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• Sym-prod

Sym-prod-fix

Fixing function for sym-prod structures.

Signature

(sym-prod-fix x) → new-x

Arguments

x — Guard (sym-prod-p x).

Returns

new-x — Type (sym-prod-p new-x).

Definitions and Theorems

Function: sym-prod-fix$inline

(defun sym-prod-fix$inline (x)
  (declare (xargs :guard (sym-prod-p x)))
  (let ((acl2::__function__ 'sym-prod-fix))
    (declare (ignorable acl2::__function__))
    (mbe :logic
         (b* ((sym (symbol-fix (cdr (std::da-nth 0 x)))))
           (list (cons 'sym sym)))
         :exec x)))

Theorem: sym-prod-p-of-sym-prod-fix

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

Theorem: sym-prod-fix-when-sym-prod-p

(defthm sym-prod-fix-when-sym-prod-p
  (implies (sym-prod-p x)
           (equal (sym-prod-fix x) x)))

Function: sym-prod-equiv$inline

(defun sym-prod-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (sym-prod-p acl2::x)
                              (sym-prod-p acl2::y))))
  (equal (sym-prod-fix acl2::x)
         (sym-prod-fix acl2::y)))

Theorem: sym-prod-equiv-is-an-equivalence

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

Theorem: sym-prod-equiv-implies-equal-sym-prod-fix-1

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

Theorem: sym-prod-fix-under-sym-prod-equiv

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

Theorem: equal-of-sym-prod-fix-1-forward-to-sym-prod-equiv

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

Theorem: equal-of-sym-prod-fix-2-forward-to-sym-prod-equiv

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

Theorem: sym-prod-equiv-of-sym-prod-fix-1-forward

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

Theorem: sym-prod-equiv-of-sym-prod-fix-2-forward

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