module Graded.Erasure.Target.Properties.Substitution where
open import Graded.Erasure.Target hiding (refl ; trans)
open import Graded.Erasure.Target.Properties.Weakening
open import Tools.Fin
open import Tools.Nat
open import Tools.PropositionalEquality hiding (subst)
open import Tools.Reasoning.PropositionalEquality
private
variable
ℓ m n : Nat
ρ ρ′ : Wk m n
σ σ′ : Subst m n
t : Term n
substVar-lift : (∀ x → σ x ≡ σ′ x) → ∀ x → liftSubst σ x ≡ liftSubst σ′ x
substVar-lift eq x0 = refl
substVar-lift eq (x +1) = cong wk1 (eq x)
substVar-lifts : (∀ x → σ x ≡ σ′ x) → ∀ n x → liftSubstn σ n x ≡ liftSubstn σ′ n x
substVar-lifts eq 0 x = eq x
substVar-lifts eq (1+ n) x0 = refl
substVar-lifts eq (1+ n) (x +1) = cong wk1 (substVar-lifts eq n x)
mutual
substVar-to-subst : ((x : Fin n) → σ x ≡ σ′ x)
→ (t : Term n) → t [ σ ] ≡ t [ σ′ ]
substVar-to-subst eq (var x) = eq x
substVar-to-subst eq (lam t) = cong lam (substVar-to-subst (substVar-lift eq) t)
substVar-to-subst eq (t ∘ u) = cong₂ _∘_ (substVar-to-subst eq t) (substVar-to-subst eq u)
substVar-to-subst eq zero = refl
substVar-to-subst eq (suc t) = cong suc (substVar-to-subst eq t)
substVar-to-subst eq (natrec z s n) = cong₃ natrec (substVar-to-subst eq z) (substVar-to-subst (substVar-lifts eq 2) s) (substVar-to-subst eq n)
substVar-to-subst eq (prod t u) = cong₂ prod (substVar-to-subst eq t) (substVar-to-subst eq u)
substVar-to-subst eq (fst t) = cong fst (substVar-to-subst eq t)
substVar-to-subst eq (snd t) = cong snd (substVar-to-subst eq t)
substVar-to-subst eq (prodrec t u) = cong₂ prodrec (substVar-to-subst eq t) (substVar-to-subst (substVar-lifts eq 2) u)
substVar-to-subst eq star = refl
substVar-to-subst eq ↯ = refl
subst-lift-id : (x : Fin (1+ n)) → (liftSubst idSubst) x ≡ idSubst x
subst-lift-id x0 = refl
subst-lift-id (x +1) = refl
subst-lifts-id : (n : Nat) → (x : Fin (n + m)) → (liftSubstn idSubst n) x ≡ idSubst x
subst-lifts-id 0 x = refl
subst-lifts-id (1+ n) x0 = refl
subst-lifts-id (1+ n) (x +1) = cong wk1 (subst-lifts-id n x)
mutual
subst-id : (t : Term n) → t [ idSubst ] ≡ t
subst-id (var x) = refl
subst-id (lam t) = cong lam (trans (substVar-to-subst subst-lift-id t) (subst-id t))
subst-id (t ∘ u) = cong₂ _∘_ (subst-id t) (subst-id u)
subst-id zero = refl
subst-id (suc t) = cong suc (subst-id t)
subst-id (natrec z s n) = cong₃ natrec (subst-id z) (trans (substVar-to-subst (subst-lifts-id 2) s) (subst-id s)) (subst-id n)
subst-id (prod t u) = cong₂ prod (subst-id t) (subst-id u)
subst-id (fst t) = cong fst (subst-id t)
subst-id (snd t) = cong snd (subst-id t)
subst-id (prodrec t u) = cong₂ prodrec (subst-id t) (trans (substVar-to-subst (subst-lifts-id 2) u) (subst-id u))
subst-id star = refl
subst-id ↯ = refl
subst-lift-•ₛ : ∀ t
→ t [ lift ρ •ₛ liftSubst σ ]
≡ t [ liftSubst (ρ •ₛ σ) ]
subst-lift-•ₛ =
substVar-to-subst (λ { x0 → refl ; (x +1) → sym (wk1-wk≡lift-wk1 _ _)})
helper1 : (n : Nat) (x : Fin (1+ n + m)) →
(lift (liftn ρ n) •ₛ liftSubst (liftSubstn σ n)) x ≡
liftSubst (liftSubstn (ρ •ₛ σ) n) x
helper1 0 x0 = refl
helper1 0 (x +1) = sym (wk1-wk≡lift-wk1 _ _)
helper1 (1+ n) x0 = refl
helper1 (1+ n) (x +1) = trans (sym (wk1-wk≡lift-wk1 _ _)) (cong wk1 (helper1 n x))
subst-lifts-•ₛ : ∀ n t
→ t [ liftn ρ n •ₛ liftSubstn σ n ]
≡ t [ liftSubstn (ρ •ₛ σ) n ]
subst-lifts-•ₛ 0 t = refl
subst-lifts-•ₛ (1+ n) t = substVar-to-subst (helper1 n) t
subst-lift-ₛ• : ∀ t
→ t [ liftSubst σ ₛ• lift ρ ]
≡ t [ liftSubst (σ ₛ• ρ) ]
subst-lift-ₛ• = substVar-to-subst (λ { x0 → refl ; (x +1) → refl})
helper2 : (n : Nat) → (x : Fin (1+ n + m))
→ liftSubst (liftSubstn σ n) (wkVar (lift (liftn ρ n)) x) ≡
liftSubst (liftSubstn (λ x₁ → σ (wkVar ρ x₁)) n) x
helper2 0 x0 = refl
helper2 0 (x +1) = refl
helper2 (1+ n) x0 = refl
helper2 (1+ n) (x +1) = cong wk1 (helper2 n x)
subst-lifts-ₛ• : ∀ n t
→ t [ liftSubstn σ n ₛ• liftn ρ n ]
≡ t [ liftSubstn (σ ₛ• ρ) n ]
subst-lifts-ₛ• 0 t = refl
subst-lifts-ₛ• (1+ n) t = substVar-to-subst (helper2 n) t
mutual
wk-subst : ∀ t → wk ρ (t [ σ ]) ≡ t [ ρ •ₛ σ ]
wk-subst (var x) = refl
wk-subst (lam t) = cong lam (trans (wk-subst t) (subst-lift-•ₛ t))
wk-subst (t ∘ u) = cong₂ _∘_ (wk-subst t) (wk-subst u)
wk-subst zero = refl
wk-subst (suc t) = cong suc (wk-subst t)
wk-subst (natrec z s n) = cong₃ natrec (wk-subst z) (trans (wk-subst s) (subst-lifts-•ₛ 2 s)) (wk-subst n)
wk-subst (prod t u) = cong₂ prod (wk-subst t) (wk-subst u)
wk-subst (fst t) = cong fst (wk-subst t)
wk-subst (snd t) = cong snd (wk-subst t)
wk-subst (prodrec t u) = cong₂ prodrec (wk-subst t) (trans (wk-subst u) (subst-lifts-•ₛ 2 u))
wk-subst star = refl
wk-subst ↯ = refl
mutual
subst-wk : ∀ t → wk ρ t [ σ ] ≡ t [ σ ₛ• ρ ]
subst-wk (var x) = refl
subst-wk (lam t) = cong lam (trans (subst-wk t) (subst-lift-ₛ• t))
subst-wk (t ∘ u) = cong₂ _∘_ (subst-wk t) (subst-wk u)
subst-wk zero = refl
subst-wk (suc t) = cong suc (subst-wk t)
subst-wk (natrec z s n) = cong₃ natrec (subst-wk z) (trans (subst-wk s) (subst-lifts-ₛ• 2 s)) (subst-wk n)
subst-wk (prod t u) = cong₂ prod (subst-wk t) (subst-wk u)
subst-wk (fst t) = cong fst (subst-wk t)
subst-wk (snd t) = cong snd (subst-wk t)
subst-wk (prodrec t u) = cong₂ prodrec (subst-wk t) (trans (subst-wk u) (subst-lifts-ₛ• 2 u))
subst-wk star = refl
subst-wk ↯ = refl
wk-subst-lift : (G : Term (1+ n))
→ wk (lift ρ) (G [ liftSubst σ ])
≡ G [ liftSubst (ρ •ₛ σ) ]
wk-subst-lift G = trans (wk-subst G) (subst-lift-•ₛ G)
wk≡subst : (ρ : Wk m n) (t : Term n) → wk ρ t ≡ t [ toSubst ρ ]
wk≡subst ρ t = trans (cong (wk ρ) (sym (subst-id t))) (wk-subst t)
substCompLift : ∀ x
→ (liftSubst σ ₛ•ₛ liftSubst σ′) x
≡ (liftSubst (σ ₛ•ₛ σ′)) x
substCompLift x0 = refl
substCompLift {σ = σ} {σ′ = σ′} (x +1) = trans (subst-wk (σ′ x)) (sym (wk-subst (σ′ x)))
substCompLifts : ∀ n x
→ (liftSubstn σ n ₛ•ₛ liftSubstn σ′ n) x
≡ (liftSubstn (σ ₛ•ₛ σ′) n) x
substCompLifts 0 x = refl
substCompLifts (1+ n) x0 = refl
substCompLifts {σ = σ} {σ′ = σ′} (1+ n) (x +1) =
trans (substCompLift {σ = liftSubstn σ n} {σ′ = liftSubstn σ′ n} (x +1))
(cong wk1 (substCompLifts n x))
mutual
substCompEq : ∀ (t : Term n)
→ t [ σ′ ] [ σ ] ≡ t [ σ ₛ•ₛ σ′ ]
substCompEq (var x) = refl
substCompEq (lam t) = cong lam (trans (substCompEq t) (substVar-to-subst substCompLift t))
substCompEq (t ∘ u) = cong₂ _∘_ (substCompEq t) (substCompEq u)
substCompEq zero = refl
substCompEq (suc t) = cong suc (substCompEq t)
substCompEq (natrec z s n) = cong₃ natrec (substCompEq z) (trans (substCompEq s) (substVar-to-subst (substCompLifts 2) s)) (substCompEq n)
substCompEq (prod t u) = cong₂ prod (substCompEq t) (substCompEq u)
substCompEq (fst t) = cong fst (substCompEq t)
substCompEq (snd t) = cong snd (substCompEq t)
substCompEq (prodrec t u) = cong₂ prodrec (substCompEq t) (trans (substCompEq u) (substVar-to-subst (substCompLifts 2) u))
substCompEq star = refl
substCompEq ↯ = refl
wk-comp-subst : ∀ {a : Term m} (ρ : Wk m ℓ) (ρ′ : Wk ℓ n) G
→ wk (lift (ρ • ρ′)) G [ a ]₀ ≡ wk (lift ρ) (wk (lift ρ′) G) [ a ]₀
wk-comp-subst {a = a} ρ ρ′ G =
cong (λ x → x [ a ]₀) (sym (wk-comp (lift ρ) (lift ρ′) G))
wk-β : ∀ {a : Term m} t → wk ρ (t [ a ]₀) ≡ wk (lift ρ) t [ wk ρ a ]₀
wk-β t = trans (wk-subst t) (sym (trans (subst-wk t)
(substVar-to-subst (λ { x0 → refl ; (x +1) → refl}) t)))
wk-lift-β :
∀ u →
wk (lift ρ) (u [ liftSubst (sgSubst t) ]) ≡
wk (lift (lift ρ)) u [ liftSubst (sgSubst (wk ρ t)) ]
wk-lift-β {ρ = ρ} {t = t} u =
wk (lift ρ) (u [ liftSubst (sgSubst t) ]) ≡⟨ wk-subst u ⟩
u [ lift ρ •ₛ liftSubst (sgSubst t) ] ≡˘⟨ substVar-to-subst
(λ where
x0 → refl
(x0 +1) → wk1-wk≡lift-wk1 _ _
((_ +1) +1) → refl)
u ⟩
u [ liftSubst (sgSubst (wk ρ t)) ₛ• lift (lift ρ) ] ≡˘⟨ subst-wk u ⟩
wk (lift (lift ρ)) u [ liftSubst (sgSubst (wk ρ t)) ] ∎
wk-β-doubleSubst : ∀ (ρ : Wk m n) (s : Term (1+ (1+ n))) (t u : Term n)
→ wk ρ (s [ u , t ])
≡ wk (lift (lift ρ)) s [ wk ρ u , wk ρ t ]
wk-β-doubleSubst ρ s t u =
begin
wk ρ (s [ σₜ t u ])
≡⟨ wk-subst s ⟩
s [ ρ •ₛ (σₜ t u) ]
≡⟨ substVar-to-subst (λ { x0 → refl ; (x0 +1) → refl ; (x +1 +1) → refl}) s ⟩
s [ (σₜ (wk ρ t) (wk ρ u)) ₛ• (lift (lift ρ)) ]
≡⟨ sym (subst-wk s) ⟩
wk (lift (lift ρ)) s [ wk ρ u , wk ρ t ] ∎
where
σₜ : (x y : Term ℓ) → Subst ℓ (1+ (1+ ℓ))
σₜ x y = consSubst (consSubst idSubst y) x
substVarSingletonComp : ∀ {u} (x : Fin (1+ n))
→ (sgSubst u ₛ•ₛ liftSubst σ) x ≡ (consSubst σ u) x
substVarSingletonComp x0 = refl
substVarSingletonComp {σ = σ} (x +1) = trans (subst-wk (σ x)) (subst-id (σ x))
substSingletonComp : ∀ {a} t
→ t [ sgSubst a ₛ•ₛ liftSubst σ ] ≡ t [ consSubst σ a ]
substSingletonComp = substVar-to-subst substVarSingletonComp
singleSubstComp : ∀ t (σ : Subst m n) G
→ G [ liftSubst σ ] [ t ]₀
≡ G [ consSubst σ t ]
singleSubstComp t σ G = trans (substCompEq G) (substSingletonComp G)
singleSubstWkComp : ∀ t (σ : Subst m n) G
→ wk (lift ρ) (G [ liftSubst σ ]) [ t ]₀
≡ G [ consSubst (ρ •ₛ σ) t ]
singleSubstWkComp t σ G =
trans (cong (_[ t ]₀)
(trans (wk-subst G) (subst-lift-•ₛ G)))
(trans (substCompEq G) (substSingletonComp G))
singleSubstLift : ∀ G t
→ G [ t ]₀ [ σ ]
≡ G [ liftSubst σ ] [ t [ σ ] ]₀
singleSubstLift G t =
trans (substCompEq G)
(trans (trans (substVar-to-subst (λ { x0 → refl ; (x +1) → refl}) G)
(sym (substSingletonComp G)))
(sym (substCompEq G)))
doubleSubstLift : (σ : Subst m n) (G : Term (1+ (1+ n))) (t u : Term n)
→ G [ t , u ] [ σ ]
≡ G [ liftSubstn σ 2 ] [ t [ σ ] , u [ σ ] ]
doubleSubstLift {n = n} σ G t u = begin
G [ t , u ] [ σ ]
≡⟨⟩
G [ consSubst (sgSubst t) u ] [ σ ]
≡⟨ substCompEq G ⟩
G [ σ ₛ•ₛ consSubst (sgSubst t) u ]
≡⟨ substVar-to-subst eq G ⟩
G [ (consSubst (sgSubst (t [ σ ])) (u [ σ ])) ₛ•ₛ (liftSubst (liftSubst σ)) ]
≡˘⟨ substCompEq G ⟩
G [ liftSubstn σ 2 ] [ consSubst (sgSubst (t [ σ ])) (u [ σ ]) ]
≡⟨⟩
G [ liftSubstn σ 2 ] [ t [ σ ] , u [ σ ] ] ∎
where
σ₁ = consSubst (sgSubst t) u
σ₂ = consSubst (sgSubst (t [ σ ])) (u [ σ ])
eq : (x : Fin (1+ (1+ n))) → (σ ₛ•ₛ σ₁) x ≡ (σ₂ ₛ•ₛ (liftSubstn σ 2)) x
eq x0 = refl
eq (_+1 x0) = refl
eq (x +1 +1) = begin
(σ ₛ•ₛ σ₁) (x +1 +1) ≡⟨⟩
σ x ≡˘⟨ subst-id (σ x) ⟩
(σ x) [ idSubst ] ≡⟨⟩
(σ x) [ σ₂ ₛ• (step id • step id) ] ≡˘⟨ subst-wk (σ x) ⟩
wk ((step id) • (step id)) (σ x) [ σ₂ ] ≡˘⟨ cong (_[ σ₂ ]) (wk-comp (step id) (step id) (σ x)) ⟩
wk1 (wk1 (σ x)) [ σ₂ ] ≡⟨⟩
(σ₂ ₛ•ₛ (liftSubst (liftSubst σ))) (x +1 +1) ∎
wk1-tail : (t : Term n) → wk1 t [ σ ] ≡ t [ tail σ ]
wk1-tail {σ = σ} t = begin
wk1 t [ σ ] ≡⟨⟩
wk (step id) t [ σ ] ≡⟨ subst-wk t ⟩
t [ σ ₛ• step id ] ≡⟨⟩
t [ tail σ ] ∎
wk1-tailId : (t : Term n) → wk1 t ≡ t [ tail idSubst ]
wk1-tailId t = trans (sym (subst-id (wk1 t))) (subst-wk t)
wk1-sgSubst : ∀ (t : Term n) t' → (wk1 t) [ t' ]₀ ≡ t
wk1-sgSubst t t' rewrite wk1-tailId t =
let substVar-sgSubst-tail : ∀ a n → (sgSubst a ₛ•ₛ tail idSubst) n ≡ idSubst n
substVar-sgSubst-tail a n = refl
in trans (trans
(substCompEq t)
(substVar-to-subst (substVar-sgSubst-tail t') t))
(subst-id t)
doubleSubstComp : (A : Term (1+ (1+ n))) (t u : Term m) (σ : Subst m n)
→ A [ liftSubstn σ 2 ] [ t , u ]
≡ A [ consSubst (consSubst σ t) u ]
doubleSubstComp {n = n} A t u σ = begin
A [ liftSubstn σ 2 ] [ t , u ]
≡⟨ substCompEq A ⟩
A [ consSubst (consSubst idSubst t) u ₛ•ₛ liftSubstn σ 2 ]
≡⟨ substVar-to-subst varEq A ⟩
A [ consSubst (consSubst σ t) u ] ∎
where
varEq : (x : Fin (1+ (1+ n)))
→ (consSubst (consSubst idSubst t) u ₛ•ₛ liftSubstn σ 2) x
≡ consSubst (consSubst σ t) u x
varEq x0 = refl
varEq (x0 +1) = refl
varEq (x +1 +1) = trans (wk1-tail (wk1 (σ x)))
(trans (wk1-tail (σ x)) (subst-id (σ x)))
subst-liftSubst-sgSubst :
∀ u →
u [ liftSubst (sgSubst t) ] [ liftSubst σ ] ≡
u [ liftSubstn σ 2 ] [ liftSubst (sgSubst (t [ σ ])) ]
subst-liftSubst-sgSubst {t = t} {σ = σ} u =
u [ liftSubst (sgSubst t) ] [ liftSubst σ ] ≡⟨ substCompEq u ⟩
u [ liftSubst σ ₛ•ₛ liftSubst (sgSubst t) ] ≡⟨ substVar-to-subst substCompLift u ⟩
u [ liftSubst (σ ₛ•ₛ sgSubst t) ] ≡˘⟨ substVar-to-subst
(substVar-lift λ where
x0 → refl
(_ +1) → wk1-sgSubst _ _)
u ⟩
u [ liftSubst (sgSubst (t [ σ ]) ₛ•ₛ liftSubst σ) ] ≡˘⟨ substVar-to-subst substCompLift u ⟩
u [ liftSubst (sgSubst (t [ σ ])) ₛ•ₛ liftSubst (liftSubst σ) ] ≡˘⟨ substCompEq u ⟩
u [ liftSubstn σ 2 ] [ liftSubst (sgSubst (t [ σ ])) ] ∎