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• Span

Span-fix

Fixing function for span structures.

Signature

(span-fix x) → new-x

Arguments

x — Guard (spanp x).

Returns

new-x — Type (spanp new-x).

Definitions and Theorems

Function: span-fix$inline

(defun span-fix$inline (x)
  (declare (xargs :guard (spanp x)))
  (mbe :logic
       (b* ((start (position-fix (car x)))
            (end (position-fix (cdr x))))
         (cons start end))
       :exec x))

Theorem: spanp-of-span-fix

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

Theorem: span-fix-when-spanp

(defthm span-fix-when-spanp
  (implies (spanp x)
           (equal (span-fix x) x)))

Function: span-equiv$inline

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

Theorem: span-equiv-is-an-equivalence

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

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

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

Theorem: span-fix-under-span-equiv

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

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

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

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

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

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

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

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

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

Theorem: span-fix$inline-of-span-fix-x

(defthm span-fix$inline-of-span-fix-x
  (equal (span-fix$inline (span-fix x))
         (span-fix$inline x)))

Theorem: span-fix$inline-span-equiv-congruence-on-x

(defthm span-fix$inline-span-equiv-congruence-on-x
  (implies (span-equiv x x-equiv)
           (equal (span-fix$inline x)
                  (span-fix$inline x-equiv)))
  :rule-classes :congruence)