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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: Div.v 8642 2006-03-17 10:09:02Z notin $ i*)
(** Euclidean division *)
V7only [Import nat_scope.].
Open Local Scope nat_scope.
Require Le.
Require Euclid_def.
Require Compare_dec.
Implicit Variables Type n,a,b,q,r:nat.
Fixpoint inf_dec [n:nat] : nat->bool :=
[m:nat] Cases n m of
O _ => true
| (S n') O => false
| (S n') (S m') => (inf_dec n' m')
end.
Theorem div1 : (b:nat)(gt b O)->(a:nat)(diveucl a b).
Realizer Fix div1 {div1/2: nat->nat->diveucl :=
[b,a]Cases a of
O => (O,O)
| (S n) =>
let (q,r) = (div1 b n) in
if (le_gt_dec b (S r)) then ((S q),O)
else (q,(S r))
end}.
Program_all.
Rewrite e.
Replace b with (S r).
Simpl.
Elim plus_n_O; Auto with arith.
Apply le_antisym; Auto with arith.
Elim plus_n_Sm; Auto with arith.
Qed.
Theorem div2 : (b:nat)(gt b O)->(a:nat)(diveucl a b).
Realizer Fix div1 {div1/2: nat->nat->diveucl :=
[b,a]Cases a of
O => (O,O)
| (S n) =>
let (q,r) = (div1 b n) in
if (inf_dec b (S r)) :: :: { {(le b (S r))}+{(gt b (S r))} }
then ((S q),O)
else (q,(S r))
end}.
Program_all.
Rewrite e.
Replace b with (S r).
Simpl.
Elim plus_n_O; Auto with arith.
Apply le_antisym; Auto with arith.
Elim plus_n_Sm; Auto with arith.
Qed.
|
