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• Amb?-expr/tyname

Amb?-expr/tyname-fix

Fixing function for amb?-expr/tyname structures.

Signature

(amb?-expr/tyname-fix x) → new-x

Arguments

x — Guard (amb?-expr/tyname-p x).

Returns

new-x — Type (amb?-expr/tyname-p new-x).

Definitions and Theorems

Function: amb?-expr/tyname-fix$inline

(defun amb?-expr/tyname-fix$inline (x)
  (declare (xargs :guard (amb?-expr/tyname-p x)))
  (mbe :logic
       (case (amb?-expr/tyname-kind x)
         (:expr (b* ((expr (expr-fix (cdr x))))
                  (cons :expr expr)))
         (:tyname (b* ((tyname (tyname-fix (cdr x))))
                    (cons :tyname tyname)))
         (:ambig (b* ((expr/tyname (amb-expr/tyname-fix (cdr x))))
                   (cons :ambig expr/tyname))))
       :exec x))

Theorem: amb?-expr/tyname-p-of-amb?-expr/tyname-fix

(defthm amb?-expr/tyname-p-of-amb?-expr/tyname-fix
  (b* ((new-x (amb?-expr/tyname-fix$inline x)))
    (amb?-expr/tyname-p new-x))
  :rule-classes :rewrite)

Theorem: amb?-expr/tyname-fix-when-amb?-expr/tyname-p

(defthm amb?-expr/tyname-fix-when-amb?-expr/tyname-p
  (implies (amb?-expr/tyname-p x)
           (equal (amb?-expr/tyname-fix x) x)))

Function: amb?-expr/tyname-equiv$inline

(defun amb?-expr/tyname-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (amb?-expr/tyname-p acl2::x)
                              (amb?-expr/tyname-p acl2::y))))
  (equal (amb?-expr/tyname-fix acl2::x)
         (amb?-expr/tyname-fix acl2::y)))

Theorem: amb?-expr/tyname-equiv-is-an-equivalence

(defthm amb?-expr/tyname-equiv-is-an-equivalence
  (and (booleanp (amb?-expr/tyname-equiv x y))
       (amb?-expr/tyname-equiv x x)
       (implies (amb?-expr/tyname-equiv x y)
                (amb?-expr/tyname-equiv y x))
       (implies (and (amb?-expr/tyname-equiv x y)
                     (amb?-expr/tyname-equiv y z))
                (amb?-expr/tyname-equiv x z)))
  :rule-classes (:equivalence))

Theorem: amb?-expr/tyname-equiv-implies-equal-amb?-expr/tyname-fix-1

(defthm amb?-expr/tyname-equiv-implies-equal-amb?-expr/tyname-fix-1
  (implies (amb?-expr/tyname-equiv acl2::x x-equiv)
           (equal (amb?-expr/tyname-fix acl2::x)
                  (amb?-expr/tyname-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: amb?-expr/tyname-fix-under-amb?-expr/tyname-equiv

(defthm amb?-expr/tyname-fix-under-amb?-expr/tyname-equiv
  (amb?-expr/tyname-equiv (amb?-expr/tyname-fix acl2::x)
                          acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-amb?-expr/tyname-fix-1-forward-to-amb?-expr/tyname-equiv

(defthm
  equal-of-amb?-expr/tyname-fix-1-forward-to-amb?-expr/tyname-equiv
  (implies (equal (amb?-expr/tyname-fix acl2::x)
                  acl2::y)
           (amb?-expr/tyname-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-amb?-expr/tyname-fix-2-forward-to-amb?-expr/tyname-equiv

(defthm
  equal-of-amb?-expr/tyname-fix-2-forward-to-amb?-expr/tyname-equiv
  (implies (equal acl2::x (amb?-expr/tyname-fix acl2::y))
           (amb?-expr/tyname-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: amb?-expr/tyname-equiv-of-amb?-expr/tyname-fix-1-forward

(defthm amb?-expr/tyname-equiv-of-amb?-expr/tyname-fix-1-forward
  (implies (amb?-expr/tyname-equiv (amb?-expr/tyname-fix acl2::x)
                                   acl2::y)
           (amb?-expr/tyname-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: amb?-expr/tyname-equiv-of-amb?-expr/tyname-fix-2-forward

(defthm amb?-expr/tyname-equiv-of-amb?-expr/tyname-fix-2-forward
 (implies
     (amb?-expr/tyname-equiv acl2::x (amb?-expr/tyname-fix acl2::y))
     (amb?-expr/tyname-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: amb?-expr/tyname-kind$inline-of-amb?-expr/tyname-fix-x

(defthm amb?-expr/tyname-kind$inline-of-amb?-expr/tyname-fix-x
  (equal (amb?-expr/tyname-kind$inline (amb?-expr/tyname-fix x))
         (amb?-expr/tyname-kind$inline x)))

Theorem: amb?-expr/tyname-kind$inline-amb?-expr/tyname-equiv-congruence-on-x

(defthm
 amb?-expr/tyname-kind$inline-amb?-expr/tyname-equiv-congruence-on-x
 (implies (amb?-expr/tyname-equiv x x-equiv)
          (equal (amb?-expr/tyname-kind$inline x)
                 (amb?-expr/tyname-kind$inline x-equiv)))
 :rule-classes :congruence)

Theorem: consp-of-amb?-expr/tyname-fix

(defthm consp-of-amb?-expr/tyname-fix
  (consp (amb?-expr/tyname-fix x))
  :rule-classes :type-prescription)

Theorem: amb?-expr/tyname-fix$inline-of-amb?-expr/tyname-fix-x

(defthm amb?-expr/tyname-fix$inline-of-amb?-expr/tyname-fix-x
  (equal (amb?-expr/tyname-fix$inline (amb?-expr/tyname-fix x))
         (amb?-expr/tyname-fix$inline x)))

Theorem: amb?-expr/tyname-fix$inline-amb?-expr/tyname-equiv-congruence-on-x

(defthm
 amb?-expr/tyname-fix$inline-amb?-expr/tyname-equiv-congruence-on-x
 (implies (amb?-expr/tyname-equiv x x-equiv)
          (equal (amb?-expr/tyname-fix$inline x)
                 (amb?-expr/tyname-fix$inline x-equiv)))
 :rule-classes :congruence)