46 lines
1.7 KiB
Text
46 lines
1.7 KiB
Text
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/-
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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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Author: Floris van Doorn
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Coherence conditions for operations on pathovers
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-/
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open function
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namespace eq
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variables {A A' A'' : Type} {a a' a₂ : A} (B : A → Type) {p : a = a₂}
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{b b' : B a} {b₂ : B a₂}
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definition pathover_ap_id (q : b =[p] b₂) : pathover_ap B id q = change_path (ap_id p)⁻¹ q :=
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by induction q; reflexivity
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definition pathover_ap_compose (B : A'' → Type) (g : A' → A'') (f : A → A')
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{b : B (g (f a))} {b₂ : B (g (f a₂))} (q : b =[p] b₂) : pathover_ap B (g ∘ f) q
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= change_path (ap_compose g f p)⁻¹ (pathover_ap B g (pathover_ap (B ∘ g) f q)) :=
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by induction q; reflexivity
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definition pathover_ap_compose_rev (B : A'' → Type) (g : A' → A'') (f : A → A')
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{b : B (g (f a))} {b₂ : B (g (f a₂))} (q : b =[p] b₂) :
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pathover_ap B g (pathover_ap (B ∘ g) f q)
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= change_path (ap_compose g f p) (pathover_ap B (g ∘ f) q) :=
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by induction q; reflexivity
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definition apdo_eq_apdo_ap {B : A' → Type} (g : Πb, B b) (f : A → A') (p : a = a')
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: apdo (λx, g (f x)) p = pathover_of_pathover_ap B f (apdo g (ap f p)) :=
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by induction p; reflexivity
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definition pathover_of_tr_eq_idp (r : b = b') : pathover_of_tr_eq r = pathover_idp_of_eq r :=
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idp
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definition pathover_of_tr_eq_eq_concato (r : p ▸ b = b₂)
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: pathover_of_tr_eq r = pathover_tr p b ⬝o pathover_idp_of_eq r :=
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by induction r; induction p; reflexivity
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definition apo011_eq_apo11_apdo (f : Πa, B a → A') (p : a = a₂) (q : b =[p] b₂)
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: apo011 f p q = eq_of_pathover (apo11 (apdo f p) q) :=
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by induction q; reflexivity
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end eq
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