src/HOL/Induct/Mutil.thy

 (*  Title:      HOL/Induct/Mutil.thy
     ID:         $Id$
     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
-    Copyright   1996  University of Cambridge
+                Tobias Nipkow, TUM (part 2)
 *)
 
 header {* The Mutilated Chess Board Problem *}
 
 theory Mutil imports Main begin
+
+subsection{* The Mutilated Chess Board Cannot be Tiled by Dominoes *}
 
 text {*
   The Mutilated Chess Board Problem, formalized inductively.
 
   Originator is Max Black, according to J A Robinson.  Popularized as

   apply (induct set: tiling)
   apply (auto simp add: Un_assoc)
   done
 
 
+lemma tiling_Diff1E [intro]:
+assumes "t-a \<in> tiling A"  "a \<in> A"  "a \<subseteq> t"
+shows "t \<in> tiling A"
+proof -
+  from assms(2-3) have  "EX r. t = r Un a & r Int a = {}"
+    by (metis Diff_disjoint Int_commute Un_Diff_cancel Un_absorb1 Un_commute)
+  thus ?thesis using assms(1,2)
+    by (auto simp:Un_Diff)
+       (metis Compl_Diff_eq Diff_Compl Diff_empty Int_commute Un_Diff_cancel
+              Un_commute double_complement tiling.Un)
+qed
+
+
 text {* \medskip Chess boards *}
 
 lemma Sigma_Suc1 [simp]:
-     "lessThan (Suc n) \<times> B = ({n} \<times> B) \<union> ((lessThan n) \<times> B)"
+     "{0..< Suc n} \<times> B = ({n} \<times> B) \<union> ({0..<n} \<times> B)"
   by auto
 
 lemma Sigma_Suc2 [simp]:
-     "A \<times> lessThan (Suc n) = (A \<times> {n}) \<union> (A \<times> (lessThan n))"
+     "A \<times> {0..< Suc n} = (A \<times> {n}) \<union> (A \<times> {0..<n})"
   by auto
 
-lemma sing_Times_lemma: "({i} \<times> {n}) \<union> ({i} \<times> {m}) = {(i, m), (i, n)}"
-  by auto
-
-lemma dominoes_tile_row [intro!]: "{i} \<times> lessThan (2 * n) \<in> tiling domino"
-  apply (induct n)
-   apply (simp_all add: Un_assoc [symmetric])
-  apply (rule tiling.Un)
-    apply (auto simp add: sing_Times_lemma)
-  done
-
-lemma dominoes_tile_matrix: "(lessThan m) \<times> lessThan (2 * n) \<in> tiling domino"
+lemma dominoes_tile_row [intro!]: "{i} \<times> {0..< 2*n} \<in> tiling domino"
+apply (induct n)
+apply (simp_all del:Un_insert_left add: Un_assoc [symmetric])
+done
+
+lemma dominoes_tile_matrix: "{0..<m} \<times> {0..< 2*n} \<in> tiling domino"
   by (induct m) auto
 
 
 text {* \medskip @{term coloured} and Dominoes *}
 

 
 
 text {* \medskip Final argument is surprisingly complex *}
 
 theorem gen_mutil_not_tiling:
-       "t \<in> tiling domino ==>
-	 (i + j) mod 2 = 0 ==> (m + n) mod 2 = 0 ==>
-	 {(i, j), (m, n)} \<subseteq> t
-       ==> (t - {(i, j)} - {(m, n)}) \<notin> tiling domino"
-  apply (rule notI)
-  apply (subgoal_tac
-          "card (whites \<inter> (t - {(i, j)} - {(m, n)})) <
-           card (blacks \<inter> (t - {(i, j)} - {(m, n)}))")
-   apply (force simp only: tiling_domino_0_1)
-  apply (simp add: tiling_domino_0_1 [symmetric])
-  apply (simp add: coloured_def card_Diff2_less)
-  done
+  "t \<in> tiling domino ==>
+  (i + j) mod 2 = 0 ==> (m + n) mod 2 = 0 ==>
+  {(i, j), (m, n)} \<subseteq> t
+  ==> (t - {(i,j)} - {(m,n)}) \<notin> tiling domino"
+apply (rule notI)
+apply (subgoal_tac
+  "card (whites \<inter> (t - {(i,j)} - {(m,n)})) <
+   card (blacks \<inter> (t - {(i,j)} - {(m,n)}))")
+ apply (force simp only: tiling_domino_0_1)
+apply (simp add: tiling_domino_0_1 [symmetric])
+apply (simp add: coloured_def card_Diff2_less)
+done
 
 text {* Apply the general theorem to the well-known case *}
 
 theorem mutil_not_tiling:
-       "t = lessThan (2 * Suc m) \<times> lessThan (2 * Suc n)
-	 ==> t - {(0, 0)} - {(Suc (2 * m), Suc (2 * n))} \<notin> tiling domino"
-  apply (rule gen_mutil_not_tiling)
-     apply (blast intro!: dominoes_tile_matrix)
-    apply auto
-  done
+  "t = {0..< 2 * Suc m} \<times> {0..< 2 * Suc n}
+   ==> t - {(0,0)} - {(Suc(2 * m), Suc(2 * n))} \<notin> tiling domino"
+apply (rule gen_mutil_not_tiling)
+ apply (blast intro!: dominoes_tile_matrix)
+apply auto
+done
+
+
+subsection{* The Mutilated Chess Board Can be Tiled by Ls *}
+
+text{* We remove any square from a chess board of size $2^n \times 2^n$.
+The result can be tiled by L-shaped tiles.
+Found in Velleman's \emph{How to Prove it}.
+
+The four possible L-shaped tiles are obtained by dropping
+one of the four squares from $\{(x,y),(x+1,y),(x,y+1),(x+1,y+1)\}$: *}
+
+definition "L2 (x::nat) (y::nat) = {(x,y), (x+1,y), (x, y+1)}"
+definition "L3 (x::nat) (y::nat) = {(x,y), (x+1,y), (x+1, y+1)}"
+definition "L0 (x::nat) (y::nat) = {(x+1,y), (x,y+1), (x+1, y+1)}"
+definition "L1 (x::nat) (y::nat) = {(x,y), (x,y+1), (x+1, y+1)}"
+
+text{* All tiles: *}
+
+definition Ls :: "(nat * nat) set set" where
+"Ls \<equiv> { L0 x y | x y. True} \<union> { L1 x y | x y. True} \<union>
+      { L2 x y | x y. True} \<union> { L3 x y | x y. True}"
+
+lemma LinLs: "L0 i j : Ls & L1 i j : Ls & L2 i j : Ls & L3 i j : Ls"
+by(fastsimp simp:Ls_def)
+
+
+text{* Square $2^n \times 2^n$ grid, shifted by $i$ and $j$: *}
+
+definition "square2 (n::nat) (i::nat) (j::nat) = {i..< 2^n+i} \<times> {j..< 2^n+j}"
+
+lemma in_square2[simp]:
+  "(a,b) : square2 n i j \<longleftrightarrow> i\<le>a \<and> a<2^n+i \<and> j\<le>b \<and> b<2^n+j"
+by(simp add:square2_def)
+
+lemma square2_Suc: "square2 (Suc n) i j =
+  square2 n i j \<union> square2 n (2^n + i) j \<union> square2 n i (2^n + j) \<union>
+  square2 n (2^n + i) (2^n + j)"
+by(auto simp:square2_def)
+
+lemma square2_disj: "square2 n i j \<inter> square2 n x y = {} \<longleftrightarrow>
+  (2^n+i \<le> x \<or> 2^n+x \<le> i) \<or> (2^n+j \<le> y \<or> 2^n+y \<le> j)" (is "?A = ?B")
+proof-
+  { assume ?B hence ?A by(auto simp:square2_def) }
+  moreover
+  { assume "\<not> ?B"
+    hence "(max i x, max j y) : square2 n i j \<inter> square2 n x y" by simp
+    hence "\<not> ?A" by blast }
+  ultimately show ?thesis by blast
+qed
+
+text{* Some specific lemmas: *}
+
+lemma pos_pow2: "(0::nat) < 2^(n::nat)"
+by simp
+
+declare nat_zero_less_power_iff[simp del] zero_less_power[simp del]
+
+lemma Diff_insert_if: shows
+  "B \<noteq> {} \<Longrightarrow> a:A \<Longrightarrow> A - insert a B = (A-B - {a})" and
+  "B \<noteq> {} \<Longrightarrow> a ~: A \<Longrightarrow> A - insert a B = A-B"
+by auto
+
+lemma DisjI1: "A Int B = {} \<Longrightarrow> (A-X) Int B = {}"
+by blast
+lemma DisjI2: "A Int B = {} \<Longrightarrow> A Int (B-X) = {}"
+by blast
+
+text{* The main theorem: *}
+
+theorem Ls_can_tile: "i \<le> a \<Longrightarrow> a < 2^n + i \<Longrightarrow> j \<le> b \<Longrightarrow> b < 2^n + j
+  \<Longrightarrow> square2 n i j - {(a,b)} : tiling Ls"
+proof(induct n arbitrary: a b i j)
+  case 0 thus ?case by (simp add:square2_def)
+next
+  case (Suc n) note IH = Suc(1) and a = Suc(2-3) and b = Suc(4-5)
+  hence "a<2^n+i \<and> b<2^n+j \<or>
+         2^n+i\<le>a \<and> a<2^(n+1)+i \<and> b<2^n+j \<or>
+         a<2^n+i \<and> 2^n+j\<le>b \<and> b<2^(n+1)+j \<or>
+         2^n+i\<le>a \<and> a<2^(n+1)+i \<and> 2^n+j\<le>b \<and> b<2^(n+1)+j" (is "?A|?B|?C|?D")
+    by simp arith
+  moreover
+  { assume "?A"
+    hence "square2 n i j - {(a,b)} : tiling Ls" using IH a b by auto
+    moreover have "square2 n (2^n+i) j - {(2^n+i,2^n+j - 1)} : tiling Ls"
+      by(rule IH)(insert pos_pow2[of n], auto)
+    moreover have "square2 n i (2^n+j) - {(2^n+i - 1, 2^n+j)} : tiling Ls"
+      by(rule IH)(insert pos_pow2[of n], auto)
+    moreover have "square2 n (2^n+i) (2^n+j) - {(2^n+i, 2^n+j)} : tiling Ls"
+      by(rule IH)(insert pos_pow2[of n], auto)
+    ultimately
+    have "square2 (n+1) i j - {(a,b)} - L0 (2^n+i - 1) (2^n+j - 1) \<in> tiling Ls"
+      using  a b `?A`
+      by (clarsimp simp: square2_Suc L0_def Un_Diff Diff_insert_if)
+         (fastsimp intro!: tiling_UnI DisjI1 DisjI2 square2_disj[THEN iffD2]
+                   simp:Int_Un_distrib2)
+  } moreover
+  { assume "?B"
+    hence "square2 n (2^n+i) j - {(a,b)} : tiling Ls" using IH a b by auto
+    moreover have "square2 n i j - {(2^n+i - 1,2^n+j - 1)} : tiling Ls"
+      by(rule IH)(insert pos_pow2[of n], auto)
+    moreover have "square2 n i (2^n+j) - {(2^n+i - 1, 2^n+j)} : tiling Ls"
+      by(rule IH)(insert pos_pow2[of n], auto)
+    moreover have "square2 n (2^n+i) (2^n+j) - {(2^n+i, 2^n+j)} : tiling Ls"
+      by(rule IH)(insert pos_pow2[of n], auto)
+    ultimately
+    have "square2 (n+1) i j - {(a,b)} - L1 (2^n+i - 1) (2^n+j - 1) \<in> tiling Ls"
+      using  a b `?B`
+      by (simp add: square2_Suc L1_def Un_Diff Diff_insert_if le_diff_conv2)
+         (fastsimp intro!: tiling_UnI DisjI1 DisjI2 square2_disj[THEN iffD2]
+                   simp:Int_Un_distrib2)
+  } moreover
+  { assume "?C"
+    hence "square2 n i (2^n+j) - {(a,b)} : tiling Ls" using IH a b by auto
+    moreover have "square2 n i j - {(2^n+i - 1,2^n+j - 1)} : tiling Ls"
+      by(rule IH)(insert pos_pow2[of n], auto)
+    moreover have "square2 n (2^n+i) j - {(2^n+i, 2^n+j - 1)} : tiling Ls"
+      by(rule IH)(insert pos_pow2[of n], auto)
+    moreover have "square2 n (2^n+i) (2^n+j) - {(2^n+i, 2^n+j)} : tiling Ls"
+      by(rule IH)(insert pos_pow2[of n], auto)
+    ultimately
+    have "square2 (n+1) i j - {(a,b)} - L3 (2^n+i - 1) (2^n+j - 1) \<in> tiling Ls"
+      using  a b `?C`
+      by (simp add: square2_Suc L3_def Un_Diff Diff_insert_if le_diff_conv2)
+         (fastsimp intro!: tiling_UnI DisjI1 DisjI2 square2_disj[THEN iffD2]
+                   simp:Int_Un_distrib2)
+  } moreover
+  { assume "?D"
+    hence "square2 n (2^n+i) (2^n+j) -{(a,b)} : tiling Ls" using IH a b by auto
+    moreover have "square2 n i j - {(2^n+i - 1,2^n+j - 1)} : tiling Ls"
+      by(rule IH)(insert pos_pow2[of n], auto)
+    moreover have "square2 n (2^n+i) j - {(2^n+i, 2^n+j - 1)} : tiling Ls"
+      by(rule IH)(insert pos_pow2[of n], auto)
+    moreover have "square2 n i (2^n+j) - {(2^n+i - 1, 2^n+j)} : tiling Ls"
+      by(rule IH)(insert pos_pow2[of n], auto)
+    ultimately
+    have "square2 (n+1) i j - {(a,b)} - L2 (2^n+i - 1) (2^n+j - 1) \<in> tiling Ls"
+      using  a b `?D`
+      by (simp add: square2_Suc L2_def Un_Diff Diff_insert_if le_diff_conv2)
+         (fastsimp intro!: tiling_UnI DisjI1 DisjI2 square2_disj[THEN iffD2]
+                   simp:Int_Un_distrib2)
+  } moreover
+  have "?A \<Longrightarrow> L0 (2^n + i - 1) (2^n + j - 1) \<subseteq> square2 (n+1) i j - {(a, b)}"
+    using a b by(simp add:L0_def) arith moreover
+  have "?B \<Longrightarrow> L1 (2^n + i - 1) (2^n + j - 1) \<subseteq> square2 (n+1) i j - {(a, b)}"
+    using a b by(simp add:L1_def) arith moreover
+  have "?C \<Longrightarrow> L3 (2^n + i - 1) (2^n + j - 1) \<subseteq> square2 (n+1) i j - {(a, b)}"
+    using a b by(simp add:L3_def) arith moreover
+  have "?D \<Longrightarrow> L2 (2^n + i - 1) (2^n + j - 1) \<subseteq> square2 (n+1) i j - {(a, b)}"
+    using a b by(simp add:L2_def) arith
+  ultimately show ?case
+    apply simp
+    apply(erule disjE)
+    apply (metis LinLs tiling_Diff1E)
+    apply (metis LinLs tiling_Diff1E)
+    done
+qed
+
+corollary Ls_can_tile00:
+  "a < 2^n \<Longrightarrow> b < 2^n \<Longrightarrow> square2 n 0 0 - {(a, b)} \<in> tiling Ls"
+by(rule Ls_can_tile) auto
 
 end