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2 changed files with 234 additions and 272 deletions
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@ -200,10 +200,10 @@ A variable `x` appears _free_ in a term `M` if `M` contains an
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occurrence of `x` that is not bound in an enclosing lambda abstraction.
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For example:
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- `x` appears free, but `f` does not, in `λ[ f ∶ (𝔹 ⇒ 𝔹) ] ` f · ` x`
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- both `f` and `x` appear free in `(λ[ f ∶ (𝔹 ⇒ 𝔹) ] ` f · ` x) · ` f`;
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indeed, `f` appears both bound and free in this term
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- no variables appear free in `λ[ f ∶ (𝔹 ⇒ 𝔹) ] λ[ x ∶ 𝔹 ] ` f · ` x`
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- Variable `x` appears free, but `f` does not, in ``λ[ f ∶ (𝔹 ⇒ 𝔹) ] ` f · ` x``.
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- Both `f` and `x` appear free in ``(λ[ f ∶ (𝔹 ⇒ 𝔹) ] ` f · ` x) · ` f``.
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Indeed, `f` appears both bound and free in this term.
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- No variables appear free in ``λ[ f ∶ (𝔹 ⇒ 𝔹) ] λ[ x ∶ 𝔹 ] ` f · ` x``.
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Formally:
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498
src/Typed.lagda
498
src/Typed.lagda
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@ -34,8 +34,6 @@ pattern [_,_,_] x y z = x ∷ y ∷ z ∷ []
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## Syntax
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\begin{code}
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infix 4 _wf
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infix 4 _∉_
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infix 4 _∋_`:_
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infix 4 _⊢_`:_
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infixl 5 _,_`:_
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@ -44,7 +42,6 @@ infix 6 `λ_`→_
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infixl 7 `if0_then_else_
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infix 8 `suc_ `pred_ `Y_
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infixl 9 _·_
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infix 10 S_
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Id : Set
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Id = String
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@ -73,14 +70,12 @@ data _∋_`:_ : Env → Id → Type → Set where
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--------------------
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→ Γ , x `: A ∋ x `: A
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S_ : ∀ {Γ A B x w}
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S : ∀ {Γ A B x w}
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→ w ≢ x
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→ Γ ∋ w `: B
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--------------------
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→ Γ , x `: A ∋ w `: B
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_∉_ : Id → Env → Set
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x ∉ Γ = ∀ {A} → ¬ (Γ ∋ x `: A)
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data _⊢_`:_ : Env → Term → Type → Set where
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Ax : ∀ {Γ A x}
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@ -89,7 +84,6 @@ data _⊢_`:_ : Env → Term → Type → Set where
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→ Γ ⊢ ` x `: A
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⊢λ : ∀ {Γ x N A B}
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→ x ∉ Γ
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→ Γ , x `: A ⊢ N `: B
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--------------------------
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→ Γ ⊢ (`λ x `→ N) `: A `→ B
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@ -125,48 +119,51 @@ data _⊢_`:_ : Env → Term → Type → Set where
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→ Γ ⊢ M `: A `→ A
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----------------
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→ Γ ⊢ `Y M `: A
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data _wf : Env → Set where
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empty :
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-----
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ε wf
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extend : ∀ {Γ x A}
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→ Γ wf
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→ x ∉ Γ
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-------------------------
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→ (Γ , x `: A) wf
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\end{code}
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## Test examples
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\begin{code}
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s≢z : "s" ≢ "z"
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s≢z ()
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n≢z : "n" ≢ "z"
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n≢z ()
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n≢s : "n" ≢ "s"
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n≢s ()
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m≢z : "m" ≢ "z"
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m≢z ()
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m≢s : "m" ≢ "s"
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m≢s ()
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m≢n : "m" ≢ "n"
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m≢n ()
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p≢n : "p" ≢ "n"
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p≢n ()
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p≢m : "p" ≢ "m"
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p≢m ()
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two : Term
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two = `suc `suc `zero
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⊢two : ε ⊢ two `: `ℕ
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⊢two = (⊢suc (⊢suc ⊢zero))
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⊢two = ⊢suc (⊢suc ⊢zero)
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plus : Term
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plus = `Y (`λ "p" `→ `λ "m" `→ `λ "n" `→ `if0 ` "m" then ` "n" else ` "p" · (`pred ` "m") · ` "n")
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plus = `Y (`λ "p" `→ `λ "m" `→ `λ "n" `→
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`if0 ` "m" then ` "n" else `suc (` "p" · (`pred (` "m")) · ` "n"))
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⊢plus : ε ⊢ plus `: `ℕ `→ `ℕ `→ `ℕ
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⊢plus = (⊢Y (⊢λ p∉ (⊢λ m∉ (⊢λ n∉
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(⊢if0 (Ax ⊢m) (Ax ⊢n) (Ax ⊢p · (⊢pred (Ax ⊢m)) · Ax ⊢n))))))
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⊢plus = (⊢Y (⊢λ (⊢λ (⊢λ (⊢if0 (Ax ⊢m) (Ax ⊢n) (⊢suc (Ax ⊢p · (⊢pred (Ax ⊢m)) · Ax ⊢n)))))))
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where
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⊢p = S S Z
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⊢m = S Z
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⊢p = S p≢n (S p≢m Z)
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⊢m = S m≢n Z
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⊢n = Z
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Γ₀ = ε
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Γ₁ = Γ₀ , "p" `: `ℕ `→ `ℕ `→ `ℕ
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Γ₂ = Γ₁ , "m" `: `ℕ
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p∉ : "p" ∉ Γ₀
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p∉ ()
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m∉ : "m" ∉ Γ₁
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m∉ (S ())
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n∉ : "n" ∉ Γ₂
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n∉ (S S ())
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four : Term
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four = plus · two · two
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@ -181,55 +178,31 @@ twoCh : Term
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twoCh = `λ "s" `→ `λ "z" `→ (` "s" · (` "s" · ` "z"))
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⊢twoCh : ε ⊢ twoCh `: Ch
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⊢twoCh = (⊢λ s∉ (⊢λ z∉ (Ax ⊢s · (Ax ⊢s · Ax ⊢z))))
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⊢twoCh = (⊢λ (⊢λ (Ax ⊢s · (Ax ⊢s · Ax ⊢z))))
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where
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⊢s = S Z
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⊢s = S s≢z Z
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⊢z = Z
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Γ₀ = ε
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Γ₁ = Γ₀ , "s" `: `ℕ `→ `ℕ
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s∉ : "s" ∉ ε
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s∉ ()
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z∉ : "z" ∉ Γ₁
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z∉ (S ())
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plusCh : Term
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plusCh = `λ "m" `→ `λ "n" `→ `λ "s" `→ `λ "z" `→
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` "m" · ` "s" · (` "n" · ` "s" · ` "z")
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⊢plusCh : ε ⊢ plusCh `: Ch `→ Ch `→ Ch
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⊢plusCh = (⊢λ m∉ (⊢λ n∉ (⊢λ s∉ (⊢λ z∉ (Ax ⊢m · Ax ⊢s · (Ax ⊢n · Ax ⊢s · Ax ⊢z))))))
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⊢plusCh = (⊢λ (⊢λ (⊢λ (⊢λ (Ax ⊢m · Ax ⊢s · (Ax ⊢n · Ax ⊢s · Ax ⊢z))))))
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where
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⊢m = S S S Z
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⊢n = S S Z
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⊢s = S Z
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⊢m = S m≢z (S m≢s (S m≢n Z))
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⊢n = S n≢z (S n≢s Z)
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⊢s = S s≢z Z
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⊢z = Z
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Γ₀ = ε
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Γ₁ = Γ₀ , "m" `: Ch
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Γ₂ = Γ₁ , "n" `: Ch
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Γ₃ = Γ₂ , "s" `: `ℕ `→ `ℕ
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m∉ : "m" ∉ Γ₀
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m∉ ()
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n∉ : "n" ∉ Γ₁
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n∉ (S ())
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s∉ : "s" ∉ Γ₂
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s∉ (S S ())
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z∉ : "z" ∉ Γ₃
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z∉ (S S S ())
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fromCh : Term
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fromCh = `λ "m" `→ ` "m" · (`λ "s" `→ `suc ` "s") · `zero
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⊢fromCh : ε ⊢ fromCh `: Ch `→ `ℕ
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⊢fromCh = (⊢λ m∉ (Ax ⊢m · (⊢λ s∉ (⊢suc (Ax ⊢s))) · ⊢zero))
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⊢fromCh = (⊢λ (Ax ⊢m · (⊢λ (⊢suc (Ax ⊢s))) · ⊢zero))
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where
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⊢m = Z
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⊢s = Z
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Γ₀ = ε
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Γ₁ = Γ₀ , "m" `: Ch
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m∉ : "m" ∉ Γ₀
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m∉ ()
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s∉ : "s" ∉ Γ₁
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s∉ (S ())
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fourCh : Term
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fourCh = fromCh · (plusCh · twoCh · twoCh)
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@ -243,18 +216,18 @@ fourCh = fromCh · (plusCh · twoCh · twoCh)
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\begin{code}
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lookup : ∀ {Γ x A} → Γ ∋ x `: A → Id
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lookup {Γ , x `: A} Z = x
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lookup {Γ , x `: A} (S ⊢w) = lookup {Γ} ⊢w
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lookup {Γ , x `: A} Z = x
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lookup {Γ , x `: A} (S w≢ ⊢w) = lookup {Γ} ⊢w
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erase : ∀ {Γ M A} → Γ ⊢ M `: A → Term
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erase (Ax ⊢w) = ` lookup ⊢w
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erase (⊢λ {x = x} x∉ ⊢N) = `λ x `→ erase ⊢N
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erase (⊢L · ⊢M) = erase ⊢L · erase ⊢M
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erase (⊢zero) = `zero
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erase (⊢suc ⊢M) = `suc (erase ⊢M)
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erase (⊢pred ⊢M) = `pred (erase ⊢M)
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erase (⊢if0 ⊢L ⊢M ⊢N) = `if0 (erase ⊢L) then (erase ⊢M) else (erase ⊢N)
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erase (⊢Y ⊢M) = `Y (erase ⊢M)
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erase (Ax ⊢w) = ` lookup ⊢w
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erase (⊢λ {x = x} ⊢N) = `λ x `→ erase ⊢N
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erase (⊢L · ⊢M) = erase ⊢L · erase ⊢M
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erase (⊢zero) = `zero
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erase (⊢suc ⊢M) = `suc (erase ⊢M)
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erase (⊢pred ⊢M) = `pred (erase ⊢M)
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erase (⊢if0 ⊢L ⊢M ⊢N) = `if0 (erase ⊢L) then (erase ⊢M) else (erase ⊢N)
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erase (⊢Y ⊢M) = `Y (erase ⊢M)
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\end{code}
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### Properties of erasure
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@ -265,19 +238,19 @@ cong₃ : ∀ {A B C D : Set} (f : A → B → C → D) {s t u v x y} →
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cong₃ f refl refl refl = refl
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lookup-lemma : ∀ {Γ x A} → (⊢x : Γ ∋ x `: A) → lookup ⊢x ≡ x
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lookup-lemma Z = refl
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lookup-lemma (S ⊢w) = lookup-lemma ⊢w
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lookup-lemma Z = refl
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lookup-lemma (S w≢ ⊢w) = lookup-lemma ⊢w
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erase-lemma : ∀ {Γ M A} → (⊢M : Γ ⊢ M `: A) → erase ⊢M ≡ M
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erase-lemma (Ax ⊢x) = cong `_ (lookup-lemma ⊢x)
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erase-lemma (⊢λ {x = x} x∉ ⊢N) = cong (`λ x `→_) (erase-lemma ⊢N)
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erase-lemma (⊢L · ⊢M) = cong₂ _·_ (erase-lemma ⊢L) (erase-lemma ⊢M)
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erase-lemma (⊢zero) = refl
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erase-lemma (⊢suc ⊢M) = cong `suc_ (erase-lemma ⊢M)
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erase-lemma (⊢pred ⊢M) = cong `pred_ (erase-lemma ⊢M)
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erase-lemma (⊢if0 ⊢L ⊢M ⊢N) = cong₃ `if0_then_else_
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(erase-lemma ⊢L) (erase-lemma ⊢M) (erase-lemma ⊢N)
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erase-lemma (⊢Y ⊢M) = cong `Y_ (erase-lemma ⊢M)
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erase-lemma (Ax ⊢x) = cong `_ (lookup-lemma ⊢x)
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erase-lemma (⊢λ {x = x} ⊢N) = cong (`λ x `→_) (erase-lemma ⊢N)
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erase-lemma (⊢L · ⊢M) = cong₂ _·_ (erase-lemma ⊢L) (erase-lemma ⊢M)
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erase-lemma (⊢zero) = refl
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erase-lemma (⊢suc ⊢M) = cong `suc_ (erase-lemma ⊢M)
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erase-lemma (⊢pred ⊢M) = cong `pred_ (erase-lemma ⊢M)
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erase-lemma (⊢if0 ⊢L ⊢M ⊢N) = cong₃ `if0_then_else_
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(erase-lemma ⊢L) (erase-lemma ⊢M) (erase-lemma ⊢N)
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erase-lemma (⊢Y ⊢M) = cong `Y_ (erase-lemma ⊢M)
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\end{code}
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@ -349,7 +322,7 @@ _ : (`λ "m" `→ ` "m" · ` "n") [ "n" := ` "p" · ` "q" ]
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≡ `λ "m" `→ ` "m" · (` "p" · ` "q")
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_ = refl
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_ : subst (∅ , "m" ↦ ` "p" , "n" ↦ ` "q") (` "m" · ` "n") ≡ (` "p" · ` "q")
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_ : subst (∅ , "m" ↦ two , "n" ↦ four) (` "m" · ` "n") ≡ (two · four)
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_ = refl
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\end{code}
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@ -434,11 +407,10 @@ data _⟶_ : Term → Term → Set where
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---------------
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→ `Y M ⟶ `Y M′
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β-Y : ∀ {V x N}
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→ Value V
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→ V ≡ `λ x `→ N
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β-Y : ∀ {F x N}
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→ F ≡ `λ x `→ N
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-------------------------
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→ `Y V ⟶ N [ x := `Y V ]
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→ `Y F ⟶ N [ x := `Y F ]
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\end{code}
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## Reflexive and transitive closure
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@ -497,7 +469,7 @@ canonical : ∀ {V A}
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→ Canonical V A
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canonical ⊢zero Zero = Zero
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canonical (⊢suc ⊢V) (Suc VV) = Suc (canonical ⊢V VV)
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canonical (⊢λ x∉ ⊢N) Fun = Fun ⊢N
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canonical (⊢λ ⊢N) Fun = Fun ⊢N
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\end{code}
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Every canonical form has a type and a value.
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@ -509,10 +481,7 @@ type : ∀ {V A}
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→ ε ⊢ V `: A
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type Zero = ⊢zero
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type (Suc CV) = ⊢suc (type CV)
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type (Fun {x = x} ⊢N) = ⊢λ x∉ ⊢N
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where
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x∉ : x ∉ ε
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x∉ ()
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type (Fun {x = x} ⊢N) = ⊢λ ⊢N
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value : ∀ {V A}
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→ Canonical V A
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@ -523,6 +492,54 @@ value (Suc CV) = Suc (value CV)
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value (Fun ⊢N) = Fun
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\end{code}
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## Values do not reduce
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Values do not reduce.
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\begin{code}
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Val-⟶ : ∀ {M N} → Value M → ¬ (M ⟶ N)
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Val-⟶ Fun ()
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Val-⟶ Zero ()
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Val-⟶ (Suc VM) (ξ-suc M⟶N) = Val-⟶ VM M⟶N
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\end{code}
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As a corollary, terms that reduce are not values.
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\begin{code}
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⟶-Val : ∀ {M N} → (M ⟶ N) → ¬ Value M
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⟶-Val M⟶N VM = Val-⟶ VM M⟶N
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\end{code}
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## Reduction is deterministic
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\begin{code}
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det : ∀ {M M′ M″}
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→ (M ⟶ M′)
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→ (M ⟶ M″)
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----------
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→ M′ ≡ M″
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det (ξ-·₁ L⟶L′) (ξ-·₁ L⟶L″) = cong₂ _·_ (det L⟶L′ L⟶L″) refl
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det (ξ-·₁ L⟶L′) (ξ-·₂ VL _) = ⊥-elim (Val-⟶ VL L⟶L′)
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det (ξ-·₁ L⟶L′) (β-`→ _) = ⊥-elim (Val-⟶ Fun L⟶L′)
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det (ξ-·₂ VL _) (ξ-·₁ L⟶L″) = ⊥-elim (Val-⟶ VL L⟶L″)
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det (ξ-·₂ _ M⟶M′) (ξ-·₂ _ M⟶M″) = cong₂ _·_ refl (det M⟶M′ M⟶M″)
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det (ξ-·₂ _ M⟶M′) (β-`→ VM) = ⊥-elim (Val-⟶ VM M⟶M′)
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det (β-`→ VM) (ξ-·₁ L⟶L″) = ⊥-elim (Val-⟶ Fun L⟶L″)
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det (β-`→ VM) (ξ-·₂ _ M⟶M″) = ⊥-elim (Val-⟶ VM M⟶M″)
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det (β-`→ _) (β-`→ _) = refl
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det (ξ-suc M⟶M′) (ξ-suc M⟶M″) = cong `suc_ (det M⟶M′ M⟶M″)
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det (ξ-pred M⟶M′) (ξ-pred M⟶M″) = cong `pred_ (det M⟶M′ M⟶M″)
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det (ξ-pred M⟶M′) β-pred-zero = {!!}
|
||||
det (ξ-pred M⟶M′) (β-pred-suc x) = {!!}
|
||||
det β-pred-zero L⟶N = {!!}
|
||||
det (β-pred-suc x) L⟶N = {!!}
|
||||
det (ξ-if0 L⟶M) L⟶N = {!!}
|
||||
det β-if0-zero L⟶N = {!!}
|
||||
det (β-if0-suc x) L⟶N = {!!}
|
||||
det (ξ-Y L⟶M) L⟶N = {!!}
|
||||
det (β-Y x₁) L⟶N = {!!}
|
||||
|
||||
\end{code}
|
||||
|
||||
## Progress
|
||||
|
||||
\begin{code}
|
||||
|
|
@ -538,7 +555,7 @@ data Progress (M : Term) (A : Type) : Set where
|
|||
|
||||
progress : ∀ {M A} → ε ⊢ M `: A → Progress M A
|
||||
progress (Ax ())
|
||||
progress (⊢λ x∉ ⊢N) = done (Fun ⊢N)
|
||||
progress (⊢λ ⊢N) = done (Fun ⊢N)
|
||||
progress (⊢L · ⊢M) with progress ⊢L
|
||||
... | step L⟶L′ = step (ξ-·₁ L⟶L′)
|
||||
... | done (Fun _) with progress ⊢M
|
||||
|
|
@ -558,7 +575,7 @@ progress (⊢if0 ⊢L ⊢M ⊢N) with progress ⊢L
|
|||
... | done (Suc CM) = step (β-if0-suc (value CM))
|
||||
progress (⊢Y ⊢M) with progress ⊢M
|
||||
... | step M⟶M′ = step (ξ-Y M⟶M′)
|
||||
... | done (Fun _) = step (β-Y Fun refl)
|
||||
... | done (Fun _) = step (β-Y refl)
|
||||
\end{code}
|
||||
|
||||
|
||||
|
|
@ -599,204 +616,104 @@ free-lemma (⊢Y ⊢M) w∈ = free-lemma ⊢M w∈
|
|||
|
||||
### Renaming
|
||||
|
||||
Let's try an example. The result I want to prove is:
|
||||
|
||||
⊢subst : ∀ {Γ Δ ρ}
|
||||
→ (∀ {x A} → Γ ∋ x `: A → Δ ⊢ ρ x `: A)
|
||||
-----------------------------------------------
|
||||
→ (∀ {M A} → Γ ⊢ M `: A → Δ ⊢ subst ρ M `: A)
|
||||
|
||||
For this to work, I need to know that neither `Δ` or any of the
|
||||
bound variables in `ρ x` will collide with any bound variable in `M`.
|
||||
How can I establish this?
|
||||
|
||||
In particular, I need to check that the conditions for ordinary
|
||||
substitution are sufficient to establish the required invariants.
|
||||
In that case we have:
|
||||
|
||||
⊢substitution : ∀ {Γ x A N B M} →
|
||||
Γ , x `: A ⊢ N `: B →
|
||||
Γ ⊢ M `: A →
|
||||
--------------------
|
||||
Γ ⊢ N [ x := M ] `: B
|
||||
|
||||
Here, since `N` is well-typed, none of it's bound variables collide
|
||||
with `Γ`, and hence cannot collide with any free variable of `M`.
|
||||
*But* we can't make a similar guarantee for the *bound* variables
|
||||
of `M`, so substitution may break the invariants. Here are examples:
|
||||
|
||||
(`λ "x" `→ `λ "y" `→ ` "x") (`λ "y" `→ ` "y")
|
||||
⟶
|
||||
(`λ "y" → (`λ "y" `→ ` "y"))
|
||||
|
||||
ε , "z" `: `ℕ ⊢ (`λ "x" `→ `λ "y" → ` "x" · ` "y" · ` "z") (`λ "y" `→ ` "y" · ` "z")
|
||||
⟶
|
||||
ε , "z" `: `ℕ ⊢ (`λ "y" → (`λ "y" `→ ` "y" · ` "z") · ` "y" · ` "z")
|
||||
|
||||
This doesn't maintain the invariant, but doesn't break either.
|
||||
But I don't know how to prove it never breaks. Maybe I can come
|
||||
up with an example that does break after a few steps. Or, maybe
|
||||
I don't need the nested variables to be unique. Maybe all I need
|
||||
is for the free variables in each `ρ x` to be distinct from any
|
||||
of the bound variables in `N`. But this requires every bound
|
||||
variable in `N` to not appear in `Γ`. Not clear how to maintain
|
||||
such a condition without the invariant, so I don't know how
|
||||
the proof works. Bugger!
|
||||
|
||||
Consider a term with free variables, where every bound
|
||||
variable of the term is distinct from any free variable.
|
||||
(This is trivially true for a closed term.) Question: if
|
||||
I never reduce under lambda, do I ever need
|
||||
to perform renaming?
|
||||
|
||||
It's easy to come up with a counter-example if I allow
|
||||
reduction under lambda.
|
||||
|
||||
(λ y → (λ x → λ y → x y) y) ⟶ (λ y → (λ y′ → y y′))
|
||||
|
||||
The above requires renaming. But if I remove the outer lambda
|
||||
|
||||
(λ x → λ y → x y) y ⟶ (λ y → (λ y′ → y y′))
|
||||
|
||||
then the term on the left violates the condition on free
|
||||
variables, and any term I can think of that causes problems
|
||||
also violates the condition. So I may be able to do something
|
||||
here.
|
||||
|
||||
|
||||
\begin{code}
|
||||
{-
|
||||
⊢rename : ∀ {Γ Δ xs}
|
||||
⊢rename : ∀ {Γ Δ}
|
||||
→ (∀ {x A} → Γ ∋ x `: A → Δ ∋ x `: A)
|
||||
--------------------------------------------------
|
||||
→ (∀ {M A} → Γ ⊢ M `: A → Δ ⊢ M `: A)
|
||||
⊢rename ⊢σ (Ax ⊢x) = Ax (⊢σ ⊢x)
|
||||
⊢rename {Γ} {Δ} ⊢σ (⊢λ {x = x} {N = N} {A = A} x∉Γ ⊢N)
|
||||
= ⊢λ x∉Δ (⊢rename {Γ′} {Δ′} ⊢σ′ ⊢N)
|
||||
⊢rename ⊢σ (Ax ⊢x) = Ax (⊢σ ⊢x)
|
||||
⊢rename {Γ} {Δ} ⊢σ (⊢λ {x = x} {N = N} {A = A} ⊢N)
|
||||
= ⊢λ (⊢rename {Γ′} {Δ′} ⊢σ′ ⊢N)
|
||||
where
|
||||
Γ′ = Γ , x `: A
|
||||
Δ′ = Δ , x `: A
|
||||
xs′ = x ∷ xs
|
||||
|
||||
⊢σ′ : ∀ {w B} → w ∈ xs′ → Γ′ ∋ w `: B → Δ′ ∋ w `: B
|
||||
⊢σ′ w∈′ Z = Z
|
||||
⊢σ′ w∈′ (S w≢ ⊢w) = S w≢ (⊢σ ∈w ⊢w)
|
||||
where
|
||||
∈w = there⁻¹ w∈′ w≢
|
||||
⊢σ′ : ∀ {w B} → Γ′ ∋ w `: B → Δ′ ∋ w `: B
|
||||
⊢σ′ Z = Z
|
||||
⊢σ′ (S w≢ ⊢w) = S w≢ (⊢σ ⊢w)
|
||||
|
||||
⊆xs′ : free N ⊆ xs′
|
||||
⊆xs′ = \\-to-∷ ⊆xs
|
||||
⊢rename ⊢σ ⊆xs (⊢L · ⊢M) = ⊢rename ⊢σ L⊆ ⊢L · ⊢rename ⊢σ M⊆ ⊢M
|
||||
where
|
||||
L⊆ = trans-⊆ ⊆-++₁ ⊆xs
|
||||
M⊆ = trans-⊆ ⊆-++₂ ⊆xs
|
||||
⊢rename ⊢σ ⊆xs (⊢zero) = ⊢zero
|
||||
⊢rename ⊢σ ⊆xs (⊢suc ⊢M) = ⊢suc (⊢rename ⊢σ ⊆xs ⊢M)
|
||||
⊢rename ⊢σ ⊆xs (⊢pred ⊢M) = ⊢pred (⊢rename ⊢σ ⊆xs ⊢M)
|
||||
⊢rename ⊢σ ⊆xs (⊢if0 {L = L} ⊢L ⊢M ⊢N)
|
||||
= ⊢if0 (⊢rename ⊢σ L⊆ ⊢L) (⊢rename ⊢σ M⊆ ⊢M) (⊢rename ⊢σ N⊆ ⊢N)
|
||||
where
|
||||
L⊆ = trans-⊆ ⊆-++₁ ⊆xs
|
||||
M⊆ = trans-⊆ ⊆-++₁ (trans-⊆ (⊆-++₂ {free L}) ⊆xs)
|
||||
N⊆ = trans-⊆ ⊆-++₂ (trans-⊆ (⊆-++₂ {free L}) ⊆xs)
|
||||
⊢rename ⊢σ ⊆xs (⊢Y ⊢M) = ⊢Y (⊢rename ⊢σ ⊆xs ⊢M)
|
||||
-}
|
||||
⊢rename ⊢σ (⊢L · ⊢M) = ⊢rename ⊢σ ⊢L · ⊢rename ⊢σ ⊢M
|
||||
⊢rename ⊢σ (⊢zero) = ⊢zero
|
||||
⊢rename ⊢σ (⊢suc ⊢M) = ⊢suc (⊢rename ⊢σ ⊢M)
|
||||
⊢rename ⊢σ (⊢pred ⊢M) = ⊢pred (⊢rename ⊢σ ⊢M)
|
||||
⊢rename ⊢σ (⊢if0 ⊢L ⊢M ⊢N)
|
||||
= ⊢if0 (⊢rename ⊢σ ⊢L) (⊢rename ⊢σ ⊢M) (⊢rename ⊢σ ⊢N)
|
||||
⊢rename ⊢σ (⊢Y ⊢M) = ⊢Y (⊢rename ⊢σ ⊢M)
|
||||
\end{code}
|
||||
|
||||
|
||||
### Substitution preserves types
|
||||
|
||||
\begin{code}
|
||||
{-
|
||||
⊢subst : ∀ {Γ Δ xs ys ρ}
|
||||
→ (∀ {x} → x ∈ xs → free (ρ x) ⊆ ys)
|
||||
→ (∀ {x A} → x ∈ xs → Γ ∋ x `: A → Δ ⊢ ρ x `: A)
|
||||
-------------------------------------------------------------
|
||||
→ (∀ {M A} → free M ⊆ xs → Γ ⊢ M `: A → Δ ⊢ subst ys ρ M `: A)
|
||||
⊢subst Σ ⊢ρ ⊆xs (Ax ⊢x)
|
||||
= ⊢ρ (⊆xs here) ⊢x
|
||||
⊢subst {Γ} {Δ} {xs} {ys} {ρ} Σ ⊢ρ ⊆xs (⊢λ {x = x} {N = N} {A = A} ⊢N)
|
||||
= ⊢λ {x = y} {A = A} (⊢subst {Γ′} {Δ′} {xs′} {ys′} {ρ′} Σ′ ⊢ρ′ ⊆xs′ ⊢N)
|
||||
⊢subst : ∀ {Γ Δ ρ}
|
||||
→ (∀ {x A} → Γ ∋ x `: A → Δ ⊢ ρ x `: A)
|
||||
-------------------------------------------------
|
||||
→ (∀ {M A} → Γ ⊢ M `: A → Δ ⊢ subst ρ M `: A)
|
||||
⊢subst ⊢ρ (Ax ⊢x) = ⊢ρ ⊢x
|
||||
⊢subst {Γ} {Δ} {ρ} ⊢ρ (⊢λ {x = x} {N = N} {A = A} ⊢N)
|
||||
= ⊢λ {x = x} {A = A} (⊢subst {Γ′} {Δ′} {ρ′} ⊢ρ′ ⊢N)
|
||||
where
|
||||
y = fresh ys
|
||||
Γ′ = Γ , x `: A
|
||||
Δ′ = Δ , y `: A
|
||||
xs′ = x ∷ xs
|
||||
ys′ = y ∷ ys
|
||||
ρ′ = ρ , x ↦ ` y
|
||||
Δ′ = Δ , x `: A
|
||||
ρ′ = ρ , x ↦ ` x
|
||||
|
||||
Σ′ : ∀ {w} → w ∈ xs′ → free (ρ′ w) ⊆ ys′
|
||||
Σ′ {w} w∈′ with w ≟ x
|
||||
... | yes refl = ⊆-++₁
|
||||
... | no w≢ = ⊆-++₂ ∘ Σ (there⁻¹ w∈′ w≢)
|
||||
|
||||
⊆xs′ : free N ⊆ xs′
|
||||
⊆xs′ = \\-to-∷ ⊆xs
|
||||
⊢σ : ∀ {w C} → Δ ∋ w `: C → Δ′ ∋ w `: C
|
||||
⊢σ ⊢w = S {!!} ⊢w
|
||||
|
||||
⊢σ : ∀ {w C} → w ∈ ys → Δ ∋ w `: C → Δ′ ∋ w `: C
|
||||
⊢σ w∈ ⊢w = S (fresh-lemma w∈) ⊢w
|
||||
|
||||
⊢ρ′ : ∀ {w C} → w ∈ xs′ → Γ′ ∋ w `: C → Δ′ ⊢ ρ′ w `: C
|
||||
⊢ρ′ {w} _ Z with w ≟ x
|
||||
⊢ρ′ : ∀ {w C} → Γ′ ∋ w `: C → Δ′ ⊢ ρ′ w `: C
|
||||
⊢ρ′ {w} Z with w ≟ x
|
||||
... | yes _ = Ax Z
|
||||
... | no w≢ = ⊥-elim (w≢ refl)
|
||||
⊢ρ′ {w} w∈′ (S w≢ ⊢w) with w ≟ x
|
||||
⊢ρ′ {w} (S w≢ ⊢w) with w ≟ x
|
||||
... | yes refl = ⊥-elim (w≢ refl)
|
||||
... | no _ = ⊢rename {Δ} {Δ′} {ys} ⊢σ (Σ w∈) (⊢ρ w∈ ⊢w)
|
||||
where
|
||||
w∈ = there⁻¹ w∈′ w≢
|
||||
... | no _ = ⊢rename {Δ} {Δ′} ⊢σ (⊢ρ ⊢w)
|
||||
|
||||
⊢subst Σ ⊢ρ ⊆xs (⊢L · ⊢M)
|
||||
= ⊢subst Σ ⊢ρ L⊆ ⊢L · ⊢subst Σ ⊢ρ M⊆ ⊢M
|
||||
where
|
||||
L⊆ = trans-⊆ ⊆-++₁ ⊆xs
|
||||
M⊆ = trans-⊆ ⊆-++₂ ⊆xs
|
||||
⊢subst Σ ⊢ρ ⊆xs ⊢zero = ⊢zero
|
||||
⊢subst Σ ⊢ρ ⊆xs (⊢suc ⊢M) = ⊢suc (⊢subst Σ ⊢ρ ⊆xs ⊢M)
|
||||
⊢subst Σ ⊢ρ ⊆xs (⊢pred ⊢M) = ⊢pred (⊢subst Σ ⊢ρ ⊆xs ⊢M)
|
||||
⊢subst Σ ⊢ρ ⊆xs (⊢if0 {L = L} ⊢L ⊢M ⊢N)
|
||||
= ⊢if0 (⊢subst Σ ⊢ρ L⊆ ⊢L) (⊢subst Σ ⊢ρ M⊆ ⊢M) (⊢subst Σ ⊢ρ N⊆ ⊢N)
|
||||
where
|
||||
L⊆ = trans-⊆ ⊆-++₁ ⊆xs
|
||||
M⊆ = trans-⊆ ⊆-++₁ (trans-⊆ (⊆-++₂ {free L}) ⊆xs)
|
||||
N⊆ = trans-⊆ ⊆-++₂ (trans-⊆ (⊆-++₂ {free L}) ⊆xs)
|
||||
⊢subst Σ ⊢ρ ⊆xs (⊢Y ⊢M) = ⊢Y (⊢subst Σ ⊢ρ ⊆xs ⊢M)
|
||||
⊢subst ⊢ρ (⊢L · ⊢M) = ⊢subst ⊢ρ ⊢L · ⊢subst ⊢ρ ⊢M
|
||||
⊢subst ⊢ρ ⊢zero = ⊢zero
|
||||
⊢subst ⊢ρ (⊢suc ⊢M) = ⊢suc (⊢subst ⊢ρ ⊢M)
|
||||
⊢subst ⊢ρ (⊢pred ⊢M) = ⊢pred (⊢subst ⊢ρ ⊢M)
|
||||
⊢subst ⊢ρ (⊢if0 ⊢L ⊢M ⊢N)
|
||||
= ⊢if0 (⊢subst ⊢ρ ⊢L) (⊢subst ⊢ρ ⊢M) (⊢subst ⊢ρ ⊢N)
|
||||
⊢subst ⊢ρ (⊢Y ⊢M) = ⊢Y (⊢subst ⊢ρ ⊢M)
|
||||
\end{code}
|
||||
|
||||
⊢substitution : ∀ {Γ x A N B M} →
|
||||
Γ , x `: A ⊢ N `: B →
|
||||
Γ ⊢ M `: A →
|
||||
--------------------
|
||||
Γ ⊢ N [ x := M ] `: B
|
||||
Let's look at examples. Assume `M` is closed. Example 1.
|
||||
|
||||
subst (∅ , "x" ↦ M) (`λ "y" `→ ` "x") ≡ `λ "y" `→ M
|
||||
|
||||
Example 2.
|
||||
|
||||
subst (∅ , "y" ↦ ` "y" , "x" ↦ M) (`λ "y" `→ ` "x" · ` "y")
|
||||
≡
|
||||
`λ "y" `→ subst (∅ , "y" ↦ ` "y" , "x" ↦ M , "y" ↦ ` "y") (` "x" · ` "y")
|
||||
≡
|
||||
`λ "y" `→ (M · ` "y")
|
||||
|
||||
The hypotheses of the theorem appear to be violated. Drat!
|
||||
|
||||
\begin{code}
|
||||
⊢substitution : ∀ {Γ x A N B M}
|
||||
→ Γ , x `: A ⊢ N `: B
|
||||
→ Γ ⊢ M `: A
|
||||
----------------------
|
||||
→ Γ ⊢ N [ x := M ] `: B
|
||||
⊢substitution {Γ} {x} {A} {N} {B} {M} ⊢N ⊢M =
|
||||
⊢subst {Γ′} {Γ} {xs} {ys} {ρ} Σ ⊢ρ {N} {B} ⊆xs ⊢N
|
||||
⊢subst {Γ′} {Γ} {ρ} ⊢ρ {N} {B} ⊢N
|
||||
where
|
||||
Γ′ = Γ , x `: A
|
||||
xs = free N
|
||||
ys = free M ++ (free N \\ x)
|
||||
ρ = ∅ , x ↦ M
|
||||
|
||||
Σ : ∀ {w} → w ∈ xs → free (ρ w) ⊆ ys
|
||||
Σ {w} w∈ y∈ with w ≟ x
|
||||
... | yes _ = ⊆-++₁ y∈
|
||||
... | no w≢ rewrite ∈-[_] y∈ = ⊆-++₂ (∈-≢-to-\\ w∈ w≢)
|
||||
|
||||
⊢ρ : ∀ {w B} → w ∈ xs → Γ′ ∋ w `: B → Γ ⊢ ρ w `: B
|
||||
⊢ρ {w} w∈ Z with w ≟ x
|
||||
... | yes _ = ⊢M
|
||||
... | no w≢ = ⊥-elim (w≢ refl)
|
||||
⊢ρ {w} w∈ (S w≢ ⊢w) with w ≟ x
|
||||
... | yes refl = ⊥-elim (w≢ refl)
|
||||
... | no _ = Ax ⊢w
|
||||
|
||||
⊆xs : free N ⊆ xs
|
||||
⊆xs x∈ = x∈
|
||||
-}
|
||||
⊢ρ : ∀ {w B} → Γ′ ∋ w `: B → Γ ⊢ ρ w `: B
|
||||
⊢ρ {w} Z with w ≟ x
|
||||
... | yes _ = ⊢M
|
||||
... | no w≢ = ⊥-elim (w≢ refl)
|
||||
⊢ρ {w} (S w≢ ⊢w) with w ≟ x
|
||||
... | yes refl = ⊥-elim (w≢ refl)
|
||||
... | no _ = Ax ⊢w
|
||||
\end{code}
|
||||
|
||||
### Preservation
|
||||
|
||||
\begin{code}
|
||||
{-
|
||||
preservation : ∀ {Γ M N A}
|
||||
→ Γ ⊢ M `: A
|
||||
→ M ⟶ N
|
||||
|
|
@ -816,14 +733,12 @@ preservation (⊢if0 ⊢L ⊢M ⊢N) (ξ-if0 L⟶) = ⊢if0 (preservat
|
|||
preservation (⊢if0 ⊢zero ⊢M ⊢N) β-if0-zero = ⊢M
|
||||
preservation (⊢if0 (⊢suc ⊢V) ⊢M ⊢N) (β-if0-suc _) = ⊢N
|
||||
preservation (⊢Y ⊢M) (ξ-Y M⟶) = ⊢Y (preservation ⊢M M⟶)
|
||||
preservation (⊢Y (⊢λ ⊢N)) (β-Y _ refl) = ⊢substitution ⊢N (⊢Y (⊢λ ⊢N))
|
||||
-}
|
||||
preservation (⊢Y (⊢λ ⊢N)) (β-Y refl) = ⊢substitution ⊢N (⊢Y (⊢λ ⊢N))
|
||||
\end{code}
|
||||
|
||||
## Normalise
|
||||
|
||||
\begin{code}
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{-
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data Normalise {M A} (⊢M : ε ⊢ M `: A) : Set where
|
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out-of-gas : ∀ {N} → M ⟶* N → ε ⊢ N `: A → Normalise ⊢M
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normal : ∀ {V} → ℕ → Canonical V A → M ⟶* V → Normalise ⊢M
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@ -836,6 +751,53 @@ normalise {L} (suc m) ⊢L with progress ⊢L
|
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... | ⊢M with normalise m ⊢M
|
||||
... | out-of-gas M⟶*N ⊢N = out-of-gas (L ⟶⟨ L⟶M ⟩ M⟶*N) ⊢N
|
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... | normal n CV M⟶*V = normal n CV (L ⟶⟨ L⟶M ⟩ M⟶*V)
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-}
|
||||
\end{code}
|
||||
|
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## Sample execution
|
||||
|
||||
\begin{code}
|
||||
_ : plus · two · two ⟶* (`suc (`suc (`suc (`suc `zero))))
|
||||
_ =
|
||||
begin
|
||||
plus · two · two
|
||||
⟶⟨ ξ-·₁ (ξ-·₁ (β-Y refl)) ⟩
|
||||
(`λ "m" `→ (`λ "n" `→ `if0 ` "m" then ` "n" else
|
||||
`suc (plus · (`pred (` "m")) · (` "n")))) · two · two
|
||||
⟶⟨ ξ-·₁ (β-`→ (Suc (Suc Zero))) ⟩
|
||||
(`λ "n" `→ `if0 two then ` "n" else
|
||||
`suc (plus · (`pred two) · (` "n"))) · two
|
||||
⟶⟨ β-`→ (Suc (Suc Zero)) ⟩
|
||||
`if0 two then two else
|
||||
`suc (plus · (`pred two) · two)
|
||||
⟶⟨ β-if0-suc (Suc Zero) ⟩
|
||||
`suc (plus · (`pred two) · two)
|
||||
⟶⟨ ξ-suc (ξ-·₁ (ξ-·₁ (β-Y refl))) ⟩
|
||||
`suc ((`λ "m" `→ (`λ "n" `→ `if0 ` "m" then ` "n" else
|
||||
`suc (plus · (`pred (` "m")) · (` "n")))) · (`pred two) · two)
|
||||
⟶⟨ ξ-suc (ξ-·₁ (ξ-·₂ Fun (β-pred-suc (Suc Zero)))) ⟩
|
||||
`suc ((`λ "m" `→ (`λ "n" `→ `if0 ` "m" then ` "n" else
|
||||
`suc (plus · (`pred (` "m")) · (` "n")))) · (`suc `zero) · two)
|
||||
⟶⟨ ξ-suc (ξ-·₁ (β-`→ (Suc Zero))) ⟩
|
||||
`suc ((`λ "n" `→ `if0 `suc `zero then ` "n" else
|
||||
`suc (plus · (`pred (`suc `zero)) · (` "n")))) · two
|
||||
⟶⟨ ξ-suc (β-`→ (Suc (Suc Zero))) ⟩
|
||||
`suc (`if0 `suc `zero then two else
|
||||
`suc (plus · (`pred (`suc `zero)) · two))
|
||||
⟶⟨ ξ-suc (β-if0-suc Zero) ⟩
|
||||
`suc (`suc (plus · (`pred (`suc `zero)) · two))
|
||||
⟶⟨ ξ-suc (ξ-suc (ξ-·₁ (ξ-·₁ (β-Y refl)))) ⟩
|
||||
`suc (`suc ((`λ "m" `→ (`λ "n" `→ `if0 ` "m" then ` "n" else
|
||||
`suc (plus · (`pred (` "m")) · (` "n")))) · (`pred (`suc `zero)) · two))
|
||||
⟶⟨ ξ-suc (ξ-suc (ξ-·₁ (ξ-·₂ Fun (β-pred-suc Zero)))) ⟩
|
||||
`suc (`suc ((`λ "m" `→ (`λ "n" `→ `if0 ` "m" then ` "n" else
|
||||
`suc (plus · (`pred (` "m")) · (` "n")))) · `zero · two))
|
||||
⟶⟨ ξ-suc (ξ-suc (ξ-·₁ (β-`→ Zero))) ⟩
|
||||
`suc (`suc ((`λ "n" `→ `if0 `zero then ` "n" else
|
||||
`suc (plus · (`pred `zero) · (` "n"))) · two))
|
||||
⟶⟨ ξ-suc (ξ-suc (β-`→ (Suc (Suc Zero)))) ⟩
|
||||
`suc (`suc (`if0 `zero then two else
|
||||
`suc (plus · (`pred `zero) · two)))
|
||||
⟶⟨ ξ-suc (ξ-suc β-if0-zero) ⟩
|
||||
`suc (`suc (`suc (`suc `zero)))
|
||||
∎
|
||||
\end{code}
|
||||
|
|
|
|||
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Reference in a new issue