src/HOL/Isar_Examples/Mutilated_Checkerboard.thy
 author blanchet Thu Sep 11 18:54:36 2014 +0200 (2014-09-11) changeset 58306 117ba6cbe414 parent 55656 eb07b0acbebc child 58614 7338eb25226c permissions -rw-r--r--
renamed 'rep_datatype' to 'old_rep_datatype' (HOL)
```     1 (*  Title:      HOL/Isar_Examples/Mutilated_Checkerboard.thy
```
```     2     Author:     Markus Wenzel, TU Muenchen (Isar document)
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory (original scripts)
```
```     4 *)
```
```     5
```
```     6 header {* The Mutilated Checker Board Problem *}
```
```     7
```
```     8 theory Mutilated_Checkerboard
```
```     9 imports Main
```
```    10 begin
```
```    11
```
```    12 text {* The Mutilated Checker Board Problem, formalized inductively.
```
```    13   See \cite{paulson-mutilated-board} for the original tactic script version. *}
```
```    14
```
```    15 subsection {* Tilings *}
```
```    16
```
```    17 inductive_set tiling :: "'a set set \<Rightarrow> 'a set set"
```
```    18   for A :: "'a set set"
```
```    19 where
```
```    20   empty: "{} \<in> tiling A"
```
```    21 | Un: "a \<in> A \<Longrightarrow> t \<in> tiling A \<Longrightarrow> a \<subseteq> - t \<Longrightarrow> a \<union> t \<in> tiling A"
```
```    22
```
```    23
```
```    24 text "The union of two disjoint tilings is a tiling."
```
```    25
```
```    26 lemma tiling_Un:
```
```    27   assumes "t \<in> tiling A"
```
```    28     and "u \<in> tiling A"
```
```    29     and "t \<inter> u = {}"
```
```    30   shows "t \<union> u \<in> tiling A"
```
```    31 proof -
```
```    32   let ?T = "tiling A"
```
```    33   from `t \<in> ?T` and `t \<inter> u = {}`
```
```    34   show "t \<union> u \<in> ?T"
```
```    35   proof (induct t)
```
```    36     case empty
```
```    37     with `u \<in> ?T` show "{} \<union> u \<in> ?T" by simp
```
```    38   next
```
```    39     case (Un a t)
```
```    40     show "(a \<union> t) \<union> u \<in> ?T"
```
```    41     proof -
```
```    42       have "a \<union> (t \<union> u) \<in> ?T"
```
```    43         using `a \<in> A`
```
```    44       proof (rule tiling.Un)
```
```    45         from `(a \<union> t) \<inter> u = {}` have "t \<inter> u = {}" by blast
```
```    46         then show "t \<union> u \<in> ?T" by (rule Un)
```
```    47         from `a \<subseteq> - t` and `(a \<union> t) \<inter> u = {}`
```
```    48         show "a \<subseteq> - (t \<union> u)" by blast
```
```    49       qed
```
```    50       also have "a \<union> (t \<union> u) = (a \<union> t) \<union> u"
```
```    51         by (simp only: Un_assoc)
```
```    52       finally show ?thesis .
```
```    53     qed
```
```    54   qed
```
```    55 qed
```
```    56
```
```    57
```
```    58 subsection {* Basic properties of ``below'' *}
```
```    59
```
```    60 definition below :: "nat \<Rightarrow> nat set"
```
```    61   where "below n = {i. i < n}"
```
```    62
```
```    63 lemma below_less_iff [iff]: "i \<in> below k \<longleftrightarrow> i < k"
```
```    64   by (simp add: below_def)
```
```    65
```
```    66 lemma below_0: "below 0 = {}"
```
```    67   by (simp add: below_def)
```
```    68
```
```    69 lemma Sigma_Suc1: "m = n + 1 \<Longrightarrow> below m \<times> B = ({n} \<times> B) \<union> (below n \<times> B)"
```
```    70   by (simp add: below_def less_Suc_eq) blast
```
```    71
```
```    72 lemma Sigma_Suc2:
```
```    73   "m = n + 2 \<Longrightarrow>
```
```    74     A \<times> below m = (A \<times> {n}) \<union> (A \<times> {n + 1}) \<union> (A \<times> below n)"
```
```    75   by (auto simp add: below_def)
```
```    76
```
```    77 lemmas Sigma_Suc = Sigma_Suc1 Sigma_Suc2
```
```    78
```
```    79
```
```    80 subsection {* Basic properties of ``evnodd'' *}
```
```    81
```
```    82 definition evnodd :: "(nat \<times> nat) set \<Rightarrow> nat \<Rightarrow> (nat \<times> nat) set"
```
```    83   where "evnodd A b = A \<inter> {(i, j). (i + j) mod 2 = b}"
```
```    84
```
```    85 lemma evnodd_iff: "(i, j) \<in> evnodd A b \<longleftrightarrow> (i, j) \<in> A  \<and> (i + j) mod 2 = b"
```
```    86   by (simp add: evnodd_def)
```
```    87
```
```    88 lemma evnodd_subset: "evnodd A b \<subseteq> A"
```
```    89   unfolding evnodd_def by (rule Int_lower1)
```
```    90
```
```    91 lemma evnoddD: "x \<in> evnodd A b \<Longrightarrow> x \<in> A"
```
```    92   by (rule subsetD) (rule evnodd_subset)
```
```    93
```
```    94 lemma evnodd_finite: "finite A \<Longrightarrow> finite (evnodd A b)"
```
```    95   by (rule finite_subset) (rule evnodd_subset)
```
```    96
```
```    97 lemma evnodd_Un: "evnodd (A \<union> B) b = evnodd A b \<union> evnodd B b"
```
```    98   unfolding evnodd_def by blast
```
```    99
```
```   100 lemma evnodd_Diff: "evnodd (A - B) b = evnodd A b - evnodd B b"
```
```   101   unfolding evnodd_def by blast
```
```   102
```
```   103 lemma evnodd_empty: "evnodd {} b = {}"
```
```   104   by (simp add: evnodd_def)
```
```   105
```
```   106 lemma evnodd_insert: "evnodd (insert (i, j) C) b =
```
```   107     (if (i + j) mod 2 = b
```
```   108       then insert (i, j) (evnodd C b) else evnodd C b)"
```
```   109   by (simp add: evnodd_def)
```
```   110
```
```   111
```
```   112 subsection {* Dominoes *}
```
```   113
```
```   114 inductive_set domino :: "(nat \<times> nat) set set"
```
```   115 where
```
```   116   horiz: "{(i, j), (i, j + 1)} \<in> domino"
```
```   117 | vertl: "{(i, j), (i + 1, j)} \<in> domino"
```
```   118
```
```   119 lemma dominoes_tile_row:
```
```   120   "{i} \<times> below (2 * n) \<in> tiling domino"
```
```   121   (is "?B n \<in> ?T")
```
```   122 proof (induct n)
```
```   123   case 0
```
```   124   show ?case by (simp add: below_0 tiling.empty)
```
```   125 next
```
```   126   case (Suc n)
```
```   127   let ?a = "{i} \<times> {2 * n + 1} \<union> {i} \<times> {2 * n}"
```
```   128   have "?B (Suc n) = ?a \<union> ?B n"
```
```   129     by (auto simp add: Sigma_Suc Un_assoc)
```
```   130   also have "\<dots> \<in> ?T"
```
```   131   proof (rule tiling.Un)
```
```   132     have "{(i, 2 * n), (i, 2 * n + 1)} \<in> domino"
```
```   133       by (rule domino.horiz)
```
```   134     also have "{(i, 2 * n), (i, 2 * n + 1)} = ?a" by blast
```
```   135     finally show "\<dots> \<in> domino" .
```
```   136     show "?B n \<in> ?T" by (rule Suc)
```
```   137     show "?a \<subseteq> - ?B n" by blast
```
```   138   qed
```
```   139   finally show ?case .
```
```   140 qed
```
```   141
```
```   142 lemma dominoes_tile_matrix:
```
```   143   "below m \<times> below (2 * n) \<in> tiling domino"
```
```   144   (is "?B m \<in> ?T")
```
```   145 proof (induct m)
```
```   146   case 0
```
```   147   show ?case by (simp add: below_0 tiling.empty)
```
```   148 next
```
```   149   case (Suc m)
```
```   150   let ?t = "{m} \<times> below (2 * n)"
```
```   151   have "?B (Suc m) = ?t \<union> ?B m" by (simp add: Sigma_Suc)
```
```   152   also have "\<dots> \<in> ?T"
```
```   153   proof (rule tiling_Un)
```
```   154     show "?t \<in> ?T" by (rule dominoes_tile_row)
```
```   155     show "?B m \<in> ?T" by (rule Suc)
```
```   156     show "?t \<inter> ?B m = {}" by blast
```
```   157   qed
```
```   158   finally show ?case .
```
```   159 qed
```
```   160
```
```   161 lemma domino_singleton:
```
```   162   assumes "d \<in> domino"
```
```   163     and "b < 2"
```
```   164   shows "\<exists>i j. evnodd d b = {(i, j)}"  (is "?P d")
```
```   165   using assms
```
```   166 proof induct
```
```   167   from `b < 2` have b_cases: "b = 0 \<or> b = 1" by arith
```
```   168   fix i j
```
```   169   note [simp] = evnodd_empty evnodd_insert mod_Suc
```
```   170   from b_cases show "?P {(i, j), (i, j + 1)}" by rule auto
```
```   171   from b_cases show "?P {(i, j), (i + 1, j)}" by rule auto
```
```   172 qed
```
```   173
```
```   174 lemma domino_finite:
```
```   175   assumes "d \<in> domino"
```
```   176   shows "finite d"
```
```   177   using assms
```
```   178 proof induct
```
```   179   fix i j :: nat
```
```   180   show "finite {(i, j), (i, j + 1)}" by (intro finite.intros)
```
```   181   show "finite {(i, j), (i + 1, j)}" by (intro finite.intros)
```
```   182 qed
```
```   183
```
```   184
```
```   185 subsection {* Tilings of dominoes *}
```
```   186
```
```   187 lemma tiling_domino_finite:
```
```   188   assumes t: "t \<in> tiling domino"  (is "t \<in> ?T")
```
```   189   shows "finite t"  (is "?F t")
```
```   190   using t
```
```   191 proof induct
```
```   192   show "?F {}" by (rule finite.emptyI)
```
```   193   fix a t assume "?F t"
```
```   194   assume "a \<in> domino"
```
```   195   then have "?F a" by (rule domino_finite)
```
```   196   from this and `?F t` show "?F (a \<union> t)" by (rule finite_UnI)
```
```   197 qed
```
```   198
```
```   199 lemma tiling_domino_01:
```
```   200   assumes t: "t \<in> tiling domino"  (is "t \<in> ?T")
```
```   201   shows "card (evnodd t 0) = card (evnodd t 1)"
```
```   202   using t
```
```   203 proof induct
```
```   204   case empty
```
```   205   show ?case by (simp add: evnodd_def)
```
```   206 next
```
```   207   case (Un a t)
```
```   208   let ?e = evnodd
```
```   209   note hyp = `card (?e t 0) = card (?e t 1)`
```
```   210     and at = `a \<subseteq> - t`
```
```   211   have card_suc:
```
```   212     "\<And>b. b < 2 \<Longrightarrow> card (?e (a \<union> t) b) = Suc (card (?e t b))"
```
```   213   proof -
```
```   214     fix b :: nat
```
```   215     assume "b < 2"
```
```   216     have "?e (a \<union> t) b = ?e a b \<union> ?e t b" by (rule evnodd_Un)
```
```   217     also obtain i j where e: "?e a b = {(i, j)}"
```
```   218     proof -
```
```   219       from `a \<in> domino` and `b < 2`
```
```   220       have "\<exists>i j. ?e a b = {(i, j)}" by (rule domino_singleton)
```
```   221       then show ?thesis by (blast intro: that)
```
```   222     qed
```
```   223     also have "\<dots> \<union> ?e t b = insert (i, j) (?e t b)" by simp
```
```   224     also have "card \<dots> = Suc (card (?e t b))"
```
```   225     proof (rule card_insert_disjoint)
```
```   226       from `t \<in> tiling domino` have "finite t"
```
```   227         by (rule tiling_domino_finite)
```
```   228       then show "finite (?e t b)"
```
```   229         by (rule evnodd_finite)
```
```   230       from e have "(i, j) \<in> ?e a b" by simp
```
```   231       with at show "(i, j) \<notin> ?e t b" by (blast dest: evnoddD)
```
```   232     qed
```
```   233     finally show "?thesis b" .
```
```   234   qed
```
```   235   then have "card (?e (a \<union> t) 0) = Suc (card (?e t 0))" by simp
```
```   236   also from hyp have "card (?e t 0) = card (?e t 1)" .
```
```   237   also from card_suc have "Suc \<dots> = card (?e (a \<union> t) 1)"
```
```   238     by simp
```
```   239   finally show ?case .
```
```   240 qed
```
```   241
```
```   242
```
```   243 subsection {* Main theorem *}
```
```   244
```
```   245 definition mutilated_board :: "nat \<Rightarrow> nat \<Rightarrow> (nat \<times> nat) set"
```
```   246   where
```
```   247     "mutilated_board m n =
```
```   248       below (2 * (m + 1)) \<times> below (2 * (n + 1))
```
```   249         - {(0, 0)} - {(2 * m + 1, 2 * n + 1)}"
```
```   250
```
```   251 theorem mutil_not_tiling: "mutilated_board m n \<notin> tiling domino"
```
```   252 proof (unfold mutilated_board_def)
```
```   253   let ?T = "tiling domino"
```
```   254   let ?t = "below (2 * (m + 1)) \<times> below (2 * (n + 1))"
```
```   255   let ?t' = "?t - {(0, 0)}"
```
```   256   let ?t'' = "?t' - {(2 * m + 1, 2 * n + 1)}"
```
```   257
```
```   258   show "?t'' \<notin> ?T"
```
```   259   proof
```
```   260     have t: "?t \<in> ?T" by (rule dominoes_tile_matrix)
```
```   261     assume t'': "?t'' \<in> ?T"
```
```   262
```
```   263     let ?e = evnodd
```
```   264     have fin: "finite (?e ?t 0)"
```
```   265       by (rule evnodd_finite, rule tiling_domino_finite, rule t)
```
```   266
```
```   267     note [simp] = evnodd_iff evnodd_empty evnodd_insert evnodd_Diff
```
```   268     have "card (?e ?t'' 0) < card (?e ?t' 0)"
```
```   269     proof -
```
```   270       have "card (?e ?t' 0 - {(2 * m + 1, 2 * n + 1)})
```
```   271         < card (?e ?t' 0)"
```
```   272       proof (rule card_Diff1_less)
```
```   273         from _ fin show "finite (?e ?t' 0)"
```
```   274           by (rule finite_subset) auto
```
```   275         show "(2 * m + 1, 2 * n + 1) \<in> ?e ?t' 0" by simp
```
```   276       qed
```
```   277       then show ?thesis by simp
```
```   278     qed
```
```   279     also have "\<dots> < card (?e ?t 0)"
```
```   280     proof -
```
```   281       have "(0, 0) \<in> ?e ?t 0" by simp
```
```   282       with fin have "card (?e ?t 0 - {(0, 0)}) < card (?e ?t 0)"
```
```   283         by (rule card_Diff1_less)
```
```   284       then show ?thesis by simp
```
```   285     qed
```
```   286     also from t have "\<dots> = card (?e ?t 1)"
```
```   287       by (rule tiling_domino_01)
```
```   288     also have "?e ?t 1 = ?e ?t'' 1" by simp
```
```   289     also from t'' have "card \<dots> = card (?e ?t'' 0)"
```
```   290       by (rule tiling_domino_01 [symmetric])
```
```   291     finally have "\<dots> < \<dots>" . then show False ..
```
```   292   qed
```
```   293 qed
```
```   294
```
```   295 end
```