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  • Math
  • Testing-utilities

• Token+span

Token+span-equiv

Basic equivalence relation for token+span structures.

Definitions and Theorems

Function: token+span-equiv$inline

(defun token+span-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (token+span-p acl2::x)
                              (token+span-p acl2::y))))
  (equal (token+span-fix acl2::x)
         (token+span-fix acl2::y)))

Theorem: token+span-equiv-is-an-equivalence

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

Theorem: token+span-equiv-implies-equal-token+span-fix-1

(defthm token+span-equiv-implies-equal-token+span-fix-1
  (implies (token+span-equiv acl2::x x-equiv)
           (equal (token+span-fix acl2::x)
                  (token+span-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: token+span-fix-under-token+span-equiv

(defthm token+span-fix-under-token+span-equiv
  (token+span-equiv (token+span-fix acl2::x)
                    acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-token+span-fix-1-forward-to-token+span-equiv

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

Theorem: equal-of-token+span-fix-2-forward-to-token+span-equiv

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

Theorem: token+span-equiv-of-token+span-fix-1-forward

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

Theorem: token+span-equiv-of-token+span-fix-2-forward

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