Conv.EXISTS_IMP

STRUCTURE

SYNOPSIS

Moves an existential quantification inwards through an implication.

DESCRIPTION

When applied to a term of the form ?x. P ==> Q, where x is not free in both P and Q, EXISTS_IMP_CONV returns a theorem of one of three forms, depending on occurrences of the variable x in P and Q. If x is free in P but not in Q, then the theorem:

   |- (?x. P ==> Q) = (!x.P) ==> Q

is returned. If x is free in Q but not in P, then the result is:

   |- (?x. P ==> Q) = P ==> (?x.Q)

And if x is free in neither P nor Q, then the result is:

   |- (?x. P ==> Q) = (!x.P) ==> (?x.Q)

FAILURE

EXISTS_IMP_CONV fails if it is applied to a term not of the form ?x. P ==> Q, or if it is applied to a term ?x. P ==> Q in which the variable x is free in both P and Q.

SEEALSO

LEFT_IMP_FORALL_CONV, RIGHT_IMP_EXISTS_CONV