first draft of proof for subst with holes
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src/Typed.lagda
160
src/Typed.lagda
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@ -216,15 +216,44 @@ erase-lemma (⊢L · ⊢M) = cong₂ _·_ (erase-lemma ⊢L) (er
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### Lists as sets
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\begin{code}
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infix 4 _∈_
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infix 4 _⊆_
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infixl 5 _∪_
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_∈_ : Id → List Id → Set
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x ∈ xs = Any (x ≡_) xs
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_⊆_ : List Id → List Id → Set
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xs ⊆ ys = ∀ {x} → x ∈ xs → x ∈ ys
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xs ⊆ ys = ∀ {x} → x ∈ xs → x ∈ ys
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∷⊆ : ∀ {x xs ys} → xs ⊆ ys → (x ∷ xs) ⊆ (x ∷ ys)
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∷⊆ xs⊆ (here refl) = here refl
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∷⊆ xs⊆ (there ∈xs) = there (xs⊆ ∈xs)
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_∪_ : List Id → List Id → List Id
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xs ∪ ys = xs ++ ys
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left : ∀ {xs ys} → xs ⊆ xs ∪ ys
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left (here refl) = here refl
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left (there x∈) = there (left x∈)
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right : ∀ {xs ys} → ys ⊆ xs ∪ ys
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right {[]} y∈ = y∈
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right {x ∷ xs} y∈ = there (right {xs} y∈)
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prev : ∀ {z y xs} → y ≢ z → z ∈ y ∷ xs → z ∈ xs
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prev y≢z (here z≡y) = ⊥-elim (y≢z (sym z≡y))
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prev _ (there z∈) = z∈
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⊆∷ : ∀ {x xs} → xs ⊆ x ∷ xs
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⊆∷ y∈ = there (y∈)
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∷⊆∷ : ∀ {x xs ys} → xs ⊆ ys → (x ∷ xs) ⊆ (x ∷ ys)
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∷⊆∷ xs⊆ (here refl) = here refl
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∷⊆∷ xs⊆ (there ∈xs) = there (xs⊆ ∈xs)
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[_]⊆ : ∀ {x xs} → [ x ] ⊆ xs → x ∈ xs
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[_]⊆ ⊆xs = ⊆xs (here refl)
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⊆[_] : ∀ {x xs} → x ∈ xs → [ x ] ⊆ xs
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⊆[_] x∈ (here refl) = x∈
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⊆[_] x∈ (there ())
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\end{code}
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### Free variables
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@ -276,7 +305,7 @@ subst xs ρ (ƛ x ⦂ A ⇒ N) = ƛ y ⦂ A ⇒ subst (y ∷ xs) (ρ , x ↦
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subst xs ρ (L · M) = subst xs ρ L · subst xs ρ M
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_[_:=_] : Term → Id → Term → Term
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N [ x := M ] = subst (free M) (∅ , x ↦ M) N
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N [ x := M ] = subst (free M ∪ free N) (∅ , x ↦ M) N
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\end{code}
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@ -373,32 +402,48 @@ free-lemma : ∀ {Γ M A} → Γ ⊢ M ⦂ A → free M ⊆ dom Γ
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free-lemma = {!!}
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\end{code}
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### Weakening
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### Renaming
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\begin{code}
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⊢weaken : ∀ {Γ Δ} → (∀ {z C} → Γ ∋ z ⦂ C → Δ ∋ z ⦂ C) →
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(∀ {M C} → Γ ⊢ M ⦂ C → Δ ⊢ M ⦂ C)
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⊢weaken ⊢σ (⌊ ⊢x ⌋) = ⌊ ⊢σ ⊢x ⌋
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⊢weaken {Γ} {Δ} ⊢σ (ƛ_ {x = x} {A = A} N)
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= ƛ (⊢weaken {Γ , x ⦂ A} {Δ , x ⦂ A} ⊢σ′ N)
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⊢rename : ∀ {Γ Δ xs} → (∀ {x A} → x ∈ xs → Γ ∋ x ⦂ A → Δ ∋ x ⦂ A) →
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(∀ {M A} → free M ⊆ xs → Γ ⊢ M ⦂ A → Δ ⊢ M ⦂ A)
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⊢rename ⊢σ ⊆xs (⌊ ⊢x ⌋) = ⌊ ⊢σ ∈xs ⊢x ⌋
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where
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⊢σ′ : ∀ {z C} → Γ , x ⦂ A ∋ z ⦂ C → Δ , x ⦂ A ∋ z ⦂ C
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⊢σ′ Z = Z
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⊢σ′ (S x≢y k) = S x≢y (⊢σ k)
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⊢weaken ⊢σ (L · M) = ⊢weaken ⊢σ L · ⊢weaken ⊢σ M
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∈xs = [_]⊆ ⊆xs
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⊢rename {Γ} {Δ} {xs} ⊢σ ⊆xs (ƛ_ {x = x} {A = A} {N = N} ⊢N)
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= ƛ (⊢rename {Γ′} {Δ′} {xs′} ⊢σ′ ⊆xs′ ⊢N)
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where
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Γ′ = Γ , x ⦂ A
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Δ′ = Δ , x ⦂ A
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xs′ = x ∷ xs
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⊢σ′ : ∀ {y B} → y ∈ xs′ → Γ′ ∋ y ⦂ B → Δ′ ∋ y ⦂ B
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⊢σ′ ∈xs′ Z = Z
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⊢σ′ ∈xs′ (S x≢y k) = S x≢y (⊢σ ∈xs k)
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where
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∈xs = {!!}
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⊆xs′ : free N ⊆ xs′
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⊆xs′ = {!!}
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⊢rename {xs = xs} ⊢σ {L · M} ⊆xs (⊢L · ⊢M) = ⊢rename ⊢σ L⊆xs ⊢L · ⊢rename ⊢σ M⊆xs ⊢M
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where
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L⊆xs : free L ⊆ xs
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L⊆xs = {!!}
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M⊆xs : free M ⊆ xs
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M⊆xs = {!!}
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\end{code}
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### Substitution preserves types
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\begin{code}
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⊢subst : ∀ {Γ Δ xs ρ} → (dom Δ ⊆ xs) →
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(∀ {x A} → Γ ∋ x ⦂ A → Δ ⊢ ρ x ⦂ A) →
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(∀ {M A} → Γ ⊢ M ⦂ A → Δ ⊢ subst xs ρ M ⦂ A)
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⊢subst ⊆xs ⊢ρ ⌊ ⊢x ⌋ = ⊢ρ ⊢x
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⊢subst {Γ} {Δ} {xs} {ρ} ⊆xs ⊢ρ (ƛ_ {x = x} {A = A} ⊢N)
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= ƛ ⊢subst {Γ′} {Δ′} {xs′} {ρ′} ⊆xs′ ⊢ρ′ ⊢N
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⊢subst : ∀ {Γ Δ xs ρ} →
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(∀ {x} → x ∈ xs → free (ρ x) ⊆ xs) →
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(∀ {x A} → x ∈ xs → Γ ∋ x ⦂ A → Δ ⊢ ρ x ⦂ A) →
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(∀ {M A} → free M ⊆ xs → Γ ⊢ M ⦂ A → Δ ⊢ subst xs ρ M ⦂ A)
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⊢subst Σ ⊢ρ ⊆xs ⌊ ⊢x ⌋ = ⊢ρ {!!} ⊢x
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⊢subst {Γ} {Δ} {xs} {ρ} Σ ⊢ρ ⊆xs (ƛ_ {x = x} {A = A} {N = N} ⊢N)
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= ƛ ⊢subst {Γ′} {Δ′} {xs′} {ρ′} Σ′ ⊢ρ′ ⊆xs′ ⊢N
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where
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y = fresh xs
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Γ′ = Γ , x ⦂ A
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@ -406,42 +451,59 @@ free-lemma = {!!}
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xs′ = y ∷ xs
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ρ′ = ρ , x ↦ ⌊ y ⌋
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⊆xs′ : dom Δ′ ⊆ xs′
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⊆xs′ = ∷⊆ ⊆xs
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Σ′ : ∀ {z} → z ∈ xs′ → free (ρ′ z) ⊆ xs′
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Σ′ (here refl) = {!!}
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Σ′ (there x∈) = {!!}
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⊆xs′ : free N ⊆ xs′
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⊆xs′ = {!!}
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y≢ : ∀ {z C} → Δ ∋ z ⦂ C → y ≢ z
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y≢ ⊢z = fresh-lemma (⊆xs (dom-lemma ⊢z))
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⊢σ : ∀ {z C} → z ∈ xs → Δ ∋ z ⦂ C → Δ′ ∋ z ⦂ C
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⊢σ z∈ ⊢z = S (fresh-lemma z∈) ⊢z
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⊢σ : ∀ {z C} → Δ ∋ z ⦂ C → Δ′ ∋ z ⦂ C
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⊢σ ⊢z = S (y≢ ⊢z) ⊢z
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⊢ρ′ : ∀ {z C} → z ∈ xs′ → Γ′ ∋ z ⦂ C → Δ′ ⊢ ρ′ z ⦂ C
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⊢ρ′ _ Z with x ≟ x
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... | yes _ = ⌊ Z ⌋
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... | no x≢x = ⊥-elim (x≢x refl)
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⊢ρ′ {z} z∈ (S x≢z ⊢z) with x ≟ z
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... | yes x≡z = ⊥-elim (x≢z x≡z)
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... | no _ = ⊢rename {Δ} {Δ′} {xs} ⊢σ (Σ (prev {!!} z∈)) (⊢ρ {!!} ⊢z)
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-- ⊢rename {Δ} {Δ′} {xs} (Σ (prev z∈)) ⊢σ (⊢ρ ? ⊢z)̄̄
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⊢ρ′ : ∀ {z C} → Γ′ ∋ z ⦂ C → Δ′ ⊢ ρ′ z ⦂ C
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⊢ρ′ Z with x ≟ x
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... | yes _ = ⌊ Z ⌋
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... | no x≢x = ⊥-elim (x≢x refl)
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⊢ρ′ {z} (S x≢z ⊢z) with x ≟ z
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... | yes x≡z = ⊥-elim (x≢z x≡z)
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... | no _ = ⊢weaken {Δ} {Δ′} ⊢σ (⊢ρ ⊢z)
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⊢subst {xs = xs} Σ ⊢ρ {L · M} ⊆xs (⊢L · ⊢M) = ⊢subst Σ ⊢ρ L⊆xs ⊢L · ⊢subst Σ ⊢ρ M⊆xs ⊢M
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where
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L⊆xs : free L ⊆ xs
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L⊆xs = {!!}
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M⊆xs : free M ⊆ xs
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M⊆xs = {!!}
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⊢subst ⊆xs ⊢ρ (⊢L · ⊢M) = ⊢subst ⊆xs ⊢ρ ⊢L · ⊢subst ⊆xs ⊢ρ ⊢M
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⊢substitution : ∀ {Γ x A M B N} →
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⊢substitution : ∀ {Γ x A N B M} →
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Γ , x ⦂ A ⊢ N ⦂ B →
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Γ ⊢ M ⦂ A →
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--------------------
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Γ ⊢ N [ x := M ] ⦂ B
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⊢substitution {Γ} {x} {A} {M} ⊢N ⊢M =
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{!!} -- ⊢subst {Γ , x ⦂ A} {Γ} {xs} {ρ} ⊢ρ ⊢N
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⊢substitution {Γ} {x} {A} {N} {B} {M} ⊢N ⊢M =
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⊢subst {Γ′} {Γ} {xs} {ρ} Σ ⊢ρ {N} {B} ⊆xs ⊢N
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where
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xs = dom Γ
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Γ′ = Γ , x ⦂ A
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xs = free M ∪ free N
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ρ = ∅ , x ↦ M
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⊢ρ : ∀ {z C} → Γ , x ⦂ A ∋ z ⦂ C → Γ ⊢ ρ z ⦂ C
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⊢ρ {.x} Z with x ≟ x
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... | yes _ = ⊢M
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... | no x≢x = ⊥-elim (x≢x refl)
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⊢ρ {z} (S x≢z ⊢x) with x ≟ z
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... | yes x≡z = ⊥-elim (x≢z x≡z)
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... | no _ = {!!}
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Σ : ∀ {x} → x ∈ xs → free (ρ x) ⊆ xs
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Σ {y} y∈ with x ≟ y
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... | no _ = ⊆[_] y∈
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... | yes _ = {!!} -- left
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⊢ρ : ∀ {z C} → z ∈ xs → Γ′ ∋ z ⦂ C → Γ ⊢ ρ z ⦂ C
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⊢ρ {.x} z∈ Z with x ≟ x
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... | yes _ = ⊢M
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... | no x≢x = ⊥-elim (x≢x refl)
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⊢ρ {z} z∈ (S x≢z ⊢z) with x ≟ z
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... | yes x≡z = ⊥-elim (x≢z x≡z)
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... | no _ = ⌊ ⊢z ⌋
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⊆xs : free N ⊆ xs
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⊆xs = {!!}
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\end{code}
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Can I falsify the theorem? Consider the case where the renamed variable
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@ -467,12 +529,14 @@ Then `y≢` in the body of `⊢subst` is falsified, which could be an issue!
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### Preservation
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\begin{code}
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{-
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preservation : ∀ {Γ M N A} → Γ ⊢ M ⦂ A → M ⟹ N → Γ ⊢ N ⦂ A
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preservation ⌊ ⊢x ⌋ ()
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preservation (ƛ ⊢N) ()
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preservation (⊢L · ⊢M) (ξ-⇒₁ L⟹L′) = preservation ⊢L L⟹L′ · ⊢M
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preservation (⊢V · ⊢M) (ξ-⇒₂ valV M⟹M′) = ⊢V · preservation ⊢M M⟹M′
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preservation ((ƛ ⊢N) · ⊢W) (β-⇒ valW) = ⊢substitution ⊢N ⊢W
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-}
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\end{code}
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