(* Title: ZF/ex/BinEx.ML ID: $Id: BinEx.thy,v 1.2 2005/06/17 14:15:11 haftmann Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1994 University of Cambridge Examples of performing binary arithmetic by simplification *) theory BinEx imports Main begin (*All runtimes below are on a 300MHz Pentium*) lemma "#13 $+ #19 = #32" by simp (*0 secs*) lemma "#1234 $+ #5678 = #6912" by simp (*190 msec*) lemma "#1359 $+ #-2468 = #-1109" by simp (*160 msec*) lemma "#93746 $+ #-46375 = #47371" by simp (*300 msec*) lemma "$- #65745 = #-65745" by simp (*80 msec*) (* negation of ~54321 *) lemma "$- #-54321 = #54321" by simp (*90 msec*) lemma "#13 $* #19 = #247" by simp (*110 msec*) lemma "#-84 $* #51 = #-4284" by simp (*210 msec*) (*The worst case for 8-bit operands *) lemma "#255 $* #255 = #65025" by simp (*730 msec*) lemma "#1359 $* #-2468 = #-3354012" by simp (*1.04 secs*) (** Comparisons **) lemma "(#89) $* #10 ≠ #889" by simp lemma "(#13) $< #18 $- #4" by simp lemma "(#-345) $< #-242 $+ #-100" by simp lemma "(#13557456) $< #18678654" by simp lemma "(#999999) $<= (#1000001 $+ #1) $- #2" by simp lemma "(#1234567) $<= #1234567" by simp (*** Quotient and remainder!! [they could be faster] ***) lemma "#23 zdiv #3 = #7" by simp lemma "#23 zmod #3 = #2" by simp (** negative dividend **) lemma "#-23 zdiv #3 = #-8" by simp lemma "#-23 zmod #3 = #1" by simp (** negative divisor **) lemma "#23 zdiv #-3 = #-8" by simp lemma "#23 zmod #-3 = #-1" by simp (** Negative dividend and divisor **) lemma "#-23 zdiv #-3 = #7" by simp lemma "#-23 zmod #-3 = #-2" by simp end
lemma
#13 $+ #19 = #32
lemma
#1234 $+ #5678 = #6912
lemma
#1359 $+ #-2468 = #-1109
lemma
#93746 $+ #-46375 = #47371
lemma
$- #65745 = #-65745
lemma
$- #-54321 = #54321
lemma
#13 $× #19 = #247
lemma
#-84 $× #51 = #-4284
lemma
#255 $× #255 = #65025
lemma
#1359 $× #-2468 = #-3354012
lemma
#89 $× #10 ≠ #889
lemma
#13 $< #18 $- #4
lemma
#-345 $< #-242 $+ #-100
lemma
#13557456 $< #18678654
lemma
#999999 $≤ #1000001 $+ #1 $- #2
lemma
#1234567 $≤ #1234567
lemma
#23 zdiv #3 = #7
lemma
#23 zmod #3 = #2
lemma
#-23 zdiv #3 = #-8
lemma
#-23 zmod #3 = #1
lemma
#23 zdiv #-3 = #-8
lemma
#23 zmod #-3 = #-1
lemma
#-23 zdiv #-3 = #7
lemma
#-23 zmod #-3 = #-2