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• Truth

Env-swap-polarity

Signature

(env-swap-polarity n env) → new-env

Arguments

n — Guard (natp n).

env — Guard (natp env).

Returns

new-env — Type (natp new-env).

Definitions and Theorems

Function: env-swap-polarity

(defun env-swap-polarity (n env)
  (declare (xargs :guard (and (natp n) (natp env))))
  (let ((__function__ 'env-swap-polarity))
    (declare (ignorable __function__))
    (env-update n (not (env-lookup n env))
                env)))

Theorem: natp-of-env-swap-polarity

(defthm natp-of-env-swap-polarity
  (b* ((new-env (env-swap-polarity n env)))
    (natp new-env))
  :rule-classes :type-prescription)

Theorem: lookup-in-env-swap-polarity-same

(defthm lookup-in-env-swap-polarity-same
  (b* ((?new-env (env-swap-polarity n env)))
    (equal (env-lookup n new-env)
           (not (env-lookup n env)))))

Theorem: lookup-in-env-swap-polarity-diff

(defthm lookup-in-env-swap-polarity-diff
  (b* ((?new-env (env-swap-polarity n env)))
    (implies (not (nat-equiv n m))
             (equal (env-lookup m new-env)
                    (env-lookup m env)))))

Theorem: lookup-in-env-swap-polarity-split

(defthm lookup-in-env-swap-polarity-split
  (b* ((?new-env (env-swap-polarity n env)))
    (equal (env-lookup m new-env)
           (if (nat-equiv n m)
               (not (env-lookup n env))
             (env-lookup m env)))))

Theorem: env-swap-polarity-reverse

(defthm env-swap-polarity-reverse
  (b* ((?new-env (env-swap-polarity n env)))
    (equal (env-swap-polarity n new-env)
           (nfix env))))

Theorem: env-swap-polarity-commute

(defthm env-swap-polarity-commute
  (b* nil
    (equal (env-swap-polarity n (env-swap-polarity m env))
           (env-swap-polarity m (env-swap-polarity n env))))
  :rule-classes ((:rewrite :loop-stopper ((n m)))))

Theorem: env-swap-polarity-of-nfix-n

(defthm env-swap-polarity-of-nfix-n
  (equal (env-swap-polarity (nfix n) env)
         (env-swap-polarity n env)))

Theorem: env-swap-polarity-nat-equiv-congruence-on-n

(defthm env-swap-polarity-nat-equiv-congruence-on-n
  (implies (nat-equiv n n-equiv)
           (equal (env-swap-polarity n env)
                  (env-swap-polarity n-equiv env)))
  :rule-classes :congruence)

Theorem: env-swap-polarity-of-nfix-env

(defthm env-swap-polarity-of-nfix-env
  (equal (env-swap-polarity n (nfix env))
         (env-swap-polarity n env)))

Theorem: env-swap-polarity-nat-equiv-congruence-on-env

(defthm env-swap-polarity-nat-equiv-congruence-on-env
  (implies (nat-equiv env env-equiv)
           (equal (env-swap-polarity n env)
                  (env-swap-polarity n env-equiv)))
  :rule-classes :congruence)