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