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improved substitution

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This commit is contained in:
wadler 2018-07-01 18:39:31 -03:00
parent aefe4ba95c
commit a25ebe0e97
2 changed files with 41 additions and 28 deletions
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12
src/plta/Lambda.lagda
View file

@ -400,19 +400,19 @@ Here is the formal definition of substitution by closed terms in Agda.
\begin{code}
infix 9 _[_:=_]
infix 9 _[[_][_:=_]]
infix 9 _⟨_⟩[_:=_]
_[_:=_] : Term → Id → Term → Term
_[[_][_:=_]] : Term → Id → Id → Term → Term
N [[ x ][ y := V ]] with x ≟ y
_⟨_⟩[_:=_] : Term → Id → Id → Term → Term
N ⟨ x ⟩[ y := V ] with x ≟ y
... | yes _ = N
... | no _ = N [ y := V ]
(` x) [ y := V ] with x ≟ y
... | yes _ = V
... | no _ = ` x
(ƛ x ⇒ N) [ y := V ] = ƛ x ⇒ N [[ x ][ y := V ]]
(ƛ x ⇒ N) [ y := V ] = ƛ x ⇒ N ⟨ x ⟩[ y := V ]
(L · M) [ y := V ] = (L [ y := V ]) · (M [ y := V ])
(`zero) [ y := V ] = `zero
(`suc M) [ y := V ] = `suc (M [ y := V ])
@ -420,8 +420,8 @@ N [[ x ][ y := V ]] with x ≟ y
[zero⇒ M
|suc x ⇒ N ]) [ y := V ] = `case L [ y := V ]
[zero⇒ M [ y := V ]
|suc x ⇒ N [[ x ][ y := V ]] ]
(μ x ⇒ N) [ y := V ] = μ x ⇒ (N [[ x ][ y := V ]])
|suc x ⇒ N ⟨ x ⟩[ y := V ] ]
(μ x ⇒ N) [ y := V ] = μ x ⇒ (N ⟨ x ⟩[ y := V ])
{-

55
src/plta/Properties.lagda
View file

@ -667,41 +667,54 @@ we require an arbitrary context `Γ`, as in the statement of the lemma.
Here is the formal statement and proof that substitution preserves types.
\begin{code}
subst : ∀ {Γ x N V A B}
→ ∅ ⊢ V ⦂ A
→ Γ , x ⦂ A ⊢ N ⦂ B
subst : ∀ {Γ y N V A B}
→ ∅ ⊢ V ⦂ B
→ Γ , y ⦂ B ⊢ N ⦂ A
--------------------
→ Γ ⊢ N [ x := V ] ⦂ B
→ Γ ⊢ N [ y := V ] ⦂ A
substvar : ∀ {Γ x y V A B}
→ ∅ ⊢ V ⦂ B
→ Γ , y ⦂ B ∋ x ⦂ A
------------------------
→ Γ ⊢ (` x) [ y := V ] ⦂ A
{-
substbind : ∀ {Γ x y N V A B C}
→ ∅ ⊢ V ⦂ B
→ Γ , x ⦂ A , y ⦂ B ⊢ N ⦂ C
-----------------------------------
→ Γ , x ⦂ A ⊢ N [[ x ][ y := V ]] ⦂ C
substbind {x = x} {y = y} ⊢V ⊢N with x ≟ y
... | yes refl = drop ⊢N
... | no x≢y = subst ⊢V (swap x≢y ⊢N)
-}
→ Γ , y ⦂ B , x ⦂ A ⊢ N ⦂ C
---------------------------------
→ Γ , x ⦂ A ⊢ N ⟨ x ⟩[ y := V ] ⦂ C
subst {x = y} ⊢V (Ax {x = x} Z) with x ≟ y
... | yes refl = weaken ⊢V
... | no x≢y = ⊥-elim (x≢y refl)
subst {x = y} ⊢V (Ax {x = x} (S x≢y ∋x)) with x ≟ y
... | yes refl = ⊥-elim (x≢y refl)
... | no _ = Ax ∋x
subst {x = y} ⊢V (⊢ƛ {x = x} ⊢N) with x ≟ y
... | yes refl = ⊢ƛ (drop ⊢N)
... | no x≢y = ⊢ƛ (subst ⊢V (swap x≢y ⊢N))
subst ⊢V (Ax ∋x) = substvar ⊢V ∋x
subst ⊢V (⊢ƛ ⊢N) = ⊢ƛ (substbind ⊢V ⊢N)
subst ⊢V (⊢L · ⊢M) = subst ⊢V ⊢L · subst ⊢V ⊢M
subst ⊢V ⊢zero = ⊢zero
subst ⊢V (⊢suc ⊢M) = ⊢suc (subst ⊢V ⊢M)
subst ⊢V (⊢case ⊢L ⊢M ⊢N) = ⊢case (subst ⊢V ⊢L) (subst ⊢V ⊢M) (substbind ⊢V ⊢N)
subst ⊢V (⊢μ ⊢N) = ⊢μ (substbind ⊢V ⊢N)
substvar {x = x} {y = y} ⊢V Z with x ≟ y
... | yes refl = weaken ⊢V
... | no x≢y = ⊥-elim (x≢y refl)
substvar {x = x} {y = y} ⊢V (S x≢y ∋x) with x ≟ y
... | yes refl = ⊥-elim (x≢y refl)
... | no _ = Ax ∋x
substbind {x = x} {y = y} ⊢V ⊢N with x ≟ y
... | yes refl = drop ⊢N
... | no x≢y = subst ⊢V (swap x≢y ⊢N)
{-
subst {x = y} ⊢V (⊢ƛ {x = x} ⊢N) with x ≟ y
... | yes refl = ⊢ƛ (drop ⊢N)
... | no x≢y = ⊢ƛ (subst ⊢V (swap x≢y ⊢N))
subst {x = y} ⊢V (⊢case {x = x} ⊢L ⊢M ⊢N) with x ≟ y
... | yes refl = ⊢case (subst ⊢V ⊢L) (subst ⊢V ⊢M) (drop ⊢N)
... | no x≢y = ⊢case (subst ⊢V ⊢L) (subst ⊢V ⊢M) (subst ⊢V (swap x≢y ⊢N))
subst {x = y} ⊢V (⊢μ {x = x} ⊢M) with x ≟ y
... | yes refl = ⊢μ (drop ⊢M)
... | no x≢y = ⊢μ (subst ⊢V (swap x≢y ⊢M))
-}
{-