{-# OPTIONS --without-K --safe #-}
open import Level
open import Axiom.Extensionality.Propositional
module NonCumulative.Completeness.Terms (fext : Extensionality 0ℓ (suc 0ℓ)) where
open import Lib
open import Data.Nat.Properties
open import NonCumulative.Completeness.LogRel
open import NonCumulative.Semantics.Properties.PER fext
⊨-lookup-gen : ∀ {x i j}
(Γ≈Δ : ⊨ Γ ≈ Δ) →
x ∶[ i ] T ∈! Γ →
x ∶[ j ] T′ ∈! Δ →
ρ ≈ ρ′ ∈ ⟦ Γ≈Δ ⟧ρ →
Σ (RelTyp i T ρ T′ ρ′)
λ rel → RelExp (v x) ρ (v x) ρ′ (El _ (RelTyp.T≈T′ rel))
⊨-lookup-gen (∷-cong _ rel refl) here here (ρ≈ρ′ , ρ0≈ρ′0)
= record
{ RelTyp (rel ρ≈ρ′)
; ↘⟦T⟧ = ⟦[]⟧ ⟦wk⟧ ↘⟦T⟧
; ↘⟦T′⟧ = ⟦[]⟧ ⟦wk⟧ ↘⟦T′⟧
}
, record
{ ↘⟦t⟧ = ⟦v⟧ 0
; ↘⟦t′⟧ = ⟦v⟧ 0
; t≈t′ = ρ0≈ρ′0
}
where open RelTyp (rel ρ≈ρ′)
⊨-lookup-gen (∷-cong Γ≈Δ rel refl) (there T∈Γ) (there T′∈Δ) (ρ≈ρ′ , ρ0≈ρ′0)
with rT , record { ↘⟦t⟧ = ⟦v⟧ _ ; ↘⟦t′⟧ = ⟦v⟧ _ ; t≈t′ = t≈t′ } ← ⊨-lookup-gen Γ≈Δ T∈Γ T′∈Δ ρ≈ρ′
= record
{ RelTyp rT
; ↘⟦T⟧ = ⟦[]⟧ ⟦wk⟧ ↘⟦T⟧
; ↘⟦T′⟧ = ⟦[]⟧ ⟦wk⟧ ↘⟦T′⟧ }
, record
{ ↘⟦t⟧ = ⟦v⟧ _
; ↘⟦t′⟧ = ⟦v⟧ _
; t≈t′ = t≈t′
}
where open RelTyp rT
⊨-lookup : ∀ {x i}
(⊨Γ : ⊨ Γ) →
x ∶[ i ] T ∈! Γ →
ρ ≈ ρ′ ∈ ⟦ ⊨Γ ⟧ρ →
Σ (RelTyp i T ρ T ρ′)
λ rel → RelExp (v x) ρ (v x) ρ′ (El _ (RelTyp.T≈T′ rel))
⊨-lookup ⊨Γ T∈Γ = ⊨-lookup-gen ⊨Γ T∈Γ T∈Γ
v-≈′ : ∀ {x i} →
⊨ Γ →
x ∶[ i ] T ∈! Γ →
Γ ⊨ v x ≈ v x ∶[ i ] T
v-≈′ ⊨Γ T∈Γ = ⊨Γ , ⊨-lookup ⊨Γ T∈Γ
[]-cong′ : ∀ {i} →
Δ ⊨ t ≈ t′ ∶[ i ] T →
Γ ⊨s σ ≈ σ′ ∶ Δ →
Γ ⊨ t [ σ ] ≈ t′ [ σ′ ] ∶[ i ] T [ σ ]
[]-cong′ {_} {t} {t′} {T} {_} {σ} {σ′} (⊨Δ , t≈t′) (⊨Γ , ⊨Δ₁ , σ≈σ′) = ⊨Γ , helper
where helper : ρ ≈ ρ′ ∈ ⟦ ⊨Γ ⟧ρ →
Σ (RelTyp _ (T [ σ ]) ρ (T [ σ ]) ρ′)
λ rel → RelExp (t [ σ ]) ρ (t′ [ σ′ ]) ρ′ (El _ (RelTyp.T≈T′ rel))
helper {ρ} {ρ′} ρ≈ρ′
with ρ≈ρ ← ⟦⟧ρ-refl ⊨Γ ⊨Γ ρ≈ρ′
| ρ′≈ρ ← ⟦⟧ρ-sym′ ⊨Γ ρ≈ρ′
with σ≈σ′ ρ≈ρ
| σ≈σ′ ρ′≈ρ
| σ≈σ′ ρ≈ρ′
... | record { ⟦σ⟧ = ⟦σ⟧ ; ↘⟦σ⟧ = ↘⟦σ⟧ ; ↘⟦δ⟧ = ↘⟦σ′⟧ ; σ≈δ = ⟦σ≈σ′⟧ }
| record { ⟦σ⟧ = ⟦σ⟧′ ; ↘⟦σ⟧ = ↘⟦σ⟧′ ; ↘⟦δ⟧ = ↘⟦σ′⟧′ ; σ≈δ = ⟦σ≈σ′⟧₁ }
| record { ⟦σ⟧ = ⟦σ⟧″ ; ↘⟦σ⟧ = ↘⟦σ⟧″ ; ↘⟦δ⟧ = ↘⟦σ′⟧″ ; σ≈δ = ⟦σ≈σ′⟧₂ }
rewrite ⟦⟧s-det ↘⟦σ′⟧ ↘⟦σ′⟧′
| ⟦⟧s-det ↘⟦σ⟧″ ↘⟦σ⟧
with σ≈σ ← ⟦⟧ρ-trans′ ⊨Δ₁ ⟦σ≈σ′⟧ (⟦⟧ρ-sym′ ⊨Δ₁ ⟦σ≈σ′⟧₁) = help
where help : Σ (RelTyp _ (T [ σ ]) ρ (T [ σ ]) ρ′)
λ rel → RelExp (t [ σ ]) ρ (t′ [ σ′ ]) ρ′ (El _ (RelTyp.T≈T′ rel))
help
with t≈t′ (⊨-irrel ⊨Δ₁ ⊨Δ σ≈σ)
| t≈t′ (⊨-irrel ⊨Δ₁ ⊨Δ ⟦σ≈σ′⟧₂)
... | record { ↘⟦T⟧ = ↘⟦T⟧ ; ↘⟦T′⟧ = ↘⟦T′⟧ ; T≈T′ = T≈T′ } , _
| record { ↘⟦T⟧ = ↘⟦T⟧′ ; T≈T′ = T≈T′₁ } , re
rewrite ⟦⟧-det ↘⟦T⟧′ ↘⟦T⟧ = record
{ ↘⟦T⟧ = ⟦[]⟧ ↘⟦σ⟧ ↘⟦T⟧
; ↘⟦T′⟧ = ⟦[]⟧ ↘⟦σ⟧′ ↘⟦T′⟧
; T≈T′ = T≈T′
}
, record
{ ↘⟦t⟧ = ⟦[]⟧ ↘⟦σ⟧ ↘⟦t⟧
; ↘⟦t′⟧ = ⟦[]⟧ ↘⟦σ′⟧″ ↘⟦t′⟧
; t≈t′ = El-one-sided T≈T′₁ T≈T′ re.t≈t′
}
where module re = RelExp re
open re
[I]′ : ∀ {i} →
Γ ⊨ t ∶[ i ] T →
Γ ⊨ t [ I ] ≈ t ∶[ i ] T
[I]′ {_} {t} {T} (⊨Γ , ⊨t) = ⊨Γ , helper
where helper : ρ ≈ ρ′ ∈ ⟦ ⊨Γ ⟧ρ →
Σ (RelTyp _ T ρ T ρ′)
λ rel → RelExp (t [ I ]) ρ t ρ′ (El _ (RelTyp.T≈T′ rel))
helper ρ≈ρ′
with rt , re ← ⊨t ρ≈ρ′ = rt
, record
{ RelExp re
; ↘⟦t⟧ = ⟦[]⟧ ⟦I⟧ ↘⟦t⟧
}
where open RelExp re
[wk]′ : ∀ {x i j} →
⊨ (S ↙ j) ∷ Γ →
x ∶[ i ] T ∈! Γ →
(S ↙ j) ∷ Γ ⊨ v x [ wk ] ≈ v (1 + x) ∶[ i ] T [ wk ]
[wk]′ {S} {_} {T} {x} ⊨SΓ@(∷-cong ⊨Γ _ _) T∈Γ = ⊨SΓ , helper
where helper : ρ ≈ ρ′ ∈ ⟦ ⊨SΓ ⟧ρ →
Σ (RelTyp _ (T [ wk ]) ρ (sub T wk) ρ′)
λ rel → RelExp (v x [ wk ]) ρ (v (1 + x)) ρ′ (El _ (RelTyp.T≈T′ rel))
helper {ρ} {ρ′} (ρ≈ρ′ , ρ0≈ρ′0)
with ⊨-lookup ⊨Γ T∈Γ ρ≈ρ′
... | rt
, record { ↘⟦t⟧ = ↘⟦t⟧ ; ↘⟦t′⟧ = ⟦v⟧ _ ; t≈t′ = t≈t′ }
= record
{ RelTyp rt
; ↘⟦T⟧ = ⟦[]⟧ ⟦wk⟧ ↘⟦T⟧
; ↘⟦T′⟧ = ⟦[]⟧ ⟦wk⟧ ↘⟦T′⟧
}
, record
{ ↘⟦t⟧ = ⟦[]⟧ ⟦wk⟧ ↘⟦t⟧
; ↘⟦t′⟧ = ⟦v⟧ _
; t≈t′ = t≈t′
}
where open RelTyp rt
[∘]′ : ∀ {i} →
Γ ⊨s τ ∶ Γ′ →
Γ′ ⊨s σ ∶ Γ″ →
Γ″ ⊨ t ∶[ i ] T →
Γ ⊨ t [ σ ∘ τ ] ≈ t [ σ ] [ τ ] ∶[ i ] T [ σ ∘ τ ]
[∘]′ {_} {τ} {_} {σ} {_} {t} {T} (⊨Γ , ⊨Γ′ , ⊨τ) (⊨Γ′₁ , ⊨Γ″ , ⊨σ) (⊨Γ″₁ , ⊨t) = ⊨Γ , helper
where helper : ρ ≈ ρ′ ∈ ⟦ ⊨Γ ⟧ρ →
Σ (RelTyp _ (T [ σ ∘ τ ]) ρ (T [ σ ∘ τ ]) ρ′)
λ rel → RelExp (t [ σ ∘ τ ]) ρ (t [ σ ] [ τ ]) ρ′ (El _ (RelTyp.T≈T′ rel))
helper ρ≈ρ′ = record
{ rt
; ↘⟦T⟧ = ⟦[]⟧ (⟦∘⟧ τ.↘⟦σ⟧ σ.↘⟦σ⟧) rt.↘⟦T⟧
; ↘⟦T′⟧ = ⟦[]⟧ (⟦∘⟧ τ.↘⟦δ⟧ σ.↘⟦δ⟧) rt.↘⟦T′⟧
}
, record
{ re
; ↘⟦t⟧ = ⟦[]⟧ (⟦∘⟧ τ.↘⟦σ⟧ σ.↘⟦σ⟧) re.↘⟦t⟧
; ↘⟦t′⟧ = ⟦[]⟧ τ.↘⟦δ⟧ (⟦[]⟧ σ.↘⟦δ⟧ re.↘⟦t′⟧)
}
where module τ = RelSubst (⊨τ ρ≈ρ′)
module σ = RelSubst (⊨σ (⊨-irrel ⊨Γ′ ⊨Γ′₁ τ.σ≈δ))
⊨tρ = ⊨t (⊨-irrel ⊨Γ″ ⊨Γ″₁ σ.σ≈δ)
rt = proj₁ ⊨tρ
re = proj₂ ⊨tρ
module rt = RelTyp rt
module re = RelExp re
[,]-v-ze′ : ∀ {i} →
Γ ⊨s σ ∶ Δ →
Δ ⊨ S ∶[ 1 + i ] Se i →
Γ ⊨ s ∶[ i ] (S [ σ ]) →
Γ ⊨ v 0 [ σ , s ∶ S ↙ i ] ≈ s ∶[ i ] S [ σ ]
[,]-v-ze′ {_} {σ} {_} {S} {s} (⊨Γ , ⊨Δ , ⊨σ) (⊨Δ₁ , ⊨S) (⊨Γ₁ , ⊨s) = ⊨Γ , helper
where helper : ρ ≈ ρ′ ∈ ⟦ ⊨Γ ⟧ρ →
Σ (RelTyp _ (S [ σ ]) ρ (S [ σ ]) ρ′)
λ rel → RelExp (v 0 [ σ , s ∶ S ↙ _ ]) ρ s ρ′ (El _ (RelTyp.T≈T′ rel))
helper {ρ} {ρ′} ρ≈ρ′ = help
where module σ = RelSubst (⊨σ ρ≈ρ′)
help : Σ (RelTyp _ (S [ σ ]) ρ (S [ σ ]) ρ′)
λ rel → RelExp (v 0 [ σ , s ∶ S ↙ _ ]) ρ s ρ′ (El _ (RelTyp.T≈T′ rel))
help
with ⊨S (⊨-irrel ⊨Δ ⊨Δ₁ σ.σ≈δ)
| ⊨s (⊨-irrel ⊨Γ ⊨Γ₁ ρ≈ρ′)
... | record { ↘⟦T⟧ = ⟦Se⟧ _ ; ↘⟦T′⟧ = ⟦Se⟧ _ ; T≈T′ = U _ _ }
, record { ↘⟦t⟧ = ↘⟦t⟧ ; ↘⟦t′⟧ = ↘⟦t′⟧ ; t≈t′ = t≈t′ }
| record { ↘⟦T⟧ = ⟦[]⟧ ↘⟦σ⟧ ↘⟦T⟧ ; ↘⟦T′⟧ = ⟦[]⟧ ↘⟦σ⟧′ ↘⟦T⟧′ ; T≈T′ = T≈T′ }
, re
rewrite ⟦⟧s-det ↘⟦σ⟧ σ.↘⟦σ⟧
| ⟦⟧s-det ↘⟦σ⟧′ σ.↘⟦δ⟧
| ⟦⟧-det ↘⟦t⟧ ↘⟦T⟧
| ⟦⟧-det ↘⟦t′⟧ ↘⟦T⟧′ = record
{ ↘⟦T⟧ = ⟦[]⟧ σ.↘⟦σ⟧ ↘⟦T⟧
; ↘⟦T′⟧ = ⟦[]⟧ σ.↘⟦δ⟧ ↘⟦T⟧′
; T≈T′ = T≈T′
}
, record
{ re
; ↘⟦t⟧ = ⟦[]⟧ (⟦,⟧ σ.↘⟦σ⟧ re.↘⟦t⟧) (⟦v⟧ 0)
}
where module re = RelExp re
[,]-v-su′ : ∀ {i j x} →
Γ ⊨s σ ∶ Δ →
Δ ⊨ S ∶[ 1 + i ] Se i →
Γ ⊨ s ∶[ i ] S [ σ ] →
x ∶[ j ] T ∈! Δ →
Γ ⊨ v (1 + x) [ σ , s ∶ S ↙ i ] ≈ v x [ σ ] ∶[ j ] T [ σ ]
[,]-v-su′ {_} {σ} {_} {S} {s} {T} {_} {_} {x} (⊨Γ , ⊨Δ , ⊨σ) (⊨Δ₁ , ⊨S) (⊨Γ₁ , ⊨s) T∈Δ = ⊨Γ , helper
where helper : ρ ≈ ρ′ ∈ ⟦ ⊨Γ ⟧ρ →
Σ (RelTyp _ (T [ σ ]) ρ (T [ σ ]) ρ′)
λ rel → RelExp (v (1 + x) [ σ , s ∶ S ↙ _ ]) ρ (v x [ σ ]) ρ′ (El _ (RelTyp.T≈T′ rel))
helper {ρ} {ρ′} ρ≈ρ′ = help
where module σ = RelSubst (⊨σ ρ≈ρ′)
module re = RelExp (proj₂ (⊨s (⊨-irrel ⊨Γ ⊨Γ₁ ρ≈ρ′)))
help : Σ (RelTyp _ (T [ σ ]) ρ (T [ σ ]) ρ′)
λ rel → RelExp (v (1 + x) [ σ , s ∶ S ↙ _ ]) ρ (v x [ σ ]) ρ′ (El _ (RelTyp.T≈T′ rel))
help
with ⊨-lookup ⊨Δ T∈Δ σ.σ≈δ
... | rt′
, record { ↘⟦t⟧ = ⟦v⟧ _ ; ↘⟦t′⟧ = ⟦v⟧ _ ; t≈t′ = t≈t′ }
= record
{ rt′
; ↘⟦T⟧ = ⟦[]⟧ σ.↘⟦σ⟧ rt′.↘⟦T⟧
; ↘⟦T′⟧ = ⟦[]⟧ σ.↘⟦δ⟧ rt′.↘⟦T′⟧
}
, record
{ ↘⟦t⟧ = ⟦[]⟧ (⟦,⟧ σ.↘⟦σ⟧ re.↘⟦t⟧) (⟦v⟧ _)
; ↘⟦t′⟧ = ⟦[]⟧ σ.↘⟦δ⟧ (⟦v⟧ _)
; t≈t′ = t≈t′
}
where module rt′ = RelTyp rt′
≈-conv′ : ∀ {i} →
Γ ⊨ s ≈ t ∶[ i ] S →
Γ ⊨ S ≈ T ∶[ 1 + i ] Se i →
Γ ⊨ s ≈ t ∶[ i ] T
≈-conv′ {_} {s} {t} {S} {T} (⊨Γ , s≈t) (⊨Γ₁ , S≈T) = ⊨Γ , helper
where helper : ρ ≈ ρ′ ∈ ⟦ ⊨Γ ⟧ρ →
Σ (RelTyp _ T ρ T ρ′)
λ rel → RelExp s ρ t ρ′ (El _ (RelTyp.T≈T′ rel))
helper ρ≈ρ′
with S≈T (⊨-irrel ⊨Γ ⊨Γ₁ ρ≈ρ′)
| S≈T (⟦⟧ρ-refl ⊨Γ ⊨Γ₁ ρ≈ρ′)
| s≈t ρ≈ρ′
... | record { ↘⟦T⟧ = ⟦Se⟧ _ ; ↘⟦T′⟧ = ⟦Se⟧ _ ; T≈T′ = U eq _ }
, record { ↘⟦t⟧ = ↘⟦S⟧ ; ↘⟦t′⟧ = ↘⟦T⟧ ; t≈t′ = ⟦S≈T⟧ }
| record { ↘⟦T⟧ = ⟦Se⟧ _ ; ↘⟦T′⟧ = ⟦Se⟧ _ ; T≈T′ = U eq′ _ }
, record { ↘⟦t⟧ = ↘⟦S⟧′ ; ↘⟦t′⟧ = ↘⟦T⟧′ ; t≈t′ = ⟦S≈T⟧′ }
| record { ↘⟦T⟧ = ↘⟦S⟧″ ; ↘⟦T′⟧ = ↘⟦T⟧″ ; T≈T′ = T≈T′ }
, re
rewrite 𝕌-wellfounded-≡-𝕌 _ (≤-reflexive (sym eq′))
| 𝕌-wellfounded-≡-𝕌 _ (≤-reflexive (sym eq))
| ⟦⟧-det ↘⟦S⟧′ ↘⟦S⟧
| ⟦⟧-det ↘⟦S⟧″ ↘⟦S⟧ = record
{ ↘⟦T⟧ = ↘⟦T⟧′
; ↘⟦T′⟧ = ↘⟦T⟧
; T≈T′ = 𝕌-trans (𝕌-sym ⟦S≈T⟧′) ⟦S≈T⟧
}
, record
{ RelExp re
; t≈t′ = El-one-sided′ ⟦S≈T⟧ (𝕌-trans (𝕌-sym ⟦S≈T⟧′) ⟦S≈T⟧) (El-one-sided T≈T′ ⟦S≈T⟧ (RelExp.t≈t′ re))
}
≈-sym′ : ∀ {i} →
Γ ⊨ t ≈ t′ ∶[ i ] T →
Γ ⊨ t′ ≈ t ∶[ i ] T
≈-sym′ {_} {t} {t′} {T} (⊨Γ , t≈t′) = ⊨Γ , helper
where helper : ρ ≈ ρ′ ∈ ⟦ ⊨Γ ⟧ρ →
Σ (RelTyp _ T ρ T ρ′)
λ rel → RelExp t′ ρ t ρ′ (El _ (RelTyp.T≈T′ rel))
helper ρ≈ρ′
with rt , re ← t≈t′ (⟦⟧ρ-sym′ ⊨Γ ρ≈ρ′) = record
{ ↘⟦T⟧ = ↘⟦T′⟧
; ↘⟦T′⟧ = ↘⟦T⟧
; T≈T′ = 𝕌-sym T≈T′
}
, record
{ ↘⟦t⟧ = ↘⟦t′⟧
; ↘⟦t′⟧ = ↘⟦t⟧
; t≈t′ = El-sym T≈T′ (𝕌-sym T≈T′) re.t≈t′
}
where module rt = RelTyp rt
module re = RelExp re
open rt
open re
≈-trans′ : ∀ {i} →
Γ ⊨ t ≈ t′ ∶[ i ] T →
Γ ⊨ t′ ≈ t″ ∶[ i ] T →
Γ ⊨ t ≈ t″ ∶[ i ] T
≈-trans′ {_} {t} {t′} {T} {t″} (⊨Γ , t≈t′) (⊨Γ₁ , t′≈t″) = ⊨Γ , helper
where helper : ρ ≈ ρ′ ∈ ⟦ ⊨Γ ⟧ρ →
Σ (RelTyp _ T ρ T ρ′)
λ rel → RelExp t ρ t″ ρ′ (El _ (RelTyp.T≈T′ rel))
helper ρ≈ρ′
with t≈t′ (⟦⟧ρ-refl ⊨Γ ⊨Γ ρ≈ρ′)
| t′≈t″ (⊨-irrel ⊨Γ ⊨Γ₁ ρ≈ρ′)
... | record { ↘⟦T⟧ = ↘⟦T⟧ ; T≈T′ = T≈T′ }
, record { ↘⟦t⟧ = ↘⟦t⟧ ; ↘⟦t′⟧ = ↘⟦t′⟧ ; t≈t′ = t≈t′ }
| rt@record { ↘⟦T⟧ = ↘⟦T⟧′ ; T≈T′ = T≈T′₁ }
, record { ↘⟦t⟧ = ↘⟦t′⟧₁ ; ↘⟦t′⟧ = ↘⟦t″⟧ ; t≈t′ = t′≈t″ }
rewrite ⟦⟧-det ↘⟦t′⟧₁ ↘⟦t′⟧
| ⟦⟧-det ↘⟦T⟧ ↘⟦T⟧′ = rt
, record
{ ↘⟦t⟧ = ↘⟦t⟧
; ↘⟦t′⟧ = ↘⟦t″⟧
; t≈t′ = El-trans′ T≈T′₁ (El-one-sided T≈T′ T≈T′₁ t≈t′) t′≈t″
}