|
2222
|
1 |
(* Title: HOL/ex/Puzzle.thy
|
|
969
|
2 |
ID: $Id$
|
|
1476
|
3 |
Author: Tobias Nipkow
|
|
969
|
4 |
Copyright 1993 TU Muenchen
|
|
|
5 |
|
|
2222
|
6 |
A question from "Bundeswettbewerb Mathematik"
|
|
13116
|
7 |
|
|
|
8 |
Proof due to Herbert Ehler
|
|
969
|
9 |
*)
|
|
|
10 |
|
|
13116
|
11 |
theory Puzzle = Main:
|
|
|
12 |
|
|
|
13 |
consts f :: "nat => nat"
|
|
|
14 |
|
|
|
15 |
axioms f_ax [intro!]: "f(f(n)) < f(Suc(n))"
|
|
|
16 |
|
|
|
17 |
|
|
|
18 |
lemma lemma0 [rule_format]: "\<forall>n. k=f(n) --> n <= f(n)"
|
|
|
19 |
apply (induct_tac "k" rule: nat_less_induct)
|
|
|
20 |
apply (rule allI)
|
|
|
21 |
apply (rename_tac "i")
|
|
|
22 |
apply (case_tac "i")
|
|
|
23 |
apply simp
|
|
|
24 |
apply (blast intro!: Suc_leI intro: le_less_trans)
|
|
|
25 |
done
|
|
|
26 |
|
|
|
27 |
lemma lemma1: "n <= f(n)"
|
|
|
28 |
by (blast intro: lemma0)
|
|
|
29 |
|
|
|
30 |
lemma lemma2: "f(n) < f(Suc(n))"
|
|
|
31 |
by (blast intro: le_less_trans lemma1)
|
|
|
32 |
|
|
|
33 |
lemma f_mono [rule_format (no_asm)]: "m <= n --> f(m) <= f(n)"
|
|
|
34 |
apply (induct_tac "n")
|
|
|
35 |
apply simp
|
|
|
36 |
apply (rule impI)
|
|
|
37 |
apply (erule le_SucE)
|
|
|
38 |
apply (cut_tac n = n in lemma2, auto)
|
|
|
39 |
done
|
|
|
40 |
|
|
|
41 |
lemma f_id: "f(n) = n"
|
|
|
42 |
apply (rule order_antisym)
|
|
|
43 |
apply (rule_tac [2] lemma1)
|
|
|
44 |
apply (blast intro: leI dest: leD f_mono Suc_leI)
|
|
|
45 |
done
|
|
|
46 |
|
|
969
|
47 |
end
|
|
13116
|
48 |
|