author | wenzelm |
Tue, 21 Sep 1999 17:30:55 +0200 | |
changeset 7565 | bfa85f429629 |
parent 7480 | 0a0e0dbe1269 |
child 7761 | 7fab9592384f |
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
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The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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The Mutilated Checker Board Problem, formalized inductively. |
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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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Popularized as the Mutilated Checkerboard Problem by J McCarthy. |
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See also HOL/Induct/Mutil for the original Isabelle tactic scripts. |
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7382
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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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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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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inductive "tiling A" |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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22 |
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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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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text "The union of two disjoint tilings is a tiling"; |
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The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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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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proof; |
7480 | 31 |
assume "t : tiling A" (is "_ : ?T"); |
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thus "u : ?T --> t Int u = {} --> t Un u : ?T" (is "?P t"); |
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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33 |
proof (induct t set: tiling); |
7480 | 34 |
show "?P {}"; by simp; |
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The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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35 |
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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fix a t; |
7480 | 37 |
assume "a : A" "t : ?T" "?P t" "a <= - t"; |
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show "?P (a Un t)"; |
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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proof (intro impI); |
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assume "u : ?T" "(a Un t) Int u = {}"; |
7565 | 41 |
have hyp: "t Un u: ?T"; by (blast!); |
42 |
have "a <= - (t Un u)"; by (blast!); |
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7480 | 43 |
with _ hyp; have "a Un (t Un u) : ?T"; by (rule tiling.Un); |
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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44 |
also; have "a Un (t Un u) = (a Un t) Un u"; by (simp only: Un_assoc); |
7480 | 45 |
finally; show "... : ?T"; .; |
7382
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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48 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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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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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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by (simp add: below_def); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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7385 | 60 |
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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61 |
by (simp add: below_def); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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62 |
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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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64 |
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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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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67 |
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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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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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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71 |
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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 evnodd *}; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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73 |
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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constdefs |
7385 | 75 |
evnodd :: "(nat * nat) set => nat => (nat * nat) set" |
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33c01075d343
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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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77 |
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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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79 |
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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80 |
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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81 |
lemma evnodd_subset: "evnodd A b <= A"; |
7385 | 82 |
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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83 |
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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84 |
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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85 |
by (rule subsetD, rule evnodd_subset); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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86 |
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7385 | 87 |
lemma evnodd_finite: "finite A ==> finite (evnodd A b)"; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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88 |
by (rule finite_subset, rule evnodd_subset); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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89 |
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7385 | 90 |
lemma evnodd_Un: "evnodd (A Un B) b = evnodd A b Un evnodd B b"; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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91 |
by (unfold evnodd_def) blast; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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92 |
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7385 | 93 |
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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94 |
by (unfold evnodd_def) blast; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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95 |
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7385 | 96 |
lemma evnodd_empty: "evnodd {} b = {}"; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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97 |
by (simp add: evnodd_def); |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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98 |
|
7385 | 99 |
lemma evnodd_insert: "evnodd (insert (i, j) C) b = |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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100 |
(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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101 |
by (simp add: evnodd_def) blast; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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102 |
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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103 |
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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section {* Dominoes *}; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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105 |
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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106 |
consts |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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107 |
domino :: "(nat * nat) set set"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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108 |
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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109 |
inductive domino |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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110 |
intrs |
7385 | 111 |
horiz: "{(i, j), (i, j + 1)} : domino" |
112 |
vertl: "{(i, j), (i + 1, j)} : domino"; |
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7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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113 |
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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114 |
|
7385 | 115 |
lemma dominoes_tile_row: "{i} Times below (2 * n) : tiling domino" |
7480 | 116 |
(is "?P n" is "?B n : ?T"); |
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33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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117 |
proof (induct n); |
7480 | 118 |
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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119 |
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7480 | 120 |
fix n; assume hyp: "?P n"; |
121 |
let ?a = "{i} Times {2 * n + 1} Un {i} Times {2 * n}"; |
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7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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122 |
|
7480 | 123 |
have "?B (Suc n) = ?a Un ?B n"; by (simp add: Sigma_Suc Un_assoc); |
124 |
also; have "... : ?T"; |
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7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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125 |
proof (rule tiling.Un); |
7385 | 126 |
have "{(i, 2 * n), (i, 2 * n + 1)} : domino"; by (rule domino.horiz); |
7480 | 127 |
also; have "{(i, 2 * n), (i, 2 * n + 1)} = ?a"; by blast; |
7385 | 128 |
finally; show "... : domino"; .; |
7480 | 129 |
from hyp; show "?B n : ?T"; .; |
130 |
show "?a <= - ?B n"; by force; |
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7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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131 |
qed; |
7480 | 132 |
finally; show "?P (Suc n)"; .; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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133 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
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134 |
|
7385 | 135 |
lemma dominoes_tile_matrix: "below m Times below (2 * n) : tiling domino" |
7480 | 136 |
(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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parents:
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137 |
proof (induct m); |
7480 | 138 |
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:
diff
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139 |
|
7480 | 140 |
fix m; assume hyp: "?P m"; |
141 |
let ?t = "{m} Times below (2 * n)"; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
142 |
|
7480 | 143 |
have "?B (Suc m) = ?t Un ?B m"; by (simp add: Sigma_Suc); |
144 |
also; have "... : ?T"; |
|
7385 | 145 |
proof (rule tiling_Un [rulify]); |
7480 | 146 |
show "?t : ?T"; by (rule dominoes_tile_row); |
147 |
from hyp; show "?B m : ?T"; .; |
|
148 |
show "?t Int ?B m = {}"; by blast; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
149 |
qed; |
7480 | 150 |
finally; show "?P (Suc m)"; .; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
151 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
152 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
153 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
154 |
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
|
155 |
proof -; |
7565 | 156 |
assume b: "b < 2"; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
157 |
assume "d : domino"; |
7480 | 158 |
thus ?thesis (is "?P d"); |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
159 |
proof (induct d set: domino); |
7565 | 160 |
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;
wenzelm
parents:
diff
changeset
|
161 |
fix i j; |
7385 | 162 |
note [simp] = evnodd_empty evnodd_insert mod_Suc; |
7480 | 163 |
from b_cases; show "?P {(i, j), (i, j + 1)}"; by rule auto; |
164 |
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;
wenzelm
parents:
diff
changeset
|
165 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
166 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
167 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
168 |
lemma domino_finite: "d: domino ==> finite d"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
169 |
proof (induct set: domino); |
7434 | 170 |
fix i j :: nat; |
7385 | 171 |
show "finite {(i, j), (i, j + 1)}"; by (intro Finites.intrs); |
172 |
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;
wenzelm
parents:
diff
changeset
|
173 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
174 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
175 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
176 |
section {* Tilings of dominoes *}; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
177 |
|
7480 | 178 |
lemma tiling_domino_finite: "t : tiling domino ==> finite t" (is "t : ?T ==> ?F t"); |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
179 |
proof -; |
7480 | 180 |
assume "t : ?T"; |
181 |
thus "?F t"; |
|
7385 | 182 |
proof (induct t set: tiling); |
7480 | 183 |
show "?F {}"; by (rule Finites.emptyI); |
184 |
fix a t; assume "?F t"; |
|
185 |
assume "a : domino"; hence "?F a"; by (rule domino_finite); |
|
186 |
thus "?F (a Un t)"; by (rule finite_UnI); |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
187 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
188 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
189 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
190 |
lemma tiling_domino_01: "t : tiling domino ==> card (evnodd t 0) = card (evnodd t 1)" |
7480 | 191 |
(is "t : ?T ==> ?P t"); |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
192 |
proof -; |
7480 | 193 |
assume "t : ?T"; |
194 |
thus "?P t"; |
|
7385 | 195 |
proof (induct t set: tiling); |
7480 | 196 |
show "?P {}"; by (simp add: evnodd_def); |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
197 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
198 |
fix a t; |
7480 | 199 |
let ?e = evnodd; |
200 |
assume "a : domino" "t : ?T" |
|
201 |
and hyp: "card (?e t 0) = card (?e t 1)" |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
202 |
and "a <= - t"; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
203 |
|
7480 | 204 |
have card_suc: "!!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;
wenzelm
parents:
diff
changeset
|
205 |
proof -; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
206 |
fix b; assume "b < 2"; |
7480 | 207 |
have "EX i j. ?e a b = {(i, j)}"; by (rule domino_singleton); |
208 |
thus "?thesis b"; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
209 |
proof (elim exE); |
7480 | 210 |
have "?e (a Un t) b = ?e a b Un ?e t b"; by (rule evnodd_Un); |
7565 | 211 |
also; fix i j; assume e: "?e a b = {(i, j)}"; |
7480 | 212 |
also; have "... Un ?e t b = insert (i, j) (?e t b)"; by simp; |
213 |
also; have "card ... = Suc (card (?e t b))"; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
214 |
proof (rule card_insert_disjoint); |
7480 | 215 |
show "finite (?e t b)"; by (rule evnodd_finite, rule tiling_domino_finite); |
7565 | 216 |
have "(i, j) : ?e a b"; by (simp!); |
217 |
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
|
218 |
qed; |
7480 | 219 |
finally; show ?thesis; .; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
220 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
221 |
qed; |
7480 | 222 |
hence "card (?e (a Un t) 0) = Suc (card (?e t 0))"; by simp; |
223 |
also; from hyp; have "card (?e t 0) = card (?e t 1)"; .; |
|
224 |
also; from card_suc; have "Suc ... = card (?e (a Un t) 1)"; by simp; |
|
225 |
finally; show "?P (a Un t)"; .; |
|
7382
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 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
229 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
230 |
section {* Main theorem *}; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
231 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
232 |
constdefs |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
233 |
mutilated_board :: "nat => nat => (nat * nat) set" |
7385 | 234 |
"mutilated_board m n == below (2 * (m + 1)) Times below (2 * (n + 1)) |
235 |
- {(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
|
236 |
|
7385 | 237 |
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
|
238 |
proof (unfold mutilated_board_def); |
7480 | 239 |
let ?T = "tiling domino"; |
240 |
let ?t = "below (2 * (m + 1)) Times below (2 * (n + 1))"; |
|
241 |
let ?t' = "?t - {(0, 0)}"; |
|
242 |
let ?t'' = "?t' - {(2 * m + 1, 2 * n + 1)}"; |
|
243 |
show "?t'' ~: ?T"; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
244 |
proof; |
7480 | 245 |
have t: "?t : ?T"; by (rule dominoes_tile_matrix); |
246 |
assume t'': "?t'' : ?T"; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
247 |
|
7480 | 248 |
let ?e = evnodd; |
249 |
have fin: "finite (?e ?t 0)"; 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
|
250 |
|
7385 | 251 |
note [simp] = evnodd_iff evnodd_empty evnodd_insert evnodd_Diff; |
7480 | 252 |
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
|
253 |
proof -; |
7480 | 254 |
have "card (?e ?t' 0 - {(2 * m + 1, 2 * n + 1)}) < card (?e ?t' 0)"; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
255 |
proof (rule card_Diff1_less); |
7480 | 256 |
show "finite (?e ?t' 0)"; by (rule finite_subset, rule fin) force; |
257 |
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
|
258 |
qed; |
7480 | 259 |
thus ?thesis; by simp; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
260 |
qed; |
7480 | 261 |
also; have "... < card (?e ?t 0)"; |
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
262 |
proof -; |
7480 | 263 |
have "(0, 0) : ?e ?t 0"; by simp; |
264 |
with fin; have "card (?e ?t 0 - {(0, 0)}) < card (?e ?t 0)"; by (rule card_Diff1_less); |
|
265 |
thus ?thesis; by simp; |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
266 |
qed; |
7480 | 267 |
also; from t; have "... = card (?e ?t 1)"; by (rule tiling_domino_01); |
268 |
also; have "?e ?t 1 = ?e ?t'' 1"; by simp; |
|
269 |
also; from t''; have "card ... = card (?e ?t'' 0)"; by (rule tiling_domino_01 [RS sym]); |
|
7382
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
270 |
finally; show False; ..; |
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 |
qed; |
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
273 |
|
33c01075d343
The Mutilated Chess Board Problem -- Isar'ized version of HOL/Inductive/Mutil;
wenzelm
parents:
diff
changeset
|
274 |
|
7383 | 275 |
end; |