1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| A.Asperti, C.Sacerdoti Coen, *)
8 (* ||A|| E.Tassi, S.Zacchiroli *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU Lesser General Public License Version 2.1 *)
13 (**************************************************************************)
15 set "baseuri" "cic:/matita/nat/orders".
18 include "higher_order_defs/ordering.ma".
21 inductive le (n:nat) : nat \to Prop \def
23 | le_S : \forall m:nat. le n m \to le n (S m).
25 interpretation "natural 'less or equal to'" 'leq x y = (cic:/matita/nat/orders/le.ind#xpointer(1/1) x y).
27 interpretation "natural 'neither less nor equal to'" 'nleq x y =
28 (cic:/matita/logic/connectives/Not.con
29 (cic:/matita/nat/orders/le.ind#xpointer(1/1) x y)).
31 definition lt: nat \to nat \to Prop \def
32 \lambda n,m:nat.(S n) \leq m.
34 interpretation "natural 'less than'" 'lt x y = (cic:/matita/nat/orders/lt.con x y).
36 interpretation "natural 'not less than'" 'nless x y =
37 (cic:/matita/logic/connectives/Not.con (cic:/matita/nat/orders/lt.con x y)).
39 definition ge: nat \to nat \to Prop \def
40 \lambda n,m:nat.m \leq n.
42 interpretation "natural 'greater or equal to'" 'geq x y = (cic:/matita/nat/orders/ge.con x y).
44 definition gt: nat \to nat \to Prop \def
47 interpretation "natural 'greater than'" 'gt x y = (cic:/matita/nat/orders/gt.con x y).
49 interpretation "natural 'not greater than'" 'ngtr x y =
50 (cic:/matita/logic/connectives/Not.con (cic:/matita/nat/orders/gt.con x y)).
52 theorem transitive_le : transitive nat le.
53 unfold transitive.intros.elim H1.
55 apply le_S.assumption.
58 theorem trans_le: \forall n,m,p:nat. n \leq m \to m \leq p \to n \leq p
61 theorem transitive_lt: transitive nat lt.
62 unfold transitive.unfold lt.intros.elim H1.
63 apply le_S. assumption.
64 apply le_S.assumption.
67 theorem trans_lt: \forall n,m,p:nat. lt n m \to lt m p \to lt n p
70 theorem le_S_S: \forall n,m:nat. n \leq m \to S n \leq S m.
73 apply le_S.assumption.
76 theorem le_O_n : \forall n:nat. O \leq n.
82 theorem le_n_Sn : \forall n:nat. n \leq S n.
83 intros. apply le_S.apply le_n.
86 theorem le_pred_n : \forall n:nat. pred n \leq n.
89 simplify.apply le_n_Sn.
92 theorem le_S_S_to_le : \forall n,m:nat. S n \leq S m \to n \leq m.
93 intros.change with (pred (S n) \leq pred (S m)).
94 elim H.apply le_n.apply (trans_le ? (pred n1)).assumption.
98 theorem lt_S_S_to_lt: \forall n,m.
100 intros. apply le_S_S_to_le. assumption.
103 theorem leS_to_not_zero : \forall n,m:nat. S n \leq m \to not_zero m.
104 intros.elim H.exact I.exact I.
108 theorem not_le_Sn_O: \forall n:nat. S n \nleq O.
109 intros.unfold Not.simplify.intros.apply (leS_to_not_zero ? ? H).
112 theorem not_le_Sn_n: \forall n:nat. S n \nleq n.
113 intros.elim n.apply not_le_Sn_O.unfold Not.simplify.intros.cut (S n1 \leq n1).
115 apply le_S_S_to_le.assumption.
118 theorem lt_pred: \forall n,m.
119 O < n \to n < m \to pred n < pred m.
121 [intros.apply False_ind.apply (not_le_Sn_O ? H)
122 |intros.apply False_ind.apply (not_le_Sn_O ? H1)
123 |intros.simplify.unfold.apply le_S_S_to_le.assumption
128 theorem le_to_or_lt_eq : \forall n,m:nat.
129 n \leq m \to n < m \lor n = m.
132 left.unfold lt.apply le_S_S.assumption.
136 theorem lt_to_not_eq : \forall n,m:nat. n<m \to n \neq m.
137 unfold Not.intros.cut ((le (S n) m) \to False).
138 apply Hcut.assumption.rewrite < H1.
143 theorem lt_to_le : \forall n,m:nat. n<m \to n \leq m.
144 simplify.intros.unfold lt in H.elim H.
145 apply le_S. apply le_n.
146 apply le_S. assumption.
149 theorem lt_S_to_le : \forall n,m:nat. n < S m \to n \leq m.
151 apply le_S_S_to_le.assumption.
154 theorem not_le_to_lt: \forall n,m:nat. n \nleq m \to m<n.
156 apply (nat_elim2 (\lambda n,m.n \nleq m \to m<n)).
157 intros.apply (absurd (O \leq n1)).apply le_O_n.assumption.
158 unfold Not.unfold lt.intros.apply le_S_S.apply le_O_n.
159 unfold Not.unfold lt.intros.apply le_S_S.apply H.intros.apply H1.apply le_S_S.
163 theorem lt_to_not_le: \forall n,m:nat. n<m \to m \nleq n.
164 unfold Not.unfold lt.intros 3.elim H.
165 apply (not_le_Sn_n n H1).
166 apply H2.apply lt_to_le. apply H3.
169 theorem not_lt_to_le: \forall n,m:nat. Not (lt n m) \to le m n.
172 apply not_le_to_lt.exact H.
175 theorem le_to_not_lt: \forall n,m:nat. le n m \to Not (lt m n).
176 intros.unfold Not.unfold lt.
177 apply lt_to_not_le.unfold lt.
178 apply le_S_S.assumption.
182 theorem le_n_O_to_eq : \forall n:nat. n \leq O \to O=n.
183 intro.elim n.reflexivity.
189 theorem le_n_O_elim: \forall n:nat.n \leq O \to \forall P: nat \to Prop.
194 apply (not_le_Sn_O ? H1).
197 theorem le_n_Sm_elim : \forall n,m:nat.n \leq S m \to
198 \forall P:Prop. (S n \leq S m \to P) \to (n=S m \to P) \to P.
200 apply H2.reflexivity.
201 apply H3. apply le_S_S. assumption.
205 lemma le_to_le_to_eq: \forall n,m. n \le m \to m \le n \to n = m.
207 [intros.apply le_n_O_to_eq.assumption
208 |intros.apply sym_eq.apply le_n_O_to_eq.assumption
209 |intros.apply eq_f.apply H
210 [apply le_S_S_to_le.assumption
211 |apply le_S_S_to_le.assumption
216 (* lt and le trans *)
217 theorem lt_O_S : \forall n:nat. O < S n.
218 intro. unfold. apply le_S_S. apply le_O_n.
221 theorem lt_to_le_to_lt: \forall n,m,p:nat. lt n m \to le m p \to lt n p.
223 assumption.unfold lt.apply le_S.assumption.
226 theorem le_to_lt_to_lt: \forall n,m,p:nat. le n m \to lt m p \to lt n p.
228 assumption.apply H2.unfold lt.
229 apply lt_to_le.assumption.
232 theorem lt_S_to_lt: \forall n,m. S n < m \to n < m.
234 apply (trans_lt ? (S n))
235 [apply le_n|assumption]
238 theorem ltn_to_ltO: \forall n,m:nat. lt n m \to lt O m.
239 intros.apply (le_to_lt_to_lt O n).
240 apply le_O_n.assumption.
243 theorem lt_O_n_elim: \forall n:nat. lt O n \to
244 \forall P:nat\to Prop. (\forall m:nat.P (S m)) \to P n.
245 intro.elim n.apply False_ind.exact (not_le_Sn_O O H).
249 (* other abstract properties *)
250 theorem antisymmetric_le : antisymmetric nat le.
251 unfold antisymmetric.intros 2.
252 apply (nat_elim2 (\lambda n,m.(n \leq m \to m \leq n \to n=m))).
253 intros.apply le_n_O_to_eq.assumption.
254 intros.apply False_ind.apply (not_le_Sn_O ? H).
255 intros.apply eq_f.apply H.
256 apply le_S_S_to_le.assumption.
257 apply le_S_S_to_le.assumption.
260 theorem antisym_le: \forall n,m:nat. n \leq m \to m \leq n \to n=m
261 \def antisymmetric_le.
263 theorem decidable_le: \forall n,m:nat. decidable (n \leq m).
265 apply (nat_elim2 (\lambda n,m.decidable (n \leq m))).
266 intros.unfold decidable.left.apply le_O_n.
267 intros.unfold decidable.right.exact (not_le_Sn_O n1).
268 intros 2.unfold decidable.intro.elim H.
269 left.apply le_S_S.assumption.
270 right.unfold Not.intro.apply H1.apply le_S_S_to_le.assumption.
273 theorem decidable_lt: \forall n,m:nat. decidable (n < m).
274 intros.exact (decidable_le (S n) m).
277 (* well founded induction principles *)
279 theorem nat_elim1 : \forall n:nat.\forall P:nat \to Prop.
280 (\forall m.(\forall p. (p \lt m) \to P p) \to P m) \to P n.
281 intros.cut (\forall q:nat. q \le n \to P q).
282 apply (Hcut n).apply le_n.
283 elim n.apply (le_n_O_elim q H1).
285 intros.apply False_ind.apply (not_le_Sn_O p H2).
286 apply H.intros.apply H1.
288 apply lt_S_to_le.assumption.
289 apply (lt_to_le_to_lt p q (S n1) H3 H2).
292 (* some properties of functions *)
294 definition increasing \def \lambda f:nat \to nat.
295 \forall n:nat. f n < f (S n).
297 theorem increasing_to_monotonic: \forall f:nat \to nat.
298 increasing f \to monotonic nat lt f.
299 unfold monotonic.unfold lt.unfold increasing.unfold lt.intros.elim H1.apply H.
300 apply (trans_le ? (f n1)).
301 assumption.apply (trans_le ? (S (f n1))).
306 theorem le_n_fn: \forall f:nat \to nat. (increasing f)
307 \to \forall n:nat. n \le (f n).
310 apply (trans_le ? (S (f n1))).
311 apply le_S_S.apply H1.
312 simplify in H. unfold increasing in H.unfold lt in H.apply H.
315 theorem increasing_to_le: \forall f:nat \to nat. (increasing f)
316 \to \forall m:nat. \exists i. m \le (f i).
318 apply (ex_intro ? ? O).apply le_O_n.
320 apply (ex_intro ? ? (S a)).
321 apply (trans_le ? (S (f a))).
322 apply le_S_S.assumption.
323 simplify in H.unfold increasing in H.unfold lt in H.
327 theorem increasing_to_le2: \forall f:nat \to nat. (increasing f)
328 \to \forall m:nat. (f O) \le m \to
329 \exists i. (f i) \le m \land m <(f (S i)).
331 apply (ex_intro ? ? O).
332 split.apply le_n.apply H.
334 cut ((S n1) < (f (S a)) \lor (S n1) = (f (S a))).
336 apply (ex_intro ? ? a).
337 split.apply le_S. assumption.assumption.
338 apply (ex_intro ? ? (S a)).
339 split.rewrite < H7.apply le_n.
342 apply le_to_or_lt_eq.apply H6.