idk what this progress was but here's progress

src/2023-05-14-biinv-equiv.agda

@@ -13,11 +13,20 @@ open import Data.Product.Base
 id : ∀ {ℓ} {A : Type ℓ} → A → A
 id x = x
 
+linv′ : {A B : Type} (f : A → B) → Type
+linv′ {A} {B} f = Σ[ g ∈ (B → A) ] (g ∘ f ≡ id)
+
+rinv′ : {A B : Type} (f : A → B) → Type
+rinv′ {A} {B} f = Σ[ g ∈ (B → A) ] (f ∘ g ≡ id)
+
+-- Using this definition will need funExt everywhere
+-- Use this:
+
 linv : {A B : Type} (f : A → B) → Type
-linv {A} {B} f = Σ[ g ∈ (B → A) ] (g ∘ f ≡ id)
+linv {A} {B} f = Σ[ g ∈ (B → A) ] ((x : A) → g (f x) ≡ x)
 
 rinv : {A B : Type} (f : A → B) → Type
-rinv {A} {B} f = Σ[ g ∈ (B → A) ] (f ∘ g ≡ id)
+rinv {A} {B} f = Σ[ g ∈ (B → A) ] ((y : B) → f (g y) ≡ y)
 
 biinv : {A B : Type} (f : A → B) → Type
 biinv f = linv f × rinv f
@@ -32,12 +41,12 @@ Bool-id-pointwise : (b : Bool) → Bool-id (Bool-id b) ≡ b
 Bool-id-pointwise true = refl
 Bool-id-pointwise false = refl
 
-Bool-id-equiv : Bool-id ∘ Bool-id ≡ id
-Bool-id-equiv = funExt Bool-id-pointwise
+-- Bool-id-equiv : Bool-id ∘ Bool-id ≡ id
+-- Bool-id-equiv = funExt Bool-id-pointwise
 
 Bool-linv : linv Bool-id
 Bool-linv .fst = Bool-id
-Bool-linv .snd = funExt Bool-id-pointwise
+Bool-linv .snd = ?
 
 Bool-rinv : rinv Bool-id
 Bool-rinv .fst = Bool-id

src/2023-05-15-equiv-try-again.agda

@@ -34,7 +34,7 @@ convert-fiber Top = true , refl
 convert-fiber Bottom = false , refl
 
 convert-fiber-is-contr : (y : Other) → (fz : fiber convert y) → convert-fiber y ≡ fz
-convert-fiber-is-contr y fz i =
+convert-fiber-is-contr y fz =
   let
     fx : fiber convert y
     fx = convert-fiber y
@@ -71,15 +71,22 @@ convert-fiber-is-contr y fz i =
     sfx = snd fx
     sfz = snd fz
 
-    what : sfx ≡ ?
+    what : sfx ≡ ? -- sfz
     what = ?
   in
+
   -- z = fst fz
   -- convert-fiber y ≡ fz
-  -- (x , convert x₁ ≡ y) ≡ (z , convert z₁ ≡ y)
+  -- fx ≡ fz
+  -- Look into this more:
+    -- (x , convert x₁ ≡ y) ≡ (z , convert z₁ ≡ y)
+    -- Σ[ x ∈ A ] (convert x ≡ y)
+    -- (x , _)
   --  - x ≡ z
-  --  - (convert x₁ ≡ y) ≡ (convert z₁ ≡ y)
-  eqv7 i , (λ j → ?)
+  --  ------ - (convert x₁ ≡ y) ≡ (convert z₁ ≡ y)
+  --
+  -- eqv7 ?
+  ?
 
 convert-is-equiv : isEquiv convert
 convert-is-equiv .equiv-proof y = convert-fiber y , convert-fiber-is-contr y