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