1606
|
1 |
(* Title: ZF/ex/Mutil
|
|
2 |
ID: $Id$
|
|
3 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
|
|
4 |
Copyright 1996 University of Cambridge
|
|
5 |
|
|
6 |
The Mutilated Checkerboard Problem, formalized inductively
|
|
7 |
*)
|
|
8 |
|
|
9 |
open Mutil;
|
|
10 |
|
|
11 |
|
|
12 |
(** Basic properties of evnodd **)
|
|
13 |
|
1624
|
14 |
goalw thy [evnodd_def] "<i,j>: evnodd(A,b) <-> <i,j>: A & (i#+j) mod 2 = b";
|
|
15 |
by (fast_tac eq_cs 1);
|
|
16 |
qed "evnodd_iff";
|
1606
|
17 |
|
|
18 |
goalw thy [evnodd_def] "evnodd(A, b) <= A";
|
2469
|
19 |
by (Fast_tac 1);
|
1606
|
20 |
qed "evnodd_subset";
|
|
21 |
|
|
22 |
(* Finite(X) ==> Finite(evnodd(X,b)) *)
|
|
23 |
bind_thm("Finite_evnodd", evnodd_subset RS subset_imp_lepoll RS lepoll_Finite);
|
|
24 |
|
|
25 |
goalw thy [evnodd_def] "evnodd(A Un B, b) = evnodd(A,b) Un evnodd(B,b)";
|
2469
|
26 |
by (simp_tac (!simpset addsimps [Collect_Un]) 1);
|
1606
|
27 |
qed "evnodd_Un";
|
|
28 |
|
|
29 |
goalw thy [evnodd_def] "evnodd(A - B, b) = evnodd(A,b) - evnodd(B,b)";
|
2469
|
30 |
by (simp_tac (!simpset addsimps [Collect_Diff]) 1);
|
1606
|
31 |
qed "evnodd_Diff";
|
|
32 |
|
|
33 |
goalw thy [evnodd_def]
|
|
34 |
"evnodd(cons(<i,j>,C), b) = \
|
|
35 |
\ if((i#+j) mod 2 = b, cons(<i,j>, evnodd(C,b)), evnodd(C,b))";
|
2469
|
36 |
by (asm_simp_tac (!simpset addsimps [evnodd_def, Collect_cons]
|
1606
|
37 |
setloop split_tac [expand_if]) 1);
|
|
38 |
qed "evnodd_cons";
|
|
39 |
|
|
40 |
goalw thy [evnodd_def] "evnodd(0, b) = 0";
|
2469
|
41 |
by (simp_tac (!simpset addsimps [evnodd_def]) 1);
|
1606
|
42 |
qed "evnodd_0";
|
|
43 |
|
2469
|
44 |
Addsimps [evnodd_cons, evnodd_0];
|
1606
|
45 |
|
|
46 |
(*** Dominoes ***)
|
|
47 |
|
|
48 |
goal thy "!!d. d:domino ==> Finite(d)";
|
2469
|
49 |
by (fast_tac (!claset addSIs [Finite_cons, Finite_0] addEs [domino.elim]) 1);
|
1606
|
50 |
qed "domino_Finite";
|
|
51 |
|
|
52 |
goal thy "!!d. [| d:domino; b<2 |] ==> EX i' j'. evnodd(d,b) = {<i',j'>}";
|
|
53 |
by (eresolve_tac [domino.elim] 1);
|
|
54 |
by (res_inst_tac [("k1", "i#+j")] (mod2_cases RS disjE) 2);
|
|
55 |
by (res_inst_tac [("k1", "i#+j")] (mod2_cases RS disjE) 1);
|
|
56 |
by (REPEAT_FIRST (ares_tac [add_type]));
|
|
57 |
(*Four similar cases: case (i#+j) mod 2 = b, 2#-b, ...*)
|
2469
|
58 |
by (REPEAT (asm_simp_tac (!simpset addsimps [mod_succ, succ_neq_self]
|
1606
|
59 |
setloop split_tac [expand_if]) 1
|
2469
|
60 |
THEN fast_tac (!claset addDs [ltD]) 1));
|
1606
|
61 |
qed "domino_singleton";
|
|
62 |
|
|
63 |
|
|
64 |
(*** Tilings ***)
|
|
65 |
|
|
66 |
(** The union of two disjoint tilings is a tiling **)
|
|
67 |
|
|
68 |
goal thy "!!t. t: tiling(A) ==> \
|
1630
|
69 |
\ u: tiling(A) --> t Int u = 0 --> t Un u : tiling(A)";
|
1606
|
70 |
by (etac tiling.induct 1);
|
2469
|
71 |
by (simp_tac (!simpset addsimps tiling.intrs) 1);
|
|
72 |
by (fast_tac (!claset addIs tiling.intrs
|
|
73 |
addss (!simpset addsimps [Un_assoc,
|
|
74 |
subset_empty_iff RS iff_sym])) 1);
|
1630
|
75 |
bind_thm ("tiling_UnI", result() RS mp RS mp);
|
1606
|
76 |
|
|
77 |
goal thy "!!t. t:tiling(domino) ==> Finite(t)";
|
|
78 |
by (eresolve_tac [tiling.induct] 1);
|
|
79 |
by (resolve_tac [Finite_0] 1);
|
2469
|
80 |
by (fast_tac (!claset addIs [domino_Finite, Finite_Un]) 1);
|
1606
|
81 |
qed "tiling_domino_Finite";
|
|
82 |
|
|
83 |
goal thy "!!t. t: tiling(domino) ==> |evnodd(t,0)| = |evnodd(t,1)|";
|
|
84 |
by (eresolve_tac [tiling.induct] 1);
|
2469
|
85 |
by (simp_tac (!simpset addsimps [evnodd_def]) 1);
|
1624
|
86 |
by (res_inst_tac [("b1","0")] (domino_singleton RS exE) 1);
|
2469
|
87 |
by (Simp_tac 2 THEN assume_tac 1);
|
1624
|
88 |
by (res_inst_tac [("b1","1")] (domino_singleton RS exE) 1);
|
2469
|
89 |
by (Simp_tac 2 THEN assume_tac 1);
|
|
90 |
by (step_tac (!claset) 1);
|
1624
|
91 |
by (subgoal_tac "ALL p b. p:evnodd(a,b) --> p~:evnodd(ta,b)" 1);
|
2469
|
92 |
by (asm_simp_tac (!simpset addsimps [evnodd_Un, Un_cons, tiling_domino_Finite,
|
|
93 |
evnodd_subset RS subset_Finite,
|
|
94 |
Finite_imp_cardinal_cons]) 1);
|
|
95 |
by (fast_tac (!claset addSDs [evnodd_subset RS subsetD] addEs [equalityE]) 1);
|
1606
|
96 |
qed "tiling_domino_0_1";
|
|
97 |
|
|
98 |
goal thy "!!i n. [| i: nat; n: nat |] ==> {i} * (n #+ n) : tiling(domino)";
|
|
99 |
by (nat_ind_tac "n" [] 1);
|
2469
|
100 |
by (simp_tac (!simpset addsimps tiling.intrs) 1);
|
|
101 |
by (asm_simp_tac (!simpset addsimps [Un_assoc RS sym, Sigma_succ2]) 1);
|
1606
|
102 |
by (resolve_tac tiling.intrs 1);
|
|
103 |
by (assume_tac 2);
|
2469
|
104 |
by (subgoal_tac (*seems the easiest way of turning one to the other*)
|
1606
|
105 |
"{i}*{succ(n1#+n1)} Un {i}*{n1#+n1} = {<i,n1#+n1>, <i,succ(n1#+n1)>}" 1);
|
|
106 |
by (fast_tac eq_cs 2);
|
2469
|
107 |
by (asm_simp_tac (!simpset addsimps [domino.horiz]) 1);
|
1606
|
108 |
by (fast_tac (eq_cs addEs [mem_irrefl, mem_asym]) 1);
|
|
109 |
qed "dominoes_tile_row";
|
|
110 |
|
|
111 |
goal thy "!!m n. [| m: nat; n: nat |] ==> m * (n #+ n) : tiling(domino)";
|
|
112 |
by (nat_ind_tac "m" [] 1);
|
2469
|
113 |
by (simp_tac (!simpset addsimps tiling.intrs) 1);
|
|
114 |
by (asm_simp_tac (!simpset addsimps [Sigma_succ1]) 1);
|
1606
|
115 |
by (fast_tac (eq_cs addIs [tiling_UnI, dominoes_tile_row]
|
|
116 |
addEs [mem_irrefl]) 1);
|
|
117 |
qed "dominoes_tile_matrix";
|
|
118 |
|
|
119 |
|
|
120 |
goal thy "!!m n. [| m: nat; n: nat; \
|
|
121 |
\ t = (succ(m)#+succ(m))*(succ(n)#+succ(n)); \
|
|
122 |
\ t' = t - {<0,0>} - {<succ(m#+m), succ(n#+n)>} |] ==> \
|
|
123 |
\ t' ~: tiling(domino)";
|
|
124 |
by (resolve_tac [notI] 1);
|
|
125 |
by (dresolve_tac [tiling_domino_0_1] 1);
|
|
126 |
by (subgoal_tac "|evnodd(t',0)| < |evnodd(t',1)|" 1);
|
2469
|
127 |
by (asm_full_simp_tac (!simpset addsimps [lt_not_refl]) 1);
|
1606
|
128 |
by (subgoal_tac "t : tiling(domino)" 1);
|
1624
|
129 |
(*Requires a small simpset that won't move the succ applications*)
|
1606
|
130 |
by (asm_simp_tac (ZF_ss addsimps [nat_succI, add_type,
|
2469
|
131 |
dominoes_tile_matrix]) 2);
|
1606
|
132 |
by (subgoal_tac "(m#+m)#+(n#+n) = (m#+n)#+(m#+n)" 1);
|
2469
|
133 |
by (asm_simp_tac (!simpset addsimps add_ac) 2);
|
1606
|
134 |
by (asm_full_simp_tac
|
2469
|
135 |
(!simpset addsimps [evnodd_Diff, mod2_add_self,
|
|
136 |
mod2_succ_succ, tiling_domino_0_1 RS sym]) 1);
|
1606
|
137 |
by (resolve_tac [lt_trans] 1);
|
|
138 |
by (REPEAT
|
|
139 |
(rtac Finite_imp_cardinal_Diff 1
|
|
140 |
THEN
|
2469
|
141 |
asm_simp_tac (!simpset addsimps [tiling_domino_Finite, Finite_evnodd,
|
|
142 |
Finite_Diff]) 1
|
1606
|
143 |
THEN
|
2469
|
144 |
asm_simp_tac (!simpset addsimps [evnodd_iff, nat_0_le RS ltD,
|
|
145 |
mod2_add_self]) 1));
|
1606
|
146 |
qed "mutil_not_tiling";
|