{-# OPTIONS --rewriting #-}
module A201801.StdCMTTTerms where
open import A201801.Prelude
open import A201801.Fin
open import A201801.Vec
open import A201801.AllVec
Scopes : Nat → Set
Scopes d = Vec Nat d
mutual
data Term : ∀ {d} → Scopes d → Nat → Set
where
VAR : ∀ {d g} → {σ : Scopes d}
→ Fin g
→ Term σ g
LAM : ∀ {d g} → {σ : Scopes d}
→ Term σ (suc g)
→ Term σ g
APP : ∀ {d g} → {σ : Scopes d}
→ Term σ g → Term σ g
→ Term σ g
MVAR : ∀ {m d g i} → {σ : Scopes d}
→ σ ∋⟨ i ⟩ m → Terms σ g m
→ Term σ g
BOX : ∀ {m d g} → {σ : Scopes d}
→ Term σ m
→ Term σ g
LETBOX : ∀ {m d g} → {σ : Scopes d}
→ Term σ g → Term (σ , m) g
→ Term σ g
Terms : ∀ {d} → Scopes d → Nat → Nat → Set
Terms σ g n = Vec (Term σ g) n
mutual
REN : ∀ {d g g′} → {σ : Scopes d}
→ g′ ≥ g → Term σ g
→ Term σ g′
REN e (VAR i) = VAR (REN∋ e i)
REN e (LAM M) = LAM (REN (keep e) M)
REN e (APP M N) = APP (REN e M) (REN e N)
REN e (MVAR 𝒾 y) = MVAR 𝒾 (RENS e y)
REN e (BOX M) = BOX M
REN e (LETBOX M N) = LETBOX (REN e M) (REN e N)
RENS : ∀ {d g g′ n} → {σ : Scopes d}
→ g′ ≥ g → Terms σ g n
→ Terms σ g′ n
RENS e ∙ = ∙
RENS e (x , M) = RENS e x , REN e M
WK : ∀ {d g} → {σ : Scopes d}
→ Term σ g
→ Term σ (suc g)
WK M = REN (drop id≥) M
VZ : ∀ {d g} → {σ : Scopes d}
→ Term σ (suc g)
VZ = VAR zero
WKS : ∀ {d g n} → {σ : Scopes d}
→ Terms σ g n
→ Terms σ (suc g) n
WKS x = RENS (drop id≥) x
LIFTS : ∀ {d g n} → {σ : Scopes d}
→ Terms σ g n
→ Terms σ (suc g) (suc n)
LIFTS x = WKS x , VZ
VARS : ∀ {d g g′} → {σ : Scopes d}
→ g′ ≥ g
→ Terms σ g′ g
VARS done = ∙
VARS (drop e) = WKS (VARS e)
VARS (keep e) = LIFTS (VARS e)
IDS : ∀ {d g} → {σ : Scopes d}
→ Terms σ g g
IDS = VARS id≥
mutual
MREN : ∀ {d d′ e g} → {σ : Scopes d} {σ′ : Scopes d′}
→ σ′ ⊇⟨ e ⟩ σ → Term σ g
→ Term σ′ g
MREN η (VAR i) = VAR i
MREN η (LAM M) = LAM (MREN η M)
MREN η (APP M N) = APP (MREN η M) (MREN η N)
MREN η (MVAR 𝒾 y) = MVAR (ren∋ η 𝒾) (MRENS η y)
MREN η (BOX M) = BOX (MREN η M)
MREN η (LETBOX M N) = LETBOX (MREN η M) (MREN (keep η) N)
MRENS : ∀ {d d′ e g n} → {σ : Scopes d} {σ′ : Scopes d′}
→ σ′ ⊇⟨ e ⟩ σ → Terms σ g n
→ Terms σ′ g n
MRENS η ∙ = ∙
MRENS η (x , M) = MRENS η x , MREN η M
MWK : ∀ {m d g} → {σ : Scopes d}
→ Term σ g
→ Term (σ , m) g
MWK M = MREN (drop id⊇) M
MWKS : ∀ {m d g n} → {σ : Scopes d}
→ Terms σ g n
→ Terms (σ , m) g n
MWKS x = MRENS (drop id⊇) x
MVZ : ∀ {m d g} → {σ : Scopes d}
→ Terms σ g m
→ Term (σ , m) g
MVZ y = MVAR zero (MWKS y)
mutual
SUB : ∀ {d g n} → {σ : Scopes d}
→ Terms σ g n → Term σ n
→ Term σ g
SUB x (VAR i) = GET x i
SUB x (LAM M) = LAM (SUB (LIFTS x) M)
SUB x (APP M N) = APP (SUB x M) (SUB x N)
SUB x (MVAR 𝒾 y) = MVAR 𝒾 (SUBS x y)
SUB x (BOX M) = BOX M
SUB x (LETBOX M N) = LETBOX (SUB x M) (SUB (MWKS x) N)
SUBS : ∀ {d g n m} → {σ : Scopes d}
→ Terms σ g n → Terms σ n m
→ Terms σ g m
SUBS x ∙ = ∙
SUBS x (y , M) = SUBS x y , SUB x M
CUT : ∀ {d g} → {σ : Scopes d}
→ Term σ g → Term σ (suc g)
→ Term σ g
CUT M N = SUB (IDS , M) N
Term₁ : ∀ {d} → Scopes d → Nat → Set
Term₁ σ m = Term σ m
Terms₁ : ∀ {d n} → Scopes d → Scopes n → Set
Terms₁ σ τ = All (Term₁ σ) τ
MRENS₁ : ∀ {d d′ e n} → {σ : Scopes d} {σ′ : Scopes d′} {τ : Scopes n}
→ σ′ ⊇⟨ e ⟩ σ → Terms₁ σ τ
→ Terms₁ σ′ τ
MRENS₁ e x = maps (MREN e) x
MWKS₁ : ∀ {m d n} → {σ : Scopes d} {τ : Scopes n}
→ Terms₁ σ τ
→ Terms₁ (σ , m) τ
MWKS₁ x = MRENS₁ (drop id⊇) x
MLIFTS₁ : ∀ {m d n} → {σ : Scopes d} {τ : Scopes n}
→ Terms₁ σ τ
→ Terms₁ (σ , m) (τ , m)
MLIFTS₁ x = MWKS₁ x , MVZ IDS
MVARS₁ : ∀ {d d′ e} → {σ : Scopes d} {σ′ : Scopes d′}
→ σ′ ⊇⟨ e ⟩ σ
→ Terms₁ σ′ σ
MVARS₁ done = ∙
MVARS₁ (drop η) = MWKS₁ (MVARS₁ η)
MVARS₁ (keep η) = MLIFTS₁ (MVARS₁ η)
MIDS₁ : ∀ {d} → {σ : Scopes d}
→ Terms₁ σ σ
MIDS₁ = MVARS₁ id⊇
mutual
MSUB : ∀ {d g n} → {σ : Scopes d} {τ : Scopes n}
→ Terms₁ σ τ → Term τ g
→ Term σ g
MSUB x (VAR i) = VAR i
MSUB x (LAM M) = LAM (MSUB x M)
MSUB x (APP M N) = APP (MSUB x M) (MSUB x N)
MSUB x (MVAR 𝒾 y) = SUB (MSUBS x y) (get x 𝒾)
MSUB x (BOX M) = BOX (MSUB x M)
MSUB x (LETBOX M N) = LETBOX (MSUB x M) (MSUB (MLIFTS₁ x) N)
MSUBS : ∀ {d g n m} → {σ : Scopes d} {τ : Scopes n}
→ Terms₁ σ τ → Terms τ g m
→ Terms σ g m
MSUBS x ∙ = ∙
MSUBS x (y , M) = MSUBS x y , MSUB x M
MCUT : ∀ {m d g} → {σ : Scopes d}
→ Term₁ σ m → Term (σ , m) g
→ Term σ g
MCUT M N = MSUB (MIDS₁ , M) N
UNLAM : ∀ {d g} → {σ : Scopes d}
→ Term σ g
→ Term σ (suc g)
UNLAM M = APP (WK M) VZ
EX : ∀ {d g} → {σ : Scopes d}
→ Term σ (suc (suc g))
→ Term σ (suc (suc g))
EX M = APP (APP (WK (WK (LAM (LAM M)))) VZ) (WK VZ)
UP : ∀ {m d g} → {σ : Scopes d}
→ Term σ (suc g)
→ Term (σ , m) g
UP M = APP (LAM (MWK M)) (BOX (MVZ IDS))
DOWN : ∀ {m d g} → {σ : Scopes d}
→ Term (σ , m) g
→ Term σ (suc g)
DOWN M = LETBOX VZ (WK M)
MEX : ∀ {m o d g} → {σ : Scopes d}
→ Term ((σ , m) , o) g
→ Term ((σ , o) , m) g
MEX M = UP (UP (EX (DOWN (DOWN M))))