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Sig-path

Sig-path-fix

(sig-path-fix x) is a usual ACL2::fty list fixing function.

Signature
(sig-path-fix x) → fty::newx
Arguments: x — Guard (sig-path-p x).
Returns: fty::newx — Type (sig-path-p fty::newx).

In the logic, we apply sig-fix to each member of the x. In the execution, none of that is actually necessary and this is just an inlined identity function.

Definitions and Theorems

Function: sig-path-fix$inline

(defun sig-path-fix$inline (x)
  (declare (xargs :guard (sig-path-p x)))
  (let ((acl2::__function__ 'sig-path-fix))
    (declare (ignorable acl2::__function__))
    (mbe :logic
         (if (atom x)
             nil
           (cons (sig-fix (car x))
                 (sig-path-fix (cdr x))))
         :exec x)))

Theorem: sig-path-p-of-sig-path-fix

(defthm sig-path-p-of-sig-path-fix
  (b* ((fty::newx (sig-path-fix$inline x)))
    (sig-path-p fty::newx))
  :rule-classes :rewrite)

Theorem: sig-path-fix-when-sig-path-p

(defthm sig-path-fix-when-sig-path-p
  (implies (sig-path-p x)
           (equal (sig-path-fix x) x)))

Function: sig-path-equiv$inline

(defun sig-path-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (sig-path-p acl2::x)
                              (sig-path-p acl2::y))))
  (equal (sig-path-fix acl2::x)
         (sig-path-fix acl2::y)))

Theorem: sig-path-equiv-is-an-equivalence

(defthm sig-path-equiv-is-an-equivalence
  (and (booleanp (sig-path-equiv x y))
       (sig-path-equiv x x)
       (implies (sig-path-equiv x y)
                (sig-path-equiv y x))
       (implies (and (sig-path-equiv x y)
                     (sig-path-equiv y z))
                (sig-path-equiv x z)))
  :rule-classes (:equivalence))

Theorem: sig-path-equiv-implies-equal-sig-path-fix-1

(defthm sig-path-equiv-implies-equal-sig-path-fix-1
  (implies (sig-path-equiv acl2::x x-equiv)
           (equal (sig-path-fix acl2::x)
                  (sig-path-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: sig-path-fix-under-sig-path-equiv

(defthm sig-path-fix-under-sig-path-equiv
  (sig-path-equiv (sig-path-fix acl2::x)
                  acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-sig-path-fix-1-forward-to-sig-path-equiv

(defthm equal-of-sig-path-fix-1-forward-to-sig-path-equiv
  (implies (equal (sig-path-fix acl2::x) acl2::y)
           (sig-path-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-sig-path-fix-2-forward-to-sig-path-equiv

(defthm equal-of-sig-path-fix-2-forward-to-sig-path-equiv
  (implies (equal acl2::x (sig-path-fix acl2::y))
           (sig-path-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: sig-path-equiv-of-sig-path-fix-1-forward

(defthm sig-path-equiv-of-sig-path-fix-1-forward
  (implies (sig-path-equiv (sig-path-fix acl2::x)
                           acl2::y)
           (sig-path-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: sig-path-equiv-of-sig-path-fix-2-forward

(defthm sig-path-equiv-of-sig-path-fix-2-forward
  (implies (sig-path-equiv acl2::x (sig-path-fix acl2::y))
           (sig-path-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: car-of-sig-path-fix-x-under-sig-equiv

(defthm car-of-sig-path-fix-x-under-sig-equiv
  (sig-equiv (car (sig-path-fix acl2::x))
             (car acl2::x)))

Theorem: car-sig-path-equiv-congruence-on-x-under-sig-equiv

(defthm car-sig-path-equiv-congruence-on-x-under-sig-equiv
  (implies (sig-path-equiv acl2::x x-equiv)
           (sig-equiv (car acl2::x) (car x-equiv)))
  :rule-classes :congruence)

Theorem: cdr-of-sig-path-fix-x-under-sig-path-equiv

(defthm cdr-of-sig-path-fix-x-under-sig-path-equiv
  (sig-path-equiv (cdr (sig-path-fix acl2::x))
                  (cdr acl2::x)))

Theorem: cdr-sig-path-equiv-congruence-on-x-under-sig-path-equiv

(defthm cdr-sig-path-equiv-congruence-on-x-under-sig-path-equiv
  (implies (sig-path-equiv acl2::x x-equiv)
           (sig-path-equiv (cdr acl2::x)
                           (cdr x-equiv)))
  :rule-classes :congruence)

Theorem: cons-of-sig-fix-x-under-sig-path-equiv

(defthm cons-of-sig-fix-x-under-sig-path-equiv
  (sig-path-equiv (cons (sig-fix acl2::x) acl2::y)
                  (cons acl2::x acl2::y)))

Theorem: cons-sig-equiv-congruence-on-x-under-sig-path-equiv

(defthm cons-sig-equiv-congruence-on-x-under-sig-path-equiv
  (implies (sig-equiv acl2::x x-equiv)
           (sig-path-equiv (cons acl2::x acl2::y)
                           (cons x-equiv acl2::y)))
  :rule-classes :congruence)

Theorem: cons-of-sig-path-fix-y-under-sig-path-equiv

(defthm cons-of-sig-path-fix-y-under-sig-path-equiv
  (sig-path-equiv (cons acl2::x (sig-path-fix acl2::y))
                  (cons acl2::x acl2::y)))

Theorem: cons-sig-path-equiv-congruence-on-y-under-sig-path-equiv

(defthm cons-sig-path-equiv-congruence-on-y-under-sig-path-equiv
  (implies (sig-path-equiv acl2::y y-equiv)
           (sig-path-equiv (cons acl2::x acl2::y)
                           (cons acl2::x y-equiv)))
  :rule-classes :congruence)

Theorem: consp-of-sig-path-fix

(defthm consp-of-sig-path-fix
  (equal (consp (sig-path-fix acl2::x))
         (consp acl2::x)))

Theorem: sig-path-fix-under-iff

(defthm sig-path-fix-under-iff
  (iff (sig-path-fix acl2::x)
       (consp acl2::x)))

Theorem: sig-path-fix-of-cons

(defthm sig-path-fix-of-cons
  (equal (sig-path-fix (cons a x))
         (cons (sig-fix a) (sig-path-fix x))))

Theorem: len-of-sig-path-fix

(defthm len-of-sig-path-fix
  (equal (len (sig-path-fix acl2::x))
         (len acl2::x)))

Theorem: sig-path-fix-of-append

(defthm sig-path-fix-of-append
  (equal (sig-path-fix (append std::a std::b))
         (append (sig-path-fix std::a)
                 (sig-path-fix std::b))))

Theorem: sig-path-fix-of-repeat

(defthm sig-path-fix-of-repeat
  (equal (sig-path-fix (acl2::repeat acl2::n acl2::x))
         (acl2::repeat acl2::n (sig-fix acl2::x))))

Theorem: list-equiv-refines-sig-path-equiv

(defthm list-equiv-refines-sig-path-equiv
  (implies (acl2::list-equiv acl2::x acl2::y)
           (sig-path-equiv acl2::x acl2::y))
  :rule-classes :refinement)

Theorem: nth-of-sig-path-fix

(defthm nth-of-sig-path-fix
  (equal (nth acl2::n (sig-path-fix acl2::x))
         (if (< (nfix acl2::n) (len acl2::x))
             (sig-fix (nth acl2::n acl2::x))
           nil)))

Theorem: sig-path-equiv-implies-sig-path-equiv-append-1

(defthm sig-path-equiv-implies-sig-path-equiv-append-1
  (implies (sig-path-equiv acl2::x fty::x-equiv)
           (sig-path-equiv (append acl2::x acl2::y)
                           (append fty::x-equiv acl2::y)))
  :rule-classes (:congruence))

Theorem: sig-path-equiv-implies-sig-path-equiv-append-2

(defthm sig-path-equiv-implies-sig-path-equiv-append-2
  (implies (sig-path-equiv acl2::y fty::y-equiv)
           (sig-path-equiv (append acl2::x acl2::y)
                           (append acl2::x fty::y-equiv)))
  :rule-classes (:congruence))

Theorem: sig-path-equiv-implies-sig-path-equiv-nthcdr-2

(defthm sig-path-equiv-implies-sig-path-equiv-nthcdr-2
  (implies (sig-path-equiv acl2::l l-equiv)
           (sig-path-equiv (nthcdr acl2::n acl2::l)
                           (nthcdr acl2::n l-equiv)))
  :rule-classes (:congruence))

Theorem: sig-path-equiv-implies-sig-path-equiv-take-2

(defthm sig-path-equiv-implies-sig-path-equiv-take-2
  (implies (sig-path-equiv acl2::l l-equiv)
           (sig-path-equiv (take acl2::n acl2::l)
                           (take acl2::n l-equiv)))
  :rule-classes (:congruence))