author | berghofe |
Tue, 25 Jun 1996 13:11:29 +0200 | |
changeset 1824 | 44254696843a |
parent 1820 | e381e1c51689 |
child 2017 | dd3e2a91aeca |
permissions | -rw-r--r-- |
1465 | 1 |
(* Title: HOL/ex/puzzle.ML |
969 | 2 |
ID: $Id$ |
1465 | 3 |
Author: Tobias Nipkow |
969 | 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 |
qed "nat_exh"; |
|
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); |
|
1266 | 19 |
by (Simp_tac 1); |
969 | 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); |
|
1820 | 24 |
by (best_tac (!claset addIs [lessD,Puzzle.f_ax,le_less_trans,le_trans]) 1); |
25 |
by (fast_tac (!claset addIs [Puzzle.f_ax]) 1); |
|
1266 | 26 |
val lemma = result() RS spec RS mp; |
969 | 27 |
|
28 |
goal Puzzle.thy "n <= f(n)"; |
|
1820 | 29 |
by (fast_tac (!claset addIs [lemma]) 1); |
969 | 30 |
qed "lemma1"; |
31 |
||
32 |
goal Puzzle.thy "f(n) < f(Suc(n))"; |
|
1820 | 33 |
by (fast_tac (!claset addIs [Puzzle.f_ax,le_less_trans,lemma1]) 1); |
969 | 34 |
qed "lemma2"; |
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); |
|
1266 | 38 |
by (Simp_tac 1); |
1673
d22110ddd0af
repaired critical proofs depending on the order inside non-confluent SimpSets
oheimb
parents:
1465
diff
changeset
|
39 |
by (simp_tac (!simpset addsimps [less_Suc_eq]) 1); |
1820 | 40 |
by (fast_tac (!claset addIs (le_trans::prems)) 1); |
969 | 41 |
bind_thm("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); |
|
1820 | 47 |
by (fast_tac (!claset addIs [le_refl]) 1); |
969 | 48 |
qed "mono_lemma"; |
49 |
||
50 |
val prems = goal Puzzle.thy "m <= n ==> f(m) <= f(n)"; |
|
1820 | 51 |
by (fast_tac (!claset addIs ([mono_lemma,less_imp_le,lemma2]@prems)) 1); |
969 | 52 |
qed "f_mono"; |
53 |
||
54 |
goal Puzzle.thy "f(n) = n"; |
|
55 |
by (rtac le_anti_sym 1); |
|
56 |
by (rtac lemma1 2); |
|
1820 | 57 |
by (fast_tac (!claset addIs [Puzzle.f_ax,leI] addDs [leD,f_mono,lessD]) 1); |
969 | 58 |
result(); |