From b5b90ddd374807c14ec8ed4a79bb5c1a4faa6c31 Mon Sep 17 00:00:00 2001
From: depsterr <depsterr@protonmail.com>
Date: Fri, 28 Oct 2022 16:12:38 +0200
Subject: [PATCH] intensionality hell

---
 src/CwF/Base.lagda.md       |  4 ++--
 src/CwF/Structures.lagda.md | 21 +++++++++++++++++----
 2 files changed, 19 insertions(+), 6 deletions(-)

diff --git a/src/CwF/Base.lagda.md b/src/CwF/Base.lagda.md
index 60acfdb..e4dfbb6 100644
--- a/src/CwF/Base.lagda.md
+++ b/src/CwF/Base.lagda.md
@@ -152,7 +152,7 @@ For every context $\Gamma$ and $A ∈ Ty(\Gamma)$ there is a projection map $p_{
 Dubbing $p$ a projection is no coincidence, it projects a map out of a weakened map.
 ```
   field
-    p-∘ : {Δ Γ : Ob} {γ : Hom Δ Γ} {A : ∣ Ty Γ ∣} (a : ∣ Tr Δ (A ⟦ γ ⟧) ∣)
+    p-∘ : {Δ Γ : Ob} (γ : Hom Δ Γ) {A : ∣ Ty Γ ∣} (a : ∣ Tr Δ (A ⟦ γ ⟧) ∣)
         → p ∘ ⟨ γ , a ⟩ ≡ γ
 ```
 
@@ -162,7 +162,7 @@ Likewise, $q$ "projects" an object out of a weakened substitution.
             → ∣ Tr Δ (A ⟦ p ⟧ ⟦ ⟨ γ , a ⟩ ⟧) ∣ ≡ ∣ Tr Δ (A ⟦ γ ⟧) ∣
   q-∘-pathp {Δ} {Γ} {γ} {A} a =
     ∣ Tr Δ (A ⟦ p ⟧ ⟦ ⟨ γ , a ⟩ ⟧) ∣ ≡˘⟨ ⟦⟧-tr-comp p ⟨ γ , a ⟩ A ⟩
-    ∣ Tr Δ (A ⟦ p ∘ ⟨ γ , a ⟩ ⟧)   ∣ ≡⟨ (λ i → ∣ Tr Δ (A ⟦ p-∘ a i ⟧) ∣) ⟩
+    ∣ Tr Δ (A ⟦ p ∘ ⟨ γ , a ⟩ ⟧)   ∣ ≡⟨ (λ i → ∣ Tr Δ (A ⟦ p-∘ γ a i ⟧) ∣) ⟩
     ∣ Tr Δ (A ⟦ γ ⟧)               ∣ ∎
               
   field
diff --git a/src/CwF/Structures.lagda.md b/src/CwF/Structures.lagda.md
index 9f54e52..96cae3b 100644
--- a/src/CwF/Structures.lagda.md
+++ b/src/CwF/Structures.lagda.md
@@ -57,8 +57,7 @@ Substitution must also be closed under all of these.
     B ⟦ ⟨ id, x ⟩ ⟧ ⟦ γ ⟧
     typeof app f x [ γ ]
 
-    it would seem CwFs really don't want to be formalized :/
-    not enough definitional equalities...
+    Welcome to intensionality hell.
 
 ```
   app-subst-pathp : {Δ Γ : Ob} (γ : Hom Δ Γ) {A : ∣ Ty Γ ∣} {B : ∣ Ty (Γ ; A) ∣} 
@@ -72,10 +71,23 @@ Substitution must also be closed under all of these.
       ≡⟨⟩
     B ⟦ ⟨ γ ∘ p , transport (λ i → ⟦⟧-tr-comp γ p A (~ i)) q ⟩ ∘ ⟨id, x [ γ ] ⟩ ⟧
       ≡⟨ (λ i → B ⟦ ⟨⟩-∘  ⟨id, x [ γ ] ⟩ (γ ∘ p) A (transport (λ i → ⟦⟧-tr-comp γ (p {Δ} {A ⟦ γ ⟧}) A (~ i)) q) i ⟧) ⟩
-    B ⟦ ⟨ γ ∘ p ∘∘ ⟨id, x [ γ ] ⟩ , transport (λ i → ⟦⟧-tr-comp γ (p {Δ} {A ⟦ γ ⟧}) A (~ i)) q [ ⟨id, x [ γ ] ⟩ ] ⟩ ⟧
-      ≡⟨⟩
     B ⟦ ⟨ (γ ∘ p) ∘ ⟨id, x [ γ ] ⟩ , transport (λ i →  ⟦⟧-tr-comp (γ ∘ p) ⟨id, x [ γ ] ⟩ A (~ i))
                                                (transport (λ j → ⟦⟧-tr-comp γ (p {Δ} {A ⟦ γ ⟧}) A (~ j)) q [ ⟨id, x [ γ ] ⟩ ]) ⟩ ⟧
+      ≡⟨ ((λ i → B ⟦ ⟨ assoc γ p ⟨id, x [ γ ] ⟩ (~ i) , {!   !} ⟩ ⟧)) ⟩
+    B ⟦ ⟨ γ ∘ (p ∘ ⟨id, x [ γ ] ⟩) , transport (λ i →  ⟦⟧-tr-comp γ (p ∘ ⟨id, x [ γ ] ⟩) A (~ i))
+                                               (transport (λ j → ⟦⟧-tr-comp p ⟨id, x [ γ ] ⟩ (A ⟦ γ ⟧) (~ j)) (q [ ⟨id, x [ γ ] ⟩ ])) ⟩ ⟧
+      ≡⟨⟩
+    B ⟦ ⟨ γ ∘∘ p ∘ ⟨id, x [ γ ] ⟩ , transport (λ i → ⟦⟧-tr-comp p ⟨id, x [ γ ] ⟩ (A ⟦ γ ⟧) (~ i)) (q [ ⟨id, x [ γ ] ⟩ ]) ⟩ ⟧
+      ≡⟨ (λ i → B ⟦ ⟨ γ ∘∘ p-∘ id (transport (λ i → ⟦⟧-tr-id (A ⟦ γ ⟧) (~ i)) (x [ γ ])) i , {!   !} ⟩ ⟧) ⟩
+    B ⟦ ⟨ γ ∘∘ id , transport (λ i → ⟦⟧-tr-id (A ⟦ γ ⟧) (~ i)) (x [ γ ]) ⟩ ⟧
+      ≡⟨⟩
+    B ⟦ ⟨ γ ∘ id , transport (λ i → ⟦⟧-tr-comp γ id A (~ i)) (transport (λ j → ⟦⟧-tr-id (A ⟦ γ ⟧) (~ j)) (x [ γ ])) ⟩ ⟧
+      ≡⟨ {!   !} ⟩
+    B ⟦ ⟨ γ ∘ id , transport ((λ j → ⟦⟧-tr-id (A ⟦ γ ⟧) (~ j)) ∙ (λ i → ⟦⟧-tr-comp γ id A (~ i))) (x [ γ ]) ⟩ ⟧
+      ≡⟨ {!   !} ⟩
+    B ⟦ ⟨ γ , x [ γ ] ⟩ ⟧
+      ≡⟨ {!   !} ⟩
+    B ⟦ ⟨ id ∘∘ γ , transport (λ i → ∣ Tr Δ (⟦⟧-id A (~ i) ⟦ γ ⟧) ∣) (x [ γ ]) ⟩ ⟧
       ≡⟨ {!   !} ⟩
     B ⟦ ⟨id, x ⟩ ⟧ ⟦ γ ⟧
       ∎
@@ -95,3 +107,4 @@ Lastly comes the star of the show, $\beta$ reduction.
          (a : ∣ Tr Γ A ∣)
        → app (Cλ A B b) a ≡ b [ ⟨id, a ⟩ ]
 ```
+ 
\ No newline at end of file