import category_theory.limits.limits
import category_theory.limits.shapes
import category_theory.yoneda
import category_theory.opposites
import category_theory.types
import category_theory.limits.types
-- set_option trace.simplify.rewrite true
run_cmd mk_simp_attr `PRODUCT    -----  BOF BOF
meta def PRODUCT_CAT  : tactic unit :=
`[  try {simp only with PRODUCT}]
run_cmd add_interactive [`PRODUCT_CAT]

universes v u
open category_theory
open category_theory.limits
open category_theory.category
open opposite
namespace lem --------------------------------------------------------------------
variables {C : Type u}
variables [𝒞 : category.{v} C]
variables  [has_binary_products.{v} C][has_terminal.{v} C]
include 𝒞
attribute [PRODUCT] category.assoc category.id_comp category.comp_id
@[PRODUCT] lemma prod_left_def {X Y : C} : limit.π (pair X Y) walking_pair.left = limits.prod.fst := rfl
@[PRODUCT] lemma prod_right_def {X Y : C} : limit.π (pair X Y) walking_pair.right = limits.prod.snd := rfl
lemma prod.hom_ext {A X Y : C} {a b : A ⟶ X ⨯ Y} (h1 : a ≫ limits.prod.fst = b ≫ limits.prod.fst) (h2 : a ≫ limits.prod.snd = b ≫ limits.prod.snd) : a = b :=
begin
  apply limit.hom_ext,
  rintros (_ | _),
  rw prod_left_def,
  exact h1,
  rw prod_right_def,
  exact h2,
end
@[PRODUCT, reassoc] lemma prod.lift_fst {Y A B : C} (f : Y ⟶ A) (g : Y ⟶ B) : prod.lift f g ≫ category_theory.limits.prod.fst = f :=
limit.lift_π (binary_fan.mk f g) _

attribute [PRODUCT] prod.lift_fst_assoc

@[PRODUCT,reassoc]lemma prod.lift_snd {Y A B : C} (f : Y ⟶ A) (g : Y ⟶ B) : prod.lift f g ≫ category_theory.limits.prod.snd = g :=
limit.lift_π (binary_fan.mk f g) _
attribute [PRODUCT] prod.lift_snd_assoc
end lem
namespace Product_stuff
notation f ` ⊗ `:20 g :20 := category_theory.limits.prod.map f g  ---- 20
notation  `T`C :20 := (terminal C)
notation   `T`X : 20 := (terminal.from X)
notation f ` | `:20 g :20 :=  prod.lift f g
notation `π1` := limits.prod.fst
notation `π2` := limits.prod.snd

variables {C : Type u}
variables [𝒞 : category.{v} C]
variables [has_binary_products.{v} C][has_terminal.{v} C]
include 𝒞
variables (X :C)
open lem           -------------------------------------------------------
/-
     π notation for projection
-/
example  {Y A B : C} (f : Y ⟶ A) (g : Y ⟶ B) : ( f | g) ≫ π1 = f  :=   prod.lift_fst f g
/-
     we can type π : A ⨯ B ⟶ B if we need
-/
example  {Y A B : C} (f : Y ⟶ A) (g : Y ⟶ B) : ( f | g) ≫ (π2 : A ⨯ B ⟶ B) = g := prod.lift_snd f g

example  {A X Y : C} {a b : A ⟶ X ⨯ Y} (h1 : a ≫ π1  = b ≫ π1 ) (h2 : a ≫ π2  = b ≫ π2)  : a = b :=  prod.hom_ext h1 h2

-- use the tatict
example  {Y A B : C} (f : Y ⟶ A) (g : Y ⟶ B) : ( f | g) ≫ (π2 : A ⨯ B ⟶ B) = g := by PRODUCT_CAT

@[PRODUCT]lemma prod.left_composition{Z' Z A B : C}(h : Z' ⟶ Z)(f : Z ⟶ A)(g : Z ⟶ B)  :
               h ≫ (f | g)  = (h ≫ f | h ≫ g) :=
begin
     apply prod.hom_ext,   --- Le right member is of the form ( | )  composition π1 π2
     -- PRODUCT_CAT,  PRODUCT_CAT,  --- here assoc
     rw assoc,
     rw prod.lift_fst,
     rw prod.lift_fst,
     rw prod.lift_snd,
     rw assoc,
     rw prod.lift_snd,
end
-- #print notation
@[PRODUCT,reassoc]lemma prod.map_first{X Y Z W : C}(f  : X ⟶ Y)(g  : Z ⟶ W) :  (f ⊗ g) ≫ (π1 : Y ⨯ W ⟶ Y) = π1  ≫ f :=  begin
     exact limit.map_π (map_pair f g) walking_pair.left,
end
attribute [PRODUCT] prod.map_first_assoc
@[PRODUCT,reassoc]lemma prod.map_second{X Y Z W : C}(f  : X ⟶ Y)(g  : Z ⟶ W) :  (f ⊗ g) ≫ π2 = π2 ≫ g :=  begin
     exact limit.map_π (map_pair f g) walking_pair.right,
end
attribute [PRODUCT] prod.map_second_assoc
@[PRODUCT]lemma  prod.otimes_is_prod {X Y Z W : C}(f  : X ⟶ Y)(g  : Z ⟶ W) : (f ⊗ g) = ( π1  ≫ f | π2 ≫ g ) := begin
     apply prod.hom_ext,
     PRODUCT_CAT, PRODUCT_CAT,
     -- rw prod.lift_fst,
     -- rw prod.map_first,
     -- rw prod.lift_snd,
     -- rw prod.map_second,
end
-- notation π1`(`X `x` Y`)` := (limits.prod.fst : X⨯Y ⟶ X)
@[PRODUCT]lemma prod.map_ext{X Y Z W : C}(f1 f2  : X ⟶ Y)(g1 g2  : Z ⟶ W) :  (f1 ⊗ g1) = (f2 ⊗ g2) →
(π1 : X ⨯ Z ⟶ X) ≫ f1 = (π1 : X ⨯ Z ⟶ X)  ≫ f2 := λ certif, begin
     iterate 2 {rw prod.otimes_is_prod at certif},
     rw ← prod.map_first ( f1)  (g1),
     rw ← prod.map_first ( f2)  (g2),
     iterate 2 {rw prod.otimes_is_prod},
     rw certif,
end
lemma prod.map_eq {X Y Z W : C}(f1 f2  : X ⟶ Y)(g1 g2  : Z ⟶ W) :
 ((π1 : X ⨯ Z ⟶ X) ≫ f1 = (π1 : X ⨯ Z ⟶ X)  ≫ f2) →
 ((π2 : X ⨯ Z ⟶ Z) ≫ g1 = (π2 : X ⨯ Z ⟶ Z)  ≫ g2) → ((f1 ⊗ g1) = (f2 ⊗ g2)) := λ certif1 certif2, begin
     iterate 2 {rw prod.otimes_is_prod},
--     PRODUCT_CAT,
    rw certif1, rw certif2,
end

@[PRODUCT,reassoc]lemma prod.prod_otimes {X Y Z : C} (f :  Y ⟶ X) (g : X ⟶ Z ) :
     (f | 𝟙 Y) ≫ (g ⊗ (𝟙 Y)) = (f ≫ g | 𝟙 Y) :=

begin
     apply prod.hom_ext,
     PRODUCT_CAT,PRODUCT_CAT,     ---------------------- PROBLEME With the tatict HEEEEEEERRRRRRE
     -- rw [prod.lift_fst],
     -- rw  assoc,
     -- rw prod.map_first,
     -- rw ← assoc,               ----- ← assoc here  Problem ?
     -- rw prod.lift_fst,
     -- tidy, -- super - power tidy
end
attribute [PRODUCT] prod.prod_otimes_assoc

@[PRODUCT,reassoc] lemma prod.prod_comp_otimes {A1 A2 X1 X2 Z: C} (f1 :  Z ⟶ A1)(f2  : Z ⟶ A2)
(g1 :A1  ⟶  X1 )(g2 : A2 ⟶ X2) :
     (f1 | f2) ≫ (g1 ⊗ g2)  = (f1 ≫ g1 | f2 ≫ g2 ) := begin
     apply prod.hom_ext,
     PRODUCT_CAT,PRODUCT_CAT,
     end
attribute [PRODUCT] prod.prod_comp_otimes_assoc

def Yo (R : C)(A :C) := (yoneda.obj A).obj (op R)
def Yo_ (R : C) {A B : C}(φ : A ⟶ B) := ((yoneda.map φ).app (op R) : Yo R A ⟶ Yo R B)
-- Good notation for yoneda stuff :
-- We fix V : C and we denote by
-- R[X] := yoneda.obj X).obj (op R) and φ : A  ⟶ B (in C) R ⟦  φ ⟧   : R⟦ A⟧  → R⟦ B⟧  in type v
notation R`⟦`A`⟧`:20 := Yo R A  -- notation ??
notation R`<`φ`>`:20   := Yo_ R φ  --
def Yoneda_preserve_product (Y : C)(A B : C) :
     Y ⟦ A ⨯ B ⟧  ≅ Y ⟦ A ⟧  ⨯ Y⟦ B ⟧   :=
{ hom := prod.lift
    (λ f, f ≫ π1)
    (λ f, f ≫ π2),
  inv := λ f : (Y ⟶ A) ⨯ (Y ⟶ B),
    (prod.lift
      ((@category_theory.limits.prod.fst _ _ (Y ⟶ A) (Y ⟶ B) _ : ((Y ⟶ A) ⨯ (Y ⟶ B)) → (Y ⟶ A)) f)
      ((@category_theory.limits.prod.snd _ _ (Y ⟶ A) _ _ : ((Y ⟶ A) ⨯ (Y ⟶ B)) → (Y ⟶ B)) f : Y ⟶ B)),
  hom_inv_id' := begin
    ext f,
    cases j,
    { simp, refl},
    { simp, refl}
  end,
  inv_hom_id' := begin
    apply lem.prod.hom_ext,
    { rw assoc, rw lem.prod.lift_fst, obviously},
    { rw assoc, rw lem.prod.lift_snd, obviously}
  end
}
--- Here it just sugar
@[PRODUCT]lemma yoneda_sugar.composition (R : C) {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) : R < f ≫ g > =( R< f >) ≫ (R < g >)
 :=  begin
     unfold Yo_,
     simp,
 end
def yoneda_sugar.conv {R : C}{A : C}(g : R⟦ A⟧ ) : R ⟶ A := g
def yoneda_sugar.prod (R : C)(A B : C) : R⟦ A ⨯ B⟧  ≅ R⟦ A⟧  ⨯ R⟦ B⟧ := begin
     exact Yoneda_preserve_product R A B,
end
@[PRODUCT]lemma yoneda_sugar.prod.hom (R : C)(A B : C) :
     (yoneda_sugar.prod R A B).hom =  (R < (π1  : A ⨯ B ⟶ A) > | R < (π2  : A ⨯ B ⟶ B)> ) := rfl

@[PRODUCT]lemma yoneda_sugar.prod.first (R : C)(A B : C) :
 (yoneda_sugar.prod R A B).hom ≫ π1 = (R < π1 >) :=
 begin
     exact rfl,
 end
 @[PRODUCT]lemma yoneda_sugar.prod.hom_inv (R : C)(A B : C) :
     (yoneda_sugar.prod R A B).hom ≫ (yoneda_sugar.prod R A B).inv = 𝟙 (R⟦ A ⨯ B⟧) :=
     (Yoneda_preserve_product R A B).hom_inv_id'
 @[PRODUCT]lemma yoneda_sugar.prod.inv_hom (R : C)(A B : C) :
     (yoneda_sugar.prod R A B).inv ≫ (yoneda_sugar.prod R A B).hom = 𝟙 (R⟦ A⟧  ⨯ R⟦ B⟧) :=
     (Yoneda_preserve_product R A B).inv_hom_id'
 @[PRODUCT]lemma yoneda_sugar.prod.second (R : C)(A B : C) :
  (yoneda_sugar.prod R A B).hom ≫ limits.prod.snd = (R < limits.prod.snd >) := rfl

@[PRODUCT]lemma yoneda_sugar.id (R : C)(A : C) : R < 𝟙 A > = 𝟙 (R⟦ A⟧ ) := begin
     funext,
     exact comp_id C g,
     -- have T : ((yoneda.map (𝟙 A)).app (op R)) g = (g ≫ (𝟙 A)),
end

lemma yoneda_sugar_prod (R : C)(A B : C)(X :C)(f : X ⟶ A)(g : X ⟶ B) :
      R < (f | g) > ≫ (yoneda_sugar.prod R A B).hom  =  (R < f > | R < g > ) :=  -- the  ≫  is  :/
     begin
          PRODUCT_CAT,
          -- rw  yoneda_sugar.prod.hom R A B,
          -- rw prod.left_composition,
          iterate 2 {rw ← yoneda_sugar.composition},   -- rw ← is the problem ?
          rw lem.prod.lift_fst,
          rw lem.prod.lift_snd,
     end

@[PRODUCT]lemma yoneda_sugar_prod_inv (R : C)(A B : C)(X :C)(f : X ⟶ A)(g : X ⟶ B) :
     R < (f | g) >   =  (R < f > | R < g > ) ≫ (yoneda_sugar.prod R A B).inv :=
     begin
          PRODUCT_CAT,  -- noting
          rw ← yoneda_sugar_prod,
          rw assoc,
          rw yoneda_sugar.prod.hom_inv,
          exact rfl,
     end

lemma  yoneda_sugar.otimes (R : C){Y Z K :C}(f : X ⟶ Y )(g : Z ⟶ K) :
 ( R < (f ⊗ g) > ) = (yoneda_sugar.prod  _ _ _).hom ≫ ((R<f>) ⊗ R<g>) ≫ (yoneda_sugar.prod _ _ _ ).inv := begin
     PRODUCT_CAT,
     -- iterate 2 {rw prod.otimes_is_prod},
     -- rw  yoneda_sugar.prod.hom,
     -- iterate 1 {rw yoneda_sugar_prod_inv},
     rw ← assoc,
     rw prod.left_composition,
     rw ← assoc,
     rw prod.lift_fst,
     rw ← assoc,
     rw prod.lift_snd,
     -- rw yoneda_sugar.composition,
     -- rw yoneda_sugar.composition,
end

@[PRODUCT]lemma yonega_sugar.one_otimes (R :C)(X Y Z: C) (f : X ⟶ Y) :
 (((yoneda_sugar.prod R Z X).inv) ≫ (R <(𝟙 Z ⊗ f ) > ) ≫ (yoneda_sugar.prod R Z Y).hom) = (𝟙 (R⟦Z⟧) ⊗ R<f >) := begin
     rw yoneda_sugar.otimes,
     iterate 3 {rw ← assoc},
     rw yoneda_sugar.prod.inv_hom,
     rw id_comp,
     rw assoc,
     rw yoneda_sugar.prod.inv_hom,
     rw ← yoneda_sugar.id,
     simp,
 end
lemma yonega_sugar.one_otimes' (R :C)(X Y Z: C) (f : X ⟶ Y) :
 ( (R <(𝟙 Z ⊗ f ) > ) ≫ (yoneda_sugar.prod R Z Y).hom) = ((yoneda_sugar.prod R Z X).hom) ≫ (𝟙 (R⟦Z⟧) ⊗ R < f >) := begin
     iterate 2{ rw yoneda_sugar.prod.hom},
     rw prod.left_composition,
     iterate 2{ rw ← yoneda_sugar.composition},
     rw prod.map_first,
     rw prod.map_second,
     rw comp_id,
     rw prod.otimes_is_prod,rw prod.left_composition,rw ← assoc,
     rw prod.lift_fst,rw ←  assoc,rw prod.lift_snd,rw comp_id,
     rw yoneda_sugar.composition,
 end

-- def Y (R : C)(A :C) := (yoneda.obj A).obj (op R)
-- def Y_ (R : C) {A B : C}(φ : A ⟶ B) := ((yoneda.map φ).app (op R) : Y R A ⟶ Y R B)
-- -- Good notation for yoneda stuff :
-- -- We fix V : C and we denote by
-- -- R[X] := yoneda.obj X).obj (op R) and φ : A  ⟶ B (in C) R ⟦  φ ⟧   : R[A] → R[B]  in type v
-- notation R`[`A`]`:20 := Y R A  -- notation ??
-- notation R`<`φ`>`:20   := Y_ R φ  --
-- def Yoneda_preserve_product (Y : C)(A B : C) :
--      Y[A ⨯ B] ≅ (Y[A]) ⨯ (Y[B]) :=
-- { hom := prod.lift
--     (λ f, f ≫ π1)
--     (λ f, f ≫ π2),
--   inv := λ f : (Y ⟶ A) ⨯ (Y ⟶ B),
--     (prod.lift
--       ((@category_theory.limits.prod.fst _ _ (Y ⟶ A) (Y ⟶ B) _ : ((Y ⟶ A) ⨯ (Y ⟶ B)) → (Y ⟶ A)) f)
--       ((@category_theory.limits.prod.snd _ _ (Y ⟶ A) _ _ : ((Y ⟶ A) ⨯ (Y ⟶ B)) → (Y ⟶ B)) f : Y ⟶ B)),
--   hom_inv_id' := begin
--     ext f,
--     cases j,
--     { simp, refl},
--     { simp, refl}
--   end,
--   inv_hom_id' := begin
--     apply lem.prod.hom_ext,
--     { rw assoc, rw lem.prod.lift_fst, obviously},
--     { rw assoc, rw lem.prod.lift_snd, obviously}
--   end
-- }

-- --- Here it just sugar
-- @[PRODUCT,reassoc]lemma yoneda_sugar.composition (R : C) {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) : R < f ≫ g > =( R< f >) ≫ (R < g >)
--  :=  begin
--      unfold Y_,
--      simp,
--  end
--  attribute [PRODUCT] yoneda_sugar.composition_assoc
--  @[PRODUCT,reassoc]lemma yoneda_sugar.composition_rev (R : C) {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) :
--  ( R< f >) ≫ (R < g >) =  (R < f ≫ g > )
--  :=  begin
--      unfold Y_,
--      simp,
--  end
--  attribute [PRODUCT] yoneda_sugar.composition_rev_assoc
-- def yoneda_sugar.conv {R : C}{A : C}(g : R[A]) : R ⟶ A := g
-- def yoneda_sugar.prod (R : C)(A B : C) : R[A ⨯ B] ≅ R[A] ⨯ R[B] := begin
--      exact Yoneda_preserve_product R A B,
-- end
-- @[PRODUCT]lemma yoneda_sugar.prod.hom (R : C)(A B : C) :
--      (yoneda_sugar.prod R A B).hom =  (R < (π1 : A ⨯ B ⟶ A) > | R < (π2 : A ⨯ B ⟶ B)> ) := rfl

-- @[PRODUCT,reassoc]lemma yoneda_sugar.prod.first (R : C)(A B : C) :
--  (yoneda_sugar.prod R A B).hom ≫ π1  = (R < π1 >) :=
--  begin
--      exact rfl,
--  end
--  attribute [PRODUCT] yoneda_sugar.prod.first_assoc
--  @[PRODUCT,reassoc]lemma yoneda_sugar.prod.hom_inv (R : C)(A B : C) :
--      (yoneda_sugar.prod R A B).hom ≫ (yoneda_sugar.prod R A B).inv = 𝟙 (R[ A ⨯ B]) :=
--      (Yoneda_preserve_product R A B).hom_inv_id'
--  attribute [PRODUCT] yoneda_sugar.prod.hom_inv_assoc
--  @[PRODUCT,reassoc]lemma yoneda_sugar.prod.inv_hom (R : C)(A B : C) :
--      (yoneda_sugar.prod R A B).inv ≫ (yoneda_sugar.prod R A B).hom = 𝟙 ( R [A]  ⨯ R[B]) :=
--      (Yoneda_preserve_product R A B).inv_hom_id'
--       attribute [PRODUCT] yoneda_sugar.prod.inv_hom_assoc
--  @[PRODUCT,reassoc]lemma yoneda_sugar.prod.second (R : C)(A B : C) :
--   (yoneda_sugar.prod R A B).hom ≫ π2 = (R < (π2 : A ⨯ B ⟶ B) >) := rfl
-- attribute [PRODUCT] yoneda_sugar.prod.second_assoc
-- @[PRODUCT]lemma yoneda_sugar.id (R : C)(A : C) : R < 𝟙 A > = 𝟙 ( R [A] ) := begin
--      funext,
--      exact comp_id C g,
--      -- have T : ((yoneda.map (𝟙 A)).app (op R)) g = (g ≫ (𝟙 A)),
-- end
-- @[PRODUCT,reassoc,refl]lemma yoneda_sugar_prod (R : C)(A B : C)(X :C)(f : X ⟶ A)(g : X ⟶ B) :
--       R < (f | g) > ≫ (yoneda_sugar.prod R A B).hom  =  (R < f > | R < g > ) :=  -- the  ≫  is  :/
--      begin
--            PRODUCT_CAT,
--           -- rw  yoneda_sugar.prod.hom R A B,
--           -- rw prod.left_composition,
--           -- iterate 2 {rw ← yoneda_sugar.composition},   -- rw ← is the problem ?
--           -- PRODUCT_CAT,
--           -- rw lem.prod.lift_fst,
--           -- rw lem.prod.lift_snd,
--      end
-- attribute [PRODUCT] yoneda_sugar_prod_assoc
-- @[PRODUCT]lemma yoneda_sugar_prod_inv (R : C)(A B : C)(X :C)(f : X ⟶ A)(g : X ⟶ B) :
--      R < (f | g) >   =  (R < f > | R < g > ) ≫ (yoneda_sugar.prod R A B).inv :=
--      begin
--           PRODUCT_CAT,  -- noting   HERE PROBLEM the tatic do nothing
--           rw ← yoneda_sugar_prod,
--           rw assoc,
--           rw yoneda_sugar.prod.hom_inv,
--           exact rfl,
--      end
-- @[PRODUCT]lemma  yoneda_sugar.otimes (R : C){Y Z K :C}(f : X ⟶ Y )(g : Z ⟶ K) :
--  ( R < (f ⊗ g) > ) = (yoneda_sugar.prod  _ _ _).hom ≫ ((R<f>) ⊗ R<g>) ≫ (yoneda_sugar.prod _ _ _ ).inv := begin
--      -- PRODUCT_CAT,
--      iterate 2 {rw prod.otimes_is_prod},
--      rw  yoneda_sugar.prod.hom,
--      iterate 1 {rw yoneda_sugar_prod_inv},
--      rw ← assoc,
--      rw prod.left_composition,
--      rw ← assoc,
--      rw prod.lift_fst,
--      rw ← assoc,
--      rw prod.lift_snd,
--      rw yoneda_sugar.composition,
--      rw yoneda_sugar.composition,
--      exact rfl,
-- end
-- @[PRODUCT]lemma yonega_sugar.one_otimes (R :C)(X Y Z: C) (f : X ⟶ Y) :
--  (((yoneda_sugar.prod R Z X).inv) ≫ (R <(𝟙 Z ⊗ f ) > ) ≫ (yoneda_sugar.prod R Z Y).hom) = (𝟙 (R[Z]) ⊗ R < f >) := begin
--      rw yoneda_sugar.otimes,
--      iterate 3 {rw ← assoc},
--      rw yoneda_sugar.prod.inv_hom,
--      rw id_comp,
--      rw assoc,
--      rw yoneda_sugar.prod.inv_hom,
--      rw ← yoneda_sugar.id,
--      simp,
--  end
-- lemma yonega_sugar.one_otimes' (R :C)(X Y Z: C) (f : X ⟶ Y) :
--  ( (R <(𝟙 Z ⊗ f ) > ) ≫ (yoneda_sugar.prod R Z Y).hom) = ((yoneda_sugar.prod R Z X).hom) ≫ (𝟙 (R[Z]) ⊗ R < f >) := begin
--      iterate 2{ rw yoneda_sugar.prod.hom},
--      rw prod.left_composition,
--      iterate 2{ rw ← yoneda_sugar.composition},
--      rw prod.map_first,
--      rw prod.map_second,
--      rw comp_id,
--      rw prod.otimes_is_prod,rw prod.left_composition,rw ← assoc,
--      rw prod.lift_fst,rw ←  assoc,rw prod.lift_snd,rw comp_id,
--      rw yoneda_sugar.composition,
--      exact rfl,
--  end
--  end Product_stuff
 end Product_stuff


import model_theory.substructures
import model_theory.bundled

--OUTPUT 1
theorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=
begin
  -- For each $n$, let $\mathbf A_n$ be the formula:
  let An : L.formula := λ n, ∃ (x1 : L.var) (x2 : L.var) (x3 : L.var) (x4 : L.var) (x5 : L.var),
    -- $\{x_1 \ne x_2 \land x_1 \ne x_3 \land \ldots \land x_{n - 1} \ne x_n\}$
    (ne (L.var.app x1 []) (L.var.app x2 [])) ∧
    (ne (L.var.app x1 []) (L.var.app x3 [])) ∧
    (ne (L.var.app x1 []) (L.var.app x4 [])) ∧
    (ne (L.var.app x1 []) (L.var.app x5 [])) ∧
    (ne (L.var.app x2 []) (L.var.app x3 [])) ∧
    (ne (L.var.app x2 []) (L.var.app x4 [])) ∧
    (ne (L.var.app x2 []) (L.var.app x5 [])) ∧
    (ne (L.var.app x3 []) (L.var.app x4 [])) ∧
    (ne (L.var.app x3 []) (L.var.app x5 [])) ∧
    (ne (L.var.app x4 []) (L.var.app x5 [])),
  have h1 : ∀ (n : ℕ), ∃ (M : F.Model) [fintype M], n ≤ @fintype.card M fintype, from h,

  -- Then $\mathbf A_i$ is true in a structure $\AA$ iff $\AA$ has at least $n$ elements.
  have h2 : ∀ (n : ℕ) (M : F.Model) [fintype M], n ≤ @fintype.card M fintype → F.Model.satisfies M (An n), from
    assume (n : ℕ) (M : F.Model) [fintype M] (hle : n ≤ @fintype.card M fintype),
    have h3 : ∃ (s : M.Interpretation), F.Model.satisfies M (An n) (s : M.Interpretation), from by {
      use (M.Interpretation.mk (λ (v : L.var), M.Interpretation.mk (λ (a : L.param), M.Interpretation.mk (λ (b : L.param), M.Interpretation.mk (λ (c : L.param), M.Interpretation.mk (λ (d : L.param), M.Interpretation.mk (λ (e : L.param),
        M.Interpretation.mk (λ (x1 : L.var), L.var.app x1 []) (M.Interpretation.mk (λ (x2 : L.var), L.var.app x2 []) (M.Interpretation.mk (λ (x3 : L.var), L.var.app x3 []) (M.Interpretation.mk (λ (x4 : L.var), L.var.app x4 []) (M.Interpretation.mk (λ (x5 : L.var), L.var.app x5 []) (M.Interpretation.mk (λ (x6 : L.var), L.var.app x6 []) (M.Interpretation.mk (λ (x7 : L.var), L.var.app x7 []) (M.Interpretation.mk (λ (x8 : L.var), L.var.app x8 []) (M.Interpretation.mk (λ (x9 : L.var), L.var.app x9 []) (M.Interpretation.mk (λ (x10 : L.var), L.var.app x10 []) (M.Interpretation.mk (λ (x11 : L.var), L.var.app x11 []) (M.Interpretation.mk (λ (x12 : L.var), L.var.app x12 []) (M.Interpretation.mk (λ (x13 : L.var), L.var.app x13 []) (M.Interpretation.mk (λ (x14 : L.var), L.var.app x14 []) (M.Interpretation.mk (λ (x15 : L.var), L.var.app x15 []) (M.Interpretation.mk (λ (x16 : L.var), L.var.app x16 []) (M.Interpretation.mk (λ (x17 : L.var), L.var.app x17 []) (M.Interpretation.mk (λ (x18 : L.var), L.var.app x18 []) (M.Interpretation.mk (λ (x19 : L.var), L.var.app x19 []) (M.Interpretation.mk (λ (x20 : L.var), L.var.app x20 []) (M.Interpretation.mk (λ (x21 : L.var), L.var.app x21 []) (M.Interpretation.mk (λ (x22 : L.var), L.var.app x22 []) (M.Interpretation.mk (λ (x23 : L.var), L.var.app x23 []) (M.Interpretation.mk (λ (x24 : L.var), L.var.app x24 []) (M.Interpretation.mk (λ (x25 : L.var), L.var.app x25 []) (M.Interpretation.mk (λ (x26 : L.var), L.var.app x26 []) (M.Interpretation.mk (λ (x27 : L.var), L.var.app x27 []) (M.Interpretation.mk (λ (x28 : L.var), L.var.app x28 []) (M.Interpretation.mk (λ (x29 : L.var), L.var.app x29 []) (M.Interpretation.mk (λ (x30 : L.var), L.var.app x30 []) (M.Interpretation.mk (λ (x31 : L.var), L.var.app x31 []) (M.Interpretation.mk (λ (x32 : L.var), L.var.app x32 []) (M.Interpretation.mk (λ (x33 : L.var), L.var.app x33 []) (M.Interpretation.mk (λ (x34 : L.var), L.var.app x34 []) (M.Interpretation.mk (λ (x35 : L.var), L.var.app x35 []) (M.Interpretation.mk (λ (x36 : L.var), L.var.app x36 []) (M.Interpretation.mk (λ (x37 : L.var), L.var.app x37 []) (M.Interpretation.mk (λ (x38 : L.var), L.var.app x38 []) (M.Interpretation.mk (λ (x39 : L.var), L.var.app x39 []) (M.Interpretation.mk (λ (x40 : L.var), L.var.app x40 []) (M.Interpretation.mk (λ (x41 : L.var), L.var.app x41 []) (M.Interpretation.mk (λ (x42 : L.var), L.var.app x42 []) (M.Interpretation.mk (λ (x43 : L.var), L.var.app x43 []) (M.Interpretation.mk (λ (x44 : L.var), L.var.app x44 []) (M.Interpretation.mk (λ (x45 : L.var), L.var.app x45 []) (M.Interpretation.mk (λ (x46 : L.var), L.var.app x46 []) (M.Interpretation.mk (λ (x47 : L.var), L.var.app x47 []) (M.Interpretation.mk (λ (x48 : L.var), L.var.app x48 []) (M.Interpretation.mk (
end --Needs more than 2000 tokens!

--OUTPUT 2
theorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=
begin
  -- For each $n$, let $\mathbf A_n$ be the formula:
  let A_n : L.formula := ∃ (x : fin n), ¬ (∀ (i j : fin n), i ≠ j → L.interp M x i ≠ L.interp M x j),

  -- Then $\mathbf A_i$ is true in a structure $\AA$ iff $\AA$ has at least $n$ elements.
  have h1 : ∀ (n : ℕ) (M : F.Model), (∃ (x : fin n), ¬ (∀ (i j : fin n), i ≠ j → L.interp M x i ≠ L.interp M x j)) ↔ n ≤ fintype.card M, from assume (n : ℕ) (M : F.Model) [mfin : fintype M],
  begin
    -- First, assume that $\mathbf A_n$ is true in $\AA$.
    assume (h2 : ∃ (x : fin n), ¬ (∀ (i j : fin n), i ≠ j → L.interp M x i ≠ L.interp M x j)),
    -- Then there exists $x$ such that $x_i \ne x_j$ for some $i \ne j$.
    have h3 : ∃ (x : fin n), ∃ (i j : fin n), i ≠ j ∧ L.interp M x i = L.interp M x j, from by {
      have h3 : ∃ (x : fin n), ∃ (i j : fin n), i ≠ j ∧ L.interp M x i ≠ L.interp M x j, from by {
        have h3 : ∃ (x : fin n), ¬ (∀ (i j : fin n), i ≠ j → L.interp M x i ≠ L.interp M x j), from by {
          exact h2,
        },
        have h4 : ∃ (x : fin n), ¬ (∃ (i j : fin n), i ≠ j ∧ L.interp M x i ≠ L.interp M x j), from by {
          cases h3,
          use x,
          show ¬ (∃ (i j : fin n), i ≠ j ∧ L.interp M x i ≠ L.interp M x j), from not_not.mp (mt not_exists_not.mp a),
        },
        have h5 : ∃ (x : fin n), ∀ (i j : fin n), i ≠ j → L.interp M x i = L.interp M x j, from by {
          cases h4,
          use x,
          show ∀ (i j : fin n), i ≠ j → L.interp M x i = L.interp M x j, from mt not_implies_iff.mp a,
        },
        have h6 : ∃ (x : fin n), ∃ (i j : fin n), i ≠ j ∧ L.interp M x i = L.interp M x j, from by {
          cases h5,
          use x,
          use i,
          use i,
          split,
          exact i.symm,
          exact a i i (i.symm.trans i.ne.symm),
        },
        show ∃ (x : fin n), ∃ (i j : fin n), i ≠ j ∧ L.interp M x i = L.interp M x j, from by {
          exact h6,
        },
      },
      show ∃ (x : fin n), ∃ (i j : fin n), i ≠ j ∧ L.interp M x i = L.interp M x j, from by {
        exact h3,
      },
    },
    -- Then there exists $x$ such that $x_i = x_j$ for some $i \ne j$.
    have h4 : ∃ (x : fin n), ∃ (i j : fin n), i ≠ j ∧ L.interp M x i ≠ L.interp M x j, from by {
      cases h3,
      use x,
      use i,
      use j,
      split,
      exact a,
      exact a.left.symm.trans a.right.symm,
    },
    -- Then there exists $x$ such that $x_i = x_j$ for some $i \ne j$.
    have h5 : ∃ (x : fin n), ∃ (i j : fin n), i ≠ j ∧ L.interp M x i = L.interp M x j, from by {
      cases h4,
      use x,
      use i,
      use j,
      split,
      exact a,
      exact a.left.symm.trans a.right.symm,
    },
    -- Since $x_i = x_j$, $x_i$ and $x_j$ are the same element of $\AA$.
    have h6 : ∃ (x : fin n), ∃ (i j : fin n), i ≠ j ∧ x i = x j, from by {
      cases h5,
      use x,
      use i,
      use j,
      split,
      exact a,
      exact a.left.symm.trans a.right.symm,
    },
    -- So there are at least $n$ elements of $\AA$.
    have h7 : n ≤ fintype.card M, from by {
      cases h6,
      use x,
      use i,
      use j,
      have h8 : x i = x j, from by {
        exact a.right,
      },
      have h9 : x i.val = x j.val, from by {
        rw h8,
      },
      have h10 : i.val = j.val, from by {
        rw ← inj_eq,
        exact h9,
      },
      have h11 : i = j, from by {
        exact fin.eq_of_veq h10,
      },
      have h12 : i ≠ j, from by {
        exact a.left,
      },
      show false, from by {
        exact h12 h11,
      },
    },
    show n ≤ fintype.card M, from by {
      exact h7,
    },
  end,

  -- Take:
  have h2 : ∀ (n : ℕ), n ≤ fintype.card M → M ⊨ A_n, from by {
    assume (n : ℕ),
    assume h2 : n ≤ fintype.card M,
    have h3 : ∃ (x : fin n), ¬ (∀ (i j : fin n), i ≠ j → L.interp M x i ≠ L.interp M x j), from by {
      use (λ (i : fin n), i),
      show ¬ (∀ (i j : fin n), i ≠ j → L.interp M (λ (i : fin n), i) i ≠ L.interp M (λ (i : fin n), i) j), from by {
        assume h4 : ∀ (i j : fin n), i ≠ j → L.interp M (λ (i : fin n), i) i ≠ L.interp M (λ (i : fin n), i) j,
        have h5 : fin n → fin n → Prop, from by {
          assume i j : fin n,
          show i ≠ j → L.interp M (λ (i : fin n), i) i ≠ L.interp M (λ (i : fin n), i) j, from by {
            exact h4 i j,
          },
        },
        have h6 : ∀ (i j : fin n), i ≠ j → L.interp M (λ (i : fin n), i) i ≠ L.interp M (λ (i : fin n), i) j, from by {
          rw function.funext_iff,
          show ∀ x y : fin n, x ≠ y → L.interp M (λ (i : fin n), i) x ≠ L.interp M (λ (i : fin n), i) y, from by {
            assume x y : fin n
end --Needs more than 2000 tokens!

--OUTPUT 3
theorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=
begin
  -- For each $n$, let $\mathbf A_n$ be the formula:
  let A : ℕ → L.Theory,
  -- $\exists x_1 \exists x_2 \ldots \exists x_n: \{x_1 \ne x_2 \land x_1 \ne x_3 \land \ldots \land x_{n - 1} \ne x_n\}$
  A n := λ m, ∃ x1 : m, ∃ x2 : m, ⋯ ∃ xn : m, ∀ i : fin n, ∀ j : fin n, i ≠ j → xi ≠ xj,
  have ha : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  -- Then $\mathbf A_i$ is true in a structure $\AA$ iff $\AA$ has at least $n$ elements.
  have h1 : ∀ (n : ℕ) (m : F.Model) [mfin : fintype m], (∀ (i : fin n), ∀ (j : fin n), i ≠ j → xi ≠ xj) → n ≤ fintype.card m, from by {
    assume (n : ℕ) (m : F.Model) [mfin : fintype m] (h : ∀ (i : fin n), ∀ (j : fin n), i ≠ j → xi ≠ xj),
    induction n with n hn,
    have h1 : ∃ (x : m), true, from exists.intro m.default trivial,
    have h2 : 1 ≤ fintype.card m, from by {apply nat.le_of_succ_le_succ,apply nat.succ_le_of_lt,apply mfin.eq_of_veq,apply h1},
    exact h2,
    have h1 : ∃ (x1 : m) (x2 : m), x1 ≠ x2, from by {
      have h2 : ∃ (x1 : m), x1 ≠ x2, from exists.intro m.default (h (fin.mk n dec_trivial) (fin.mk 0 dec_trivial) (ne_of_gt (nat.lt_succ_self 0))),
      exact h2,
    },
    have h2 : 1 + n ≤ fintype.card m, from by {apply nat.le_of_succ_le_succ,apply nat.succ_le_of_lt,apply mfin.eq_of_veq,apply h1},
    exact h2,
  },

  -- Take:
  -- $$ \Gamma := F \cup \bigcup_{i \mathop = 1}^\infty A_i $$
  let Γ : L.Theory, Γ := (λ m, F m ∧ ∀ n : ℕ, A n m),
  -- Since $F$ has models of arbitrarily large size, every finite subset of $\Gamma$ is satisfiable.
  have h2 : ∀ (n : ℕ), ∃ (m : F.Model), Γ n m, from by {
    assume (n : ℕ),
    have h3 : ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from ha n,
    have h4 : ∃ (m : F.Model) [mfin : fintype m], Γ n m, from by {
      cases h3 with (m : F.Model) h3,
      cases h3 with (mfin : fintype m) h3,
      use m,
      use mfin,
      obviously,
    },
    exact h4,
  },
  -- From the Compactness Theorem, $\Gamma$ is satisfiable in some model $\mathbf{M}$.
  have h3 : ∃ (M : F.Model), Γ M, from by {
    apply first_order.compactness h2,
  },
  cases h3 with (M : F.Model) h3,
  -- But since $\mathbf{M} \models A_i$ for each $i$, $\mathbf{M}$ must be infinite.
  have h4 : infinite M, from by {
    have h5 : ∀ n : ℕ, ∃ (x1 : M) (x2 : M), x1 ≠ x2, from by {
      assume n : ℕ,
      have h6 : ∃ (x1 : M) (x2 : M), x1 ≠ x2, from by {
        cases h3 (n+1) with h3 h3,
        cases h3 with h3 h3,
        cases h3 with h3 h3,
        use x1,
        use x2,
        exact ne_of_gt (nat.lt_succ_self 0),
      },
      exact h6,
    },
    exact h5,
  },
  -- So $F$ has an infinite model.
  exact h4,
end

--OUTPUT 4
theorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=
begin
  -- For each $n$, let $\mathbf A_n$ be the formula:
  let A := (λ (n : ℕ), ∃ x₁, ∃ x₂, ∃ x₃, ⋀ i j, i ≠ j → x i ≠ x j) in

  -- Then $\mathbf A_i$ is true in a structure $\AA$ iff $\AA$ has at least $n$ elements.
  have h1 : ∀ n : ℕ, @fintype.card (F.Model.restrict (A n)) (fintype.restrict _ _) ≤ n, from
    assume (n : ℕ) (m : F.Model) [mfin : fintype m] (h2 : (A n) m),
    begin
      have h3 : nonempty (F.Model.restrict (A n) m), from by {
        rcases h2 with ⟨⟨x₁,x₂⟩,h3⟩,
        use ⟨⟨x₁,x₂⟩,h3.right⟩,
      },
      have h4 : fintype (F.Model.restrict (A n) m), from by {
        rcases h2 with ⟨⟨x₁,x₂⟩,h3⟩,
        apply fintype.restrict _ _,
        apply fintype.of_injective _ _ _,
        assume ⟨x,hx⟩ ⟨y,hy⟩ heq,
        have h5 : x = y, from by {
          cases hx with h5 h5,
          cases hy with hy hy,
          have h6 : x ≠ y, from by {
            rw heq,
            apply h3.left,
          },
          have h7 : x ≠ y, from by {
            rw heq at h5,
            apply h5,
          },
          apply ne.elim h7,
          apply ne.elim h6,
        },
        rw h5,
        apply exists.intro x,
        apply exists.intro hx.left,
      },
      have h5 : @fintype.card (F.Model.restrict (A n) m) h4 ≤ n, from by {
        rcases h2 with ⟨⟨x₁,x₂⟩,h3⟩,
        rw ← fintype.card_le_iff_subsingleton,
        apply finset.subsingleton_of_le,
        assume x y hx hy,
        have h6 : x ≠ y, from by {
          cases hx with h6 h6,
          cases hy with hy hy,
          have h7 : x ≠ y, from by {
            apply h3.left,
          },
          have h8 : x ≠ y, from by {
            rw ← h6 at hy,
            apply hy,
          },
          apply ne.elim h7,
          apply ne.elim h8,
        },
        have h7 : x ≠ y, from by {
          cases hx with h7 h7,
          cases hy with hy hy,
          have h8 : x ≠ y, from by {
            apply h3.left,
          },
          have h9 : x ≠ y, from by {
            rw ← h7 at hy,
            apply hy,
          },
          apply ne.elim h8,
          apply ne.elim h9,
        },
        apply ne.elim h6,
        apply ne.elim h7,
      },
      apply le_trans h5 (le_of_lt (lt_of_le_of_lt mfin.2 (nat.lt_succ_self n))),
    end,

  -- Take:
  -- $$ \Gamma := F \cup \bigcup_{i \mathop = 1}^\infty A_i $$
  let Γ := (λ (n : ℕ), ∃ x₁, ∃ x₂, ∃ x₃, ⋀ i j, i ≠ j → x i ≠ x j) in

  -- Since $F$ has models of arbitrarily large size, every finite subset of $\Gamma$ is satisfiable.
  have h2 : ∀ {G : set (L.Theory.Formula)}, finite G → ∃ (m : F.Model), ∀ (ϕ : L.Theory.Formula), ϕ ∈ G → F.Model.satisfies m ϕ, from
    assume (G : set (L.Theory.Formula)) (h3 : finite G),
    begin
      -- For each $n$, let $\mathbf A_n$ be the formula:
      let A := (λ (n : ℕ), ∃ x₁, ∃ x₂, ∃ x₃, ⋀ i j, i ≠ j → x i ≠ x j) in

      -- Then $\mathbf A_i$ is true in a structure $\AA$ iff $\AA$ has at least $n$ elements.
      have h4 : ∀ n : ℕ, @fintype.card (F.Model.restrict (A n)) (fintype.restrict _ _) ≤ n, from
        assume (n : ℕ) (m : F.Model) [mfin : fintype m] (h2 : (A n) m),
        begin
          have h3 : nonempty (F.Model.restrict (A n) m), from by {
            rcases h2 with ⟨⟨x₁,x₂⟩,h3⟩,
            use ⟨⟨x₁,x₂⟩,h3.right⟩,
          },
          have h4 : fintype (F.Model.restrict (A n) m), from by {
            rcases h2 with ⟨⟨x₁,x₂⟩,h3⟩,
            apply fintype.restrict _ _,
            apply fintype.of_injective _ _ _,
            assume ⟨x,hx⟩ ⟨y,hy⟩ heq,
            have h5 : x = y, from by {
              cases hx with h5 h5,
              cases hy with hy hy,
              have h6 : x ≠ y, from by {
                rw heq,
                apply h3.left,
              },
              have h7 : x ≠ y, from by {
                rw heq at h5,
                apply h5,
              },
              apply ne.elim h7,
              apply ne.elim h6,
            },
            rw h5,
            apply exists.intro x,
            apply exists.intro hx.left,
          },
          have h5 : @fintype.card (F.Model.restrict (A n) m) h4 ≤ n, from by {
            rcases h2 with ⟨⟨x₁,x₂⟩,h3⟩,
            rw ← fintype.card_le_iff_subsingleton,
            apply finset.subsingleton_of_le,
            assume x y hx hy,
            have h6 : x ≠ y, from by {
              cases hx with h6 h6,
              cases hy with hy hy,
              have h7 : x ≠ y, from by {
                apply h3.left,
              },
              have h8 : x ≠ y, from by {
                rw ← h6 at hy,
                apply hy,
              },
              apply ne.elim h7,
              apply ne.elim h8,
            },
            have h7 : x ≠ y, from by {
              cases hx with h7 h7,
              cases hy with hy hy,
              have h8 : x ≠ y, from by {
                apply h3.left,
              },
              have h9 : x ≠ y, from by {
                rw ← h7 at hy,
end --Needs more than 2000 tokens!

--OUTPUT 5
theorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=
begin
  -- For each $n$, let $\mathbf A_n$ be the formula
  let A := λ n : ℕ, ∃ x y z : L.sorts.1, (x ≠ y ∧ x ≠ z ∧ y ≠ z),
  -- Then $\mathbf A_i$ is true in a structure $\AA$ iff $\AA$ has at least $n$ elements.
  have h1 : ∀ n : ℕ, fintype (L.sorts.1), from by {
    assume (n : ℕ), use L.sorts.1, use L.sorts, rw fintype.card_le_one,
    exact nat.succ_pos n, },
  have h2 : ∀ n : ℕ, ∀ m : F.Model [mfin : fintype m], n ≤ fintype.card (L.sorts.1) → ∃ x y z : m, (x ≠ y ∧ x ≠ z ∧ y ≠ z), from
    assume (n : ℕ) (m : F.Model) [mfin : fintype m] (hn : n ≤ fintype.card (L.sorts.1)),
    have h4 : n ≤ fintype.card m, from by {
      apply le_of_lt, apply fintype.card_lt_card,
      exact mfin, },
    have h5 : n ≤ fintype.card (L.sorts.1), from by {
      apply nat.le_trans h4,
      apply fintype.card_le_card,
      exact h1 _, },
      exists.elim (h n) (λ m hm, exists.elim hm (λ mfin hmfin, ⟨m.1,m.2,m.3,hmfin⟩)),

  -- Take:
  -- $$ \Gamma := F \cup \bigcup_{i \mathop = 1}^\infty A_i $$
  let Γ := (F.set ∪ set.range A),

  -- Since $F$ has models of arbitrarily large size, every finite subset of $\Gamma$ is satisfiable.
  have h3 : ∀ {φ : L.Formula} {S : set L.Formula}, finite S → S ⊆ Γ → F.formula_true φ m → F.formula_true φ (F.Model.mk S), from by {
    assume φ S hS hSsub hφ,
    induction hS with φ' S hS' ih,
    show F.formula_true φ (F.Model.mk ∅), from hφ,
    have h1 : φ' ∈ Γ, from by {
      apply set.mem_union_left,
      apply set.mem_range_self,
    },
    have h2 : S ⊆ Γ, from by {
      apply set.subset_union_right,
      apply set.subset.trans hSsub h1,
    },
    show F.formula_true φ (F.Model.mk (set.insert φ' S)), from by {
      apply F.formula_true.insert,
      apply ih,
      exact hS',
      exact hS',
      exact h2,
      apply hφ,
    },
  },

  -- From the Compactness Theorem, $\Gamma$ is satisfiable in some model $\mathbf{M}$.
  have h4 : ∃ (M : F.Model), ∀ φ : L.Formula, φ ∈ Γ → F.formula_true φ M, from by {
    have h5 : ∀ φ : L.Formula, φ ∈ Γ → ∃ (M : F.Model), F.formula_true φ M, from assume (φ : L.Formula) (hφ : φ ∈ Γ),
      exists.elim (set.mem_union.mp hφ) (λ hφ' hφ'', exists.elim hφ' (λ hφ''' hφ'''') ),
    apply F.Compactness h5,
  },

  -- But since $\mathbf{M} \models A_i$ for each $i$, $\mathbf{M}$ must be infinite.
  have h5 : ∀ n : ℕ, ∃ (x y z : M), (x ≠ y ∧ x ≠ z ∧ y ≠ z), from assume (n : ℕ),
    exists.elim (h4 (A n)) (λ hM hM', exists.elim (h2 n hM hM') (λ x hx, exists.elim hx (λ y hxy, exists.elim hxy (λ z hxyz, ⟨x,y,z,hxyz⟩)))),

  -- So $F$ has an infinite model.
  show ∃ (M : F.Model), infinite M, from
    exists.elim h4 (λ M hM, exists.elim (fintype.card_infinite M) (λ hM' hM'', ⟨M,hM'⟩)),
end

--OUTPUT 6
theorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=
begin
  -- for each $n$, let $\mathbf A_n$ be the formula:
  let A : ℕ → L.formula,
  have h1 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from by {
    assume n : ℕ,
    exact h n, },
  have h2 : ∀ n : ℕ, ∃ (m : F.Model), n ≤ @fintype.card m mfin, from by {
    assume n : ℕ,
    cases h1 n with m mfin,
    use m,
    exact mfin, },
  have h3 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n < @fintype.card m mfin, from by {
    assume n : ℕ,
    have h4 : ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h2 n,
    cases h4 with m mfin,
    use m,
    use mfin,
    have h5 : n ≤ @fintype.card m mfin, from mfin,
    have h6 : ∃ (k : ℕ), n < k, from nat.find_lt h5,
    cases h6 with k hlt,
    exact hlt,
  },
  have h4 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin ∧ n < @fintype.card m mfin, from by {
    assume n : ℕ,
    have h5 : ∃ (m : F.Model) [mfin : fintype m], n < @fintype.card m mfin, from h3 n,
    cases h5 with m mfin,
    have h6 : ∃ (k : ℕ), n ≤ k, from nat.find_le n,
    cases h6 with k hle,
    use m,
    use mfin,
    have h7 : n < @fintype.card m mfin, from mfin,
    have h8 : ∃ (l : ℕ), n ≤ l ∧ l < @fintype.card m mfin, from nat.find_lt_ge h7 hle,
    cases h8 with l hlt,
    use l,
    exact ⟨hlt.left, hlt.right⟩,
  },
  have h5 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n < @fintype.card m mfin, from by {
    assume n : ℕ,
    have h6 : ∃ (m : F.Model) [mfin : fintype m], n < @fintype.card m mfin ∧ n ≤ @fintype.card m mfin, from h4 n,
    cases h6 with m mfin,
    use m,
    use mfin.right,
    exact mfin.left,
  },
  have h6 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin ∧ n < @fintype.card m mfin, from by {
    assume n : ℕ,
    have h7 : ∃ (m : F.Model) [mfin : fintype m], n < @fintype.card m mfin, from h5 n,
    cases h7 with m mfin,
    have h8 : ∃ (k : ℕ), n ≤ k, from nat.find_le n,
    cases h8 with k hle,
    use m,
    use mfin,
    have h9 : n < @fintype.card m mfin, from mfin,
    have h10 : ∃ (l : ℕ), n ≤ l ∧ l < @fintype.card m mfin, from nat.find_lt_ge h9 hle,
    cases h10 with l hlt,
    use l,
    exact ⟨hlt.left, hlt.right⟩,
  },
  have h7 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin ∧ n < @fintype.card m mfin, from by {
    assume n : ℕ,
    have h8 : ∃ (m : F.Model) [mfin : fintype m], n < @fintype.card m mfin, from h5 n,
    cases h8 with m mfin,
    have h9 : ∃ (k : ℕ), n ≤ k, from nat.find_le n,
    cases h9 with k hle,
    use m,
    use mfin,
    have h10 : n < @fintype.card m mfin, from mfin,
    have h11 : ∃ (l : ℕ), n ≤ l ∧ l < @fintype.card m mfin, from nat.find_lt_ge h10 hle,
    cases h11 with l hlt,
    use l,
    exact ⟨hlt.left, hlt.right⟩,
  },
  have h8 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n < @fintype.card m mfin, from by {
    assume n : ℕ,
    have h9 : ∃ (m : F.Model) [mfin : fintype m], n < @fintype.card m mfin ∧ n ≤ @fintype.card m mfin, from h7 n,
    cases h9 with m mfin,
    use m,
    use mfin.right,
    exact mfin.left,
  },
  have h9 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n < @fintype.card m mfin, from by {
    assume n : ℕ,
    have h10 : ∃ (m : F.Model) [mfin : fintype m], n < @fintype.card m mfin ∧ n ≤ @fintype.card m mfin, from h7 n,
    cases h10 with m mfin,
    use m,
    use mfin.right,
    exact mfin.left,
  },
  have h10 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n < @fintype.card m mfin, from by {
    assume n : ℕ,
    have h11 : ∃ (m : F.Model) [mfin : fintype m], n < @fintype.card m mfin ∧ n ≤ @fintype.card m mfin, from h7 n,
    cases h11 with m mfin,
    use m,
    use mfin.right,
    exact mfin.left,
  },
  have h11 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n < @fintype.card m mfin, from by {
    assume n : ℕ,
    have h12 : ∃ (m : F.Model) [mfin : fintype m], n < @fintype.card m mfin ∧ n ≤ @fintype.card m mfin, from h7 n,
    cases h12 with m mfin,
    use m,
    use mfin.right,
    exact mfin.left,
  },
  have h12 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n < @fintype.card m mfin, from by {
    assume n : ℕ,
    have h13 : ∃ (m : F.Model) [mfin : fintype m], n < @fintype.card m mfin ∧ n ≤ @fintype.card m mfin, from h7
end --Needs more than 2000 tokens!

--OUTPUT 7
theorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=
begin
  let A : L.sentence := ∃ x₁, ∃ x₂, x₁ ≠ x₂,
  let A₁ : L.sentence := ∃ x₁, ∃ x₂, ∃ x₃, (x₁ ≠ x₂ ∧ x₁ ≠ x₃ ∧ x₂ ≠ x₃),
  let A₂ : L.sentence := ∃ x₁, ∃ x₂, ∃ x₃, ∃ x₄, (x₁ ≠ x₂ ∧ x₁ ≠ x₃ ∧ x₁ ≠ x₄ ∧ x₂ ≠ x₃ ∧ x₂ ≠ x₄ ∧ x₃ ≠ x₄),
  let A₃ : L.sentence := ∃ x₁, ∃ x₂, ∃ x₃, ∃ x₄, ∃ x₅, (x₁ ≠ x₂ ∧ x₁ ≠ x₃ ∧ x₁ ≠ x₄ ∧ x₁ ≠ x₅ ∧ x₂ ≠ x₃ ∧ x₂ ≠ x₄ ∧ x₂ ≠ x₅ ∧ x₃ ≠ x₄ ∧ x₃ ≠ x₅ ∧ x₄ ≠ x₅),
  let A₄ : L.sentence := ∃ x₁, ∃ x₂, ∃ x₃, ∃ x₄, ∃ x₅, ∃ x₆, (x₁ ≠ x₂ ∧ x₁ ≠ x₃ ∧ x₁ ≠ x₄ ∧ x₁ ≠ x₅ ∧ x₁ ≠ x₆ ∧ x₂ ≠ x₃ ∧ x₂ ≠ x₄ ∧ x₂ ≠ x₅ ∧ x₂ ≠ x₆ ∧ x₃ ≠ x₄ ∧ x₃ ≠ x₅ ∧ x₃ ≠ x₆ ∧ x₄ ≠ x₅ ∧ x₄ ≠ x₆ ∧ x₅ ≠ x₆),
  let A₅ : L.sentence := ∃ x₁, ∃ x₂, ∃ x₃, ∃ x₄, ∃ x₅, ∃ x₆, ∃ x₇, (x₁ ≠ x₂ ∧ x₁ ≠ x₃ ∧ x₁ ≠ x₄ ∧ x₁ ≠ x₅ ∧ x₁ ≠ x₆ ∧ x₁ ≠ x₇ ∧ x₂ ≠ x₃ ∧ x₂ ≠ x₄ ∧ x₂ ≠ x₅ ∧ x₂ ≠ x₆ ∧ x₂ ≠ x₇ ∧ x₃ ≠ x₄ ∧ x₃ ≠ x₅ ∧ x₃ ≠ x₆ ∧ x₃ ≠ x₇ ∧ x₄ ≠ x₅ ∧ x₄ ≠ x₆ ∧ x₄ ≠ x₇ ∧ x₅ ≠ x₆ ∧ x₅ ≠ x₇ ∧ x₆ ≠ x₇),
  let A₆ : L.sentence := ∃ x₁, ∃ x₂, ∃ x₃, ∃ x₄, ∃ x₅, ∃ x₆, ∃ x₇, ∃ x₈, (x₁ ≠ x₂ ∧ x₁ ≠ x₃ ∧ x₁ ≠ x₄ ∧ x₁ ≠ x₅ ∧ x₁ ≠ x₆ ∧ x₁ ≠ x₇ ∧ x₁ ≠ x₈ ∧ x₂ ≠ x₃ ∧ x₂ ≠ x₄ ∧ x₂ ≠ x₅ ∧ x₂ ≠ x₆ ∧ x₂ ≠ x₇ ∧ x₂ ≠ x₈ ∧ x₃ ≠ x₄ ∧ x₃ ≠ x₅ ∧ x₃ ≠ x₆ ∧ x₃ ≠ x₇ ∧ x₃ ≠ x₈ ∧ x₄ ≠ x₅ ∧ x₄ ≠ x₆ ∧ x₄ ≠ x₇ ∧ x₄ ≠ x₈ ∧ x₅ ≠ x₆ ∧ x₅ ≠ x₇ ∧ x₅ ≠ x₈ ∧ x₆ ≠ x₇ ∧ x₆ ≠ x₈ ∧ x₇ ≠ x₈),
  let A₇ : L.sentence := ∃ x₁, ∃ x₂, ∃ x₃, ∃ x₄, ∃ x₅, ∃ x₆, ∃ x₇, ∃ x₈, ∃ x₉, (x₁ ≠ x₂ ∧ x₁ ≠ x₃ ∧ x₁ ≠ x₄ ∧ x₁ ≠ x₅ ∧ x₁ ≠ x₆ ∧ x₁ ≠ x₇ ∧ x₁ ≠ x₈ ∧ x₁ ≠ x₉ ∧ x₂ ≠ x₃ ∧ x₂ ≠ x₄ ∧ x₂ ≠ x₅ ∧ x₂ ≠ x₆ ∧ x₂ ≠ x₇ ∧ x₂ ≠ x₈ ∧ x₂ ≠ x₉ ∧ x₃ ≠ x₄ ∧ x₃ ≠ x₅ ∧ x₃ ≠ x₆ ∧ x₃ ≠ x₇ ∧ x₃ ≠ x₈ ∧ x₃ ≠ x₉ ∧ x₄ ≠ x₅ ∧ x₄ ≠ x₆ ∧ x₄ ≠ x₇ ∧ x₄ ≠ x₈ ∧ x₄ ≠ x₉ ∧ x₅ ≠ x₆ ∧ x₅ ≠ x₇ ∧ x₅ ≠ x₈ ∧ x₅ ≠ x₉ ∧ x₆ ≠ x₇ ∧ x₆ ≠ x₈ ∧ x₆ ≠ x₉ ∧ x₇ ≠ x₈ ∧ x₇ ≠ x₉ ∧ x₈ ≠ x₉),
  let A₈ : L.sentence := ∃ x₁, ∃ x₂, ∃ x₃, ∃ x₄, ∃ x₅, ∃ x₆, ∃ x₇, ∃ x₈, ∃ x₉, ∃ x₁₀, (x₁ ≠ x₂ ∧ x₁ ≠ x₃ ∧ x₁ ≠ x₄ ∧ x₁ ≠ x₅ ∧ x₁ ≠ x₆ ∧ x
end --Needs more than 2000 tokens!

--OUTPUT 8
theorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=
begin
  -- Let $F$ be a set of first-order formulas which has finite models of arbitrarily large size.
  assume (F : L.Theory),
  assume h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin,

  -- Then $F$ has an infinite model.
  show ∃ (M : F.Model), infinite M, from
  begin
    -- For each $n$, let $\mathbf A_n$ be the formula:
    let A (n : ℕ) : L.formula := ∃ (x1 : L.sort 0), ∃ (x2 : L.sort 0),
      ∃ (x3 : L.sort 0), ∃ (x4 : L.sort 0), ∃ (x5 : L.sort 0),
      ∃ (x6 : L.sort 0), ∃ (x7 : L.sort 0), ∃ (x8 : L.sort 0),
      ∃ (x9 : L.sort 0), ∃ (x10 : L.sort 0),
      ∃ (x11 : L.sort 0), ∃ (x12 : L.sort 0),
      ∃ (x13 : L.sort 0), ∃ (x14 : L.sort 0),
      ∃ (x15 : L.sort 0), ∃ (x16 : L.sort 0),
      ∃ (x17 : L.sort 0), ∃ (x18 : L.sort 0),
      ∃ (x19 : L.sort 0), ∃ (x20 : L.sort 0),
      ∃ (x21 : L.sort 0), ∃ (x22 : L.sort 0),
      ∃ (x23 : L.sort 0), ∃ (x24 : L.sort 0),
      ∃ (x25 : L.sort 0), ∃ (x26 : L.sort 0),
      ∃ (x27 : L.sort 0), ∃ (x28 : L.sort 0),
      ∃ (x29 : L.sort 0), ∃ (x30 : L.sort 0),
      ∃ (x31 : L.sort 0), ∃ (x32 : L.sort 0),
      ∃ (x33 : L.sort 0), ∃ (x34 : L.sort 0),
      ∃ (x35 : L.sort 0), ∃ (x36 : L.sort 0),
      ∃ (x37 : L.sort 0), ∃ (x38 : L.sort 0),
      ∃ (x39 : L.sort 0), ∃ (x40 : L.sort 0),
      ∃ (x41 : L.sort 0), ∃ (x42 : L.sort 0),
      ∃ (x43 : L.sort 0), ∃ (x44 : L.sort 0),
      ∃ (x45 : L.sort 0), ∃ (x46 : L.sort 0),
      ∃ (x47 : L.sort 0), ∃ (x48 : L.sort 0),
      ∃ (x49 : L.sort 0), ∃ (x50 : L.sort 0),
      ∃ (x51 : L.sort 0), ∃ (x52 : L.sort 0),
      ∃ (x53 : L.sort 0), ∃ (x54 : L.sort 0),
      ∃ (x55 : L.sort 0), ∃ (x56 : L.sort 0),
      ∃ (x57 : L.sort 0), ∃ (x58 : L.sort 0),
      ∃ (x59 : L.sort 0), ∃ (x60 : L.sort 0),
      ∃ (x61 : L.sort 0), ∃ (x62 : L.sort 0),
      ∃ (x63 : L.sort 0), ∃ (x64 : L.sort 0),
      ∃ (x65 : L.sort 0), ∃ (x66 : L.sort 0),
      ∃ (x67 : L.sort 0), ∃ (x68 : L.sort 0),
      ∃ (x69 : L.sort 0), ∃ (x70 : L.sort 0),
      ∃ (x71 : L.sort 0), ∃ (x72 : L.sort 0),
      ∃ (x73 : L.sort 0), ∃ (x74 : L.sort 0),
      ∃ (x75 : L.sort 0), ∃ (x76 : L.sort 0),
      ∃ (x77 : L.sort 0), ∃ (x78 : L.sort 0),
      ∃ (x79 : L.sort 0), ∃ (x80 : L.sort 0),
      ∃ (x81 : L.sort 0), ∃ (x82 : L.sort 0),
      ∃ (x83 : L.sort 0), ∃ (x84 : L.sort 0),
      ∃ (x85 : L.sort 0), ∃ (x86 : L.sort 0),
      ∃ (x87 : L.sort 0), ∃ (x88 : L.sort 0),
      ∃ (x89 : L.sort 0), ∃ (x90 : L.sort 0),
      ∃ (x91 : L.sort 0), ∃ (x92 : L.sort 0),
      ∃ (x93 : L.sort 0), ∃ (x94 : L.sort 0),
      ∃ (x95 : L.sort 0), ∃ (x96 : L.sort 0),
      ∃ (x97 : L.sort 0), ∃ (x98 : L.sort 0),
      ∃ (x99 : L.sort 0), ∃ (x100 : L.sort 0),
      (∀ (x101 : L.sort 0), ¬L.eq x1 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x2 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x3 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x4 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x5 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x6 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x7 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x8 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x9 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x10 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x11 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x12 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x13 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x14 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x15 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x16 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x17 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x18 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x19 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x20 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x21 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x22 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x23 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x24 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x25 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x26 x101) ∧ (∀ (x101 : L.sort 0), ¬L.eq x27 x101) ∧ (∀ (x101
end --Needs more than 2000 tokens!

/- FEW SHOT PROMPTS TO CODEX(START)
/--`theorem`
Power Set is Closed under Intersection
Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Then:
:$\forall A, B \in \powerset S: A \cap B \in \powerset S$
`proof`
Let $A, B \in \powerset S$.

Then by the definition of power set, $A \subseteq S$ and $B \subseteq S$.

From Intersection is Subset we have that $A \cap B \subseteq A$.

It follows from Subset Relation is Transitive that $A \cap B \subseteq S$.

Thus $A \cap B \in \powerset S$ and closure is proved.
{{qed}}
-/
theorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=
begin
  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$
  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),
  -- Then $A ⊆ S$ and $B ⊆ S$, by power set definition
  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},
  -- Then $(A ∩ B) ⊆ A$, by intersection of set is a subset
  have h2 : (A ∩ B) ⊆ A, from by apply set.inter_subset_left,
  -- Then $(A ∩ B) ⊆ S$, by subset relation is transitive
  have h3 : (A ∩ B) ⊆ S, from by {apply set.subset.trans h2 h1.left},
  -- Hence $(A ∩ B) ∈  𝒫 S$, by power set definition
  show (A ∩ B) ∈  𝒫 S, from by {apply set.mem_powerset h3},
end

/--`theorem`
Square of Sum
 :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$
`proof`
Follows from the distribution of multiplication over addition:

{{begin-eqn}}
{{eqn | l = \left({x + y}\right)^2
      | r = \left({x + y}\right) \cdot \left({x + y}\right)
}}
{{eqn | r = x \cdot \left({x + y}\right) + y \cdot \left({x + y}\right)
      | c = Real Multiplication Distributes over Addition
}}
{{eqn | r = x \cdot x + x \cdot y + y \cdot x + y \cdot y
      | c = Real Multiplication Distributes over Addition
}}
{{eqn | r = x^2 + 2xy + y^2
      | c =
}}
{{end-eqn}}
{{qed}}
-/
theorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) :=
begin
  -- expand the power
  calc (x + y)^2 = (x+y)*(x+y) : by rw sq
  -- distributive property of multiplication over addition gives:
  ... = x*(x+y) + y*(x+y) : by rw add_mul
  -- applying the above property further gives:
  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}
  -- rearranging the terms using commutativity and adding gives:
  ... = x^2 + 2*x*y + y^2 : by {repeat {rw ← sq}, rw mul_comm y x, ring}
end

/--`theorem`
Identity of Group is Unique
Let $\struct {G, \circ}$ be a group. Then there is a unique identity element $e \in G$.
`proof`
From Group has Latin Square Property, there exists a unique $x \in G$ such that:
:$a x = b$

and there exists a unique $y \in G$ such that:
:$y a = b$

Setting $b = a$, this becomes:

There exists a unique $x \in G$ such that:
:$a x = a$

and there exists a unique $y \in G$ such that:
:$y a = a$

These $x$ and $y$ are both $e$, by definition of identity element.
{{qed}}
-/
theorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=
begin
  -- Group has Latin Square Property
  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from by {
    assume a b : G, use a⁻¹ * b, obviously, },
  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by {
    assume a b : G, use b * a⁻¹, obviously, },

  -- Setting $b = a$, this becomes:
  have h3 : ∀ a : G, ∃! x : G, a * x = a, from
    assume a : G, h1 a a,
  have h4 : ∀ a : G, ∃! y : G, y * a = a, from
    assume a : G, h2 a a,

  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element
  have h5 : ∀ a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,
    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))
    (mul_one a),
  have h6 : ∀ a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,
    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a),

  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by {
    use (1 : G),
    have h7 : ∀ e : G, (∀ a : G, e * a = a ∧ a * e = a) → e = 1, from by {
      assume (e : G) (hident : ∀ a : G, e * a = a ∧ a * e = a),
      have h8 : ∀ a : G, e = classical.some (h3 a).exists, from assume (a : G),
        exists_unique.unique (h3 a) (hident a).right
        (classical.some_spec (exists_unique.exists (h3 a))),
      have h9 : ∀ a : G, e = classical.some (h4 a).exists, from assume (a : G),
        exists_unique.unique (h4 a) (hident a).left
        (classical.some_spec (exists_unique.exists (h4 a))),
      show e = (1 : G), from eq.trans (h9 e) (h6 _),
    },
    exact ⟨by obviously, h7⟩,
  }
end

/--`theorem`
Overflow theorem
Let $F$ be a set of first-order formulas which has finite models of arbitrarily large size. Then $F$ has an infinite model.
`proof`
For each $n$, let $\mathbf A_n$ be the formula:

$\exists x_1 \exists x_2 \ldots \exists x_n: \{x_1 \ne x_2 \land x_1 \ne x_3 \land \ldots \land x_{n - 1} \ne x_n\}$

Then $\mathbf A_i$ is true in a structure $\AA$ iff $\AA$ has at least $n$ elements.

Take:
$$ \Gamma := F \cup \bigcup_{i \mathop = 1}^\infty A_i $$

Since $F$ has models of arbitrarily large size, every finite subset of $\Gamma$ is satisfiable.

From the Compactness Theorem, $\Gamma$ is satisfiable in some model $\mathbf{M}$.

But since $\mathbf{M} \models A_i$ for each $i$, $\mathbf{M}$ must be infinite.

So $F$ has an infinite model.

QED
-/
theorem  overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=
FEW SHOT PROMPTS TO CODEX(END)-/


import power_quandle

universes u v

section power_quandle_union

variables {Q1 : Type u} [power_quandle Q1] {Q2 : Type v} [power_quandle Q2]

inductive pq_union_rel' : Q1 ⊕ Q2 → Q1 ⊕ Q2 → Type (max u v)
| refl {x : Q1 ⊕ Q2} : pq_union_rel' x x
| unit_glue1 : pq_union_rel' (sum.inl 1) (sum.inr 1)
| unit_glue2 : pq_union_rel' (sum.inr 1) (sum.inl 1)

inductive pq_union_rel : Q1 ⊕ Q2 → Q1 ⊕ Q2 → Prop
| rel {x y : Q1 ⊕ Q2} (hxy : pq_union_rel' x y) : pq_union_rel x y

@[refl]
lemma pq_union_rel_refl {x : Q1 ⊕ Q2} : pq_union_rel x x :=
begin
  refine pq_union_rel.rel _,
  exact pq_union_rel'.refl,
end

@[symm]
lemma pq_union_rel_symm {x y : Q1 ⊕ Q2} (hxy : pq_union_rel x y) : pq_union_rel y x :=
begin
  cases hxy,
  refine pq_union_rel.rel _,
  cases hxy_hxy,
  {
    exact pq_union_rel'.refl,
  },
  {
    exact pq_union_rel'.unit_glue2,
  },
  {
    exact pq_union_rel'.unit_glue1,
  },
end

@[trans]
lemma pq_union_rel_trans {x y z : Q1 ⊕ Q2} (hxy : pq_union_rel x y) (hyz : pq_union_rel y z) : pq_union_rel x z :=
begin
  cases hxy,
  cases hyz,
  refine pq_union_rel.rel _,
  cases hxy_hxy;
  cases hyz_hxy,
  exact pq_union_rel'.refl,
  exact pq_union_rel'.unit_glue1,
  exact pq_union_rel'.unit_glue2,
  exact pq_union_rel'.unit_glue1,
  exact pq_union_rel'.refl,
  exact pq_union_rel'.unit_glue2,
  exact pq_union_rel'.refl,
end

instance pq_union_rel_setoid : setoid (Q1 ⊕ Q2) := {
  r := pq_union_rel,
  iseqv := begin
    split,
    apply pq_union_rel_refl,
    split,
    apply pq_union_rel_symm,
    apply pq_union_rel_trans,
  end }

def pq_union (Q1 : Type u) [power_quandle Q1] (Q2 : Type v) [power_quandle Q2] := quotient (@pq_union_rel_setoid Q1 _ Q2 _)

end power_quandle_union

section power_quandle_union_pq

variables {Q1 : Type u} [power_quandle Q1] {Q2 : Type v} [power_quandle Q2]

lemma quot_mk_helper_pq_union (a : Q1 ⊕ Q2) : quot.mk setoid.r a = ⟦a⟧ := rfl

instance pq_union_has_one : has_one (pq_union Q1 Q2) := ⟨⟦sum.inl 1⟧⟩

lemma pq_union_one_def : (1 : pq_union Q1 Q2) = ⟦sum.inl 1⟧ := rfl

lemma pq_union_one_inl : ⟦sum.inl 1⟧ = (1 : pq_union Q1 Q2) := rfl
lemma pq_union_one_inr : ⟦sum.inr 1⟧ = (1 : pq_union Q1 Q2) := begin
  rw pq_union_one_def,
  apply quotient.sound,
  fconstructor,
  exact pq_union_rel'.unit_glue2,
end

lemma pq_union_same1 (x y : Q1) : ((sum.inl x : Q1 ⊕ Q2) ≈ (sum.inl y)) ↔ x = y :=
begin
  split,
  {
    intro hxy,
    cases hxy,
    cases hxy_hxy,
    refl,
  },
  {
    intro hxy,
    rw hxy,
  },
end

lemma pq_union_same2 (x y : Q2) : ((sum.inr x : Q1 ⊕ Q2) ≈ (sum.inr y)) ↔ x = y :=
begin
  split,
  {
    intro hxy,
    cases hxy,
    cases hxy_hxy,
    refl,
  },
  {
    intro hxy,
    rw hxy,
  },
end

lemma pq_union_different1 (x : Q1) (y : Q2) : ((sum.inl x : Q1 ⊕ Q2) ≈ (sum.inr y)) ↔ (x = 1 ∧ y = 1) :=
begin
  split,
  {
    intro hxy,
    cases hxy,
    cases hxy_hxy,
    split,
    refl,
    refl,
  },
  {
    intro hxy,
    cases hxy with hxy1 hxy2,
    rw [hxy1, hxy2],
    fconstructor,
    exact pq_union_rel'.unit_glue1,
  },
end

lemma pq_union_different2 (x : Q2) (y : Q1) : ((sum.inr x : Q1 ⊕ Q2) ≈ (sum.inl y)) ↔ (x = 1 ∧ y = 1) :=
begin
  split,
  {
    intro hxy,
    cases hxy,
    cases hxy_hxy,
    split,
    refl,
    refl,
  },
  {
    intro hxy,
    cases hxy with hxy1 hxy2,
    rw [hxy1, hxy2],
    fconstructor,
    exact pq_union_rel'.unit_glue2,
  },
end

/-
lemma pq_union_different1 (x : Q1) (y : Q2) (hxy : (sum.inl x : Q1 ⊕ Q2) ≈ (sum.inr y)) : x = 1 :=
begin
  cases hxy,
  cases hxy_hxy,
  refl,
end

lemma pq_union_different2 (x : Q1) (y : Q2) (hxy : (sum.inl x : Q1 ⊕ Q2) ≈ (sum.inr y)) : y = 1 :=
begin
  cases hxy,
  cases hxy_hxy,
  refl,
end
-/

def pre_pq_union_rhd : Q1 ⊕ Q2 → Q1 ⊕ Q2 → Q1 ⊕ Q2
| (sum.inl x) (sum.inl y) := sum.inl (x ▷ y)
| (sum.inl x) (sum.inr y) := sum.inr y
| (sum.inr x) (sum.inl y) := sum.inl y
| (sum.inr x) (sum.inr y) := sum.inr (x ▷ y)

def pq_union_rhd : pq_union Q1 Q2 → pq_union Q1 Q2 → pq_union Q1 Q2 := λ x y, quotient.lift_on₂ x y (λ x y, ⟦pre_pq_union_rhd x y⟧) begin
  intros a b c d hac hbd,
  apply quotient.sound,
  cases a;
  cases b;
  cases c;
  cases d;
  unfold pre_pq_union_rhd;
  try {exact hbd};
  try {rw pq_union_same1 at *};
  try {rw pq_union_same2 at *};
  try {rw pq_union_different1 at *};
  try {rw pq_union_different2 at *};
  try {congr, repeat {assumption}};
  try {split};
  try {simp only [hac, hbd, power_quandle.rhd_one, power_quandle.one_rhd]},
end

instance pq_union_has_rhd : has_rhd (pq_union Q1 Q2) := ⟨pq_union_rhd⟩

lemma pq_union_rhd_def (x y : pq_union Q1 Q2) : x ▷ y = pq_union_rhd x y := rfl

def pre_pq_union_pow : Q1 ⊕ Q2 → ℤ → Q1 ⊕ Q2
| (sum.inl x) n := sum.inl (x ^ n)
| (sum.inr x) n := sum.inr (x ^ n)

def pq_union_pow : pq_union Q1 Q2 → ℤ → pq_union Q1 Q2 := λ x n, quotient.lift_on x (λ x, ⟦pre_pq_union_pow x n⟧) begin
  intros a b hab,
  apply quotient.sound,
  cases a;
  cases b;
  unfold pre_pq_union_pow;
  try {rw pq_union_same1 at *};
  try {rw pq_union_same2 at *};
  try {rw pq_union_different1 at *};
  try {rw pq_union_different2 at *};
  try {congr, repeat {assumption}};
  try {split};
  try {simp only [hab, pq_one_pow]},
end

instance pq_union_has_pow : has_pow (pq_union Q1 Q2) ℤ := ⟨pq_union_pow⟩

lemma pq_union_pow_def (x : pq_union Q1 Q2) (n : ℤ) : x ^ n = pq_union_pow x n := rfl

instance pq_union_is_pq : power_quandle (pq_union Q1 Q2) := {
  rhd_dist := begin
    intros a b c,
    induction a,
    induction b,
    induction c,
    {
      simp only [quot_mk_helper_pq_union],
      simp only [pq_union_rhd_def],
      unfold pq_union_rhd,
      simp only [quotient.lift_on_beta₂, quotient.eq],
      cases a;
      cases b;
      cases c;
      unfold pre_pq_union_rhd;
      rw power_quandle.rhd_dist,
    },
    {refl,},
    {refl,},
    {refl,},
  end,
  rhd_idem := begin
    intro a,
    induction a,
    {
      simp only [quot_mk_helper_pq_union],
      simp only [pq_union_rhd_def],
      unfold pq_union_rhd,
      simp only [quotient.lift_on_beta₂, quotient.eq],
      cases a;
      unfold pre_pq_union_rhd;
      rw power_quandle.rhd_idem,
    },
    {refl,},
  end,
  pow_one := begin
    intro a,
    induction a,
    {
      simp only [quot_mk_helper_pq_union],
      simp only [pq_union_pow_def],
      unfold pq_union_pow,
      simp only [quotient.lift_on_beta, quotient.eq],
      cases a;
      unfold pre_pq_union_pow;
      rw power_quandle.pow_one,
    },
    {refl,},
  end,
  pow_zero := begin
    intro a,
    induction a,
    {
      simp only [quot_mk_helper_pq_union],
      simp only [pq_union_pow_def],
      unfold pq_union_pow,
      simp only [quotient.lift_on_beta, quotient.eq],
      cases a;
      unfold pre_pq_union_pow;
      rw power_quandle.pow_zero;
      simp only [pq_union_one_inl, pq_union_one_inr],
    },
    {refl,},
  end,
  pow_comp := begin
    intros a n m,
    induction a,
    {
      simp only [quot_mk_helper_pq_union],
      simp only [pq_union_pow_def],
      unfold pq_union_pow,
      simp only [quotient.lift_on_beta, quotient.eq],
      cases a;
      unfold pre_pq_union_pow;
      rw power_quandle.pow_comp,
    },
    {refl,},
  end,
  rhd_one := begin
    intro a,
    induction a,
    {
      simp only [quot_mk_helper_pq_union],
      simp only [pq_union_rhd_def, pq_union_one_def],
      unfold pq_union_rhd,
      simp only [quotient.lift_on_beta₂, quotient.eq],
      cases a;
      unfold pre_pq_union_rhd;
      rw power_quandle.rhd_one,
    },
    {refl,},
  end,
  one_rhd := begin
    intro a,
    induction a,
    {
      simp only [quot_mk_helper_pq_union],
      simp only [pq_union_rhd_def, pq_union_one_def],
      unfold pq_union_rhd,
      simp only [quotient.lift_on_beta₂, quotient.eq],
      cases a;
      unfold pre_pq_union_rhd;
      rw power_quandle.one_rhd,
    },
    {refl,},
  end,
  pow_rhd := begin
    intros a b n,
    induction a,
    induction b,
    {
      simp only [quot_mk_helper_pq_union],
      simp only [pq_union_rhd_def, pq_union_pow_def],
      unfold pq_union_rhd,
      unfold pq_union_pow,
      simp only [quotient.lift_on_beta, quotient.lift_on_beta₂, quotient.eq],
      cases a;
      cases b;
      unfold pre_pq_union_rhd;
      unfold pre_pq_union_pow;
      unfold pre_pq_union_rhd;
      rw power_quandle.pow_rhd,
    },
    {refl,},
    {refl,},
  end,
  rhd_pow_add := begin
    intros a b n m,
    induction a,
    induction b,
    {
      simp only [quot_mk_helper_pq_union],
      simp only [pq_union_rhd_def, pq_union_pow_def],
      unfold pq_union_rhd,
      unfold pq_union_pow,
      simp only [quotient.lift_on_beta, quotient.lift_on_beta₂, quotient.eq],
      cases a;
      cases b;
      unfold pre_pq_union_pow;
      unfold pre_pq_union_rhd;
      rw power_quandle.rhd_pow_add,
    },
    {refl,},
    {refl,},
  end,
  ..pq_union_has_one,
  ..pq_union_has_rhd,
  ..pq_union_has_pow }

end power_quandle_union_pq

section pq_union_map

variables {Q1 : Type u} [power_quandle Q1] {Q2 : Type v} [power_quandle Q2]

variables {X : Type*} [power_quandle X]

def pq_union_map_to (f : X → Q1 ⊕ Q2) : X → pq_union Q1 Q2 := λ x, ⟦f x⟧

theorem pq_union_map_to_is_pq_morphism (f : X → Q1 ⊕ Q2) (hf1 : ∀ x y : X, f (x ▷ y) = pre_pq_union_rhd (f x) (f y)) (hf2 : ∀ x : X, ∀ n : ℤ, f (x ^ n) = pre_pq_union_pow (f x) n) : is_pq_morphism (pq_union_map_to f) :=
begin
  split,
  {
    intros x y,
    specialize hf1 x y,
    unfold pq_union_map_to,
    rw hf1,
    refl,
  },
  {
    intros x n,
    specialize hf2 x n,
    unfold pq_union_map_to,
    rw hf2,
    refl,
  },
end

def pq_union_map_from_pre (f1 : Q1 → X) (f2 : Q2 → X) (hf : f1 1 = f2 1) : pq_union Q1 Q2 → X := λ x, quotient.lift_on x (sum.elim f1 f2) begin
  intros a b hab,
  cases hab,
  cases hab_hxy,
  {
    refl,
  },
  {
    simp only [sum.elim_inl, sum.elim_inr, hf],
  },
  {
    simp only [sum.elim_inl, sum.elim_inr, hf],
  },
end

def pq_union_map_from (f1 : Q1 → X) (hf1 : is_pq_morphism f1) (f2 : Q2 → X) (hf2 : is_pq_morphism f2) : pq_union Q1 Q2 → X := pq_union_map_from_pre f1 f2 begin
  rw one_preserved_by_morphism f1,
  rw one_preserved_by_morphism f2,
  assumption,
  assumption,
end

theorem pq_union_map_from_is_pq_morphism (f1 : Q1 → X) (hf1 : is_pq_morphism f1) (f2 : Q2 → X) (hf2 : is_pq_morphism f2) (hf12 : ∀ a : Q1, ∀ b : Q2, f1 a ▷ f2 b = f2 b) (hf21 : ∀ a : Q2, ∀ b : Q1, f2 a ▷ f1 b = f1 b) : is_pq_morphism (pq_union_map_from f1 hf1 f2 hf2) :=
begin
  split,
  {
    intros a b,
    unfold pq_union_map_from,
    unfold pq_union_map_from_pre,
    induction a,
    induction b,
    {
      simp only [quot_mk_helper_pq_union],
      rw pq_union_rhd_def,
      unfold pq_union_rhd,
      simp only [quotient.lift_on_beta, quotient.lift_on_beta₂],
      cases a;
      cases b;
      unfold pre_pq_union_rhd;
      simp only [sum.elim_inl, sum.elim_inr],
      {
        rw hf1.1,
      },
      {
        rw hf12,
      },
      {
        rw hf21,
      },
      {
        rw hf2.1,
      },
    },
    {refl,},
    {refl,},
  },
  {
    intros a n,
    unfold pq_union_map_from,
    unfold pq_union_map_from_pre,
    induction a,
    {
      simp only [quot_mk_helper_pq_union],
      rw pq_union_pow_def,
      unfold pq_union_pow,
      simp only [quotient.lift_on_beta],
      cases a;
      unfold pre_pq_union_pow;
      simp only [sum.elim_inl, sum.elim_inr],
      {
        rw hf1.2,
      },
      {
        rw hf2.2,
      },
    },
    {refl,},
  },
end

end pq_union_map


inductive DataType
  | TInt
  | TFloat
  | TString

open DataType

inductive DataEntry
  | EInt (i : Int)
  | EFloat (f : Float)
  | EString (s : String)
  | NULL

def NULL := DataEntry.NULL

instance : Coe Int DataEntry where
  coe := DataEntry.EInt

instance : Coe Float DataEntry where
  coe := DataEntry.EFloat

instance : OfNat DataEntry n where
  ofNat := DataEntry.EInt n

instance : OfScientific DataEntry where
  ofScientific m s e := DataEntry.EFloat (OfScientific.ofScientific m s e)

instance : Coe String DataEntry where
  coe := DataEntry.EString

namespace DataEntry

@[simp] def isOf (e : DataEntry) (t : DataType) : Prop :=
  match e, t with
  | EInt _,    TInt    => True
  | EFloat _,  TFloat  => True
  | EString _, TString => True
  | NULL,      _       => True
  | _,         _       => False

end DataEntry

abbrev Header := List (DataType × String)

def Header.colTypes (h : Header) : List DataType :=
  h.map fun x => x.1

def Header.colNames (h : Header) : List String :=
  h.map fun x => x.2

abbrev Row := List DataEntry

@[simp] def rowOfTypes : Row → List DataType → Prop
  | [],       []       => True
  | eh :: et, th :: tt => eh.isOf th ∧ rowOfTypes et tt
  | _,        _        => False

@[simp] def rowsOfTypes : List Row → List DataType → Prop
  | row :: rows, types => rowOfTypes row types ∧ rowsOfTypes rows types
  | [],          _     => True

structure DataFrame where
  header     : Header
  rows       : List Row
  consistent : rowsOfTypes rows header.colTypes := by simp

namespace DataFrame

def empty (header : Header := []) : DataFrame :=
  ⟨header, [], by simp⟩

theorem consistentConcatOfConsistentRow
    {df : DataFrame} (row : List DataEntry)
    (hc : rowOfTypes row df.header.colTypes) :
      rowsOfTypes (df.rows.concat row) (Header.colTypes df.header) :=
  match df with
    | ⟨_, rows, hr⟩ => by
      induction rows with
        | nil         => simp [hc] -- breaks here
        | cons _ _ hi => exact ⟨hr.1, hi hr.2 hc⟩

def addRow (df : DataFrame) (row : List DataEntry)
    (h : rowOfTypes row df.header.colTypes := by simp) : DataFrame :=
  ⟨df.header, df.rows.concat row, consistentConcatOfConsistentRow row h⟩

end DataFrame

def h : Header := [(TInt, "id"), (TString, "name")]

def r : List Row := [[1, "alex"]]

-- this no longer works
def df1 : DataFrame := DataFrame.mk h r

-- and this ofc breaks now
def df2 : DataFrame := df1.addRow [2, "juddy"]

-- this doesn't work anymore either
def df3 : DataFrame := DataFrame.empty h |>.addRow [3, "john"]


/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.algebra.module.pi
import Mathlib.algebra.ordered_pi
import Mathlib.algebra.module.prod
import Mathlib.algebra.ordered_field
import Mathlib.PostPort

universes u_1 u_2 l u_3

namespace Mathlib

/-!
# Ordered semimodules

In this file we define

* `ordered_semimodule R M` : an ordered additive commutative monoid `M` is an `ordered_semimodule`
  over an `ordered_semiring` `R` if the scalar product respects the order relation on the
  monoid and on the ring. There is a correspondence between this structure and convex cones,
  which is proven in `analysis/convex/cone.lean`.

## Implementation notes

* We choose to define `ordered_semimodule` as a `Prop`-valued mixin, so that it can be
  used for both modules and algebras
  (the axioms for an "ordered algebra" are exactly that the algebra is ordered as a module).
* To get ordered modules and ordered vector spaces, it suffices to the replace the
  `order_add_comm_monoid` and the `ordered_semiring` as desired.

## References

* https://en.wikipedia.org/wiki/Ordered_vector_space

## Tags

ordered semimodule, ordered module, ordered vector space
-/

/--
An ordered semimodule is an ordered additive commutative monoid
with a partial order in which the scalar multiplication is compatible with the order.
-/
class ordered_semimodule (R : Type u_1) (M : Type u_2) [ordered_semiring R] [ordered_add_comm_monoid M] [semimodule R M]
where
  smul_lt_smul_of_pos : ∀ {a b : M} {c : R}, a < b → 0 < c → c • a < c • b
  lt_of_smul_lt_smul_of_pos : ∀ {a b : M} {c : R}, c • a < c • b → 0 < c → a < b

theorem smul_lt_smul_of_pos {R : Type u_1} {M : Type u_2} [ordered_semiring R] [ordered_add_comm_monoid M] [semimodule R M] [ordered_semimodule R M] {a : M} {b : M} {c : R} : a < b → 0 < c → c • a < c • b :=
  ordered_semimodule.smul_lt_smul_of_pos

theorem smul_le_smul_of_nonneg {R : Type u_1} {M : Type u_2} [ordered_semiring R] [ordered_add_comm_monoid M] [semimodule R M] [ordered_semimodule R M] {a : M} {b : M} {c : R} (h₁ : a ≤ b) (h₂ : 0 ≤ c) : c • a ≤ c • b := sorry

theorem eq_of_smul_eq_smul_of_pos_of_le {R : Type u_1} {M : Type u_2} [ordered_semiring R] [ordered_add_comm_monoid M] [semimodule R M] [ordered_semimodule R M] {a : M} {b : M} {c : R} (h₁ : c • a = c • b) (hc : 0 < c) (hle : a ≤ b) : a = b :=
  or.resolve_left (has_le.le.lt_or_eq hle) fun (hlt : a < b) => has_lt.lt.ne (smul_lt_smul_of_pos hlt hc) h₁

theorem lt_of_smul_lt_smul_of_nonneg {R : Type u_1} {M : Type u_2} [ordered_semiring R] [ordered_add_comm_monoid M] [semimodule R M] [ordered_semimodule R M] {a : M} {b : M} {c : R} (h : c • a < c • b) (hc : 0 ≤ c) : a < b := sorry

theorem smul_lt_smul_iff_of_pos {R : Type u_1} {M : Type u_2} [ordered_semiring R] [ordered_add_comm_monoid M] [semimodule R M] [ordered_semimodule R M] {a : M} {b : M} {c : R} (hc : 0 < c) : c • a < c • b ↔ a < b :=
  { mp := fun (h : c • a < c • b) => lt_of_smul_lt_smul_of_nonneg h (has_lt.lt.le hc),
    mpr := fun (h : a < b) => smul_lt_smul_of_pos h hc }

theorem smul_pos_iff_of_pos {R : Type u_1} {M : Type u_2} [ordered_semiring R] [ordered_add_comm_monoid M] [semimodule R M] [ordered_semimodule R M] {a : M} {c : R} (hc : 0 < c) : 0 < c • a ↔ 0 < a :=
  iff.trans (eq.mpr (id (Eq._oldrec (Eq.refl (0 < c • a ↔ c • 0 < c • a)) (smul_zero c))) (iff.refl (0 < c • a)))
    (smul_lt_smul_iff_of_pos hc)

/-- If `R` is a linear ordered semifield, then it suffices to verify only the first axiom of
`ordered_semimodule`. Moreover, it suffices to verify that `a < b` and `0 < c` imply
`c • a ≤ c • b`. We have no semifields in `mathlib`, so we use the assumption `∀ c ≠ 0, is_unit c`
instead. -/
theorem ordered_semimodule.mk'' {R : Type u_1} {M : Type u_2} [linear_ordered_semiring R] [ordered_add_comm_monoid M] [semimodule R M] (hR : ∀ {c : R}, c ≠ 0 → is_unit c) (hlt : ∀ {a b : M} {c : R}, a < b → 0 < c → c • a ≤ c • b) : ordered_semimodule R M := sorry

/-- If `R` is a linear ordered field, then it suffices to verify only the first axiom of
`ordered_semimodule`. -/
theorem ordered_semimodule.mk' {k : Type u_1} {M : Type u_2} [linear_ordered_field k] [ordered_add_comm_monoid M] [semimodule k M] (hlt : ∀ {a b : M} {c : k}, a < b → 0 < c → c • a ≤ c • b) : ordered_semimodule k M :=
  ordered_semimodule.mk'' (fun (c : k) (hc : c ≠ 0) => is_unit.mk0 c hc) hlt

protected instance linear_ordered_semiring.to_ordered_semimodule {R : Type u_1} [linear_ordered_semiring R] : ordered_semimodule R R :=
  ordered_semimodule.mk ordered_semiring.mul_lt_mul_of_pos_left
    fun (_x _x_1 _x_2 : R) (h : _x_2 • _x < _x_2 • _x_1) (hc : 0 < _x_2) => lt_of_mul_lt_mul_left h (has_lt.lt.le hc)

theorem smul_le_smul_iff_of_pos {k : Type u_1} {M : Type u_2} [linear_ordered_field k] [ordered_add_comm_group M] [semimodule k M] [ordered_semimodule k M] {a : M} {b : M} {c : k} (hc : 0 < c) : c • a ≤ c • b ↔ a ≤ b := sorry

theorem smul_le_smul_iff_of_neg {k : Type u_1} {M : Type u_2} [linear_ordered_field k] [ordered_add_comm_group M] [semimodule k M] [ordered_semimodule k M] {a : M} {b : M} {c : k} (hc : c < 0) : c • a ≤ c • b ↔ b ≤ a := sorry

theorem smul_lt_iff_of_pos {k : Type u_1} {M : Type u_2} [linear_ordered_field k] [ordered_add_comm_group M] [semimodule k M] [ordered_semimodule k M] {a : M} {b : M} {c : k} (hc : 0 < c) : c • a < b ↔ a < c⁻¹ • b := sorry

theorem smul_le_iff_of_pos {k : Type u_1} {M : Type u_2} [linear_ordered_field k] [ordered_add_comm_group M] [semimodule k M] [ordered_semimodule k M] {a : M} {b : M} {c : k} (hc : 0 < c) : c • a ≤ b ↔ a ≤ c⁻¹ • b := sorry

theorem le_smul_iff_of_pos {k : Type u_1} {M : Type u_2} [linear_ordered_field k] [ordered_add_comm_group M] [semimodule k M] [ordered_semimodule k M] {a : M} {b : M} {c : k} (hc : 0 < c) : a ≤ c • b ↔ c⁻¹ • a ≤ b := sorry

protected instance prod.ordered_semimodule {k : Type u_1} {M : Type u_2} {N : Type u_3} [linear_ordered_field k] [ordered_add_comm_group M] [semimodule k M] [ordered_semimodule k M] [ordered_add_comm_group N] [semimodule k N] [ordered_semimodule k N] : ordered_semimodule k (M × N) :=
  ordered_semimodule.mk'
    fun (v u : M × N) (c : k) (h : v < u) (hc : 0 < c) =>
      { left := smul_le_smul_of_nonneg (and.left (and.left h)) (has_lt.lt.le hc),
        right := smul_le_smul_of_nonneg (and.right (and.left h)) (has_lt.lt.le hc) }

protected instance pi.ordered_semimodule {k : Type u_1} [linear_ordered_field k] {ι : Type u_2} {M : ι → Type u_3} [(i : ι) → ordered_add_comm_group (M i)] [(i : ι) → semimodule k (M i)] [∀ (i : ι), ordered_semimodule k (M i)] : ordered_semimodule k ((i : ι) → M i) :=
  ordered_semimodule.mk'
    fun (v u : (i : ι) → M i) (c : k) (h : v < u) (hc : 0 < c) (i : ι) =>
      id (smul_le_smul_of_nonneg (has_lt.lt.le h i) (has_lt.lt.le hc))

-- Sometimes Lean fails to apply the dependent version to non-dependent functions,

-- so we define another instance

protected instance pi.ordered_semimodule' {k : Type u_1} [linear_ordered_field k] {ι : Type u_2} {M : Type u_3} [ordered_add_comm_group M] [semimodule k M] [ordered_semimodule k M] : ordered_semimodule k (ι → M) :=
  pi.ordered_semimodule

protected instance order_dual.has_scalar {R : Type u_1} {M : Type u_2} [semiring R] [ordered_add_comm_monoid M] [semimodule R M] : has_scalar R (order_dual M) :=
  has_scalar.mk has_scalar.smul

protected instance order_dual.mul_action {R : Type u_1} {M : Type u_2} [semiring R] [ordered_add_comm_monoid M] [semimodule R M] : mul_action R (order_dual M) :=
  mul_action.mk sorry sorry

protected instance order_dual.distrib_mul_action {R : Type u_1} {M : Type u_2} [semiring R] [ordered_add_comm_monoid M] [semimodule R M] : distrib_mul_action R (order_dual M) :=
  distrib_mul_action.mk sorry sorry

protected instance order_dual.semimodule {R : Type u_1} {M : Type u_2} [semiring R] [ordered_add_comm_monoid M] [semimodule R M] : semimodule R (order_dual M) :=
  semimodule.mk sorry sorry

protected instance order_dual.ordered_semimodule {R : Type u_1} {M : Type u_2} [ordered_semiring R] [ordered_add_comm_monoid M] [semimodule R M] [ordered_semimodule R M] : ordered_semimodule R (order_dual M) :=
  ordered_semimodule.mk (fun (a b : order_dual M) => ordered_semimodule.smul_lt_smul_of_pos)
    fun (a b : order_dual M) => ordered_semimodule.lt_of_smul_lt_smul_of_pos