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• Scspecseq

Scspecseq-equiv

Basic equivalence relation for scspecseq structures.

Definitions and Theorems

Function: scspecseq-equiv$inline

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

Theorem: scspecseq-equiv-is-an-equivalence

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

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

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

Theorem: scspecseq-fix-under-scspecseq-equiv

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

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

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

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

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

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

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

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

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