src/HOL/ex/Puzzle.thy
author obua
Mon Apr 10 16:00:34 2006 +0200 (2006-04-10)
changeset 19404 9bf2cdc9e8e8
parent 17388 495c799df31d
child 23813 5440f9f5522c
permissions -rw-r--r--
Moved stuff from Ring_and_Field to Matrix
     1 (*  Title:      HOL/ex/Puzzle.thy
     2     ID:         $Id$
     3     Author:     Tobias Nipkow
     4     Copyright   1993 TU Muenchen
     5 
     6 A question from "Bundeswettbewerb Mathematik"
     7 
     8 Proof due to Herbert Ehler
     9 *)
    10 
    11 header {* A question from ``Bundeswettbewerb Mathematik'' *}
    12 
    13 theory Puzzle imports Main begin
    14 
    15 consts f :: "nat => nat"
    16 
    17 specification (f)
    18   f_ax [intro!]: "f(f(n)) < f(Suc(n))"
    19     by (rule exI [of _ id], simp)
    20 
    21 
    22 lemma lemma0 [rule_format]: "\<forall>n. k=f(n) --> n <= f(n)"
    23 apply (induct_tac "k" rule: nat_less_induct)
    24 apply (rule allI)
    25 apply (rename_tac "i")
    26 apply (case_tac "i")
    27  apply simp
    28 apply (blast intro!: Suc_leI intro: le_less_trans)
    29 done
    30 
    31 lemma lemma1: "n <= f(n)"
    32 by (blast intro: lemma0)
    33 
    34 lemma lemma2: "f(n) < f(Suc(n))"
    35 by (blast intro: le_less_trans lemma1)
    36 
    37 lemma f_mono [rule_format (no_asm)]: "m <= n --> f(m) <= f(n)"
    38 apply (induct_tac "n")
    39  apply simp
    40 apply (rule impI)
    41 apply (erule le_SucE)
    42  apply (cut_tac n = n in lemma2, auto) 
    43 done
    44 
    45 lemma f_id: "f(n) = n"
    46 apply (rule order_antisym)
    47 apply (rule_tac [2] lemma1) 
    48 apply (blast intro: leI dest: leD f_mono Suc_leI)
    49 done
    50 
    51 end
    52