replace more usage of J with just refl pattern matching
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@ -19,23 +19,15 @@ open import HottBook.Util
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### Lemma 2.1.1
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```
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sym : {l : Level} {A : Set l} {x y : A}
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→ (x ≡ y) → y ≡ x
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sym {l} {A} {x} {y} p = J (λ x y p → y ≡ x) (λ x → refl) x y p
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sym : {l : Level} {A : Set l} {x y : A} → (x ≡ y) → y ≡ x
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sym {l} {A} {x} {y} refl = refl
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```
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### Lemma 2.1.2
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```
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trans : {l : Level} {A : Set l} {x y z : A}
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→ (x ≡ y)
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→ (y ≡ z)
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→ (x ≡ z)
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trans {l} {A} {x} {y} {z} p q =
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let
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func1 x y p = J (λ a b p → a ≡ b) (λ x → refl) x y p
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func2 = J (λ a b p → (z : A) → (q : b ≡ z) → a ≡ z) func1 x y p
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in func2 z q
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trans : {l : Level} {A : Set l} {x y z : A} → (x ≡ y) → (y ≡ z) → (x ≡ z)
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trans {l} {A} {x} {y} {z} refl refl = refl
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infixr 30 _∙_
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_∙_ = trans
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@ -67,33 +59,22 @@ module ≡-Reasoning where
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```
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module lemma2∙1∙4 {l : Level} {A : Set l} where
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i1 : {x y : A} (p : x ≡ y) → p ≡ p ∙ refl
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i1 {x} {y} p = J (λ x y p → p ≡ p ∙ refl) (λ _ → refl) x y p
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i1 {x} {y} refl = refl
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i2 : {x y : A} (p : x ≡ y) → p ≡ refl ∙ p
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i2 {x} {y} p = J (λ x y p → p ≡ refl ∙ p) (λ _ → refl) x y p
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i2 {x} {y} refl = refl
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ii1 : {x y : A} (p : x ≡ y) → (sym p) ∙ p ≡ refl
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ii1 {x} {y} p = J (λ x y p → (sym p) ∙ p ≡ refl) (λ _ → refl) x y p
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ii1 {x} {y} refl = refl
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ii2 : {x y : A} (p : x ≡ y) → p ∙ (sym p) ≡ refl
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ii2 {x} {y} p = J (λ x y p → p ∙ (sym p) ≡ refl) (λ _ → refl) x y p
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ii2 {x} {y} refl = refl
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iii : {x y : A} (p : x ≡ y) → sym (sym p) ≡ p
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iii {x} {y} p = J (λ x y p → sym (sym p) ≡ p) (λ _ → refl) x y p
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iii {x} {y} refl = refl
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iv : {x y z w : A} (p : x ≡ y) (q : y ≡ z) (r : z ≡ w) → p ∙ (q ∙ r) ≡ (p ∙ q) ∙ r
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iv {x} {y} {z} {w} p q r = let
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open ≡-Reasoning
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C : (x y : A) → (p : x ≡ y) → Set l
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C x y p = (q : y ≡ z) → p ∙ (q ∙ r) ≡ (p ∙ q) ∙ r
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c : (x : A) → C x x refl
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c x q1 = begin
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refl ∙ (q1 ∙ r) ≡⟨ sym (i2 (q1 ∙ r)) ⟩
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(q1 ∙ r) ≡⟨ ap (λ p → p ∙ r) (i2 q1) ⟩
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(refl ∙ q1) ∙ r ∎
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in J C c x y p q
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iv {x} {y} {z} {w} refl refl refl = refl
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```
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### Definition 2.1.7 (pointed type)
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@ -144,21 +125,16 @@ module lemma2∙2∙2 {A B C : Set} where
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open ≡-Reasoning
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i : {f : A → B} {x y z : A} (p : x ≡ y) (q : y ≡ z) → ap f (p ∙ q) ≡ ap f p ∙ ap f q
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i {f} {x} {y} {z} p q = J
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((λ x1 y1 p1 → (q1 : y1 ≡ z) → ap f (p1 ∙ q1) ≡ ap f p1 ∙ ap f q1))
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(λ x1 q1 → J
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(λ x2 y2 p2 → ap f (refl ∙ p2) ≡ ap f refl ∙ ap f p2)
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(λ x2 → refl) x1 z q1
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) x y p q
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i {f} {x} {y} {z} refl refl = refl
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ii : {f : A → B} {x y : A} (p : x ≡ y) → ap f (sym p) ≡ sym (ap f p)
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ii {f} {x} {y} p = J (λ x1 y1 p1 → ap f (sym p1) ≡ sym (ap f p1)) (λ x1 → refl) x y p
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ii {f} {x} {y} refl = refl
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iii : (f : A → B) (g : B → C) {x y : A} (p : x ≡ y) → ap g (ap f p) ≡ ap (g ∘ f) p
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iii f g {x} {y} p = J (λ x1 y1 p1 → ap g (ap f p1) ≡ ap (g ∘ f) p1) (λ x1 → refl) x y p
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iii f g {x} {y} refl = refl
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iv : {x y : A} (p : x ≡ y) → ap id p ≡ p
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iv {x} {y} p = J (λ x1 y1 p1 → ap id p1 ≡ p1) (λ x1 → refl) x y p
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iv {x} {y} refl = refl
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```
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## 2.3 Type families are fibrations
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@ -170,24 +146,19 @@ transport : {l₁ l₂ : Level} {A : Set l₁} {x y : A}
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→ (P : A → Set l₂)
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→ (p : x ≡ y)
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→ P x → P y
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transport {l₁} {l₂} {A} {x} {y} P p = J (λ x y p → P x → P y) (λ x → id) x y p
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transport {l₁} {l₂} {A} {x} {y} P refl = id
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```
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### Lemma 2.3.2 (Path lifting property)
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```
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lift : {A : Set} {P : A → Set} {x y : A}
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→ (u : P x)
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→ (p : x ≡ y)
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→ (x , u) ≡ (y , transport P p u)
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lift {A} {P} {x} {y} u p =
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J (λ x1 y1 p1 → (u1 : P x1) → (x1 , u1) ≡ (y1 , transport P p1 u1)) (λ x1 u1 → refl) x y p u
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```
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Verifying its property:
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```
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module lemma2∙3∙2 where
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lift : {A : Set} {P : A → Set} {x y : A}
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→ (u : P x)
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→ (p : x ≡ y)
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→ (x , u) ≡ (y , transport P p u)
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lift {A} {P} {x} {y} u refl = refl
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pr₁ : {A : Set} {P : A → Set}
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{x y : A}
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{u : P x} {v : P y}
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@ -199,8 +170,7 @@ module lemma2∙3∙2 where
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lift-prop : {A : Set} {P : A → Set} {x y : A} (u : P x) → (p : x ≡ y)
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→ pr₁ (lift {A} {P} u p) ≡ p
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lift-prop {A} {P} {x} {y} u p =
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J (λ x1 y1 p1 → (u1 : P x1) → pr₁ (lift {A} {P} u1 p1) ≡ p1) (λ x1 u1 → refl) x y p u
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lift-prop {A} {P} {x} {y} u refl = refl
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```
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### Lemma 2.3.4 (Dependent map)
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@ -210,8 +180,7 @@ apd : {l₁ l₂ : Level} {A : Set l₁} {P : A → Set l₂} {x y : A}
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→ (f : (x : A) → P x)
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→ (p : x ≡ y)
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→ transport P p (f x) ≡ f y
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apd {l₁} {l₂} {A} {P} {x} {y} f p =
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J (λ x y p → transport P p (f x) ≡ f y) (λ x → refl) x y p
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apd {l₁} {l₂} {A} {P} {x} {y} f refl = refl
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```
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### Lemma 2.3.5
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@ -222,8 +191,7 @@ transportconst : {l₁ l₂ : Level} {A : Set l₁} {x y : A}
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→ (p : x ≡ y)
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→ (b : B)
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→ transport (λ _ → B) p b ≡ b
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transportconst {l₁} {l₂} {A} {x} {y} B p b =
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J (λ x y p → transport (λ _ → B) p b ≡ b) (λ x → refl) x y p
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transportconst {l₁} {l₂} {A} {x} {y} B refl b = refl
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```
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### Equation 2.3.6
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@ -252,8 +220,7 @@ lemma2∙3∙8 : {l : Level} {A B : Set l}
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→ {x y : A}
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→ (p : x ≡ y)
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→ apd f p ≡ transportconst B p (f x) ∙ ap f p
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lemma2∙3∙8 {l} {A} {B} f {x} {y} p =
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J (λ x y p → apd f p ≡ transportconst B p (f x) ∙ ap f p) (λ x → refl) x y p
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lemma2∙3∙8 {l} {A} {B} f {x} {y} refl = refl
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```
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### Lemma 2.3.9
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@ -264,13 +231,7 @@ lemma2∙3∙9 : {A : Set}
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→ (P : A → Set)
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→ (p : x ≡ y) → (q : y ≡ z) → (u : P x)
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→ transport P q (transport P p u) ≡ transport P (p ∙ q) u
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lemma2∙3∙9 {A} {x} {y} {z} P p q u =
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let
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f2 : (x1 : A) → (q1 : x1 ≡ z) (u1 : P x1) → transport P q1 u1 ≡ transport P (refl ∙ q1) u1
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f2 x1 q1 u1 =
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ap (λ x → transport P x u1) (lemma2∙1∙4.i2 q1)
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f1 = J (λ x1 y1 p1 → (q1 : y1 ≡ z) → (u1 : P x1) → transport P q1 (transport P p1 u1) ≡ transport P (p1 ∙ q1) u1) f2 x y p
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in f1 q u
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lemma2∙3∙9 {A} {x} {y} {z} P refl refl u = refl
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```
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### Lemma 2.3.10
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@ -283,9 +244,7 @@ lemma2∙3∙10 : {l1 l2 : Level} {A B : Set l1}
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→ (p : x ≡ y)
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→ (u : P (f x))
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→ transport (P ∘ f) p u ≡ transport P (ap f p) u
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lemma2∙3∙10 {l} {A} {B} f P {x} {y} p u =
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J (λ x1 y1 p1 → (u1 : P (f x1)) → transport (P ∘ f) p1 u1 ≡ transport P (ap f p1) u1)
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(λ x1 u1 → refl) x y p u
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lemma2∙3∙10 {l} {A} {B} f P {x} {y} refl u = refl
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```
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### Lemma 2.3.11
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@ -295,9 +254,7 @@ lemma2∙3∙11 : {A : Set} {P Q : A → Set}
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→ (f : (x : A) → P x → Q x)
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→ {x y : A} (p : x ≡ y) → (u : P x)
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→ transport Q p (f x u) ≡ f y (transport P p u)
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lemma2∙3∙11 {A} {P} {Q} f {x} {y} p u =
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J (λ x1 y1 p1 → (u1 : P x1) → transport Q p1 (f x1 u1) ≡ f y1 (transport P p1 u1))
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(λ x1 u1 → refl) x y p u
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lemma2∙3∙11 {A} {P} {Q} f {x} {y} refl u = refl
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```
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## 2.4 Homotopies and equivalences
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@ -340,16 +297,14 @@ Homotopy forms an equivalence relation:
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```
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lemma2∙4∙3 : {A B : Set} {f g : A → B} (H : f ∼ g) {x y : A} (p : x ≡ y)
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→ H x ∙ ap g p ≡ ap f p ∙ H y
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lemma2∙4∙3 {A} {B} {f} {g} H {x} {y} p = let
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open ≡-Reasoning
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in
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J (λ x1 y1 p1 → (H x1 ∙ ap g p1 ≡ ap f p1 ∙ H y1)) (λ x1 → begin
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H x1 ∙ ap g refl ≡⟨⟩
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H x1 ∙ refl ≡⟨ sym (lemma2∙1∙4.i1 (H x1)) ⟩
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H x1 ≡⟨ lemma2∙1∙4.i2 (H x1) ⟩
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refl ∙ H x1 ≡⟨⟩
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ap f refl ∙ H x1 ∎
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) x y p
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lemma2∙4∙3 {A} {B} {f} {g} H {x} {y} refl =
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let open ≡-Reasoning in
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begin
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H x ∙ ap g refl ≡⟨⟩
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H x ∙ refl ≡⟨ sym (lemma2∙1∙4.i1 (H x)) ⟩
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H x ≡⟨ lemma2∙1∙4.i2 (H x) ⟩
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refl ∙ H x ≡⟨⟩
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ap f refl ∙ H x ∎
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```
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### Corollary 2.4.4
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@ -704,9 +659,7 @@ equation2∙9∙4 : {l1 l2 : Level} {X : Set l1} {x1 x2 : X}
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→ (f : A x1 → B x1)
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→ (p : x1 ≡ x2)
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→ transport (λ x → A x → B x) p f ≡ λ x → transport B p (f (transport A (sym p) x))
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equation2∙9∙4 {l1} {l2} {X} {x1} {x2} {A} {B} f p =
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J (λ x3 y3 p1 → (f1 : A x3 → B x3) → (transport (λ x → A x → B x) p1 f1 ≡ λ x → transport B p1 (f1 (transport A (sym p1) x))))
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(λ x3 f1 → refl) x1 x2 p f
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equation2∙9∙4 {l1} {l2} {X} {x1} {x2} {A} {B} f refl = refl
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```
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### Equation 2.9.5
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@ -730,9 +683,7 @@ module equation2∙9∙5 {X : Set} {x1 x2 : X} where
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→ (p : x1 ≡ x2)
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→ (a : A x2)
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→ transport (Π A B) p f a ≡ transport (hat B) (sym (pair-≡-d (sym p) refl)) (f (transport A (sym p) a))
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equation2∙9∙5 A B f p a =
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J (λ x1' x2' p' → (f' : (a : A x1') → B x1' a) → (a' : A x2') → transport (Π A B) p' f' a' ≡ transport (hat B) (sym (pair-≡-d (sym p') refl)) (f' (transport A (sym p') a')))
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(λ x' f' a' → refl) x1 x2 p f a
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equation2∙9∙5 A B f refl a = refl
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```
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### Lemma 2.9.6
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@ -761,13 +712,6 @@ idtoeqv : {l : Level} {A B : Set l}
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→ (A ≡ B)
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→ (A ≃ B)
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idtoeqv {l} {A} {B} refl = lemma2∙4∙12.id-equiv A
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-- idtoeqv {l} {A} {B} p = J C c A B p
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-- where
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-- C : (x y : Set l) (p : x ≡ y) → Set l
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-- C x y p = x ≃ y
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-- c : (x : Set l) → C x x refl
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-- c x = id , qinv-to-isequiv id-qinv
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```
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### Axiom 2.10.3 (Univalence)
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@ -863,13 +807,9 @@ theorem2∙11∙3 : {A B : Set} → {f g : A → B} → {a a' : A}
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→ (p : a ≡ a')
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→ (q : f a ≡ g a)
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→ transport (λ x → f x ≡ g x) p q ≡ sym (ap f p) ∙ q ∙ (ap g p)
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theorem2∙11∙3 {A} {B} {f} {g} {a} {a'} p q =
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J (λ x y p → (q1 : f x ≡ g x) → transport (λ x → f x ≡ g x) p q1 ≡ sym (ap f p) ∙ q1 ∙ (ap g p))
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(λ x q1 →
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J (λ x y p → transport (λ z → f z ≡ g z) refl q1 ≡ sym (ap f refl) ∙ q1 ∙ ap g refl)
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(λ x → let helper = ap (λ p → sym refl ∙ p) (lemma2∙1∙4.i1 q1) in lemma2∙1∙4.i2 q1 ∙ helper)
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(f a) (g a) q)
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a a' p q
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theorem2∙11∙3 {A} {B} {f} {g} {a} {a'} refl q =
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let helper = ap (λ p → sym refl ∙ p) (lemma2∙1∙4.i1 q) in
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lemma2∙1∙4.i2 q ∙ helper
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```
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### Theorem 2.11.4
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@ -879,25 +819,18 @@ theorem2∙11∙4 : {A : Set} {B : A → Set} {f g : (x : A) → B x} {a a' : A}
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→ (p : a ≡ a')
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→ (q : f a ≡ g a)
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→ transport (λ x → f x ≡ g x) p q ≡ sym (apd f p) ∙ ap (transport B p) q ∙ apd g p
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theorem2∙11∙4 {A} {B} {f} {g} {a} {a'} p q =
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let open ≡-Reasoning in
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J (λ x y p → (q1 : f x ≡ g x) → transport (λ x → f x ≡ g x) p q1 ≡ sym (apd f p) ∙ ap (transport B p) q1 ∙ apd g p)
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(λ x q1 →
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J (λ x y p → transport (λ z → f z ≡ g z) refl q1 ≡ sym (apd f refl) ∙ ap (transport B refl) q1 ∙ apd g refl)
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(λ y →
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let
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q2 = ap id q1
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q1≡q2 : q1 ≡ q2
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q1≡q2 = J (λ a b p → p ≡ ap id p) (λ x → refl) (f x) (g x) q1
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in
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begin
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transport (λ z → f z ≡ g z) refl q1 ≡⟨⟩
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q1 ≡⟨ q1≡q2 ⟩
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q2 ≡⟨ (let helper = ap (λ p → refl ∙ p) (lemma2∙1∙4.i1 q2) in lemma2∙1∙4.i2 q2 ∙ helper) ⟩
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refl ∙ ap (transport B refl) q1 ∙ refl ∎
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)
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(f a) (g a) q)
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a a' p q
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theorem2∙11∙4 {A} {B} {f} {g} {a} {a'} refl q =
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let
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open ≡-Reasoning
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q2 = ap id q
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q≡q2 : q ≡ q2
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q≡q2 = J (λ a b p → p ≡ ap id p) (λ x → refl) (f a) (g a) q
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in
|
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begin
|
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transport (λ z → f z ≡ g z) refl q ≡⟨⟩
|
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q ≡⟨ q≡q2 ⟩
|
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q2 ≡⟨ (let helper = ap (λ p → refl ∙ p) (lemma2∙1∙4.i1 q2) in lemma2∙1∙4.i2 q2 ∙ helper) ⟩
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refl ∙ ap (transport B refl) q ∙ refl ∎
|
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```
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|
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### Theorem 2.11.5
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|
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Loading…
Reference in a new issue