`prove_rec_fn_exists : thm -> term -> thm`
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