src/HOL/Isar_examples/MutilatedCheckerboard.thy
 author haftmann Mon Jan 30 08:20:56 2006 +0100 (2006-01-30) changeset 18851 9502ce541f01 parent 18241 afdba6b3e383 child 22273 9785397cc344 permissions -rw-r--r--
```     1 (*  Title:      HOL/Isar_examples/MutilatedCheckerboard.thy
```
```     2     ID:         \$Id\$
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```     3     Author:     Markus Wenzel, TU Muenchen (Isar document)
```
```     4                 Lawrence C Paulson, Cambridge University Computer Laboratory (original scripts)
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```     5 *)
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```     6
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```     7 header {* The Mutilated Checker Board Problem *}
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```     8
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```     9 theory MutilatedCheckerboard imports Main begin
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```    10
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```    11 text {*
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```    12  The Mutilated Checker Board Problem, formalized inductively.  See
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```    13  \cite{paulson-mutilated-board} and
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```    14  \url{http://isabelle.in.tum.de/library/HOL/Induct/Mutil.html} for the
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```    15  original tactic script version.
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```    16 *}
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```    17
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```    18 subsection {* Tilings *}
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```    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
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```    28
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```    29 text "The union of two disjoint tilings is a tiling."
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```    30
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```    31 lemma tiling_Un:
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```    32   assumes "t : tiling A" and "u : tiling A" and "t Int u = {}"
```
```    33   shows "t Un u : tiling A"
```
```    34 proof -
```
```    35   let ?T = "tiling A"
```
```    36   from `t : ?T` and `t Int u = {}`
```
```    37   show "t Un u : ?T"
```
```    38   proof (induct t)
```
```    39     case empty
```
```    40     with `u : ?T` show "{} Un u : ?T" by simp
```
```    41   next
```
```    42     case (Un a t)
```
```    43     show "(a Un t) Un u : ?T"
```
```    44     proof -
```
```    45       have "a Un (t Un u) : ?T"
```
```    46       proof (rule tiling.Un)
```
```    47         show "a : A" .
```
```    48         from `(a Un t) Int u = {}` have "t Int u = {}" by blast
```
```    49         then show "t Un u: ?T" by (rule Un)
```
```    50         have "a <= - t" .
```
```    51         with `(a Un t) Int u = {}` show "a <= - (t Un u)" by blast
```
```    52       qed
```
```    53       also have "a Un (t Un u) = (a Un t) Un u"
```
```    54         by (simp only: Un_assoc)
```
```    55       finally show ?thesis .
```
```    56     qed
```
```    57   qed
```
```    58 qed
```
```    59
```
```    60
```
```    61 subsection {* Basic properties of ``below'' *}
```
```    62
```
```    63 constdefs
```
```    64   below :: "nat => nat set"
```
```    65   "below n == {i. i < n}"
```
```    66
```
```    67 lemma below_less_iff [iff]: "(i: below k) = (i < k)"
```
```    68   by (simp add: below_def)
```
```    69
```
```    70 lemma below_0: "below 0 = {}"
```
```    71   by (simp add: below_def)
```
```    72
```
```    73 lemma Sigma_Suc1:
```
```    74     "m = n + 1 ==> below m <*> B = ({n} <*> B) Un (below n <*> B)"
```
```    75   by (simp add: below_def less_Suc_eq) blast
```
```    76
```
```    77 lemma Sigma_Suc2:
```
```    78     "m = n + 2 ==> A <*> below m =
```
```    79       (A <*> {n}) Un (A <*> {n + 1}) Un (A <*> below n)"
```
```    80   by (auto simp add: below_def)
```
```    81
```
```    82 lemmas Sigma_Suc = Sigma_Suc1 Sigma_Suc2
```
```    83
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```    84
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```    85 subsection {* Basic properties of ``evnodd'' *}
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```    86
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```    87 constdefs
```
```    88   evnodd :: "(nat * nat) set => nat => (nat * nat) set"
```
```    89   "evnodd A b == A Int {(i, j). (i + j) mod 2 = b}"
```
```    90
```
```    91 lemma evnodd_iff:
```
```    92     "(i, j): evnodd A b = ((i, j): A  & (i + j) mod 2 = b)"
```
```    93   by (simp add: evnodd_def)
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```    94
```
```    95 lemma evnodd_subset: "evnodd A b <= A"
```
```    96   by (unfold evnodd_def, rule Int_lower1)
```
```    97
```
```    98 lemma evnoddD: "x : evnodd A b ==> x : A"
```
```    99   by (rule subsetD, rule evnodd_subset)
```
```   100
```
```   101 lemma evnodd_finite: "finite A ==> finite (evnodd A b)"
```
```   102   by (rule finite_subset, rule evnodd_subset)
```
```   103
```
```   104 lemma evnodd_Un: "evnodd (A Un B) b = evnodd A b Un evnodd B b"
```
```   105   by (unfold evnodd_def) blast
```
```   106
```
```   107 lemma evnodd_Diff: "evnodd (A - B) b = evnodd A b - evnodd B b"
```
```   108   by (unfold evnodd_def) blast
```
```   109
```
```   110 lemma evnodd_empty: "evnodd {} b = {}"
```
```   111   by (simp add: evnodd_def)
```
```   112
```
```   113 lemma evnodd_insert: "evnodd (insert (i, j) C) b =
```
```   114     (if (i + j) mod 2 = b
```
```   115       then insert (i, j) (evnodd C b) else evnodd C b)"
```
```   116   by (simp add: evnodd_def) blast
```
```   117
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```   118
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```   119 subsection {* Dominoes *}
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```   120
```
```   121 consts
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```   122   domino :: "(nat * nat) set set"
```
```   123
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```   124 inductive domino
```
```   125   intros
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```   126     horiz: "{(i, j), (i, j + 1)} : domino"
```
```   127     vertl: "{(i, j), (i + 1, j)} : domino"
```
```   128
```
```   129 lemma dominoes_tile_row:
```
```   130   "{i} <*> below (2 * n) : tiling domino"
```
```   131   (is "?B n : ?T")
```
```   132 proof (induct n)
```
```   133   case 0
```
```   134   show ?case by (simp add: below_0 tiling.empty)
```
```   135 next
```
```   136   case (Suc n)
```
```   137   let ?a = "{i} <*> {2 * n + 1} Un {i} <*> {2 * n}"
```
```   138   have "?B (Suc n) = ?a Un ?B n"
```
```   139     by (auto simp add: Sigma_Suc Un_assoc)
```
```   140   also have "... : ?T"
```
```   141   proof (rule tiling.Un)
```
```   142     have "{(i, 2 * n), (i, 2 * n + 1)} : domino"
```
```   143       by (rule domino.horiz)
```
```   144     also have "{(i, 2 * n), (i, 2 * n + 1)} = ?a" by blast
```
```   145     finally show "... : domino" .
```
```   146     show "?B n : ?T" by (rule Suc)
```
```   147     show "?a <= - ?B n" by blast
```
```   148   qed
```
```   149   finally show ?case .
```
```   150 qed
```
```   151
```
```   152 lemma dominoes_tile_matrix:
```
```   153   "below m <*> below (2 * n) : tiling domino"
```
```   154   (is "?B m : ?T")
```
```   155 proof (induct m)
```
```   156   case 0
```
```   157   show ?case by (simp add: below_0 tiling.empty)
```
```   158 next
```
```   159   case (Suc m)
```
```   160   let ?t = "{m} <*> below (2 * n)"
```
```   161   have "?B (Suc m) = ?t Un ?B m" by (simp add: Sigma_Suc)
```
```   162   also have "... : ?T"
```
```   163   proof (rule tiling_Un)
```
```   164     show "?t : ?T" by (rule dominoes_tile_row)
```
```   165     show "?B m : ?T" by (rule Suc)
```
```   166     show "?t Int ?B m = {}" by blast
```
```   167   qed
```
```   168   finally show ?case .
```
```   169 qed
```
```   170
```
```   171 lemma domino_singleton:
```
```   172   assumes d: "d : domino" and "b < 2"
```
```   173   shows "EX i j. evnodd d b = {(i, j)}"  (is "?P d")
```
```   174   using d
```
```   175 proof induct
```
```   176   from `b < 2` have b_cases: "b = 0 | b = 1" by arith
```
```   177   fix i j
```
```   178   note [simp] = evnodd_empty evnodd_insert mod_Suc
```
```   179   from b_cases show "?P {(i, j), (i, j + 1)}" by rule auto
```
```   180   from b_cases show "?P {(i, j), (i + 1, j)}" by rule auto
```
```   181 qed
```
```   182
```
```   183 lemma domino_finite:
```
```   184   assumes d: "d: domino"
```
```   185   shows "finite d"
```
```   186   using d
```
```   187 proof induct
```
```   188   fix i j :: nat
```
```   189   show "finite {(i, j), (i, j + 1)}" by (intro Finites.intros)
```
```   190   show "finite {(i, j), (i + 1, j)}" by (intro Finites.intros)
```
```   191 qed
```
```   192
```
```   193
```
```   194 subsection {* Tilings of dominoes *}
```
```   195
```
```   196 lemma tiling_domino_finite:
```
```   197   assumes t: "t : tiling domino"  (is "t : ?T")
```
```   198   shows "finite t"  (is "?F t")
```
```   199   using t
```
```   200 proof induct
```
```   201   show "?F {}" by (rule Finites.emptyI)
```
```   202   fix a t assume "?F t"
```
```   203   assume "a : domino" then have "?F a" by (rule domino_finite)
```
```   204   then show "?F (a Un t)" by (rule finite_UnI)
```
```   205 qed
```
```   206
```
```   207 lemma tiling_domino_01:
```
```   208   assumes t: "t : tiling domino"  (is "t : ?T")
```
```   209   shows "card (evnodd t 0) = card (evnodd t 1)"
```
```   210   using t
```
```   211 proof induct
```
```   212   case empty
```
```   213   show ?case by (simp add: evnodd_def)
```
```   214 next
```
```   215   case (Un a t)
```
```   216   let ?e = evnodd
```
```   217   note hyp = `card (?e t 0) = card (?e t 1)`
```
```   218     and at = `a <= - t`
```
```   219   have card_suc:
```
```   220     "!!b. b < 2 ==> card (?e (a Un t) b) = Suc (card (?e t b))"
```
```   221   proof -
```
```   222     fix b :: nat assume "b < 2"
```
```   223     have "?e (a Un t) b = ?e a b Un ?e t b" by (rule evnodd_Un)
```
```   224     also obtain i j where e: "?e a b = {(i, j)}"
```
```   225     proof -
```
```   226       have "EX i j. ?e a b = {(i, j)}" by (rule domino_singleton)
```
```   227       then show ?thesis by (blast intro: that)
```
```   228     qed
```
```   229     also have "... Un ?e t b = insert (i, j) (?e t b)" by simp
```
```   230     also have "card ... = Suc (card (?e t b))"
```
```   231     proof (rule card_insert_disjoint)
```
```   232       show "finite (?e t b)"
```
```   233         by (rule evnodd_finite, rule tiling_domino_finite)
```
```   234       from e have "(i, j) : ?e a b" by simp
```
```   235       with at show "(i, j) ~: ?e t b" by (blast dest: evnoddD)
```
```   236     qed
```
```   237     finally show "?thesis b" .
```
```   238   qed
```
```   239   then have "card (?e (a Un t) 0) = Suc (card (?e t 0))" by simp
```
```   240   also from hyp have "card (?e t 0) = card (?e t 1)" .
```
```   241   also from card_suc have "Suc ... = card (?e (a Un t) 1)"
```
```   242     by simp
```
```   243   finally show ?case .
```
```   244 qed
```
```   245
```
```   246
```
```   247 subsection {* Main theorem *}
```
```   248
```
```   249 constdefs
```
```   250   mutilated_board :: "nat => nat => (nat * nat) set"
```
```   251   "mutilated_board m n ==
```
```   252     below (2 * (m + 1)) <*> below (2 * (n + 1))
```
```   253       - {(0, 0)} - {(2 * m + 1, 2 * n + 1)}"
```
```   254
```
```   255 theorem mutil_not_tiling: "mutilated_board m n ~: tiling domino"
```
```   256 proof (unfold mutilated_board_def)
```
```   257   let ?T = "tiling domino"
```
```   258   let ?t = "below (2 * (m + 1)) <*> below (2 * (n + 1))"
```
```   259   let ?t' = "?t - {(0, 0)}"
```
```   260   let ?t'' = "?t' - {(2 * m + 1, 2 * n + 1)}"
```
```   261
```
```   262   show "?t'' ~: ?T"
```
```   263   proof
```
```   264     have t: "?t : ?T" by (rule dominoes_tile_matrix)
```
```   265     assume t'': "?t'' : ?T"
```
```   266
```
```   267     let ?e = evnodd
```
```   268     have fin: "finite (?e ?t 0)"
```
```   269       by (rule evnodd_finite, rule tiling_domino_finite, rule t)
```
```   270
```
```   271     note [simp] = evnodd_iff evnodd_empty evnodd_insert evnodd_Diff
```
```   272     have "card (?e ?t'' 0) < card (?e ?t' 0)"
```
```   273     proof -
```
```   274       have "card (?e ?t' 0 - {(2 * m + 1, 2 * n + 1)})
```
```   275         < card (?e ?t' 0)"
```
```   276       proof (rule card_Diff1_less)
```
```   277         from _ fin show "finite (?e ?t' 0)"
```
```   278           by (rule finite_subset) auto
```
```   279         show "(2 * m + 1, 2 * n + 1) : ?e ?t' 0" by simp
```
```   280       qed
```
```   281       then show ?thesis by simp
```
```   282     qed
```
```   283     also have "... < card (?e ?t 0)"
```
```   284     proof -
```
```   285       have "(0, 0) : ?e ?t 0" by simp
```
```   286       with fin have "card (?e ?t 0 - {(0, 0)}) < card (?e ?t 0)"
```
```   287         by (rule card_Diff1_less)
```
```   288       then show ?thesis by simp
```
```   289     qed
```
```   290     also from t have "... = card (?e ?t 1)"
```
```   291       by (rule tiling_domino_01)
```
```   292     also have "?e ?t 1 = ?e ?t'' 1" by simp
```
```   293     also from t'' have "card ... = card (?e ?t'' 0)"
```
```   294       by (rule tiling_domino_01 [symmetric])
```
```   295     finally have "... < ..." . then show False ..
```
```   296   qed
```
```   297 qed
```
```   298
```
```   299 end
```