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