author | haftmann |
Fri, 17 Jun 2005 16:12:49 +0200 | |
changeset 16417 | 9bc16273c2d4 |
parent 14276 | 950c12139016 |
child 18242 | 2215049cd29c |
permissions | -rw-r--r-- |
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(* Title: HOL/Induct/Mutil.thy |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1996 University of Cambridge |
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*) |
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header {* The Mutilated Chess Board Problem *} |
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theory Mutil imports Main begin |
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text {* |
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The Mutilated Chess Board Problem, formalized inductively. |
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Originator is Max Black, according to J A Robinson. Popularized as |
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the Mutilated Checkerboard Problem by J McCarthy. |
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*} |
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consts tiling :: "'a set set => 'a set set" |
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inductive "tiling A" |
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intros |
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empty [simp, intro]: "{} \<in> tiling A" |
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Un [simp, intro]: "[| a \<in> A; t \<in> tiling A; a \<inter> t = {} |] |
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==> a \<union> t \<in> tiling A" |
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consts domino :: "(nat \<times> nat) set set" |
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inductive domino |
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intros |
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horiz [simp]: "{(i, j), (i, Suc j)} \<in> domino" |
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vertl [simp]: "{(i, j), (Suc i, j)} \<in> domino" |
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text {* \medskip Sets of squares of the given colour*} |
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constdefs |
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coloured :: "nat => (nat \<times> nat) set" |
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"coloured b == {(i, j). (i + j) mod 2 = b}" |
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syntax whites :: "(nat \<times> nat) set" |
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blacks :: "(nat \<times> nat) set" |
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translations |
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"whites" == "coloured 0" |
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"blacks" == "coloured (Suc 0)" |
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text {* \medskip The union of two disjoint tilings is a tiling *} |
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lemma tiling_UnI [intro]: |
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"[|t \<in> tiling A; u \<in> tiling A; t \<inter> u = {} |] ==> t \<union> u \<in> tiling A" |
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apply (induct set: tiling) |
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apply (auto simp add: Un_assoc) |
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done |
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text {* \medskip Chess boards *} |
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lemma Sigma_Suc1 [simp]: |
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"lessThan (Suc n) \<times> B = ({n} \<times> B) \<union> ((lessThan n) \<times> B)" |
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by (auto simp add: lessThan_def) |
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lemma Sigma_Suc2 [simp]: |
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"A \<times> lessThan (Suc n) = (A \<times> {n}) \<union> (A \<times> (lessThan n))" |
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by (auto simp add: lessThan_def) |
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lemma sing_Times_lemma: "({i} \<times> {n}) \<union> ({i} \<times> {m}) = {(i, m), (i, n)}" |
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by auto |
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lemma dominoes_tile_row [intro!]: "{i} \<times> lessThan (2 * n) \<in> tiling domino" |
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apply (induct n) |
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apply (simp_all add: Un_assoc [symmetric]) |
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apply (rule tiling.Un) |
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apply (auto simp add: sing_Times_lemma) |
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done |
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lemma dominoes_tile_matrix: "(lessThan m) \<times> lessThan (2 * n) \<in> tiling domino" |
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by (induct m, auto) |
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text {* \medskip @{term coloured} and Dominoes *} |
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lemma coloured_insert [simp]: |
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"coloured b \<inter> (insert (i, j) t) = |
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(if (i + j) mod 2 = b then insert (i, j) (coloured b \<inter> t) |
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else coloured b \<inter> t)" |
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by (auto simp add: coloured_def) |
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lemma domino_singletons: |
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"d \<in> domino ==> |
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(\<exists>i j. whites \<inter> d = {(i, j)}) \<and> |
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(\<exists>m n. blacks \<inter> d = {(m, n)})"; |
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apply (erule domino.cases) |
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apply (auto simp add: mod_Suc) |
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done |
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lemma domino_finite [simp]: "d \<in> domino ==> finite d" |
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by (erule domino.cases, auto) |
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text {* \medskip Tilings of dominoes *} |
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lemma tiling_domino_finite [simp]: "t \<in> tiling domino ==> finite t" |
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by (induct set: tiling, auto) |
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declare |
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Int_Un_distrib [simp] |
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Diff_Int_distrib [simp] |
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lemma tiling_domino_0_1: |
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"t \<in> tiling domino ==> card(whites \<inter> t) = card(blacks \<inter> t)" |
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apply (induct set: tiling) |
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apply (drule_tac [2] domino_singletons) |
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apply auto |
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apply (subgoal_tac "\<forall>p C. C \<inter> a = {p} --> p \<notin> t") |
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-- {* this lemma tells us that both ``inserts'' are non-trivial *} |
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apply (simp (no_asm_simp)) |
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apply blast |
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done |
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text {* \medskip Final argument is surprisingly complex *} |
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theorem gen_mutil_not_tiling: |
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"t \<in> tiling domino ==> |
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(i + j) mod 2 = 0 ==> (m + n) mod 2 = 0 ==> |
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{(i, j), (m, n)} \<subseteq> t |
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==> (t - {(i, j)} - {(m, n)}) \<notin> tiling domino" |
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apply (rule notI) |
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apply (subgoal_tac |
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"card (whites \<inter> (t - {(i, j)} - {(m, n)})) < |
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card (blacks \<inter> (t - {(i, j)} - {(m, n)}))") |
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apply (force simp only: tiling_domino_0_1) |
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apply (simp add: tiling_domino_0_1 [symmetric]) |
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apply (simp add: coloured_def card_Diff2_less) |
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done |
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text {* Apply the general theorem to the well-known case *} |
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theorem mutil_not_tiling: |
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"t = lessThan (2 * Suc m) \<times> lessThan (2 * Suc n) |
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==> t - {(0, 0)} - {(Suc (2 * m), Suc (2 * n))} \<notin> tiling domino" |
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apply (rule gen_mutil_not_tiling) |
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apply (blast intro!: dominoes_tile_matrix) |
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apply auto |
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done |
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end |