fix free_abelian_group and direct_sum
This commit is contained in:
Ulrik Buchholtz
parent
2f16008700
commit
0c4baacfcc
3 changed files with 67 additions and 40 deletions
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@ -17,8 +17,11 @@ namespace group
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parameters {I : Type} [is_set I] (Y : I → AbGroup)
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variables {A' : AbGroup} {Y' : I → AbGroup}
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definition dirsum_carrier : AbGroup := free_ab_group (Σi, Y i)
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local abbreviation ι [constructor] := @free_ab_group_inclusion
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open option pointed
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definition dirsum_carrier : AbGroup := free_ab_group (Σi, Y i)₊
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local abbreviation ι [constructor] := (@free_ab_group_inclusion (Σi, Y i)₊ _ ∘ some)
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inductive dirsum_rel : dirsum_carrier → Type :=
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| rmk : Πi y₁ y₂, dirsum_rel (ι ⟨i, y₁⟩ * ι ⟨i, y₂⟩ * (ι ⟨i, y₁ * y₂⟩)⁻¹)
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@ -38,19 +41,24 @@ namespace group
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refine @set_quotient.rec_prop _ _ _ H _,
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refine @set_quotient.rec_prop _ _ _ (λx, !H) _,
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esimp, intro l, induction l with s l ih,
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exact h₂,
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induction s with v v,
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induction v with i y,
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exact h₃ _ _ (h₁ i y) ih,
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induction v with i y,
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refine h₃ (gqg_map _ _ (class_of [inr ⟨i, y⟩])) _ _ ih,
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refine transport P _ (h₁ i y⁻¹),
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refine _ ⬝ !one_mul,
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refine _ ⬝ ap (λx, mul x _) (to_respect_zero (dirsum_incl i)),
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apply gqg_eq_of_rel',
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apply tr, esimp,
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refine transport dirsum_rel _ (dirsum_rel.rmk i y⁻¹ y),
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rewrite [mul.left_inv, mul.assoc],
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{ exact h₂ },
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{ induction s with z z,
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{ induction z with v,
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{ refine transport P _ ih, apply ap class_of, symmetry,
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exact eq_of_rel (tr (free_ab_group.fcg_rel.resp_append !free_ab_group.fcg_rel.cancelpt1 (free_ab_group.fcg_rel.rrefl l))) },
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{ induction v with i y, exact h₃ _ _ (h₁ i y) ih } },
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{ induction z with v,
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{ refine transport P _ ih, apply ap class_of, symmetry,
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exact eq_of_rel (tr (free_ab_group.fcg_rel.resp_append !free_ab_group.fcg_rel.cancelpt2 (free_ab_group.fcg_rel.rrefl l))) },
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{ induction v with i y,
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refine h₃ (gqg_map _ _ (class_of [inr (some ⟨i, y⟩)])) _ _ ih,
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refine transport P _ (h₁ i y⁻¹),
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refine _ ⬝ !one_mul,
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refine _ ⬝ ap (λx, mul x _) (to_respect_zero (dirsum_incl i)),
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apply gqg_eq_of_rel',
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apply tr, esimp,
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refine transport dirsum_rel _ (dirsum_rel.rmk i y⁻¹ y),
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rewrite [mul.left_inv, mul.assoc]} } }
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end
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definition dirsum_homotopy {φ ψ : dirsum →g A'}
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@ -63,15 +71,16 @@ namespace group
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end
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definition dirsum_elim_resp_quotient (f : Πi, Y i →g A') (g : dirsum_carrier)
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(r : ∥dirsum_rel g∥) : free_ab_group_elim (λv, f v.1 v.2) g = 1 :=
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(r : ∥dirsum_rel g∥) : free_ab_group_elim ((pmap_equiv_left (Σi, Y i) A')⁻¹ (λv, f v.1 v.2)) g = 1 :=
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begin
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induction r with r, induction r,
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rewrite [to_respect_mul, to_respect_inv, to_respect_mul, ▸*, ↑foldl, *one_mul,
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to_respect_mul], apply mul.right_inv
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rewrite [to_respect_mul, to_respect_inv, to_respect_mul, ▸*, ↑foldl, *one_mul],
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rewrite [↑pmap_equiv_left], esimp,
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rewrite [-to_respect_mul], apply mul.right_inv
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end
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definition dirsum_elim [constructor] (f : Πi, Y i →g A') : dirsum →g A' :=
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gqg_elim _ (free_ab_group_elim (λv, f v.1 v.2)) (dirsum_elim_resp_quotient f)
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gqg_elim _ (free_ab_group_elim ((pmap_equiv_left (Σi, Y i) A')⁻¹ (λv, f v.1 v.2))) (dirsum_elim_resp_quotient f)
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definition dirsum_elim_compute (f : Πi, Y i →g A') (i : I) (y : Y i) :
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dirsum_elim f (dirsum_incl i y) = f i y :=
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@ -84,7 +93,10 @@ namespace group
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begin
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apply gqg_elim_unique,
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apply free_ab_group_elim_unique,
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intro x, induction x with i y, exact H i y
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intro x, induction x with z,
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{ esimp, refine _ ⬝ to_respect_zero k, apply ap k, apply ap class_of,
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exact eq_of_rel (tr !free_ab_group.fcg_rel.cancelpt1) },
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{ induction z with i y, exact H i y }
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end
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end
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@ -9,22 +9,24 @@ Constructions with groups
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import algebra.group_theory hit.set_quotient types.list types.sum .free_group
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open eq algebra is_trunc set_quotient relation sigma sigma.ops prod sum list trunc function equiv trunc_index
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group
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group pointed
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namespace group
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variables {G G' : Group} {g g' h h' k : G} {A B : AbGroup}
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variables (X : Type) {Y : Type} [is_set X] [is_set Y] {l l' : list (X ⊎ X)}
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variables (X : Type*) {Y : Type*} [is_set X] [is_set Y] {l l' : list (X ⊎ X)}
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/- Free Abelian Group of a set -/
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/- Free Abelian Group on a pointed set -/
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namespace free_ab_group
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inductive fcg_rel : list (X ⊎ X) → list (X ⊎ X) → Type :=
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| rrefl : Πl, fcg_rel l l
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| cancel1 : Πx, fcg_rel [inl x, inr x] []
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| cancel2 : Πx, fcg_rel [inr x, inl x] []
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| rflip : Πx y, fcg_rel [x, y] [y, x]
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| rrefl : Πl, fcg_rel l l
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| cancel1 : Πx, fcg_rel [inl x, inr x] []
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| cancel2 : Πx, fcg_rel [inr x, inl x] []
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| cancelpt1 : fcg_rel [inl pt] []
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| cancelpt2 : fcg_rel [inr pt] []
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| rflip : Πx y, fcg_rel [x, y] [y, x]
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| resp_append : Π{l₁ l₂ l₃ l₄}, fcg_rel l₁ l₂ → fcg_rel l₃ l₄ →
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fcg_rel (l₁ ++ l₃) (l₂ ++ l₄)
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| rtrans : Π{l₁ l₂ l₃}, fcg_rel l₁ l₂ → fcg_rel l₂ l₃ →
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@ -49,6 +51,8 @@ namespace group
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{ reflexivity},
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{ repeat esimp [map], exact cancel2 x},
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{ repeat esimp [map], exact cancel1 x},
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{ exact cancelpt2 X },
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{ exact cancelpt1 X },
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{ repeat esimp [map], apply fcg_rel.rflip},
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{ rewrite [+map_append], exact resp_append IH₁ IH₂},
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{ exact rtrans IH₁ IH₂}
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@ -60,6 +64,8 @@ namespace group
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{ reflexivity},
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{ repeat esimp [map], exact cancel2 x},
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{ repeat esimp [map], exact cancel1 x},
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{ exact cancelpt1 X },
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{ exact cancelpt2 X },
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{ repeat esimp [map], apply fcg_rel.rflip},
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{ rewrite [+reverse_append], exact resp_append IH₂ IH₁},
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{ exact rtrans IH₁ IH₂}
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@ -146,22 +152,24 @@ namespace group
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/- The universal property of the free commutative group -/
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variables {X A}
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definition free_ab_group_inclusion [constructor] (x : X) : free_ab_group X :=
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class_of [inl x]
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definition free_ab_group_inclusion [constructor] : X →* free_ab_group X :=
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ppi.mk (λ x, class_of [inl x]) (eq_of_rel (tr (fcg_rel.cancelpt1 X)))
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theorem fgh_helper_respect_fcg_rel (f : X → A) (r : fcg_rel X l l')
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theorem fgh_helper_respect_fcg_rel (f : X →* A) (r : fcg_rel X l l')
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: Π(g : A), foldl (fgh_helper f) g l = foldl (fgh_helper f) g l' :=
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begin
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induction r with l x x x y l₁ l₂ l₃ l₄ r₁ r₂ IH₁ IH₂ l₁ l₂ l₃ r₁ r₂ IH₁ IH₂: intro g,
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{ reflexivity},
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{ unfold [foldl], apply mul_inv_cancel_right},
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{ unfold [foldl], apply inv_mul_cancel_right},
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{ unfold [foldl], rewrite (respect_pt f), apply mul_one },
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{ unfold [foldl], rewrite [respect_pt f, one_inv], apply mul_one },
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{ unfold [foldl, fgh_helper], apply mul.right_comm},
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{ rewrite [+foldl_append, IH₁, IH₂]},
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{ exact !IH₁ ⬝ !IH₂}
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end
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definition free_ab_group_elim [constructor] (f : X → A) : free_ab_group X →g A :=
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definition free_ab_group_elim [constructor] (f : X →* A) : free_ab_group X →g A :=
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begin
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fapply homomorphism.mk,
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{ intro g, refine set_quotient.elim _ _ g,
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@ -172,10 +180,15 @@ namespace group
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esimp, refine !foldl_append ⬝ _, esimp, apply fgh_helper_mul}
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end
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definition fn_of_free_ab_group_elim [unfold_full] (φ : free_ab_group X →g A) : X → A :=
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φ ∘ free_ab_group_inclusion
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definition fn_of_free_ab_group_elim [unfold_full] (φ : free_ab_group X →g A) : X →* A :=
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ppi.mk (φ ∘ free_ab_group_inclusion)
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begin
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refine (_ ⬝ @respect_one _ _ _ _ φ (homomorphism.p φ)),
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apply ap φ, apply eq_of_rel, apply tr,
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exact (fcg_rel.cancelpt1 X)
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end
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definition free_ab_group_elim_unique [constructor] (f : X → A) (k : free_ab_group X →g A)
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definition free_ab_group_elim_unique [constructor] (f : X →* A) (k : free_ab_group X →g A)
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(H : k ∘ free_ab_group_inclusion ~ f) : k ~ free_ab_group_elim f :=
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begin
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refine set_quotient.rec_prop _, intro l, esimp,
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@ -188,18 +201,20 @@ namespace group
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end
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variables (X A)
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definition free_ab_group_elim_equiv_fn [constructor] : (free_ab_group X →g A) ≃ (X → A) :=
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definition free_ab_group_elim_equiv_fn [constructor] : (free_ab_group X →g A) ≃ (X →* A) :=
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begin
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fapply equiv.MK,
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{ exact fn_of_free_ab_group_elim},
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{ exact free_ab_group_elim},
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{ intro f, apply eq_of_homotopy, intro x, esimp, unfold [foldl], apply one_mul},
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{ intro f, apply eq_of_phomotopy, fapply phomotopy.mk,
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{ intro x, esimp, unfold [foldl], apply one_mul },
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{ apply is_prop.elim } },
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{ intro k, symmetry, apply homomorphism_eq, apply free_ab_group_elim_unique,
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reflexivity }
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end
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definition free_ab_group_functor (f : X → Y) : free_ab_group X →g free_ab_group Y :=
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free_ab_group_elim (free_ab_group_inclusion ∘ f)
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definition free_ab_group_functor (f : X →* Y) : free_ab_group X →g free_ab_group Y :=
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free_ab_group_elim (free_ab_group_inclusion ∘* f)
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-- set_option pp.all true
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-- definition free_ab_group.rec {P : free_ab_group X → Type} [H : Πg, is_prop (P g)]
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@ -129,8 +129,8 @@ namespace group
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/- The universal property of the free group -/
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variables {X G}
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definition free_group_inclusion [constructor] (x : X) : free_group X :=
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class_of [inl x]
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definition free_group_inclusion [constructor] : X →* free_group X :=
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ppi.mk (λ x, class_of [inl x]) (eq_of_rel (tr (free_group_rel.cancelpt1 X)))
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definition fgh_helper [unfold 6] (f : X → G) (g : G) (x : X ⊎ X) : G :=
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g * sum.rec (λz, f z) (λz, (f z)⁻¹) x
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