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Congruence-rule

Congruence-rule-fix

Fixing function for congruence-rule structures.

Signature
(congruence-rule-fix x) → new-x
Arguments: x — Guard (congruence-rule-p x).
Returns: new-x — Type (congruence-rule-p new-x).

Definitions and Theorems

Function: congruence-rule-fix$inline

(defun congruence-rule-fix$inline (x)
 (declare (xargs :guard (congruence-rule-p x)))
 (let ((__function__ 'congruence-rule-fix))
  (declare (ignorable __function__))
  (mbe
      :logic
      (b* ((equiv-req (pseudo-fnsym-fix (std::da-nth 0 x)))
           (fn (pseudo-fnsym-fix (std::da-nth 1 x)))
           (arg-contexts (equiv-contextslist-fix (std::da-nth 2 x)))
           (arity (nfix (std::da-nth 3 x))))
        (list equiv-req fn arg-contexts arity))
      :exec x)))

Theorem: congruence-rule-p-of-congruence-rule-fix

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

Theorem: congruence-rule-fix-when-congruence-rule-p

(defthm congruence-rule-fix-when-congruence-rule-p
  (implies (congruence-rule-p x)
           (equal (congruence-rule-fix x) x)))

Function: congruence-rule-equiv$inline

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

Theorem: congruence-rule-equiv-is-an-equivalence

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

Theorem: congruence-rule-equiv-implies-equal-congruence-rule-fix-1

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

Theorem: congruence-rule-fix-under-congruence-rule-equiv

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

Theorem: equal-of-congruence-rule-fix-1-forward-to-congruence-rule-equiv

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

Theorem: equal-of-congruence-rule-fix-2-forward-to-congruence-rule-equiv

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

Theorem: congruence-rule-equiv-of-congruence-rule-fix-1-forward

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

Theorem: congruence-rule-equiv-of-congruence-rule-fix-2-forward

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