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Var/const-sort

Var/const-sort-equiv

Basic equivalence relation for var/const-sort structures.

Definitions and Theorems

Function: var/const-sort-equiv$inline

(defun var/const-sort-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (var/const-sortp acl2::x)
                              (var/const-sortp acl2::y))))
  (equal (var/const-sort-fix acl2::x)
         (var/const-sort-fix acl2::y)))

Theorem: var/const-sort-equiv-is-an-equivalence

(defthm var/const-sort-equiv-is-an-equivalence
  (and (booleanp (var/const-sort-equiv x y))
       (var/const-sort-equiv x x)
       (implies (var/const-sort-equiv x y)
                (var/const-sort-equiv y x))
       (implies (and (var/const-sort-equiv x y)
                     (var/const-sort-equiv y z))
                (var/const-sort-equiv x z)))
  :rule-classes (:equivalence))

Theorem: var/const-sort-equiv-implies-equal-var/const-sort-fix-1

(defthm var/const-sort-equiv-implies-equal-var/const-sort-fix-1
  (implies (var/const-sort-equiv acl2::x x-equiv)
           (equal (var/const-sort-fix acl2::x)
                  (var/const-sort-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: var/const-sort-fix-under-var/const-sort-equiv

(defthm var/const-sort-fix-under-var/const-sort-equiv
  (var/const-sort-equiv (var/const-sort-fix acl2::x)
                        acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-var/const-sort-fix-1-forward-to-var/const-sort-equiv

(defthm
      equal-of-var/const-sort-fix-1-forward-to-var/const-sort-equiv
  (implies (equal (var/const-sort-fix acl2::x)
                  acl2::y)
           (var/const-sort-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-var/const-sort-fix-2-forward-to-var/const-sort-equiv

(defthm
      equal-of-var/const-sort-fix-2-forward-to-var/const-sort-equiv
  (implies (equal acl2::x (var/const-sort-fix acl2::y))
           (var/const-sort-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: var/const-sort-equiv-of-var/const-sort-fix-1-forward

(defthm var/const-sort-equiv-of-var/const-sort-fix-1-forward
  (implies (var/const-sort-equiv (var/const-sort-fix acl2::x)
                                 acl2::y)
           (var/const-sort-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: var/const-sort-equiv-of-var/const-sort-fix-2-forward

(defthm var/const-sort-equiv-of-var/const-sort-fix-2-forward
  (implies
       (var/const-sort-equiv acl2::x (var/const-sort-fix acl2::y))
       (var/const-sort-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)