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• Ipasir$a

Ipasir$a-equiv

Basic equivalence relation for ipasir$a structures.

Definitions and Theorems

Function: ipasir$a-equiv$inline

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

Theorem: ipasir$a-equiv-is-an-equivalence

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

Theorem: ipasir$a-equiv-implies-equal-ipasir$a-fix-1

(defthm ipasir$a-equiv-implies-equal-ipasir$a-fix-1
  (implies (ipasir$a-equiv acl2::x x-equiv)
           (equal (ipasir$a-fix acl2::x)
                  (ipasir$a-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: ipasir$a-fix-under-ipasir$a-equiv

(defthm ipasir$a-fix-under-ipasir$a-equiv
  (ipasir$a-equiv (ipasir$a-fix acl2::x)
                  acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-ipasir$a-fix-1-forward-to-ipasir$a-equiv

(defthm equal-of-ipasir$a-fix-1-forward-to-ipasir$a-equiv
  (implies (equal (ipasir$a-fix acl2::x) acl2::y)
           (ipasir$a-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-ipasir$a-fix-2-forward-to-ipasir$a-equiv

(defthm equal-of-ipasir$a-fix-2-forward-to-ipasir$a-equiv
  (implies (equal acl2::x (ipasir$a-fix acl2::y))
           (ipasir$a-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: ipasir$a-equiv-of-ipasir$a-fix-1-forward

(defthm ipasir$a-equiv-of-ipasir$a-fix-1-forward
  (implies (ipasir$a-equiv (ipasir$a-fix acl2::x)
                           acl2::y)
           (ipasir$a-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: ipasir$a-equiv-of-ipasir$a-fix-2-forward

(defthm ipasir$a-equiv-of-ipasir$a-fix-2-forward
  (implies (ipasir$a-equiv acl2::x (ipasir$a-fix acl2::y))
           (ipasir$a-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)