(* Title: HOL/Isar_Examples/Mutilated_Checkerboard.thy
Author: Markus Wenzel, TU Muenchen (Isar document)
Author: Lawrence C Paulson, Cambridge University Computer Laboratory (original scripts)
*)
section \<open>The Mutilated Checker Board Problem\<close>
theory Mutilated_Checkerboard
imports Main
begin
text \<open>
The Mutilated Checker Board Problem, formalized inductively. See @{cite
"paulson-mutilated-board"} for the original tactic script version.
\<close>
subsection \<open>Tilings\<close>
inductive_set tiling :: "'a set set \<Rightarrow> 'a set set"
for A :: "'a set set"
where
empty: "{} \<in> tiling A"
| Un: "a \<in> A \<Longrightarrow> t \<in> tiling A \<Longrightarrow> a \<subseteq> - t \<Longrightarrow> a \<union> t \<in> tiling A"
text \<open>The union of two disjoint tilings is a tiling.\<close>
lemma tiling_Un:
assumes "t \<in> tiling A"
and "u \<in> tiling A"
and "t \<inter> u = {}"
shows "t \<union> u \<in> tiling A"
proof -
let ?T = "tiling A"
from \<open>t \<in> ?T\<close> and \<open>t \<inter> u = {}\<close>
show "t \<union> u \<in> ?T"
proof (induct t)
case empty
with \<open>u \<in> ?T\<close> show "{} \<union> u \<in> ?T" by simp
next
case (Un a t)
show "(a \<union> t) \<union> u \<in> ?T"
proof -
have "a \<union> (t \<union> u) \<in> ?T"
using \<open>a \<in> A\<close>
proof (rule tiling.Un)
from \<open>(a \<union> t) \<inter> u = {}\<close> have "t \<inter> u = {}" by blast
then show "t \<union> u \<in> ?T" by (rule Un)
from \<open>a \<subseteq> - t\<close> and \<open>(a \<union> t) \<inter> u = {}\<close>
show "a \<subseteq> - (t \<union> u)" by blast
qed
also have "a \<union> (t \<union> u) = (a \<union> t) \<union> u"
by (simp only: Un_assoc)
finally show ?thesis .
qed
qed
qed
subsection \<open>Basic properties of ``below''\<close>
definition below :: "nat \<Rightarrow> nat set"
where "below n = {i. i < n}"
lemma below_less_iff [iff]: "i \<in> below k \<longleftrightarrow> i < k"
by (simp add: below_def)
lemma below_0: "below 0 = {}"
by (simp add: below_def)
lemma Sigma_Suc1: "m = n + 1 \<Longrightarrow> below m \<times> B = ({n} \<times> B) \<union> (below n \<times> B)"
by (simp add: below_def less_Suc_eq) blast
lemma Sigma_Suc2:
"m = n + 2 \<Longrightarrow>
A \<times> below m = (A \<times> {n}) \<union> (A \<times> {n + 1}) \<union> (A \<times> below n)"
by (auto simp add: below_def)
lemmas Sigma_Suc = Sigma_Suc1 Sigma_Suc2
subsection \<open>Basic properties of ``evnodd''\<close>
definition evnodd :: "(nat \<times> nat) set \<Rightarrow> nat \<Rightarrow> (nat \<times> nat) set"
where "evnodd A b = A \<inter> {(i, j). (i + j) mod 2 = b}"
lemma evnodd_iff: "(i, j) \<in> evnodd A b \<longleftrightarrow> (i, j) \<in> A \<and> (i + j) mod 2 = b"
by (simp add: evnodd_def)
lemma evnodd_subset: "evnodd A b \<subseteq> A"
unfolding evnodd_def by (rule Int_lower1)
lemma evnoddD: "x \<in> evnodd A b \<Longrightarrow> x \<in> A"
by (rule subsetD) (rule evnodd_subset)
lemma evnodd_finite: "finite A \<Longrightarrow> finite (evnodd A b)"
by (rule finite_subset) (rule evnodd_subset)
lemma evnodd_Un: "evnodd (A \<union> B) b = evnodd A b \<union> evnodd B b"
unfolding evnodd_def by blast
lemma evnodd_Diff: "evnodd (A - B) b = evnodd A b - evnodd B b"
unfolding evnodd_def by blast
lemma evnodd_empty: "evnodd {} b = {}"
by (simp add: evnodd_def)
lemma evnodd_insert: "evnodd (insert (i, j) C) b =
(if (i + j) mod 2 = b
then insert (i, j) (evnodd C b) else evnodd C b)"
by (simp add: evnodd_def)
subsection \<open>Dominoes\<close>
inductive_set domino :: "(nat \<times> nat) set set"
where
horiz: "{(i, j), (i, j + 1)} \<in> domino"
| vertl: "{(i, j), (i + 1, j)} \<in> domino"
lemma dominoes_tile_row:
"{i} \<times> below (2 * n) \<in> tiling domino"
(is "?B n \<in> ?T")
proof (induct n)
case 0
show ?case by (simp add: below_0 tiling.empty)
next
case (Suc n)
let ?a = "{i} \<times> {2 * n + 1} \<union> {i} \<times> {2 * n}"
have "?B (Suc n) = ?a \<union> ?B n"
by (auto simp add: Sigma_Suc Un_assoc)
also have "\<dots> \<in> ?T"
proof (rule tiling.Un)
have "{(i, 2 * n), (i, 2 * n + 1)} \<in> domino"
by (rule domino.horiz)
also have "{(i, 2 * n), (i, 2 * n + 1)} = ?a" by blast
finally show "\<dots> \<in> domino" .
show "?B n \<in> ?T" by (rule Suc)
show "?a \<subseteq> - ?B n" by blast
qed
finally show ?case .
qed
lemma dominoes_tile_matrix:
"below m \<times> below (2 * n) \<in> tiling domino"
(is "?B m \<in> ?T")
proof (induct m)
case 0
show ?case by (simp add: below_0 tiling.empty)
next
case (Suc m)
let ?t = "{m} \<times> below (2 * n)"
have "?B (Suc m) = ?t \<union> ?B m" by (simp add: Sigma_Suc)
also have "\<dots> \<in> ?T"
proof (rule tiling_Un)
show "?t \<in> ?T" by (rule dominoes_tile_row)
show "?B m \<in> ?T" by (rule Suc)
show "?t \<inter> ?B m = {}" by blast
qed
finally show ?case .
qed
lemma domino_singleton:
assumes "d \<in> domino"
and "b < 2"
shows "\<exists>i j. evnodd d b = {(i, j)}" (is "?P d")
using assms
proof induct
from \<open>b < 2\<close> have b_cases: "b = 0 \<or> b = 1" by arith
fix i j
note [simp] = evnodd_empty evnodd_insert mod_Suc
from b_cases show "?P {(i, j), (i, j + 1)}" by rule auto
from b_cases show "?P {(i, j), (i + 1, j)}" by rule auto
qed
lemma domino_finite:
assumes "d \<in> domino"
shows "finite d"
using assms
proof induct
fix i j :: nat
show "finite {(i, j), (i, j + 1)}" by (intro finite.intros)
show "finite {(i, j), (i + 1, j)}" by (intro finite.intros)
qed
subsection \<open>Tilings of dominoes\<close>
lemma tiling_domino_finite:
assumes t: "t \<in> tiling domino" (is "t \<in> ?T")
shows "finite t" (is "?F t")
using t
proof induct
show "?F {}" by (rule finite.emptyI)
fix a t assume "?F t"
assume "a \<in> domino"
then have "?F a" by (rule domino_finite)
from this and \<open>?F t\<close> show "?F (a \<union> t)" by (rule finite_UnI)
qed
lemma tiling_domino_01:
assumes t: "t \<in> tiling domino" (is "t \<in> ?T")
shows "card (evnodd t 0) = card (evnodd t 1)"
using t
proof induct
case empty
show ?case by (simp add: evnodd_def)
next
case (Un a t)
let ?e = evnodd
note hyp = \<open>card (?e t 0) = card (?e t 1)\<close>
and at = \<open>a \<subseteq> - t\<close>
have card_suc: "card (?e (a \<union> t) b) = Suc (card (?e t b))" if "b < 2" for b :: nat
proof -
have "?e (a \<union> t) b = ?e a b \<union> ?e t b" by (rule evnodd_Un)
also obtain i j where e: "?e a b = {(i, j)}"
proof -
from \<open>a \<in> domino\<close> and \<open>b < 2\<close>
have "\<exists>i j. ?e a b = {(i, j)}" by (rule domino_singleton)
then show ?thesis by (blast intro: that)
qed
also have "\<dots> \<union> ?e t b = insert (i, j) (?e t b)" by simp
also have "card \<dots> = Suc (card (?e t b))"
proof (rule card_insert_disjoint)
from \<open>t \<in> tiling domino\<close> have "finite t"
by (rule tiling_domino_finite)
then show "finite (?e t b)"
by (rule evnodd_finite)
from e have "(i, j) \<in> ?e a b" by simp
with at show "(i, j) \<notin> ?e t b" by (blast dest: evnoddD)
qed
finally show ?thesis .
qed
then have "card (?e (a \<union> t) 0) = Suc (card (?e t 0))" by simp
also from hyp have "card (?e t 0) = card (?e t 1)" .
also from card_suc have "Suc \<dots> = card (?e (a \<union> t) 1)"
by simp
finally show ?case .
qed
subsection \<open>Main theorem\<close>
definition mutilated_board :: "nat \<Rightarrow> nat \<Rightarrow> (nat \<times> nat) set"
where
"mutilated_board m n =
below (2 * (m + 1)) \<times> below (2 * (n + 1))
- {(0, 0)} - {(2 * m + 1, 2 * n + 1)}"
theorem mutil_not_tiling: "mutilated_board m n \<notin> tiling domino"
proof (unfold mutilated_board_def)
let ?T = "tiling domino"
let ?t = "below (2 * (m + 1)) \<times> below (2 * (n + 1))"
let ?t' = "?t - {(0, 0)}"
let ?t'' = "?t' - {(2 * m + 1, 2 * n + 1)}"
show "?t'' \<notin> ?T"
proof
have t: "?t \<in> ?T" by (rule dominoes_tile_matrix)
assume t'': "?t'' \<in> ?T"
let ?e = evnodd
have fin: "finite (?e ?t 0)"
by (rule evnodd_finite, rule tiling_domino_finite, rule t)
note [simp] = evnodd_iff evnodd_empty evnodd_insert evnodd_Diff
have "card (?e ?t'' 0) < card (?e ?t' 0)"
proof -
have "card (?e ?t' 0 - {(2 * m + 1, 2 * n + 1)})
< card (?e ?t' 0)"
proof (rule card_Diff1_less)
from _ fin show "finite (?e ?t' 0)"
by (rule finite_subset) auto
show "(2 * m + 1, 2 * n + 1) \<in> ?e ?t' 0" by simp
qed
then show ?thesis by simp
qed
also have "\<dots> < card (?e ?t 0)"
proof -
have "(0, 0) \<in> ?e ?t 0" by simp
with fin have "card (?e ?t 0 - {(0, 0)}) < card (?e ?t 0)"
by (rule card_Diff1_less)
then show ?thesis by simp
qed
also from t have "\<dots> = card (?e ?t 1)"
by (rule tiling_domino_01)
also have "?e ?t 1 = ?e ?t'' 1" by simp
also from t'' have "card \<dots> = card (?e ?t'' 0)"
by (rule tiling_domino_01 [symmetric])
finally have "\<dots> < \<dots>" . then show False ..
qed
qed
end