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Lhs.lisp

Lhs-normp

Signature
(lhs-normp x) → *
Arguments: x — Guard (lhs-p x).

Definitions and Theorems

Function: lhs-normp

(defun lhs-normp (x)
       (declare (xargs :guard (lhs-p x)))
       (let ((__function__ 'lhs-normp))
            (declare (ignorable __function__))
            (equal x (lhs-norm x))))

Function: lhs-norm-equiv

(defun lhs-norm-equiv (x y)
       (declare (xargs :non-executable t))
       (prog2$ (acl2::throw-nonexec-error 'lhs-norm-equiv
                                          (list x y))
               (equal (lhs-norm x) (lhs-norm y))))

Theorem: lhs-norm-equiv-is-an-equivalence

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

Theorem: lhs-norm-equiv-implies-equal-lhs-norm-1

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

Theorem: lhs-norm-under-lhs-norm-equiv

(defthm lhs-norm-under-lhs-norm-equiv
        (lhs-norm-equiv (lhs-norm x) x)
        :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-lhs-norm-1-forward-to-lhs-norm-equiv

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

Theorem: equal-of-lhs-norm-2-forward-to-lhs-norm-equiv

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

Theorem: lhs-norm-equiv-of-lhs-norm-1-forward

(defthm lhs-norm-equiv-of-lhs-norm-1-forward
        (implies (lhs-norm-equiv (lhs-norm x) y)
                 (lhs-norm-equiv x y))
        :rule-classes :forward-chaining)

Theorem: lhs-norm-equiv-of-lhs-norm-2-forward

(defthm lhs-norm-equiv-of-lhs-norm-2-forward
        (implies (lhs-norm-equiv x (lhs-norm y))
                 (lhs-norm-equiv x y))
        :rule-classes :forward-chaining)

Theorem: lhs-equiv-refines-lhs-norm-equiv

(defthm lhs-equiv-refines-lhs-norm-equiv
        (implies (lhs-equiv x y)
                 (lhs-norm-equiv x y))
        :rule-classes (:refinement))

Theorem: lhs-cons-of-lhs-norm-y-under-lhs-norm-equiv

(defthm lhs-cons-of-lhs-norm-y-under-lhs-norm-equiv
        (lhs-norm-equiv (lhs-cons x (lhs-norm y))
                        (lhs-cons x y)))

Theorem: lhs-cons-lhs-norm-equiv-congruence-on-y-under-lhs-norm-equiv

(defthm lhs-cons-lhs-norm-equiv-congruence-on-y-under-lhs-norm-equiv
        (implies (lhs-norm-equiv y y-equiv)
                 (lhs-norm-equiv (lhs-cons x y)
                                 (lhs-cons x y-equiv)))
        :rule-classes :congruence)

Theorem: lhs-eval-of-lhs-norm-x

(defthm lhs-eval-of-lhs-norm-x
        (equal (lhs-eval (lhs-norm x) env)
               (lhs-eval x env)))

Theorem: lhs-eval-lhs-norm-equiv-congruence-on-x

(defthm lhs-eval-lhs-norm-equiv-congruence-on-x
        (implies (lhs-norm-equiv x x-equiv)
                 (equal (lhs-eval x env)
                        (lhs-eval x-equiv env)))
        :rule-classes :congruence)

Theorem: lhs-eval-zero-of-lhs-norm-x

(defthm lhs-eval-zero-of-lhs-norm-x
        (equal (lhs-eval-zero (lhs-norm x) env)
               (lhs-eval-zero x env)))

Theorem: lhs-eval-zero-lhs-norm-equiv-congruence-on-x

(defthm lhs-eval-zero-lhs-norm-equiv-congruence-on-x
        (implies (lhs-norm-equiv x x-equiv)
                 (equal (lhs-eval-zero x env)
                        (lhs-eval-zero x-equiv env)))
        :rule-classes :congruence)

Theorem: lhs-norm-cdr-lhs-norm

(defthm lhs-norm-cdr-lhs-norm
        (implies (lhs-normp x)
                 (lhs-normp (cdr x))))

Theorem: lhs-normp-of-lhs-norm

(defthm lhs-normp-of-lhs-norm
        (lhs-normp (lhs-norm x)))

Theorem: lhs-normp-of-lhs-cons

(defthm lhs-normp-of-lhs-cons
        (implies (lhs-normp x)
                 (lhs-normp (lhs-cons a x))))

Theorem: lhs-norm-when-lhs-normp

(defthm lhs-norm-when-lhs-normp
        (implies (lhs-normp x)
                 (equal (lhs-norm x) x)))

Theorem: lhs-normp-of-lhs-fix

(defthm lhs-normp-of-lhs-fix
        (implies (lhs-normp x)
                 (lhs-normp (lhs-fix x))))