		Search-engine friendly clone of the ACL2 documentation.

Bit-blasting

Nat-bool-a4env-upper-boundp

Signature
(nat-bool-a4env-upper-boundp bound x) → *
Arguments: bound — Guard (natp bound).; x — Guard (nat-bool-a4env-p! x).

Definitions and Theorems

Function: nat-bool-a4env-upper-boundp

(defun nat-bool-a4env-upper-boundp (bound x)
  (declare (xargs :guard (and (natp bound)
                              (nat-bool-a4env-p! x))))
  (let ((__function__ 'nat-bool-a4env-upper-boundp))
    (declare (ignorable __function__))
    (if (atom x)
        t
      (and (if (mbt (consp (car x)))
               (nat-bool-a4vec-upper-boundp bound (cdar x))
             t)
           (nat-bool-a4env-upper-boundp bound (cdr x))))))

Theorem: nat-bool-a4env-vars-nonmember-by-upper-bound

(defthm nat-bool-a4env-vars-nonmember-by-upper-bound
  (implies (and (nat-bool-a4env-upper-boundp bound x)
                (<= (nfix bound) v)
                (nat-bool-a4env-p x))
           (not (member v (nat-bool-a4env-vars x)))))

Theorem: nat-bool-a4vec/env-no-intersection-by-bounds

(defthm nat-bool-a4vec/env-no-intersection-by-bounds
  (implies (and (nat-bool-a4vec-upper-boundp bound0 x0)
                (nat-bool-a4env-lower-boundp bound1 x1)
                (<= (nfix bound0) (nfix bound1))
                (nat-bool-a4vec-p x0)
                (nat-bool-a4env-p x1))
           (not (intersectp (nat-bool-a4vec-vars x0)
                            (nat-bool-a4env-vars x1)))))

Theorem: nat-bool-a4env-no-intersection-by-bounds

(defthm nat-bool-a4env-no-intersection-by-bounds
  (implies (and (nat-bool-a4env-upper-boundp bound0 x0)
                (nat-bool-a4env-lower-boundp bound1 x1)
                (<= (nfix bound0) (nfix bound1))
                (nat-bool-a4env-p x0)
                (nat-bool-a4env-p x1))
           (not (intersectp (nat-bool-a4env-vars x0)
                            (nat-bool-a4env-vars x1)))))

Theorem: nat-bool-a4env-a4vec-no-intersection-by-bounds

(defthm nat-bool-a4env-a4vec-no-intersection-by-bounds
  (implies (and (nat-bool-a4env-upper-boundp bound0 x0)
                (nat-bool-a4vec-lower-boundp bound1 x1)
                (<= (nfix bound0) (nfix bound1))
                (nat-bool-a4env-p x0)
                (nat-bool-a4vec-p x1))
           (not (intersectp (nat-bool-a4env-vars x0)
                            (nat-bool-a4vec-vars x1)))))

Theorem: nat-bool-a4env-upper-boundp-higher

(defthm nat-bool-a4env-upper-boundp-higher
  (implies (and (nat-bool-a4env-upper-boundp bound x)
                (<= (nfix bound) (nfix n)))
           (nat-bool-a4env-upper-boundp n x)))

Theorem: nat-bool-a4env-upper-boundp-of-nfix-bound

(defthm nat-bool-a4env-upper-boundp-of-nfix-bound
  (equal (nat-bool-a4env-upper-boundp (nfix bound)
                                      x)
         (nat-bool-a4env-upper-boundp bound x)))

Theorem: nat-bool-a4env-upper-boundp-nat-equiv-congruence-on-bound

(defthm nat-bool-a4env-upper-boundp-nat-equiv-congruence-on-bound
  (implies (nat-equiv bound bound-equiv)
           (equal (nat-bool-a4env-upper-boundp bound x)
                  (nat-bool-a4env-upper-boundp bound-equiv x)))
  :rule-classes :congruence)

Theorem: nat-bool-a4env-upper-boundp-of-svex-a4vec-env-fix-x

(defthm nat-bool-a4env-upper-boundp-of-svex-a4vec-env-fix-x
  (equal (nat-bool-a4env-upper-boundp bound (svex-a4vec-env-fix x))
         (nat-bool-a4env-upper-boundp bound x)))

Theorem: nat-bool-a4env-upper-boundp-svex-a4vec-env-equiv-congruence-on-x

(defthm
   nat-bool-a4env-upper-boundp-svex-a4vec-env-equiv-congruence-on-x
  (implies (svex-a4vec-env-equiv x x-equiv)
           (equal (nat-bool-a4env-upper-boundp bound x)
                  (nat-bool-a4env-upper-boundp bound x-equiv)))
  :rule-classes :congruence)