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| {-# OPTIONS --prop --rewriting #-} | ||
| module IFPA where | ||
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| open import Lib hiding (id; _∘_) | ||
| open import StrictLib | ||
| open import IF | ||
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| _ᵖᵃS : ∀{ℓ} → TyS → Set (suc ℓ) | ||
| _ᵖᵃS {ℓ} U = Prop ℓ | ||
| _ᵖᵃS {ℓ} (T ⇒̂S B) = T → _ᵖᵃS {ℓ} B | ||
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| _ᵖᵃc : ∀{ℓ} → SCon → Set (suc ℓ) | ||
| ∙c ᵖᵃc = Lift _ ⊤ | ||
| _ᵖᵃc {ℓ} (Γc ▶c B) = (_ᵖᵃc {ℓ} Γc) × _ᵖᵃS {ℓ} B | ||
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| _ᵖᵃt : ∀{ℓ Γc B} → Tm Γc B → _ᵖᵃc {ℓ} Γc → _ᵖᵃS {ℓ} B | ||
| (var vvz ᵖᵃt) (γ , α) = α | ||
| (var (vvs t) ᵖᵃt) (γ , α) = (var t ᵖᵃt) γ | ||
| ((t $S α) ᵖᵃt) γ = (t ᵖᵃt) γ α | ||
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| _ᵖᵃP : ∀{ℓ Γc} → TyP Γc → _ᵖᵃc {ℓ} Γc → Prop ℓ | ||
| (El a ᵖᵃP) γ = (a ᵖᵃt) γ | ||
| (Π̂P T A ᵖᵃP) γ = (τ : T) → ((A τ) ᵖᵃP) γ | ||
| ((a ⇒P A) ᵖᵃP) γ = (a ᵖᵃt) γ → (A ᵖᵃP) γ | ||
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| _ᵖᵃC : ∀{ℓ Γc} → Con Γc → _ᵖᵃc {ℓ} Γc → Prop ℓ | ||
| (∙ ᵖᵃC) γ = P⊤ | ||
| ((Γ ▶P A) ᵖᵃC) γ = (Γ ᵖᵃC) γ ∧ (A ᵖᵃP) γ | ||
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| _ᵖᵃs : ∀{ℓ}{Γc Δc} → Sub Γc Δc → _ᵖᵃc {ℓ} Γc → _ᵖᵃc {ℓ} Δc | ||
| (ε ᵖᵃs) γ = lift tt | ||
| ((σ , t) ᵖᵃs) γ = (σ ᵖᵃs) γ , (t ᵖᵃt) γ | ||
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| []tᵖᵃ : ∀{ℓ Γc Δc B}{t : Tm Δc B}{σ : Sub Γc Δc} | ||
| → (γc : _ᵖᵃc {ℓ} Γc) → ((t [ σ ]t) ᵖᵃt) γc ≡ (t ᵖᵃt) ((σ ᵖᵃs) γc) | ||
| []tᵖᵃ {t = var vvz} {σ , x} γc = refl | ||
| []tᵖᵃ {t = var (vvs a)}{σ , x} γc = []tᵖᵃ {t = var a} γc | ||
| []tᵖᵃ {t = t $S α} {σ} γc = happly ([]tᵖᵃ {t = t} γc) α | ||
| {-# REWRITE []tᵖᵃ #-} | ||
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| []Tᵖᵃ : ∀{ℓ Γc Δc A}{σ : Sub Γc Δc} → (γc : _ᵖᵃc {ℓ} Γc) → ((A [ σ ]T) ᵖᵃP) γc ≡ (A ᵖᵃP) ((σ ᵖᵃs) γc) | ||
| []Tᵖᵃ {A = El a} γc = []tᵖᵃ {t = a} γc | ||
| []Tᵖᵃ {A = Π̂P T A} γc = PΠ≡ refl λ τ → []Tᵖᵃ {A = A τ} γc | ||
| []Tᵖᵃ {A = a ⇒P A} γc = (λ p → _ → p) & []Tᵖᵃ {A = A} γc | ||
| {-# REWRITE []Tᵖᵃ #-} | ||
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| []Cᵖᵃ : ∀{ℓ Γc Δc}{σ : Sub Γc Δc}{Δ : Con Δc}{γc : _ᵖᵃc {ℓ} Γc} → ((Δ [ σ ]C) ᵖᵃC) γc ≡ (Δ ᵖᵃC) ((σ ᵖᵃs) γc) | ||
| []Cᵖᵃ {Δ = ∙} = refl | ||
| []Cᵖᵃ {Δ = Δ ▶P A} = (λ p → p ∧ _) & []Cᵖᵃ | ||
| {-# REWRITE []Cᵖᵃ #-} | ||
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| vs,ᵖᵃ : ∀{ℓ Γc B B'}{x : Tm Γc B}{γc}{α : _ᵖᵃS {ℓ} B'} → (vs x ᵖᵃt) (γc , α) ≡ (x ᵖᵃt) γc | ||
| vs,ᵖᵃ {x = var x} = refl | ||
| vs,ᵖᵃ {x = x $S α} = happly (vs,ᵖᵃ {x = x}) α | ||
| {-# REWRITE vs,ᵖᵃ #-} | ||
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| wk,ᵖᵃ : ∀{ℓ Γc Δc B γc}{α : B ᵖᵃS}{σ : Sub Γc Δc} → _ᵖᵃs {ℓ} (wk {B = B} σ) (γc , α) ≡ (σ ᵖᵃs) γc | ||
| wk,ᵖᵃ {σ = ε} = refl | ||
| wk,ᵖᵃ {σ = σ , t} = ,≡ wk,ᵖᵃ refl | ||
| {-# REWRITE wk,ᵖᵃ #-} | ||
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| idᵖᵃ : ∀{ℓ Γc} → (γc : _ᵖᵃc {ℓ} Γc) → (id ᵖᵃs) γc ≡ γc | ||
| idᵖᵃ {ℓ}{∙c} γc = refl | ||
| idᵖᵃ {ℓ}{Γc ▶c B} (γc , α) = ,≡ (idᵖᵃ γc) refl | ||
| {-# REWRITE idᵖᵃ #-} | ||
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| wkid : ∀{ℓ Γc B} γc → _ᵖᵃs {ℓ} (wk {B = B} (id {Γc = Γc})) γc ≡ ₁ γc | ||
| wkid (γc , α) = refl | ||
| {-# REWRITE wkid #-} | ||
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| ∘ᵖᵃ : ∀{ℓ Γc Δc Σc}{σ : Sub Δc Σc}{δ : Sub Γc Δc}{γc} → _ᵖᵃs {ℓ} (σ ∘ δ) γc ≡ (σ ᵖᵃs) ((δ ᵖᵃs) γc) | ||
| ∘ᵖᵃ {σ = ε} = refl | ||
| ∘ᵖᵃ {σ = σ , t} {δ = δ}{γc = γc} = happly2 _,_ (∘ᵖᵃ {σ = σ} {δ = δ}) ((t ᵖᵃt) ((δ ᵖᵃs) γc)) | ||
| {-# REWRITE ∘ᵖᵃ #-} | ||
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| π₁ᵃ : ∀{ℓ Γc Δc A}{σ : Sub Γc (Δc ▶c A)}{γc} → _ᵖᵃs {ℓ} (π₁ σ) γc ≡ ₁ ((σ ᵖᵃs) γc) | ||
| π₁ᵃ {σ = σ , x} = refl | ||
| {-# REWRITE π₁ᵃ #-} | ||
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| π₂ᵃ : ∀{ℓ Γc Δc A}{σ : Sub Γc (Δc ▶c A)}{γc} → _ᵖᵃt {ℓ} (π₂ σ) γc ≡ ₂ ((σ ᵖᵃs) γc) | ||
| π₂ᵃ {σ = σ , x} = refl | ||
| {-# REWRITE π₂ᵃ #-} | ||
| {- | ||
| Twkᵃ : ∀{ℓ Γc}{B A γc T} → _ᵖᵃP {ℓ} (Twk {Γc = Γc}{B = B} A) (γc , T) ≡ _ᵖᵃP A γc | ||
| Twkᵃ {A = El x} = refl | ||
| Twkᵃ {A = Π̂P T B} = Π≡ refl λ τ → Twkᵃ {A = B τ} | ||
| Twkᵃ {A = a ⇒P A} = Π≡ refl λ α → Twkᵃ {A = A} | ||
| {-# REWRITE Twkᵃ #-} | ||
| -} | ||
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| _ᵖᵃtP : ∀{ℓ Γc}{Γ : Con Γc}{A}(tP : TmP Γ A){γc}(γ : _ᵖᵃC {ℓ} Γ γc) → _ᵖᵃP {ℓ} A γc | ||
| (varP vvzP ᵖᵃtP) (γ , α) = α | ||
| (varP (vvsP x) ᵖᵃtP) (γ , α) = (varP x ᵖᵃtP) γ | ||
| ((tP $P sP) ᵖᵃtP) γ = (tP ᵖᵃtP) γ ((sP ᵖᵃtP) γ) | ||
| ((tP $̂P τ) ᵖᵃtP) γ = (tP ᵖᵃtP) γ τ | ||
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| _ᵖᵃsP : ∀{ℓ Γc}{Γ Δ : Con Γc}(σP : SubP Γ Δ){γc} | ||
| → _ᵖᵃC {ℓ} Γ γc → _ᵖᵃC {ℓ} Δ γc | ||
| (εP ᵖᵃsP) γ = ptt | ||
| ((σP ,P tP) ᵖᵃsP) γ = (σP ᵖᵃsP) γ , (tP ᵖᵃtP) γ | ||
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| --TODO define ∘Pᵃ, π₁Pᵃ, π₂Pᵃ |