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
134 theorem le_to_or_lt_eq : \forall n,m:nat.
135 n \leq m \to n < m \lor n = m.
138 left.unfold lt.apply le_S_S.assumption.
142 theorem lt_to_not_eq : \forall n,m:nat. n<m \to n \neq m.
143 unfold Not.intros.cut ((le (S n) m) \to False).
144 apply Hcut.assumption.rewrite < H1.
149 theorem eq_to_not_lt: \forall a,b:nat.
155 apply (lt_to_not_eq b b)
162 theorem lt_to_le : \forall n,m:nat. n<m \to n \leq m.
163 simplify.intros.unfold lt in H.elim H.
164 apply le_S. apply le_n.
165 apply le_S. assumption.
168 theorem lt_S_to_le : \forall n,m:nat. n < S m \to n \leq m.
170 apply le_S_S_to_le.assumption.
173 theorem not_le_to_lt: \forall n,m:nat. n \nleq m \to m<n.
175 apply (nat_elim2 (\lambda n,m.n \nleq m \to m<n)).
176 intros.apply (absurd (O \leq n1)).apply le_O_n.assumption.
177 unfold Not.unfold lt.intros.apply le_S_S.apply le_O_n.
178 unfold Not.unfold lt.intros.apply le_S_S.apply H.intros.apply H1.apply le_S_S.
182 theorem lt_to_not_le: \forall n,m:nat. n<m \to m \nleq n.
183 unfold Not.unfold lt.intros 3.elim H.
184 apply (not_le_Sn_n n H1).
185 apply H2.apply lt_to_le. apply H3.
188 theorem not_lt_to_le: \forall n,m:nat. Not (lt n m) \to le m n.
191 apply not_le_to_lt.exact H.
194 theorem le_to_not_lt: \forall n,m:nat. le n m \to Not (lt m n).
195 intros.unfold Not.unfold lt.
196 apply lt_to_not_le.unfold lt.
197 apply le_S_S.assumption.
201 theorem le_n_O_to_eq : \forall n:nat. n \leq O \to O=n.
202 intro.elim n.reflexivity.
205 [2: apply H1 | skip].
208 theorem le_n_O_elim: \forall n:nat.n \leq O \to \forall P: nat \to Prop.
213 apply (not_le_Sn_O ? H1).
216 theorem le_n_Sm_elim : \forall n,m:nat.n \leq S m \to
217 \forall P:Prop. (S n \leq S m \to P) \to (n=S m \to P) \to P.
219 apply H2.reflexivity.
220 apply H3. apply le_S_S. assumption.
224 lemma le_to_le_to_eq: \forall n,m. n \le m \to m \le n \to n = m.
226 [intros.apply le_n_O_to_eq.assumption
227 |intros.apply sym_eq.apply le_n_O_to_eq.assumption
228 |intros.apply eq_f.apply H
229 [apply le_S_S_to_le.assumption
230 |apply le_S_S_to_le.assumption
235 (* lt and le trans *)
236 theorem lt_O_S : \forall n:nat. O < S n.
237 intro. unfold. apply le_S_S. apply le_O_n.
240 theorem lt_to_le_to_lt: \forall n,m,p:nat. lt n m \to le m p \to lt n p.
242 assumption.unfold lt.apply le_S.assumption.
245 theorem le_to_lt_to_lt: \forall n,m,p:nat. le n m \to lt m p \to lt n p.
247 assumption.apply H2.unfold lt.
248 apply lt_to_le.assumption.
251 theorem lt_S_to_lt: \forall n,m. S n < m \to n < m.
253 apply (trans_lt ? (S n))
254 [apply le_n|assumption]
257 theorem ltn_to_ltO: \forall n,m:nat. lt n m \to lt O m.
258 intros.apply (le_to_lt_to_lt O n).
259 apply le_O_n.assumption.
262 theorem lt_O_n_elim: \forall n:nat. lt O n \to
263 \forall P:nat\to Prop. (\forall m:nat.P (S m)) \to P n.
264 intro.elim n.apply False_ind.exact (not_le_Sn_O O H).
268 (* other abstract properties *)
269 theorem antisymmetric_le : antisymmetric nat le.
270 unfold antisymmetric.intros 2.
271 apply (nat_elim2 (\lambda n,m.(n \leq m \to m \leq n \to n=m))).
272 intros.apply le_n_O_to_eq.assumption.
273 intros.apply False_ind.apply (not_le_Sn_O ? H).
274 intros.apply eq_f.apply H.
275 apply le_S_S_to_le.assumption.
276 apply le_S_S_to_le.assumption.
279 theorem antisym_le: \forall n,m:nat. n \leq m \to m \leq n \to n=m
280 \def antisymmetric_le.
282 theorem decidable_le: \forall n,m:nat. decidable (n \leq m).
284 apply (nat_elim2 (\lambda n,m.decidable (n \leq m))).
285 intros.unfold decidable.left.apply le_O_n.
286 intros.unfold decidable.right.exact (not_le_Sn_O n1).
287 intros 2.unfold decidable.intro.elim H.
288 left.apply le_S_S.assumption.
289 right.unfold Not.intro.apply H1.apply le_S_S_to_le.assumption.
292 theorem decidable_lt: \forall n,m:nat. decidable (n < m).
293 intros.exact (decidable_le (S n) m).
296 (* well founded induction principles *)
298 theorem nat_elim1 : \forall n:nat.\forall P:nat \to Prop.
299 (\forall m.(\forall p. (p \lt m) \to P p) \to P m) \to P n.
300 intros.cut (\forall q:nat. q \le n \to P q).
301 apply (Hcut n).apply le_n.
302 elim n.apply (le_n_O_elim q H1).
304 intros.apply False_ind.apply (not_le_Sn_O p H2).
305 apply H.intros.apply H1.
307 apply lt_S_to_le.assumption.
308 apply (lt_to_le_to_lt p q (S n1) H3 H2).
311 (* some properties of functions *)
313 definition increasing \def \lambda f:nat \to nat.
314 \forall n:nat. f n < f (S n).
316 theorem increasing_to_monotonic: \forall f:nat \to nat.
317 increasing f \to monotonic nat lt f.
318 unfold monotonic.unfold lt.unfold increasing.unfold lt.intros.elim H1.apply H.
319 apply (trans_le ? (f n1)).
320 assumption.apply (trans_le ? (S (f n1))).
325 theorem le_n_fn: \forall f:nat \to nat. (increasing f)
326 \to \forall n:nat. n \le (f n).
329 apply (trans_le ? (S (f n1))).
330 apply le_S_S.apply H1.
331 simplify in H. unfold increasing in H.unfold lt in H.apply H.
334 theorem increasing_to_le: \forall f:nat \to nat. (increasing f)
335 \to \forall m:nat. \exists i. m \le (f i).
337 apply (ex_intro ? ? O).apply le_O_n.
339 apply (ex_intro ? ? (S a)).
340 apply (trans_le ? (S (f a))).
341 apply le_S_S.assumption.
342 simplify in H.unfold increasing in H.unfold lt in H.
346 theorem increasing_to_le2: \forall f:nat \to nat. (increasing f)
347 \to \forall m:nat. (f O) \le m \to
348 \exists i. (f i) \le m \land m <(f (S i)).
350 apply (ex_intro ? ? O).
351 split.apply le_n.apply H.
353 cut ((S n1) < (f (S a)) \lor (S n1) = (f (S a))).
355 apply (ex_intro ? ? a).
356 split.apply le_S. assumption.assumption.
357 apply (ex_intro ? ? (S a)).
358 split.rewrite < H7.apply le_n.
361 apply le_to_or_lt_eq.apply H6.