\DOC Induct_on
\TYPE {Induct_on : term -> tactic}
\SYNOPSIS
Performs structural induction, using the type of the given term.
\KEYWORDS
tactic, induction
\LIBRARY
SingleStep.
\DESCRIBE
Given a term {M}, {Induct_on} attempts to perform an induction based on
the type of {M}. The induction theorem to be used is extracted from the
{TypeBase} database, which holds useful facts about the system's defined
types.
{Induct_on} can be used to specify variables that are buried in the
quantifier prefix, i.e., not the leading quantified
variable. {Induct_on} can also perform induction on non-variable
terms. If {M} is a non-variable term that does not occur bound in the
goal, then {Induct_on} equates {M} to a new variable {v} (one not
occurring in the goal), moves all hypotheses in which free variables of
{M} occur to the conclusion of the goal, adds the antecedent {v = M},
and quantifies all free variables of {M} before universally quantifying
{v} and then finally inducting on {v}.
{Induct_on} may also be used to apply an induction theorem coming from
declaration of a mutually recursive datattype.
\FAILURE
{Induct_on} fails if an induction theorem corresponding to the type of
{M} is not found in the {TypeBase} database.
\EXAMPLE
If attempting to prove
{
!x. LENGTH (REVERSE x) = LENGTH x
}
one can apply {Induct_on `x`} to begin a proof by induction on the
list structure of {x}. In this case, {Induct_on} serves as an explicit
version of {Induct}.
\SEEALSO
bossLib.Induct, bossLib.completeInduct_on,
bossLib.measureInduct_on, Prim_rec.INDUCT_THEN, bossLib.Cases,
bossLib.Hol_datatype, goalstackLib.g, goalstackLib.e.
\ENDDOC