src/ZF/Integ/IntDiv.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/ZF/Integ/IntDiv.thy	Fri Aug 11 13:27:17 2000 +0200
@@ -0,0 +1,62 @@
+(*  Title:      ZF/IntDiv.thy
+    ID:         $Id$
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1999  University of Cambridge
+
+The division operators div, mod
+*)
+
+IntDiv = IntArith + 
+
+constdefs
+  quorem :: "[i,i] => o"
+    "quorem == %<a,b> <q,r>.
+                      a = b$*q $+ r &
+                      (#0$<b & #0$<=r & r$<b | ~(#0$<b) & b$<r & r $<= #0)"
+
+  adjust :: "[i,i,i] => i"
+    "adjust(a,b) == %<q,r>. if #0 $<= r$-b then <#2$*q $+ #1,r$-b>
+                            else <#2$*q,r>"
+
+(**
+(** the division algorithm **)
+
+(*for the case a>=0, b>0*)
+consts posDivAlg :: "int*int => int*int"
+recdef posDivAlg "inv_image less_than (%<a,b>. nat(a $- b $+ #1))"
+    "posDivAlg <a,b> =
+       (if (a$<b | b$<=#0) then <#0,a>
+        else adjust a b (posDivAlg<a,#2$*b>))"
+
+(*for the case a<0, b>0*)
+consts negDivAlg :: "int*int => int*int"
+recdef negDivAlg "inv_image less_than (%<a,b>. nat(- a $- b))"
+    "negDivAlg <a,b> =
+       (if (#0$<=a$+b | b$<=#0) then <#-1,a$+b>
+        else adjust a b (negDivAlg<a,#2$*b>))"
+
+(*for the general case b~=0*)
+
+constdefs
+  negateSnd :: "int*int => int*int"
+    "negateSnd == %<q,r>. <q,-r>"
+
+  (*The full division algorithm considers all possible signs for a, b
+    including the special case a=0, b<0, because negDivAlg requires a<0*)
+  divAlg :: "int*int => int*int"
+    "divAlg ==
+       %<a,b>. if #0$<=a then
+                  if #0$<=b then posDivAlg <a,b>
+                  else if a=#0 then <#0,#0>
+                       else negateSnd (negDivAlg <-a,-b>)
+               else 
+                  if #0$<b then negDivAlg <a,b>
+                  else         negateSnd (posDivAlg <-a,-b>)"
+
+defs
+  div_def   "a div b == fst (divAlg <a,b>)"
+  mod_def   "a mod b == snd (divAlg <a,b>)"
+
+**)
+
+end