author | wenzelm |
Sat, 30 Oct 1999 20:20:48 +0200 | |
changeset 7982 | d534b897ce39 |
parent 7968 | 964b65b4e433 |
child 8281 | 188e2924433e |
permissions | -rw-r--r-- |
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The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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(* Title: HOL/Isar_examples/MutilatedCheckerboard.thy |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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ID: $Id$ |
7385 | 3 |
Author: Markus Wenzel, TU Muenchen (Isar document) |
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Lawrence C Paulson, Cambridge University Computer Laboratory (original scripts) |
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7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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*) |
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The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
7761 | 7 |
header {* The Mutilated Checker Board Problem *}; |
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||
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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theory MutilatedCheckerboard = Main:; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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7968 | 11 |
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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15 |
original tactic script version. |
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*}; |
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The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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7761 | 18 |
subsection {* Tilings *}; |
7382
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The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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19 |
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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consts |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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tiling :: "'a set set => 'a set set"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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22 |
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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23 |
inductive "tiling A" |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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24 |
intrs |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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25 |
empty: "{} : tiling A" |
7800 | 26 |
Un: "[| a : A; t : tiling A; a <= - t |] |
27 |
==> a Un t : tiling A"; |
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7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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28 |
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The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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29 |
|
7800 | 30 |
text "The union of two disjoint tilings is a tiling."; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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31 |
|
7761 | 32 |
lemma tiling_Un: |
7800 | 33 |
"t : tiling A --> u : tiling A --> t Int u = {} |
34 |
--> t Un u : tiling A"; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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35 |
proof; |
7480 | 36 |
assume "t : tiling A" (is "_ : ?T"); |
37 |
thus "u : ?T --> t Int u = {} --> t Un u : ?T" (is "?P t"); |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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38 |
proof (induct t set: tiling); |
7480 | 39 |
show "?P {}"; by simp; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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40 |
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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41 |
fix a t; |
7480 | 42 |
assume "a : A" "t : ?T" "?P t" "a <= - t"; |
43 |
show "?P (a Un t)"; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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44 |
proof (intro impI); |
7480 | 45 |
assume "u : ?T" "(a Un t) Int u = {}"; |
7565 | 46 |
have hyp: "t Un u: ?T"; by (blast!); |
47 |
have "a <= - (t Un u)"; by (blast!); |
|
7480 | 48 |
with _ hyp; have "a Un (t Un u) : ?T"; by (rule tiling.Un); |
7761 | 49 |
also; have "a Un (t Un u) = (a Un t) Un u"; |
50 |
by (simp only: Un_assoc); |
|
7480 | 51 |
finally; show "... : ?T"; .; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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52 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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53 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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54 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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55 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
7874 | 57 |
subsection {* Basic properties of ``below'' *}; |
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The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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58 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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59 |
constdefs |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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below :: "nat => nat set" |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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61 |
"below n == {i. i < n}"; |
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The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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62 |
|
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The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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lemma below_less_iff [iff]: "(i: below k) = (i < k)"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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64 |
by (simp add: below_def); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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65 |
|
7385 | 66 |
lemma below_0: "below 0 = {}"; |
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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67 |
by (simp add: below_def); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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68 |
|
7761 | 69 |
lemma Sigma_Suc1: |
70 |
"below (Suc n) Times B = ({n} Times B) Un (below n Times B)"; |
|
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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71 |
by (simp add: below_def less_Suc_eq) blast; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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72 |
|
7761 | 73 |
lemma Sigma_Suc2: |
74 |
"A Times below (Suc n) = (A Times {n}) Un (A Times (below n))"; |
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The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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75 |
by (simp add: below_def less_Suc_eq) blast; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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76 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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77 |
lemmas Sigma_Suc = Sigma_Suc1 Sigma_Suc2; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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78 |
|
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The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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79 |
|
7874 | 80 |
subsection {* Basic properties of ``evnodd'' *}; |
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81 |
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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82 |
constdefs |
7385 | 83 |
evnodd :: "(nat * nat) set => nat => (nat * nat) set" |
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84 |
"evnodd A b == A Int {(i, j). (i + j) mod 2 = b}"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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85 |
|
7761 | 86 |
lemma evnodd_iff: |
87 |
"(i, j): evnodd A b = ((i, j): A & (i + j) mod 2 = b)"; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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88 |
by (simp add: evnodd_def); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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89 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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90 |
lemma evnodd_subset: "evnodd A b <= A"; |
7385 | 91 |
by (unfold evnodd_def, rule Int_lower1); |
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The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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92 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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93 |
lemma evnoddD: "x : evnodd A b ==> x : A"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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94 |
by (rule subsetD, rule evnodd_subset); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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95 |
|
7385 | 96 |
lemma evnodd_finite: "finite A ==> finite (evnodd A b)"; |
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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97 |
by (rule finite_subset, rule evnodd_subset); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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98 |
|
7385 | 99 |
lemma evnodd_Un: "evnodd (A Un B) b = evnodd A b Un evnodd B b"; |
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The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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100 |
by (unfold evnodd_def) blast; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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101 |
|
7385 | 102 |
lemma evnodd_Diff: "evnodd (A - B) b = evnodd A b - evnodd B b"; |
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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103 |
by (unfold evnodd_def) blast; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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104 |
|
7385 | 105 |
lemma evnodd_empty: "evnodd {} b = {}"; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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106 |
by (simp add: evnodd_def); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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parents:
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107 |
|
7385 | 108 |
lemma evnodd_insert: "evnodd (insert (i, j) C) b = |
7761 | 109 |
(if (i + j) mod 2 = b |
110 |
then insert (i, j) (evnodd C b) else evnodd C b)"; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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111 |
by (simp add: evnodd_def) blast; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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112 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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113 |
|
7761 | 114 |
subsection {* Dominoes *}; |
7382
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The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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115 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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116 |
consts |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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117 |
domino :: "(nat * nat) set set"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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118 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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119 |
inductive domino |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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120 |
intrs |
7385 | 121 |
horiz: "{(i, j), (i, j + 1)} : domino" |
122 |
vertl: "{(i, j), (i + 1, j)} : domino"; |
|
7382
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The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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123 |
|
7800 | 124 |
lemma dominoes_tile_row: |
125 |
"{i} Times below (2 * n) : tiling domino" |
|
7480 | 126 |
(is "?P n" is "?B n : ?T"); |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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127 |
proof (induct n); |
7480 | 128 |
show "?P 0"; by (simp add: below_0 tiling.empty); |
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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129 |
|
7480 | 130 |
fix n; assume hyp: "?P n"; |
131 |
let ?a = "{i} Times {2 * n + 1} Un {i} Times {2 * n}"; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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132 |
|
7480 | 133 |
have "?B (Suc n) = ?a Un ?B n"; by (simp add: Sigma_Suc Un_assoc); |
134 |
also; have "... : ?T"; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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135 |
proof (rule tiling.Un); |
7761 | 136 |
have "{(i, 2 * n), (i, 2 * n + 1)} : domino"; |
137 |
by (rule domino.horiz); |
|
7480 | 138 |
also; have "{(i, 2 * n), (i, 2 * n + 1)} = ?a"; by blast; |
7385 | 139 |
finally; show "... : domino"; .; |
7480 | 140 |
from hyp; show "?B n : ?T"; .; |
141 |
show "?a <= - ?B n"; by force; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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142 |
qed; |
7480 | 143 |
finally; show "?P (Suc n)"; .; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
144 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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parents:
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145 |
|
7761 | 146 |
lemma dominoes_tile_matrix: |
147 |
"below m Times below (2 * n) : tiling domino" |
|
7480 | 148 |
(is "?P m" is "?B m : ?T"); |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
149 |
proof (induct m); |
7480 | 150 |
show "?P 0"; by (simp add: below_0 tiling.empty); |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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parents:
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|
151 |
|
7480 | 152 |
fix m; assume hyp: "?P m"; |
153 |
let ?t = "{m} Times below (2 * n)"; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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parents:
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|
154 |
|
7480 | 155 |
have "?B (Suc m) = ?t Un ?B m"; by (simp add: Sigma_Suc); |
156 |
also; have "... : ?T"; |
|
7385 | 157 |
proof (rule tiling_Un [rulify]); |
7480 | 158 |
show "?t : ?T"; by (rule dominoes_tile_row); |
159 |
from hyp; show "?B m : ?T"; .; |
|
160 |
show "?t Int ?B m = {}"; by blast; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
161 |
qed; |
7480 | 162 |
finally; show "?P (Suc m)"; .; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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163 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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164 |
|
7761 | 165 |
lemma domino_singleton: |
166 |
"[| d : domino; b < 2 |] ==> EX i j. evnodd d b = {(i, j)}"; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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parents:
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|
167 |
proof -; |
7565 | 168 |
assume b: "b < 2"; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
169 |
assume "d : domino"; |
7480 | 170 |
thus ?thesis (is "?P d"); |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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parents:
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|
171 |
proof (induct d set: domino); |
7565 | 172 |
from b; have b_cases: "b = 0 | b = 1"; by arith; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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parents:
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|
173 |
fix i j; |
7385 | 174 |
note [simp] = evnodd_empty evnodd_insert mod_Suc; |
7480 | 175 |
from b_cases; show "?P {(i, j), (i, j + 1)}"; by rule auto; |
176 |
from b_cases; show "?P {(i, j), (i + 1, j)}"; by rule auto; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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changeset
|
177 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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parents:
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changeset
|
178 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
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changeset
|
179 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
180 |
lemma domino_finite: "d: domino ==> finite d"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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parents:
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|
181 |
proof (induct set: domino); |
7434 | 182 |
fix i j :: nat; |
7385 | 183 |
show "finite {(i, j), (i, j + 1)}"; by (intro Finites.intrs); |
184 |
show "finite {(i, j), (i + 1, j)}"; by (intro Finites.intrs); |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
185 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
186 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
187 |
|
7761 | 188 |
subsection {* Tilings of dominoes *}; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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changeset
|
189 |
|
7761 | 190 |
lemma tiling_domino_finite: |
191 |
"t : tiling domino ==> finite t" (is "t : ?T ==> ?F t"); |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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parents:
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changeset
|
192 |
proof -; |
7480 | 193 |
assume "t : ?T"; |
194 |
thus "?F t"; |
|
7385 | 195 |
proof (induct t set: tiling); |
7480 | 196 |
show "?F {}"; by (rule Finites.emptyI); |
197 |
fix a t; assume "?F t"; |
|
198 |
assume "a : domino"; hence "?F a"; by (rule domino_finite); |
|
199 |
thus "?F (a Un t)"; by (rule finite_UnI); |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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changeset
|
200 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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parents:
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changeset
|
201 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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parents:
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changeset
|
202 |
|
7761 | 203 |
lemma tiling_domino_01: |
204 |
"t : tiling domino ==> card (evnodd t 0) = card (evnodd t 1)" |
|
7480 | 205 |
(is "t : ?T ==> ?P t"); |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
206 |
proof -; |
7480 | 207 |
assume "t : ?T"; |
208 |
thus "?P t"; |
|
7385 | 209 |
proof (induct t set: tiling); |
7480 | 210 |
show "?P {}"; by (simp add: evnodd_def); |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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parents:
diff
changeset
|
211 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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parents:
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|
212 |
fix a t; |
7480 | 213 |
let ?e = evnodd; |
214 |
assume "a : domino" "t : ?T" |
|
215 |
and hyp: "card (?e t 0) = card (?e t 1)" |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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216 |
and "a <= - t"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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parents:
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changeset
|
217 |
|
7761 | 218 |
have card_suc: |
219 |
"!!b. b < 2 ==> card (?e (a Un t) b) = Suc (card (?e t b))"; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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parents:
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|
220 |
proof -; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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parents:
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changeset
|
221 |
fix b; assume "b < 2"; |
7480 | 222 |
have "EX i j. ?e a b = {(i, j)}"; by (rule domino_singleton); |
223 |
thus "?thesis b"; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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parents:
diff
changeset
|
224 |
proof (elim exE); |
7480 | 225 |
have "?e (a Un t) b = ?e a b Un ?e t b"; by (rule evnodd_Un); |
7874 | 226 |
also; fix i j; assume "?e a b = {(i, j)}"; |
7480 | 227 |
also; have "... Un ?e t b = insert (i, j) (?e t b)"; by simp; |
228 |
also; have "card ... = Suc (card (?e t b))"; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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parents:
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changeset
|
229 |
proof (rule card_insert_disjoint); |
7761 | 230 |
show "finite (?e t b)"; |
231 |
by (rule evnodd_finite, rule tiling_domino_finite); |
|
7565 | 232 |
have "(i, j) : ?e a b"; by (simp!); |
233 |
thus "(i, j) ~: ?e t b"; by (force! dest: evnoddD); |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
234 |
qed; |
7480 | 235 |
finally; show ?thesis; .; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
236 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
237 |
qed; |
7480 | 238 |
hence "card (?e (a Un t) 0) = Suc (card (?e t 0))"; by simp; |
239 |
also; from hyp; have "card (?e t 0) = card (?e t 1)"; .; |
|
7761 | 240 |
also; from card_suc; have "Suc ... = card (?e (a Un t) 1)"; |
241 |
by simp; |
|
7480 | 242 |
finally; show "?P (a Un t)"; .; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
243 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
244 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
245 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
246 |
|
7761 | 247 |
subsection {* Main theorem *}; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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parents:
diff
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|
248 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
249 |
constdefs |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
250 |
mutilated_board :: "nat => nat => (nat * nat) set" |
7761 | 251 |
"mutilated_board m n == |
252 |
below (2 * (m + 1)) Times below (2 * (n + 1)) |
|
253 |
- {(0, 0)} - {(2 * m + 1, 2 * n + 1)}"; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
254 |
|
7385 | 255 |
theorem mutil_not_tiling: "mutilated_board m n ~: tiling domino"; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
256 |
proof (unfold mutilated_board_def); |
7480 | 257 |
let ?T = "tiling domino"; |
258 |
let ?t = "below (2 * (m + 1)) Times below (2 * (n + 1))"; |
|
259 |
let ?t' = "?t - {(0, 0)}"; |
|
260 |
let ?t'' = "?t' - {(2 * m + 1, 2 * n + 1)}"; |
|
7761 | 261 |
|
7480 | 262 |
show "?t'' ~: ?T"; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
263 |
proof; |
7480 | 264 |
have t: "?t : ?T"; by (rule dominoes_tile_matrix); |
265 |
assume t'': "?t'' : ?T"; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
266 |
|
7480 | 267 |
let ?e = evnodd; |
7761 | 268 |
have fin: "finite (?e ?t 0)"; |
269 |
by (rule evnodd_finite, rule tiling_domino_finite, rule t); |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
270 |
|
7385 | 271 |
note [simp] = evnodd_iff evnodd_empty evnodd_insert evnodd_Diff; |
7480 | 272 |
have "card (?e ?t'' 0) < card (?e ?t' 0)"; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
273 |
proof -; |
7800 | 274 |
have "card (?e ?t' 0 - {(2 * m + 1, 2 * n + 1)}) |
275 |
< card (?e ?t' 0)"; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
276 |
proof (rule card_Diff1_less); |
7800 | 277 |
show "finite (?e ?t' 0)"; |
278 |
by (rule finite_subset, rule fin) force; |
|
7480 | 279 |
show "(2 * m + 1, 2 * n + 1) : ?e ?t' 0"; by simp; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
280 |
qed; |
7480 | 281 |
thus ?thesis; by simp; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
282 |
qed; |
7480 | 283 |
also; have "... < card (?e ?t 0)"; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
284 |
proof -; |
7480 | 285 |
have "(0, 0) : ?e ?t 0"; by simp; |
7761 | 286 |
with fin; have "card (?e ?t 0 - {(0, 0)}) < card (?e ?t 0)"; |
287 |
by (rule card_Diff1_less); |
|
7480 | 288 |
thus ?thesis; by simp; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
289 |
qed; |
7800 | 290 |
also; from t; have "... = card (?e ?t 1)"; |
291 |
by (rule tiling_domino_01); |
|
7480 | 292 |
also; have "?e ?t 1 = ?e ?t'' 1"; by simp; |
7761 | 293 |
also; from t''; have "card ... = card (?e ?t'' 0)"; |
294 |
by (rule tiling_domino_01 [RS sym]); |
|
7874 | 295 |
finally; have "... < ..."; .; thus False; ..; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
296 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
297 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
298 |
|
7383 | 299 |
end; |