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painful pathp

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Rachel Lambda Samuelsson 2022-10-09 15:57:21 +02:00
parent 6d3af73b9f
commit 9c5ae69cbd
2 changed files with 31 additions and 8 deletions
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4
src/CwF/Base.lagda.md
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@ -179,7 +179,7 @@ Further motivating the use of the word projection, pairing up the two projection
Lastly, it is required that the weakening map behaves as expected under composition. Lastly, it is required that the weakening map behaves as expected under composition.
``` ```
⟨⟩-∘ : {Δ Γ Θ : Ob} {γ : Hom Γ Δ} {δ : Hom Δ Θ} {A : ∣ Ty Θ ∣} ⟨⟩-∘ : {Δ Γ Θ : Ob} (γ : Hom Γ Δ) (δ : Hom Δ Θ) (A : ∣ Ty Θ ∣)
{a : ∣ Tr Δ (A ⟦ δ ⟧) ∣} (a : ∣ Tr Δ (A ⟦ δ ⟧) ∣)
→ ⟨ δ , a ⟩ ∘ γ ≡ ⟨ δ ∘∘ γ , a [ γ ] ⟩ → ⟨ δ , a ⟩ ∘ γ ≡ ⟨ δ ∘∘ γ , a [ γ ] ⟩
``` ```

29
src/CwF/Structures.lagda.md
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@ -49,7 +49,7 @@ Substitution must also be closed under all of these.
There's quite a hefty pathp here... There's quite a hefty pathp here...
typeof app (transport (λ i → ∣ Tr Δ (∏-subst γ A B i) ∣) (f [ γ ])) (x [ γ ]) typeof app (transport (λ i → ∣ Tr Δ (∏-subst γ A B i) ∣) (f [ γ ])) (x [ γ ])
B ⟦ ⟨ γ ∘ p , q ⟩ ⟧ ⟦ ⟨id, x [ γ ] ⟩ ⟧ B ⟦ ⟨ γ ∘ p , q ⟩ ⟧ ⟦ ⟨ id, x [ γ ] ⟩ ⟧
B ⟦ ⟨ γ ∘ p , q ⟩ ∘ ⟨id, x [ γ ] ⟩ ⟧ B ⟦ ⟨ γ ∘ p , q ⟩ ∘ ⟨id, x [ γ ] ⟩ ⟧
B ⟦ ⟨ γ ∘ p ∘ ⟨id, x [ γ ] ⟩, q [ ⟨id, x [ γ ] ⟩ ] ⟩ ⟧ B ⟦ ⟨ γ ∘ p ∘ ⟨id, x [ γ ] ⟩, q [ ⟨id, x [ γ ] ⟩ ] ⟩ ⟧
B ⟦ ⟨ γ ∘ id, x [ γ ] ⟩ ⟧ B ⟦ ⟨ γ ∘ id, x [ γ ] ⟩ ⟧
@ -57,11 +57,34 @@ 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 :/
not enough definitional equalities...
```
app-subst-pathp : {Δ Γ : Ob} (γ : Hom Δ Γ) {A : ∣ Ty Γ ∣} {B : ∣ Ty (Γ ; A) ∣}
(f : ∣ Tr Γ (∏ A B) ∣) (x : ∣ Tr Γ A ∣)
→ B ⟦ ⟨ γ ∘∘ p , q ⟩ ⟧ ⟦ ⟨id, x [ γ ] ⟩ ⟧
≡ B ⟦ ⟨id, x ⟩ ⟧ ⟦ γ ⟧
app-subst-pathp {Δ} {Γ} γ {A} {B} f x =
B ⟦ ⟨ γ ∘∘ p , q ⟩ ⟧ ⟦ ⟨id, x [ γ ] ⟩ ⟧
≡˘⟨ ⟦⟧-compose ⟨ γ ∘∘ p , q ⟩ ⟨id, x [ γ ] ⟩ B ⟩
B ⟦ ⟨ γ ∘∘ p , 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 ⟧) ⟩
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 ] ⟩ ]) ⟩ ⟧
≡⟨ {! !} ⟩
B ⟦ ⟨id, x ⟩ ⟧ ⟦ γ ⟧
∎
field
app-subst : {Δ Γ : Ob} (γ : Hom Δ Γ) {A : ∣ Ty Γ ∣} {B : ∣ Ty (Γ ; A) ∣} app-subst : {Δ Γ : Ob} (γ : Hom Δ Γ) {A : ∣ Ty Γ ∣} {B : ∣ Ty (Γ ; A) ∣}
(f : ∣ Tr Γ (∏ A B) ∣) (x : ∣ Tr Γ A ∣) (f : ∣ Tr Γ (∏ A B) ∣) (x : ∣ Tr Γ A ∣)
→ PathP {! !} → PathP (λ i → ∣ Tr Δ (app-subst-pathp γ f x (~ i)) ∣ )
(app f x [ γ ]) (app f x [ γ ])
(app (transport (λ i → ∣ Tr Δ (∏-subst γ A B i) ∣) (f [ γ ])) (x [ γ ])) (app (transport (λ i → ∣ Tr Δ (∏-subst γ A B i) ∣) (f [ γ ])) (x [ γ ]))
``` ```