693 lines
24 KiB
Common Lisp
693 lines
24 KiB
Common Lisp
;
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; Copyright (c) 2008-2016 Stefan Krah. All rights reserved.
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;
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; Redistribution and use in source and binary forms, with or without
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; modification, are permitted provided that the following conditions
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; are met:
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;
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; 1. Redistributions of source code must retain the above copyright
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; notice, this list of conditions and the following disclaimer.
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;
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; 2. Redistributions in binary form must reproduce the above copyright
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; notice, this list of conditions and the following disclaimer in the
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; documentation and/or other materials provided with the distribution.
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;
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; THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS "AS IS" AND
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; ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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; IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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; ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
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; FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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; DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
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; OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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; HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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; LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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; OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
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; SUCH DAMAGE.
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;
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(in-package "ACL2")
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(include-book "arithmetic/top-with-meta" :dir :system)
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(include-book "arithmetic-2/floor-mod/floor-mod" :dir :system)
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;; =====================================================================
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;; Proofs for several functions in umodarith.h
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;; =====================================================================
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;; =====================================================================
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;; Helper theorems
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;; =====================================================================
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(defthm elim-mod-m<x<2*m
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(implies (and (<= m x)
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(< x (* 2 m))
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(rationalp x) (rationalp m))
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(equal (mod x m)
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(+ x (- m)))))
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(defthm modaux-1a
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(implies (and (< x m) (< 0 x) (< 0 m)
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(rationalp x) (rationalp m))
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(equal (mod (- x) m)
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(+ (- x) m))))
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(defthm modaux-1b
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(implies (and (< (- x) m) (< x 0) (< 0 m)
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(rationalp x) (rationalp m))
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(equal (mod x m)
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(+ x m)))
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:hints (("Goal" :use ((:instance modaux-1a
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(x (- x)))))))
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(defthm modaux-1c
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(implies (and (< x m) (< 0 x) (< 0 m)
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(rationalp x) (rationalp m))
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(equal (mod x m)
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x)))
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(defthm modaux-2a
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(implies (and (< 0 b) (< b m)
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(natp x) (natp b) (natp m)
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(< (mod (+ b x) m) b))
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(equal (mod (+ (- m) b x) m)
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(+ (- m) b (mod x m)))))
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(defthm modaux-2b
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(implies (and (< 0 b) (< b m)
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(natp x) (natp b) (natp m)
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(< (mod (+ b x) m) b))
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(equal (mod (+ b x) m)
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(+ (- m) b (mod x m))))
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:hints (("Goal" :use (modaux-2a))))
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(defthm linear-mod-1
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(implies (and (< x m) (< b m)
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(natp x) (natp b)
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(rationalp m))
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(equal (< x (mod (+ (- b) x) m))
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(< x b)))
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:hints (("Goal" :use ((:instance modaux-1a
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(x (+ b (- x))))))))
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(defthm linear-mod-2
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(implies (and (< 0 b) (< b m)
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(natp x) (natp b)
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(natp m))
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(equal (< (mod x m)
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(mod (+ (- b) x) m))
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(< (mod x m) b))))
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(defthm linear-mod-3
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(implies (and (< x m) (< b m)
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(natp x) (natp b)
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(rationalp m))
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(equal (<= b (mod (+ b x) m))
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(< (+ b x) m)))
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:hints (("Goal" :use ((:instance elim-mod-m<x<2*m
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(x (+ b x)))))))
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(defthm modaux-2c
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(implies (and (< 0 b) (< b m)
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(natp x) (natp b) (natp m)
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(<= b (mod (+ b x) m)))
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(equal (mod (+ b x) m)
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(+ b (mod x m))))
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:hints (("Subgoal *1/8''" :use (linear-mod-3))))
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(defthmd modaux-2d
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(implies (and (< x m) (< 0 x) (< 0 m)
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(< (- m) b) (< b 0) (rationalp m)
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(<= x (mod (+ b x) m))
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(rationalp x) (rationalp b))
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(equal (+ (- m) (mod (+ b x) m))
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(+ b x)))
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:hints (("Goal" :cases ((<= 0 (+ b x))))
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("Subgoal 2'" :use ((:instance modaux-1b
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(x (+ b x)))))))
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(defthm mod-m-b
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(implies (and (< 0 x) (< 0 b) (< 0 m)
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(< x b) (< b m)
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(natp x) (natp b) (natp m))
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(equal (mod (+ (mod (- x) m) b) m)
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(mod (- x) b))))
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;; =====================================================================
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;; addmod, submod
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;; =====================================================================
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(defun addmod (a b m base)
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(let* ((s (mod (+ a b) base))
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(s (if (< s a) (mod (- s m) base) s))
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(s (if (>= s m) (mod (- s m) base) s)))
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s))
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(defthmd addmod-correct
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(implies (and (< 0 m) (< m base)
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(< a m) (<= b m)
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(natp m) (natp base)
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(natp a) (natp b))
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(equal (addmod a b m base)
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(mod (+ a b) m)))
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:hints (("Goal" :cases ((<= base (+ a b))))
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("Subgoal 2.1'" :use ((:instance elim-mod-m<x<2*m
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(x (+ a b)))))))
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(defun submod (a b m base)
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(let* ((d (mod (- a b) base))
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(d (if (< a d) (mod (+ d m) base) d)))
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d))
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(defthmd submod-aux1
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(implies (and (< a (mod (+ a (- b)) base))
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(< 0 base) (< a base) (<= b base)
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(natp base) (natp a) (natp b))
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(< a b))
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:rule-classes :forward-chaining)
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(defthmd submod-aux2
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(implies (and (<= (mod (+ a (- b)) base) a)
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(< 0 base) (< a base) (< b base)
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(natp base) (natp a) (natp b))
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(<= b a))
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:rule-classes :forward-chaining)
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(defthmd submod-correct
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(implies (and (< 0 m) (< m base)
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(< a m) (<= b m)
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(natp m) (natp base)
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(natp a) (natp b))
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(equal (submod a b m base)
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(mod (- a b) m)))
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:hints (("Goal" :cases ((<= base (+ a b))))
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("Subgoal 2.2" :use ((:instance submod-aux1)))
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("Subgoal 2.2'''" :cases ((and (< 0 (+ a (- b) m))
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(< (+ a (- b) m) m))))
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("Subgoal 2.1" :use ((:instance submod-aux2)))
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("Subgoal 1.2" :use ((:instance submod-aux1)))
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("Subgoal 1.1" :use ((:instance submod-aux2)))))
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(defun submod-2 (a b m base)
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(let* ((d (mod (- a b) base))
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(d (if (< a b) (mod (+ d m) base) d)))
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d))
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(defthm submod-2-correct
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(implies (and (< 0 m) (< m base)
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(< a m) (<= b m)
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(natp m) (natp base)
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(natp a) (natp b))
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(equal (submod-2 a b m base)
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(mod (- a b) m)))
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:hints (("Subgoal 2'" :cases ((and (< 0 (+ a (- b) m))
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(< (+ a (- b) m) m))))))
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;; =========================================================================
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;; ext-submod is correct
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;; =========================================================================
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; a < 2*m, b < 2*m
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(defun ext-submod (a b m base)
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(let* ((a (if (>= a m) (- a m) a))
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(b (if (>= b m) (- b m) b))
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(d (mod (- a b) base))
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(d (if (< a b) (mod (+ d m) base) d)))
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d))
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; a < 2*m, b < 2*m
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(defun ext-submod-2 (a b m base)
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(let* ((a (mod a m))
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(b (mod b m))
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(d (mod (- a b) base))
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(d (if (< a b) (mod (+ d m) base) d)))
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d))
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(defthmd ext-submod-ext-submod-2-equal
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(implies (and (< 0 m) (< m base)
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(< a (* 2 m)) (< b (* 2 m))
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(natp m) (natp base)
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(natp a) (natp b))
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(equal (ext-submod a b m base)
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(ext-submod-2 a b m base))))
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(defthmd ext-submod-2-correct
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(implies (and (< 0 m) (< m base)
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(< a (* 2 m)) (< b (* 2 m))
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(natp m) (natp base)
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(natp a) (natp b))
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(equal (ext-submod-2 a b m base)
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(mod (- a b) m))))
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;; =========================================================================
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;; dw-reduce is correct
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;; =========================================================================
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(defun dw-reduce (hi lo m base)
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(let* ((r1 (mod hi m))
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(r2 (mod (+ (* r1 base) lo) m)))
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r2))
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(defthmd dw-reduce-correct
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(implies (and (< 0 m) (< m base)
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(< hi base) (< lo base)
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(natp m) (natp base)
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(natp hi) (natp lo))
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(equal (dw-reduce hi lo m base)
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(mod (+ (* hi base) lo) m))))
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(defthmd <=-multiply-both-sides-by-z
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(implies (and (rationalp x) (rationalp y)
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(< 0 z) (rationalp z))
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(equal (<= x y)
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(<= (* z x) (* z y)))))
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(defthmd dw-reduce-aux1
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(implies (and (< 0 m) (< m base)
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(natp m) (natp base)
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(< lo base) (natp lo)
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(< x m) (natp x))
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(< (+ lo (* base x)) (* base m)))
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:hints (("Goal" :cases ((<= (+ x 1) m)))
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("Subgoal 1''" :cases ((<= (* base (+ x 1)) (* base m))))
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("subgoal 1.2" :use ((:instance <=-multiply-both-sides-by-z
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(x (+ 1 x))
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(y m)
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(z base))))))
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(defthm dw-reduce-aux2
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(implies (and (< x (* base m))
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(< 0 m) (< m base)
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(natp m) (natp base) (natp x))
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(< (floor x m) base)))
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;; This is the necessary condition for using _mpd_div_words().
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(defthmd dw-reduce-second-quotient-fits-in-single-word
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(implies (and (< 0 m) (< m base)
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(< hi base) (< lo base)
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(natp m) (natp base)
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(natp hi) (natp lo)
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(equal r1 (mod hi m)))
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(< (floor (+ (* r1 base) lo) m)
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base))
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:hints (("Goal" :cases ((< r1 m)))
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("Subgoal 1''" :cases ((< (+ lo (* base (mod hi m))) (* base m))))
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("Subgoal 1.2" :use ((:instance dw-reduce-aux1
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(x (mod hi m)))))))
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;; =========================================================================
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;; dw-submod is correct
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;; =========================================================================
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(defun dw-submod (a hi lo m base)
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(let* ((r (dw-reduce hi lo m base))
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(d (mod (- a r) base))
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(d (if (< a r) (mod (+ d m) base) d)))
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d))
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(defthmd dw-submod-aux1
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(implies (and (natp a) (< 0 m) (natp m)
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(natp x) (equal r (mod x m)))
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(equal (mod (- a x) m)
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(mod (- a r) m))))
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(defthmd dw-submod-correct
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(implies (and (< 0 m) (< m base)
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(natp a) (< a m)
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(< hi base) (< lo base)
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(natp m) (natp base)
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(natp hi) (natp lo))
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(equal (dw-submod a hi lo m base)
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(mod (- a (+ (* base hi) lo)) m)))
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:hints (("Goal" :in-theory (disable dw-reduce)
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:use ((:instance dw-submod-aux1
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(x (+ lo (* base hi)))
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(r (dw-reduce hi lo m base)))
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(:instance dw-reduce-correct)))))
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;; =========================================================================
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;; ANSI C arithmetic for uint64_t
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;; =========================================================================
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(defun add (a b)
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(mod (+ a b)
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(expt 2 64)))
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(defun sub (a b)
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(mod (- a b)
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(expt 2 64)))
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(defun << (w n)
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(mod (* w (expt 2 n))
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(expt 2 64)))
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(defun >> (w n)
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(floor w (expt 2 n)))
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;; join upper and lower half of a double word, yielding a 128 bit number
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(defun join (hi lo)
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(+ (* (expt 2 64) hi) lo))
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;; =============================================================================
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;; Fast modular reduction
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;; =============================================================================
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;; These are the three primes used in the Number Theoretic Transform.
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;; A fast modular reduction scheme exists for all of them.
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(defmacro p1 ()
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(+ (expt 2 64) (- (expt 2 32)) 1))
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(defmacro p2 ()
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(+ (expt 2 64) (- (expt 2 34)) 1))
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(defmacro p3 ()
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(+ (expt 2 64) (- (expt 2 40)) 1))
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;; reduce the double word number hi*2**64 + lo (mod p1)
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(defun simple-mod-reduce-p1 (hi lo)
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(+ (* (expt 2 32) hi) (- hi) lo))
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;; reduce the double word number hi*2**64 + lo (mod p2)
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(defun simple-mod-reduce-p2 (hi lo)
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(+ (* (expt 2 34) hi) (- hi) lo))
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;; reduce the double word number hi*2**64 + lo (mod p3)
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(defun simple-mod-reduce-p3 (hi lo)
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(+ (* (expt 2 40) hi) (- hi) lo))
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; ----------------------------------------------------------
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; The modular reductions given above are correct
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; ----------------------------------------------------------
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(defthmd congruence-p1-aux
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(equal (* (expt 2 64) hi)
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(+ (* (p1) hi)
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(* (expt 2 32) hi)
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(- hi))))
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(defthmd congruence-p2-aux
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(equal (* (expt 2 64) hi)
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(+ (* (p2) hi)
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(* (expt 2 34) hi)
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(- hi))))
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(defthmd congruence-p3-aux
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(equal (* (expt 2 64) hi)
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(+ (* (p3) hi)
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(* (expt 2 40) hi)
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(- hi))))
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(defthmd mod-augment
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(implies (and (rationalp x)
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(rationalp y)
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(rationalp m))
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(equal (mod (+ x y) m)
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(mod (+ x (mod y m)) m))))
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(defthmd simple-mod-reduce-p1-congruent
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(implies (and (integerp hi)
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(integerp lo))
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(equal (mod (simple-mod-reduce-p1 hi lo) (p1))
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(mod (join hi lo) (p1))))
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:hints (("Goal''" :use ((:instance congruence-p1-aux)
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(:instance mod-augment
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(m (p1))
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(x (+ (- hi) lo (* (expt 2 32) hi)))
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(y (* (p1) hi)))))))
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(defthmd simple-mod-reduce-p2-congruent
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(implies (and (integerp hi)
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(integerp lo))
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(equal (mod (simple-mod-reduce-p2 hi lo) (p2))
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(mod (join hi lo) (p2))))
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:hints (("Goal''" :use ((:instance congruence-p2-aux)
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(:instance mod-augment
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(m (p2))
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(x (+ (- hi) lo (* (expt 2 34) hi)))
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(y (* (p2) hi)))))))
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(defthmd simple-mod-reduce-p3-congruent
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(implies (and (integerp hi)
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(integerp lo))
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(equal (mod (simple-mod-reduce-p3 hi lo) (p3))
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(mod (join hi lo) (p3))))
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:hints (("Goal''" :use ((:instance congruence-p3-aux)
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(:instance mod-augment
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(m (p3))
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(x (+ (- hi) lo (* (expt 2 40) hi)))
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(y (* (p3) hi)))))))
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; ---------------------------------------------------------------------
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; We need a number less than 2*p, so that we can use the trick from
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; elim-mod-m<x<2*m for the final reduction.
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; For p1, two modular reductions are sufficient, for p2 and p3 three.
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; ---------------------------------------------------------------------
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;; p1: the first reduction is less than 2**96
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(defthmd simple-mod-reduce-p1-<-2**96
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(implies (and (< hi (expt 2 64))
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(< lo (expt 2 64))
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(natp hi) (natp lo))
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(< (simple-mod-reduce-p1 hi lo)
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(expt 2 96))))
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;; p1: the second reduction is less than 2*p1
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(defthmd simple-mod-reduce-p1-<-2*p1
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(implies (and (< hi (expt 2 64))
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(< lo (expt 2 64))
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(< (join hi lo) (expt 2 96))
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(natp hi) (natp lo))
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(< (simple-mod-reduce-p1 hi lo)
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(* 2 (p1)))))
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;; p2: the first reduction is less than 2**98
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(defthmd simple-mod-reduce-p2-<-2**98
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(implies (and (< hi (expt 2 64))
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(< lo (expt 2 64))
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(natp hi) (natp lo))
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(< (simple-mod-reduce-p2 hi lo)
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(expt 2 98))))
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;; p2: the second reduction is less than 2**69
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(defthmd simple-mod-reduce-p2-<-2*69
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(implies (and (< hi (expt 2 64))
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(< lo (expt 2 64))
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(< (join hi lo) (expt 2 98))
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(natp hi) (natp lo))
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(< (simple-mod-reduce-p2 hi lo)
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|
(expt 2 69))))
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|
|
|
;; p3: the third reduction is less than 2*p2
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|
(defthmd simple-mod-reduce-p2-<-2*p2
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|
(implies (and (< hi (expt 2 64))
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|
(< lo (expt 2 64))
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|
(< (join hi lo) (expt 2 69))
|
|
(natp hi) (natp lo))
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|
(< (simple-mod-reduce-p2 hi lo)
|
|
(* 2 (p2)))))
|
|
|
|
|
|
;; p3: the first reduction is less than 2**104
|
|
(defthmd simple-mod-reduce-p3-<-2**104
|
|
(implies (and (< hi (expt 2 64))
|
|
(< lo (expt 2 64))
|
|
(natp hi) (natp lo))
|
|
(< (simple-mod-reduce-p3 hi lo)
|
|
(expt 2 104))))
|
|
|
|
;; p3: the second reduction is less than 2**81
|
|
(defthmd simple-mod-reduce-p3-<-2**81
|
|
(implies (and (< hi (expt 2 64))
|
|
(< lo (expt 2 64))
|
|
(< (join hi lo) (expt 2 104))
|
|
(natp hi) (natp lo))
|
|
(< (simple-mod-reduce-p3 hi lo)
|
|
(expt 2 81))))
|
|
|
|
;; p3: the third reduction is less than 2*p3
|
|
(defthmd simple-mod-reduce-p3-<-2*p3
|
|
(implies (and (< hi (expt 2 64))
|
|
(< lo (expt 2 64))
|
|
(< (join hi lo) (expt 2 81))
|
|
(natp hi) (natp lo))
|
|
(< (simple-mod-reduce-p3 hi lo)
|
|
(* 2 (p3)))))
|
|
|
|
|
|
; -------------------------------------------------------------------------
|
|
; The simple modular reductions, adapted for compiler friendly C
|
|
; -------------------------------------------------------------------------
|
|
|
|
(defun mod-reduce-p1 (hi lo)
|
|
(let* ((y hi)
|
|
(x y)
|
|
(hi (>> hi 32))
|
|
(x (sub lo x))
|
|
(hi (if (> x lo) (+ hi -1) hi))
|
|
(y (<< y 32))
|
|
(lo (add y x))
|
|
(hi (if (< lo y) (+ hi 1) hi)))
|
|
(+ (* hi (expt 2 64)) lo)))
|
|
|
|
(defun mod-reduce-p2 (hi lo)
|
|
(let* ((y hi)
|
|
(x y)
|
|
(hi (>> hi 30))
|
|
(x (sub lo x))
|
|
(hi (if (> x lo) (+ hi -1) hi))
|
|
(y (<< y 34))
|
|
(lo (add y x))
|
|
(hi (if (< lo y) (+ hi 1) hi)))
|
|
(+ (* hi (expt 2 64)) lo)))
|
|
|
|
(defun mod-reduce-p3 (hi lo)
|
|
(let* ((y hi)
|
|
(x y)
|
|
(hi (>> hi 24))
|
|
(x (sub lo x))
|
|
(hi (if (> x lo) (+ hi -1) hi))
|
|
(y (<< y 40))
|
|
(lo (add y x))
|
|
(hi (if (< lo y) (+ hi 1) hi)))
|
|
(+ (* hi (expt 2 64)) lo)))
|
|
|
|
|
|
; -------------------------------------------------------------------------
|
|
; The compiler friendly versions are equal to the simple versions
|
|
; -------------------------------------------------------------------------
|
|
|
|
(defthm mod-reduce-aux1
|
|
(implies (and (<= 0 a) (natp a) (natp m)
|
|
(< (- m) b) (<= b 0)
|
|
(integerp b)
|
|
(< (mod (+ b a) m)
|
|
(mod a m)))
|
|
(equal (mod (+ b a) m)
|
|
(+ b (mod a m))))
|
|
:hints (("Subgoal 2" :use ((:instance modaux-1b
|
|
(x (+ a b)))))))
|
|
|
|
(defthm mod-reduce-aux2
|
|
(implies (and (<= 0 a) (natp a) (natp m)
|
|
(< b m) (natp b)
|
|
(< (mod (+ b a) m)
|
|
(mod a m)))
|
|
(equal (+ m (mod (+ b a) m))
|
|
(+ b (mod a m)))))
|
|
|
|
|
|
(defthm mod-reduce-aux3
|
|
(implies (and (< 0 a) (natp a) (natp m)
|
|
(< (- m) b) (< b 0)
|
|
(integerp b)
|
|
(<= (mod a m)
|
|
(mod (+ b a) m)))
|
|
(equal (+ (- m) (mod (+ b a) m))
|
|
(+ b (mod a m))))
|
|
:hints (("Subgoal 1.2'" :use ((:instance modaux-1b
|
|
(x b))))
|
|
("Subgoal 1''" :use ((:instance modaux-2d
|
|
(x I))))))
|
|
|
|
|
|
(defthm mod-reduce-aux4
|
|
(implies (and (< 0 a) (natp a) (natp m)
|
|
(< b m) (natp b)
|
|
(<= (mod a m)
|
|
(mod (+ b a) m)))
|
|
(equal (mod (+ b a) m)
|
|
(+ b (mod a m)))))
|
|
|
|
|
|
(defthm mod-reduce-p1==simple-mod-reduce-p1
|
|
(implies (and (< hi (expt 2 64))
|
|
(< lo (expt 2 64))
|
|
(natp hi) (natp lo))
|
|
(equal (mod-reduce-p1 hi lo)
|
|
(simple-mod-reduce-p1 hi lo)))
|
|
:hints (("Goal" :in-theory (disable expt)
|
|
:cases ((< 0 hi)))
|
|
("Subgoal 1.2.2'" :use ((:instance mod-reduce-aux1
|
|
(m (expt 2 64))
|
|
(b (+ (- HI) LO))
|
|
(a (* (expt 2 32) hi)))))
|
|
("Subgoal 1.2.1'" :use ((:instance mod-reduce-aux3
|
|
(m (expt 2 64))
|
|
(b (+ (- HI) LO))
|
|
(a (* (expt 2 32) hi)))))
|
|
("Subgoal 1.1.2'" :use ((:instance mod-reduce-aux2
|
|
(m (expt 2 64))
|
|
(b (+ (- HI) LO))
|
|
(a (* (expt 2 32) hi)))))
|
|
("Subgoal 1.1.1'" :use ((:instance mod-reduce-aux4
|
|
(m (expt 2 64))
|
|
(b (+ (- HI) LO))
|
|
(a (* (expt 2 32) hi)))))))
|
|
|
|
|
|
(defthm mod-reduce-p2==simple-mod-reduce-p2
|
|
(implies (and (< hi (expt 2 64))
|
|
(< lo (expt 2 64))
|
|
(natp hi) (natp lo))
|
|
(equal (mod-reduce-p2 hi lo)
|
|
(simple-mod-reduce-p2 hi lo)))
|
|
:hints (("Goal" :cases ((< 0 hi)))
|
|
("Subgoal 1.2.2'" :use ((:instance mod-reduce-aux1
|
|
(m (expt 2 64))
|
|
(b (+ (- HI) LO))
|
|
(a (* (expt 2 34) hi)))))
|
|
("Subgoal 1.2.1'" :use ((:instance mod-reduce-aux3
|
|
(m (expt 2 64))
|
|
(b (+ (- HI) LO))
|
|
(a (* (expt 2 34) hi)))))
|
|
("Subgoal 1.1.2'" :use ((:instance mod-reduce-aux2
|
|
(m (expt 2 64))
|
|
(b (+ (- HI) LO))
|
|
(a (* (expt 2 34) hi)))))
|
|
("Subgoal 1.1.1'" :use ((:instance mod-reduce-aux4
|
|
(m (expt 2 64))
|
|
(b (+ (- HI) LO))
|
|
(a (* (expt 2 34) hi)))))))
|
|
|
|
|
|
(defthm mod-reduce-p3==simple-mod-reduce-p3
|
|
(implies (and (< hi (expt 2 64))
|
|
(< lo (expt 2 64))
|
|
(natp hi) (natp lo))
|
|
(equal (mod-reduce-p3 hi lo)
|
|
(simple-mod-reduce-p3 hi lo)))
|
|
:hints (("Goal" :cases ((< 0 hi)))
|
|
("Subgoal 1.2.2'" :use ((:instance mod-reduce-aux1
|
|
(m (expt 2 64))
|
|
(b (+ (- HI) LO))
|
|
(a (* (expt 2 40) hi)))))
|
|
("Subgoal 1.2.1'" :use ((:instance mod-reduce-aux3
|
|
(m (expt 2 64))
|
|
(b (+ (- HI) LO))
|
|
(a (* (expt 2 40) hi)))))
|
|
("Subgoal 1.1.2'" :use ((:instance mod-reduce-aux2
|
|
(m (expt 2 64))
|
|
(b (+ (- HI) LO))
|
|
(a (* (expt 2 40) hi)))))
|
|
("Subgoal 1.1.1'" :use ((:instance mod-reduce-aux4
|
|
(m (expt 2 64))
|
|
(b (+ (- HI) LO))
|
|
(a (* (expt 2 40) hi)))))))
|
|
|
|
|
|
|