(* Title: HOL/IntDiv.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1999 University of Cambridge
The division operators div, mod and the divides relation "dvd"
*)
IntDiv = IntArith + Recdef +
constdefs
quorem :: "(int*int) * (int*int) => bool"
"quorem == %((a,b), (q,r)).
a = b*q + r &
(if 0 < b then 0<=r & r<b else b<r & r <= 0)"
adjust :: "[int, int*int] => int*int"
"adjust 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 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 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))"
instance
int :: "Divides.div" (*avoid clash with 'div' token*)
defs
div_def "a div b == fst (divAlg (a,b))"
mod_def "a mod b == snd (divAlg (a,b))"
end