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

    Env-mismatch

    Signature
    (env-mismatch truth env1 env2 numvars) → var
    Arguments
    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

    (defun env-mismatch (truth env1 env2 numvars)
      (declare (xargs :guard (and (integerp truth)
                                  (natp env1)
                                  (natp env2)
                                  (natp numvars))))
      (let ((__function__ 'env-mismatch))
        (declare (ignorable __function__))
        (env-mismatch-aux 0 truth env1 env2 numvars)))

    Theorem: natp-of-env-mismatch

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

    Theorem: env-mismatch-bound

    (defthm env-mismatch-bound
      (b* ((?var (env-mismatch truth env1 env2 numvars)))
        (implies (posp numvars)
                 (< var numvars)))
      :rule-classes (:rewrite :linear))

    Theorem: env-mismatch-when-mismatch

    (defthm env-mismatch-when-mismatch
      (b* ((?var (env-mismatch truth env1 env2 numvars)))
        (implies (and (depends-on m truth numvars)
                      (xor (env-lookup m env1)
                           (env-lookup m env2))
                      (< (nfix m) (nfix numvars)))
                 (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-when-eval-mismatch

    (defthm env-mismatch-when-eval-mismatch
      (b* ((?var (env-mismatch truth env1 env2 numvars)))
        (implies (and (not (equal (truth-eval truth env1 numvars)
                                  (truth-eval truth env2 numvars))))
                 (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-of-ifix-truth

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

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

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

    Theorem: env-mismatch-of-nfix-env1

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

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

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

    Theorem: env-mismatch-of-nfix-env2

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

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

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

    Theorem: env-mismatch-of-nfix-numvars

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

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

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