• Top
  • Documentation
  • Books
  • Recursion-and-induction
  • Boolean-reasoning
    • Ipasir
    • Aignet
    • Aig
    • Satlink
    • Truth
      • Index-permute-shrink
      • Permute-stretch
      • Env-mismatch-aux
      • Permute-shrink
      • Permute-polarity
      • Env-permute-polarity
      • Env-permute-shrink
      • Permute-var-down
      • Swap-vars-aux
      • Env-permute-stretch
      • Swap-vars
      • Permute-var-up
      • Negative-cofactor
      • Truth-perm-rev
      • Index-permute-stretch
      • Env-mismatch
      • Truth-perm
      • Swap-polarity
      • Positive-cofactor
      • Index-perm-rev
      • Env-perm-rev
      • Nth-set-bit-pos
      • Index-swap
      • Env-move-var-down
      • Is-xor-with-var
      • Index-perm
      • Env-swap-vars
      • Var
      • Truth-eval
      • Index-move-down
      • Env-update
      • Env-perm
      • Depends-on-witness
      • Env-swap-polarity
      • Var-repetitions
      • Env-move-var-up
      • Index-move-up
      • Depends-on
      • Truth-norm
      • Index-listp
      • Env-diff-index
      • Env-lookup
      • True
      • False
    • Ubdds
    • Bdd
    • Faig
    • Bed
    • 4v
  • Debugging
  • Projects
  • Std
  • Proof-automation
  • Macro-libraries
  • ACL2
  • Interfacing-tools
  • Hardware-verification
  • Software-verification
  • Testing-utilities
  • Math

• 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)