aboutsummaryrefslogtreecommitdiffhomepage
path: root/theories/Arith/Compare_dec.v
blob: 573f54e9f7b3a66c43b0cf7d88734ca5590255f2 (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
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
(************************************************************************)
(*  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 $Id$ i*)

Require Import Le.
Require Import Lt.
Require Import Gt.
Require Import Decidable.

Open Local Scope nat_scope.

Implicit Types m n x y : nat.

Definition zerop n : {n = 0} + {0 < n}.
  destruct n; auto with arith.
Defined.

Definition lt_eq_lt_dec n m : {n < m} + {n = m} + {m < n}.
  induction n; simple destruct m; auto with arith.
  intros m0; elim (IHn m0); auto with arith.
  induction 1; auto with arith.
Defined.

Definition gt_eq_gt_dec n m : {m > n} + {n = m} + {n > m}.
  exact lt_eq_lt_dec.
Defined.

Definition le_lt_dec n m : {n <= m} + {m < n}.
  induction n.
  auto with arith.
  destruct m.
  auto with arith.
  elim (IHn m); auto with arith.
Defined.

Definition le_le_S_dec n m : {n <= m} + {S m <= n}.
  exact le_lt_dec.
Defined.

Definition le_ge_dec n m : {n <= m} + {n >= m}.
  intros; elim (le_lt_dec n m); auto with arith.
Defined.

Definition le_gt_dec n m : {n <= m} + {n > m}.
  exact le_lt_dec.
Defined.

Definition le_lt_eq_dec n m : n <= m -> {n < m} + {n = m}.
  intros; elim (lt_eq_lt_dec n m); auto with arith.
  intros; absurd (m < n); auto with arith.
Defined.

(** Proofs of decidability *)

Theorem dec_le : forall n m, decidable (n <= m).
Proof.
  intros x y; unfold decidable in |- *; elim (le_gt_dec x y);
    [ auto with arith | intro; right; apply gt_not_le; assumption ].
Qed.

Theorem dec_lt : forall n m, decidable (n < m).
Proof.
  intros x y; unfold lt in |- *; apply dec_le.
Qed.

Theorem dec_gt : forall n m, decidable (n > m).
Proof.
  intros x y; unfold gt in |- *; apply dec_lt.
Qed.

Theorem dec_ge : forall n m, decidable (n >= m).
Proof.
  intros x y; unfold ge in |- *; apply dec_le.
Qed.

Theorem not_eq : forall n m, n <> m -> n < m \/ m < n.
Proof.
  intros x y H; elim (lt_eq_lt_dec x y);
    [ intros H1; elim H1;
      [ auto with arith | intros H2; absurd (x = y); assumption ]
      | auto with arith ].
Qed.


Theorem not_le : forall n m, ~ n <= m -> n > m.
Proof.
  intros x y H; elim (le_gt_dec x y);
    [ intros H1; absurd (x <= y); assumption | trivial with arith ].
Qed.

Theorem not_gt : forall n m, ~ n > m -> n <= m.
Proof.
  intros x y H; elim (le_gt_dec x y);
    [ trivial with arith | intros H1; absurd (x > y); assumption ].
Qed.

Theorem not_ge : forall n m, ~ n >= m -> n < m.
Proof.
  intros x y H; exact (not_le y x H).
Qed.

Theorem not_lt : forall n m, ~ n < m -> n >= m.
Proof.
  intros x y H; exact (not_gt y x H). 
Qed.


(** A ternary comparison function in the spirit of [Zcompare]. *)

Fixpoint nat_compare n m :=
  match n, m with
   | O, O => Eq
   | O, S _ => Lt
   | S _, O => Gt
   | S n', S m' => nat_compare n' m'
  end.

Lemma nat_compare_S : forall n m, nat_compare (S n) (S m) = nat_compare n m.
Proof.
 reflexivity.
Qed.

Lemma nat_compare_eq_iff : forall n m, nat_compare n m = Eq <-> n = m.
Proof.
  induction n; destruct m; simpl; split; auto; try discriminate;
   destruct (IHn m); auto.
Qed.

Lemma nat_compare_eq : forall n m, nat_compare n m = Eq -> n = m.
Proof.
  intros; apply -> nat_compare_eq_iff; auto.
Qed.

Lemma nat_compare_lt : forall n m, n<m <-> nat_compare n m = Lt.
Proof.
  induction n; destruct m; simpl; split; auto with arith;
   try solve [inversion 1].
  destruct (IHn m); auto with arith.
  destruct (IHn m); auto with arith.
Qed.

Lemma nat_compare_gt : forall n m, n>m <-> nat_compare n m = Gt.
Proof.
  induction n; destruct m; simpl; split; auto with arith;
   try solve [inversion 1].
  destruct (IHn m); auto with arith.
  destruct (IHn m); auto with arith.
Qed.

Lemma nat_compare_le : forall n m, n<=m <-> nat_compare n m <> Gt.
Proof.
  split.
  intros LE; contradict LE.
   apply lt_not_le. apply <- nat_compare_gt; auto.
  intros NGT. apply not_lt. contradict NGT.
   apply -> nat_compare_gt; auto.
Qed.

Lemma nat_compare_ge : forall n m, n>=m <-> nat_compare n m <> Lt.
Proof.
  split.
  intros GE; contradict GE.
   apply lt_not_le. apply <- nat_compare_lt; auto.
  intros NLT. apply not_lt. contradict NLT.
   apply -> nat_compare_lt; auto.
Qed.

(** Some projections of the above equivalences, used in OrderedTypeEx. *)

Lemma nat_compare_Lt_lt : forall n m, nat_compare n m = Lt -> n<m.
Proof.
  intros; apply <- nat_compare_lt; auto.
Qed.

Lemma nat_compare_Gt_gt : forall n m, nat_compare n m = Gt -> n>m.
Proof.
  intros; apply <- nat_compare_gt; auto.
Qed.

(** A previous definition of [nat_compare] in terms of [lt_eq_lt_dec].
    The new version avoids the creation of proof parts. *)

Definition nat_compare_alt (n m:nat) :=
  match lt_eq_lt_dec n m with
    | inleft (left _) => Lt
    | inleft (right _) => Eq
    | inright _ => Gt
  end.

Lemma nat_compare_equiv: forall n m,
 nat_compare n m = nat_compare_alt n m.
Proof.
  intros; unfold nat_compare_alt; destruct lt_eq_lt_dec as [[LT|EQ]|GT].
  apply -> nat_compare_lt; auto.
  apply <- nat_compare_eq_iff; auto.
  apply -> nat_compare_gt; auto.
Qed.


(** A boolean version of [le] over [nat]. *)

Fixpoint leb (m:nat) : nat -> bool :=
  match m with
    | O => fun _:nat => true
    | S m' =>
      fun n:nat => match n with
                     | O => false
                     | S n' => leb m' n'
                   end
  end.

Lemma leb_correct : forall m n, m <= n -> leb m n = true.
Proof.
  induction m as [| m IHm]. trivial.
  destruct n. intro H. elim (le_Sn_O _ H).
  intros. simpl in |- *. apply IHm. apply le_S_n. assumption.
Qed.

Lemma leb_complete : forall m n, leb m n = true -> m <= n.
Proof.
  induction m. trivial with arith.
  destruct n. intro H. discriminate H.
  auto with arith.
Qed.

Lemma leb_iff : forall m n, leb m n = true <-> m <= n.
Proof.
  split; auto using leb_correct, leb_complete.
Qed.

Lemma leb_correct_conv : forall m n, m < n -> leb n m = false.
Proof.
  intros.
  generalize (leb_complete n m).
  destruct (leb n m); auto.
  intros; elim (lt_not_le m n); auto.
Qed.

Lemma leb_complete_conv : forall m n, leb n m = false -> m < n.
Proof.
  intros m n EQ. apply not_le.
  intro LE. apply leb_correct in LE. rewrite LE in EQ; discriminate.
Qed.

Lemma leb_iff_conv : forall m n, leb n m = false <-> m < n.
Proof.
  split; auto using leb_complete_conv, leb_correct_conv.
Qed.

Lemma leb_compare : forall n m, leb n m = true <-> nat_compare n m <> Gt.
Proof.
 split; intros.
 apply -> nat_compare_le. auto using leb_complete.
 apply leb_correct. apply <- nat_compare_le; auto.
Qed.