Explicit Contexts in LF

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Explicit Contexts in LF

• Karl Crary
• Carnegie Mellon University

Workshop on Mechanized Metatheory, 9/21/06
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The LF methodology

• Construct isomorphisms between
• Syntactic classes and LF types
• Expressions and LF terms(of appropriate type)
• Judgements and LF types
• Derivations and LF terms(of appropriate type)

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The LF methodology

• Isomorphisms must commute with substitution.
• To do so
• Identify OL variables with LF variables.
• Identify OL assumptions with LF assumptions.
• Consequently, identify OL contexts with
  (fragments of) LF contexts.

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LF example Syntax
tp type. exp type. o tp. arrow
tp -gt tp -gt tp. b exp. lam tp -gt
(exp -gt exp) -gt exp. app exp -gt exp -gt exp.
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LF example Semantics
of exp -gt tp -gt type. of/b of b o. of/lam
of (lam A (x M x)) (arrow A B)
lt- (x of x A -gt of (M x) B). of/app
of (app M N) B lt- of M (arrow A B)
lt- of N A.
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The problem

• Contexts are implicit.
• Cannot be manipulated by proofs.
• Can be a problem for theorems involving a
  distinguished bound variable.

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Distinguished bound variable

• If of M Aand (x of x A -gt of (N x) B)then
  of (N M) B
• In Twelf syntax

subst of M A -gt (x of x A -gt of (N x) B)
-gt of (N M) B -gt type. mode subst D1 D2
-D3.
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Important caveat

• Theres no reason to do this!
• Substitution lemma is free in LF.
• If D1 of M Aand D2 (x of x A -gt of (N x)
  B)then D2 M D1 of (N M) B
• Illustrative example, not a motivating one.

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Motivating examples

• Substitution lemma for languages with different
  judgements on left and right
• Sequent calculus, imperative type systems
• Narrowing in F-sub (Poplmark challenge)
• Functionality
• Defined notions of substitution
• Hereditary substitution
• Linear and modal logic

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Substitution theorem

• If of M Aand (x of x A -gt of (N x) B)then
  of (N M) B
• By induction on the second argument.

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A proof case

• SupposeD1 of M AD2 x dof x A
  of/lam (y e D x d y e) x of x A
  -gt of (lam B (y N x y)) (arrow B
  C)D x of x A -gt y of y B -gt of (N x
  y) C

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A proof case

• Let yexp and eof y B be arbitrary.
• Thereforex dof x A D x d y e x of x A
  -gt of (N x y) C
• By induction there existsD y e of (N M y) C
• Thereforeof/lam D of (lam B (y N M y))
  (arrow B C)

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A proof case in Twelf
- subst D1 (x dof x A of/lam
(y eof y B D x d y e)) (of/lam D') lt-
(y eof y B subst D1 (x
dof x A D x d y e) (D' y e of (N M
y) C)).
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Assumption permutation

• Proof permutes x d with y e.
• No room for bindings between distinguished
  variable and its scope.
• Undistinguished variables go in context.
• In essence, the distinguished variable must
  appear last.
• Permute assumptions to preserve this condition.

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Uh oh!

• With dependent types, we cannot permute
  assumptions.
• When es type depends on x, it cannot be pulled
  outside.

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The problem

• When
• doing an inductive proof in Twelf
• that involves a distinguished bound variable,
• and the setting includes dependent types,
• You have a problem
• Cannot keep the designated bound variable last.

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Explicit contexts

• Make the context into an explicit object that the
  proof can manipulate.
• This allows us to place the variable of interest
  anywhere in the context.
• Proof technique only!
• No change to LF or Twelf.
• No change to syntax!
• Still using higher-order abstract syntax.
• Can convert from and to implicit contexts.

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Methodology

• Syntax is still entirely higher-order.
• Give two versions of the semantics.
• Implicit and explicit context.
• Convert derivations to use explicit contexts when
  necessary.

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Contexts
ctx type.nil ctx.cons ctx -gt exp -gt tp
-gt ctx.
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First cut lookup
lookup ctx -gt exp -gt tp -gt type.lookup/hit
lookup (cons G X A) X A.lookup/miss lookup
(cons G Y _) X A lt- lookup G X A.
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First cut semantics
ofe ctx -gt exp -gt tp -gt type. ofe/var ofe
G X A lt- lookup G X A. ofe/closed
ofe G M A lt- of M A.
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First cut semantics
ofe/lam ofe G (lam A (x M x))
(arrow A B) lt- (xexp ofe (cons G x
A) (M x) B). ofe/app ofe G (app M N) B lt-
ofe G M (arrow A B) lt- ofe G N A.
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Problem bad contexts

• Contexts are merely association lists of terms
  and types.
• Syntax permits
• Binding of non-variables.
• Multiple bindings of a single variable.
• Need a context formation judgement
• Each term should be a distinct variable.

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Context formation

• Distinguish variables using a hypothetical
  judgement.
• Also assigns an ordering to variables.
• Context formation judgement
• Only variables may appear.
• Variables must be ordered.(Hence, no duplicates.)

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Variable ordering
isvar exp -gt nat -gt type. Assumption "isvar
X I" indicates (1) X is a variable, and (2) x
carries order stamp I. precedes exp -gt exp
-gt type. precedes/i precedes X Y lt-
isvar X I lt- isvar Y J lt- lt I J.
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Context formation
bounded ctx -gt exp -gt type.bounded/nil
bounded nil X lt- isvar X
_.bounded/cons bounded (cons G Y _) X
lt- precedes Y X lt- bounded
G Y. ordered ctx -gt type.ordered/nil
ordered nil.ordered/cons ordered (cons G X _)
lt- bounded G X.
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Lookup
lookup ctx -gt exp -gt tp -gt type.lookup/hit
lookup (cons G X A) X A lt- bounded G
X. lookup/miss lookup (cons G Y _) X A lt-
bounded G Y lt- lookup G X A.

• Lemma if lookup G X A then ordered G.

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Semantics
ofe/closed ofe G M A lt- of M A lt-
ordered G. ofe/lam ofe G (lam A (x M x))
(arrow A B) lt- (xexp isvar x I
-gt ofe (cons G x A) (M x) B).
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Usage

• When combined, these theorems allow us to do
  proofs for the implicit system.
• Convert to explicit form.
• Perform the desired proof.
• Convert back to implicit form.

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Substitution theorem

• Before
• If ? ? M Aand ?, xA ? N Bthen ? ? M/x N
  B
• Now
• If ?1 ? M Aand ?1, xA, ?2 ? N Bthen ?1, ?2
  ? M/x N B

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Substitution theorem in Twelf

• If (x append (cons G1 x A) G2 (G
  x))and append G1 G2 Gand ofe G1 M Aand (x
  isvar x I -gt ofe (G x) (N x) B)then
  ofe G (N M) B

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Context Lemmas

• If lookup G X A then X is not a lambda or
  application.
• That is, contexts bind only variables.
• If (x append (cons G1 x A) G2 (G
  x))and (x isvar x I -gt lookup (G x) x
  B)then tp-eq A B.
• That is, contexts bind distinct variables.

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Translation to implicit form

• If ofe nil M Athen of M A
• Proof is not very hard.

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Translation to explicit form

• If of M Athen ofe nil M A
• Proof is tricky.
• This is the enabling technical achievement.

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Cut elimination

• Main lemma is a form of cut elimination.
• Cut explicit-context lookup againstimplicit-con
  text of assumption.
• Prove simultaneously for cuts into of and ofe.
• If (x of x A -gt of (M x) B)and (x isvar x I
  -gt lookup (G x) x A)then (x isvar x I
  -gt ofe (G x) (M x) B)

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Conclusion

• General technique for proofs involving
• A distinguished bound variable
• Dependent types
• Used in type safety proof for SML IL.
• See Daniel Lees talk this afternoon.
• Not an extension to LF.
• Not a new representation technique
• Still use higher-order syntax and judgements.