src/HOL/Induct/Mutil.ML
 author paulson Tue, 27 May 1997 13:25:00 +0200 changeset 3357 c224dddc5f71 parent 3120 c58423c20740 child 3414 804c8a178a7f permissions -rw-r--r--
Removal of card_insert_disjoint, which is now a default rewrite rule
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(*  Title:      HOL/ex/Mutil
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

The Mutilated Chess Board Problem, formalized inductively
*)

open Mutil;

(** The union of two disjoint tilings is a tiling **)

goal thy "!!t. t: tiling A ==> \
\              u: tiling A --> t <= Compl u --> t Un u : tiling A";
by (etac tiling.induct 1);
by (Simp_tac 1);
by (simp_tac (!simpset addsimps [Un_assoc]) 1);
by (blast_tac (!claset addIs tiling.intrs) 1);
qed_spec_mp "tiling_UnI";

(*** Chess boards ***)

val [below_0, below_Suc] = nat_recs below_def;

goal thy "ALL i. (i: below k) = (i<k)";
by (nat_ind_tac "k" 1);
by (ALLGOALS (asm_simp_tac (!simpset addsimps [less_Suc_eq])));
by (Blast_tac 1);
qed_spec_mp "below_less_iff";

goal thy "below(Suc n) Times B = ({n} Times B) Un ((below n) Times B)";
by (Simp_tac 1);
by (Blast_tac 1);
qed "Sigma_Suc1";

goal thy "A Times below(Suc n) = (A Times {n}) Un (A Times (below n))";
by (Simp_tac 1);
by (Blast_tac 1);
qed "Sigma_Suc2";

(*Deletion is essential to allow use of Sigma_Suc1,2*)
Delsimps [below_Suc];

goal thy "{i} Times below(n + n) : tiling domino";
by (nat_ind_tac "n" 1);
by (ALLGOALS (asm_simp_tac (!simpset addsimps [Un_assoc RS sym, Sigma_Suc2])));
by (resolve_tac tiling.intrs 1);
by (assume_tac 2);
by (subgoal_tac    (*seems the easiest way of turning one to the other*)
"({i} Times {Suc(n+n)}) Un ({i} Times {n+n}) = \
\    {(i, n+n), (i, Suc(n+n))}" 1);
by (Blast_tac 2);
by (asm_simp_tac (!simpset addsimps [domino.horiz]) 1);
by (blast_tac (!claset addEs  [less_irrefl, less_asym]
qed "dominoes_tile_row";

goal thy "(below m) Times below(n + n) : tiling domino";
by (nat_ind_tac "m" 1);
by (ALLGOALS (asm_simp_tac (!simpset addsimps [Sigma_Suc1])));
by (blast_tac (!claset addSIs [tiling_UnI, dominoes_tile_row]
addSEs [below_less_iff RS iffD1 RS less_irrefl]) 1);
qed "dominoes_tile_matrix";

(*** Basic properties of evnodd ***)

goalw thy [evnodd_def] "(i,j): evnodd A b = ((i,j): A  &  (i+j) mod 2 = b)";
by (Simp_tac 1);
qed "evnodd_iff";

goalw thy [evnodd_def] "evnodd A b <= A";
by (rtac Int_lower1 1);
qed "evnodd_subset";

(* finite X ==> finite(evnodd X b) *)
bind_thm("finite_evnodd", evnodd_subset RS finite_subset);

goalw thy [evnodd_def] "evnodd (A Un B) b = evnodd A b Un evnodd B b";
by (Blast_tac 1);
qed "evnodd_Un";

goalw thy [evnodd_def] "evnodd (A - B) b = evnodd A b - evnodd B b";
by (Blast_tac 1);
qed "evnodd_Diff";

goalw thy [evnodd_def] "evnodd {} b = {}";
by (Simp_tac 1);
qed "evnodd_empty";

goalw thy [evnodd_def]
"evnodd (insert (i,j) C) b = \
\    (if (i+j) mod 2 = b then insert (i,j) (evnodd C b) else evnodd C b)";
setloop (split_tac [expand_if] THEN' Step_tac)) 1);
qed "evnodd_insert";

(*** Dominoes ***)

goal thy "!!d. [| d:domino; b<2 |] ==> EX i j. evnodd d b = {(i,j)}";
by (eresolve_tac [domino.elim] 1);
by (res_inst_tac [("k1", "i+j")] (mod2_cases RS disjE) 2);
by (res_inst_tac [("k1", "i+j")] (mod2_cases RS disjE) 1);
by (REPEAT_FIRST assume_tac);
(*Four similar cases: case (i+j) mod 2 = b, 2#-b, ...*)
[less_Suc_eq, evnodd_insert, evnodd_empty, mod_Suc]
setloop split_tac [expand_if]) 1
THEN Blast_tac 1));
qed "domino_singleton";

goal thy "!!d. d:domino ==> finite d";
by (blast_tac (!claset addSIs [finite_insertI, finite_emptyI]
qed "domino_finite";

(*** Tilings of dominoes ***)

goal thy "!!t. t:tiling domino ==> finite t";
by (eresolve_tac [tiling.induct] 1);
by (rtac finite_emptyI 1);
qed "tiling_domino_finite";

goal thy "!!t. t: tiling domino ==> card(evnodd t 0) = card(evnodd t 1)";
by (eresolve_tac [tiling.induct] 1);
by (simp_tac (!simpset addsimps [evnodd_def]) 1);
by (res_inst_tac [("b1","0")] (domino_singleton RS exE) 1);
by (Simp_tac 2 THEN assume_tac 1);
by (res_inst_tac [("b1","1")] (domino_singleton RS exE) 1);
by (Simp_tac 2 THEN assume_tac 1);
by (Step_tac 1);
by (subgoal_tac "ALL p b. p : evnodd a b --> p ~: evnodd ta b" 1);
by (asm_simp_tac (!simpset addsimps [evnodd_Un, Un_insert_left,
tiling_domino_finite,
evnodd_subset RS finite_subset]) 1);
qed "tiling_domino_0_1";

goal thy "!!m n. [| t = below(Suc m + Suc m) Times below(Suc n + Suc n);   \
\                   t' = t - {(0,0)} - {(Suc(m+m), Suc(n+n))}              \
\                |] ==> t' ~: tiling domino";
by (rtac notI 1);
by (dtac tiling_domino_0_1 1);
by (subgoal_tac "card(evnodd t' 0) < card(evnodd t' 1)" 1);
by (Asm_full_simp_tac 1);
by (subgoal_tac "t : tiling domino" 1);
(*Requires a small simpset that won't move the Suc applications*)
by (asm_simp_tac (HOL_ss addsimps [dominoes_tile_matrix]) 2);
by (subgoal_tac "(m+m)+(n+n) = (m+n)+(m+n)" 1);