0
|
1 |
(* Title: HOL/ex/puzzle.ML
|
|
2 |
ID: $Id$
|
|
3 |
Author: Tobias Nipkow
|
|
4 |
Copyright 1993 TU Muenchen
|
|
5 |
|
|
6 |
For puzzle.thy. A question from "Bundeswettbewerb Mathematik"
|
|
7 |
|
|
8 |
Proof due to Herbert Ehler
|
|
9 |
*)
|
|
10 |
|
|
11 |
(*specialized form of induction needed below*)
|
|
12 |
val prems = goal Nat.thy "[| P(0); !!n. P(Suc(n)) |] ==> !n.P(n)";
|
|
13 |
by (EVERY1 [rtac (nat_induct RS allI), resolve_tac prems, resolve_tac prems]);
|
|
14 |
val nat_exh = result();
|
|
15 |
|
|
16 |
goal Puzzle.thy "! n. k=f(n) --> n <= f(n)";
|
|
17 |
by (res_inst_tac [("n","k")] less_induct 1);
|
|
18 |
by (rtac nat_exh 1);
|
|
19 |
by (simp_tac nat_ss 1);
|
|
20 |
by (rtac impI 1);
|
|
21 |
by (rtac classical 1);
|
|
22 |
by (dtac not_leE 1);
|
|
23 |
by (subgoal_tac "f(na) <= f(f(na))" 1);
|
|
24 |
by (best_tac (HOL_cs addIs [lessD,Puzzle.f_ax,le_less_trans,le_trans]) 1);
|
|
25 |
by (fast_tac (HOL_cs addIs [Puzzle.f_ax]) 1);
|
|
26 |
val lemma = result() RS spec RS mp;
|
|
27 |
|
|
28 |
goal Puzzle.thy "n <= f(n)";
|
|
29 |
by (fast_tac (HOL_cs addIs [lemma]) 1);
|
|
30 |
val lemma1 = result();
|
|
31 |
|
|
32 |
goal Puzzle.thy "f(n) < f(Suc(n))";
|
|
33 |
by (fast_tac (HOL_cs addIs [Puzzle.f_ax,le_less_trans,lemma1]) 1);
|
|
34 |
val lemma2 = result();
|
|
35 |
|
|
36 |
val prems = goal Puzzle.thy "(!!n.f(n) <= f(Suc(n))) ==> m<n --> f(m) <= f(n)";
|
|
37 |
by (res_inst_tac[("n","n")]nat_induct 1);
|
|
38 |
by (simp_tac nat_ss 1);
|
|
39 |
by (simp_tac nat_ss 1);
|
|
40 |
by (fast_tac (HOL_cs addIs (le_trans::prems)) 1);
|
|
41 |
val mono_lemma1 = result() RS mp;
|
|
42 |
|
|
43 |
val [p1,p2] = goal Puzzle.thy
|
|
44 |
"[| !! n. f(n)<=f(Suc(n)); m<=n |] ==> f(m) <= f(n)";
|
|
45 |
by (rtac (p2 RS le_imp_less_or_eq RS disjE) 1);
|
|
46 |
by (etac (p1 RS mono_lemma1) 1);
|
|
47 |
by (fast_tac (HOL_cs addIs [le_refl]) 1);
|
|
48 |
val mono_lemma = result();
|
|
49 |
|
|
50 |
val prems = goal Puzzle.thy "m <= n ==> f(m) <= f(n)";
|
|
51 |
by (fast_tac (HOL_cs addIs ([mono_lemma,less_imp_le,lemma2]@prems)) 1);
|
|
52 |
val f_mono = result();
|
|
53 |
|
|
54 |
goal Puzzle.thy "f(n) = n";
|
|
55 |
by (rtac le_anti_sym 1);
|
|
56 |
by (rtac lemma1 2);
|
|
57 |
by (fast_tac (HOL_cs addIs [Puzzle.f_ax,leI] addDs [leD,f_mono,lessD]) 1);
|
|
58 |
result();
|