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Equiv-contexts

Equiv-contexts-fix

(equiv-contexts-fix x) is a usual ACL2::fty list fixing function.

Signature
(equiv-contexts-fix x) → fty::newx
Arguments: x — Guard (equiv-contextsp x).
Returns: fty::newx — Type (equiv-contextsp fty::newx).

In the logic, we apply pseudo-fnsym-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: equiv-contexts-fix$inline

(defun equiv-contexts-fix$inline (x)
  (declare (xargs :guard (equiv-contextsp x)))
  (let ((__function__ 'equiv-contexts-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             nil
           (cons (pseudo-fnsym-fix (car x))
                 (equiv-contexts-fix (cdr x))))
         :exec x)))

Theorem: equiv-contextsp-of-equiv-contexts-fix

(defthm equiv-contextsp-of-equiv-contexts-fix
  (b* ((fty::newx (equiv-contexts-fix$inline x)))
    (equiv-contextsp fty::newx))
  :rule-classes :rewrite)

Theorem: equiv-contexts-fix-when-equiv-contextsp

(defthm equiv-contexts-fix-when-equiv-contextsp
  (implies (equiv-contextsp x)
           (equal (equiv-contexts-fix x) x)))

Function: equiv-contexts-equiv$inline

(defun equiv-contexts-equiv$inline (x y)
  (declare (xargs :guard (and (equiv-contextsp x)
                              (equiv-contextsp y))))
  (equal (equiv-contexts-fix x)
         (equiv-contexts-fix y)))

Theorem: equiv-contexts-equiv-is-an-equivalence

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

Theorem: equiv-contexts-equiv-implies-equal-equiv-contexts-fix-1

(defthm equiv-contexts-equiv-implies-equal-equiv-contexts-fix-1
  (implies (equiv-contexts-equiv x x-equiv)
           (equal (equiv-contexts-fix x)
                  (equiv-contexts-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: equiv-contexts-fix-under-equiv-contexts-equiv

(defthm equiv-contexts-fix-under-equiv-contexts-equiv
  (equiv-contexts-equiv (equiv-contexts-fix x)
                        x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-equiv-contexts-fix-1-forward-to-equiv-contexts-equiv

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

Theorem: equal-of-equiv-contexts-fix-2-forward-to-equiv-contexts-equiv

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

Theorem: equiv-contexts-equiv-of-equiv-contexts-fix-1-forward

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

Theorem: equiv-contexts-equiv-of-equiv-contexts-fix-2-forward

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

Theorem: car-of-equiv-contexts-fix-x-under-pseudo-fnsym-equiv

(defthm car-of-equiv-contexts-fix-x-under-pseudo-fnsym-equiv
  (pseudo-fnsym-equiv (car (equiv-contexts-fix x))
                      (car x)))

Theorem: car-equiv-contexts-equiv-congruence-on-x-under-pseudo-fnsym-equiv

(defthm
  car-equiv-contexts-equiv-congruence-on-x-under-pseudo-fnsym-equiv
  (implies (equiv-contexts-equiv x x-equiv)
           (pseudo-fnsym-equiv (car x)
                               (car x-equiv)))
  :rule-classes :congruence)

Theorem: cdr-of-equiv-contexts-fix-x-under-equiv-contexts-equiv

(defthm cdr-of-equiv-contexts-fix-x-under-equiv-contexts-equiv
  (equiv-contexts-equiv (cdr (equiv-contexts-fix x))
                        (cdr x)))

Theorem: cdr-equiv-contexts-equiv-congruence-on-x-under-equiv-contexts-equiv

(defthm
 cdr-equiv-contexts-equiv-congruence-on-x-under-equiv-contexts-equiv
 (implies (equiv-contexts-equiv x x-equiv)
          (equiv-contexts-equiv (cdr x)
                                (cdr x-equiv)))
 :rule-classes :congruence)

Theorem: cons-of-pseudo-fnsym-fix-x-under-equiv-contexts-equiv

(defthm cons-of-pseudo-fnsym-fix-x-under-equiv-contexts-equiv
  (equiv-contexts-equiv (cons (pseudo-fnsym-fix x) y)
                        (cons x y)))

Theorem: cons-pseudo-fnsym-equiv-congruence-on-x-under-equiv-contexts-equiv

(defthm
 cons-pseudo-fnsym-equiv-congruence-on-x-under-equiv-contexts-equiv
 (implies (pseudo-fnsym-equiv x x-equiv)
          (equiv-contexts-equiv (cons x y)
                                (cons x-equiv y)))
 :rule-classes :congruence)

Theorem: cons-of-equiv-contexts-fix-y-under-equiv-contexts-equiv

(defthm cons-of-equiv-contexts-fix-y-under-equiv-contexts-equiv
  (equiv-contexts-equiv (cons x (equiv-contexts-fix y))
                        (cons x y)))

Theorem: cons-equiv-contexts-equiv-congruence-on-y-under-equiv-contexts-equiv

(defthm
 cons-equiv-contexts-equiv-congruence-on-y-under-equiv-contexts-equiv
 (implies (equiv-contexts-equiv y y-equiv)
          (equiv-contexts-equiv (cons x y)
                                (cons x y-equiv)))
 :rule-classes :congruence)

Theorem: consp-of-equiv-contexts-fix

(defthm consp-of-equiv-contexts-fix
  (equal (consp (equiv-contexts-fix x))
         (consp x)))

Theorem: equiv-contexts-fix-under-iff

(defthm equiv-contexts-fix-under-iff
  (iff (equiv-contexts-fix x) (consp x)))

Theorem: equiv-contexts-fix-of-cons

(defthm equiv-contexts-fix-of-cons
  (equal (equiv-contexts-fix (cons a x))
         (cons (pseudo-fnsym-fix a)
               (equiv-contexts-fix x))))

Theorem: len-of-equiv-contexts-fix

(defthm len-of-equiv-contexts-fix
  (equal (len (equiv-contexts-fix x))
         (len x)))

Theorem: equiv-contexts-fix-of-append

(defthm equiv-contexts-fix-of-append
  (equal (equiv-contexts-fix (append std::a std::b))
         (append (equiv-contexts-fix std::a)
                 (equiv-contexts-fix std::b))))

Theorem: equiv-contexts-fix-of-repeat

(defthm equiv-contexts-fix-of-repeat
  (equal (equiv-contexts-fix (repeat n x))
         (repeat n (pseudo-fnsym-fix x))))

Theorem: list-equiv-refines-equiv-contexts-equiv

(defthm list-equiv-refines-equiv-contexts-equiv
  (implies (list-equiv x y)
           (equiv-contexts-equiv x y))
  :rule-classes :refinement)

Theorem: nth-of-equiv-contexts-fix

(defthm nth-of-equiv-contexts-fix
  (equal (nth n (equiv-contexts-fix x))
         (if (< (nfix n) (len x))
             (pseudo-fnsym-fix (nth n x))
           nil)))

Theorem: equiv-contexts-equiv-implies-equiv-contexts-equiv-append-1

(defthm equiv-contexts-equiv-implies-equiv-contexts-equiv-append-1
  (implies (equiv-contexts-equiv x fty::x-equiv)
           (equiv-contexts-equiv (append x y)
                                 (append fty::x-equiv y)))
  :rule-classes (:congruence))

Theorem: equiv-contexts-equiv-implies-equiv-contexts-equiv-append-2

(defthm equiv-contexts-equiv-implies-equiv-contexts-equiv-append-2
  (implies (equiv-contexts-equiv y fty::y-equiv)
           (equiv-contexts-equiv (append x y)
                                 (append x fty::y-equiv)))
  :rule-classes (:congruence))

Theorem: equiv-contexts-equiv-implies-equiv-contexts-equiv-nthcdr-2

(defthm equiv-contexts-equiv-implies-equiv-contexts-equiv-nthcdr-2
  (implies (equiv-contexts-equiv l l-equiv)
           (equiv-contexts-equiv (nthcdr n l)
                                 (nthcdr n l-equiv)))
  :rule-classes (:congruence))

Theorem: equiv-contexts-equiv-implies-equiv-contexts-equiv-take-2

(defthm equiv-contexts-equiv-implies-equiv-contexts-equiv-take-2
  (implies (equiv-contexts-equiv l l-equiv)
           (equiv-contexts-equiv (take n l)
                                 (take n l-equiv)))
  :rule-classes (:congruence))