feat(types): incorporate pathovers in the files of the types folder
Conflicts: hott/cubical/pathover.hlean
This commit is contained in:
Floris van Doorn
Leonardo de Moura
parent
0b12d51b25
commit
4a29f4bdd4
13 changed files with 260 additions and 212 deletions
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@ -70,9 +70,8 @@ namespace nat_trans
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intro η, apply nat_trans_eq, intro a, apply idp,
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intro S,
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fapply sigma_eq,
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apply eq_of_homotopy, intro a,
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apply idp,
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apply is_hprop.elim,
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{ apply eq_of_homotopy, intro a, apply idp},
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{ apply is_hprop.elimo}
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end
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definition is_hset_nat_trans [instance] : is_hset (F ⟹ G) :=
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@ -41,7 +41,7 @@ namespace algebra
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cases ps, cases ph1, cases ph2, cases ph3, cases ph4, reflexivity
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end
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definition group_pathover {G : group A} {H : group B} (f : A ≃ B) : (Π(g h : A), f (g * h) = f g * f h) → G =[ua f] H :=
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definition group_pathover {G : group A} {H : group B} {f : A ≃ B} : (Π(g h : A), f (g * h) = f g * f h) → G =[ua f] H :=
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begin
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revert H,
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eapply (rec_on_ua_idp' f),
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@ -7,6 +7,5 @@ The core of the HoTT library
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-/
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import types
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import cubical.square
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import hit.circle
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import algebra.hott
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@ -261,6 +261,11 @@ namespace equiv
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protected definition trans [trans] (f : A ≃ B) (g: B ≃ C) : A ≃ C :=
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equiv.mk (g ∘ f) !is_equiv_compose
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infixl `⬝e`:75 := equiv.trans
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postfix [parsing-only] `⁻¹ᵉ`:(max + 1) := equiv.symm
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-- notation for inverse which is not overloaded
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abbreviation erfl [constructor] := @equiv.refl
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definition equiv_of_eq_fn_of_equiv (f : A ≃ B) {f' : A → B} (Heq : f = f') : A ≃ B :=
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equiv.mk f' (is_equiv_eq_closed Heq)
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@ -287,11 +292,7 @@ namespace equiv
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definition equiv_of_eq_of_equiv {A B C : Type} (p : A ≃ B) (q : B = C) : A ≃ C := q ▸ p
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namespace ops
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infixl `⬝e`:75 := equiv.trans
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postfix `⁻¹` := equiv.symm -- overloaded notation for inverse
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postfix [parsing-only] `⁻¹ᵉ`:(max + 1) := equiv.symm
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-- notation for inverse which is not overloaded
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abbreviation erfl [constructor] := @equiv.refl
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end ops
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end equiv
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@ -7,7 +7,7 @@ Theorems about W-types (well-founded trees)
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-/
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import .sigma .pi
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open eq sigma sigma.ops equiv is_equiv
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open eq equiv is_equiv sigma sigma.ops
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inductive Wtype.{l k} {A : Type.{l}} (B : A → Type.{k}) : Type.{max l k} :=
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sup : Π (a : A), (B a → Wtype.{l k} B) → Wtype.{l k} B
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@ -35,71 +35,71 @@ namespace Wtype
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protected definition eta (w : W a, B a) : ⟨w.1 , w.2⟩ = w :=
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by cases w; exact idp
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definition sup_eq_sup (p : a = a') (q : p ▸ f = f') : ⟨a, f⟩ = ⟨a', f'⟩ :=
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by cases p; cases q; exact idp
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definition sup_eq_sup (p : a = a') (q : f =[p] f') : ⟨a, f⟩ = ⟨a', f'⟩ :=
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by cases q; exact idp
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definition Wtype_eq (p : w.1 = w'.1) (q : p ▸ w.2 = w'.2) : w = w' :=
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definition Wtype_eq (p : w.1 = w'.1) (q : w.2 =[p] w'.2) : w = w' :=
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by cases w; cases w';exact (sup_eq_sup p q)
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definition Wtype_eq_pr1 (p : w = w') : w.1 = w'.1 :=
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by cases p;exact idp
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definition Wtype_eq_pr2 (p : w = w') : Wtype_eq_pr1 p ▸ w.2 = w'.2 :=
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by cases p;exact idp
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definition Wtype_eq_pr2 (p : w = w') : w.2 =[Wtype_eq_pr1 p] w'.2 :=
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by cases p;exact idpo
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namespace ops
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postfix `..1`:(max+1) := Wtype_eq_pr1
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postfix `..2`:(max+1) := Wtype_eq_pr2
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end ops open ops open sigma
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definition sup_path_W (p : w.1 = w'.1) (q : p ▸ w.2 = w'.2)
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definition sup_path_W (p : w.1 = w'.1) (q : w.2 =[p] w'.2)
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: ⟨(Wtype_eq p q)..1, (Wtype_eq p q)..2⟩ = ⟨p, q⟩ :=
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by cases w; cases w'; cases p; cases q; exact idp
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by cases w; cases w'; cases q; exact idp
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definition pr1_path_W (p : w.1 = w'.1) (q : p ▸ w.2 = w'.2) : (Wtype_eq p q)..1 = p :=
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definition pr1_path_W (p : w.1 = w'.1) (q : w.2 =[p] w'.2) : (Wtype_eq p q)..1 = p :=
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!sup_path_W..1
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definition pr2_path_W (p : w.1 = w'.1) (q : p ▸ w.2 = w'.2)
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: pr1_path_W p q ▸ (Wtype_eq p q)..2 = q :=
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definition pr2_path_W (p : w.1 = w'.1) (q : w.2 =[p] w'.2)
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: (Wtype_eq p q)..2 =[pr1_path_W p q] q :=
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!sup_path_W..2
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definition eta_path_W (p : w = w') : Wtype_eq (p..1) (p..2) = p :=
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by cases p; cases w; exact idp
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definition transport_pr1_path_W {B' : A → Type} (p : w.1 = w'.1) (q : p ▸ w.2 = w'.2)
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definition transport_pr1_path_W {B' : A → Type} (p : w.1 = w'.1) (q : w.2 =[p] w'.2)
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: transport (λx, B' x.1) (Wtype_eq p q) = transport B' p :=
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by cases w; cases w'; cases p; cases q; exact idp
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by cases w; cases w'; cases q; exact idp
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definition path_W_uncurried (pq : Σ(p : w.1 = w'.1), p ▸ w.2 = w'.2) : w = w' :=
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definition path_W_uncurried (pq : Σ(p : w.1 = w'.1), w.2 =[p] w'.2) : w = w' :=
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by cases pq with p q; exact (Wtype_eq p q)
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definition sup_path_W_uncurried (pq : Σ(p : w.1 = w'.1), p ▸ w.2 = w'.2)
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definition sup_path_W_uncurried (pq : Σ(p : w.1 = w'.1), w.2 =[p] w'.2)
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: ⟨(path_W_uncurried pq)..1, (path_W_uncurried pq)..2⟩ = pq :=
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by cases pq with p q; exact (sup_path_W p q)
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definition pr1_path_W_uncurried (pq : Σ(p : w.1 = w'.1), p ▸ w.2 = w'.2)
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definition pr1_path_W_uncurried (pq : Σ(p : w.1 = w'.1), w.2 =[p] w'.2)
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: (path_W_uncurried pq)..1 = pq.1 :=
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!sup_path_W_uncurried..1
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definition pr2_path_W_uncurried (pq : Σ(p : w.1 = w'.1), p ▸ w.2 = w'.2)
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: (pr1_path_W_uncurried pq) ▸ (path_W_uncurried pq)..2 = pq.2 :=
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definition pr2_path_W_uncurried (pq : Σ(p : w.1 = w'.1), w.2 =[p] w'.2)
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: (path_W_uncurried pq)..2 =[pr1_path_W_uncurried pq] pq.2 :=
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!sup_path_W_uncurried..2
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definition eta_path_W_uncurried (p : w = w') : path_W_uncurried ⟨p..1, p..2⟩ = p :=
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!eta_path_W
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definition transport_pr1_path_W_uncurried {B' : A → Type} (pq : Σ(p : w.1 = w'.1), p ▸ w.2 = w'.2)
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definition transport_pr1_path_W_uncurried {B' : A → Type} (pq : Σ(p : w.1 = w'.1), w.2 =[p] w'.2)
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: transport (λx, B' x.1) (@path_W_uncurried A B w w' pq) = transport B' pq.1 :=
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by cases pq with p q; exact (transport_pr1_path_W p q)
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definition isequiv_path_W /-[instance]-/ (w w' : W a, B a)
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: is_equiv (@path_W_uncurried A B w w') :=
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: is_equiv (path_W_uncurried : (Σ(p : w.1 = w'.1), w.2 =[p] w'.2) → w = w') :=
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adjointify path_W_uncurried
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(λp, ⟨p..1, p..2⟩)
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eta_path_W_uncurried
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sup_path_W_uncurried
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definition equiv_path_W (w w' : W a, B a) : (Σ(p : w.1 = w'.1), p ▸ w.2 = w'.2) ≃ (w = w') :=
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definition equiv_path_W (w w' : W a, B a) : (Σ(p : w.1 = w'.1), w.2 =[p] w'.2) ≃ (w = w') :=
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equiv.mk path_W_uncurried !isequiv_path_W
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definition double_induction_on {P : (W a, B a) → (W a, B a) → Type} (w w' : W a, B a)
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@ -115,7 +115,7 @@ namespace Wtype
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end
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/- truncatedness -/
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open is_trunc
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open is_trunc pi
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definition trunc_W [instance] (n : trunc_index)
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[HA : is_trunc (n.+1) A] : is_trunc (n.+1) (W a, B a) :=
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begin
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@ -123,9 +123,9 @@ namespace Wtype
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eapply (double_induction_on w w'), intro a a' f f' IH,
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fapply is_trunc_equiv_closed,
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{ apply equiv_path_W},
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{ fapply is_trunc_sigma,
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intro p, cases p, esimp,
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apply pi.is_trunc_eq_pi}
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{ apply is_trunc_sigma,
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intro p, cases p, esimp, apply is_trunc_equiv_closed_rev,
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apply pathover_idp}
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end
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end Wtype
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@ -4,5 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn
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-/
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import .sigma .prod .pi .equiv .fiber .eq .trunc .arrow .pointed .function .trunc .bool
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import .sigma .prod .pi .equiv .fiber .trunc .arrow .pointed .function .trunc .bool
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import .eq .square
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import .nat .int
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@ -3,11 +3,11 @@ Copyright (c) 2014 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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Author: Floris van Doorn
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Ported from Coq HoTT
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Partially ported from Coq HoTT
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Theorems about path types (identity types)
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-/
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open eq sigma sigma.ops equiv is_equiv
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open eq sigma sigma.ops equiv is_equiv equiv.ops
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namespace eq
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/- Path spaces -/
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@ -22,31 +22,31 @@ namespace eq
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definition whisker_left_con_right (p : a1 = a2) {q q' q'' : a2 = a3} (r : q = q') (s : q' = q'')
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: whisker_left p (r ⬝ s) = whisker_left p r ⬝ whisker_left p s :=
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begin
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cases p, cases r, cases s, apply idp
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cases p, cases r, cases s, exact idp
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end
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definition whisker_right_con_right {p p' p'' : a1 = a2} (q : a2 = a3) (r : p = p') (s : p' = p'')
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: whisker_right (r ⬝ s) q = whisker_right r q ⬝ whisker_right s q :=
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begin
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cases q, cases r, cases s, apply idp
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cases q, cases r, cases s, exact idp
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end
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definition whisker_left_con_left (p : a1 = a2) (p' : a2 = a3) {q q' : a3 = a4} (r : q = q')
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: whisker_left (p ⬝ p') r = !con.assoc ⬝ whisker_left p (whisker_left p' r) ⬝ !con.assoc' :=
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begin
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cases p', cases p, cases r, cases q, apply idp
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cases p', cases p, cases r, cases q, exact idp
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end
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definition whisker_right_con_left {p p' : a1 = a2} (q : a2 = a3) (q' : a3 = a4) (r : p = p')
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: whisker_right r (q ⬝ q') = !con.assoc' ⬝ whisker_right (whisker_right r q) q' ⬝ !con.assoc :=
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begin
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cases q', cases q, cases r, cases p, apply idp
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cases q', cases q, cases r, cases p, exact idp
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end
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definition whisker_left_inv_left (p : a2 = a1) {q q' : a2 = a3} (r : q = q')
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: !con_inv_cancel_left⁻¹ ⬝ whisker_left p (whisker_left p⁻¹ r) ⬝ !con_inv_cancel_left = r :=
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begin
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cases p, cases r, cases q, apply idp
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cases p, cases r, cases q, exact idp
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end
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/- Transporting in path spaces.
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@ -68,61 +68,66 @@ namespace eq
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definition transport_eq_lr (p : a1 = a2) (q : a1 = a1)
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: transport (λx, x = x) p q = p⁻¹ ⬝ q ⬝ p :=
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begin
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cases p,
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symmetry, transitivity (refl a1)⁻¹ ⬝ q,
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apply con_idp,
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apply idp_con
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end
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by cases p; rewrite [▸*,idp_con]
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definition transport_eq_Fl (p : a1 = a2) (q : f a1 = b)
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: transport (λx, f x = b) p q = (ap f p)⁻¹ ⬝ q :=
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by cases p; cases q; apply idp
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by cases p; cases q; reflexivity
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definition transport_eq_Fr (p : a1 = a2) (q : b = f a1)
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: transport (λx, b = f x) p q = q ⬝ (ap f p) :=
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by cases p; apply idp
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by cases p; reflexivity
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definition transport_eq_FlFr (p : a1 = a2) (q : f a1 = g a1)
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: transport (λx, f x = g x) p q = (ap f p)⁻¹ ⬝ q ⬝ (ap g p) :=
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begin
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cases p,
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symmetry, transitivity (ap f (refl a1))⁻¹ ⬝ q,
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apply con_idp,
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apply idp_con
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end
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by cases p; rewrite [▸*,idp_con]
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definition transport_eq_FlFr_D {B : A → Type} {f g : Πa, B a}
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(p : a1 = a2) (q : f a1 = g a1)
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: transport (λx, f x = g x) p q = (apd f p)⁻¹ ⬝ ap (transport B p) q ⬝ (apd g p) :=
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begin
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cases p,
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symmetry,
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transitivity _,
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apply con_idp,
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transitivity _,
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apply idp_con,
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apply ap_id
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end
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by cases p; rewrite [▸*,idp_con,ap_id]
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definition transport_eq_FFlr (p : a1 = a2) (q : h (f a1) = a1)
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: transport (λx, h (f x) = x) p q = (ap h (ap f p))⁻¹ ⬝ q ⬝ p :=
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begin
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cases p,
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symmetry,
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transitivity (ap h (ap f (refl a1)))⁻¹ ⬝ q,
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apply con_idp,
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apply idp_con,
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end
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by cases p; rewrite [▸*,idp_con]
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definition transport_eq_lFFr (p : a1 = a2) (q : a1 = h (f a1))
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: transport (λx, x = h (f x)) p q = p⁻¹ ⬝ q ⬝ (ap h (ap f p)) :=
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begin
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cases p, symmetry,
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transitivity (refl a1)⁻¹ ⬝ q,
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apply con_idp,
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apply idp_con,
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end
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by cases p; rewrite [▸*,idp_con]
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/- Pathovers -/
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-- In the comment we give the fibration of the pathover
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definition pathover_eq_l (p : a1 = a2) (q : a1 = a3) : q =[p] p⁻¹ ⬝ q := /-(λx, x = a3)-/
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by cases p; cases q; exact idpo
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definition pathover_eq_r (p : a2 = a3) (q : a1 = a2) : q =[p] q ⬝ p := /-(λx, a1 = x)-/
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by cases p; cases q; exact idpo
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definition pathover_eq_lr (p : a1 = a2) (q : a1 = a1) : q =[p] p⁻¹ ⬝ q ⬝ p := /-(λx, x = x)-/
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by cases p; rewrite [▸*,idp_con]; exact idpo
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definition pathover_eq_Fl (p : a1 = a2) (q : f a1 = b) : q =[p] (ap f p)⁻¹ ⬝ q := /-(λx, f x = b)-/
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by cases p; cases q; exact idpo
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definition pathover_eq_Fr (p : a1 = a2) (q : b = f a1) : q =[p] q ⬝ (ap f p) := /-(λx, b = f x)-/
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by cases p; exact idpo
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definition pathover_eq_FlFr (p : a1 = a2) (q : f a1 = g a1) : q =[p] (ap f p)⁻¹ ⬝ q ⬝ (ap g p) :=
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/-(λx, f x = g x)-/
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by cases p; rewrite [▸*,idp_con]; exact idpo
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definition pathover_eq_FlFr_D {B : A → Type} {f g : Πa, B a} (p : a1 = a2) (q : f a1 = g a1)
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: q =[p] (apd f p)⁻¹ ⬝ ap (transport B p) q ⬝ (apd g p) := /-(λx, f x = g x)-/
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by cases p; rewrite [▸*,idp_con,ap_id];exact idpo
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definition pathover_eq_FFlr (p : a1 = a2) (q : h (f a1) = a1) : q =[p] (ap h (ap f p))⁻¹ ⬝ q ⬝ p :=
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/-(λx, h (f x) = x)-/
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by cases p; rewrite [▸*,idp_con];exact idpo
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definition pathover_eq_lFFr (p : a1 = a2) (q : a1 = h (f a1)) : q =[p] p⁻¹ ⬝ q ⬝ (ap h (ap f p)) :=
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/-(λx, x = h (f x))-/
|
||||
by cases p; rewrite [▸*,idp_con];exact idpo
|
||||
|
||||
-- The Functorial action of paths is [ap].
|
||||
|
||||
|
|
@ -151,7 +156,7 @@ namespace eq
|
|||
(λq, by cases p;cases q;exact idp)
|
||||
local attribute is_equiv_concat_left [instance]
|
||||
|
||||
definition equiv_eq_closed_left (p : a1 = a2) (a3 : A) : (a1 = a3) ≃ (a2 = a3) :=
|
||||
definition equiv_eq_closed_left (a3 : A) (p : a1 = a2) : (a1 = a3) ≃ (a2 = a3) :=
|
||||
equiv.mk (concat p⁻¹) _
|
||||
|
||||
definition is_equiv_concat_right [instance] (p : a2 = a3) (a1 : A)
|
||||
|
|
@ -162,11 +167,11 @@ namespace eq
|
|||
(λq, by cases p;cases q;exact idp)
|
||||
local attribute is_equiv_concat_right [instance]
|
||||
|
||||
definition equiv_eq_closed_right (p : a2 = a3) (a1 : A) : (a1 = a2) ≃ (a1 = a3) :=
|
||||
definition equiv_eq_closed_right (a1 : A) (p : a2 = a3) : (a1 = a2) ≃ (a1 = a3) :=
|
||||
equiv.mk (λq, q ⬝ p) _
|
||||
|
||||
definition eq_equiv_eq_closed (p : a1 = a2) (q : a3 = a4) : (a1 = a3) ≃ (a2 = a4) :=
|
||||
equiv.trans (equiv_eq_closed_left p a3) (equiv_eq_closed_right q a2)
|
||||
equiv.trans (equiv_eq_closed_left a3 p) (equiv_eq_closed_right a2 q)
|
||||
|
||||
definition is_equiv_whisker_left (p : a1 = a2) (q r : a2 = a3)
|
||||
: is_equiv (@whisker_left A a1 a2 a3 p q r) :=
|
||||
|
|
@ -179,10 +184,10 @@ namespace eq
|
|||
apply concat2,
|
||||
{apply concat, {apply whisker_left_con_right},
|
||||
apply concat2,
|
||||
{cases p, cases q, apply idp},
|
||||
{apply idp}},
|
||||
{cases p, cases r, apply idp}},
|
||||
{intro s, cases s, cases q, cases p, apply idp}
|
||||
{cases p, cases q, exact idp},
|
||||
{exact idp}},
|
||||
{cases p, cases r, exact idp}},
|
||||
{intro s, cases s, cases q, cases p, exact idp}
|
||||
end
|
||||
|
||||
definition eq_equiv_con_eq_con_left (p : a1 = a2) (q r : a2 = a3) : (q = r) ≃ (p ⬝ q = p ⬝ r) :=
|
||||
|
|
@ -266,6 +271,44 @@ namespace eq
|
|||
: (q ⬝ p⁻¹ = r) ≃ (q = r ⬝ p) :=
|
||||
equiv.mk _ !is_equiv_eq_con_of_con_inv_eq
|
||||
|
||||
/- Pathover Equivalences -/
|
||||
|
||||
definition pathover_eq_equiv_l (p : a1 = a2) (q : a1 = a3) (r : a2 = a3) : q =[p] r ≃ q = p ⬝ r :=
|
||||
/-(λx, x = a3)-/
|
||||
by cases p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹
|
||||
|
||||
definition pathover_eq_equiv_r (p : a2 = a3) (q : a1 = a2) (r : a1 = a3) : q =[p] r ≃ q ⬝ p = r :=
|
||||
/-(λx, a1 = x)-/
|
||||
by cases p; apply pathover_idp
|
||||
|
||||
definition pathover_eq_equiv_lr (p : a1 = a2) (q : a1 = a1) (r : a2 = a2)
|
||||
: q =[p] r ≃ q ⬝ p = p ⬝ r := /-(λx, x = x)-/
|
||||
by cases p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹
|
||||
|
||||
definition pathover_eq_equiv_Fl (p : a1 = a2) (q : f a1 = b) (r : f a2 = b)
|
||||
: q =[p] r ≃ q = ap f p ⬝ r := /-(λx, f x = b)-/
|
||||
by cases p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹
|
||||
|
||||
definition pathover_eq_equiv_Fr (p : a1 = a2) (q : b = f a1) (r : b = f a2)
|
||||
: q =[p] r ≃ q ⬝ ap f p = r := /-(λx, b = f x)-/
|
||||
by cases p; apply pathover_idp
|
||||
|
||||
definition pathover_eq_equiv_FlFr (p : a1 = a2) (q : f a1 = g a1) (r : f a2 = g a2)
|
||||
: q =[p] r ≃ q ⬝ ap g p = ap f p ⬝ r := /-(λx, f x = g x)-/
|
||||
by cases p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹
|
||||
|
||||
definition pathover_eq_equiv_FFlr (p : a1 = a2) (q : h (f a1) = a1) (r : h (f a2) = a2)
|
||||
: q =[p] r ≃ q ⬝ p = ap h (ap f p) ⬝ r :=
|
||||
/-(λx, h (f x) = x)-/
|
||||
by cases p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹
|
||||
|
||||
definition pathover_eq_equiv_lFFr (p : a1 = a2) (q : a1 = h (f a1)) (r : a2 = h (f a2))
|
||||
: q =[p] r ≃ q ⬝ ap h (ap f p) = p ⬝ r :=
|
||||
/-(λx, x = h (f x))-/
|
||||
by cases p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹
|
||||
|
||||
-- a lot of this library still needs to be ported from Coq HoTT
|
||||
|
||||
|
||||
|
||||
end eq
|
||||
|
|
|
|||
|
|
@ -36,10 +36,7 @@ namespace fiber
|
|||
{apply equiv.symm, apply equiv_sigma_eq},
|
||||
apply sigma_equiv_sigma_id,
|
||||
intro p,
|
||||
apply equiv_of_equiv_of_eq,
|
||||
rotate 1,
|
||||
apply inv_con_eq_equiv_eq_con,
|
||||
{apply (ap (λx, x = _)), rewrite transport_eq_Fl}
|
||||
apply pathover_eq_equiv_Fl,
|
||||
end
|
||||
|
||||
definition fiber_eq {x y : fiber f b} (p : point x = point y)
|
||||
|
|
|
|||
|
|
@ -44,7 +44,7 @@ namespace is_trunc
|
|||
{ apply equiv.to_is_equiv, apply is_contr.sigma_char},
|
||||
apply (@is_hprop.mk), intros,
|
||||
fapply sigma_eq, {apply x.2},
|
||||
apply (@is_hprop.elim)},
|
||||
apply (@is_hprop.elimo)},
|
||||
{ intro A,
|
||||
apply is_trunc_is_equiv_closed,
|
||||
apply equiv.to_is_equiv,
|
||||
|
|
|
|||
|
|
@ -3,13 +3,13 @@ Copyright (c) 2014-15 Floris van Doorn. All rights reserved.
|
|||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Author: Floris van Doorn
|
||||
|
||||
Ported from Coq HoTT
|
||||
Partially ported from Coq HoTT
|
||||
Theorems about pi-types (dependent function spaces)
|
||||
-/
|
||||
|
||||
import types.sigma
|
||||
|
||||
open eq equiv is_equiv funext
|
||||
open eq equiv is_equiv funext sigma
|
||||
|
||||
namespace pi
|
||||
variables {A A' : Type} {B : A → Type} {B' : A' → Type} {C : Πa, B a → Type}
|
||||
|
|
@ -59,6 +59,26 @@ namespace pi
|
|||
: (transport (λa, Π(b : A'), C a b) p f) b = transport (λa, C a b) p (f b) :=
|
||||
eq.rec_on p idp
|
||||
|
||||
/- Pathovers -/
|
||||
|
||||
definition pi_pathover {f : Πb, C a b} {g : Πb', C a' b'} {p : a = a'}
|
||||
(r : Π(b : B a) (b' : B a') (q : b =[p] b'), f b =[apo011 C p q] g b') : f =[p] g :=
|
||||
begin
|
||||
cases p, apply pathover_idp_of_eq,
|
||||
apply eq_of_homotopy, intro b,
|
||||
apply eq_of_pathover_idp, apply r
|
||||
end
|
||||
|
||||
definition pi_pathover' {C : (Σa, B a) → Type} {f : Πb, C ⟨a, b⟩} {g : Πb', C ⟨a', b'⟩}
|
||||
{p : a = a'} (r : Π(b : B a) (b' : B a') (q : pathover B b p b'), f b =[dpair_eq_dpair p q] g b')
|
||||
: f =[p] g :=
|
||||
begin
|
||||
cases p, apply pathover_idp_of_eq,
|
||||
apply eq_of_homotopy, intro b,
|
||||
apply (@eq_of_pathover_idp _ C), exact (r b b (pathover.idpatho b)),
|
||||
end
|
||||
|
||||
|
||||
/- Maps on paths -/
|
||||
|
||||
/- The action of maps given by lambda. -/
|
||||
|
|
@ -91,7 +111,8 @@ namespace pi
|
|||
(Π(b : B a), transportD B (λ(a : A) (b : B a), C ⟨a, b⟩) p b (f b) = g (transport B p b)) -/
|
||||
definition heq_pi_sigma {C : (Σa, B a) → Type} (p : a = a')
|
||||
(f : Π(b : B a), C ⟨a, b⟩) (g : Π(b' : B a'), C ⟨a', b'⟩) :
|
||||
(Π(b : B a), (sigma_eq p idp) ▸ (f b) = g (p ▸ b)) ≃ (Π(b : B a), p ▸D (f b) = g (p ▸ b)) :=
|
||||
(Π(b : B a), (sigma_eq p !pathover_tr) ▸ (f b) = g (p ▸ b)) ≃
|
||||
(Π(b : B a), p ▸D (f b) = g (p ▸ b)) :=
|
||||
eq.rec_on p (λg, !equiv.refl) g
|
||||
end
|
||||
|
||||
|
|
|
|||
|
|
@ -3,7 +3,7 @@ Copyright (c) 2014-15 Floris van Doorn. All rights reserved.
|
|||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Author: Floris van Doorn
|
||||
|
||||
Ported from Coq HoTT
|
||||
Partially ported from Coq HoTT
|
||||
Theorems about sigma-types (dependent sums)
|
||||
-/
|
||||
|
||||
|
|
@ -26,11 +26,11 @@ namespace sigma
|
|||
definition eta3 : Π (u : Σa b c, D a b c), ⟨u.1, u.2.1, u.2.2.1, u.2.2.2⟩ = u
|
||||
| eta3 ⟨u₁, u₂, u₃, u₄⟩ := idp
|
||||
|
||||
definition dpair_eq_dpair (p : a = a') (q : p ▸ b = b') : ⟨a, b⟩ = ⟨a', b'⟩ :=
|
||||
by cases p; cases q; reflexivity
|
||||
definition dpair_eq_dpair (p : a = a') (q : b =[p] b') : ⟨a, b⟩ = ⟨a', b'⟩ :=
|
||||
by cases q; reflexivity
|
||||
|
||||
definition sigma_eq (p : u.1 = v.1) (q : p ▸ u.2 = v.2) : u = v :=
|
||||
by cases u; cases v; apply (dpair_eq_dpair p q)
|
||||
definition sigma_eq (p : u.1 = v.1) (q : u.2 =[p] v.2) : u = v :=
|
||||
by cases u; cases v; exact (dpair_eq_dpair p q)
|
||||
|
||||
/- Projections of paths from a total space -/
|
||||
|
||||
|
|
@ -39,80 +39,79 @@ namespace sigma
|
|||
|
||||
postfix `..1`:(max+1) := eq_pr1
|
||||
|
||||
definition eq_pr2 (p : u = v) : p..1 ▸ u.2 = v.2 :=
|
||||
by cases p; reflexivity
|
||||
definition eq_pr2 (p : u = v) : u.2 =[p..1] v.2 :=
|
||||
by cases p; exact idpo
|
||||
|
||||
postfix `..2`:(max+1) := eq_pr2
|
||||
|
||||
private definition dpair_sigma_eq (p : u.1 = v.1) (q : p ▸ u.2 = v.2)
|
||||
private definition dpair_sigma_eq (p : u.1 = v.1) (q : u.2 =[p] v.2)
|
||||
: ⟨(sigma_eq p q)..1, (sigma_eq p q)..2⟩ = ⟨p, q⟩ :=
|
||||
by cases u; cases v; cases p; cases q; apply idp
|
||||
by cases u; cases v; cases q; apply idp
|
||||
|
||||
definition sigma_eq_pr1 (p : u.1 = v.1) (q : p ▸ u.2 = v.2) : (sigma_eq p q)..1 = p :=
|
||||
definition sigma_eq_pr1 (p : u.1 = v.1) (q : u.2 =[p] v.2) : (sigma_eq p q)..1 = p :=
|
||||
(dpair_sigma_eq p q)..1
|
||||
|
||||
definition sigma_eq_pr2 (p : u.1 = v.1) (q : p ▸ u.2 = v.2)
|
||||
: sigma_eq_pr1 p q ▸ (sigma_eq p q)..2 = q :=
|
||||
definition sigma_eq_pr2 (p : u.1 = v.1) (q : u.2 =[p] v.2)
|
||||
: (sigma_eq p q)..2 =[sigma_eq_pr1 p q] q :=
|
||||
(dpair_sigma_eq p q)..2
|
||||
|
||||
definition sigma_eq_eta (p : u = v) : sigma_eq (p..1) (p..2) = p :=
|
||||
by cases p; cases u; reflexivity
|
||||
|
||||
definition tr_pr1_sigma_eq {B' : A → Type} (p : u.1 = v.1) (q : p ▸ u.2 = v.2)
|
||||
definition tr_pr1_sigma_eq {B' : A → Type} (p : u.1 = v.1) (q : u.2 =[p] v.2)
|
||||
: transport (λx, B' x.1) (sigma_eq p q) = transport B' p :=
|
||||
by cases u; cases v; cases p; cases q; reflexivity
|
||||
by cases u; cases v; cases q; reflexivity
|
||||
|
||||
/- the uncurried version of sigma_eq. We will prove that this is an equivalence -/
|
||||
|
||||
definition sigma_eq_uncurried : Π (pq : Σ(p : u.1 = v.1), p ▸ u.2 = v.2), u = v
|
||||
| sigma_eq_uncurried ⟨pq₁, pq₂⟩ := sigma_eq pq₁ pq₂
|
||||
definition sigma_eq_unc : Π (pq : Σ(p : u.1 = v.1), u.2 =[p] v.2), u = v
|
||||
| sigma_eq_unc ⟨pq₁, pq₂⟩ := sigma_eq pq₁ pq₂
|
||||
|
||||
definition dpair_sigma_eq_uncurried : Π (pq : Σ(p : u.1 = v.1), p ▸ u.2 = v.2),
|
||||
⟨(sigma_eq_uncurried pq)..1, (sigma_eq_uncurried pq)..2⟩ = pq
|
||||
| dpair_sigma_eq_uncurried ⟨pq₁, pq₂⟩ := dpair_sigma_eq pq₁ pq₂
|
||||
definition dpair_sigma_eq_unc : Π (pq : Σ(p : u.1 = v.1), u.2 =[p] v.2),
|
||||
⟨(sigma_eq_unc pq)..1, (sigma_eq_unc pq)..2⟩ = pq
|
||||
| dpair_sigma_eq_unc ⟨pq₁, pq₂⟩ := dpair_sigma_eq pq₁ pq₂
|
||||
|
||||
definition sigma_eq_pr1_uncurried (pq : Σ(p : u.1 = v.1), p ▸ u.2 = v.2)
|
||||
: (sigma_eq_uncurried pq)..1 = pq.1 :=
|
||||
(dpair_sigma_eq_uncurried pq)..1
|
||||
definition sigma_eq_pr1_unc (pq : Σ(p : u.1 = v.1), u.2 =[p] v.2)
|
||||
: (sigma_eq_unc pq)..1 = pq.1 :=
|
||||
(dpair_sigma_eq_unc pq)..1
|
||||
|
||||
definition sigma_eq_pr2_uncurried (pq : Σ(p : u.1 = v.1), p ▸ u.2 = v.2)
|
||||
: (sigma_eq_pr1_uncurried pq) ▸ (sigma_eq_uncurried pq)..2 = pq.2 :=
|
||||
(dpair_sigma_eq_uncurried pq)..2
|
||||
definition sigma_eq_pr2_unc (pq : Σ(p : u.1 = v.1), u.2 =[p] v.2) :
|
||||
(sigma_eq_unc pq)..2 =[sigma_eq_pr1_unc pq] pq.2 :=
|
||||
(dpair_sigma_eq_unc pq)..2
|
||||
|
||||
definition sigma_eq_eta_uncurried (p : u = v) : sigma_eq_uncurried ⟨p..1, p..2⟩ = p :=
|
||||
definition sigma_eq_eta_unc (p : u = v) : sigma_eq_unc ⟨p..1, p..2⟩ = p :=
|
||||
sigma_eq_eta p
|
||||
|
||||
definition tr_sigma_eq_pr1_uncurried {B' : A → Type}
|
||||
(pq : Σ(p : u.1 = v.1), p ▸ u.2 = v.2)
|
||||
: transport (λx, B' x.1) (@sigma_eq_uncurried A B u v pq) = transport B' pq.1 :=
|
||||
definition tr_sigma_eq_pr1_unc {B' : A → Type}
|
||||
(pq : Σ(p : u.1 = v.1), u.2 =[p] v.2)
|
||||
: transport (λx, B' x.1) (@sigma_eq_unc A B u v pq) = transport B' pq.1 :=
|
||||
destruct pq tr_pr1_sigma_eq
|
||||
|
||||
definition is_equiv_sigma_eq [instance] (u v : Σa, B a)
|
||||
: is_equiv (@sigma_eq_uncurried A B u v) :=
|
||||
adjointify sigma_eq_uncurried
|
||||
: is_equiv (@sigma_eq_unc A B u v) :=
|
||||
adjointify sigma_eq_unc
|
||||
(λp, ⟨p..1, p..2⟩)
|
||||
sigma_eq_eta_uncurried
|
||||
dpair_sigma_eq_uncurried
|
||||
sigma_eq_eta_unc
|
||||
dpair_sigma_eq_unc
|
||||
|
||||
definition equiv_sigma_eq (u v : Σa, B a) : (Σ(p : u.1 = v.1), p ▸ u.2 = v.2) ≃ (u = v) :=
|
||||
equiv.mk sigma_eq_uncurried !is_equiv_sigma_eq
|
||||
definition equiv_sigma_eq (u v : Σa, B a) : (Σ(p : u.1 = v.1), u.2 =[p] v.2) ≃ (u = v) :=
|
||||
equiv.mk sigma_eq_unc !is_equiv_sigma_eq
|
||||
|
||||
definition dpair_eq_dpair_con (p1 : a = a' ) (q1 : p1 ▸ b = b' )
|
||||
(p2 : a' = a'') (q2 : p2 ▸ b' = b'') :
|
||||
dpair_eq_dpair (p1 ⬝ p2) (con_tr p1 p2 b ⬝ ap (transport B p2) q1 ⬝ q2)
|
||||
= dpair_eq_dpair p1 q1 ⬝ dpair_eq_dpair p2 q2 :=
|
||||
by cases p1; cases p2; cases q1; cases q2; reflexivity
|
||||
definition dpair_eq_dpair_con (p1 : a = a' ) (q1 : b =[p1] b' )
|
||||
(p2 : a' = a'') (q2 : b' =[p2] b'') :
|
||||
dpair_eq_dpair (p1 ⬝ p2) (q1 ⬝o q2) = dpair_eq_dpair p1 q1 ⬝ dpair_eq_dpair p2 q2 :=
|
||||
by cases q1; cases q2; reflexivity
|
||||
|
||||
definition sigma_eq_con (p1 : u.1 = v.1) (q1 : p1 ▸ u.2 = v.2)
|
||||
(p2 : v.1 = w.1) (q2 : p2 ▸ v.2 = w.2) :
|
||||
sigma_eq (p1 ⬝ p2) (con_tr p1 p2 u.2 ⬝ ap (transport B p2) q1 ⬝ q2)
|
||||
= sigma_eq p1 q1 ⬝ sigma_eq p2 q2 :=
|
||||
definition sigma_eq_con (p1 : u.1 = v.1) (q1 : u.2 =[p1] v.2)
|
||||
(p2 : v.1 = w.1) (q2 : v.2 =[p2] w.2) :
|
||||
sigma_eq (p1 ⬝ p2) (q1 ⬝o q2) = sigma_eq p1 q1 ⬝ sigma_eq p2 q2 :=
|
||||
by cases u; cases v; cases w; apply dpair_eq_dpair_con
|
||||
|
||||
local attribute dpair_eq_dpair [reducible]
|
||||
definition dpair_eq_dpair_con_idp (p : a = a') (q : p ▸ b = b') :
|
||||
dpair_eq_dpair p q = dpair_eq_dpair p idp ⬝ dpair_eq_dpair idp q :=
|
||||
by cases p; cases q; reflexivity
|
||||
definition dpair_eq_dpair_con_idp (p : a = a') (q : b =[p] b') :
|
||||
dpair_eq_dpair p q = dpair_eq_dpair p !pathover_tr ⬝
|
||||
dpair_eq_dpair idp (pathover_idp_of_eq (tr_eq_of_pathover q)) :=
|
||||
by cases q; reflexivity
|
||||
|
||||
/- eq_pr1 commutes with the groupoid structure. -/
|
||||
|
||||
|
|
@ -122,21 +121,22 @@ namespace sigma
|
|||
|
||||
/- Applying dpair to one argument is the same as dpair_eq_dpair with reflexivity in the first place. -/
|
||||
|
||||
definition ap_dpair (q : b₁ = b₂) : ap (sigma.mk a) q = dpair_eq_dpair idp q :=
|
||||
definition ap_dpair (q : b₁ = b₂) :
|
||||
ap (sigma.mk a) q = dpair_eq_dpair idp (pathover_idp_of_eq q) :=
|
||||
by cases q; reflexivity
|
||||
|
||||
/- Dependent transport is the same as transport along a sigma_eq. -/
|
||||
|
||||
definition transportD_eq_transport (p : a = a') (c : C a b) :
|
||||
p ▸D c = transport (λu, C (u.1) (u.2)) (dpair_eq_dpair p idp) c :=
|
||||
p ▸D c = transport (λu, C (u.1) (u.2)) (dpair_eq_dpair p !pathover_tr) c :=
|
||||
by cases p; reflexivity
|
||||
|
||||
definition sigma_eq_eq_sigma_eq {p1 q1 : a = a'} {p2 : p1 ▸ b = b'} {q2 : q1 ▸ b = b'}
|
||||
(r : p1 = q1) (s : r ▸ p2 = q2) : sigma_eq p1 p2 = sigma_eq q1 q2 :=
|
||||
by cases r; cases s; reflexivity
|
||||
definition sigma_eq_eq_sigma_eq {p1 q1 : a = a'} {p2 : b =[p1] b'} {q2 : b =[q1] b'}
|
||||
(r : p1 = q1) (s : p2 =[r] q2) : sigma_eq p1 p2 = sigma_eq q1 q2 :=
|
||||
by cases s; reflexivity
|
||||
|
||||
/- A path between paths in a total space is commonly shown component wise. -/
|
||||
definition sigma_eq2 {p q : u = v} (r : p..1 = q..1) (s : r ▸ p..2 = q..2)
|
||||
definition sigma_eq2 {p q : u = v} (r : p..1 = q..1) (s : p..2 =[r] q..2)
|
||||
: p = q :=
|
||||
begin
|
||||
revert q r s,
|
||||
|
|
@ -147,9 +147,7 @@ namespace sigma
|
|||
apply sigma_eq_eta,
|
||||
end
|
||||
|
||||
/- In Coq they often have to give u and v explicitly when using the following definition -/
|
||||
definition sigma_eq2_uncurried {p q : u = v}
|
||||
(rs : Σ(r : p..1 = q..1), transport (λx, transport B x u.2 = v.2) r p..2 = q..2) : p = q :=
|
||||
definition sigma_eq2_unc {p q : u = v} (rs : Σ(r : p..1 = q..1), p..2 =[r] q..2) : p = q :=
|
||||
destruct rs sigma_eq2
|
||||
|
||||
/- Transport -/
|
||||
|
|
@ -175,72 +173,68 @@ namespace sigma
|
|||
cases p, cases bcd with b cd, cases cd, reflexivity
|
||||
end
|
||||
|
||||
/- Pathovers -/
|
||||
|
||||
definition eta_pathover (p : a = a') (bc : Σ(b : B a), C a b)
|
||||
: bc =[p] ⟨p ▸ bc.1, p ▸D bc.2⟩ :=
|
||||
by cases p; cases bc; apply idpo
|
||||
|
||||
definition sigma_pathover (p : a = a') (u : Σ(b : B a), C a b) (v : Σ(b : B a'), C a' b)
|
||||
(r : u.1 =[p] v.1) (s : u.2 =[apo011 C p r] v.2) : u =[p] v :=
|
||||
begin cases u, cases v, cases r, esimp [apo011] at s, induction s using idp_rec_on, apply idpo end
|
||||
|
||||
/- TODO:
|
||||
* define the projections from the type u =[p] v
|
||||
* show that the uncurried version of sigma_pathover is an equivalence
|
||||
-/
|
||||
|
||||
/- Functorial action -/
|
||||
variables (f : A → A') (g : Πa, B a → B' (f a))
|
||||
|
||||
definition sigma_functor (u : Σa, B a) : Σa', B' a' :=
|
||||
definition sigma_functor [unfold-c 7] (u : Σa, B a) : Σa', B' a' :=
|
||||
⟨f u.1, g u.1 u.2⟩
|
||||
|
||||
/- Equivalences -/
|
||||
|
||||
definition is_equiv_sigma_functor [H1 : is_equiv f] [H2 : Π a, is_equiv (g a)]
|
||||
: is_equiv (sigma_functor f g) :=
|
||||
adjointify (sigma_functor f g)
|
||||
(sigma_functor f⁻¹ (λ(a' : A') (b' : B' a'),
|
||||
((g (f⁻¹ a'))⁻¹ (transport B' (right_inv f a')⁻¹ b'))))
|
||||
begin
|
||||
intro u',
|
||||
cases u' with a' b',
|
||||
intro u', cases u' with a' b',
|
||||
apply sigma_eq (right_inv f a'),
|
||||
-- rewrite right_inv,
|
||||
-- end
|
||||
-- "rewrite right_inv (g (f⁻¹ a'))"
|
||||
apply concat, apply (ap (λx, (transport B' (right_inv f a') x))), apply (right_inv (g (f⁻¹ a'))),
|
||||
show right_inv f a' ▸ ((right_inv f a')⁻¹ ▸ b') = b',
|
||||
from tr_inv_tr (right_inv f a') b'
|
||||
rewrite [▸*,right_inv (g (f⁻¹ a')),▸*],
|
||||
apply tr_pathover
|
||||
end
|
||||
begin
|
||||
intro u,
|
||||
cases u with a b,
|
||||
apply (sigma_eq (left_inv f a)),
|
||||
calc
|
||||
transport B (left_inv f a) ((g (f⁻¹ (f a)))⁻¹ (transport B' (right_inv f (f a))⁻¹ (g a b)))
|
||||
= (g a)⁻¹ (transport (B' ∘ f) (left_inv f a) (transport B' (right_inv f (f a))⁻¹ (g a b)))
|
||||
: by esimp; rewrite (fn_tr_eq_tr_fn (left_inv f a) (λ a, (g a)⁻¹))
|
||||
... = (g a)⁻¹ (transport B' (ap f (left_inv f a)) (transport B' (right_inv f (f a))⁻¹ (g a b)))
|
||||
: ap (g a)⁻¹ !transport_compose
|
||||
... = (g a)⁻¹ (transport B' (ap f (left_inv f a)) (transport B' (ap f (left_inv f a))⁻¹ (g a b)))
|
||||
: ap (λ x, (g a)⁻¹ (transport B' (ap f (left_inv f a)) (transport B' x⁻¹ (g a b)))) (adj f a)
|
||||
... = (g a)⁻¹ (g a b) : {!tr_inv_tr}
|
||||
... = b : by esimp; rewrite (left_inv (g a) b)
|
||||
apply pathover_of_tr_eq,
|
||||
rewrite [▸*,adj f,-(fn_tr_eq_tr_fn (left_inv f a) (λ a, (g a)⁻¹)),
|
||||
▸*,transport_compose B' f,tr_inv_tr,left_inv]
|
||||
end
|
||||
|
||||
definition sigma_equiv_sigma_of_is_equiv [H1 : is_equiv f] [H2 : Π a, is_equiv (g a)]
|
||||
: (Σa, B a) ≃ (Σa', B' a') :=
|
||||
equiv.mk (sigma_functor f g) !is_equiv_sigma_functor
|
||||
|
||||
section
|
||||
local attribute inv [irreducible]
|
||||
local attribute function.compose [irreducible] --this is needed for the following class inference problem
|
||||
definition sigma_equiv_sigma (Hf : A ≃ A') (Hg : Π a, B a ≃ B' (to_fun Hf a)) :
|
||||
(Σa, B a) ≃ (Σa', B' a') :=
|
||||
sigma_equiv_sigma_of_is_equiv (to_fun Hf) (λ a, to_fun (Hg a))
|
||||
end
|
||||
|
||||
definition sigma_equiv_sigma_id {B' : A → Type} (Hg : Π a, B a ≃ B' a) : (Σa, B a) ≃ Σa, B' a :=
|
||||
sigma_equiv_sigma equiv.refl Hg
|
||||
|
||||
definition ap_sigma_functor_eq_dpair (p : a = a') (q : p ▸ b = b')
|
||||
: ap (sigma.sigma_functor f g) (sigma_eq p q)
|
||||
= sigma_eq (ap f p)
|
||||
((transport_compose _ f p (g a b))⁻¹ ⬝ (fn_tr_eq_tr_fn p g b)⁻¹ ⬝ ap (g a') q) :=
|
||||
by cases p; cases q; reflexivity
|
||||
definition ap_sigma_functor_eq_dpair (p : a = a') (q : b =[p] b') :
|
||||
ap (sigma_functor f g) (sigma_eq p q) = sigma_eq (ap f p) (pathover.rec_on q idpo) :=
|
||||
by cases q; reflexivity
|
||||
|
||||
definition ap_sigma_functor_eq (p : u.1 = v.1) (q : p ▸ u.2 = v.2)
|
||||
: ap (sigma.sigma_functor f g) (sigma_eq p q) =
|
||||
sigma_eq (ap f p)
|
||||
((transport_compose B' f p (g u.1 u.2))⁻¹ ⬝ (fn_tr_eq_tr_fn p g u.2)⁻¹ ⬝ ap (g v.1) q) :=
|
||||
by cases u; cases v; apply ap_sigma_functor_eq_dpair
|
||||
-- definition ap_sigma_functor_eq (p : u.1 = v.1) (q : u.2 =[p] v.2)
|
||||
-- : ap (sigma_functor f g) (sigma_eq p q) =
|
||||
-- sigma_eq (ap f p)
|
||||
-- ((transport_compose B' f p (g u.1 u.2))⁻¹ ⬝ (fn_tr_eq_tr_fn p g u.2)⁻¹ ⬝ ap (g v.1) q) :=
|
||||
-- by cases u; cases v; apply ap_sigma_functor_eq_dpair
|
||||
|
||||
/- definition 3.11.9(i): Summing up a contractible family of types does nothing. -/
|
||||
open is_trunc
|
||||
|
|
@ -249,7 +243,7 @@ namespace sigma
|
|||
adjointify pr1
|
||||
(λa, ⟨a, !center⟩)
|
||||
(λa, idp)
|
||||
(λu, sigma_eq idp !center_eq)
|
||||
(λu, sigma_eq idp (pathover_idp_of_eq !center_eq))
|
||||
|
||||
definition sigma_equiv_of_is_contr_pr2 [H : Π a, is_contr (B a)] : (Σa, B a) ≃ A :=
|
||||
equiv.mk pr1 _
|
||||
|
|
@ -262,7 +256,7 @@ namespace sigma
|
|||
(λu, (center_eq u.1)⁻¹ ▸ u.2)
|
||||
(λb, ⟨!center, b⟩)
|
||||
(λb, ap (λx, x ▸ b) !hprop_eq_of_is_contr)
|
||||
(λu, sigma_eq !center_eq !tr_inv_tr))
|
||||
(λu, sigma_eq !center_eq !tr_pathover))
|
||||
|
||||
/- Associativity -/
|
||||
|
||||
|
|
@ -284,7 +278,7 @@ namespace sigma
|
|||
|
||||
/- Symmetry -/
|
||||
|
||||
definition comm_equiv_uncurried (C : A × A' → Type) : (Σa a', C (a, a')) ≃ (Σa' a, C (a, a')) :=
|
||||
definition comm_equiv_unc (C : A × A' → Type) : (Σa a', C (a, a')) ≃ (Σa' a, C (a, a')) :=
|
||||
calc
|
||||
(Σa a', C (a, a')) ≃ Σu, C u : assoc_equiv_prod
|
||||
... ≃ Σv, C (flip v) : sigma_equiv_sigma !prod_comm_equiv
|
||||
|
|
@ -292,7 +286,7 @@ namespace sigma
|
|||
... ≃ (Σa' a, C (a, a')) : assoc_equiv_prod
|
||||
|
||||
definition sigma_comm_equiv (C : A → A' → Type) : (Σa a', C a a') ≃ (Σa' a, C a a') :=
|
||||
comm_equiv_uncurried (λu, C (prod.pr1 u) (prod.pr2 u))
|
||||
comm_equiv_unc (λu, C (prod.pr1 u) (prod.pr2 u))
|
||||
|
||||
definition equiv_prod (A B : Type) : (Σ(a : A), B) ≃ A × B :=
|
||||
equiv.mk _ (adjointify
|
||||
|
|
@ -323,30 +317,30 @@ namespace sigma
|
|||
|
||||
/- *** The negative universal property. -/
|
||||
|
||||
protected definition coind_uncurried (fg : Σ(f : Πa, B a), Πa, C a (f a)) (a : A)
|
||||
protected definition coind_unc (fg : Σ(f : Πa, B a), Πa, C a (f a)) (a : A)
|
||||
: Σ(b : B a), C a b :=
|
||||
⟨fg.1 a, fg.2 a⟩
|
||||
|
||||
protected definition coind (f : Π a, B a) (g : Π a, C a (f a)) (a : A) : Σ(b : B a), C a b :=
|
||||
sigma.coind_uncurried ⟨f, g⟩ a
|
||||
sigma.coind_unc ⟨f, g⟩ a
|
||||
|
||||
--is the instance below dangerous?
|
||||
--in Coq this can be done without function extensionality
|
||||
definition is_equiv_coind [instance] (C : Πa, B a → Type)
|
||||
: is_equiv (@sigma.coind_uncurried _ _ C) :=
|
||||
: is_equiv (@sigma.coind_unc _ _ C) :=
|
||||
adjointify _ (λ h, ⟨λa, (h a).1, λa, (h a).2⟩)
|
||||
(λ h, proof eq_of_homotopy (λu, !sigma.eta) qed)
|
||||
(λfg, destruct fg (λ(f : Π (a : A), B a) (g : Π (x : A), C x (f x)), proof idp qed))
|
||||
|
||||
definition sigma_pi_equiv_pi_sigma : (Σ(f : Πa, B a), Πa, C a (f a)) ≃ (Πa, Σb, C a b) :=
|
||||
equiv.mk sigma.coind_uncurried _
|
||||
equiv.mk sigma.coind_unc _
|
||||
end
|
||||
|
||||
/- ** Subtypes (sigma types whose second components are hprops) -/
|
||||
|
||||
/- To prove equality in a subtype, we only need equality of the first component. -/
|
||||
definition subtype_eq [H : Πa, is_hprop (B a)] (u v : Σa, B a) : u.1 = v.1 → u = v :=
|
||||
(sigma_eq_uncurried ∘ (@inv _ _ pr1 (@is_equiv_pr1 _ _ (λp, !is_trunc.is_trunc_eq))))
|
||||
sigma_eq_unc ∘ inv pr1
|
||||
|
||||
definition is_equiv_subtype_eq [H : Πa, is_hprop (B a)] (u v : Σa, B a)
|
||||
: is_equiv (subtype_eq u v) :=
|
||||
|
|
@ -361,20 +355,12 @@ namespace sigma
|
|||
[HA : is_trunc n A] [HB : Πa, is_trunc n (B a)] : is_trunc n (Σa, B a) :=
|
||||
begin
|
||||
revert A B HA HB,
|
||||
eapply (trunc_index.rec_on n),
|
||||
intro A B HA HB,
|
||||
fapply is_trunc.is_trunc_equiv_closed,
|
||||
symmetry,
|
||||
apply sigma_equiv_of_is_contr_pr1,
|
||||
intro n IH A B HA HB,
|
||||
fapply is_trunc.is_trunc_succ_intro, intro u v,
|
||||
fapply is_trunc.is_trunc_equiv_closed,
|
||||
induction n with n IH,
|
||||
{ intro A B HA HB, fapply is_trunc_equiv_closed_rev, apply sigma_equiv_of_is_contr_pr1},
|
||||
{ intro A B HA HB, apply is_trunc_succ_intro, intro u v,
|
||||
apply is_trunc_equiv_closed,
|
||||
apply equiv_sigma_eq,
|
||||
apply IH,
|
||||
apply is_trunc.is_trunc_eq,
|
||||
intro p,
|
||||
show is_trunc n (p ▸ u .2 = v .2), from
|
||||
is_trunc.is_trunc_eq n (p ▸ u.2) (v.2),
|
||||
exact IH _ _ _ _}
|
||||
end
|
||||
|
||||
end sigma
|
||||
|
|
|
|||
|
|
@ -9,6 +9,7 @@ Various datatypes.
|
|||
* [pi](pi.hlean)
|
||||
* [arrow](arrow.hlean)
|
||||
* [eq](eq.hlean)
|
||||
* [square](square.hlean): type of squares in a type
|
||||
* [fiber](fiber.hlean)
|
||||
* [hprop_trunc](hprop_trunc.hlean): in this file we prove that `is_trunc n A` is a mere proposition. We separate this from [trunc](trunc.hlean) to avoid circularity in imports.
|
||||
* [equiv](equiv.hlean)
|
||||
|
|
|
|||
Loading…
Reference in a new issue