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intensionality hell

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Rachel Lambda Samuelsson 2022-10-28 16:12:38 +02:00
parent 9c5ae69cbd
commit b5b90ddd37
2 changed files with 19 additions and 6 deletions
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4
src/CwF/Base.lagda.md
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@ -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. Dubbing $p$ a projection is no coincidence, it projects a map out of a weakened map.
``` ```
field 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 → 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 ⟧) ∣ → ∣ Tr Δ (A ⟦ p ⟧ ⟦ ⟨ γ , a ⟩ ⟧) ∣ ≡ ∣ Tr Δ (A ⟦ γ ⟧) ∣
q-∘-pathp {Δ} {Γ} {γ} {A} a = 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 ⟩ ⟧) ∣ ≡˘⟨ ⟦⟧-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 ⟧) ∣ ∎ ∣ Tr Δ (A ⟦ γ ⟧) ∣ ∎
field field

21
src/CwF/Structures.lagda.md
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@ -57,8 +57,7 @@ Substitution must also be closed under all of these.
B ⟦ ⟨ id, x ⟩ ⟧ ⟦ γ ⟧ B ⟦ ⟨ id, x ⟩ ⟧ ⟦ γ ⟧
typeof app f x [ γ ] typeof app f x [ γ ]
it would seem CwFs really don't want to be formalized :/ Welcome to intensionality hell.
not enough definitional equalities...
``` ```
app-subst-pathp : {Δ Γ : Ob} (γ : Hom Δ Γ) {A : ∣ Ty Γ ∣} {B : ∣ Ty (Γ ; A) ∣} 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 ] ⟩ ⟧ 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 ⟧) ⟩ ≡⟨ (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)) 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 ] ⟩ ]) ⟩ ⟧ (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 ⟧ B ⟦ ⟨id, x ⟩ ⟧ ⟦ γ ⟧
∎ ∎
@ -95,3 +107,4 @@ Lastly comes the star of the show, $\beta$ reduction.
(a : ∣ Tr Γ A ∣) (a : ∣ Tr Γ A ∣)
→ app (Cλ A B b) a ≡ b [ ⟨id, a ⟩ ] → app (Cλ A B b) a ≡ b [ ⟨id, a ⟩ ]
``` ```