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• Svstmt

Svstmt-equiv

Basic equivalence relation for svstmt structures.

Definitions and Theorems

Function: svstmt-equiv$inline

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

Theorem: svstmt-equiv-is-an-equivalence

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

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

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

Theorem: svstmt-fix-under-svstmt-equiv

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

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

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

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

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

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

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

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

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