{-# OPTIONS --rewriting #-}
module A201801.IPLAlternativeDerivations where
open import A201801.Prelude
open import A201801.Category
open import A201801.List
open import A201801.ListLemmas
open import A201801.AllList
open import A201801.IPLPropositions
import A201801.IPLStandardDerivations as IPL
import A201801.IPLStandardDerivationsLemmas as IPL
infix 3 _⊢_true
data _⊢_true : List Form → Form → Set
where
vz : ∀ {A Γ} → Γ , A ⊢ A true
wk : ∀ {A B Γ} → Γ ⊢ A true
→ Γ , B ⊢ A true
cut : ∀ {A B Γ} → Γ ⊢ A true → Γ , A ⊢ B true
→ Γ ⊢ B true
lam : ∀ {A B Γ} → Γ , A ⊢ B true
→ Γ ⊢ A ⊃ B true
unlam : ∀ {A B Γ} → Γ ⊢ A ⊃ B true
→ Γ , A ⊢ B true
infix 3 _⊢_alltrue
_⊢_alltrue : List Form → List Form → Set
Γ ⊢ Ξ alltrue = All (Γ ⊢_true) Ξ
wks : ∀ {A Γ Ξ} → Γ ⊢ Ξ alltrue
→ Γ , A ⊢ Ξ alltrue
wks ξ = maps wk ξ
lifts : ∀ {A Γ Ξ} → Γ ⊢ Ξ alltrue
→ Γ , A ⊢ Ξ , A alltrue
lifts ξ = wks ξ , vz
vars : ∀ {Γ Γ′} → Γ′ ⊇ Γ
→ Γ′ ⊢ Γ alltrue
vars done = ∙
vars (drop η) = wks (vars η)
vars (keep η) = lifts (vars η)
ids : ∀ {Γ} → Γ ⊢ Γ alltrue
ids = vars id
sub : ∀ {Γ Ξ A} → Γ ⊢ Ξ alltrue → Ξ ⊢ A true
→ Γ ⊢ A true
sub (ξ , 𝒞) vz = 𝒞
sub (ξ , 𝒞) (wk 𝒟) = sub ξ 𝒟
sub ξ (cut 𝒟 ℰ) = cut (sub ξ 𝒟) (sub (lifts ξ) ℰ)
sub ξ (lam 𝒟) = lam (sub (lifts ξ) 𝒟)
sub (ξ , 𝒞) (unlam 𝒟) = cut 𝒞 (unlam (sub ξ 𝒟))
var : ∀ {Γ A} → Γ ∋ A
→ Γ ⊢ A true
var zero = vz
var (suc i) = wk (var i)
app : ∀ {Γ A B} → Γ ⊢ A ⊃ B true → Γ ⊢ A true
→ Γ ⊢ B true
app 𝒟 ℰ = cut ℰ (unlam 𝒟)
relam : ∀ {Γ A B} → Γ ⊢ A ⊃ B true
→ Γ ⊢ A ⊃ B true
relam 𝒟 = lam (unlam 𝒟)
pseudocut : ∀ {Γ A B} → Γ ⊢ A true → Γ , A ⊢ B true
→ Γ ⊢ B true
pseudocut 𝒟 ℰ = app (lam ℰ) 𝒟
↓ : ∀ {Γ A} → Γ ⊢ A true
→ Γ IPL.⊢ A true
↓ vz = IPL.vz
↓ (wk 𝒟) = IPL.wk (↓ 𝒟)
↓ (cut 𝒟 ℰ) = IPL.cut (↓ 𝒟) (↓ ℰ)
↓ (lam 𝒟) = IPL.lam (↓ 𝒟)
↓ (unlam 𝒟) = IPL.unlam (↓ 𝒟)
↑ : ∀ {Γ A} → Γ IPL.⊢ A true
→ Γ ⊢ A true
↑ (IPL.var i) = var i
↑ (IPL.lam 𝒟) = lam (↑ 𝒟)
↑ (IPL.app 𝒟 ℰ) = app (↑ 𝒟) (↑ ℰ)
lem-var : ∀ {Γ A} → (i : Γ ∋ A)
→ ↓ (var i) ≡ IPL.var i
lem-var zero = refl
lem-var (suc i) = IPL.wk & lem-var i
⋮ IPL.var ∘ suc & id-ren∋ i
id↓↑ : ∀ {Γ A} → (𝒟 : Γ IPL.⊢ A true)
→ (↓ ∘ ↑) 𝒟 ≡ 𝒟
id↓↑ (IPL.var i) = lem-var i
id↓↑ (IPL.lam 𝒟) = IPL.lam & id↓↑ 𝒟
id↓↑ (IPL.app 𝒟 ℰ) = IPL.app & ( IPL.id-cons-wk-sub IPL.ids (↓ (↑ ℰ)) (↓ (↑ 𝒟))
⋮ IPL.id-sub (↓ (↑ 𝒟))
⋮ id↓↑ 𝒟
)
⊗ id↓↑ ℰ