author | wenzelm |
Tue, 13 Sep 2005 22:19:23 +0200 | |
changeset 17339 | ab97ccef124a |
parent 16417 | 9bc16273c2d4 |
child 18153 | a084aa91f701 |
permissions | -rw-r--r-- |
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(* Title: HOL/Isar_examples/MutilatedCheckerboard.thy |
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ID: $Id$ |
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Author: Markus Wenzel, TU Muenchen (Isar document) |
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Lawrence C Paulson, Cambridge University Computer Laboratory (original scripts) |
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*) |
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header {* The Mutilated Checker Board Problem *} |
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theory MutilatedCheckerboard imports Main begin |
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text {* |
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The Mutilated Checker Board Problem, formalized inductively. See |
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\cite{paulson-mutilated-board} and |
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\url{http://isabelle.in.tum.de/library/HOL/Induct/Mutil.html} for the |
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original tactic script version. |
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*} |
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subsection {* Tilings *} |
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consts |
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tiling :: "'a set set => 'a set set" |
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inductive "tiling A" |
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intros |
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empty: "{} : tiling A" |
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Un: "a : A ==> t : tiling A ==> a <= - t ==> a Un t : tiling A" |
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text "The union of two disjoint tilings is a tiling." |
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lemma tiling_Un: |
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"t : tiling A ==> u : tiling A ==> t Int u = {} |
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==> t Un u : tiling A" |
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proof - |
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let ?T = "tiling A" |
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assume u: "u : ?T" |
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assume "t : ?T" |
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thus "t Int u = {} ==> t Un u : ?T" (is "PROP ?P t") |
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proof (induct t) |
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case empty |
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with u show "{} Un u : ?T" by simp |
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next |
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case (Un a t) |
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show "(a Un t) Un u : ?T" |
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proof - |
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have "a Un (t Un u) : ?T" |
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proof (rule tiling.Un) |
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show "a : A" . |
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have atu: "(a Un t) Int u = {}" . |
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hence "t Int u = {}" by blast |
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thus "t Un u: ?T" by (rule Un) |
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have "a <= - t" . |
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with atu show "a <= - (t Un u)" by blast |
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qed |
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also have "a Un (t Un u) = (a Un t) Un u" |
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by (simp only: Un_assoc) |
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finally show ?thesis . |
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qed |
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qed |
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qed |
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subsection {* Basic properties of ``below'' *} |
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constdefs |
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below :: "nat => nat set" |
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"below n == {i. i < n}" |
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lemma below_less_iff [iff]: "(i: below k) = (i < k)" |
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by (simp add: below_def) |
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lemma below_0: "below 0 = {}" |
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by (simp add: below_def) |
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lemma Sigma_Suc1: |
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"m = n + 1 ==> below m <*> B = ({n} <*> B) Un (below n <*> B)" |
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by (simp add: below_def less_Suc_eq) blast |
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lemma Sigma_Suc2: |
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"m = n + 2 ==> A <*> below m = |
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(A <*> {n}) Un (A <*> {n + 1}) Un (A <*> below n)" |
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by (auto simp add: below_def) |
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lemmas Sigma_Suc = Sigma_Suc1 Sigma_Suc2 |
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subsection {* Basic properties of ``evnodd'' *} |
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constdefs |
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evnodd :: "(nat * nat) set => nat => (nat * nat) set" |
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"evnodd A b == A Int {(i, j). (i + j) mod 2 = b}" |
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lemma evnodd_iff: |
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"(i, j): evnodd A b = ((i, j): A & (i + j) mod 2 = b)" |
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by (simp add: evnodd_def) |
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lemma evnodd_subset: "evnodd A b <= A" |
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by (unfold evnodd_def, rule Int_lower1) |
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lemma evnoddD: "x : evnodd A b ==> x : A" |
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by (rule subsetD, rule evnodd_subset) |
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lemma evnodd_finite: "finite A ==> finite (evnodd A b)" |
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by (rule finite_subset, rule evnodd_subset) |
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lemma evnodd_Un: "evnodd (A Un B) b = evnodd A b Un evnodd B b" |
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by (unfold evnodd_def) blast |
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lemma evnodd_Diff: "evnodd (A - B) b = evnodd A b - evnodd B b" |
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by (unfold evnodd_def) blast |
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lemma evnodd_empty: "evnodd {} b = {}" |
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by (simp add: evnodd_def) |
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lemma evnodd_insert: "evnodd (insert (i, j) C) b = |
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(if (i + j) mod 2 = b |
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then insert (i, j) (evnodd C b) else evnodd C b)" |
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by (simp add: evnodd_def) blast |
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subsection {* Dominoes *} |
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consts |
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domino :: "(nat * nat) set set" |
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inductive domino |
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intros |
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horiz: "{(i, j), (i, j + 1)} : domino" |
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vertl: "{(i, j), (i + 1, j)} : domino" |
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lemma dominoes_tile_row: |
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"{i} <*> below (2 * n) : tiling domino" |
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(is "?B n : ?T") |
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proof (induct n) |
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case 0 |
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show ?case by (simp add: below_0 tiling.empty) |
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next |
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case (Suc n) |
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let ?a = "{i} <*> {2 * n + 1} Un {i} <*> {2 * n}" |
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have "?B (Suc n) = ?a Un ?B n" |
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by (auto simp add: Sigma_Suc Un_assoc) |
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also have "... : ?T" |
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proof (rule tiling.Un) |
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have "{(i, 2 * n), (i, 2 * n + 1)} : domino" |
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by (rule domino.horiz) |
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also have "{(i, 2 * n), (i, 2 * n + 1)} = ?a" by blast |
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finally show "... : domino" . |
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show "?B n : ?T" by (rule Suc) |
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show "?a <= - ?B n" by blast |
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qed |
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finally show ?case . |
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qed |
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lemma dominoes_tile_matrix: |
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"below m <*> below (2 * n) : tiling domino" |
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(is "?B m : ?T") |
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proof (induct m) |
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case 0 |
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show ?case by (simp add: below_0 tiling.empty) |
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next |
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case (Suc m) |
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let ?t = "{m} <*> below (2 * n)" |
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have "?B (Suc m) = ?t Un ?B m" by (simp add: Sigma_Suc) |
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also have "... : ?T" |
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proof (rule tiling_Un) |
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show "?t : ?T" by (rule dominoes_tile_row) |
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show "?B m : ?T" by (rule Suc) |
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show "?t Int ?B m = {}" by blast |
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qed |
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finally show ?case . |
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qed |
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lemma domino_singleton: |
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"d : domino ==> b < 2 ==> EX i j. evnodd d b = {(i, j)}" |
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proof - |
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assume b: "b < 2" |
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assume "d : domino" |
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thus ?thesis (is "?P d") |
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proof induct |
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from b have b_cases: "b = 0 | b = 1" by arith |
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fix i j |
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note [simp] = evnodd_empty evnodd_insert mod_Suc |
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from b_cases show "?P {(i, j), (i, j + 1)}" by rule auto |
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from b_cases show "?P {(i, j), (i + 1, j)}" by rule auto |
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qed |
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qed |
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lemma domino_finite: "d: domino ==> finite d" |
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proof - |
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assume "d: domino" |
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thus ?thesis |
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proof induct |
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fix i j :: nat |
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show "finite {(i, j), (i, j + 1)}" by (intro Finites.intros) |
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show "finite {(i, j), (i + 1, j)}" by (intro Finites.intros) |
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qed |
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qed |
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subsection {* Tilings of dominoes *} |
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lemma tiling_domino_finite: |
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"t : tiling domino ==> finite t" (is "t : ?T ==> ?F t") |
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proof - |
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assume "t : ?T" |
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thus "?F t" |
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proof induct |
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show "?F {}" by (rule Finites.emptyI) |
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fix a t assume "?F t" |
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assume "a : domino" hence "?F a" by (rule domino_finite) |
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thus "?F (a Un t)" by (rule finite_UnI) |
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qed |
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qed |
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lemma tiling_domino_01: |
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"t : tiling domino ==> card (evnodd t 0) = card (evnodd t 1)" |
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(is "t : ?T ==> _") |
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proof - |
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assume "t : ?T" |
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thus ?thesis |
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proof induct |
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case empty |
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show ?case by (simp add: evnodd_def) |
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next |
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case (Un a t) |
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let ?e = evnodd |
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have hyp: "card (?e t 0) = card (?e t 1)" . |
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have at: "a <= - t" . |
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have card_suc: |
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"!!b. b < 2 ==> card (?e (a Un t) b) = Suc (card (?e t b))" |
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proof - |
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fix b :: nat assume "b < 2" |
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have "?e (a Un t) b = ?e a b Un ?e t b" by (rule evnodd_Un) |
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also obtain i j where e: "?e a b = {(i, j)}" |
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proof - |
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have "EX i j. ?e a b = {(i, j)}" by (rule domino_singleton) |
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thus ?thesis by (blast intro: that) |
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qed |
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also have "... Un ?e t b = insert (i, j) (?e t b)" by simp |
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also have "card ... = Suc (card (?e t b))" |
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proof (rule card_insert_disjoint) |
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show "finite (?e t b)" |
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by (rule evnodd_finite, rule tiling_domino_finite) |
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from e have "(i, j) : ?e a b" by simp |
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with at show "(i, j) ~: ?e t b" by (blast dest: evnoddD) |
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qed |
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finally show "?thesis b" . |
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qed |
|
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hence "card (?e (a Un t) 0) = Suc (card (?e t 0))" by simp |
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also from hyp have "card (?e t 0) = card (?e t 1)" . |
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also from card_suc have "Suc ... = card (?e (a Un t) 1)" |
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by simp |
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finally show ?case . |
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qed |
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qed |
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subsection {* Main theorem *} |
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constdefs |
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mutilated_board :: "nat => nat => (nat * nat) set" |
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"mutilated_board m n == |
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below (2 * (m + 1)) <*> below (2 * (n + 1)) |
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- {(0, 0)} - {(2 * m + 1, 2 * n + 1)}" |
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theorem mutil_not_tiling: "mutilated_board m n ~: tiling domino" |
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proof (unfold mutilated_board_def) |
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let ?T = "tiling domino" |
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let ?t = "below (2 * (m + 1)) <*> below (2 * (n + 1))" |
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let ?t' = "?t - {(0, 0)}" |
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let ?t'' = "?t' - {(2 * m + 1, 2 * n + 1)}" |
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show "?t'' ~: ?T" |
274 |
proof |
|
275 |
have t: "?t : ?T" by (rule dominoes_tile_matrix) |
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assume t'': "?t'' : ?T" |
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let ?e = evnodd |
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have fin: "finite (?e ?t 0)" |
|
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by (rule evnodd_finite, rule tiling_domino_finite, rule t) |
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note [simp] = evnodd_iff evnodd_empty evnodd_insert evnodd_Diff |
283 |
have "card (?e ?t'' 0) < card (?e ?t' 0)" |
|
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proof - |
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have "card (?e ?t' 0 - {(2 * m + 1, 2 * n + 1)}) |
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< card (?e ?t' 0)" |
287 |
proof (rule card_Diff1_less) |
|
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from _ fin show "finite (?e ?t' 0)" |
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by (rule finite_subset) auto |
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show "(2 * m + 1, 2 * n + 1) : ?e ?t' 0" by simp |
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qed |
292 |
thus ?thesis by simp |
|
293 |
qed |
|
294 |
also have "... < card (?e ?t 0)" |
|
295 |
proof - |
|
296 |
have "(0, 0) : ?e ?t 0" by simp |
|
297 |
with fin have "card (?e ?t 0 - {(0, 0)}) < card (?e ?t 0)" |
|
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by (rule card_Diff1_less) |
|
299 |
thus ?thesis by simp |
|
300 |
qed |
|
301 |
also from t have "... = card (?e ?t 1)" |
|
302 |
by (rule tiling_domino_01) |
|
303 |
also have "?e ?t 1 = ?e ?t'' 1" by simp |
|
304 |
also from t'' have "card ... = card (?e ?t'' 0)" |
|
305 |
by (rule tiling_domino_01 [symmetric]) |
|
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finally have "... < ..." . thus False .. |
|
307 |
qed |
|
308 |
qed |
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end |