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• Constraint

Constraint-equiv

Basic equivalence relation for constraint structures.

Definitions and Theorems

Function: constraint-equiv$inline

(defun constraint-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (constraintp acl2::x)
                              (constraintp acl2::y))))
  (equal (constraint-fix acl2::x)
         (constraint-fix acl2::y)))

Theorem: constraint-equiv-is-an-equivalence

(defthm constraint-equiv-is-an-equivalence
  (and (booleanp (constraint-equiv x y))
       (constraint-equiv x x)
       (implies (constraint-equiv x y)
                (constraint-equiv y x))
       (implies (and (constraint-equiv x y)
                     (constraint-equiv y z))
                (constraint-equiv x z)))
  :rule-classes (:equivalence))

Theorem: constraint-equiv-implies-equal-constraint-fix-1

(defthm constraint-equiv-implies-equal-constraint-fix-1
  (implies (constraint-equiv acl2::x x-equiv)
           (equal (constraint-fix acl2::x)
                  (constraint-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: constraint-fix-under-constraint-equiv

(defthm constraint-fix-under-constraint-equiv
  (constraint-equiv (constraint-fix acl2::x)
                    acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-constraint-fix-1-forward-to-constraint-equiv

(defthm equal-of-constraint-fix-1-forward-to-constraint-equiv
  (implies (equal (constraint-fix acl2::x) acl2::y)
           (constraint-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-constraint-fix-2-forward-to-constraint-equiv

(defthm equal-of-constraint-fix-2-forward-to-constraint-equiv
  (implies (equal acl2::x (constraint-fix acl2::y))
           (constraint-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: constraint-equiv-of-constraint-fix-1-forward

(defthm constraint-equiv-of-constraint-fix-1-forward
  (implies (constraint-equiv (constraint-fix acl2::x)
                             acl2::y)
           (constraint-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: constraint-equiv-of-constraint-fix-2-forward

(defthm constraint-equiv-of-constraint-fix-2-forward
  (implies (constraint-equiv acl2::x (constraint-fix acl2::y))
           (constraint-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)