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Constraint-db

Constraint-db-equiv

Basic equivalence relation for constraint-db structures.

Definitions and Theorems

Function: constraint-db-equiv$inline

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

Theorem: constraint-db-equiv-is-an-equivalence

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

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

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

Theorem: constraint-db-fix-under-constraint-db-equiv

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

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

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

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

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

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

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

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

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