/usr/share/acl2-7.2dfsg/books/system/hl-nat-combine-onto.lisp is in acl2-books-source 7.2dfsg-3.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 | ;; Proof that hl-nat-combine is onto the naturals
;;
;; Public Domain 2015 David Greve
;;
;; The creative work contained within this file is free and
;; unencumbered and has been released into the public domain by its
;; creator, David Greve. Anyone is free to copy, modify, publish,
;; use, compile, sell or distribute this work or any derivative of
;; this work for any purpose, commercial or non-commercial, and by any
;; means. See: http://unlicense.org/
;;
;; Author: David Greve <TheBeaNerd@gmail.com>
;;
;; On 12/26/2014 Matt Kaufmann posted to the ACL2 list a challenge
;; from Bob Boyer regarding Jared Davis' work on the function
;; hl-nat-combine. Bob is quoted as saying:
;;
;; I hope someone some day follows up on Jared's work on this and
;; checks that hl-addr-combine[sic] is also onto the naturals. Just a
;; nice little fact that tidies things up in my mind. I love that
;; function.
;;
;; Note that the helper function hl-nat-combine is, in fact, onto the
;; naturals whereas the top level hl-addr-combine function may produce
;; negative integer results by design.
;;
;; The proof below demonstrates that hl-nat-combine is onto the
;; naturals
(in-package "ACL2")
(include-book "system/hl-addr-combine" :dir :system)
(local
(encapsulate
()
(local (include-book "arithmetic-5/top" :dir :system))
(set-default-hints '((nonlinearp-default-hint++
id stable-under-simplificationp
hist nil)))
(defun even-product (odd even)
(or (and (integerp odd) (evenp even))
(and (integerp even) (evenp odd))))
(defthm even-product-plus-1
(implies (and (natp x)
(natp y)
(or (equal x (+ y 1))
(equal y (+ x 1))))
(even-product x y)))
(defthm integerp-/-2
(implies (even-product odd even)
(integerp (/ (* odd even) 2))))
(defthm natp-*
(implies (and (natp x) (natp y))
(natp (* x y))))
(defthm natp-/
(implies (and (integerp (/ (* x y) 2))
(natp x)
(natp y))
(natp (/ (* x y) 2))))
(defthm natp-plus
(implies (and (natp a) (natp b))
(natp (+ a b))))
(in-theory (enable hl-nat-combine))
(defthm natp-hl-nat-combine
(implies
(and (natp a) (natp b))
(natp (hl-nat-combine a b)))
:INSTRUCTIONS (:PRO (:DV 1)
:EXPAND :UP (:REWRITE NATP-PLUS)
(:REWRITE NATP-/)
(:REWRITE INTEGERP-/-2)
(:REWRITE EVEN-PRODUCT-PLUS-1)
:S))
(defthm hl-nat-combine-zp-b
(equal (hl-nat-combine n 0) (1+ (hl-nat-combine 0 (1- n)))))
(defthm hl-nat-combine-delta
(implies
(and (integerp x) (integerp y) (integerp a))
(equal (hl-nat-combine (+ (- a) x) (+ a y))
(+ (hl-nat-combine x y) a))))
(defthm hl-nat-combine-delta-instance
(implies
(integerp a)
(equal (hl-nat-combine (+ -1 a) 1)
(+ (hl-nat-combine a 0) 1)))
:hints (("Goal" :in-theory (disable hl-nat-combine-delta)
:use (:instance hl-nat-combine-delta (x a) (y 0) (a 1)))))
(in-theory (disable hl-nat-combine))
(defthm natp-a-minus-1
(implies
(not (zp a))
(natp (+ -1 a))))
(defthm natp-linear-fact
(implies
(natp n)
(not (< n -1))))
(defthm natp-implies-intergerp
(implies
(natp x)
(integerp x)))
(defun fnd (a b n)
(declare (xargs :measure (nfix (- (nfix n)
(hl-nat-combine (nfix a) (nfix b))))))
(let ((n (nfix n))
(a (nfix a))
(b (nfix b)))
(if (<= n (hl-nat-combine a b)) (cons a b)
(if (zp a) (fnd (+ b 1) 0 n)
(fnd (1- a) (1+ b) n)))))
(defthm open-fnd-0
(equal (fnd 0 b n)
(let ((n (nfix n))
(a 0)
(b (nfix b)))
(if (<= n (hl-nat-combine a b)) (cons a b)
(fnd (+ b 1) 0 n))))
:INSTRUCTIONS ((:DV 1) :X :TOP :S))
(defthm yeah-1
(implies
(and
(< (+ 1 x) n)
(integerp n)
(integerp x))
(not (< n (+ 2 x))))
:rule-classes (:rewrite :forward-chaining))
(defthm yeah-2
(implies
(and
(< x n)
(integerp n)
(integerp x))
(not (< n (+ 1 x))))
:rule-classes (:rewrite :forward-chaining))
(defthmd fnd-works
(implies
(<= (hl-nat-combine (nfix a) (nfix b)) (nfix n))
(equal (hl-nat-combine (car (fnd a b n))
(cdr (fnd a b n)))
(nfix n))))
(defthm fnd-finds-natps
(and
(natp (car (fnd a b n)))
(natp (cdr (fnd a b n))))
:hints (("Goal" :in-theory (disable nfix))))
))
(defthm natp-hl-nat-combine
(implies
(and (natp a) (natp b))
(natp (hl-nat-combine a b))))
(defun fnd (a b n)
(declare (xargs :measure (nfix (- (nfix n)
(hl-nat-combine (nfix a) (nfix b))))))
(let ((n (nfix n))
(a (nfix a))
(b (nfix b)))
(if (<= n (hl-nat-combine a b)) (cons a b)
(if (zp a) (fnd (+ b 1) 0 n)
(fnd (1- a) (1+ b) n)))))
(defthm onto-property
(implies
(natp n)
(let ((a (car (fnd 0 0 n)))
(b (cdr (fnd 0 0 n))))
(and (equal n (hl-nat-combine a b))
(natp a)
(natp b))))
:rule-classes nil
:hints (("Goal" :use (:instance fnd-works
(a 0) (b 0) (n n)))))
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