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2.4.8

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Michael Zhang 2024-07-21 14:20:19 -05:00
parent 4af035f713
commit fabb22e855
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src/HottBook/Chapter2.lagda.md
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@ -337,31 +337,23 @@ id-qinv = mkQinv id (λ _ → refl) (λ _ → refl)
``` ```
module example2∙4∙8 where module example2∙4∙8 where
private i : {A : Set} → {x y z : A} → (p : x ≡ y) → qinv (p ∙_)
variable i {A} {x} {y} {z} p = mkQinv g forward backward
A : Set
x y z : A
i : (p : x ≡ y) → qinv (p ∙_)
i p = mkQinv g forward backward
where where
g = (sym p) ∙_ g = (sym p) ∙_
forward : (_∙_ p ∘ g) ∼ id forward : (_∙_ p ∘ g) ∼ id
forward q = lemma2∙1∙4.iv p (sym p) q ∙ ap (_∙ q) (lemma2∙1∙4.ii2 p) ∙ sym (lemma2∙1∙4.i2 q) forward q = lemma2∙1∙4.iv p (sym p) q ∙ ap (_∙ q) (lemma2∙1∙4.ii2 p) ∙ sym (lemma2∙1∙4.i2 q)
-- backward : (g ∘ (_∙_ p)) ∼ id backward : (q : y ≡ z) → sym p ∙ (p ∙ q) ≡ q
backward : {y z : A} → (q : {! y ≡ z !}) → {! !} ≡ q backward q = lemma2∙1∙4.iv (sym p) p q ∙ ap (_∙ q) (lemma2∙1∙4.ii1 p) ∙ sym (lemma2∙1∙4.i2 q)
-- sym p ∙ (q ∙ p) ≡ q
backward q = {! !}
ii : (p : x ≡ y) → qinv (_∙ p) ii : {A : Set} → {x y z : A} → (p : x ≡ y) → qinv (_∙ p)
ii p = mkQinv g forward backward ii {A} {x} {y} {z} p = mkQinv g forward backward
where where
g : z ≡ y → z ≡ x
g = _∙ (sym p) g = _∙ (sym p)
forward : (_∙ p ∘ g) ∼ id forward : (_∙ p ∘ g) ∼ id
-- (q ∙ sym(p)) ∙ p ≡ q
forward q = sym (lemma2∙1∙4.iv q (sym p) p) ∙ ap (q ∙_) (lemma2∙1∙4.ii1 p) ∙ sym (lemma2∙1∙4.i1 q) forward q = sym (lemma2∙1∙4.iv q (sym p) p) ∙ ap (q ∙_) (lemma2∙1∙4.ii1 p) ∙ sym (lemma2∙1∙4.i1 q)
backward : (g ∘ _∙ p) ∼ id backward : (g ∘ _∙ p) ∼ id
-- (q ∙ p) ∙ (sym p) ≡ q
backward q = sym (lemma2∙1∙4.iv q p (sym p)) ∙ ap (q ∙_) (lemma2∙1∙4.ii2 p) ∙ sym (lemma2∙1∙4.i1 q) backward q = sym (lemma2∙1∙4.iv q p (sym p)) ∙ ap (q ∙_) (lemma2∙1∙4.ii2 p) ∙ sym (lemma2∙1∙4.i1 q)
``` ```