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• Sd-problem

Sd-problem-equiv

Basic equivalence relation for sd-problem structures.

Definitions and Theorems

Function: sd-problem-equiv$inline

(defun sd-problem-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (sd-problem-p acl2::x)
                              (sd-problem-p acl2::y))))
  (equal (sd-problem-fix acl2::x)
         (sd-problem-fix acl2::y)))

Theorem: sd-problem-equiv-is-an-equivalence

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

Theorem: sd-problem-equiv-implies-equal-sd-problem-fix-1

(defthm sd-problem-equiv-implies-equal-sd-problem-fix-1
  (implies (sd-problem-equiv acl2::x x-equiv)
           (equal (sd-problem-fix acl2::x)
                  (sd-problem-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: sd-problem-fix-under-sd-problem-equiv

(defthm sd-problem-fix-under-sd-problem-equiv
  (sd-problem-equiv (sd-problem-fix acl2::x)
                    acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-sd-problem-fix-1-forward-to-sd-problem-equiv

(defthm equal-of-sd-problem-fix-1-forward-to-sd-problem-equiv
  (implies (equal (sd-problem-fix acl2::x) acl2::y)
           (sd-problem-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-sd-problem-fix-2-forward-to-sd-problem-equiv

(defthm equal-of-sd-problem-fix-2-forward-to-sd-problem-equiv
  (implies (equal acl2::x (sd-problem-fix acl2::y))
           (sd-problem-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: sd-problem-equiv-of-sd-problem-fix-1-forward

(defthm sd-problem-equiv-of-sd-problem-fix-1-forward
  (implies (sd-problem-equiv (sd-problem-fix acl2::x)
                             acl2::y)
           (sd-problem-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: sd-problem-equiv-of-sd-problem-fix-2-forward

(defthm sd-problem-equiv-of-sd-problem-fix-2-forward
  (implies (sd-problem-equiv acl2::x (sd-problem-fix acl2::y))
           (sd-problem-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)