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small cosmetic stuff #12

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merged 1 commit into from
Feb 12, 2025
Merged

small cosmetic stuff #12

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30 changes: 15 additions & 15 deletions HoTT-Agda/core/lib/PathOver.agda
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Original file line number Diff line number Diff line change
Expand Up @@ -320,14 +320,12 @@ module _ {i j} {A : Type i} where

-- apd-tr conversion

from-transp-g : (B : A → Type j) {a a' : A} (p : a == a')
{u : B a} {v : B a'}
→ (transport B p u == v)
→ (u == v [ B ↓ p ])
from-transp-g : (B : A → Type j) {a a' : A} (p : a == a') {u : B a} {v : B a'}
→ transport B p u == v → u == v [ B ↓ p ]
from-transp-g B idp h = h

apd-to-tr : (B : A → Type j) (f : (a : A) → B a) {x y : A} (p : x == y)
(s : transport B p (f x) == f y)
apd-to-tr : (B : A → Type j) (f : (a : A) → B a) {x y : A}
(p : x == y) (s : transport B p (f x) == f y)
→ apd f p == from-transp-g B p s → apd-tr f p == s
apd-to-tr B f idp s h = h

Expand All @@ -341,8 +339,7 @@ module _ {i j} {A : Type i} {B : Type j} (f g : A → B) where

module _ (K : (z : A) → f z == g z) where

apd-to-hnat : {x y : A} (p : x == y)
(m : ap f p == K x ∙ ap g p ∙' ! (K y))
apd-to-hnat : {x y : A} (p : x == y) (m : ap f p == K x ∙ ap g p ∙' ! (K y))
→ apd K p == from-hmpty-nat p m → hmpty-nat-∙'-r K p == m
apd-to-hnat {x} idp m q = lemma (K x) m q
where
Expand All @@ -354,13 +351,14 @@ module _ {i j} {A : Type i} {B : Type j} (f g : A → B) where
apd-to-hnat-∙ : {x y z : A} (p₁ : x == y) (p₂ : y == z)
{m₁ : ap f p₁ == K x ∙ ap g p₁ ∙' ! (K y)} {m₂ : ap f p₂ == K y ∙ ap g p₂ ∙' ! (K z)}
(τ₁ : hmpty-nat-∙'-r K p₁ == m₁) (τ₂ : hmpty-nat-∙'-r K p₂ == m₂)
→ hmpty-nat-∙'-r K (p₁ ∙ p₂)
→
hmpty-nat-∙'-r K (p₁ ∙ p₂)
==
↯ (ap-∙ f p₁ p₂ ◃∙
ap (λ p → p ∙ ap f p₂) m₁ ◃∙
ap (λ p → (K x ∙ ap g p₁ ∙' ! (K y)) ∙ p) m₂ ◃∙
assoc-tri-!-mid (K x) (ap g p₁) (K y) (ap g p₂) (! (K z)) ◃∙
ap (λ p → K x ∙ p ∙' ! (K z)) (! (ap-∙ g p₁ p₂)) ◃∎)
↯ (ap-∙ f p₁ p₂ ◃∙
ap (λ p → p ∙ ap f p₂) m₁ ◃∙
ap (λ p → (K x ∙ ap g p₁ ∙' ! (K y)) ∙ p) m₂ ◃∙
assoc-tri-!-mid (K x) (ap g p₁) (K y) (ap g p₂) (! (K z)) ◃∙
ap (λ p → K x ∙ p ∙' ! (K z)) (! (ap-∙ g p₁ p₂)) ◃∎)
apd-to-hnat-∙ {x} idp idp idp idp = assoc-tri-!-coher (K x)

apd-to-hnat-! : {x y : A} (p : x == y)
Expand All @@ -370,7 +368,9 @@ module _ {i j} {A : Type i} {B : Type j} (f g : A → B) where

apd-to-hnat-ap! : ∀ {l} {C : Type l} (h : B → C) {x y : A} (p : x == y)
{m : ap f p == K x ∙ ap g p ∙' ! (K y)} (τ : hmpty-nat-∙'-r K p == m)
→ hmpty-nat-∙'-r (λ z → ap h (! (K z))) p ==
→
hmpty-nat-∙'-r (λ z → ap h (! (K z))) p
==
ap-∘-long h g f K p ∙
! (ap (λ q → ap h (! (K x)) ∙ ap h q ∙' ! (ap h (! (K y)))) m) ∙
! (ap (λ q → ap h (! (K x)) ∙ q ∙' ! (ap h (! (K y)))) (ap-∘ h f p))
Expand Down