blob: 24f4ba36c7ed50c5c837786faf76a0f6da6e71ca (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
|
(***********************************************************************)
(* 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 *)
(***********************************************************************)
(* $Id$ *)
(* Euclidean division *)
Require Le.
Require Euclid_def.
Require Compare_dec.
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.
Save.
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.
Save.
|
