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;; Copyright (C) 2024 Carl Kwan
;;
;; Methods for "splicing" up matrices
;;
;; Summary:
;; This file contains events for defining the matrix analogs for
;; nthcdr and append. Presently, the only definitions included are
;; - col-append, append two matrices column-wise
;; - nth-col-cdr, get the nth final columns of a matrix
;; - nth-row-cdr, get the nth final rows of a matrix
;;
;; Requires:
(in-package "ACL2")
(include-book "std/util/define" :dir :system)
(include-book "workshops/2003/hendrix/support/mmult" :dir :system)
(set-induction-depth-limit 1)
(defthm col-count-col-cons
(implies (col-cons-guard c m)
(equal (col-count (col-cons c m))
(+ 1 (col-count m)))))
(defmacro col-append-guard (m n)
`(and (matrixp ,m)
(matrixp ,n)
(or (m-emptyp ,m)
(m-emptyp ,n)
(equal (row-count ,m) (row-count ,n)))))
(define col-append ((M matrixp) (N matrixp))
:guard (col-append-guard m n)
:returns mn
:verify-guards nil
(cond ((not (col-append-guard m n))
(mzero (row-count n) (col-count n)))
((m-emptyp m) n)
((m-emptyp n) m)
(t (col-cons (col-car m)
(col-append (col-cdr M) N))))
///
(more-returns
(mn (implies (col-append-guard m n)
(equal (row-count mn)
(if (m-emptyp m) (row-count n) (row-count m))))
:name row-count-col-append)
(mn (matrixp mn))
(mn (implies (col-append-guard m n)
(equal (col-count mn)
(+ (col-count m) (col-count n))))
:name col-count-col-append))
(verify-guards col-append)
(defthm col-append-m-emptyp
(implies (col-append-guard m n)
(equal (m-emptyp (col-append m n))
(and (m-emptyp m) (m-emptyp n)))))
(defthm col-append-def
(implies (col-append-guard m n)
(equal (col-append m n)
(if (m-emptyp m)
n
(col-cons (col-car m) (col-append (col-cdr m) n)))))
:rule-classes ((:definition)))
(encapsulate
()
(local
(defthm lemma-1
(implies (and (matrixp m) (consp (row-car m)))
(equal (append (row-car m) nil)
(row-car m)))))
(local
(defthm lemma-2
(implies (matrixp m)
(equal (append (row i m) nil) (row i m)))))
;; Induction hint not necessary for ACL2 version 8.5. However, the ACL2
;; development version as of 2024-06-28 (and possibly earlier) requires
;; this hint.
(defthm row-of-col-append
(implies (col-append-guard m n)
(equal (row i (col-append m n))
(append (row i m) (row i n))))
:hints (("Goal" :induct (col-append m n)))))
(defthm sm*-col-append
(implies (col-append-guard M N)
(equal (sm* a (col-append M N))
(col-append (sm* a M) (sm* a N)))))
(defthm col*-col-append
(implies (and (col-append-guard M N) (col*-guard v M) (col*-guard v N))
(equal (col* v (col-append M N))
(append (col* v M) (col* v N)))))
(defthm mvectorp-of-col*-col-append
(implies (and (col-append-guard M N) (col*-guard v M) (col*-guard v N))
(mvectorp (col* v (col-append m n))))
:rule-classes :type-prescription)
(defthm len-of-col*-col-append
(implies (and (col-append-guard M N) (col*-guard v M) (col*-guard v N))
(equal (len (col* v (col-append m n))) (+ (col-count m) (col-count n)))))
(defthm m*-col-append
(implies (and (m*-guard A B) (col-append-guard B C) (m*-guard A C))
(equal (m* A (col-append B C))
(col-append (m* A B) (m* A C))))
:hints (("Goal" :in-theory (e/d (m*-by-col-def) (m*))))))
;; end of define col-append
;; Get final columns of a matrix
(define nth-col-cdr ((i natp) (m matrixp))
:guard (< i (col-count m))
:returns (rest matrixp)
(if (not (matrixp m))
(m-empty)
(if (zp i)
m
(nth-col-cdr (1- i) (col-cdr m))))
///
(more-returns
(rest (implies (matrixp m)
(equal (col i m)
(col-car rest)))
:name col-car-of-nth-col-cdr)
(rest (implies (and (matrixp m) (< i 0)) (equal rest m))
:name nth-col-cdr-<-0)
(rest (implies (m-emptyp m) (m-emptyp rest))
:name nth-col-cdr-when-emptyp)
(rest (implies (not (integerp i)) (equal rest (if (matrixp m) m (m-empty))))
:name nth-col-cdr-when-not-integer))
(defthm nth-col-cdr-of-zero
(equal (nth-col-cdr 0 m) (if (matrixp m) m (m-empty))))
(defthm nth-col-cdr-of-col-cons
(implies (col-cons-guard c m)
(equal (nth-col-cdr i (col-cons c m))
(if (zp i) (col-cons c m) (nth-col-cdr (1- i) m)))))
(local (set-induction-depth-limit 2))
;; Induction hint not necessary for ACL2 version 8.5. However, the ACL2
;; development version as of 2024-06-28 (and possibly earlier) requires
;; this hint.
(defthm row-of-nth-col-cdr
(implies (matrixp m)
(equal (row j (nth-col-cdr i m))
(nthcdr i (row j m))))
:hints (("Goal" :induct (nth-col-cdr i m)))))
;; end define
(in-theory (disable col-car-of-nth-col-cdr nth-col-cdr-<-0 nth-col-cdr-when-emptyp))
;; Get final rows of a matrix
(define nth-row-cdr ((i natp) (m matrixp))
:guard (< i (row-count m))
:returns (rest matrixp)
:enabled t
(if (not (matrixp m))
(m-empty)
(if (zp i)
m
(nth-row-cdr (1- i) (row-cdr m))))
///
(more-returns
(rest (implies (matrixp m)
(equal (row i m)
(row-car rest)))
:name row-car-of-nth-row-cdr)
(rest (implies (and (matrixp m) (< i 0)) (equal rest m))
:name nth-row-cdr-<-0)
(rest (implies (m-emptyp m) (m-emptyp rest))
:name nth-row-cdr-when-emptyp)
(rest (implies (matrixp m)
(equal (col-count rest) (if (< (nfix i) (row-count m)) (col-count m) 0)))
:name col-count-of-nth-row-cdr)
(rest (implies (matrixp m)
(equal (row-count rest) (if (< (nfix i) (row-count m)) (- (row-count m) (nfix i)) 0)))
:name row-count-of-nth-row-cdr)
(rest (implies (not (integerp i)) (equal rest (if (matrixp m) m (m-empty))))
:name nth-row-cdr-when-not-integer))
(defthm nth-row-cdr-of-zero
(equal (nth-row-cdr 0 m) (if (matrixp m) m (m-empty))))
(defthmd row-car-nth-row-cdr
(implies (matrixp m)
(equal (row-car (nth-row-cdr i m))
(row i m)))
:hints (("Goal" :in-theory (enable nth-row-cdr row)))))
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