Fixing function for expr/tyname structures.
(expr/tyname-fix x) → new-x
Function:
(defun expr/tyname-fix$inline (x) (declare (xargs :guard (expr/tyname-p x))) (let ((__function__ 'expr/tyname-fix)) (declare (ignorable __function__)) (mbe :logic (case (expr/tyname-kind x) (:expr (b* ((unwrap (expr-fix (std::da-nth 0 (cdr x))))) (cons :expr (list unwrap)))) (:tyname (b* ((unwrap (tyname-fix (std::da-nth 0 (cdr x))))) (cons :tyname (list unwrap))))) :exec x)))
Theorem:
(defthm expr/tyname-p-of-expr/tyname-fix (b* ((new-x (expr/tyname-fix$inline x))) (expr/tyname-p new-x)) :rule-classes :rewrite)
Theorem:
(defthm expr/tyname-fix-when-expr/tyname-p (implies (expr/tyname-p x) (equal (expr/tyname-fix x) x)))
Function:
(defun expr/tyname-equiv$inline (acl2::x acl2::y) (declare (xargs :guard (and (expr/tyname-p acl2::x) (expr/tyname-p acl2::y)))) (equal (expr/tyname-fix acl2::x) (expr/tyname-fix acl2::y)))
Theorem:
(defthm expr/tyname-equiv-is-an-equivalence (and (booleanp (expr/tyname-equiv x y)) (expr/tyname-equiv x x) (implies (expr/tyname-equiv x y) (expr/tyname-equiv y x)) (implies (and (expr/tyname-equiv x y) (expr/tyname-equiv y z)) (expr/tyname-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm expr/tyname-equiv-implies-equal-expr/tyname-fix-1 (implies (expr/tyname-equiv acl2::x x-equiv) (equal (expr/tyname-fix acl2::x) (expr/tyname-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm expr/tyname-fix-under-expr/tyname-equiv (expr/tyname-equiv (expr/tyname-fix acl2::x) acl2::x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-expr/tyname-fix-1-forward-to-expr/tyname-equiv (implies (equal (expr/tyname-fix acl2::x) acl2::y) (expr/tyname-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-expr/tyname-fix-2-forward-to-expr/tyname-equiv (implies (equal acl2::x (expr/tyname-fix acl2::y)) (expr/tyname-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm expr/tyname-equiv-of-expr/tyname-fix-1-forward (implies (expr/tyname-equiv (expr/tyname-fix acl2::x) acl2::y) (expr/tyname-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm expr/tyname-equiv-of-expr/tyname-fix-2-forward (implies (expr/tyname-equiv acl2::x (expr/tyname-fix acl2::y)) (expr/tyname-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm expr/tyname-kind$inline-of-expr/tyname-fix-x (equal (expr/tyname-kind$inline (expr/tyname-fix x)) (expr/tyname-kind$inline x)))
Theorem:
(defthm expr/tyname-kind$inline-expr/tyname-equiv-congruence-on-x (implies (expr/tyname-equiv x x-equiv) (equal (expr/tyname-kind$inline x) (expr/tyname-kind$inline x-equiv))) :rule-classes :congruence)
Theorem:
(defthm consp-of-expr/tyname-fix (consp (expr/tyname-fix x)) :rule-classes :type-prescription)