src/HOL/Induct/Mutil.ML
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(*  Title:      HOL/Induct/Mutil
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1996  University of Cambridge
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The Mutilated Chess Board Problem, formalized inductively
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*)
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Addsimps tiling.intrs;
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(** The union of two disjoint tilings is a tiling **)
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goal thy "!!t. t: tiling A ==> \
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\              u: tiling A --> t <= Compl u --> t Un u : tiling A";
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by (etac tiling.induct 1);
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by (Simp_tac 1);
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by (simp_tac (simpset() addsimps [Un_assoc]) 1);
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by (blast_tac (claset() addIs tiling.intrs) 1);
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qed_spec_mp "tiling_UnI";
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(*** Chess boards ***)
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goalw thy [below_def] "(i: below k) = (i<k)";
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by (Blast_tac 1);
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qed "below_less_iff";
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AddIffs [below_less_iff];
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goalw thy [below_def] "below 0 = {}";
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by (Simp_tac 1);
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qed "below_0";
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Addsimps [below_0];
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goalw thy [below_def]
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    "below(Suc n) Times B = ({n} Times B) Un ((below n) Times B)";
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by (simp_tac (simpset() addsimps [less_Suc_eq]) 1);
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by (Blast_tac 1);
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qed "Sigma_Suc1";
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goalw thy [below_def]
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    "A Times below(Suc n) = (A Times {n}) Un (A Times (below n))";
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by (simp_tac (simpset() addsimps [less_Suc_eq]) 1);
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by (Blast_tac 1);
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qed "Sigma_Suc2";
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goal thy "{i} Times below(n+n) : tiling domino";
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by (nat_ind_tac "n" 1);
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by (ALLGOALS (asm_simp_tac (simpset() addsimps [Un_assoc RS sym, Sigma_Suc2])));
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by (resolve_tac tiling.intrs 1);
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by (assume_tac 2);
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by (subgoal_tac    (*seems the easiest way of turning one to the other*)
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    "({i} Times {Suc(n+n)}) Un ({i} Times {n+n}) = \
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\    {(i, n+n), (i, Suc(n+n))}" 1);
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by (Blast_tac 2);
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by (asm_simp_tac (simpset() addsimps [domino.horiz]) 1);
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by (Auto_tac());
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qed "dominoes_tile_row";
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goal thy "(below m) Times below(n+n) : tiling domino";
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by (nat_ind_tac "m" 1);
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by (ALLGOALS (asm_simp_tac (simpset() addsimps [Sigma_Suc1])));
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by (blast_tac (claset() addSIs [tiling_UnI, dominoes_tile_row]
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                       addSEs [below_less_iff RS iffD1 RS less_irrefl]) 1);
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qed "dominoes_tile_matrix";
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(*** Basic properties of evnodd ***)
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goalw thy [evnodd_def] "(i,j): evnodd A b = ((i,j): A  &  (i+j) mod 2 = b)";
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by (Simp_tac 1);
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qed "evnodd_iff";
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goalw thy [evnodd_def] "evnodd A b <= A";
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by (rtac Int_lower1 1);
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qed "evnodd_subset";
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(* finite X ==> finite(evnodd X b) *)
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bind_thm("finite_evnodd", evnodd_subset RS finite_subset);
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goalw thy [evnodd_def] "evnodd (A Un B) b = evnodd A b Un evnodd B b";
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by (Blast_tac 1);
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qed "evnodd_Un";
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goalw thy [evnodd_def] "evnodd (A - B) b = evnodd A b - evnodd B b";
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by (Blast_tac 1);
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qed "evnodd_Diff";
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goalw thy [evnodd_def] "evnodd {} b = {}";
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by (Simp_tac 1);
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qed "evnodd_empty";
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goalw thy [evnodd_def]
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    "evnodd (insert (i,j) C) b = \
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\      (if (i+j) mod 2 = b then insert (i,j) (evnodd C b) else evnodd C b)";
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by (simp_tac (simpset() addsplits [expand_if]) 1);
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by (Blast_tac 1);
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qed "evnodd_insert";
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Addsimps [finite_evnodd, evnodd_Un, evnodd_Diff, evnodd_empty, evnodd_insert];
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(*** Dominoes ***)
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goal thy "!!d. [| d:domino; b<2 |] ==> EX i j. evnodd d b = {(i,j)}";
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by (eresolve_tac [domino.elim] 1);
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by (res_inst_tac [("k1", "i+j")] (mod2_cases RS disjE) 2);
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by (res_inst_tac [("k1", "i+j")] (mod2_cases RS disjE) 1);
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by (REPEAT_FIRST assume_tac);
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(*Four similar cases: case (i+j) mod 2 = b, 2#-b, ...*)
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by (REPEAT (asm_full_simp_tac (simpset() addsimps [less_Suc_eq, mod_Suc] 
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                          addsplits [expand_if]) 1
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           THEN Blast_tac 1));
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qed "domino_singleton";
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goal thy "!!d. d:domino ==> finite d";
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by (blast_tac (claset() addSEs [domino.elim]) 1);
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qed "domino_finite";
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(*** Tilings of dominoes ***)
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goal thy "!!t. t:tiling domino ==> finite t";
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by (eresolve_tac [tiling.induct] 1);
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by (rtac Finites.emptyI 1);
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by (blast_tac (claset() addSIs [finite_UnI] addIs [domino_finite]) 1);
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qed "tiling_domino_finite";
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goal thy "!!t. t: tiling domino ==> card(evnodd t 0) = card(evnodd t 1)";
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by (eresolve_tac [tiling.induct] 1);
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by (simp_tac (simpset() addsimps [evnodd_def]) 1);
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by (res_inst_tac [("b1","0")] (domino_singleton RS exE) 1);
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by (Simp_tac 2 THEN assume_tac 1);
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by (res_inst_tac [("b1","1")] (domino_singleton RS exE) 1);
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by (Simp_tac 2 THEN assume_tac 1);
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by (Clarify_tac 1);
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by (subgoal_tac "ALL p b. p : evnodd a b --> p ~: evnodd ta b" 1);
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by (asm_simp_tac (simpset() addsimps [tiling_domino_finite]) 1);
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by (blast_tac (claset() addSDs [evnodd_subset RS subsetD] 
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                        addEs  [equalityE]) 1);
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qed "tiling_domino_0_1";
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(*Final argument is surprisingly complex.  Note the use of small simpsets
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  to avoid moving Sucs, etc.*)
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goal thy "!!m n. [| t = below(Suc m + Suc m) Times below(Suc n + Suc n);   \
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\                   t' = t - {(0,0)} - {(Suc(m+m), Suc(n+n))}              \
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\                |] ==> t' ~: tiling domino";
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by (rtac notI 1);
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   148
by (subgoal_tac "card(evnodd t' 0) < card(evnodd t' 1)" 1);
4387
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   149
by (asm_full_simp_tac (HOL_ss addsimps [less_not_refl, tiling_domino_0_1]) 1);
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   150
by (subgoal_tac "t : tiling domino" 1);
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   151
by (asm_simp_tac (HOL_ss addsimps [dominoes_tile_matrix]) 2);
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   152
by (asm_full_simp_tac (simpset() 
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		       addsimps [mod_less, tiling_domino_0_1 RS sym]) 1);
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   154
(*Cardinality is smaller because of the two elements fewer*)
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   155
by (rtac less_trans 1);
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   156
by (REPEAT
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   157
    (rtac card_Diff 1 
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     THEN asm_simp_tac (simpset() addsimps [tiling_domino_finite]) 1 
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     THEN asm_simp_tac (simpset() addsimps [mod_less, evnodd_iff]) 1));
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   160
qed "mutil_not_tiling";