• 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-swap-polarity

Signature

(env-swap-polarity n env) → new-env

Arguments

n — Guard (natp n).

env — Guard (natp env).

Returns

new-env — Type (natp new-env).

Definitions and Theorems

Function: env-swap-polarity

(defun env-swap-polarity (n env)
       (declare (xargs :guard (and (natp n) (natp env))))
       (let ((__function__ 'env-swap-polarity))
            (declare (ignorable __function__))
            (env-update n (not (env-lookup n env))
                        env)))

Theorem: natp-of-env-swap-polarity

(defthm natp-of-env-swap-polarity
        (b* ((new-env (env-swap-polarity n env)))
            (natp new-env))
        :rule-classes :type-prescription)

Theorem: lookup-in-env-swap-polarity-same

(defthm lookup-in-env-swap-polarity-same
        (b* ((?new-env (env-swap-polarity n env)))
            (equal (env-lookup n new-env)
                   (not (env-lookup n env)))))

Theorem: lookup-in-env-swap-polarity-diff

(defthm lookup-in-env-swap-polarity-diff
        (b* ((?new-env (env-swap-polarity n env)))
            (implies (not (nat-equiv n m))
                     (equal (env-lookup m new-env)
                            (env-lookup m env)))))

Theorem: lookup-in-env-swap-polarity-split

(defthm lookup-in-env-swap-polarity-split
        (b* ((?new-env (env-swap-polarity n env)))
            (equal (env-lookup m new-env)
                   (if (nat-equiv n m)
                       (not (env-lookup n env))
                       (env-lookup m env)))))

Theorem: env-swap-polarity-reverse

(defthm env-swap-polarity-reverse
        (b* ((?new-env (env-swap-polarity n env)))
            (equal (env-swap-polarity n new-env)
                   (nfix env))))

Theorem: env-swap-polarity-commute

(defthm env-swap-polarity-commute
        (b* nil
            (equal (env-swap-polarity n (env-swap-polarity m env))
                   (env-swap-polarity m (env-swap-polarity n env))))
        :rule-classes ((:rewrite :loop-stopper ((n m)))))

Theorem: env-swap-polarity-of-nfix-n

(defthm env-swap-polarity-of-nfix-n
        (equal (env-swap-polarity (nfix n) env)
               (env-swap-polarity n env)))

Theorem: env-swap-polarity-nat-equiv-congruence-on-n

(defthm env-swap-polarity-nat-equiv-congruence-on-n
        (implies (nat-equiv n n-equiv)
                 (equal (env-swap-polarity n env)
                        (env-swap-polarity n-equiv env)))
        :rule-classes :congruence)

Theorem: env-swap-polarity-of-nfix-env

(defthm env-swap-polarity-of-nfix-env
        (equal (env-swap-polarity n (nfix env))
               (env-swap-polarity n env)))

Theorem: env-swap-polarity-nat-equiv-congruence-on-env

(defthm env-swap-polarity-nat-equiv-congruence-on-env
        (implies (nat-equiv env env-equiv)
                 (equal (env-swap-polarity n env)
                        (env-swap-polarity n env-equiv)))
        :rule-classes :congruence)