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Truth

Env-swap-vars

Signature
(env-swap-vars n m env) → swap-env
Arguments: n — first element to swap.
Guard (natp n).; m — second element to swap.
Guard (natp m).; env — env to apply the swap to.
Guard (natp env).
Returns: swap-env — Type (natp swap-env).

Definitions and Theorems

Function: env-swap-vars

(defun env-swap-vars (n m env)
       (declare (xargs :guard (and (natp n) (natp m) (natp env))))
       (let ((__function__ 'env-swap-vars))
            (declare (ignorable __function__))
            (b* ((env (lnfix env))
                 (n (lnfix n))
                 (m (lnfix m)))
                (env-update n (env-lookup m env)
                            (env-update m (env-lookup n env)
                                        env)))))

Theorem: natp-of-env-swap-vars

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

Theorem: env-swap-vars-commute

(defthm env-swap-vars-commute
        (equal (env-swap-vars m n x)
               (env-swap-vars n m x))
        :rule-classes ((:rewrite :loop-stopper ((n m)))))

Theorem: lookup-in-env-swap-vars

(defthm lookup-in-env-swap-vars
        (b* ((?swap-env (env-swap-vars n m env)))
            (equal (env-lookup k swap-env)
                   (env-lookup (index-swap n m k) env))))

Theorem: env-swap-vars-inverse

(defthm env-swap-vars-inverse
        (equal (env-swap-vars n m (env-swap-vars n m env))
               (nfix env)))

Theorem: env-swap-vars-self

(defthm env-swap-vars-self
        (equal (env-swap-vars n n env)
               (nfix env)))

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

(defthm env-swap-vars-of-nfix-n
        (equal (env-swap-vars (nfix n) m env)
               (env-swap-vars n m env)))

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

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

Theorem: env-swap-vars-of-nfix-m

(defthm env-swap-vars-of-nfix-m
        (equal (env-swap-vars n (nfix m) env)
               (env-swap-vars n m env)))

Theorem: env-swap-vars-nat-equiv-congruence-on-m

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

Theorem: env-swap-vars-of-nfix-env

(defthm env-swap-vars-of-nfix-env
        (equal (env-swap-vars n m (nfix env))
               (env-swap-vars n m env)))

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

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