(* Title: HOL/ex/Mutil
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
The Mutilated Chess Board Problem, formalized inductively
*)
open Mutil;
Addsimps tiling.intrs;
(** The union of two disjoint tilings is a tiling **)
goal thy "!!t. t: tiling A ==> \
\ u: tiling A --> t Int u = {} --> t Un u : tiling A";
by (etac tiling.induct 1);
by (Simp_tac 1);
by (fast_tac (!claset addIs tiling.intrs
addss (HOL_ss addsimps [Un_assoc,
subset_empty_iff RS sym])) 1);
qed_spec_mp "tiling_UnI";
(*** Chess boards ***)
val [below_0, below_Suc] = nat_recs below_def;
Addsimps [below_0, below_Suc];
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 (Fast_tac 1);
qed_spec_mp "below_less_iff";
Addsimps [below_less_iff];
goal thy "below(Suc n) Times B = ({n} Times B) Un ((below n) Times B)";
by (Simp_tac 1);
by (Fast_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 (Fast_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(n1+n1)}) Un ({i} Times {n1+n1}) = \
\ {(i, n1+n1), (i, Suc(n1+n1))}" 1);
by (Fast_tac 2);
by (asm_simp_tac (!simpset addsimps [domino.horiz]) 1);
by (fast_tac (!claset addIs [equalityI, lessI] addEs [less_irrefl, less_asym]
addDs [below_less_iff RS iffD1]) 1);
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 (fast_tac (!claset addIs [equalityI, tiling_UnI, dominoes_tile_row]
addEs [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 (Fast_tac 1);
qed "evnodd_Un";
goalw thy [evnodd_def] "evnodd (A - B) b = evnodd A b - evnodd B b";
by (Fast_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)";
by (asm_full_simp_tac (!simpset addsimps [evnodd_def]
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, ...*)
by (REPEAT (asm_full_simp_tac (!simpset addsimps
[less_Suc_eq, evnodd_insert, evnodd_empty, mod_Suc]
setloop split_tac [expand_if]) 1
THEN Fast_tac 1));
qed "domino_singleton";
goal thy "!!d. d:domino ==> finite d";
by (fast_tac (!claset addSIs [finite_insertI, finite_emptyI]
addEs [domino.elim]) 1);
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);
by (fast_tac (!claset addIs [domino_finite, finite_UnI]) 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,
card_insert_disjoint]) 1);
by (fast_tac (!claset addSDs [evnodd_subset RS subsetD] addEs [equalityE]) 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);
by (asm_simp_tac (!simpset addsimps add_ac) 2);
by (asm_full_simp_tac
(!simpset addsimps [evnodd_Diff, evnodd_insert, evnodd_empty,
mod_less, tiling_domino_0_1 RS sym]) 1);
by (rtac less_trans 1);
by (REPEAT
(rtac card_Diff 1
THEN
asm_simp_tac (!simpset addsimps [tiling_domino_finite, finite_evnodd]) 1
THEN
asm_simp_tac (!simpset addsimps [mod_less, evnodd_iff]) 1));
qed "mutil_not_tiling";