Basic equivalence relation for char+position structures.
Function:
(defun char+position-equiv$inline (acl2::x acl2::y) (declare (xargs :guard (and (char+position-p acl2::x) (char+position-p acl2::y)))) (equal (char+position-fix acl2::x) (char+position-fix acl2::y)))
Theorem:
(defthm char+position-equiv-is-an-equivalence (and (booleanp (char+position-equiv x y)) (char+position-equiv x x) (implies (char+position-equiv x y) (char+position-equiv y x)) (implies (and (char+position-equiv x y) (char+position-equiv y z)) (char+position-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm char+position-equiv-implies-equal-char+position-fix-1 (implies (char+position-equiv acl2::x x-equiv) (equal (char+position-fix acl2::x) (char+position-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm char+position-fix-under-char+position-equiv (char+position-equiv (char+position-fix acl2::x) acl2::x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-char+position-fix-1-forward-to-char+position-equiv (implies (equal (char+position-fix acl2::x) acl2::y) (char+position-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-char+position-fix-2-forward-to-char+position-equiv (implies (equal acl2::x (char+position-fix acl2::y)) (char+position-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm char+position-equiv-of-char+position-fix-1-forward (implies (char+position-equiv (char+position-fix acl2::x) acl2::y) (char+position-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm char+position-equiv-of-char+position-fix-2-forward (implies (char+position-equiv acl2::x (char+position-fix acl2::y)) (char+position-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)