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Trace-alist

Trace-alist-equiv

Basic equivalence relation for trace-alist structures.

Definitions and Theorems

Function: trace-alist-equiv$inline

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

Theorem: trace-alist-equiv-is-an-equivalence

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

Theorem: trace-alist-equiv-implies-equal-trace-alist-fix-1

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

Theorem: trace-alist-fix-under-trace-alist-equiv

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

Theorem: equal-of-trace-alist-fix-1-forward-to-trace-alist-equiv

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

Theorem: equal-of-trace-alist-fix-2-forward-to-trace-alist-equiv

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

Theorem: trace-alist-equiv-of-trace-alist-fix-1-forward

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

Theorem: trace-alist-equiv-of-trace-alist-fix-2-forward

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