{-# OPTIONS --without-K --safe #-}
open import Axiom.Extensionality.Propositional
module Mints.Soundness.Realizability (fext : ∀ {ℓ ℓ′} → Extensionality ℓ ℓ′) where
open import Lib
open import Data.List.Properties as Lₚ
open import Data.Nat.Properties as ℕₚ
open import Mints.Statics.Properties
open import Mints.Semantics.Readback
open import Mints.Semantics.Realizability fext
open import Mints.Semantics.Properties.Domain fext
open import Mints.Semantics.Properties.PER fext
open import Mints.Soundness.LogRel
open import Mints.Soundness.Properties.LogRel fext
open import Mints.Soundness.Properties.Mt fext
var-arith : ∀ Ψ″ (T : Typ) Ψ′ → len (Ψ″ ++ T ∷ Ψ′) ∸ len Ψ′ ∸ 1 ≡ len Ψ″
var-arith Ψ″ T Ψ′ = begin
len (Ψ″ ++ T ∷ Ψ′) ∸ len Ψ′ ∸ 1
≡⟨ cong (λ n → n ∸ len Ψ′ ∸ 1) (Lₚ.length-++ Ψ″) ⟩
len Ψ″ + suc (len Ψ′) ∸ len Ψ′ ∸ 1
≡⟨ cong (_∸ 1) (ℕₚ.+-∸-assoc (len Ψ″) {suc (len Ψ′)} (ℕₚ.≤-step ℕₚ.≤-refl)) ⟩
len Ψ″ + (suc (len Ψ′) ∸ len Ψ′) ∸ 1
≡⟨ cong (λ n → len Ψ″ + n ∸ 1) (ℕₚ.m+n∸n≡m 1 (len Ψ′)) ⟩
len Ψ″ + 1 ∸ 1
≡⟨ ℕₚ.m+n∸n≡m (len Ψ″) 1 ⟩
len Ψ″
∎
where open ≡-Reasoning
v0∼x-gen : ∀ Ψ → Δ ⊢r σ ∶ Γ → head Γ ≡ Ψ ++ T ∷ Ψ′ → Δ ⊢ v (len Ψ) [ σ ] ≈ v (len (head Δ) ∸ len Ψ′ ∸ 1) ∶ T [wk]* (1 + len Ψ) [ σ ]
v0∼x-gen {Δ} {σ} {Γ} {T} {Ψ′} Ψ (r-I σ≈) refl
with presup-s-≈ σ≈
... | ⊢Δ , _ , ⊢I , ⊢Γ
with ⊢≈-sym (⊢I-inv ⊢I)
... | Γ≈Δ = ≈-trans ([]-cong (v-≈ ⊢Γ n∈) σ≈)
(≈-trans ([I] (conv (ctxeq-tm Γ≈Δ (vlookup ⊢Γ n∈)) (≈-sym (≈-trans ([]-cong-Se″ ⊢T[wk]* σ≈) ([I] (ctxeq-tm Γ≈Δ ⊢T[wk]*))))))
helper)
where n∈ = n∶T[wk]n∈!ΨTΓ ⊢Γ refl
⊢T[wk]* = proj₂ (proj₂ (presup-tm (⊢vn∶T[wk]suc[n] ⊢Γ refl)))
[wkσ]≈ = []-cong-Se″ ⊢T[wk]* (s-≈-sym σ≈)
helper : Δ ⊢ v (len Ψ) ≈ v (len (head Δ) ∸ len Ψ′ ∸ 1) ∶ T [wk]* (1 + len Ψ) [ σ ]
helper
rewrite sym (⊢≈⇒len-head≡ Γ≈Δ)
| var-arith Ψ T Ψ′ = ≈-conv (ctxeq-≈ Γ≈Δ (v-≈ ⊢Γ n∈)) (≈-trans (≈-sym ([I] (ctxeq-tm Γ≈Δ ⊢T[wk]*))) [wkσ]≈)
v0∼x-gen {Δ} {σ} {_} {_} {Ψ′} Ψ (r-p {_} {τ} {T′} ⊢τ σ≈) refl
with presup-s (⊢r⇒⊢s ⊢τ)
... | _ , ⊢∺ ⊢Γ ⊢T′ = begin
v (len Ψ) [ σ ] ≈⟨ []-cong (v-≈ ⊢Γ n∈) σ≈ ⟩
v (len Ψ) [ p τ ] ≈⟨ ≈-conv ([∘] ⊢τ′ (s-wk ⊢TΓ) (vlookup ⊢Γ n∈)) [wkτ]≈ ⟩
v (len Ψ) [ wk ] [ τ ] ≈⟨ ≈-conv ([]-cong ([wk] ⊢TΓ n∈) (s-≈-refl ⊢τ′)) wkτ≈ ⟩
v (1 + len Ψ) [ τ ] ≈⟨ ≈-conv (v0∼x-gen (T′ ∷ Ψ) ⊢τ refl) wkτ≈ ⟩
v (len (head Δ) ∸ len Ψ′ ∸ 1) ∎
where open ER
n∈ = n∶T[wk]n∈!ΨTΓ ⊢Γ refl
⊢TΓ = ⊢∺ ⊢Γ ⊢T′
⊢τ′ = ⊢r⇒⊢s ⊢τ
⊢T[wk]* = proj₂ (proj₂ (presup-tm (⊢vn∶T[wk]suc[n] ⊢Γ refl)))
[wkτ]≈ = []-cong-Se″ ⊢T[wk]* (s-≈-sym σ≈)
wkτ≈ = ≈-trans ([∘]-Se ⊢T[wk]* (s-wk ⊢TΓ) ⊢τ′) [wkτ]≈
v0∼x-gen [] (r-； _ _ _ _) ()
v0∼x-gen (_ ∷ _) (r-； _ _ _ _) ()
v0∼x : ∀ {i} (A≈B : A ≈ B ∈ 𝕌 i) →
Γ ⊢ T ®[ i ] A≈B →
T ∺ Γ ⊢ v 0 ∶ T [ wk ] ®↓[ i ] l (len (head Γ)) ∈El A≈B
v0∼x {_} {_} {Γ} A≈B T∼A
with ®⇒ty A≈B T∼A
... | ⊢T
with presup-tm ⊢T
... | ⊢Γ , _ = record
{ t∶T = vlookup ⊢TΓ here
; T∼A = ®-≡ (𝕌-mon vone A≈B) A≈B (®-mon′ A≈B T∼A (r-p (⊢rI ⊢TΓ) (s-≈-sym (∘-I (s-wk ⊢TΓ))))) (D-ap-vone _)
; c∈⊥ = Bot-l (len (head Γ))
; krip = λ {Δ} {σ} ⊢σ → v0∼x-gen [] ⊢σ refl
}
where ⊢TΓ = ⊢∺ ⊢Γ ⊢T
private
module Real i (rec : ∀ j → j < i → ∀ {A B Γ T Δ σ} (A≈B : A ≈ B ∈ 𝕌 j) → Γ ⊢ T ®[ j ] A≈B → Δ ⊢r σ ∶ Γ → ∃ λ W → Rty map len Δ - A [ mt σ ] ↘ W × Δ ⊢ T [ σ ] ≈ Nf⇒Exp W ∶ Se j) where
mutual
®↓El⇒®El : (A≈B : A ≈ B ∈ 𝕌 i) → Γ ⊢ t ∶ T ®↓[ i ] c ∈El A≈B → Γ ⊢ t ∶ T ®[ i ] ↑ A c ∈El A≈B
®↓El⇒®El R@(ne C≈C′) t∼c = ne c∈⊥ , record
{ t∶T = t∶T
; ⊢T = ®⇒ty R T∼A
; krip = λ ⊢σ → proj₂ T∼A ⊢σ , krip ⊢σ
}
where open _⊢_∶_®↓[_]_∈El_ t∼c
®↓El⇒®El N t∼c = ne c∈⊥ (λ ⊢σ → ≈-conv (krip ⊢σ) (≈-trans ([]-cong-Se′ T∼A (⊢r⇒⊢s ⊢σ)) (N-[] _ (⊢r⇒⊢s ⊢σ)))) , T∼A
where open _⊢_∶_®↓[_]_∈El_ t∼c
®↓El⇒®El (U j<i eq) t∼c = record
{ t∶T = t∶T
; T≈ = T∼A
; A∈𝕌 = ne c∈⊥
; rel = subst (λ f → f _ _ _)
(sym (Glu-wellfounded-≡ j<i))
(conv t∶T T∼A , λ ⊢σ → ≈-conv (krip ⊢σ) (≈-trans ([]-cong-Se′ T∼A (⊢r⇒⊢s ⊢σ)) (lift-⊢≈-Se (Se-[] _ (⊢r⇒⊢s ⊢σ)) j<i)))
}
where open _⊢_∶_®↓[_]_∈El_ t∼c
®↓El⇒®El {□ A} {_} {Γ} {t} {_} {c} (□ A≈B) t∼c = record
{ GT = GT
; t∶T = t∶T
; a∈El = El-refl (□ A≈B) (realizability-Re (□ A≈B) c∈⊥)
; T≈ = T≈
; krip = λ {Δ} {σ} Ψs ⊢ΨsΔ ⊢σ →
let ⊢σ′ = ⊢r⇒⊢s ⊢σ
Gk = G.krip Ψs ⊢ΨsΔ ⊢σ
⊢ΨsΔ = proj₁ (presup-tm (®⇒ty _ Gk))
Aσ；≈ = A≈B (ins (mt σ) (len Ψs))
⊢t = conv t∶T T≈
⊢tσ = conv (t[σ] ⊢t ⊢σ′) (□-[] ⊢σ′ ⊢GT)
in record
{ ua = unbox′ (A [ ins (mt σ) (len Ψs) ]) (len Ψs) (c [ mt σ ])
; ↘ua = subst (λ B → unbox∙ len Ψs , ↑ (□ (A [ ins (mt σ) 1 ])) (c [ mt σ ]) ↘ unbox′ B (len Ψs) (c [ mt σ ])) (D-ins-ins A (mt σ) (len Ψs)) (unbox∙ (len Ψs))
; rel = ®El-resp-T≈ Aσ；≈
(®↓El⇒®El Aσ；≈
record
{ t∶T = □-E Ψs ⊢tσ ⊢ΨsΔ refl
; T∼A = ®-resp-≈ Aσ；≈ Gk ([]-∘-；′ Ψs ⊢ΨsΔ ⊢GT ⊢σ′)
; c∈⊥ = unbox-Bot (len Ψs) (Bot-mon (mt σ) c∈⊥)
; krip = helper Ψs ⊢t ⊢σ
})
(≈-sym ([]-∘-；′ Ψs ⊢ΨsΔ ⊢GT ⊢σ′))
}
}
where module ↓ = _⊢_∶_®↓[_]_∈El_ t∼c
open ↓
module G = Glu□ T∼A
open G
open ER
⊢GT = ®□⇒wf A≈B T∼A
helper : ∀ Ψs →
Γ ⊢ t ∶ □ GT →
Δ ⊢r σ ∶ Γ →
Δ′ ⊢r τ ∶ Ψs ++⁺ Δ →
Δ′ ⊢ (unbox (len Ψs) (t [ σ ])) [ τ ] ≈ unbox (O (mt τ) (len Ψs)) (Ne⇒Exp (proj₁ (c∈⊥ (map len Δ′ ∥ (O (mt τ) (len Ψs))) (mt σ ø mt τ ∥ len Ψs)))) ∶ GT [ σ ； 1 ] [ I ； len Ψs ] [ τ ]
helper {_} {σ} {_} {τ} Ψs ⊢t ⊢σ ⊢τ
with ⊢r-∥′ Ψs ⊢τ | presup-s (⊢r⇒⊢s ⊢σ) | presup-s (⊢r⇒⊢s ⊢τ)
... | Ψs′ , Δ″ , refl , eql , ⊢τ∥ | ⊢Δ , _ | ⊢Δ′ , ⊢ΨsΔ
with ↓.krip (⊢r-∘ ⊢σ ⊢τ∥)
... | equiv
rewrite sym (O-resp-mt τ (len Ψs))
| sym eql
| map-++⁺-commute len Ψs′ Δ″
| drop+-++⁺ (len Ψs′) (L.map len Ψs′) (map len Δ″) (Lₚ.length-map len Ψs′)
| mt-∥ τ (len Ψs) = ≈-conv
(begin
unbox (len Ψs) (t [ σ ]) [ τ ] ≈⟨ ≈-conv (unbox-[] Ψs ⊢tσ ⊢τ′ refl)
(≈-trans (≈-sym (subst (λ n → _ ⊢ GT [ _ ； _ ] ≈ GT [ σ ； 1 ] [ _ ； n ] ∶ Se _) eql
([]-∘-； Ψs′ ⊢Δ′ ⊢GT ⊢σ′ ⊢τ∥′)))
([]-∘-；′ Ψs′ ⊢Δ′ ⊢GT ⊢στ∥)) ⟩
unbox (O τ (len Ψs)) (t [ σ ] [ τ ∥ len Ψs ]) ≈⟨ subst (λ n → _ ⊢ unbox n _ ≈ unbox _ _ ∶ GT [ _ ] [ _ ]) eql
(unbox-cong Ψs′ (≈-conv (≈-sym ([∘] ⊢τ∥′ ⊢σ′ ⊢t)) (□-[] ⊢στ∥ ⊢GT)) ⊢Δ′ refl) ⟩
unbox (len Ψs′) (t [ σ ∘ τ ∥ len Ψs ]) ≈⟨ unbox-cong Ψs′ (≈-conv equiv (≈-trans ([]-cong-Se′ T≈ ⊢στ∥) (□-[] ⊢στ∥ ⊢GT))) ⊢Δ′ refl ⟩
unbox (len Ψs′)
(Ne⇒Exp (proj₁ (c∈⊥ (map len Δ″) (mt σ ø (mt τ ∥ len Ψs))))) ∎)
(begin
GT [ (σ ∘ τ ∥ len Ψs) ； 1 ] [ I ； len Ψs′ ] ≈˘⟨ []-∘-；′ Ψs′ ⊢Δ′ ⊢GT (s-∘ ⊢τ∥′ ⊢σ′) ⟩
GT [ (σ ∘ τ ∥ len Ψs) ； len Ψs′ ] ≈⟨ subst (λ n → _ ⊢ GT [ _ ； n ] ≈ _ ∶ Se _) (sym eql) ([]-；-∘ Ψs ⊢GT ⊢σ′ ⊢τ′) ⟩
GT [ σ ； len Ψs ] [ τ ] ≈⟨ []-cong-Se′ ([]-∘-；′ Ψs ⊢ΨsΔ ⊢GT ⊢σ′) ⊢τ′ ⟩
GT [ σ ； 1 ] [ I ； len Ψs ] [ τ ] ∎)
where ⊢σ′ = ⊢r⇒⊢s ⊢σ
⊢τ′ = ⊢r⇒⊢s ⊢τ
⊢τ∥′ = ⊢r⇒⊢s ⊢τ∥
⊢στ∥ = s-∘ ⊢τ∥′ ⊢σ′
⊢tσ = conv (t[σ] ⊢t ⊢σ′) (□-[] ⊢σ′ ⊢GT)
®↓El⇒®El {Π A S ρ} {_} {Γ} {t} {T} {c} (Π iA RT) t∼c = record
{ t∶T = t∶T
; a∈El = El-refl (Π iA RT) (realizability-Re (Π iA RT) c∈⊥)
; IT = IT
; OT = OT
; ⊢OT = ⊢OT
; T≈ = T≈
; krip = λ {Δ} {σ} ⊢σ → record
{ IT-rel = ΠRel.IT-rel (G.krip ⊢σ)
; ap-rel = λ s∼b b∈ →
let a , ↘a , ∼a = ap-rel ⊢σ s∼b b∈
in record
{ fa = a
; ↘fa = ↘a
; ®fa = ∼a
}
}
}
where module ↓ = _⊢_∶_®↓[_]_∈El_ t∼c
open ↓
module G = GluΠ T∼A
open G
ap-rel : Δ ⊢r σ ∶ Γ →
Δ ⊢ s ∶ IT [ σ ] ®[ i ] b ∈El (iA (mt σ)) →
(b∈ : b ∈′ El i (iA (mt σ))) →
∃ λ a → ↑ (Π A S ρ [ mt σ ]) (c [ mt σ ]) ∙ b ↘ a × Δ ⊢ t [ σ ] $ s ∶ OT [ σ , s ] ®[ i ] a ∈El (ΠRT.T≈T′ (RT (mt σ) b∈))
ap-rel {_} {σ} {s} {b} ⊢σ s∼b b∈ = [ ΠRT.⟦T⟧ (RT (mt σ) b∈) ] c [ mt σ ] $′ ↓ (A [ mt σ ]) b
, $∙ (A [ mt σ ]) (c [ mt σ ]) (ΠRT.↘⟦T⟧ (RT (mt σ) b∈))
, ®↓El⇒®El (ΠRT.T≈T′ (RT (mt σ) b∈)) record
{ t∶T = conv (Λ-E ⊢tσ ⊢s) (≈-sym ([]-q-∘-,′ ⊢OT ⊢σ′ ⊢s))
; T∼A = ΠRel.OT-rel (G.krip ⊢σ) s∼b b∈
; c∈⊥ = $-Bot (Bot-mon (mt σ) c∈⊥) (Top-trans ↑.a∈⊤ (Top-sym ↑.a∈⊤))
; krip = λ {_} {τ} ⊢τ →
let ⊢τ′ = ⊢r⇒⊢s ⊢τ
⊢στ = s-∘ ⊢τ′ ⊢σ′
eq = begin
OT [ (σ ∘ τ) , s [ τ ] ] ≈˘⟨ []-cong-Se″ ⊢OT (,-∘ ⊢σ′ ⊢IT ⊢s ⊢τ′) ⟩
OT [ σ , s ∘ τ ] ≈˘⟨ [∘]-Se ⊢OT (s-, ⊢σ′ ⊢IT ⊢s) ⊢τ′ ⟩
OT [ σ , s ] [ τ ] ∎
in begin
(t [ σ ] $ s) [ τ ] ≈⟨ ≈-conv ($-[] ⊢τ′ ⊢tσ ⊢s) (≈-trans (≈-sym ([]-q-∘-, ⊢OT ⊢σ′ ⊢τ′ (t[σ] ⊢s ⊢τ′)))
eq) ⟩
t [ σ ] [ τ ] $ s [ τ ] ≈⟨ ≈-conv ($-cong (≈-conv (≈-trans (≈-sym ([∘] ⊢τ′ ⊢σ′ t∶T)) (↓.krip (⊢r-∘ ⊢σ ⊢τ)))
(≈-trans (helper ⊢στ)
(Π-cong (≈-sym ([∘]-Se ⊢IT ⊢σ′ ⊢τ′))
(≈-refl (t[σ]-Se ⊢OT (⊢q ⊢στ ⊢IT))))))
(↑.krip ⊢τ))
(≈-trans (≈-sym ([]-q-∘-,′ ⊢OT ⊢στ (conv (t[σ] ⊢s ⊢τ′) ([∘]-Se ⊢IT ⊢σ′ ⊢τ′))))
eq) ⟩
_ $ _ ∎
}
where ⊢σ′ = ⊢r⇒⊢s ⊢σ
⊢IT = ®Π-wf iA RT T∼A
⊢s = ®El⇒tm (iA (mt σ)) s∼b
helper : ∀ {Δ σ} → Δ ⊢s σ ∶ Γ → Δ ⊢ T [ σ ] ≈ Π (IT [ σ ]) (OT [ q σ ]) ∶ Se i
helper ⊢σ = ≈-trans ([]-cong-Se′ T≈ ⊢σ) (Π-[] ⊢σ ⊢IT ⊢OT)
⊢tσ = conv (t[σ] t∶T ⊢σ′) (helper ⊢σ′)
open ER
module ↑ = _⊢_∶_®↑[_]_∈El_ (®El⇒®↑El (iA (mt σ)) s∼b)
®El⇒®↑El : (A≈B : A ≈ B ∈ 𝕌 i) → Γ ⊢ t ∶ T ®[ i ] a ∈El A≈B → Γ ⊢ t ∶ T ®↑[ i ] a ∈El A≈B
®El⇒®↑El (ne C≈C′) (ne c∈⊥ , glu) = record
{ t∶T = t∶T
; T∼A = ⊢T , λ ⊢σ → proj₁ (krip ⊢σ)
; a∈⊤ = Bot⊆Top c∈⊥
; krip = λ ⊢σ → proj₂ (krip ⊢σ)
}
where open GluNe glu
®El⇒®↑El N (t∼a , T≈N)
with presup-≈ T≈N
... | ⊢Γ , _ = record
{ t∶T = conv (®Nat⇒∶Nat t∼a ⊢Γ) (≈-sym T≈N)
; T∼A = T≈N
; a∈⊤ = ®Nat⇒∈Top t∼a
; krip = λ ⊢σ → ≈-conv (®Nat⇒≈ t∼a ⊢σ) (≈-sym (≈-trans ([]-cong-Se′ T≈N (⊢r⇒⊢s ⊢σ)) (N-[] _ (⊢r⇒⊢s ⊢σ))))
}
®El⇒®↑El (U′ j<i) t∼a = record
{ t∶T = t∶T
; T∼A = T≈
; a∈⊤ = λ ns κ → let W , ↘W , ↘W′ = realizability-Rty A∈𝕌 ns κ
in W , RU _ ↘W , RU _ ↘W′
; krip = λ {Δ} {σ} ⊢σ →
let W , ↘W , eq = rec _ j<i A∈𝕌 (subst (λ f → f _ _ _) (Glu-wellfounded-≡ j<i) rel) ⊢σ
in ≈-conv (subst (_ ⊢ _ ≈_∶ Se _) (cong Nf⇒Exp (Rty-det ↘W (proj₁ (proj₂ (realizability-Rty A∈𝕌 (map len Δ) (mt σ)))))) eq)
(≈-sym (≈-trans ([]-cong-Se′ T≈ (⊢r⇒⊢s ⊢σ)) (lift-⊢≈-Se (Se-[] _ (⊢r⇒⊢s ⊢σ)) j<i)))
}
where open GluU t∼a
®El⇒®↑El {□ A} {_} {Γ} {t} {T} (□ A≈B) t∼a = record
{ t∶T = t∶T
; T∼A = ®El⇒® (□ A≈B) t∼a
; a∈⊤ = realizability-Rf (□ A≈B) a∈El
; krip = helper
}
where open Glubox t∼a
helper : Δ ⊢r σ ∶ Γ → Δ ⊢ t [ σ ] ≈ Nf⇒Exp (proj₁ (realizability-Rf (□ A≈B) a∈El (map len Δ) (mt σ))) ∶ T [ σ ]
helper {Δ} {σ} ⊢σ = help (®El⇒®↑El (A≈B (ins (mt σ) 1)) rel)
where ⊢σ′ = ⊢r⇒⊢s ⊢σ
⊢Δ = proj₁ (presup-s ⊢σ′)
open □Krip (krip L.[ [] ] (⊢κ ⊢Δ) ⊢σ)
open ER
⊢GT = ®□⇒wf A≈B (®El⇒® (□ A≈B) t∼a)
⊢tσ = conv (t[σ] t∶T ⊢σ′) (≈-trans ([]-cong-Se′ T≈ ⊢σ′) (□-[] ⊢σ′ ⊢GT))
help : [] ∷⁺ Δ ⊢ unbox 1 (t [ σ ]) ∶ GT [ σ ； 1 ] ®↑[ i ] ua ∈El A≈B (ins (mt σ) 1) →
Δ ⊢ t [ σ ] ≈ Nf⇒Exp (proj₁ (realizability-Rf (□ A≈B) a∈El (map len Δ) (mt σ))) ∶ T [ σ ]
help record { a∈⊤ = a∈⊤ ; krip = krip }
with realizability-Rf (□ A≈B) a∈El (map len Δ) (mt σ)
| a∈⊤ (map len ([] ∷⁺ Δ)) vone
| krip (⊢rI (⊢κ ⊢Δ))
... | box w , R□ .(map len Δ) ↘ub ↘w , _
| w′ , ↘w′ , _
| equiv
rewrite unbox-det ↘ub ↘ua
| D-ap-vone (A [ ins (mt σ) 1 ])
| D-ap-vone ua
| Rf-det ↘w′ ↘w = ≈-conv (begin
t [ σ ] ≈⟨ □-η ⊢tσ ⟩
box (unbox 1 (t [ σ ])) ≈˘⟨ box-cong ([I] (conv (□-E L.[ [] ] ⊢tσ (⊢κ ⊢Δ) refl) ([I；1] ⊢GT[σ；1]))) ⟩
box (unbox 1 (t [ σ ]) [ I ]) ≈⟨ box-cong (≈-conv equiv ([I] ⊢GT[σ；1])) ⟩
box (Nf⇒Exp w) ∎)
(≈-sym (≈-trans ([]-cong-Se′ T≈ ⊢σ′) (□-[] ⊢σ′ ⊢GT)))
where ⊢GT[σ；1] = t[σ]-Se ⊢GT (s-； L.[ [] ] ⊢σ′ (⊢κ ⊢Δ) refl)
®El⇒®↑El {Π A S ρ} {_} {Γ} {t} {T} (Π iA RT) t∼a = record
{ t∶T = t∶T
; T∼A = ®El⇒® (Π iA RT) t∼a
; a∈⊤ = realizability-Rf (Π iA RT) a∈El
; krip = helper
}
where open GluΛ t∼a
⊢IT = ®Π-wf iA RT (®El⇒® (Π iA RT) t∼a)
helper : Δ ⊢r σ ∶ Γ → Δ ⊢ t [ σ ] ≈ Nf⇒Exp (proj₁ (realizability-Rf (Π iA RT) a∈El (map len Δ) (mt σ))) ∶ T [ σ ]
helper {Δ} {σ} ⊢σ
with presup-s (⊢r⇒⊢s ⊢σ)
... | ⊢Δ , _ = help
where ⊢σ′ = ⊢r⇒⊢s ⊢σ
Tσ≈ = ≈-trans ([]-cong-Se′ T≈ ⊢σ′) (Π-[] ⊢σ′ ⊢IT ⊢OT)
⊢tσ = conv (t[σ] t∶T ⊢σ′) Tσ≈
⊢ITσ = t[σ]-Se ⊢IT ⊢σ′
⊢ITσΔ = ⊢∺ ⊢Δ (t[σ]-Se ⊢IT ⊢σ′)
⊢qσ = ⊢q ⊢σ′ ⊢IT
⊢OTqσ = t[σ]-Se ⊢OT ⊢qσ
⊢σwk = s-∘ (s-wk ⊢ITσΔ) ⊢σ′
open ΛRel (krip ⊢σ) using (IT-rel)
open ΛRel (krip (⊢r-∘ ⊢σ (⊢rwk ⊢ITσΔ))) using (ap-rel)
open ER
help : Δ ⊢ t [ σ ] ≈ Nf⇒Exp (proj₁ (realizability-Rf (Π iA RT) a∈El (map len Δ) (mt σ))) ∶ T [ σ ]
help
with ap-rel
| realizability-Rf (Π iA RT) a∈El (map len Δ) (mt σ)
| ®↓El⇒®El (iA (mt σ)) (v0∼x (iA (mt σ)) IT-rel)
... | ap-rel | Λ w , RΛ .(map len Δ) ↘a ↘⟦S⟧ ↘w , _ | v∼l
rewrite ø-vone (mt σ)
with RT (mt σ) (®El⇒∈El (iA (mt σ)) v∼l)
| ap-rel (®El-resp-T≈ (iA (mt σ)) v∼l ([∘]-Se ⊢IT ⊢σ′ (s-wk ⊢ITσΔ))) (®El⇒∈El (iA (mt σ)) v∼l)
... | record { ⟦T⟧ = ⟦T⟧ ; ↘⟦T⟧ = ↘⟦T⟧ ; T≈T′ = T≈T′ }
| record { fa = fa ; ↘fa = ↘fa ; ®fa = ®fa }
with ®El⇒®↑El T≈T′ ®fa
... | record { a∈⊤ = a∈⊤ ; krip = krip }
with a∈⊤ (map len ((IT [ σ ]) ∺ Δ)) vone
| krip (⊢rI ⊢ITσΔ)
... | w′ , ↘w′ , _
| equiv
rewrite D-ap-vone ⟦T⟧
| D-ap-vone fa
| ap-det ↘a ↘fa
| ⟦⟧-det ↘⟦S⟧ ↘⟦T⟧
| Rf-det ↘w′ ↘w = ≈-conv (begin
t [ σ ] ≈⟨ Λ-η ⊢tσ ⟩
Λ (t [ σ ] [ wk ] $ v 0) ≈˘⟨ Λ-cong (≈-conv ($-cong (≈-conv ([∘] (s-wk ⊢ITσΔ) ⊢σ′ t∶T) eq)
(v-≈ ⊢ITσΔ here))
eq′) ⟩
Λ (t [ σ ∘ wk ] $ v 0) ≈˘⟨ Λ-cong ([I] (conv (Λ-E (conv (t[σ] t∶T ⊢σwk) eq) (vlookup ⊢ITσΔ here)) eq′)) ⟩
Λ ((t [ σ ∘ wk ] $ v 0) [ I ]) ≈⟨ ≈-conv (Λ-cong equiv) (Π-cong (≈-refl ⊢ITσ) ([I] ⊢OTqσ)) ⟩
Λ (Nf⇒Exp w) ∎)
(≈-sym Tσ≈)
where ITσwk≈ = [∘]-Se ⊢IT ⊢σ′ (s-wk ⊢ITσΔ)
eq = begin
T [ σ ∘ wk ] ≈⟨ []-cong-Se′ T≈ ⊢σwk ⟩
Π IT OT [ σ ∘ wk ] ≈⟨ Π-[] ⊢σwk ⊢IT ⊢OT ⟩
Π (IT [ σ ∘ wk ]) (OT [ q (σ ∘ wk) ]) ≈⟨ Π-cong (≈-sym ITσwk≈) (≈-refl (t[σ]-Se ⊢OT (⊢q ⊢σwk ⊢IT))) ⟩
Π (IT [ σ ] [ wk ]) (OT [ q (σ ∘ wk) ]) ∎
eq′ = ≈-sym ([]-q-∘-,′ ⊢OT ⊢σwk (conv (vlookup ⊢ITσΔ here) ITσwk≈))
®⇒Rty-eq : (A≈B : A ≈ B ∈ 𝕌 i) → Γ ⊢ T ®[ i ] A≈B → Δ ⊢r σ ∶ Γ → ∃ λ W → Rty map len Δ - A [ mt σ ] ↘ W × Δ ⊢ T [ σ ] ≈ Nf⇒Exp W ∶ Se i
®⇒Rty-eq {↑ _ C} {Δ = Δ} {σ} (ne C≈C′) (⊢T , rel) ⊢σ
with C≈C′ (map len Δ) (mt σ) | rel ⊢σ
... | V , ↘V , _ | r = ne V , Rne (map len Δ) ↘V , r
®⇒Rty-eq N T∼A ⊢σ = N , RN _ , ≈-trans ([]-cong-Se′ T∼A (⊢r⇒⊢s ⊢σ)) (N-[] _ (⊢r⇒⊢s ⊢σ))
®⇒Rty-eq {Δ = Δ} (U j<i eq) T∼A ⊢σ = Se _ , RU (map len Δ) , (≈-trans ([]-cong-Se′ T∼A (⊢r⇒⊢s ⊢σ)) (lift-⊢≈-Se (Se-[] _ (⊢r⇒⊢s ⊢σ)) j<i))
®⇒Rty-eq {□ A} {_} {_} {T} {Δ} {σ} (□ A≈B) T∼A ⊢σ
with presup-s (⊢r⇒⊢s ⊢σ)
... | ⊢Δ , _
with ®⇒Rty-eq (A≈B (ins (mt σ) 1)) (Glu□.krip T∼A L.[ [] ] (⊢κ ⊢Δ) ⊢σ) (⊢rI (⊢κ ⊢Δ))
... | W , ↘W , ≈W
rewrite D-ap-vone (A [ ins (mt σ) 1 ]) = □ W , R□ (map len Δ) ↘W
, (begin
T [ σ ] ≈⟨ []-cong-Se′ T≈ ⊢σ′ ⟩
□ GT [ σ ] ≈⟨ □-[] ⊢σ′ ⊢GT ⟩
□ (GT [ σ ； 1 ]) ≈˘⟨ □-cong ([I] (t[σ]-Se ⊢GT (s-； L.[ [] ] ⊢σ′ (⊢κ ⊢Δ) refl))) ⟩
□ (GT [ σ ； 1 ] [ I ]) ≈⟨ □-cong ≈W ⟩
Nf⇒Exp (□ W) ∎)
where open Glu□ T∼A
open ER
⊢σ′ = ⊢r⇒⊢s ⊢σ
⊢GT = ®□⇒wf A≈B T∼A
®⇒Rty-eq {Π A S ρ} {_} {_} {T} {Δ} {σ} (Π iA RT) T∼A ⊢σ
with presup-s (⊢r⇒⊢s ⊢σ)
... | ⊢Δ , _ = helper
where open GluΠ T∼A
⊢σ′ = ⊢r⇒⊢s ⊢σ
⊢IT = ®Π-wf iA RT T∼A
⊢ITσ = t[σ]-Se ⊢IT ⊢σ′
⊢ITσΔ = ⊢∺ ⊢Δ (t[σ]-Se ⊢IT ⊢σ′)
⊢qσ = ⊢q ⊢σ′ ⊢IT
⊢OTqσ = t[σ]-Se ⊢OT ⊢qσ
open ΠRel (krip ⊢σ) using (IT-rel)
open ΠRel (krip (⊢r-∘ ⊢σ (⊢rwk ⊢ITσΔ))) using (OT-rel)
open ER
helper : ∃ λ W → Rty map len Δ - Π A S ρ [ mt σ ] ↘ W × Δ ⊢ T [ σ ] ≈ Nf⇒Exp W ∶ Se i
helper
with ®⇒Rty-eq (iA (mt σ)) IT-rel (⊢rI ⊢Δ)
| ®↓El⇒®El (iA (mt σ)) (v0∼x (iA (mt σ)) IT-rel)
| OT-rel
... | WI , ↘WI , ≈WI
| v∼l
| OT-rel
rewrite D-ap-vone (A [ mt σ ])
| ø-vone (mt σ)
with RT (mt σ) (®El⇒∈El (iA (mt σ)) v∼l)
| OT-rel (®El-resp-T≈ (iA (mt σ)) v∼l ([∘]-Se ⊢IT ⊢σ′ (s-wk ⊢ITσΔ))) (®El⇒∈El (iA (mt σ)) v∼l)
... | record { ⟦T⟧ = ⟦S⟧ ; ↘⟦T⟧ = ↘⟦S⟧ ; T≈T′ = T≈T′ }
| rel
with ®⇒Rty-eq T≈T′ rel (⊢rI ⊢ITσΔ)
... | WO , ↘WO , ≈WO
rewrite D-ap-vone ⟦S⟧ = Π WI WO , RΠ (map len Δ) ↘WI ↘⟦S⟧ ↘WO
, (begin
T [ σ ] ≈⟨ []-cong-Se′ T≈ ⊢σ′ ⟩
Π IT OT [ σ ] ≈⟨ Π-[] ⊢σ′ ⊢IT ⊢OT ⟩
Π (IT [ σ ]) (OT [ q σ ]) ≈˘⟨ Π-cong ([I] ⊢ITσ) ([I] (ctxeq-tm (∺-cong (⊢≈-refl ⊢Δ) (≈-sym ([I] ⊢ITσ))) ⊢OTqσ)) ⟩
Π (IT [ σ ] [ I ]) (OT [ q σ ] [ I ]) ≈⟨ Π-cong ≈WI (ctxeq-≈ (∺-cong (⊢≈-refl ⊢Δ) (≈-sym ([I] ⊢ITσ))) ≈WO) ⟩
Nf⇒Exp (Π WI WO) ∎)
®⇒Rty-eq : ∀ {i} (A≈B : A ≈ B ∈ 𝕌 i) →
Γ ⊢ T ®[ i ] A≈B →
Δ ⊢r σ ∶ Γ →
∃ λ W → Rty map len Δ - A [ mt σ ] ↘ W × Δ ⊢ T [ σ ] ≈ Nf⇒Exp W ∶ Se i
®⇒Rty-eq {i = i} = <-Measure.wfRec (λ i → ∀ {A B Γ T Δ σ} →
(A≈B : A ≈ B ∈ 𝕌 i) →
Γ ⊢ T ®[ i ] A≈B →
Δ ⊢r σ ∶ Γ →
∃ λ W → Rty map len Δ - A [ mt σ ] ↘ W × Δ ⊢ T [ σ ] ≈ Nf⇒Exp W ∶ Se i)
Real.®⇒Rty-eq i
®↓El⇒®El : ∀ {i} (A≈B : A ≈ B ∈ 𝕌 i) →
Γ ⊢ t ∶ T ®↓[ i ] c ∈El A≈B →
Γ ⊢ t ∶ T ®[ i ] ↑ A c ∈El A≈B
®↓El⇒®El {i = i} = Real.®↓El⇒®El i (λ j _ → ®⇒Rty-eq {i = j})
®El⇒®↑El : ∀ {i} (A≈B : A ≈ B ∈ 𝕌 i) →
Γ ⊢ t ∶ T ®[ i ] a ∈El A≈B →
Γ ⊢ t ∶ T ®↑[ i ] a ∈El A≈B
®El⇒®↑El {i = i} = Real.®El⇒®↑El i (λ j _ → ®⇒Rty-eq {i = j})
®⇒®↑ : ∀ {i} (A≈B : A ≈ B ∈ 𝕌 i) →
Γ ⊢ T ®[ i ] A≈B →
Γ ⊢ T ®↑[ i ] A≈B
®⇒®↑ A≈B T∼A = record
{ t∶T = ®⇒ty A≈B T∼A
; A∈⊤ = realizability-Rty A≈B
; krip = λ {Δ} {σ} ⊢σ → let W , ↘W , Tσ≈ = ®⇒Rty-eq A≈B T∼A ⊢σ
in subst (λ t → _ ⊢ _ [ _ ] ≈ Nf⇒Exp t ∶ Se _)
(Rty-det ↘W (proj₁ (proj₂ (realizability-Rty A≈B (map len Δ) (mt σ)))))
Tσ≈
}
v0®x : ∀ {i} (A≈B : A ≈ B ∈ 𝕌 i) →
Γ ⊢ T ®[ i ] A≈B →
T ∺ Γ ⊢ v 0 ∶ T [ wk ] ®[ i ] ↑ A (l (len (head Γ))) ∈El A≈B
v0®x A≈B T∼A = ®↓El⇒®El A≈B (v0∼x A≈B T∼A)
®⇒≈ : ∀ {i} (A≈B : A ≈ B ∈ 𝕌 i) →
Γ ⊢ T ®[ i ] A≈B →
Γ ⊢ T′ ®[ i ] A≈B →
Γ ⊢ T ≈ T′ ∶ Se i
®⇒≈ {_} {_} {_} {T} {T′} A≈B T∼A T′∼A
with presup-tm (®⇒ty A≈B T∼A)
... | ⊢Γ , _
with ®⇒Rty-eq A≈B T∼A (⊢rI ⊢Γ) | ®⇒Rty-eq A≈B T′∼A (⊢rI ⊢Γ)
... | W , ↘W , T≈W | _ , ↘W₁ , T′≈W₁
rewrite Rty-det ↘W₁ ↘W = begin
T ≈⟨ [I]-≈ˡ-Se T≈W ⟩
Nf⇒Exp W ≈˘⟨ [I]-≈ˡ-Se T′≈W₁ ⟩
T′ ∎
where
open ER
®El⇒≈ : ∀ {i} (A≈B : A ≈ B ∈ 𝕌 i) →
Γ ⊢ t ∶ T ®[ i ] a ∈El A≈B →
Γ ⊢ t′ ∶ T ®[ i ] a ∈El A≈B →
Γ ⊢ t ≈ t′ ∶ T
®El⇒≈ {_} {_} {Γ} {t} {_} {_} {t′} A≈B t∼a t′∼a
with presup-tm (®El⇒ty A≈B t∼a)
... | ⊢Γ , _
with ®El⇒®↑El A≈B t∼a | ®El⇒®↑El A≈B t′∼a
... | record { a∈⊤ = t∈⊤ ; krip = tkrip }
| record { a∈⊤ = t′∈⊤ ; krip = t′krip }
with t∈⊤ (map len Γ) vone | tkrip (⊢rI ⊢Γ)
| t′∈⊤ (map len Γ) vone | t′krip (⊢rI ⊢Γ)
... | w , ↘w , _ | ≈w
| w′ , ↘w′ , _ | ≈w′
rewrite Rf-det ↘w′ ↘w = ≈-trans ([I]-≈ˡ ≈w) (≈-sym ([I]-≈ˡ ≈w′))
®El⇒≈-gen : ∀ {i} (A≈B : A ≈ B ∈ 𝕌 i) →
Γ ⊢ t ∶ T ®[ i ] a ∈El A≈B →
Γ ⊢ t′ ∶ T′ ®[ i ] a ∈El A≈B →
Γ ⊢ t ≈ t′ ∶ T
®El⇒≈-gen A≈B t∼a t′∼a = ®El⇒≈ A≈B t∼a (®El-resp-T≈ A≈B t′∼a (®⇒≈ A≈B (®El⇒® A≈B t′∼a) (®El⇒® A≈B t∼a)))