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• Truth

Env-diff-index

Signature

(env-diff-index env1 env2) → idx

Arguments

env1 — Guard (natp env1).

env2 — Guard (natp env2).

Returns

idx — Type (natp idx).

Definitions and Theorems

Definition: env-diff-index1

(defchoose env-diff-index1 (idx)
           (env1 env2)
           (not (equal (env-lookup idx env1)
                       (env-lookup idx env2))))

Function: env-diff-index

(defun env-diff-index (env1 env2)
       (declare (xargs :guard (and (natp env1) (natp env2))))
       (let ((__function__ 'env-diff-index))
            (declare (ignorable __function__))
            (b* ((idx (nfix (env-diff-index1 (lnfix env1)
                                             (lnfix env2)))))
                (if (equal (env-lookup idx env1)
                           (env-lookup idx env2))
                    0 idx))))

Theorem: natp-of-env-diff-index

(defthm natp-of-env-diff-index
        (b* ((idx (env-diff-index env1 env2)))
            (natp idx))
        :rule-classes :type-prescription)

Theorem: env-diff-index-correct

(defthm env-diff-index-correct
        (implies (not (equal (nfix env1) (nfix env2)))
                 (b* ((idx (env-diff-index env1 env2)))
                     (equal (env-lookup idx env1)
                            (not (env-lookup idx env2))))))

Theorem: env-diff-index-of-nfix-env1

(defthm env-diff-index-of-nfix-env1
        (equal (env-diff-index (nfix env1) env2)
               (env-diff-index env1 env2)))

Theorem: env-diff-index-nat-equiv-congruence-on-env1

(defthm env-diff-index-nat-equiv-congruence-on-env1
        (implies (nat-equiv env1 env1-equiv)
                 (equal (env-diff-index env1 env2)
                        (env-diff-index env1-equiv env2)))
        :rule-classes :congruence)

Theorem: env-diff-index-of-nfix-env2

(defthm env-diff-index-of-nfix-env2
        (equal (env-diff-index env1 (nfix env2))
               (env-diff-index env1 env2)))

Theorem: env-diff-index-nat-equiv-congruence-on-env2

(defthm env-diff-index-nat-equiv-congruence-on-env2
        (implies (nat-equiv env2 env2-equiv)
                 (equal (env-diff-index env1 env2)
                        (env-diff-index env1 env2-equiv)))
        :rule-classes :congruence)