Prim_rec.prove_rec_fn

STRUCTURE

SYNOPSIS

Proves the existence of a primitive recursive function over a concrete recursive type.

DESCRIPTION

prove_rec_fn_exists is a version of new_recursive_definition which proves only that the required function exists; it does not make a constant specification. The first argument is a primitive recursion theorem of the form generated by Hol_datatype, and the second is a user-supplied primitive recursive function definition. The theorem which is returned asserts the existence of the recursively-defined function in question (if it is primitive recursive over the type characterized by the theorem given as the first argument). See the entry for new_recursive_definition for details.

FAILURE

As for new_recursive_definition.

EXAMPLE

Given the following primitive recursion theorem for labelled binary trees:

   |- !f0 f1.
        ?fn.
          (!a. fn (LEAF a) = f0 a) /\
          !a0 a1. fn (NODE a0 a1) = f1 a0 a1 (fn a0) (fn a1) : thm

prove_rec_fn_exists can be used to prove the existence of primitive recursive functions over binary trees. Suppose the value of th is this theorem. Then the existence of a recursive function Leaves, which computes the number of leaves in a binary tree, can be proved as shown below:

   - prove_rec_fn_exists th
      ``(Leaves (LEAF (x:'a)) = 1) /\
        (Leaves (NODE t1 t2) = (Leaves t1) + (Leaves t2))``;
   > val it =
       |- ?Leaves.
            (!x. Leaves (LEAF x) = 1) /\
            !t1 t2. Leaves (NODE t1 t2) = Leaves t1 + Leaves t2 : thm

The result should be compared with the example given under new_recursive_definition.

SEEALSO

Hol_datatype, new_recursive_definition