Conv.EXISTS_AND

STRUCTURE

SYNOPSIS

Moves an existential quantification inwards through a conjunction.

DESCRIPTION

When applied to a term of the form ?x. P /\ Q, where x is not free in both P and Q, EXISTS_AND_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_AND_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

AND_EXISTS_CONV, EXISTS_AND_REORDER_CONV, LEFT_AND_EXISTS_CONV, RIGHT_AND_EXISTS_CONV