HOL: Reference

tprove

Defn.tprove : defn * tactic -> thm * thm

Prove termination of a defn.

tprove takes a defn and a tactic, and uses the tactic to prove the termination constraints of the defn. A pair of theorems (eqns,ind) is returned: eqns is the unconstrained recursion equations of the defn, and ind is the corresponding induction theorem for the equations, also unconstrained.

tprove and tgoal can be seen as analogues of prove and set_goal in the specialized domain of proving termination of recursive functions.

It is up to the user to store the results of tprove in the current theory segment.

Failure

tprove (defn,tac) fails if tac fails to prove the termination conditions of defn.

tprove (defn,tac) fails if defn represents a non-recursive or primitive recursive function.

Example

Suppose that we have defined a version of Quicksort as follows:

   - val qsort_defn =
       Hol_defn "qsort"
          `(qsort ___ [] = []) /\
           (qsort ord (x::rst) =
               APPEND (qsort ord (FILTER ($~ o ord x) rst))
                 (x :: qsort ord (FILTER (ord x) rst)))`

Also suppose that a tactic tac proves termination of qsort. (This tactic has probably been built by interactive proof after starting a goalstack with tgoal qsort_defn.) Then

   - val (qsort_eqns, qsort_ind) = tprove(qsort_defn, tac);

   > val qsort_eqns =
       |- (qsort v0 [] = []) /\
          (qsort ord (x::rst) =
             APPEND (qsort ord (FILTER ($~ o ord x) rst))
                 (x::qsort ord (FILTER (ord x) rst))) : thm

     val qsort_ind =
       |- !P.
            (!v0. P v0 []) /\
            (!ord x rst.
               P ord (FILTER ($~ o ord x) rst) /\
               P ord (FILTER (ord x) rst) ==> P ord (x::rst))
            ==>
           !v v1. P v v1 : thm

Comments

The recursion equations returned by a successful invocation of tprove are automatically added to the global compset accessed by EVAL.

See also

Defn.tgoal, Defn.Hol_defn, bossLib.EVAL