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• Truth

Env-update

Signature

(env-update var val env) → new-env

Arguments

var — Guard (natp var).

val — Guard (booleanp val).

env — Guard (natp env).

Returns

new-env — Type (natp new-env).

Definitions and Theorems

Function: env-update

(defun env-update (var val env)
       (declare (xargs :guard (and (natp var)
                                   (booleanp val)
                                   (natp env))))
       (let ((__function__ 'env-update))
            (declare (ignorable __function__))
            (install-bit var (bool->bit val)
                         (lnfix env))))

Theorem: natp-of-env-update

(defthm natp-of-env-update
        (b* ((new-env (env-update var val env)))
            (natp new-env))
        :rule-classes :type-prescription)

Theorem: env-lookup-of-env-update-split

(defthm env-lookup-of-env-update-split
        (equal (env-lookup var (env-update var1 val env))
               (if (equal (nfix var) (nfix var1))
                   (acl2::bool-fix val)
                   (env-lookup var env))))

Theorem: env-update-of-env-update

(defthm env-update-of-env-update
        (equal (env-update n x (env-update n y env))
               (env-update n x env)))

Theorem: env-update-swap

(defthm
    env-update-swap
    (implies
         (not (nat-equiv var1 var2))
         (equal (env-update var2 val2 (env-update var1 val1 env))
                (env-update var1 val1 (env-update var2 val2 env)))))

Theorem: env-update-redundant

(defthm env-update-redundant
        (implies (iff (env-lookup n env) val)
                 (equal (env-update n val env)
                        (nfix env))))

Theorem: env-update-of-nfix-var

(defthm env-update-of-nfix-var
        (equal (env-update (nfix var) val env)
               (env-update var val env)))

Theorem: env-update-nat-equiv-congruence-on-var

(defthm env-update-nat-equiv-congruence-on-var
        (implies (nat-equiv var var-equiv)
                 (equal (env-update var val env)
                        (env-update var-equiv val env)))
        :rule-classes :congruence)

Theorem: env-update-of-bool-fix-val

(defthm env-update-of-bool-fix-val
        (equal (env-update var (acl2::bool-fix val)
                           env)
               (env-update var val env)))

Theorem: env-update-iff-congruence-on-val

(defthm env-update-iff-congruence-on-val
        (implies (iff val val-equiv)
                 (equal (env-update var val env)
                        (env-update var val-equiv env)))
        :rule-classes :congruence)

Theorem: env-update-of-nfix-env

(defthm env-update-of-nfix-env
        (equal (env-update var val (nfix env))
               (env-update var val env)))

Theorem: env-update-nat-equiv-congruence-on-env

(defthm env-update-nat-equiv-congruence-on-env
        (implies (nat-equiv env env-equiv)
                 (equal (env-update var val env)
                        (env-update var val env-equiv)))
        :rule-classes :congruence)