port to non cubical hott

src/2023-05-17-equiv-no-cubical.agda

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+open import Relation.Binary.PropositionalEquality
+open import Data.Product
+open import Data.Product.Properties
+open import Data.Bool
+
+open import HottBook.Chapter2
+
+isContr : Type → Type
+isContr A = Σ[ x ∈ A ] (∀ y → x ≡ y)
+
+fiber : {A B : Type} (f : A → B) (y : B) → Type
+fiber {A = A} f y = Σ[ x ∈ A ] f x ≡ y
+
+record isEquiv {A : Type} {B : Type} (f : A → B) : Set where
+  no-eta-equality
+  field
+    equiv-proof : (y : B) → isContr (fiber f y)
+open isEquiv
+
+infix 4 _≃_
+_≃_ : (A : Set) (B : Set) → Type
+A ≃ B = Σ (A → B) \ f → (isEquiv f)
+
+--------------------------------------------------------------------------------
+
+data Other : Type where
+  Top : Other
+  Bottom : Other
+
+convert : Bool → Other
+convert true = Top
+convert false = Bottom
+
+convert-inv : Other → Bool
+convert-inv Top = true
+convert-inv Bottom = false
+
+convert-inv-id : (b : Bool) → convert-inv (convert b) ≡ b
+convert-inv-id true = refl
+convert-inv-id false = refl
+
+center-fiber : (y : Other) → fiber convert y
+center-fiber Top = true , refl
+center-fiber Bottom = false , refl
+
+center-fiber-is-contr : (y : Other) → (fz : fiber convert y) → center-fiber y ≡ fz
+center-fiber-is-contr y fz =
+  let
+    fx = center-fiber y
+    x = proj₁ fx
+    z = proj₁ fz
+
+    px : convert x ≡ y
+    px = proj₂ fx
+
+    pz : convert z ≡ y
+    pz = proj₂ fz
+
+    eqv3 : convert x ≡ convert z
+    eqv3 = px ∙ sym pz
+
+    eqv4 : convert-inv (convert x) ≡ convert-inv (convert z)
+    eqv4 = ap convert-inv eqv3
+
+    x≡z : x ≡ z
+    x≡z = sym (convert-inv-id x) ∙ eqv4 ∙ convert-inv-id z
+
+    give-me-info = ?
+  in
+  Σ-≡,≡→≡ (x≡z , ?)
+
+convert-is-equiv : isEquiv convert
+convert-is-equiv .equiv-proof y = center-fiber y , center-fiber-is-contr y
+
+convert-equiv : Bool ≃ Other
+convert-equiv = convert , convert-is-equiv