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Statement

Statement-equiv

Basic equivalence relation for statement structures.

Definitions and Theorems

Function: statement-equiv$inline

(defun statement-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (statementp acl2::x)
                              (statementp acl2::y))))
  (equal (statement-fix acl2::x)
         (statement-fix acl2::y)))

Theorem: statement-equiv-is-an-equivalence

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

Theorem: statement-equiv-implies-equal-statement-fix-1

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

Theorem: statement-fix-under-statement-equiv

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

Theorem: equal-of-statement-fix-1-forward-to-statement-equiv

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

Theorem: equal-of-statement-fix-2-forward-to-statement-equiv

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

Theorem: statement-equiv-of-statement-fix-1-forward

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

Theorem: statement-equiv-of-statement-fix-2-forward

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