blob: e5ee2682d7ee218017903d854447ae201a407816 (
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
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
|
(***********************************************************************)
(* 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 *)
(***********************************************************************)
(* $Id$ *)
Require Export ZArith_base.
Require Export ZArithRing.
Require Export Omega.
V7only [Import Z_scope.].
Open Local Scope Z_scope.
(**********************************************************************)
(** Definition and properties of square root on Z *)
(** The following tactic replaces all instances of (POS (xI ...)) by
`2*(POS ...)+1` , but only when ... is not made only with xO, XI, or xH. *)
Tactic Definition compute_POS :=
Match Context With
| [|- [(POS (xI ?1))]] ->
(Match ?1 With
| [[xH]] -> Fail
| _ -> Rewrite (POS_xI ?1))
| [|- [(POS (xO ?1))]] ->
(Match ?1 With
| [[xH]] -> Fail
| _ -> Rewrite (POS_xO ?1)).
Inductive sqrt_data [n : Z] : Set :=
c_sqrt: (s, r :Z)`n=s*s+r`->`0<=r<=2*s`->(sqrt_data n) .
Definition sqrtrempos: (p : positive) (sqrt_data (POS p)).
Refine (Fix sqrtrempos {
sqrtrempos [p : positive] : (sqrt_data (POS p)) :=
<[p : ?] (sqrt_data (POS p))> Cases p of
xH => (c_sqrt `1` `1` `0` ? ?)
| (xO xH) => (c_sqrt `2` `1` `1` ? ?)
| (xI xH) => (c_sqrt `3` `1` `2` ? ?)
| (xO (xO p')) =>
Cases (sqrtrempos p') of
(c_sqrt s' r' Heq Hint) =>
Cases (Z_le_gt_dec `4*s'+1` `4*r'`) of
(left Hle) =>
(c_sqrt (POS (xO (xO p'))) `2*s'+1` `4*r'-(4*s'+1)` ? ?)
| (right Hgt) =>
(c_sqrt (POS (xO (xO p'))) `2*s'` `4*r'` ? ?)
end
end
| (xO (xI p')) =>
Cases (sqrtrempos p') of
(c_sqrt s' r' Heq Hint) =>
Cases
(Z_le_gt_dec `4*s'+1` `4*r'+2`) of
(left Hle) =>
(c_sqrt
(POS (xO (xI p'))) `2*s'+1` `4*r'+2-(4*s'+1)` ? ?)
| (right Hgt) =>
(c_sqrt (POS (xO (xI p'))) `2*s'` `4*r'+2` ? ?)
end
end
| (xI (xO p')) =>
Cases (sqrtrempos p') of
(c_sqrt s' r' Heq Hint) =>
Cases
(Z_le_gt_dec `4*s'+1` `4*r'+1`) of
(left Hle) =>
(c_sqrt
(POS (xI (xO p'))) `2*s'+1` `4*r'+1-(4*s'+1)` ? ?)
| (right Hgt) =>
(c_sqrt (POS (xI (xO p'))) `2*s'` `4*r'+1` ? ?)
end
end
| (xI (xI p')) =>
Cases (sqrtrempos p') of
(c_sqrt s' r' Heq Hint) =>
Cases
(Z_le_gt_dec `4*s'+1` `4*r'+3`) of
(left Hle) =>
(c_sqrt
(POS (xI (xI p'))) `2*s'+1` `4*r'+3-(4*s'+1)` ? ?)
| (right Hgt) =>
(c_sqrt (POS (xI (xI p'))) `2*s'` `4*r'+3` ? ?)
end
end
end
}); Clear sqrtrempos; Repeat compute_POS;
Try (Try Rewrite Heq; Ring; Fail); Try Omega.
Defined.
(** Define with integer input, but with a strong (readable) specification. *)
Definition Zsqrt : (x:Z)`0<=x`->{s:Z & {r:Z | x=`s*s+r` /\ `s*s<=x<(s+1)*(s+1)`}}.
Refine [x]
<[x:Z]`0<=x`->{s:Z & {r:Z | x=`s*s+r` /\ `s*s<=x<(s+1)*(s+1)`}}>Cases x of
(POS p) => [h]Cases (sqrtrempos p) of
(c_sqrt s r Heq Hint) =>
(existS ? [s:Z]{r:Z | `(POS p)=s*s+r` /\
`s*s<=(POS p)<(s+1)*(s+1)`}
s
(exist Z [r:Z]((POS p)=`s*s+r` /\ `s*s<=(POS p)<(s+1)*(s+1)`)
r ?))
end
| (NEG p) => [h](False_rec
{s:Z & {r:Z |
(NEG p)=`s*s+r` /\ `s*s<=(NEG p)<(s+1)*(s+1)`}}
(h (refl_equal ? SUPERIEUR)))
| ZERO => [h](existS ? [s:Z]{r:Z | `0=s*s+r` /\ `s*s<=0<(s+1)*(s+1)`}
`0` (exist Z [r:Z](`0=0*0+r`/\`0*0<=0<(0+1)*(0+1)`)
`0` ?))
end;Try Omega.
Split;[Omega|Rewrite Heq;Ring `(s+1)*(s+1)`;Omega].
Defined.
(** Define a function of type Z->Z that computes the integer square root,
but only for positive numbers, and 0 for others. *)
Definition Zsqrt_plain : Z->Z :=
[x]Cases x of
(POS p)=>Cases (Zsqrt (POS p) (ZERO_le_POS p)) of (existS s _) => s end
|(NEG p)=>`0`
|ZERO=>`0`
end.
(** A basic theorem about Zsqrt_plain *)
Theorem Zsqrt_interval :(x:Z)`0<=x`->
`(Zsqrt_plain x)*(Zsqrt_plain x)<= x < ((Zsqrt_plain x)+1)*((Zsqrt_plain x)+1)`.
Intros x;Case x.
Unfold Zsqrt_plain;Omega.
Intros p;Unfold Zsqrt_plain;Case (Zsqrt (POS p) (ZERO_le_POS p)).
Intros s (r,(Heq,Hint)) Hle;Assumption.
Intros p Hle;Elim Hle;Auto.
Qed.
|
