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Function-specifier

Function-specifier-fix

Fixing function for function-specifier structures.

Signature
(function-specifier-fix x) → new-x
Arguments: x — Guard (function-specifierp x).
Returns: new-x — Type (function-specifierp new-x).

Definitions and Theorems

Function: function-specifier-fix$inline

(defun function-specifier-fix$inline (x)
 (declare (xargs :guard (function-specifierp x)))
 (let ((__function__ 'function-specifier-fix))
  (declare (ignorable __function__))
  (mbe
   :logic
   (case (function-specifier-kind x)
    (:regular (b* ((body (expression-fix (std::da-nth 0 (cdr x)))))
                (cons :regular (list body))))
    (:quantified
     (b*
      ((quantifier (quantifier-fix (std::da-nth 0 (cdr x))))
       (variables (typed-variable-list-fix (std::da-nth 1 (cdr x))))
       (matrix (expression-fix (std::da-nth 2 (cdr x)))))
      (cons :quantified (list quantifier variables matrix))))
    (:input-output
         (b* ((relation (expression-fix (std::da-nth 0 (cdr x)))))
           (cons :input-output (list relation)))))
   :exec x)))

Theorem: function-specifierp-of-function-specifier-fix

(defthm function-specifierp-of-function-specifier-fix
  (b* ((new-x (function-specifier-fix$inline x)))
    (function-specifierp new-x))
  :rule-classes :rewrite)

Theorem: function-specifier-fix-when-function-specifierp

(defthm function-specifier-fix-when-function-specifierp
  (implies (function-specifierp x)
           (equal (function-specifier-fix x) x)))

Function: function-specifier-equiv$inline

(defun function-specifier-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (function-specifierp acl2::x)
                              (function-specifierp acl2::y))))
  (equal (function-specifier-fix acl2::x)
         (function-specifier-fix acl2::y)))

Theorem: function-specifier-equiv-is-an-equivalence

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

Theorem: function-specifier-equiv-implies-equal-function-specifier-fix-1

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

Theorem: function-specifier-fix-under-function-specifier-equiv

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

Theorem: equal-of-function-specifier-fix-1-forward-to-function-specifier-equiv

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

Theorem: equal-of-function-specifier-fix-2-forward-to-function-specifier-equiv

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

Theorem: function-specifier-equiv-of-function-specifier-fix-1-forward

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

Theorem: function-specifier-equiv-of-function-specifier-fix-2-forward

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

Theorem: function-specifier-kind$inline-of-function-specifier-fix-x

(defthm function-specifier-kind$inline-of-function-specifier-fix-x
  (equal (function-specifier-kind$inline (function-specifier-fix x))
         (function-specifier-kind$inline x)))

Theorem: function-specifier-kind$inline-function-specifier-equiv-congruence-on-x

(defthm
 function-specifier-kind$inline-function-specifier-equiv-congruence-on-x
 (implies (function-specifier-equiv x x-equiv)
          (equal (function-specifier-kind$inline x)
                 (function-specifier-kind$inline x-equiv)))
 :rule-classes :congruence)

Theorem: consp-of-function-specifier-fix

(defthm consp-of-function-specifier-fix
  (consp (function-specifier-fix x))
  :rule-classes :type-prescription)