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
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import BinInt.
Require Import Zcompare.
Require Import Zorder.
Require Import Znat.
Require Import Zmisc.
Require Import Wf_nat.
Local Open Scope Z_scope.
(** Our purpose is to write an induction shema for {0,1,2,...}
similar to the [nat] schema (Theorem [Natlike_rec]). For that the
following implications will be used :
<<
∀n:nat, Q n == ∀n:nat, P (Z.of_nat n) ===> ∀x:Z, x <= 0 -> P x
/\
||
||
(Q O) ∧ (∀n:nat, Q n -> Q (S n)) <=== (P 0) ∧ (∀x:Z, P x -> P (Z.succ x))
<=== (Z.of_nat (S n) = Z.succ (Z.of_nat n))
<=== Z_of_nat_complete
>>
Then the diagram will be closed and the theorem proved. *)
Lemma Z_of_nat_complete (x : Z) :
0 <= x -> exists n : nat, x = Z.of_nat n.
Proof.
intros H. exists (Z.to_nat x). symmetry. now apply Z2Nat.id.
Qed.
Lemma Z_of_nat_complete_inf (x : Z) :
0 <= x -> {n : nat | x = Z_of_nat n}.
Proof.
intros H. exists (Z.to_nat x). symmetry. now apply Z2Nat.id.
Qed.
Lemma Z_of_nat_prop :
forall P:Z -> Prop,
(forall n:nat, P (Z.of_nat n)) -> forall x:Z, 0 <= x -> P x.
Proof.
intros P H x Hx. now destruct (Z_of_nat_complete x Hx) as (n,->).
Qed.
Lemma Z_of_nat_set :
forall P:Z -> Set,
(forall n:nat, P (Z_of_nat n)) -> forall x:Z, 0 <= x -> P x.
Proof.
intros P H x Hx. now destruct (Z_of_nat_complete_inf x Hx) as (n,->).
Qed.
Lemma natlike_ind :
forall P:Z -> Prop,
P 0 ->
(forall x:Z, 0 <= x -> P x -> P (Z.succ x)) ->
forall x:Z, 0 <= x -> P x.
Proof.
intros P Ho Hrec x Hx; apply Z_of_nat_prop; trivial.
induction n. exact Ho.
rewrite Nat2Z.inj_succ. apply Hrec; trivial using Nat2Z.is_nonneg.
Qed.
Lemma natlike_rec :
forall P:Z -> Set,
P 0 ->
(forall x:Z, 0 <= x -> P x -> P (Z.succ x)) ->
forall x:Z, 0 <= x -> P x.
Proof.
intros P Ho Hrec x Hx; apply Z_of_nat_set; trivial.
induction n. exact Ho.
rewrite Nat2Z.inj_succ. apply Hrec; trivial using Nat2Z.is_nonneg.
Qed.
Section Efficient_Rec.
(** [natlike_rec2] is the same as [natlike_rec], but with a different proof, designed
to give a better extracted term. *)
Let R (a b:Z) := 0 <= a /\ a < b.
Let R_wf : well_founded R.
Proof.
apply well_founded_lt_compat with Z.to_nat.
intros x y (Hx,H). apply Z2Nat.inj_lt; Z.order.
Qed.
Lemma natlike_rec2 :
forall P:Z -> Type,
P 0 ->
(forall z:Z, 0 <= z -> P z -> P (Z.succ z)) ->
forall z:Z, 0 <= z -> P z.
Proof.
intros P Ho Hrec.
induction z as [z IH] using (well_founded_induction_type R_wf).
destruct z; intros Hz.
- apply Ho.
- set (y:=Z.pred (Zpos p)).
assert (LE : 0 <= y) by (unfold y; now apply Z.lt_le_pred).
assert (EQ : Zpos p = Z.succ y) by (unfold y; now rewrite Z.succ_pred).
rewrite EQ. apply Hrec, IH; trivial.
split; trivial. unfold y; apply Z.lt_pred_l.
- now destruct Hz.
Qed.
(** A variant of the previous using [Z.pred] instead of [Z.succ]. *)
Lemma natlike_rec3 :
forall P:Z -> Type,
P 0 ->
(forall z:Z, 0 < z -> P (Z.pred z) -> P z) ->
forall z:Z, 0 <= z -> P z.
Proof.
intros P Ho Hrec.
induction z as [z IH] using (well_founded_induction_type R_wf).
destruct z; intros Hz.
- apply Ho.
- assert (EQ : 0 <= Z.pred (Zpos p)) by now apply Z.lt_le_pred.
apply Hrec. easy. apply IH; trivial. split; trivial.
apply Z.lt_pred_l.
- now destruct Hz.
Qed.
(** A more general induction principle on non-negative numbers using [Zlt]. *)
Lemma Zlt_0_rec :
forall P:Z -> Type,
(forall x:Z, (forall y:Z, 0 <= y < x -> P y) -> 0 <= x -> P x) ->
forall x:Z, 0 <= x -> P x.
Proof.
intros P Hrec.
induction x as [x IH] using (well_founded_induction_type R_wf).
destruct x; intros Hx.
- apply Hrec; trivial. intros y (Hy,Hy').
assert (0 < 0) by now apply Z.le_lt_trans with y.
discriminate.
- apply Hrec; trivial. intros y (Hy,Hy').
apply IH; trivial. now split.
- now destruct Hx.
Defined.
Lemma Zlt_0_ind :
forall P:Z -> Prop,
(forall x:Z, (forall y:Z, 0 <= y < x -> P y) -> 0 <= x -> P x) ->
forall x:Z, 0 <= x -> P x.
Proof.
exact Zlt_0_rec.
Qed.
(** Obsolete version of [Zlt] induction principle on non-negative numbers *)
Lemma Z_lt_rec :
forall P:Z -> Type,
(forall x:Z, (forall y:Z, 0 <= y < x -> P y) -> P x) ->
forall x:Z, 0 <= x -> P x.
Proof.
intros P Hrec; apply Zlt_0_rec; auto.
Qed.
Lemma Z_lt_induction :
forall P:Z -> Prop,
(forall x:Z, (forall y:Z, 0 <= y < x -> P y) -> P x) ->
forall x:Z, 0 <= x -> P x.
Proof.
exact Z_lt_rec.
Qed.
(** An even more general induction principle using [Zlt]. *)
Lemma Zlt_lower_bound_rec :
forall P:Z -> Type, forall z:Z,
(forall x:Z, (forall y:Z, z <= y < x -> P y) -> z <= x -> P x) ->
forall x:Z, z <= x -> P x.
Proof.
intros P z Hrec x Hx.
rewrite <- (Z.sub_simpl_r x z). apply Z.le_0_sub in Hx.
pattern (x - z); apply Zlt_0_rec; trivial.
clear x Hx. intros x IH Hx.
apply Hrec. intros y (Hy,Hy').
rewrite <- (Z.sub_simpl_r y z). apply IH; split.
now rewrite Z.le_0_sub.
now apply Z.lt_sub_lt_add_r.
now rewrite <- (Z.add_le_mono_r 0 x z).
Qed.
Lemma Zlt_lower_bound_ind :
forall P:Z -> Prop, forall z:Z,
(forall x:Z, (forall y:Z, z <= y < x -> P y) -> z <= x -> P x) ->
forall x:Z, z <= x -> P x.
Proof.
exact Zlt_lower_bound_rec.
Qed.
End Efficient_Rec.
|
