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• Truth

Truth-perm

Signature

(truth-perm n perm truth numvars) → perm-truth

Arguments

n — current position in the list.
    Guard (natp n).

perm — indices to permute.
    Guard (index-listp perm numvars).

truth — truth table to permute.
    Guard (integerp truth).

numvars — Guard (natp numvars).

Returns

perm-truth — Type (integerp perm-truth).

Definitions and Theorems

Function: truth-perm

(defun truth-perm (n perm truth numvars)
  (declare (xargs :guard (and (natp n)
                              (integerp truth)
                              (natp numvars)
                              (index-listp perm numvars))))
  (declare (xargs :guard (and (<= n numvars)
                              (eql (len perm) numvars))))
  (let ((__function__ 'truth-perm))
    (declare (ignorable __function__))
    (b* (((when (mbe :logic (zp (- (nfix numvars) (nfix n)))
                     :exec (eql n numvars)))
          (truth-norm truth numvars))
         (truth (swap-vars n (nth n perm)
                           truth numvars)))
      (truth-perm (1+ (lnfix n))
                  perm truth numvars))))

Theorem: integerp-of-truth-perm

(defthm integerp-of-truth-perm
  (b* ((perm-truth (truth-perm n perm truth numvars)))
    (integerp perm-truth))
  :rule-classes :type-prescription)

Theorem: eval-of-truth-perm-with-env-perm

(defthm eval-of-truth-perm-with-env-perm
  (implies (index-listp perm numvars)
           (equal (truth-eval (truth-perm n perm truth numvars)
                              (env-perm n perm env numvars)
                              numvars)
                  (truth-eval truth env numvars))))

Theorem: eval-of-truth-perm

(defthm eval-of-truth-perm
 (implies (index-listp perm numvars)
          (equal (truth-eval (truth-perm n perm truth numvars)
                             env numvars)
                 (truth-eval truth (env-perm-rev n perm env numvars)
                             numvars))))

Theorem: size-of-truth-perm-basic

(defthm size-of-truth-perm-basic
  (b* ((?perm-truth (truth-perm n perm truth numvars)))
    (unsigned-byte-p (ash 1 (nfix numvars))
                     perm-truth)))

Theorem: size-of-truth-perm

(defthm size-of-truth-perm
  (b* ((?perm-truth (truth-perm n perm truth numvars)))
    (implies (and (natp size)
                  (<= (ash 1 (nfix numvars)) size))
             (unsigned-byte-p size perm-truth))))

Theorem: truth-perm-of-nfix-n

(defthm truth-perm-of-nfix-n
  (equal (truth-perm (nfix n) perm truth numvars)
         (truth-perm n perm truth numvars)))

Theorem: truth-perm-nat-equiv-congruence-on-n

(defthm truth-perm-nat-equiv-congruence-on-n
  (implies (nat-equiv n n-equiv)
           (equal (truth-perm n perm truth numvars)
                  (truth-perm n-equiv perm truth numvars)))
  :rule-classes :congruence)

Theorem: truth-perm-of-ifix-truth

(defthm truth-perm-of-ifix-truth
  (equal (truth-perm n perm (ifix truth) numvars)
         (truth-perm n perm truth numvars)))

Theorem: truth-perm-int-equiv-congruence-on-truth

(defthm truth-perm-int-equiv-congruence-on-truth
  (implies (int-equiv truth truth-equiv)
           (equal (truth-perm n perm truth numvars)
                  (truth-perm n perm truth-equiv numvars)))
  :rule-classes :congruence)

Theorem: truth-perm-of-nfix-numvars

(defthm truth-perm-of-nfix-numvars
  (equal (truth-perm n perm truth (nfix numvars))
         (truth-perm n perm truth numvars)))

Theorem: truth-perm-nat-equiv-congruence-on-numvars

(defthm truth-perm-nat-equiv-congruence-on-numvars
  (implies (nat-equiv numvars numvars-equiv)
           (equal (truth-perm n perm truth numvars)
                  (truth-perm n perm truth numvars-equiv)))
  :rule-classes :congruence)