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Sponge

Sponge-fix

Fixing function for sponge structures.

Signature
(sponge-fix x) → new-x
Arguments: x — Guard (spongep x).
Returns: new-x — Type (spongep new-x).

Definitions and Theorems

Function: sponge-fix$inline

(defun sponge-fix$inline (x)
  (declare (xargs :guard (spongep x)))
  (let ((__function__ 'sponge-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((stat (acl2::nat-list-fix (cdr (std::da-nth 0 x))))
              (mode (mode-fix (cdr (std::da-nth 1 x))))
              (index (nfix (cdr (std::da-nth 2 x)))))
           (list (cons 'stat stat)
                 (cons 'mode mode)
                 (cons 'index index)))
         :exec x)))

Theorem: spongep-of-sponge-fix

(defthm spongep-of-sponge-fix
  (b* ((new-x (sponge-fix$inline x)))
    (spongep new-x))
  :rule-classes :rewrite)

Theorem: sponge-fix-when-spongep

(defthm sponge-fix-when-spongep
  (implies (spongep x)
           (equal (sponge-fix x) x)))

Function: sponge-equiv$inline

(defun sponge-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (spongep acl2::x)
                              (spongep acl2::y))))
  (equal (sponge-fix acl2::x)
         (sponge-fix acl2::y)))

Theorem: sponge-equiv-is-an-equivalence

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

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

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

Theorem: sponge-fix-under-sponge-equiv

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

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

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

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

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

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

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

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

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