135 lines
4.7 KiB
Text
135 lines
4.7 KiB
Text
/-
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Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn, Egbert Rijke
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Constructions with groups
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-/
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import .quotient_group .free_commutative_group
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open eq algebra is_trunc set_quotient relation sigma prod sum list trunc function equiv sigma.ops
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namespace group
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variables {G G' : AddGroup} (H : subgroup_rel G) (N : normal_subgroup_rel G) {g g' h h' k : G}
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{A B : AddAbGroup}
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variables (X : Set) {l l' : list (X ⊎ X)}
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section
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parameters {I : Set} (Y : I → AddAbGroup)
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variables {A' : AddAbGroup} {Y' : I → AddAbGroup}
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definition dirsum_carrier : AddAbGroup := free_ab_group (trunctype.mk (Σi, Y i) _)
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local abbreviation ι [constructor] := @free_ab_group_inclusion
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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⟩))
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definition dirsum : AddAbGroup := quotient_ab_group_gen dirsum_carrier (λg, ∥dirsum_rel g∥)
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-- definition dirsum_carrier_incl [constructor] (i : I) : Y i →a dirsum_carrier :=
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definition dirsum_incl [constructor] (i : I) : Y i →a dirsum :=
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add_homomorphism.mk (λy, class_of (ι ⟨i, y⟩))
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begin intro g h, symmetry, apply gqg_eq_of_rel, apply tr, apply dirsum_rel.rmk end
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parameter {Y}
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definition dirsum.rec {P : dirsum → Type} [H : Πg, is_prop (P g)]
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(h₁ : Πi (y : Y i), P (dirsum_incl i y)) (h₂ : P 0) (h₃ : Πg h, P g → P h → P (g + h)) :
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Πg, P g :=
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begin
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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)),
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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) y),
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rewrite [add.left_inv, add.assoc],
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end
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definition dirsum_homotopy {φ ψ : dirsum →a A'}
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(h : Πi (y : Y i), φ (dirsum_incl i y) = ψ (dirsum_incl i y)) : φ ~ ψ :=
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begin
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refine dirsum.rec _ _ _,
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exact h,
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refine !to_respect_zero ⬝ !to_respect_zero⁻¹,
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intro g₁ g₂ h₁ h₂, rewrite [+ to_respect_add', h₁, h₂]
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end
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definition dirsum_elim_resp_quotient (f : Πi, Y i →a 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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begin
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induction r with r, induction r,
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rewrite [to_respect_add, to_respect_neg], apply add_neg_eq_of_eq_add,
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rewrite [zero_add, to_respect_add, ▸*, ↑foldl, +one_mul, to_respect_add']
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end
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definition dirsum_elim [constructor] (f : Πi, Y i →a A') : dirsum →a 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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definition dirsum_elim_compute (f : Πi, Y i →a A') (i : I) :
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dirsum_elim f ∘g dirsum_incl i ~ f i :=
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begin
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intro g, apply zero_add
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end
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definition dirsum_elim_unique (f : Πi, Y i →a A') (k : dirsum →a A')
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(H : Πi, k ∘g dirsum_incl i ~ f i) : k ~ dirsum_elim f :=
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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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end
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end
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variables {I J : Set} {Y Y' Y'' : I → AddAbGroup}
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definition dirsum_functor [constructor] (f : Πi, Y i →a Y' i) : dirsum Y →a dirsum Y' :=
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dirsum_elim (λi, dirsum_incl Y' i ∘g f i)
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theorem dirsum_functor_compose (f' : Πi, Y' i →a Y'' i) (f : Πi, Y i →a Y' i) :
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dirsum_functor f' ∘a dirsum_functor f ~ dirsum_functor (λi, f' i ∘a f i) :=
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begin
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apply dirsum_homotopy,
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intro i y, reflexivity,
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end
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variable (Y)
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definition dirsum_functor_gid : dirsum_functor (λi, aid (Y i)) ~ aid (dirsum Y) :=
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begin
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apply dirsum_homotopy,
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intro i y, reflexivity,
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end
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variable {Y}
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definition dirsum_functor_add (f f' : Πi, Y i →a Y' i) :
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homomorphism_add (dirsum_functor f) (dirsum_functor f') ~
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dirsum_functor (λi, homomorphism_add (f i) (f' i)) :=
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begin
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apply dirsum_homotopy,
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intro i y, exact sorry
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end
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definition dirsum_functor_homotopy {f f' : Πi, Y i →a Y' i} (p : f ~2 f') :
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dirsum_functor f ~ dirsum_functor f' :=
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begin
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apply dirsum_homotopy,
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intro i y, exact sorry
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end
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definition dirsum_functor_left [constructor] (f : J → I) : dirsum (Y ∘ f) →a dirsum Y :=
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dirsum_elim (λj, dirsum_incl Y (f j))
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end group
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