src/HOL/ex/Puzzle.ML
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     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 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);
       
    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 bind_thm("lemma", result() RS spec RS mp);
       
    27 
       
    28 goal Puzzle.thy "n <= f(n)";
       
    29 by (fast_tac (HOL_cs addIs [lemma]) 1);
       
    30 qed "lemma1";
       
    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 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);
       
    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 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);
       
    47 by (fast_tac (HOL_cs addIs [le_refl]) 1);
       
    48 qed "mono_lemma";
       
    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 qed "f_mono";
       
    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();