author | wenzelm |
Sun, 29 Aug 1999 17:47:26 +0200 | |
changeset 7382 | 33c01075d343 |
child 7383 | 9c4ef0d3f36c |
permissions | -rw-r--r-- |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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1 |
(* Title: HOL/Isar_examples/MutilatedCheckerboard.thy |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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2 |
ID: $Id$ |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory (original script) |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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Markus Wenzel, TU Muenchen (Isar document) |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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The Mutilated Chess Board Problem, formalized inductively. |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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Originator is Max Black, according to J A Robinson. |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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8 |
Popularized as the Mutilated Checkerboard Problem by J McCarthy. |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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*) |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
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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|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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section {* Tilings *}; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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16 |
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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18 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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19 |
inductive "tiling A" |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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intrs |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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empty: "{} : tiling A" |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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Un: "[| a : A; t : tiling A; a <= - t |] ==> a Un t : tiling A"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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23 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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24 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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text "The union of two disjoint tilings is a tiling"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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lemma tiling_Un: "t : tiling A --> u : tiling A --> t Int u = {} --> t Un u : tiling A"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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28 |
proof; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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assume "t : tiling A" (is "_ : ??T"); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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thus "u : ??T --> t Int u = {} --> t Un u : ??T" (is "??P t"); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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31 |
proof (induct t set: tiling); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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32 |
show "??P {}"; by simp; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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33 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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fix a t; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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35 |
assume "a:A" "t : ??T" "??P t" "a <= - t"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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show "??P (a Un t)"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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37 |
proof (intro impI); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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38 |
assume "u : ??T" "(a Un t) Int u = {}"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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39 |
have hyp: "t Un u: ??T"; by blast; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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40 |
have "a <= - (t Un u)"; by blast; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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with _ hyp; have "a Un (t Un u) : ??T"; by (rule tiling.Un); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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42 |
also; have "a Un (t Un u) = (a Un t) Un u"; by (simp only: Un_assoc); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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finally; show "... : ??T"; .; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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44 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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47 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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lemma tiling_UnI: "[| t : tiling A; u : tiling A; t Int u = {} |] ==> t Un u : tiling A"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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by (rule tiling_Un [rulify]); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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50 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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section {* Basic properties of below *}; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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53 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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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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"below n == {i. i < n}"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
33c01075d343
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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59 |
by (simp add: below_def); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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lemma below_0 [simp]: "below 0 = {}"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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62 |
by (simp add: below_def); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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63 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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lemma Sigma_Suc1: "below (Suc n) Times B = ({n} Times B) Un (below n Times B)"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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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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66 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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lemma Sigma_Suc2: "A Times below (Suc n) = (A Times {n}) Un (A Times (below n))"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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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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69 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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lemmas Sigma_Suc = Sigma_Suc1 Sigma_Suc2; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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71 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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72 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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section {* Basic properties of evnodd *}; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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74 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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75 |
constdefs |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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evnodd :: "[(nat * nat) set, nat] => (nat * nat) set" |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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"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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78 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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lemma evnodd_iff: "(i, j): evnodd A b = ((i, j): A & (i + j) mod 2 = b)"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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80 |
by (simp add: evnodd_def); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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81 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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lemma evnodd_subset: "evnodd A b <= A"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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83 |
proof (unfold evnodd_def); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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84 |
show "!!B. A Int B <= A"; by (rule Int_lower1); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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85 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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86 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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87 |
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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88 |
by (rule subsetD, rule evnodd_subset); |
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_finite [simp]: "finite A ==> finite (evnodd A b)"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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91 |
by (rule finite_subset, rule evnodd_subset); |
33c01075d343
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 evnodd_Un [simp]: "evnodd (A Un B) b = evnodd A b Un evnodd B b"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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94 |
by (unfold evnodd_def) blast; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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95 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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96 |
lemma evnodd_Diff [simp]: "evnodd (A - B) b = evnodd A b - evnodd B b"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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97 |
by (unfold evnodd_def) blast; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
98 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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99 |
lemma evnodd_empty [simp]: "evnodd {} b = {}"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
100 |
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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101 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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102 |
lemma evnodd_insert [simp]: "evnodd (insert (i, j) C) b = |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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103 |
(if (i + j) mod 2 = b then insert (i, j) (evnodd C b) else evnodd C b)"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
104 |
by (simp add: evnodd_def) blast; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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105 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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106 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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107 |
section {* Dominoes *}; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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108 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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109 |
consts |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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110 |
domino :: "(nat * nat) set set"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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111 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
112 |
inductive domino |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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113 |
intrs |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
114 |
horiz: "{(i, j), (i, Suc j)} : domino" |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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|
115 |
vertl: "{(i, j), (Suc i, j)} : domino"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
116 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
117 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
118 |
lemma dominoes_tile_row: "{i} Times below (n + n) : tiling domino" |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
119 |
(is "??P n" is "??B n : ??T"); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
120 |
proof (induct n); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
121 |
have "??B 0 = {}"; by simp; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
122 |
also; have "... : ??T"; by (rule tiling.empty); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
123 |
finally; show "??P 0"; .; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
124 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
125 |
fix n; assume hyp: "??P n"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
126 |
let ??a = "{i} Times {Suc (n + n)} Un {i} Times {n + n}"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
127 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
128 |
have "??B (Suc n) = ??a Un ??B n"; by (simp add: Sigma_Suc Un_assoc); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
129 |
also; have "... : ??T"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
130 |
proof (rule tiling.Un); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
131 |
have "{(i, n + n), (i, Suc (n + n))} : domino"; by (rule domino.horiz); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
132 |
also; have "{(i, n + n), (i, Suc (n + n))} = ??a"; by blast; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
133 |
finally; show "??a : domino"; .; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
134 |
show "??B n : ??T"; by (rule hyp); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
135 |
show "??a <= - ??B n"; by force; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
136 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
137 |
finally; show "??P (Suc n)"; .; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
138 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
139 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
140 |
lemma dominoes_tile_matrix: "below m Times below (n + n) : tiling domino" |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
141 |
(is "??P m" is "??B m : ??T"); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
142 |
proof (induct m); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
143 |
show "??P 0"; by (simp add: tiling.empty) -- {* same as above *}; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
144 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
145 |
fix m; assume hyp: "??P m"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
146 |
let ??t = "{m} Times below (n + n)"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
147 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
148 |
have "??B (Suc m) = ??t Un ??B m"; by (simp add: Sigma_Suc); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
149 |
also; have "... : ??T"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
150 |
proof (rule tiling_UnI); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
151 |
show "??t : ??T"; by (rule dominoes_tile_row); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
152 |
show "??B m : ??T"; by (rule hyp); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
153 |
show "??t Int ??B m = {}"; by blast; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
154 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
155 |
finally; show "??P (Suc m)"; .; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
156 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
157 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
158 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
159 |
lemma domino_singleton: "[| d : domino; b < 2 |] ==> EX i j. evnodd d b = {(i, j)}"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
160 |
proof -; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
161 |
assume "b < 2"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
162 |
assume "d : domino"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
163 |
thus ??thesis (is "??P d"); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
164 |
proof (induct d set: domino); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
165 |
fix i j; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
166 |
have b_cases: "b = 0 | b = 1"; by arith; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
167 |
note [simp] = less_Suc_eq mod_Suc; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
168 |
from b_cases; show "??P {(i, j), (i, Suc j)}"; by rule auto; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
169 |
from b_cases; show "??P {(i, j), (Suc i, j)}"; by rule auto; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
170 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
171 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
172 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
173 |
lemma domino_finite: "d: domino ==> finite d"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
174 |
proof (induct set: domino); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
175 |
fix i j; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
176 |
show "finite {(i, j), (i, Suc j)}"; by (intro Finites.intrs); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
177 |
show "finite {(i, j), (Suc i, j)}"; by (intro Finites.intrs); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
178 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
179 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
180 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
181 |
section {* Tilings of dominoes *}; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
182 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
183 |
lemma tiling_domino_finite: "t : tiling domino ==> finite t" (is "t : ??T ==> ??F t"); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
184 |
proof -; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
185 |
assume "t : ??T"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
186 |
thus "??F t"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
187 |
proof (induct set: tiling); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
188 |
show "??F {}"; by (rule Finites.emptyI); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
189 |
fix a t; assume "??F t"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
190 |
assume "a : domino"; hence "??F a"; by (rule domino_finite); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
191 |
thus "??F (a Un t)"; by (rule finite_UnI); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
192 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
193 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
194 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
195 |
lemma tiling_domino_01: "t : tiling domino ==> card (evnodd t 0) = card (evnodd t 1)" |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
196 |
(is "t : ??T ==> ??P t"); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
197 |
proof -; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
198 |
assume "t : ??T"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
199 |
thus "??P t"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
200 |
proof (induct set: tiling); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
201 |
show "??P {}"; by (simp add: evnodd_def); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
202 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
203 |
fix a t; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
204 |
let ??e = evnodd; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
205 |
assume "a : domino" "t : ??T" |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
206 |
and hyp: "card (??e t 0) = card (??e t 1)" |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
207 |
and "a <= - t"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
208 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
209 |
have card_suc: "!!b. b < 2 ==> card (??e (a Un t) b) = Suc (card (??e t b))"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
210 |
proof -; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
211 |
fix b; assume "b < 2"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
212 |
have "EX i j. ??e a b = {(i, j)}"; by (rule domino_singleton); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
213 |
thus "??thesis b"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
214 |
proof (elim exE); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
215 |
have "??e (a Un t) b = ??e a b Un ??e t b"; by (rule evnodd_Un); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
216 |
also; fix i j; assume "??e a b = {(i, j)}"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
217 |
also; have "... Un ??e t b = insert (i, j) (??e t b)"; by simp; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
218 |
finally; have "card (??e (a Un t) b) = card (insert (i, j) (??e t b))"; by simp; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
219 |
also; have "... = Suc (card (??e t b))"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
220 |
proof (rule card_insert_disjoint); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
221 |
show "finite (??e t b)"; by (rule evnodd_finite, rule tiling_domino_finite); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
222 |
have "(i, j) : ??e a b"; by asm_simp; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
223 |
thus "(i, j) ~: ??e t b"; by (force dest: evnoddD); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
224 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
225 |
finally; show ??thesis; .; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
226 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
227 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
228 |
hence "card (??e (a Un t) 0) = Suc (card (??e t 0))"; by simp; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
229 |
also; have "card (??e t 0) = card (??e t 1)"; by (rule hyp); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
230 |
also; from card_suc; have "Suc ... = card (??e (a Un t) 1)"; by simp; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
231 |
finally; show "??P (a Un t)"; .; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
232 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
233 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
234 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
235 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
236 |
section {* Main theorem *}; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
237 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
238 |
constdefs |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
239 |
mutilated_board :: "nat => nat => (nat * nat) set" |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
240 |
"mutilated_board m n == below (Suc m + Suc m) Times below (Suc n + Suc n) |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
241 |
- {(0, 0)} - {(Suc (m + m), Suc (n + n))}"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
242 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
243 |
theorem mutil_not_tiling: "mutilated_board m n ~: tiling domino" (is "_ ~: ??T"); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
244 |
proof (unfold mutilated_board_def); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
245 |
let ??t = "below (Suc m + Suc m) Times below (Suc n + Suc n)"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
246 |
let ??t' = "??t - {(0, 0)}"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
247 |
let ??t'' = "??t' - {(Suc (m + m), Suc (n + n))}"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
248 |
show "??t'' ~: ??T"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
249 |
proof; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
250 |
let ??e = evnodd; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
251 |
note [simp] = evnodd_iff; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
252 |
assume t'': "??t'' : ??T"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
253 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
254 |
have t: "??t : ??T"; by (rule dominoes_tile_matrix); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
255 |
have fin: "finite (??e ??t 0)"; by (rule evnodd_finite, rule tiling_domino_finite, rule t); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
256 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
257 |
have "card (??e ??t'' 0) < card (??e ??t' 0)"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
258 |
proof -; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
259 |
have "card (??e ??t' 0 - {(Suc (m + m), Suc (n + n))}) < card (??e ??t' 0)"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
260 |
proof (rule card_Diff1_less); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
261 |
show "finite (??e ??t' 0)"; by (rule finite_subset, rule fin) force; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
262 |
show "(Suc (m + m), Suc (n + n)) : ??e ??t' 0"; by simp; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
263 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
264 |
thus ??thesis; by simp; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
265 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
266 |
also; have "... < card (??e ??t 0)"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
267 |
proof -; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
268 |
have "(0, 0) : ??e ??t 0"; by simp; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
269 |
with fin; have "card (??e ??t 0 - {(0, 0)}) < card (??e ??t 0)"; by (rule card_Diff1_less); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
270 |
thus ??thesis; by simp; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
271 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
272 |
also; from t; have "... = card (??e ??t 1)"; by (rule tiling_domino_01); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
273 |
also; have "??e ??t 1 = ??e ??t'' 1"; by simp; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
274 |
also; have "card ... = card (??e ??t'' 0)"; by (rule sym, rule tiling_domino_01); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
275 |
finally; show False; ..; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
276 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
277 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
278 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
279 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
280 |
end; |