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