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• Truth

Env-mismatch-aux

Signature

(env-mismatch-aux n truth env1 env2 numvars) → var

Arguments

n — Guard (natp n).

truth — Guard (integerp truth).

env1 — Guard (natp env1).

env2 — Guard (natp env2).

numvars — Guard (natp numvars).

Returns

var — Type (natp var).

Definitions and Theorems

Function: env-mismatch-aux

(defun env-mismatch-aux (n truth env1 env2 numvars)
  (declare (xargs :guard (and (natp n)
                              (integerp truth)
                              (natp env1)
                              (natp env2)
                              (natp numvars))))
  (declare (xargs :guard (<= n numvars)))
  (let ((__function__ 'env-mismatch-aux))
    (declare (ignorable __function__))
    (b* (((when (mbe :logic (zp (- (nfix numvars) (nfix n)))
                     :exec (eql n numvars)))
          0)
         ((when (and (depends-on n truth numvars)
                     (xor (env-lookup n env1)
                          (env-lookup n env2))))
          (lnfix n)))
      (env-mismatch-aux (+ 1 (lnfix n))
                        truth env1 env2 numvars))))

Theorem: natp-of-env-mismatch-aux

(defthm natp-of-env-mismatch-aux
  (b* ((var (env-mismatch-aux n truth env1 env2 numvars)))
    (natp var))
  :rule-classes :type-prescription)

Theorem: env-mismatch-aux-bound

(defthm env-mismatch-aux-bound
  (b* ((?var (env-mismatch-aux n truth env1 env2 numvars)))
    (implies (posp numvars)
             (< var numvars))))

Theorem: env-mismatch-aux-when-mismatch

(defthm env-mismatch-aux-when-mismatch
  (b* ((?var (env-mismatch-aux n truth env1 env2 numvars)))
    (implies (and (depends-on m truth numvars)
                  (xor (env-lookup m env1)
                       (env-lookup m env2))
                  (< (nfix m) (nfix numvars))
                  (<= (nfix n) (nfix m)))
             (and (depends-on var truth numvars)
                  (iff (env-lookup var env1)
                       (not (env-lookup var env2)))
                  (< var numvars)))))

Theorem: env-mismatch-aux-when-eval-mismatch

(defthm env-mismatch-aux-when-eval-mismatch
  (b* ((?var (env-mismatch-aux n truth env1 env2 numvars)))
    (implies (and (not (equal (truth-eval truth env1 numvars)
                              (truth-eval truth env2 numvars)))
                  (equal (loghead n (nfix env1))
                         (loghead n (nfix env2))))
             (and (depends-on var truth numvars)
                  (iff (env-lookup var env1)
                       (not (env-lookup var env2)))
                  (< var (nfix numvars))
                  (implies (natp numvars)
                           (< var numvars))))))

Theorem: env-mismatch-aux-of-nfix-n

(defthm env-mismatch-aux-of-nfix-n
  (equal (env-mismatch-aux (nfix n)
                           truth env1 env2 numvars)
         (env-mismatch-aux n truth env1 env2 numvars)))

Theorem: env-mismatch-aux-nat-equiv-congruence-on-n

(defthm env-mismatch-aux-nat-equiv-congruence-on-n
  (implies
       (nat-equiv n n-equiv)
       (equal (env-mismatch-aux n truth env1 env2 numvars)
              (env-mismatch-aux n-equiv truth env1 env2 numvars)))
  :rule-classes :congruence)

Theorem: env-mismatch-aux-of-ifix-truth

(defthm env-mismatch-aux-of-ifix-truth
  (equal (env-mismatch-aux n (ifix truth)
                           env1 env2 numvars)
         (env-mismatch-aux n truth env1 env2 numvars)))

Theorem: env-mismatch-aux-int-equiv-congruence-on-truth

(defthm env-mismatch-aux-int-equiv-congruence-on-truth
  (implies
       (int-equiv truth truth-equiv)
       (equal (env-mismatch-aux n truth env1 env2 numvars)
              (env-mismatch-aux n truth-equiv env1 env2 numvars)))
  :rule-classes :congruence)

Theorem: env-mismatch-aux-of-nfix-env1

(defthm env-mismatch-aux-of-nfix-env1
  (equal (env-mismatch-aux n truth (nfix env1)
                           env2 numvars)
         (env-mismatch-aux n truth env1 env2 numvars)))

Theorem: env-mismatch-aux-nat-equiv-congruence-on-env1

(defthm env-mismatch-aux-nat-equiv-congruence-on-env1
  (implies
       (nat-equiv env1 env1-equiv)
       (equal (env-mismatch-aux n truth env1 env2 numvars)
              (env-mismatch-aux n truth env1-equiv env2 numvars)))
  :rule-classes :congruence)

Theorem: env-mismatch-aux-of-nfix-env2

(defthm env-mismatch-aux-of-nfix-env2
  (equal (env-mismatch-aux n truth env1 (nfix env2)
                           numvars)
         (env-mismatch-aux n truth env1 env2 numvars)))

Theorem: env-mismatch-aux-nat-equiv-congruence-on-env2

(defthm env-mismatch-aux-nat-equiv-congruence-on-env2
  (implies
       (nat-equiv env2 env2-equiv)
       (equal (env-mismatch-aux n truth env1 env2 numvars)
              (env-mismatch-aux n truth env1 env2-equiv numvars)))
  :rule-classes :congruence)

Theorem: env-mismatch-aux-of-nfix-numvars

(defthm env-mismatch-aux-of-nfix-numvars
  (equal (env-mismatch-aux n truth env1 env2 (nfix numvars))
         (env-mismatch-aux n truth env1 env2 numvars)))

Theorem: env-mismatch-aux-nat-equiv-congruence-on-numvars

(defthm env-mismatch-aux-nat-equiv-congruence-on-numvars
  (implies
       (nat-equiv numvars numvars-equiv)
       (equal (env-mismatch-aux n truth env1 env2 numvars)
              (env-mismatch-aux n truth env1 env2 numvars-equiv)))
  :rule-classes :congruence)