Sometimes it is easier to prove a result by breaking the proof (or a part of it) into `cases.' For example, in trying to prove a sentence of the form
( n) (n > 0 => Q(n)),it might be easier to prove the following two cases separately,
( n) (n > 0 and n even => Q(n)) ( n) (n > 0 and n odd => Q(n)),than to prove the assertion in its original form. In general, there may be many different ways to break a condition like n>0 into cases. Another is
( n) (n = 1 => Q(n)) ( n) (n > 1 => Q(n)).
Examples like these suggest that cases are predicates
P1,...,PN such that
P1 and ... and PN
are equivalent, where the sentence to be proved has the
( X) P => Q;
and they further suggest that a proof by case
analysis consists of proving
( X) Pi => Q for i=1,...,N.
The rule for case analysis is then as follows:
A (P => Q) ---> A (Pi => Q) for i=1,...,N and A (P => P1 and ... and PN) .