stuck again

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

@@ -56,18 +56,50 @@ center-fiber-is-contr y fz =
     pz : convert z ≡ y
     pz = proj₂ fz
 
-    eqv3 : convert x ≡ convert z
-    eqv3 = px ∙ sym pz
+    un-ap-convert : (a b : Bool) → (p : convert a ≡ convert b) → a ≡ b
+    un-ap-convert a b p =
+      let inner = ap convert-inv p in
+      sym (convert-inv-id a) ∙ inner ∙ convert-inv-id b
 
-    eqv4 : convert-inv (convert x) ≡ convert-inv (convert z)
-    eqv4 = ap convert-inv eqv3
+    cx≡cz : convert x ≡ convert z
+    cx≡cz = px ∙ sym pz
+
+    -- eqv4 : convert-inv (convert x) ≡ convert-inv (convert z)
+    -- eqv4 = ap convert-inv cx≡cz
 
     x≡z : x ≡ z
-    x≡z = sym (convert-inv-id x) ∙ eqv4 ∙ convert-inv-id z
+    -- x≡z = sym (convert-inv-id x) ∙ eqv4 ∙ convert-inv-id z
+    x≡z = un-ap-convert x z cx≡cz
+
+    P : Bool → Type
+    P b = convert b ≡ y
+
+    -- Using subst here because that's what Σ-≡,≡→≡ uses
+    px≡pz : (subst P x≡z px) ≡ pz
+    px≡pz = J
+      (λ b p′ →
+        let
+          x≡b : x ≡ b
+          x≡b = ?
+
+          convert-the : convert b ≡ y
+          convert-the = ?
+        in
+        (subst P x≡b px) ≡ convert-the
+      ) x≡z refl
 
     give-me-info = ?
   in
-  Σ-≡,≡→≡ (x≡z , ?)
+  -- Goal type:
+  --  transport (λ cx → convert cx ≡ y)
+  --  (
+  --    sym (convert-inv-id x)
+  --    ∙ ap convert-inv (px ∙ sym pz)
+  --    ∙ convert-inv-id z
+  --  )
+  --  (proj₂ (center-fiber y))  <-- this is just px
+  --  ≡ proj₂ fz                <-- this is just pz
+  Σ-≡,≡→≡ (x≡z , px≡pz)
 
 convert-is-equiv : isEquiv convert
 convert-is-equiv .equiv-proof y = center-fiber y , center-fiber-is-contr y