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