\DOC
\TYPE {SUC_TO_NUMERAL_DEFN_CONV : conv}
\SYNOPSIS
Translates equations using {SUC n} to use numeral constructors instead.
\DESCRIBE
This conversion modifies conjunctions of universally quantified
equations so that any argument terms of the form {SUC x} on the LHS of
the equations (with {x} one of the equation's universally quantified
variables), are translated to a form appropriate for rewriting when
the argument term is a numeral.
This procedure uses the following theorem:
{
|- !f g. (!n. f (SUC n) = g n (SUC n)) =
(!n. f (NUMERAL (NUMERAL_BIT1 n)) =
g (NUMERAL (NUMERAL_BIT1 n))
(NUMERAL (NUMERAL_BIT1 n) - 1)) /\
(!n. f (NUMERAL (NUMERAL_BIT2 n)) =
g (NUMERAL (NUMERAL_BIT2 n))
(NUMERAL (NUMERAL_BIT1 n)))
}
\EXAMPLE
{
- CONV_RULE SUC_TO_NUMERAL_DEFN_CONV arithmeticTheory.FACT;
> val it =
|- (FACT 0 = 1) /\
(!n.
FACT (NUMERAL (NUMERAL_BIT1 n)) =
NUMERAL (NUMERAL_BIT1 n) *
FACT (NUMERAL (NUMERAL_BIT1 n) - 1)) /\
!n.
FACT (NUMERAL (NUMERAL_BIT2 n)) =
NUMERAL (NUMERAL_BIT2 n) *
FACT (NUMERAL (NUMERAL_BIT1 n)) : thm
}
\FAILURE
Fails if the input term is not the conjunction of universally
quantified equations, where there may be just one conjunct, and where
equations may have no quantification at all. Those conjuncts which
don't involve terms of the form {SUC x} are returned unchanged.
\COMMENTS
Useful for translating definitions over numbers (which often involve
{SUC} terms), into a form that can be used to work with numerals
easily.
\SEEALSO
numLib.num_CONV.
\ENDDOC