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• Operands

Operands-equiv

Basic equivalence relation for operands structures.

Definitions and Theorems

Function: operands-equiv$inline

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

Theorem: operands-equiv-is-an-equivalence

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

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

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

Theorem: operands-fix-under-operands-equiv

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

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

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

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

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

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

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

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

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