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