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• Mode

Mode-equiv

Basic equivalence relation for mode structures.

Definitions and Theorems

Function: mode-equiv$inline

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

Theorem: mode-equiv-is-an-equivalence

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

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

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

Theorem: mode-fix-under-mode-equiv

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

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

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

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

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

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

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

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

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