author | wenzelm |
Sat, 03 Feb 2001 17:40:16 +0100 | |
changeset 11046 | b5f5942781a0 |
parent 9930 | c02d48a47ed1 |
child 11464 | ddea204de5bc |
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 = Main: |
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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 [simp, intro] |
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empty: "{} \<in> tiling A" |
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Un: "a \<in> A ==> t \<in> tiling A ==> a \<inter> t = {} ==> 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 [simp] |
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horiz: "{(i, j), (i, Suc j)} \<in> domino" |
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vertl: "{(i, j), (Suc i, j)} \<in> domino" |
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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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text {* \medskip The union of two disjoint tilings is a tiling *} |
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lemma tiling_UnI [rule_format, 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 (erule tiling.induct) |
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prefer 2 |
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apply (simp add: Un_assoc) |
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apply auto |
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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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apply (unfold lessThan_def) |
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apply auto |
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done |
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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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apply (unfold lessThan_def) |
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apply auto |
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done |
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lemma sing_Times_lemma: "({i} \<times> {n}) \<union> ({i} \<times> {m}) = {(i, m), (i, n)}" |
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apply auto |
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done |
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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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apply (induct m) |
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apply auto |
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done |
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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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apply (unfold coloured_def) |
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apply auto |
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done |
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lemma domino_singletons: |
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"d \<in> domino ==> |
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(\<exists>i j. coloured 0 \<inter> d = {(i, j)}) \<and> |
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(\<exists>m n. coloured 1 \<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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apply (erule domino.cases) |
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apply auto |
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done |
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text {* \medskip Tilings of dominoes *} |
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|
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lemma tiling_domino_finite [simp]: "t \<in> tiling domino ==> finite t" |
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apply (erule tiling.induct) |
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apply auto |
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done |
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|
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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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|
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lemma tiling_domino_0_1: |
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"t \<in> tiling domino ==> card (coloured 0 \<inter> t) = card (coloured 1 \<inter> t)" |
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apply (erule tiling.induct) |
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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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|
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text {* \medskip Final argument is surprisingly complex *} |
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|
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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 (coloured 0 \<inter> (t - {(i, j)} - {(m, n)})) < |
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card (coloured 1 \<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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|
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text {* Apply the general theorem to the well-known case *} |
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141 |
|
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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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146 |
apply (blast intro!: dominoes_tile_matrix) |
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147 |
apply auto |
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148 |
done |
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
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149 |
|
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New directory to contain examples of (co)inductive definitions
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end |