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(***********************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
(* \VV/ *************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
(*i $Id$ i*)
(* Contribution by Claude Marché and Xavier Urbain *)
(**
Euclidean Division
Defines first of function that allows Coq to normalize.
Then only after proves the main required property.
*)
Require Export ZArith.
Require Omega.
Require ZArithRing.
(**
Euclidean division of a positive by a integer
(that is supposed to be positive).
total function than returns an arbitrary value when
divisor is not positive
*)
Fixpoint Zdiv_eucl_POS [a:positive] : Z -> Z*Z := [b:Z]
Cases a of
| xH => if `(Zge_bool b 2)` then `(0,1)` else `(1,0)`
| (xO a') =>
let (q,r) = (Zdiv_eucl_POS a' b) in
if `(Zgt_bool b (r+r))` then `(q+q,r+r)` else `(q+q+1,r+r-b)`
| (xI a') =>
let (q,r) = (Zdiv_eucl_POS a' b) in
if `(Zgt_bool b (r+r+1))` then `(q+q,r+r+1)` else `(q+q+1,r+r+1-b)`
end.
(**
Euclidean division of integers.
Total function than returns (0,0) when dividing by 0.
*)
(*
The pseudo-code is:
if b = 0 : (0,0)
if b <> 0 and a = 0 : (0,0)
if b > 0 and a < 0 : let (q,r) = div_eucl_pos (-a) b in
if r = 0 then (-q,0) else (-(q+1),b-r)
if b < 0 and a < 0 : let (q,r) = div_eucl (-a) (-b) in (q,-r)
if b < 0 and a > 0 : let (q,r) = div_eucl a (-b) in
if r = 0 then (-q,0) else (-(q+1),b+r)
*)
Definition Zdiv_eucl [a,b:Z] : Z*Z :=
Cases a b of
| ZERO _ => `(0,0)`
| _ ZERO => `(0,0)`
| (POS a') (POS _) => (Zdiv_eucl_POS a' b)
| (NEG a') (POS _) =>
let (q,r) = (Zdiv_eucl_POS a' b) in
Cases r of
| ZERO => `(-q,0)`
| _ => `(-(q+1),b-r)`
end
| (NEG a') (NEG b') =>
let (q,r) = (Zdiv_eucl_POS a' (POS b')) in `(q,-r)`
| (POS a') (NEG b') =>
let (q,r) = (Zdiv_eucl_POS a' (POS b')) in
Cases r of
| ZERO => `(-q,0)`
| _ => `(-(q+1),b+r)`
end
end.
(** Division and modulo are projections of [Zdiv_eucl] *)
Definition Zdiv [a,b:Z] : Z := let (q,_) = (Zdiv_eucl a b) in q.
Definition Zmod [a,b:Z] : Z := let (_,r) = (Zdiv_eucl a b) in r.
(* Tests:
Eval Compute in `(Zdiv_eucl 7 3)`.
Eval Compute in `(Zdiv_eucl (-7) 3)`.
Eval Compute in `(Zdiv_eucl 7 (-3))`.
Eval Compute in `(Zdiv_eucl (-7) (-3))`.
*)
(**
Main division theorem.
First a lemma for positive
*)
Lemma Z_div_mod_POS : (b:Z)`b > 0` -> (a:positive)
let (q,r)=(Zdiv_eucl_POS a b) in `(POS a) = b*q + r`/\`0<=r<b`.
Proof.
Induction a; Simpl.
Intro p.
Case (Zdiv_eucl_POS p b).
Intros q r H1.
Decompose [and] H1.
Generalize (Zgt_cases b `r+r+1`).
Case (Zgt_bool b `r+r+1`);
(Rewrite POS_xI; Rewrite H0; Split ; [ Ring | Omega ]).
Intros p.
Case (Zdiv_eucl_POS p b).
Intros q r H1.
Decompose [and] H1.
Generalize (Zgt_cases b `r+r`).
Case (Zgt_bool b `r+r`);
(Rewrite POS_xO; Rewrite H0; Split ; [ Ring | Omega ]).
Generalize (Zge_cases b `2`).
Case (Zge_bool b `2`); (Intros; Split; [Ring | Omega ]).
Omega.
Qed.
Theorem Z_div_mod : (a,b:Z)`b > 0` ->
let (q,r) = (Zdiv_eucl a b) in `a = b*q + r` /\ `0<=r<b`.
Proof.
Intros a b; Case a; Case b; Try (Simpl; Intros; Omega).
Unfold Zdiv_eucl; Intros; Apply Z_div_mod_POS; Trivial.
Intros.
Absurd `(NEG p) > 0`; [ Generalize (NEG_lt_ZERO p); Omega | Trivial ].
Intros.
Generalize (Z_div_mod_POS (POS p) H p0).
Unfold Zdiv_eucl.
Case (Zdiv_eucl_POS p0 (POS p)).
Intros z z0.
Case z0.
Intros [H1 H2].
Split; Trivial.
Replace (NEG p0) with `-(POS p0)`; [ Rewrite H1; Ring | Trivial ].
Intros p1 [H1 H2].
Split; Trivial.
Replace (NEG p0) with `-(POS p0)`; [ Rewrite H1; Ring | Trivial ].
Generalize (POS_gt_ZERO p1); Omega.
Intros p1 [H1 H2].
Split; Trivial.
Replace (NEG p0) with `-(POS p0)`; [ Rewrite H1; Ring | Trivial ].
Generalize (NEG_lt_ZERO p1); Omega.
Intros.
Absurd `(NEG p)>0`; [ Generalize (NEG_lt_ZERO p); Omega | Omega ].
Qed.
Theorem Z_div_mod_eq : (a,b:Z)`b > 0` -> `a = b * (Zdiv a b) + (Zmod a b)`.
Proof.
Unfold Zdiv Zmod.
Intros a b Hb.
Generalize (Z_div_mod a b Hb).
Pattern (Zdiv_eucl a b).
Case (Zdiv_eucl); Tauto.
Save.
(** Existence theorems *)
Set Implicit Arguments.
Theorem Zdiv_eucl_exist : (b:Z)`b > 0` -> (a:Z)
{ qr:Z*Z | let (q,r)=qr in `a=b*q+r` /\ `0 <= r < b` }.
Proof.
Intros b Hb a.
Exists (Zdiv_eucl a b).
Exact (Z_div_mod a b Hb).
Qed.
Theorem Zdiv_eucl_extended : (b:Z)`b <> 0` -> (a:Z)
{ qr:Z*Z | let (q,r)=qr in `a=b*q+r` /\ `0 <= r < |b|` }.
Proof.
Intros b Hb a.
Elim (Z_le_gt_dec `0` b);Intro Hb'.
Cut `b>0`;[Intro Hb''|Omega].
Rewrite Zabs_eq;[Apply Zdiv_eucl_exist;Assumption|Assumption].
Cut `-b>0`;[Intro Hb''|Omega].
Elim (Zdiv_eucl_exist Hb'' a);Intros qr.
Elim qr;Intros q r Hqr.
Exists (pair ? ? `-q` r).
Elim Hqr;Intros.
Split.
Rewrite <- Zmult_Zopp_left;Assumption.
Rewrite Zabs_non_eq;[Assumption|Omega].
Qed.
(** Syntax *)
Grammar znatural expr2 : constr :=
expr_div [ expr2($p) "/" expr2($c) ] -> [ (Zdiv $p $c) ]
| expr_mod [ expr2($p) "%" expr2($c) ] -> [ (Zmod $p $c) ]
.
Syntax constr
level 6:
Zdiv [ (Zdiv $n1 $n2) ]
-> [ [<hov 0> "`"(ZEXPR $n1):E "/" [0 0] (ZEXPR $n2):L "`"] ]
| Zmod [ (Zmod $n1 $n2) ]
-> [ [<hov 0> "`"(ZEXPR $n1):E "%" [0 0] (ZEXPR $n2):L "`"] ]
| Zdiv_inside
[ << (ZEXPR <<(Zdiv $n1 $n2)>>) >> ]
-> [ (ZEXPR $n1):E "/" [0 0] (ZEXPR $n2):L ]
| Zmod_inside
[ << (ZEXPR <<(Zmod $n1 $n2)>>) >> ]
-> [ (ZEXPR $n1):E "%" [0 0] (ZEXPR $n2):L ]
.
|
