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• Backtrace

Backtrace-equiv

Basic equivalence relation for backtrace structures.

Definitions and Theorems

Function: backtrace-equiv$inline

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

Theorem: backtrace-equiv-is-an-equivalence

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

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

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

Theorem: backtrace-fix-under-backtrace-equiv

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

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

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

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

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

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

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

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

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