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Pos-equiv

(pos-equiv x y) is equality for positive numbers, i.e., equality up to pos-fix.

Definitions and Theorems

Function: pos-equiv$inline

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

Theorem: pos-equiv-is-an-equivalence

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

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

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

Theorem: pos-fix-under-pos-equiv

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