shows "prefix_match_semantics pfx a \<longleftrightarrow>
            a \<in> ipset_from_netmask (pfxm_prefix pfx) (NOT pfxm_mask pfx)"
    unfolding prefix_match_semantics_wordset[OF assms]
    unfolding valid_prefix_ipset_from_netmask_ipset_from_cidr
    unfolding prefix_to_wordset_def
    apply(subst ipset_from_cidr_base_wellforemd)
     subgoal using assms by(simp add: valid_prefix_def pfxm_mask_def)
    by(simp add: pfxm_mask_def)

  lemma prefix_match_semantics_ipset_from_netmask2:
    assumes "valid_prefix pfx"
    shows "prefix_match_semantics pfx (a :: 'i::len word) \<longleftrightarrow>
            a \<in> ipset_from_cidr (pfxm_prefix pfx) (pfxm_length pfx)"
   unfolding prefix_match_semantics_ipset_from_netmask[OF assms] pfxm_mask_def ipset_from_cidr_def
   by (metis (full_types) NOT_mask_shifted_lenword word_not_not)

  lemma prefix_to_wordset_ipset_from_cidr:
    assumes "valid_prefix (pfx::'a::len prefix_match)"
    shows "prefix_to_wordset pfx = ipset_from_cidr (pfxm_prefix pfx) (pfxm_length pfx)"
  proof -
    have helper3: "(x::'a::len word) OR y = x OR y AND NOT x" for x y by (simp add: word_oa_dist2)
    have prefix_match_semantics_ipset_from_netmask:
           "(prefix_to_wordset pfx) = ipset_from_netmask (pfxm_prefix pfx) (NOT pfxm_mask pfx)"
      unfolding prefix_to_wordset_def ipset_from_netmask_def Let_def
      unfolding word_bool_alg.double_compl
      proof(goal_cases)
        case 1
        have *: "pfxm_prefix pfx AND NOT pfxm_mask pfx = pfxm_prefix pfx"
          unfolding mask_eq_0_eq_x[symmetric]
          using valid_prefix_E[OF assms] word_bw_comms(1)[of "pfxm_prefix pfx"] by simp
        hence **: "pfxm_prefix pfx AND NOT pfxm_mask pfx OR pfxm_mask pfx =
                    pfxm_prefix pfx OR pfxm_mask pfx"
          by simp
        show ?case unfolding * ** ..
      qed
      have "((mask len)::'a::len word) << len_of TYPE('a) - len = ~~ mask (len_of TYPE('a) - len)"
      for len using NOT_mask_shifted_lenword by (metis word_not_not)
      from this[of "(pfxm_length pfx)"] have mask_def2_symmetric:
        "((mask (pfxm_length pfx)::'a::len word) << len_of TYPE('a) - pfxm_length pfx) =
          NOT pfxm_mask pfx"
        unfolding pfxm_mask_def by simp

      have ipset_from_netmask_prefix:
        "ipset_from_netmask (pfxm_prefix pfx) (NOT pfxm_mask pfx) =
          ipset_from_cidr (pfxm_prefix pfx) (pfxm_length pfx)"
       unfolding ipset_from_netmask_def ipset_from_cidr_alt
       unfolding pfxm_mask_def[symmetric]
       unfolding mask_def2_symmetric
       apply(simp)
       unfolding Let_def
       using assms[unfolded valid_prefix_def]
       by (metis helper3 word_bw_comms(2))

      show ?thesis by (metis ipset_from_netmask_prefix local.prefix_match_semantics_ipset_from_netmask)
  qed

  definition prefix_to_wordinterval :: "'a::len prefix_match \<Rightarrow> 'a wordinterval" where
    "prefix_to_wordinterval pfx \<equiv> WordInterval (pfxm_prefix pfx) (pfxm_prefix pfx OR pfxm_mask pfx)"

  lemma prefix_to_wordinterval_set_eq[simp]:
    "wordinterval_to_set (prefix_to_wordinterval pfx) = prefix_to_wordset pfx"
    unfolding prefix_to_wordinterval_def prefix_to_wordset_def by simp

  lemma prefix_to_wordinterval_def2:
    "prefix_to_wordinterval pfx =
      iprange_interval ((pfxm_prefix pfx), (pfxm_prefix pfx OR pfxm_mask pfx))"
    unfolding iprange_interval.simps prefix_to_wordinterval_def by simp
  corollary prefix_to_wordinterval_ipset_from_cidr: "valid_prefix pfx \<Longrightarrow>
    wordinterval_to_set (prefix_to_wordinterval pfx) =
      ipset_from_cidr (pfxm_prefix pfx) (pfxm_length pfx)"
  using prefix_to_wordset_ipset_from_cidr prefix_to_wordinterval_set_eq by auto
end

lemma prefix_never_empty:
  fixes d:: "'a::len prefix_match"
  shows"\<not> wordinterval_empty (prefix_to_wordinterval d)"
by (simp add: le_word_or2 prefix_to_wordinterval_def)

text\<open>Getting a lowest element\<close>
  lemma ipset_from_cidr_lowest: "a \<in> ipset_from_cidr a n"
    using ip_cidr_set_def ipset_from_cidr_eq_ip_cidr_set by blast

  (*this is why I call the previous lemma 'lowest'*)
  lemma "valid_prefix (PrefixMatch a n) \<Longrightarrow> is_lowest_element a (ipset_from_cidr a n)"
    apply(simp add: is_lowest_element_def ipset_from_cidr_lowest)
    apply(simp add: ipset_from_cidr_eq_ip_cidr_set ip_cidr_set_def)
    apply(simp add: valid_prefix_def pfxm_mask_def)
    by (metis diff_zero eq_iff mask_out_sub_mask word_and_le2 word_bw_comms(1))

end


(* Property from Case-Analysis for Rippling and Inductive Proof,
   Moa Johansson, Lucas Dixon and Alan Bundy, ITP 2010.
   This Isabelle theory is produced using the TIP tool offered at the following website:
     https://github.com/tip-org/tools
   This file was originally provided as part of TIP benchmark at the following website:
     https://github.com/tip-org/benchmarks
   Yutaka Nagashima at CIIRC, CTU changed the TIP output theory file slightly
   to make it compatible with Isabelle2017.*)
  theory TIP_prop_01
imports "../../Test_Base"
begin

datatype 'a list = nil2 | cons2 "'a" "'a list"

datatype Nat = Z | S "Nat"

fun x :: "'a list => 'a list => 'a list" where
"x (nil2) z = z"
| "x (cons2 z2 xs) z = cons2 z2 (x xs z)"

fun take :: "Nat => 'a list => 'a list" where
"take (Z) z = nil2"
| "take (S z2) (nil2) = nil2"
| "take (S z2) (cons2 x2 x3) = cons2 x2 (take z2 x3)"

fun drop :: "Nat => 'a list => 'a list" where
"drop (Z) z = z"
| "drop (S z2) (nil2) = nil2"
| "drop (S z2) (cons2 x2 x3) = drop z2 x3"

theorem property0 :
  "((x (take n xs) (drop n xs)) = xs)"
  oops

end


theory tut2
imports Main

begin
(*Exercise 2*)
lemma "(\<forall>x. P x \<longrightarrow> Q) \<longrightarrow> (\<exists>x. P x \<longrightarrow> Q)"
  apply (rule impI)
  apply (erule allE)
  apply (rule exI)
  apply assumption
  done

lemma "(\<not>(\<exists> x. P x)) \<longrightarrow> (\<forall> x. \<not>P x)"
  apply (rule impI)
  apply (rule allI)
  apply (rule notI)
  apply (erule notE)
  apply (rule exI)
  apply assumption
  done

lemma "\<not>(\<forall>x. P x) \<longrightarrow> (\<exists>x. \<not>P x)"
  apply (rule impI)
  apply (rule ccontr)
  apply (erule notE)
  apply (rule allI)
  apply (rule ccontr)
  apply (erule notE)
  apply (rule exI)
  apply assumption
  done

end


(*<*) theory SV2 imports Main begin (*>*)

text {* 2005 Paper 5 Question 9 *}

lemma "((\<forall>x. (P(x) \<or> Q \<longrightarrow> \<not>R(x)) )\<and> (\<forall>x. ((Q \<longrightarrow> \<not>S(x)) \<longrightarrow> (P(x) \<and> R(x))))) \<longrightarrow> (\<forall>x. S(x))"
  apply(rule impI)
  apply(rule allI)
  apply(erule conjE)
  apply(erule allE)+
  apply (erule impE)
   apply (rule classical)
   apply(rule disjI1)
   apply(erule impE)
    apply (rule impI)
    apply (erule notE)
    apply (rule disjI2)
    apply assumption
   apply(erule conjE)
   apply assumption
  apply(rule classical)
  apply(erule impE)
  apply (rule impI)
   apply assumption
  apply(erule notE)
  apply (erule conjE)
  apply assumption
  done

lemma "\<not>( (\<not>P a)\<and> (Q a) \<and> (R b) \<and> (S b) \<and>(\<forall> x y. \<not>(Q x) \<or> (P x ) \<or> (\<not>R y) \<or>(\<not>Q y)) \<and> (\<forall>x . (\<not>S x) \<or> (\<not>R x) \<or> (Q x)) )"
  apply (rule notI)
  apply(erule conjE)+
  apply (erule notE)
  apply (erule allE)+
  apply (erule disjE)+
    apply (erule notE)
    apply assumption
   apply(erule notE)
   apply assumption
  apply (erule disjE)+
  apply (erule notE)
    apply assumption
  apply(erule notE)
   apply assumption
  apply(erule disjE)+
    apply assumption
   apply(erule notE)
   apply assumption
  apply (erule disjE)
   apply assumption
  apply (erule disjE)
  apply (erule notE)
   apply assumption
  apply(erule notE)
  apply assumption
  done