Remove all conditional expressions from a Boolean formula.
When applied to a Boolean term, CONDS_CELIM_CONV identifies subterms that are
conditional expressions of the form `if p then x else y', and eliminates
them. First they are ``pulled out'' as far as possible, e.g.
from `f (if p then x else y)' to `if p then f(x) else f(y)' and so on. When
a quantifier that binds one of the variables in the expression is reached, the
subterm is of Boolean type, say `if p then q else r', and it is replaced by a
propositional equivalent of the form `(~p \/ q) /\ (p \/ r)'.
Never fails, but will just return a reflexive theorem if the term is not
# CONDS_CELIM_CONV `y <= z ==> !x. (if x <= y then y else x) <= z`;;
val it : thm =
|- y <= z ==> (!x. (if x <= y then y else x) <= z) <=>
y <= z ==> (!x. (~(x <= y) \/ y <= z) /\ (x <= y \/ x <= z))
Mostly for initial normalization in automated rules, but may be helpful for
The function CONDS_ELIM_CONV is functionally similar, but will do the final
propositional splitting in a ``disjunctive'' rather than ``conjunctive'' way.
The disjunctive way is usually better when the term will subsequently be passed
to a refutation procedure, whereas the conjunctive form is better for
non-refutation procedures. In each case, the policy is changed in an
appropriate way after passing through quantifiers.