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Pexprs-gin

Pexprs-gin-equiv

Basic equivalence relation for pexprs-gin structures.

Definitions and Theorems

Function: pexprs-gin-equiv$inline

(defun pexprs-gin-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (pexprs-ginp acl2::x)
                              (pexprs-ginp acl2::y))))
  (equal (pexprs-gin-fix acl2::x)
         (pexprs-gin-fix acl2::y)))

Theorem: pexprs-gin-equiv-is-an-equivalence

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

Theorem: pexprs-gin-equiv-implies-equal-pexprs-gin-fix-1

(defthm pexprs-gin-equiv-implies-equal-pexprs-gin-fix-1
  (implies (pexprs-gin-equiv acl2::x x-equiv)
           (equal (pexprs-gin-fix acl2::x)
                  (pexprs-gin-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: pexprs-gin-fix-under-pexprs-gin-equiv

(defthm pexprs-gin-fix-under-pexprs-gin-equiv
  (pexprs-gin-equiv (pexprs-gin-fix acl2::x)
                    acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-pexprs-gin-fix-1-forward-to-pexprs-gin-equiv

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

Theorem: equal-of-pexprs-gin-fix-2-forward-to-pexprs-gin-equiv

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

Theorem: pexprs-gin-equiv-of-pexprs-gin-fix-1-forward

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

Theorem: pexprs-gin-equiv-of-pexprs-gin-fix-2-forward

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