Midterm
Midterm
Meeting 22, CSCI 5535, Spring 2009
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High-Level Hints
Exercise 4
• Ex 2: Language is like IMP and like λcalculus in ways
• Ex 3: Did “preservation” in class with
big-step semantics
• Ex 4: Try an example derivation in ⊢ and
“convert it” to ⊢0
• Ex5: Think about using references:
storing and reading
• Consider the simply-typed lambda
calculus with subtyping:
e ::= x | λx.e | e1 e2
τ ::= τ → τ | …
• Typing rule for λ-expression:
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Γ, x : τ ⊢ e : τ ′
Γ ⊢ λx. e : τ → τ ′
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Exercise 4
Exercise 4
• Standard subsumption
• Prove “completeness of variable-only
subsumption typing”
Γ ⊢ e : τ τ <: τ ′
Γ ⊢ e : τ′
If T:: · ⊢ e : τ, then · ⊢0 e : τ .
• Replaced with restricted variable-only
subsumption
′
Γ(x) = τ
τ <: τ
Γ ⊢0 x : τ ′
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Exercise 4
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Exercise 4
If T:: · ⊢ e : τ, then · ⊢0 e : τ .
Need generalization!
If T:: · ⊢ e : τ, then · ⊢0 e : τ .
• Try proof by induction on T.
• Case
x : τ ⊢ e : τ′
T :: · ⊢ λx. e : τ → τ ′
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Exercise 4
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Need generalization!
Corollary: If T:: · ⊢ e : τ, then · ⊢0 e : τ .
• Try
Theorem: If T:: Γ ⊢ e : τ, then Γ ⊢0 e : τ .
• Resolves issue with λ-expression
• If we can prove the theorem, then that
(trivially) implies the corollary we want
• Think about: The case where T ends in
the subsumption rule.
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