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