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• Logops-recursive-definitions-theory

Basic-logops-induction-schemes

Some typical ways to induct when proving theorems about logical operations.

Definitions and Theorems

Function: logcdr-induction-1

(defun logcdr-induction-1 (i)
  (declare (xargs :guard (integerp i)))
  (cond ((zip i) t)
        ((equal i -1) t)
        (t (logcdr-induction-1 (logcdr i)))))

Function: logcdr-induction-2

(defun logcdr-induction-2 (i j)
  (declare (xargs :guard (and (integerp i) (integerp j))))
  (cond ((zip i) t)
        ((zip j) t)
        ((equal i -1) t)
        ((equal j -1) t)
        (t (logcdr-induction-2 (logcdr i)
                               (logcdr j)))))

Function: logcdr-induction-3

(defun logcdr-induction-3 (i j k)
  (declare (xargs :guard (and (integerp i)
                              (integerp j)
                              (integerp k))))
  (cond ((zip i) t)
        ((equal i -1) t)
        (t (logcdr-induction-3 (logcdr i)
                               (logcdr j)
                               (logcdr k)))))

Function: sub1-logcdr-induction-1

(defun sub1-logcdr-induction-1 (size i)
 "The elaborate base case is for the benefit of UNSIGNED-BYTE-P, which has no guards."
 (cond ((or (not (integerp size))
            (< size 0)
            (not (integerp i)))
        t)
       ((equal size 0) t)
       (t (sub1-logcdr-induction-1 (1- size)
                                   (logcdr i)))))

Function: sub1-logcdr-induction-2

(defun sub1-logcdr-induction-2 (size i j)
 "The elaborate base case is for the benefit of UNSIGNED-BYTE-P, which has no guards."
 (cond ((or (not (integerp size))
            (< size 0)
            (not (integerp i))
            (not (integerp j)))
        t)
       ((equal size 0) t)
       (t (sub1-logcdr-induction-2 (1- size)
                                   (logcdr i)
                                   (logcdr j)))))

Function: sub1-logcdr-induction-2-w/carry

(defun sub1-logcdr-induction-2-w/carry (size i j c)
 "The elaborate base case is for the benefit of SIGNED-BYTE-P, which has no guards."
 (cond ((or (not (integerp size))
            (<= size 0)
            (not (integerp i))
            (not (integerp j))
            (not (bitp c)))
        t)
       ((equal size 1) t)
       (t (sub1-logcdr-induction-2-w/carry (1- size)
                                           (logcdr i)
                                           (logcdr j)
                                           (b-and c (logcar i))))))

Function: sub1-logcdr-induction-3

(defun sub1-logcdr-induction-3 (size i j k)
 "The elaborate base case is for the benefit of SIGNED-BYTE-P, which has no guards."
 (cond ((or (not (integerp size))
            (<= size 0)
            (not (integerp i))
            (not (integerp j))
            (not (integerp k)))
        t)
       ((equal size 1) t)
       (t (sub1-logcdr-induction-3 (1- size)
                                   (logcdr i)
                                   (logcdr j)
                                   (logcdr k)))))