blob: e6cf461030b7ccdc8267a595795e75f894202dec (
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
|
(************************************************************************)
(* 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: Wf_Z.v,v 1.1.2.1 2004/07/16 19:31:42 herbelin Exp $ i*)
Require BinInt.
Require Zcompare.
Require Zorder.
Require Znat.
Require Zmisc.
Require Zsyntax.
Require Wf_nat.
V7only [Import Z_scope.].
Open Local 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 (inject_nat n)) ===> (x:Z)`x > 0) -> (P x)
/\
||
||
(Q O) (n:nat)(Q n)->(Q (S n)) <=== (P 0) (x:Z) (P x) -> (P (Zs x))
<=== (inject_nat (S n))=(Zs (inject_nat n))
<=== inject_nat_complete
>>
Then the diagram will be closed and the theorem proved. *)
Lemma inject_nat_complete :
(x:Z)`0 <= x` -> (EX n:nat | x=(inject_nat n)).
Intro x; NewDestruct x; Intros;
[ Exists O; Auto with arith
| Specialize (ZL4 p); Intros Hp; Elim Hp; Intros;
Exists (S x); Intros; Simpl;
Specialize (bij1 x); Intro Hx0;
Rewrite <- H0 in Hx0;
Apply f_equal with f:=POS;
Apply convert_intro; Auto with arith
| Absurd `0 <= (NEG p)`;
[ Unfold Zle; Simpl; Do 2 (Unfold not); Auto with arith
| Assumption]
].
Qed.
Lemma ZL4_inf: (y:positive) { h:nat | (convert y)=(S h) }.
Intro y; NewInduction y as [p H|p H1|]; [
Elim H; Intros x H1; Exists (plus (S x) (S x));
Unfold convert ;Simpl; Rewrite ZL0; Rewrite ZL2; Unfold convert in H1;
Rewrite H1; Auto with arith
| Elim H1;Intros x H2; Exists (plus x (S x)); Unfold convert;
Simpl; Rewrite ZL0; Rewrite ZL2;Unfold convert in H2; Rewrite H2; Auto with arith
| Exists O ;Auto with arith].
Qed.
Lemma inject_nat_complete_inf :
(x:Z)`0 <= x` -> { n:nat | (x=(inject_nat n)) }.
Intro x; NewDestruct x; Intros;
[ Exists O; Auto with arith
| Specialize (ZL4_inf p); Intros Hp; Elim Hp; Intros x0 H0;
Exists (S x0); Intros; Simpl;
Specialize (bij1 x0); Intro Hx0;
Rewrite <- H0 in Hx0;
Apply f_equal with f:=POS;
Apply convert_intro; Auto with arith
| Absurd `0 <= (NEG p)`;
[ Unfold Zle; Simpl; Do 2 (Unfold not); Auto with arith
| Assumption]
].
Qed.
Lemma inject_nat_prop :
(P:Z->Prop)((n:nat)(P (inject_nat n))) ->
(x:Z) `0 <= x` -> (P x).
Intros P H x H0.
Specialize (inject_nat_complete x H0).
Intros Hn; Elim Hn; Intros.
Rewrite -> H1; Apply H.
Qed.
Lemma inject_nat_set :
(P:Z->Set)((n:nat)(P (inject_nat n))) ->
(x:Z) `0 <= x` -> (P x).
Intros P H x H0.
Specialize (inject_nat_complete_inf x H0).
Intros Hn; Elim Hn; Intros.
Rewrite -> p; Apply H.
Qed.
Lemma natlike_ind : (P:Z->Prop) (P `0`) ->
((x:Z)(`0 <= x` -> (P x) -> (P (Zs x)))) ->
(x:Z) `0 <= x` -> (P x).
Intros P H H0 x H1; Apply inject_nat_prop;
[ Induction n;
[ Simpl; Assumption
| Intros; Rewrite -> (inj_S n0);
Exact (H0 (inject_nat n0) (ZERO_le_inj n0) H2) ]
| Assumption].
Qed.
Lemma natlike_rec : (P:Z->Set) (P `0`) ->
((x:Z)(`0 <= x` -> (P x) -> (P (Zs x)))) ->
(x:Z) `0 <= x` -> (P x).
Intros P H H0 x H1; Apply inject_nat_set;
[ Induction n;
[ Simpl; Assumption
| Intros; Rewrite -> (inj_S n0);
Exact (H0 (inject_nat n0) (ZERO_le_inj n0) H2) ]
| Assumption].
Qed.
Section Efficient_Rec.
(** [natlike_rec2] is the same as [natlike_rec], but with a different proof, designed
to give a better extracted term. *)
Local R := [a,b:Z]`0<=a`/\`a<b`.
Local R_wf : (well_founded Z R).
Proof.
LetTac f := [z]Cases z of (POS p) => (convert p) | ZERO => O | (NEG _) => O end.
Apply well_founded_lt_compat with f.
Unfold R f; Clear f R.
Intros x y; Case x; Intros; Elim H; Clear H.
Case y; Intros; Apply compare_convert_O Orelse Inversion H0.
Case y; Intros; Apply compare_convert_INFERIEUR Orelse Inversion H0; Auto.
Intros; Elim H; Auto.
Qed.
Lemma natlike_rec2 : (P:Z->Type)(P `0`) ->
((z:Z)`0<=z` -> (P z) -> (P (Zs z))) -> (z:Z)`0<=z` -> (P z).
Proof.
Intros P Ho Hrec z; Pattern z; Apply (well_founded_induction_type Z R R_wf).
Intro x; Case x.
Trivial.
Intros.
Assert `0<=(Zpred (POS p))`.
Apply Zlt_ZERO_pred_le_ZERO; Unfold Zlt; Simpl; Trivial.
Rewrite Zs_pred.
Apply Hrec.
Auto.
Apply X; Auto; Unfold R; Intuition; Apply Zlt_pred_n_n.
Intros; Elim H; Simpl; Trivial.
Qed.
(** A variant of the previous using [Zpred] instead of [Zs]. *)
Lemma natlike_rec3 : (P:Z->Type)(P `0`) ->
((z:Z)`0<z` -> (P (Zpred z)) -> (P z)) -> (z:Z)`0<=z` -> (P z).
Proof.
Intros P Ho Hrec z; Pattern z; Apply (well_founded_induction_type Z R R_wf).
Intro x; Case x.
Trivial.
Intros; Apply Hrec.
Unfold Zlt; Trivial.
Assert `0<=(Zpred (POS p))`.
Apply Zlt_ZERO_pred_le_ZERO; Unfold Zlt; Simpl; Trivial.
Apply X; Auto; Unfold R; Intuition; Apply Zlt_pred_n_n.
Intros; Elim H; Simpl; Trivial.
Qed.
(** A more general induction principal using [Zlt]. *)
Lemma Z_lt_rec : (P:Z->Type)
((x:Z)((y:Z)`0 <= y < x`->(P y))->(P x)) -> (x:Z)`0 <= x`->(P x).
Proof.
Intros P Hrec z; Pattern z; Apply (well_founded_induction_type Z R R_wf).
Intro x; Case x; Intros.
Apply Hrec; Intros.
Assert H2: `0<0`.
Apply Zle_lt_trans with y; Intuition.
Inversion H2.
Firstorder.
Unfold Zle Zcompare in H; Elim H; Auto.
Defined.
Lemma Z_lt_induction :
(P:Z->Prop)
((x:Z)((y:Z)`0 <= y < x`->(P y))->(P x))
-> (x:Z)`0 <= x`->(P x).
Proof.
Exact Z_lt_rec.
Qed.
End Efficient_Rec.
|
