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Maybe-svar-fix

Signature

(maybe-svar-fix x) → xx

Arguments

x — Guard (maybe-svar-p x).

Returns

xx — Type (maybe-svar-p xx).

Definitions and Theorems

Function: maybe-svar-fix$inline

(defun maybe-svar-fix$inline (x)
  (declare (xargs :guard (maybe-svar-p x)))
  (let ((__function__ 'maybe-svar-fix))
    (declare (ignorable __function__))
    (mbe :logic (and x (svar-fix x))
         :exec x)))

Theorem: maybe-svar-p-of-maybe-svar-fix

(defthm maybe-svar-p-of-maybe-svar-fix
  (b* ((xx (maybe-svar-fix$inline x)))
    (maybe-svar-p xx))
  :rule-classes :rewrite)

Theorem: maybe-svar-fix-when-maybe-svar-p

(defthm maybe-svar-fix-when-maybe-svar-p
  (implies (maybe-svar-p x)
           (equal (maybe-svar-fix x) x)))

Function: maybe-svar-equiv$inline

(defun maybe-svar-equiv$inline (x y)
  (declare (xargs :guard (and (maybe-svar-p x)
                              (maybe-svar-p y))))
  (equal (maybe-svar-fix x)
         (maybe-svar-fix y)))

Theorem: maybe-svar-equiv-is-an-equivalence

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

Theorem: maybe-svar-equiv-implies-equal-maybe-svar-fix-1

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

Theorem: maybe-svar-fix-under-maybe-svar-equiv

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

Theorem: equal-of-maybe-svar-fix-1-forward-to-maybe-svar-equiv

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

Theorem: equal-of-maybe-svar-fix-2-forward-to-maybe-svar-equiv

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

Theorem: maybe-svar-equiv-of-maybe-svar-fix-1-forward

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

Theorem: maybe-svar-equiv-of-maybe-svar-fix-2-forward

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

Theorem: maybe-svar-equiv-refines-svar-equiv

(defthm maybe-svar-equiv-refines-svar-equiv
  (implies (maybe-svar-equiv x y)
           (svar-equiv x y))
  :rule-classes (:refinement))

Theorem: svar-p-of-maybe-svar-fix

(defthm svar-p-of-maybe-svar-fix
  (implies (maybe-svar-fix x)
           (svar-p (maybe-svar-fix x))))