/
RenamingSubstitution.lagda
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RenamingSubstitution.lagda
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\begin{code}
module Scoped.Erasure.RenamingSubstitution where
\end{code}
\begin{code}
open import Untyped
import Untyped.RenamingSubstitution as U
open import Scoped
open import Scoped.Erasure
import Scoped.RenamingSubstitution as S
open import Data.Nat
open import Data.Fin
open import Relation.Binary.PropositionalEquality hiding ([_])
open import Function
open import Data.List
\end{code}
\begin{code}
backVar : ∀{n}{w : Weirdℕ n} → Fin (len w) → WeirdFin w
backVar {w = S w} zero = Z
backVar {w = S w} (suc i) = S (backVar i)
backVar {w = T w} i = T (backVar i)
backVar-eraseVar : ∀{n}{w : Weirdℕ n}(i : WeirdFin w)
→ backVar (eraseVar i) ≡ i
backVar-eraseVar Z = refl
backVar-eraseVar (S i) = cong S (backVar-eraseVar i)
backVar-eraseVar (T i) = cong T (backVar-eraseVar i)
eraseVar-backVar : ∀{n}{w : Weirdℕ n}(i : Fin (len w))
→ eraseVar (backVar {n}{w} i) ≡ i
eraseVar-backVar {w = S w} zero = refl
eraseVar-backVar {w = S w} (suc i) = cong suc (eraseVar-backVar {w = w} i)
eraseVar-backVar {w = T w} i = eraseVar-backVar {w = w} i
erase-Ren : ∀{n n'}{w : Weirdℕ n}{w' : Weirdℕ n'}
→ S.Ren w w'
→ U.Ren (len w) (len w')
erase-Ren ρ i = eraseVar (ρ (backVar i))
lift-erase : ∀{n n'}{w : Weirdℕ n}{w' : Weirdℕ n'}
→ (ρ : S.Ren w w')
→ (α : Fin (len (S w)))
→ U.lift (erase-Ren ρ) α ≡ erase-Ren (S.lift ρ) α
lift-erase ρ zero = refl
lift-erase ρ (suc α) = refl
ren-erase : ∀{n n'}{w : Weirdℕ n}{w' : Weirdℕ n'}
→ (ρ⋆ : S.Ren⋆ n n')
→ (ρ : S.Ren w w')
→ (t : ScopedTm w)
→ U.ren (erase-Ren ρ) (eraseTm t) ≡ eraseTm (S.ren ρ⋆ ρ t)
ren-eraseList : ∀{n n'}{w : Weirdℕ n}{w' : Weirdℕ n'}
→ (ρ⋆ : S.Ren⋆ n n')
→ (ρ : S.Ren w w')
→ (ts : List (ScopedTm w))
→ U.renList (erase-Ren ρ) (eraseList ts) ≡ eraseList (S.renL ρ⋆ ρ ts)
ren-eraseList ρ⋆ ρ [] = refl
ren-eraseList ρ⋆ ρ (t ∷ ts) =
cong₂ _∷_ (ren-erase ρ⋆ ρ t) (ren-eraseList ρ⋆ ρ ts)
ren-erase ρ⋆ ρ (` x) =
cong (` ∘ eraseVar ∘ ρ) (backVar-eraseVar x)
ren-erase ρ⋆ ρ (Λ K t) = ren-erase (S.lift⋆ ρ⋆) (S.⋆lift ρ) t
ren-erase ρ⋆ ρ (t ·⋆ A) = ren-erase ρ⋆ ρ t
ren-erase ρ⋆ ρ (ƛ A t) = cong
ƛ
(trans (U.ren-cong (lift-erase ρ) (eraseTm t)) (ren-erase ρ⋆ (S.lift ρ) t))
ren-erase ρ⋆ ρ (t · u) =
cong₂ _·_ (ren-erase ρ⋆ ρ t) (ren-erase ρ⋆ ρ u)
ren-erase ρ⋆ ρ (con x) = refl
ren-erase ρ⋆ ρ (error x) = refl
ren-erase ρ⋆ ρ (builtin bn As ts) = cong (builtin bn) (ren-eraseList ρ⋆ ρ ts)
ren-erase ρ⋆ ρ (wrap pat ar t) = ren-erase ρ⋆ ρ t
ren-erase ρ⋆ ρ (unwrap t) = ren-erase ρ⋆ ρ t
--
erase-Sub : ∀{n n'}{w : Weirdℕ n}{w' : Weirdℕ n'}
→ S.Sub w w' → U.Sub (len w) (len w')
erase-Sub σ i = eraseTm (σ (backVar i))
slift-erase : ∀{n n'}{w : Weirdℕ n}{w' : Weirdℕ n'}
→ (σ : S.Sub w w')
→ (α : Fin (len (S w)))
→ U.lifts (erase-Sub σ) α ≡ erase-Sub (S.slift σ) α
slift-erase σ zero = refl
slift-erase {w' = w'} σ (suc α) = trans
(U.ren-cong (cong suc ∘ sym ∘ eraseVar-backVar {w = w'})
(eraseTm (σ (backVar α))))
(ren-erase id S (σ (backVar α)))
⋆slift-erase : ∀{n n'}{w : Weirdℕ n}{w' : Weirdℕ n'}
→ (σ : S.Sub w w')
→ (α : Fin (len w))
→ erase-Sub σ α ≡ erase-Sub (S.⋆slift σ) α
⋆slift-erase {w' = w'} σ α = trans
(trans (U.ren-id (eraseTm (σ (backVar α))))
(U.ren-cong (sym ∘ eraseVar-backVar {w = w'})
(eraseTm (σ (backVar α)))))
(ren-erase suc T (σ (backVar α)))
sub-erase : ∀{n n'}{w : Weirdℕ n}{w' : Weirdℕ n'}
→ (σ⋆ : S.Sub⋆ n n')
→ (σ : S.Sub w w')
→ (t : ScopedTm w)
→ U.sub (erase-Sub σ) (eraseTm t) ≡ eraseTm (S.sub σ⋆ σ t)
subList-erase : ∀{n n'}{w : Weirdℕ n}{w' : Weirdℕ n'}
→ (σ⋆ : S.Sub⋆ n n')
→ (σ : S.Sub w w')
→ (ts : List (ScopedTm w))
→ U.subList (erase-Sub σ) (eraseList ts) ≡ eraseList (S.subL σ⋆ σ ts)
subList-erase σ⋆ σ [] = refl
subList-erase σ⋆ σ (t ∷ ts) =
cong₂ _∷_ (sub-erase σ⋆ σ t) (subList-erase σ⋆ σ ts)
sub-erase σ⋆ σ (` x) = cong (eraseTm ∘ σ) (backVar-eraseVar x)
sub-erase σ⋆ σ (Λ K t) = trans
(U.sub-cong (⋆slift-erase σ) (eraseTm t))
(sub-erase (S.slift⋆ σ⋆) (S.⋆slift σ) t)
sub-erase σ⋆ σ (t ·⋆ A) = sub-erase σ⋆ σ t
sub-erase σ⋆ σ (ƛ A t) = cong
ƛ
(trans (U.sub-cong (slift-erase σ) (eraseTm t))
(sub-erase σ⋆ (S.slift σ) t))
sub-erase σ⋆ σ (t · u) =
cong₂ _·_ (sub-erase σ⋆ σ t) (sub-erase σ⋆ σ u)
sub-erase σ⋆ σ (con c) = refl
sub-erase σ⋆ σ (error A) = refl
sub-erase σ⋆ σ (builtin bn As ts) = cong
(builtin bn)
(subList-erase σ⋆ σ ts)
sub-erase σ⋆ σ (wrap pat arg t) = sub-erase σ⋆ σ t
sub-erase σ⋆ σ (unwrap t) = sub-erase σ⋆ σ t
erase-extend : ∀{n}{w : Weirdℕ n}
→ (u : ScopedTm w)
→ (α : Fin (suc (len w)))
→ erase-Sub (S.ext ` u) α ≡ U.extend ` (eraseTm u) α
erase-extend u zero = refl
erase-extend {w = w} u (suc α) = cong ` (eraseVar-backVar {w = w} α)
lem[] : ∀{n}{w : Weirdℕ n}(t : ScopedTm (S w))(u : ScopedTm w) →
eraseTm (t S.[ u ]) ≡ eraseTm t U.[ eraseTm u ]
lem[] t u = trans
(sym (sub-erase ` (S.ext ` u) t))
(U.sub-cong (erase-extend u) (eraseTm t))
lem[]⋆ : ∀{n}{w : Weirdℕ n}(t : ScopedTm (T w))(A : ScopedTy n) →
eraseTm (t S.[ A ]⋆) ≡ eraseTm t
lem[]⋆ {w = w} t A = trans
(sym (sub-erase (S.ext⋆ ` A) (S.⋆ext `) t))
(trans (U.sub-cong (cong ` ∘ eraseVar-backVar {w = w})
(eraseTm t))
(sym (U.sub-id (eraseTm t))))
\end{code}