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 lt_to_lt_S_S: ∀n,m. n < m → S n < S m.
106 apply (le_S_S ? ? H).
109 theorem leS_to_not_zero : \forall n,m:nat. S n \leq m \to not_zero m.
110 intros.elim H.exact I.exact I.
114 theorem not_le_Sn_O: \forall n:nat. S n \nleq O.
115 intros.unfold Not.simplify.intros.apply (leS_to_not_zero ? ? H).
118 theorem not_le_Sn_n: \forall n:nat. S n \nleq n.
119 intros.elim n.apply not_le_Sn_O.unfold Not.simplify.intros.cut (S n1 \leq n1).
121 apply le_S_S_to_le.assumption.
124 theorem lt_pred: \forall n,m.
125 O < n \to n < m \to pred n < pred m.
127 [intros.apply False_ind.apply (not_le_Sn_O ? H)
128 |intros.apply False_ind.apply (not_le_Sn_O ? H1)
129 |intros.simplify.unfold.apply le_S_S_to_le.assumption
133 theorem S_pred: \forall n:nat.lt O n \to eq nat n (S (pred n)).
134 intro.elim n.apply False_ind.exact (not_le_Sn_O O H).
135 apply eq_f.apply pred_Sn.
138 theorem le_pred_to_le:
139 ∀n,m. O < m → pred n ≤ pred m → n ≤ m.
144 rewrite > (S_pred m);
152 theorem le_to_le_pred:
153 ∀n,m. n ≤ m → pred n ≤ pred m.
159 generalize in match H1;
162 [ elim (not_le_Sn_O ? H1)
171 theorem le_to_or_lt_eq : \forall n,m:nat.
172 n \leq m \to n < m \lor n = m.
175 left.unfold lt.apply le_S_S.assumption.
178 theorem Not_lt_n_n: ∀n. n ≮ n.
183 apply (not_le_Sn_n ? H).
187 theorem lt_to_not_eq : \forall n,m:nat. n<m \to n \neq m.
188 unfold Not.intros.cut ((le (S n) m) \to False).
189 apply Hcut.assumption.rewrite < H1.
194 theorem eq_to_not_lt: \forall a,b:nat.
200 apply (lt_to_not_eq b b)
206 theorem lt_n_m_to_not_lt_m_Sn: ∀n,m. n < m → m ≮ S n.
212 generalize in match (le_S_S ? ? H);
214 generalize in match (transitive_le ? ? ? H2 H1);
216 apply (not_le_Sn_n ? H3).
220 theorem lt_to_le : \forall n,m:nat. n<m \to n \leq m.
221 simplify.intros.unfold lt in H.elim H.
222 apply le_S. apply le_n.
223 apply le_S. assumption.
226 theorem lt_S_to_le : \forall n,m:nat. n < S m \to n \leq m.
228 apply le_S_S_to_le.assumption.
231 theorem not_le_to_lt: \forall n,m:nat. n \nleq m \to m<n.
233 apply (nat_elim2 (\lambda n,m.n \nleq m \to m<n)).
234 intros.apply (absurd (O \leq n1)).apply le_O_n.assumption.
235 unfold Not.unfold lt.intros.apply le_S_S.apply le_O_n.
236 unfold Not.unfold lt.intros.apply le_S_S.apply H.intros.apply H1.apply le_S_S.
240 theorem lt_to_not_le: \forall n,m:nat. n<m \to m \nleq n.
241 unfold Not.unfold lt.intros 3.elim H.
242 apply (not_le_Sn_n n H1).
243 apply H2.apply lt_to_le. apply H3.
246 theorem not_lt_to_le: \forall n,m:nat. Not (lt n m) \to le m n.
249 apply not_le_to_lt.exact H.
252 theorem le_to_not_lt: \forall n,m:nat. le n m \to Not (lt m n).
253 intros.unfold Not.unfold lt.
254 apply lt_to_not_le.unfold lt.
255 apply le_S_S.assumption.
258 theorem not_eq_to_le_to_lt: ∀n,m. n≠m → n≤m → n<m.
260 elim (le_to_or_lt_eq ? ? H1);
267 theorem le_n_O_to_eq : \forall n:nat. n \leq O \to O=n.
268 intro.elim n.reflexivity.
271 [2: apply H1 | skip].
274 theorem le_n_O_elim: \forall n:nat.n \leq O \to \forall P: nat \to Prop.
279 apply (not_le_Sn_O ? H1).
282 theorem le_n_Sm_elim : \forall n,m:nat.n \leq S m \to
283 \forall P:Prop. (S n \leq S m \to P) \to (n=S m \to P) \to P.
285 apply H2.reflexivity.
286 apply H3. apply le_S_S. assumption.
290 lemma le_to_le_to_eq: \forall n,m. n \le m \to m \le n \to n = m.
292 [intros.apply le_n_O_to_eq.assumption
293 |intros.apply sym_eq.apply le_n_O_to_eq.assumption
294 |intros.apply eq_f.apply H
295 [apply le_S_S_to_le.assumption
296 |apply le_S_S_to_le.assumption
301 (* lt and le trans *)
302 theorem lt_O_S : \forall n:nat. O < S n.
303 intro. unfold. apply le_S_S. apply le_O_n.
306 theorem lt_to_le_to_lt: \forall n,m,p:nat. lt n m \to le m p \to lt n p.
308 assumption.unfold lt.apply le_S.assumption.
311 theorem le_to_lt_to_lt: \forall n,m,p:nat. le n m \to lt m p \to lt n p.
313 assumption.apply H2.unfold lt.
314 apply lt_to_le.assumption.
317 theorem lt_S_to_lt: \forall n,m. S n < m \to n < m.
319 apply (trans_lt ? (S n))
320 [apply le_n|assumption]
323 theorem ltn_to_ltO: \forall n,m:nat. lt n m \to lt O m.
324 intros.apply (le_to_lt_to_lt O n).
325 apply le_O_n.assumption.
328 theorem lt_O_n_elim: \forall n:nat. lt O n \to
329 \forall P:nat\to Prop. (\forall m:nat.P (S m)) \to P n.
330 intro.elim n.apply False_ind.exact (not_le_Sn_O O H).
334 (* other abstract properties *)
335 theorem antisymmetric_le : antisymmetric nat le.
336 unfold antisymmetric.intros 2.
337 apply (nat_elim2 (\lambda n,m.(n \leq m \to m \leq n \to n=m))).
338 intros.apply le_n_O_to_eq.assumption.
339 intros.apply False_ind.apply (not_le_Sn_O ? H).
340 intros.apply eq_f.apply H.
341 apply le_S_S_to_le.assumption.
342 apply le_S_S_to_le.assumption.
345 theorem antisym_le: \forall n,m:nat. n \leq m \to m \leq n \to n=m
346 \def antisymmetric_le.
348 theorem le_n_m_to_lt_m_Sn_to_eq_n_m: ∀n,m. n ≤ m → m < S n → n=m.
351 generalize in match (le_S_S_to_le ? ? H1);
357 theorem decidable_le: \forall n,m:nat. decidable (n \leq m).
359 apply (nat_elim2 (\lambda n,m.decidable (n \leq m))).
360 intros.unfold decidable.left.apply le_O_n.
361 intros.unfold decidable.right.exact (not_le_Sn_O n1).
362 intros 2.unfold decidable.intro.elim H.
363 left.apply le_S_S.assumption.
364 right.unfold Not.intro.apply H1.apply le_S_S_to_le.assumption.
367 theorem decidable_lt: \forall n,m:nat. decidable (n < m).
368 intros.exact (decidable_le (S n) m).
371 (* well founded induction principles *)
373 theorem nat_elim1 : \forall n:nat.\forall P:nat \to Prop.
374 (\forall m.(\forall p. (p \lt m) \to P p) \to P m) \to P n.
375 intros.cut (\forall q:nat. q \le n \to P q).
376 apply (Hcut n).apply le_n.
377 elim n.apply (le_n_O_elim q H1).
379 intros.apply False_ind.apply (not_le_Sn_O p H2).
380 apply H.intros.apply H1.
382 apply lt_S_to_le.assumption.
383 apply (lt_to_le_to_lt p q (S n1) H3 H2).
386 (* some properties of functions *)
388 definition increasing \def \lambda f:nat \to nat.
389 \forall n:nat. f n < f (S n).
391 theorem increasing_to_monotonic: \forall f:nat \to nat.
392 increasing f \to monotonic nat lt f.
393 unfold monotonic.unfold lt.unfold increasing.unfold lt.intros.elim H1.apply H.
394 apply (trans_le ? (f n1)).
395 assumption.apply (trans_le ? (S (f n1))).
400 theorem le_n_fn: \forall f:nat \to nat. (increasing f)
401 \to \forall n:nat. n \le (f n).
404 apply (trans_le ? (S (f n1))).
405 apply le_S_S.apply H1.
406 simplify in H. unfold increasing in H.unfold lt in H.apply H.
409 theorem increasing_to_le: \forall f:nat \to nat. (increasing f)
410 \to \forall m:nat. \exists i. m \le (f i).
412 apply (ex_intro ? ? O).apply le_O_n.
414 apply (ex_intro ? ? (S a)).
415 apply (trans_le ? (S (f a))).
416 apply le_S_S.assumption.
417 simplify in H.unfold increasing in H.unfold lt in H.
421 theorem increasing_to_le2: \forall f:nat \to nat. (increasing f)
422 \to \forall m:nat. (f O) \le m \to
423 \exists i. (f i) \le m \land m <(f (S i)).
425 apply (ex_intro ? ? O).
426 split.apply le_n.apply H.
428 cut ((S n1) < (f (S a)) \lor (S n1) = (f (S a))).
430 apply (ex_intro ? ? a).
431 split.apply le_S. assumption.assumption.
432 apply (ex_intro ? ? (S a)).
433 split.rewrite < H7.apply le_n.
436 apply le_to_or_lt_eq.apply H6.