# Theory Mutilated_Checkerboard

```(*  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 ‹The Mutilated Checker Board Problem›

theory Mutilated_Checkerboard
imports Main
begin

text ‹
The Mutilated Checker Board Problem, formalized inductively. See \<^cite>‹"paulson-mutilated-board"› for the original tactic script version.
›

subsection ‹Tilings›

inductive_set tiling :: "'a set set ⇒ 'a set set" for A :: "'a set set"
where
empty: "{} ∈ tiling A"
| Un: "a ∪ t ∈ tiling A" if "a ∈ A" and "t ∈ tiling A" and "a ⊆ - t"

text ‹The union of two disjoint tilings is a tiling.›

lemma tiling_Un:
assumes "t ∈ tiling A"
and "u ∈ tiling A"
and "t ∩ u = {}"
shows "t ∪ u ∈ tiling A"
proof -
let ?T = "tiling A"
from ‹t ∈ ?T› and ‹t ∩ u = {}›
show "t ∪ u ∈ ?T"
proof (induct t)
case empty
with ‹u ∈ ?T› show "{} ∪ u ∈ ?T" by simp
next
case (Un a t)
show "(a ∪ t) ∪ u ∈ ?T"
proof -
have "a ∪ (t ∪ u) ∈ ?T"
using ‹a ∈ A›
proof (rule tiling.Un)
from ‹(a ∪ t) ∩ u = {}› have "t ∩ u = {}" by blast
then show "t ∪ u ∈ ?T" by (rule Un)
from ‹a ⊆ - t› and ‹(a ∪ t) ∩ u = {}›
show "a ⊆ - (t ∪ u)" by blast
qed
also have "a ∪ (t ∪ u) = (a ∪ t) ∪ u"
by (simp only: Un_assoc)
finally show ?thesis .
qed
qed
qed

subsection ‹Basic properties of ``below''›

definition below :: "nat ⇒ nat set"
where "below n = {i. i < n}"

lemma below_less_iff [iff]: "i ∈ below k ⟷ i < k"

lemma below_0: "below 0 = {}"

lemma Sigma_Suc1: "m = n + 1 ⟹ below m × B = ({n} × B) ∪ (below n × B)"
by (simp add: below_def less_Suc_eq) blast

lemma Sigma_Suc2:
"m = n + 2 ⟹
A × below m = (A × {n}) ∪ (A × {n + 1}) ∪ (A × below n)"

lemmas Sigma_Suc = Sigma_Suc1 Sigma_Suc2

subsection ‹Basic properties of ``evnodd''›

definition evnodd :: "(nat × nat) set ⇒ nat ⇒ (nat × nat) set"
where "evnodd A b = A ∩ {(i, j). (i + j) mod 2 = b}"

lemma evnodd_iff: "(i, j) ∈ evnodd A b ⟷ (i, j) ∈ A  ∧ (i + j) mod 2 = b"

lemma evnodd_subset: "evnodd A b ⊆ A"
unfolding evnodd_def by (rule Int_lower1)

lemma evnoddD: "x ∈ evnodd A b ⟹ x ∈ A"
by (rule subsetD) (rule evnodd_subset)

lemma evnodd_finite: "finite A ⟹ finite (evnodd A b)"
by (rule finite_subset) (rule evnodd_subset)

lemma evnodd_Un: "evnodd (A ∪ B) b = evnodd A b ∪ 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 = {}"

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)"

subsection ‹Dominoes›

inductive_set domino :: "(nat × nat) set set"
where
horiz: "{(i, j), (i, j + 1)} ∈ domino"
| vertl: "{(i, j), (i + 1, j)} ∈ domino"

lemma dominoes_tile_row:
"{i} × below (2 * n) ∈ tiling domino"
(is "?B n ∈ ?T")
proof (induct n)
case 0
show ?case by (simp add: below_0 tiling.empty)
next
case (Suc n)
let ?a = "{i} × {2 * n + 1} ∪ {i} × {2 * n}"
have "?B (Suc n) = ?a ∪ ?B n"
by (auto simp add: Sigma_Suc Un_assoc)
also have "… ∈ ?T"
proof (rule tiling.Un)
have "{(i, 2 * n), (i, 2 * n + 1)} ∈ domino"
by (rule domino.horiz)
also have "{(i, 2 * n), (i, 2 * n + 1)} = ?a" by blast
finally show "… ∈ domino" .
show "?B n ∈ ?T" by (rule Suc)
show "?a ⊆ - ?B n" by blast
qed
finally show ?case .
qed

lemma dominoes_tile_matrix:
"below m × below (2 * n) ∈ tiling domino"
(is "?B m ∈ ?T")
proof (induct m)
case 0
show ?case by (simp add: below_0 tiling.empty)
next
case (Suc m)
let ?t = "{m} × below (2 * n)"
have "?B (Suc m) = ?t ∪ ?B m" by (simp add: Sigma_Suc)
also have "… ∈ ?T"
proof (rule tiling_Un)
show "?t ∈ ?T" by (rule dominoes_tile_row)
show "?B m ∈ ?T" by (rule Suc)
show "?t ∩ ?B m = {}" by blast
qed
finally show ?case .
qed

lemma domino_singleton:
assumes "d ∈ domino"
and "b < 2"
shows "∃i j. evnodd d b = {(i, j)}"  (is "?P d")
using assms
proof induct
from ‹b < 2› have b_cases: "b = 0 ∨ 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 ∈ 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 ‹Tilings of dominoes›

lemma tiling_domino_finite:
assumes t: "t ∈ tiling domino"  (is "t ∈ ?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 ∈ domino"
then have "?F a" by (rule domino_finite)
from this and ‹?F t› show "?F (a ∪ t)" by (rule finite_UnI)
qed

lemma tiling_domino_01:
assumes t: "t ∈ tiling domino"  (is "t ∈ ?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 = ‹card (?e t 0) = card (?e t 1)›
and at = ‹a ⊆ - t›
have card_suc: "card (?e (a ∪ t) b) = Suc (card (?e t b))" if "b < 2" for b :: nat
proof -
have "?e (a ∪ t) b = ?e a b ∪ ?e t b" by (rule evnodd_Un)
also obtain i j where e: "?e a b = {(i, j)}"
proof -
from ‹a ∈ domino› and ‹b < 2›
have "∃i j. ?e a b = {(i, j)}" by (rule domino_singleton)
then show ?thesis by (blast intro: that)
qed
also have "… ∪ ?e t b = insert (i, j) (?e t b)" by simp
also have "card … = Suc (card (?e t b))"
proof (rule card_insert_disjoint)
from ‹t ∈ tiling domino› have "finite t"
by (rule tiling_domino_finite)
then show "finite (?e t b)"
by (rule evnodd_finite)
from e have "(i, j) ∈ ?e a b" by simp
with at show "(i, j) ∉ ?e t b" by (blast dest: evnoddD)
qed
finally show ?thesis .
qed
then have "card (?e (a ∪ 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 … = card (?e (a ∪ t) 1)"
by simp
finally show ?case .
qed

subsection ‹Main theorem›

definition mutilated_board :: "nat ⇒ nat ⇒ (nat × nat) set"
where "mutilated_board m n =
below (2 * (m + 1)) × below (2 * (n + 1)) - {(0, 0)} - {(2 * m + 1, 2 * n + 1)}"

theorem mutil_not_tiling: "mutilated_board m n ∉ tiling domino"
proof (unfold mutilated_board_def)
let ?T = "tiling domino"
let ?t = "below (2 * (m + 1)) × below (2 * (n + 1))"
let ?t' = "?t - {(0, 0)}"
let ?t'' = "?t' - {(2 * m + 1, 2 * n + 1)}"

show "?t'' ∉ ?T"
proof
have t: "?t ∈ ?T" by (rule dominoes_tile_matrix)
assume t'': "?t'' ∈ ?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) ∈ ?e ?t' 0" by simp
qed
then show ?thesis by simp
qed
also have "… < card (?e ?t 0)"
proof -
have "(0, 0) ∈ ?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 "… = card (?e ?t 1)"
by (rule tiling_domino_01)
also have "?e ?t 1 = ?e ?t'' 1" by simp
also from t'' have "card … = card (?e ?t'' 0)"
by (rule tiling_domino_01 [symmetric])
finally have "… < …" . then show False ..
qed
qed

end
```