src/HOL/Isar_examples/MutilatedCheckerboard.thy
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improved theory reference in comment
```     1 (*  Title:      HOL/Isar_examples/MutilatedCheckerboard.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Markus Wenzel, TU Muenchen (Isar document)
```
```     4                 Lawrence C Paulson, Cambridge University Computer Laboratory (original scripts)
```
```     5 *)
```
```     6
```
```     7 header {* The Mutilated Checker Board Problem *}
```
```     8
```
```     9 theory MutilatedCheckerboard = Main:
```
```    10
```
```    11 text {*
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```    12  The Mutilated Checker Board Problem, formalized inductively.  See
```
```    13  \cite{paulson-mutilated-board} and
```
```    14  \url{http://isabelle.in.tum.de/library/HOL/Induct/Mutil.html} for the
```
```    15  original tactic script version.
```
```    16 *}
```
```    17
```
```    18 subsection {* Tilings *}
```
```    19
```
```    20 consts
```
```    21   tiling :: "'a set set => 'a set set"
```
```    22
```
```    23 inductive "tiling A"
```
```    24   intros
```
```    25     empty: "{} : tiling A"
```
```    26     Un: "a : A ==> t : tiling A ==> a <= - t ==> a Un t : tiling A"
```
```    27
```
```    28
```
```    29 text "The union of two disjoint tilings is a tiling."
```
```    30
```
```    31 lemma tiling_Un:
```
```    32   "t : tiling A ==> u : tiling A ==> t Int u = {}
```
```    33     ==> t Un u : tiling A"
```
```    34 proof -
```
```    35   let ?T = "tiling A"
```
```    36   assume u: "u : ?T"
```
```    37   assume "t : ?T"
```
```    38   thus "t Int u = {} ==> t Un u : ?T" (is "PROP ?P t")
```
```    39   proof (induct t)
```
```    40     from u show "{} Un u : ?T" by simp
```
```    41   next
```
```    42     fix a t
```
```    43     assume "a : A" and hyp: "PROP ?P t"
```
```    44       and at: "a <= - t" and atu: "(a Un t) Int u = {}"
```
```    45     show "(a Un t) Un u : ?T"
```
```    46     proof -
```
```    47       have "a Un (t Un u) : ?T"
```
```    48       proof (rule tiling.Un)
```
```    49         show "a : A" .
```
```    50         from atu have "t Int u = {}" by blast
```
```    51         thus "t Un u: ?T" by (rule hyp)
```
```    52         from at atu show "a <= - (t Un u)" by blast
```
```    53       qed
```
```    54       also have "a Un (t Un u) = (a Un t) Un u"
```
```    55         by (simp only: Un_assoc)
```
```    56       finally show ?thesis .
```
```    57     qed
```
```    58   qed
```
```    59 qed
```
```    60
```
```    61
```
```    62 subsection {* Basic properties of ``below'' *}
```
```    63
```
```    64 constdefs
```
```    65   below :: "nat => nat set"
```
```    66   "below n == {i. i < n}"
```
```    67
```
```    68 lemma below_less_iff [iff]: "(i: below k) = (i < k)"
```
```    69   by (simp add: below_def)
```
```    70
```
```    71 lemma below_0: "below 0 = {}"
```
```    72   by (simp add: below_def)
```
```    73
```
```    74 lemma Sigma_Suc1:
```
```    75     "m = n + 1 ==> below m <*> B = ({n} <*> B) Un (below n <*> B)"
```
```    76   by (simp add: below_def less_Suc_eq) blast
```
```    77
```
```    78 lemma Sigma_Suc2:
```
```    79     "m = n + 2 ==> A <*> below m =
```
```    80       (A <*> {n}) Un (A <*> {n + 1}) Un (A <*> below n)"
```
```    81   by (auto simp add: below_def) arith
```
```    82
```
```    83 lemmas Sigma_Suc = Sigma_Suc1 Sigma_Suc2
```
```    84
```
```    85
```
```    86 subsection {* Basic properties of ``evnodd'' *}
```
```    87
```
```    88 constdefs
```
```    89   evnodd :: "(nat * nat) set => nat => (nat * nat) set"
```
```    90   "evnodd A b == A Int {(i, j). (i + j) mod #2 = b}"
```
```    91
```
```    92 lemma evnodd_iff:
```
```    93     "(i, j): evnodd A b = ((i, j): A  & (i + j) mod #2 = b)"
```
```    94   by (simp add: evnodd_def)
```
```    95
```
```    96 lemma evnodd_subset: "evnodd A b <= A"
```
```    97   by (unfold evnodd_def, rule Int_lower1)
```
```    98
```
```    99 lemma evnoddD: "x : evnodd A b ==> x : A"
```
```   100   by (rule subsetD, rule evnodd_subset)
```
```   101
```
```   102 lemma evnodd_finite: "finite A ==> finite (evnodd A b)"
```
```   103   by (rule finite_subset, rule evnodd_subset)
```
```   104
```
```   105 lemma evnodd_Un: "evnodd (A Un B) b = evnodd A b Un evnodd B b"
```
```   106   by (unfold evnodd_def) blast
```
```   107
```
```   108 lemma evnodd_Diff: "evnodd (A - B) b = evnodd A b - evnodd B b"
```
```   109   by (unfold evnodd_def) blast
```
```   110
```
```   111 lemma evnodd_empty: "evnodd {} b = {}"
```
```   112   by (simp add: evnodd_def)
```
```   113
```
```   114 lemma evnodd_insert: "evnodd (insert (i, j) C) b =
```
```   115     (if (i + j) mod #2 = b
```
```   116       then insert (i, j) (evnodd C b) else evnodd C b)"
```
```   117   by (simp add: evnodd_def) blast
```
```   118
```
```   119
```
```   120 subsection {* Dominoes *}
```
```   121
```
```   122 consts
```
```   123   domino :: "(nat * nat) set set"
```
```   124
```
```   125 inductive domino
```
```   126   intros
```
```   127     horiz: "{(i, j), (i, j + 1)} : domino"
```
```   128     vertl: "{(i, j), (i + 1, j)} : domino"
```
```   129
```
```   130 lemma dominoes_tile_row:
```
```   131   "{i} <*> below (2 * n) : tiling domino"
```
```   132   (is "?P n" is "?B n : ?T")
```
```   133 proof (induct n)
```
```   134   show "?P 0" by (simp add: below_0 tiling.empty)
```
```   135
```
```   136   fix n assume hyp: "?P n"
```
```   137   let ?a = "{i} <*> {2 * n + 1} Un {i} <*> {2 * n}"
```
```   138
```
```   139   have "?B (Suc n) = ?a Un ?B n"
```
```   140     by (auto simp add: Sigma_Suc Un_assoc)
```
```   141   also have "... : ?T"
```
```   142   proof (rule tiling.Un)
```
```   143     have "{(i, 2 * n), (i, 2 * n + 1)} : domino"
```
```   144       by (rule domino.horiz)
```
```   145     also have "{(i, 2 * n), (i, 2 * n + 1)} = ?a" by blast
```
```   146     finally show "... : domino" .
```
```   147     from hyp show "?B n : ?T" .
```
```   148     show "?a <= - ?B n" by blast
```
```   149   qed
```
```   150   finally show "?P (Suc n)" .
```
```   151 qed
```
```   152
```
```   153 lemma dominoes_tile_matrix:
```
```   154   "below m <*> below (2 * n) : tiling domino"
```
```   155   (is "?P m" is "?B m : ?T")
```
```   156 proof (induct m)
```
```   157   show "?P 0" by (simp add: below_0 tiling.empty)
```
```   158
```
```   159   fix m assume hyp: "?P m"
```
```   160   let ?t = "{m} <*> below (2 * n)"
```
```   161
```
```   162   have "?B (Suc m) = ?t Un ?B m" by (simp add: Sigma_Suc)
```
```   163   also have "... : ?T"
```
```   164   proof (rule tiling_Un)
```
```   165     show "?t : ?T" by (rule dominoes_tile_row)
```
```   166     from hyp show "?B m : ?T" .
```
```   167     show "?t Int ?B m = {}" by blast
```
```   168   qed
```
```   169   finally show "?P (Suc m)" .
```
```   170 qed
```
```   171
```
```   172 lemma domino_singleton:
```
```   173   "d : domino ==> b < 2 ==> EX i j. evnodd d b = {(i, j)}"
```
```   174 proof -
```
```   175   assume b: "b < 2"
```
```   176   assume "d : domino"
```
```   177   thus ?thesis (is "?P d")
```
```   178   proof induct
```
```   179     from b have b_cases: "b = 0 | b = 1" by arith
```
```   180     fix i j
```
```   181     note [simp] = evnodd_empty evnodd_insert mod_Suc
```
```   182     from b_cases show "?P {(i, j), (i, j + 1)}" by rule auto
```
```   183     from b_cases show "?P {(i, j), (i + 1, j)}" by rule auto
```
```   184   qed
```
```   185 qed
```
```   186
```
```   187 lemma domino_finite: "d: domino ==> finite d"
```
```   188 proof -
```
```   189   assume "d: domino"
```
```   190   thus ?thesis
```
```   191   proof induct
```
```   192     fix i j :: nat
```
```   193     show "finite {(i, j), (i, j + 1)}" by (intro Finites.intros)
```
```   194     show "finite {(i, j), (i + 1, j)}" by (intro Finites.intros)
```
```   195   qed
```
```   196 qed
```
```   197
```
```   198
```
```   199 subsection {* Tilings of dominoes *}
```
```   200
```
```   201 lemma tiling_domino_finite:
```
```   202   "t : tiling domino ==> finite t" (is "t : ?T ==> ?F t")
```
```   203 proof -
```
```   204   assume "t : ?T"
```
```   205   thus "?F t"
```
```   206   proof induct
```
```   207     show "?F {}" by (rule Finites.emptyI)
```
```   208     fix a t assume "?F t"
```
```   209     assume "a : domino" hence "?F a" by (rule domino_finite)
```
```   210     thus "?F (a Un t)" by (rule finite_UnI)
```
```   211   qed
```
```   212 qed
```
```   213
```
```   214 lemma tiling_domino_01:
```
```   215   "t : tiling domino ==> card (evnodd t 0) = card (evnodd t 1)"
```
```   216   (is "t : ?T ==> ?P t")
```
```   217 proof -
```
```   218   assume "t : ?T"
```
```   219   thus "?P t"
```
```   220   proof induct
```
```   221     show "?P {}" by (simp add: evnodd_def)
```
```   222
```
```   223     fix a t
```
```   224     let ?e = evnodd
```
```   225     assume "a : domino" and "t : ?T"
```
```   226       and hyp: "card (?e t 0) = card (?e t 1)"
```
```   227       and at: "a <= - t"
```
```   228
```
```   229     have card_suc:
```
```   230       "!!b. b < 2 ==> card (?e (a Un t) b) = Suc (card (?e t b))"
```
```   231     proof -
```
```   232       fix b assume "b < 2"
```
```   233       have "?e (a Un t) b = ?e a b Un ?e t b" by (rule evnodd_Un)
```
```   234       also obtain i j where e: "?e a b = {(i, j)}"
```
```   235       proof -
```
```   236         have "EX i j. ?e a b = {(i, j)}" by (rule domino_singleton)
```
```   237         thus ?thesis by (blast intro: that)
```
```   238       qed
```
```   239       also have "... Un ?e t b = insert (i, j) (?e t b)" by simp
```
```   240       also have "card ... = Suc (card (?e t b))"
```
```   241       proof (rule card_insert_disjoint)
```
```   242         show "finite (?e t b)"
```
```   243           by (rule evnodd_finite, rule tiling_domino_finite)
```
```   244         from e have "(i, j) : ?e a b" by simp
```
```   245         with at show "(i, j) ~: ?e t b" by (blast dest: evnoddD)
```
```   246       qed
```
```   247       finally show "?thesis b" .
```
```   248     qed
```
```   249     hence "card (?e (a Un t) 0) = Suc (card (?e t 0))" by simp
```
```   250     also from hyp have "card (?e t 0) = card (?e t 1)" .
```
```   251     also from card_suc have "Suc ... = card (?e (a Un t) 1)"
```
```   252       by simp
```
```   253     finally show "?P (a Un t)" .
```
```   254   qed
```
```   255 qed
```
```   256
```
```   257
```
```   258 subsection {* Main theorem *}
```
```   259
```
```   260 constdefs
```
```   261   mutilated_board :: "nat => nat => (nat * nat) set"
```
```   262   "mutilated_board m n ==
```
```   263     below (2 * (m + 1)) <*> below (2 * (n + 1))
```
```   264       - {(0, 0)} - {(2 * m + 1, 2 * n + 1)}"
```
```   265
```
```   266 theorem mutil_not_tiling: "mutilated_board m n ~: tiling domino"
```
```   267 proof (unfold mutilated_board_def)
```
```   268   let ?T = "tiling domino"
```
```   269   let ?t = "below (2 * (m + 1)) <*> below (2 * (n + 1))"
```
```   270   let ?t' = "?t - {(0, 0)}"
```
```   271   let ?t'' = "?t' - {(2 * m + 1, 2 * n + 1)}"
```
```   272
```
```   273   show "?t'' ~: ?T"
```
```   274   proof
```
```   275     have t: "?t : ?T" by (rule dominoes_tile_matrix)
```
```   276     assume t'': "?t'' : ?T"
```
```   277
```
```   278     let ?e = evnodd
```
```   279     have fin: "finite (?e ?t 0)"
```
```   280       by (rule evnodd_finite, rule tiling_domino_finite, rule t)
```
```   281
```
```   282     note [simp] = evnodd_iff evnodd_empty evnodd_insert evnodd_Diff
```
```   283     have "card (?e ?t'' 0) < card (?e ?t' 0)"
```
```   284     proof -
```
```   285       have "card (?e ?t' 0 - {(2 * m + 1, 2 * n + 1)})
```
```   286         < card (?e ?t' 0)"
```
```   287       proof (rule card_Diff1_less)
```
```   288         from _ fin show "finite (?e ?t' 0)"
```
```   289           by (rule finite_subset) auto
```
```   290         show "(2 * m + 1, 2 * n + 1) : ?e ?t' 0" by simp
```
```   291       qed
```
```   292       thus ?thesis by simp
```
```   293     qed
```
```   294     also have "... < card (?e ?t 0)"
```
```   295     proof -
```
```   296       have "(0, 0) : ?e ?t 0" by simp
```
```   297       with fin have "card (?e ?t 0 - {(0, 0)}) < card (?e ?t 0)"
```
```   298         by (rule card_Diff1_less)
```
```   299       thus ?thesis by simp
```
```   300     qed
```
```   301     also from t have "... = card (?e ?t 1)"
```
```   302       by (rule tiling_domino_01)
```
```   303     also have "?e ?t 1 = ?e ?t'' 1" by simp
```
```   304     also from t'' have "card ... = card (?e ?t'' 0)"
```
```   305       by (rule tiling_domino_01 [symmetric])
```
```   306     finally have "... < ..." . thus False ..
```
```   307   qed
```
```   308 qed
```
```   309
```
```   310 end
```