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Wire

Wire-equiv

Basic equivalence relation for wire structures.

Definitions and Theorems

Function: wire-equiv$inline

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

Theorem: wire-equiv-is-an-equivalence

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

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

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

Theorem: wire-fix-under-wire-equiv

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

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

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

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

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

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

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

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

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