{-# OPTIONS --rewriting #-}
module A201801.CMLAlternativeDerivations where
open import A201801.Prelude
open import A201801.Category
open import A201801.List
open import A201801.ListLemmas
open import A201801.AllList
open import A201801.CMLPropositions
import A201801.CMLStandardDerivations as CML
import A201801.CMLStandardDerivationsLemmas as CML
mutual
infix 3 _⊢_valid[_]
data _⊢_valid[_] : List Assert → Form → List Form → Set
where
vz : ∀ {A Δ Γ} → Δ ⊢ A valid[ Γ , A ]
wk : ∀ {A B Δ Γ} → Δ ⊢ A valid[ Γ ]
→ Δ ⊢ A valid[ Γ , B ]
cut : ∀ {A B Δ Γ} → Δ ⊢ A valid[ Γ ] → Δ ⊢ B valid[ Γ , A ]
→ Δ ⊢ B valid[ Γ ]
lam : ∀ {A B Δ Γ} → Δ ⊢ B valid[ Γ , A ]
→ Δ ⊢ A ⊃ B valid[ Γ ]
unlam : ∀ {A B Δ Γ} → Δ ⊢ A ⊃ B valid[ Γ ]
→ Δ ⊢ B valid[ Γ , A ]
box : ∀ {A Ψ Δ Γ} → Δ ⊢ A valid[ Ψ ]
→ Δ ⊢ [ Ψ ] A valid[ Γ ]
unbox : ∀ {Δ Γ Ψ A} → Δ ⊢ [ Ψ ] A valid[ Γ ] → Δ ⊢ Ψ allvalid[ Γ ]
→ Δ ⊢ A valid[ Γ ]
vau : ∀ {A B Ψ Δ Γ} → Δ , ⟪ Ψ ⊫ A ⟫ ⊢ B valid[ Γ ]
→ Δ ⊢ B valid[ Γ , [ Ψ ] A ]
unvau : ∀ {A B Ψ Δ Γ} → Δ ⊢ B valid[ Γ , [ Ψ ] A ]
→ Δ , ⟪ Ψ ⊫ A ⟫ ⊢ B valid[ Γ ]
infix 3 _⊢_allvalid[_]
_⊢_allvalid[_] : List Assert → List Form → List Form → Set
Δ ⊢ Ξ allvalid[ Γ ] = All (\ A → Δ ⊢ A valid[ Γ ]) Ξ
infix 3 _⊢_allvalid*
_⊢_allvalid* : List Assert → List Assert → Set
Δ ⊢ Ξ allvalid* = All (\ { ⟪ Γ ⊫ A ⟫ → Δ ⊢ A valid[ Γ ] }) Ξ
wks : ∀ {A Δ Γ Ξ} → Δ ⊢ Ξ allvalid[ Γ ]
→ Δ ⊢ Ξ allvalid[ Γ , A ]
wks ξ = maps wk ξ
lifts : ∀ {A Δ Γ Ξ} → Δ ⊢ Ξ allvalid[ Γ ]
→ Δ ⊢ Ξ , A allvalid[ Γ , A ]
lifts ξ = wks ξ , vz
vars : ∀ {Δ Γ Γ′} → Γ′ ⊇ Γ
→ Δ ⊢ Γ allvalid[ Γ′ ]
vars done = ∙
vars (drop η) = wks (vars η)
vars (keep η) = lifts (vars η)
ids : ∀ {Δ Γ} → Δ ⊢ Γ allvalid[ Γ ]
ids = vars id
mwk : ∀ {B Ψ A Δ Γ} → Δ ⊢ A valid[ Γ ]
→ Δ , ⟪ Ψ ⊫ B ⟫ ⊢ A valid[ Γ ]
mwk 𝒟 = unvau (wk 𝒟)
mwks : ∀ {A Ψ Δ Γ Ξ} → Δ ⊢ Ξ allvalid[ Γ ]
→ Δ , ⟪ Ψ ⊫ A ⟫ ⊢ Ξ allvalid[ Γ ]
mwks ξ = maps mwk ξ
mwks* : ∀ {A Ψ Δ Ξ} → Δ ⊢ Ξ allvalid*
→ Δ , ⟪ Ψ ⊫ A ⟫ ⊢ Ξ allvalid*
mwks* ξ = maps mwk ξ
mvz : ∀ {A Ψ Δ Γ} → Δ , ⟪ Ψ ⊫ A ⟫ ⊢ Ψ allvalid[ Γ ]
→ Δ , ⟪ Ψ ⊫ A ⟫ ⊢ A valid[ Γ ]
mvz ψ = unbox (unvau vz) ψ
mlifts* : ∀ {A Ψ Δ Ξ} → Δ ⊢ Ξ allvalid*
→ Δ , ⟪ Ψ ⊫ A ⟫ ⊢ Ξ , ⟪ Ψ ⊫ A ⟫ allvalid*
mlifts* ξ = mwks* ξ , mvz ids
mvars* : ∀ {Δ Δ′} → Δ′ ⊇ Δ
→ Δ′ ⊢ Δ allvalid*
mvars* done = ∙
mvars* (drop η) = mwks* (mvars* η)
mvars* (keep η) = mlifts* (mvars* η)
mids* : ∀ {Δ} → Δ ⊢ Δ allvalid*
mids* = mvars* id
vaus : ∀ {Δ Γ Ξ Ψ A} → Δ , ⟪ Ψ ⊫ A ⟫ ⊢ Ξ allvalid[ Γ ]
→ Δ ⊢ Ξ allvalid[ Γ , [ Ψ ] A ]
vaus 𝒟 = maps vau 𝒟
unnamed : ∀ {Δ Γ Ξ Ψ A} → Δ , ⟪ Ψ ⊫ A ⟫ ⊢ Ξ allvalid[ Γ ]
→ Δ ⊢ Ξ , [ Ψ ] A allvalid[ Γ , [ Ψ ] A ]
unnamed 𝒟 = vaus 𝒟 , vz
mutual
sub : ∀ {Δ Γ Ξ A} → Δ ⊢ Ξ allvalid[ Γ ] → Δ ⊢ A valid[ Ξ ]
→ Δ ⊢ A valid[ Γ ]
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))
sub ξ (box 𝒟) = box 𝒟
sub ξ (unbox 𝒟 ψ) = unbox (sub ξ 𝒟) (subs ξ ψ)
sub (ξ , 𝒞) (vau 𝒟) = cut 𝒞 (vau (sub (mwks ξ) 𝒟))
sub ξ (unvau 𝒟) = unvau (sub (unnamed ξ) 𝒟)
subs : ∀ {Δ Γ Ξ Ψ} → Δ ⊢ Ξ allvalid[ Γ ] → Δ ⊢ Ψ allvalid[ Ξ ]
→ Δ ⊢ Ψ allvalid[ Γ ]
subs ξ ∙ = ∙
subs ξ (ψ , 𝒟) = subs ξ ψ , sub ξ 𝒟
mutual
msub : ∀ {Δ Γ Ξ A} → Δ ⊢ Ξ allvalid* → Ξ ⊢ A valid[ Γ ]
→ Δ ⊢ A valid[ Γ ]
msub ξ vz = vz
msub ξ (wk 𝒟) = wk (msub ξ 𝒟)
msub ξ (cut 𝒟 ℰ) = cut (msub ξ 𝒟) (msub ξ ℰ)
msub ξ (lam 𝒟) = lam (msub ξ 𝒟)
msub ξ (unlam 𝒟) = unlam (msub ξ 𝒟)
msub ξ (box 𝒟) = box (msub ξ 𝒟)
msub ξ (unbox 𝒟 ψ) = unbox (msub ξ 𝒟) (msubs ξ ψ)
msub ξ (vau 𝒟) = vau (msub (mlifts* ξ) 𝒟)
msub (ξ , 𝒞) (unvau 𝒟) = cut (box 𝒞) (msub ξ 𝒟)
msubs : ∀ {Δ Γ Ξ Ψ} → Δ ⊢ Ξ allvalid* → Ξ ⊢ Ψ allvalid[ Γ ]
→ Δ ⊢ Ψ allvalid[ Γ ]
msubs ξ ∙ = ∙
msubs ξ (ψ , 𝒟) = msubs ξ ψ , msub ξ 𝒟
msubs* : ∀ {Δ Ξ Ψ} → Δ ⊢ Ξ allvalid* → Ξ ⊢ Ψ allvalid*
→ Δ ⊢ Ψ allvalid*
msubs* ξ ∙ = ∙
msubs* ξ (ψ , 𝒟) = msubs* ξ ψ , msub ξ 𝒟
var : ∀ {Δ Γ A} → Γ ∋ A
→ Δ ⊢ A valid[ Γ ]
var zero = vz
var (suc i) = wk (var i)
app : ∀ {Δ Γ A B} → Δ ⊢ A ⊃ B valid[ Γ ] → Δ ⊢ A valid[ Γ ]
→ Δ ⊢ B valid[ Γ ]
app 𝒟 ℰ = cut ℰ (unlam 𝒟)
mvar : ∀ {Δ Γ Ψ A} → Δ ∋ ⟪ Ψ ⊫ A ⟫ → Δ ⊢ Ψ allvalid[ Γ ]
→ Δ ⊢ A valid[ Γ ]
mvar zero ψ = mvz ψ
mvar (suc i) ψ = unvau (mvar i (vaus ψ))
letbox : ∀ {Δ Γ Ψ A B} → Δ ⊢ [ Ψ ] A valid[ Γ ] → Δ , ⟪ Ψ ⊫ A ⟫ ⊢ B valid[ Γ ]
→ Δ ⊢ B valid[ Γ ]
letbox 𝒟 ℰ = cut 𝒟 (vau ℰ)
rebox : ∀ {Δ Γ Ψ A} → Δ ⊢ [ Ψ ] A valid[ Γ ]
→ Δ ⊢ [ Ψ ] A valid[ Γ ]
rebox 𝒟 = letbox 𝒟 (box (mvz ids))
pseudomcut : ∀ {Δ Γ Ψ A B} → Δ ⊢ A valid[ Ψ ] → Δ , ⟪ Ψ ⊫ A ⟫ ⊢ B valid[ Γ ]
→ Δ ⊢ B valid[ Γ ]
pseudomcut 𝒟 ℰ = letbox (box 𝒟) ℰ
mutual
↓ : ∀ {Δ Γ A} → Δ ⊢ A valid[ Γ ]
→ Δ CML.⊢ A valid[ Γ ]
↓ vz = CML.vz
↓ (wk 𝒟) = CML.wk (↓ 𝒟)
↓ (cut 𝒟 ℰ) = CML.cut (↓ 𝒟) (↓ ℰ)
↓ (lam 𝒟) = CML.lam (↓ 𝒟)
↓ (unlam 𝒟) = CML.unlam (↓ 𝒟)
↓ (box 𝒟) = CML.box (↓ 𝒟)
↓ (unbox 𝒟 ψ) = CML.unbox (↓ 𝒟) (↓ⁿ ψ)
↓ (vau 𝒟) = CML.vau (↓ 𝒟)
↓ (unvau 𝒟) = CML.unvau (↓ 𝒟)
↓ⁿ : ∀ {Δ Γ Ξ} → Δ ⊢ Ξ allvalid[ Γ ]
→ Δ CML.⊢ Ξ allvalid[ Γ ]
↓ⁿ ∙ = ∙
↓ⁿ (ξ , 𝒟) = ↓ⁿ ξ , ↓ 𝒟
mutual
↑ : ∀ {Δ Γ A} → Δ CML.⊢ A valid[ Γ ]
→ Δ ⊢ A valid[ Γ ]
↑ (CML.var i) = var i
↑ (CML.lam 𝒟) = lam (↑ 𝒟)
↑ (CML.app 𝒟 ℰ) = app (↑ 𝒟) (↑ ℰ)
↑ (CML.mvar i ψ) = mvar i (↑ⁿ ψ)
↑ (CML.box 𝒟) = box (↑ 𝒟)
↑ (CML.letbox 𝒟 ℰ) = letbox (↑ 𝒟) (↑ ℰ)
↑ⁿ : ∀ {Δ Γ Ξ} → Δ CML.⊢ Ξ allvalid[ Γ ]
→ Δ ⊢ Ξ allvalid[ Γ ]
↑ⁿ ∙ = ∙
↑ⁿ (ξ , 𝒟) = ↑ⁿ ξ , ↑ 𝒟
lem-var : ∀ {Δ Γ A} → (i : Γ ∋ A)
→ ↓ {Δ} (var i) ≡ CML.var i
lem-var zero = refl
lem-var (suc i) = CML.wk & lem-var i
⋮ CML.var ∘ suc & id-ren∋ i