(* Title: HOL/Isar_examples/MutilatedCheckerboard.thy
ID: $Id$
Author: Markus Wenzel, TU Muenchen (Isar document)
Lawrence C Paulson, Cambridge University Computer Laboratory (original scripts)
*)
header {* The Mutilated Checker Board Problem *}
theory MutilatedCheckerboard imports Main begin
text {*
The Mutilated Checker Board Problem, formalized inductively. See
\cite{paulson-mutilated-board} and
\url{http://isabelle.in.tum.de/library/HOL/Induct/Mutil.html} 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 : A ==> t : tiling A ==> a <= - t ==> a Un t : tiling A"
text "The union of two disjoint tilings is a tiling."
lemma tiling_Un:
assumes "t : tiling A" and "u : tiling A" and "t Int u = {}"
shows "t Un u : tiling A"
proof -
let ?T = "tiling A"
from `t : ?T` and `t Int u = {}`
show "t Un u : ?T"
proof (induct t)
case empty
with `u : ?T` show "{} Un u : ?T" by simp
next
case (Un a t)
show "(a Un t) Un u : ?T"
proof -
have "a Un (t Un u) : ?T"
using `a : A`
proof (rule tiling.Un)
from `(a Un t) Int u = {}` have "t Int u = {}" by blast
then show "t Un u: ?T" by (rule Un)
from `a <= - t` and `(a Un t) Int u = {}`
show "a <= - (t Un u)" by blast
qed
also have "a Un (t Un u) = (a Un t) Un u"
by (simp only: Un_assoc)
finally show ?thesis .
qed
qed
qed
subsection {* Basic properties of ``below'' *}
constdefs
below :: "nat => nat set"
"below n == {i. i < n}"
lemma below_less_iff [iff]: "(i: below k) = (i < k)"
by (simp add: below_def)
lemma below_0: "below 0 = {}"
by (simp add: below_def)
lemma Sigma_Suc1:
"m = n + 1 ==> below m <*> B = ({n} <*> B) Un (below n <*> B)"
by (simp add: below_def less_Suc_eq) blast
lemma Sigma_Suc2:
"m = n + 2 ==> A <*> below m =
(A <*> {n}) Un (A <*> {n + 1}) Un (A <*> below n)"
by (auto simp add: below_def)
lemmas Sigma_Suc = Sigma_Suc1 Sigma_Suc2
subsection {* Basic properties of ``evnodd'' *}
constdefs
evnodd :: "(nat * nat) set => nat => (nat * nat) set"
"evnodd A b == A Int {(i, j). (i + j) mod 2 = b}"
lemma evnodd_iff:
"(i, j): evnodd A b = ((i, j): A & (i + j) mod 2 = b)"
by (simp add: evnodd_def)
lemma evnodd_subset: "evnodd A b <= A"
by (unfold evnodd_def, 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 Un B) b = evnodd A b Un evnodd B b"
by (unfold evnodd_def) blast
lemma evnodd_Diff: "evnodd (A - B) b = evnodd A b - evnodd B b"
by (unfold evnodd_def) 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) blast
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} Un {i} <*> {2 * n}"
have "?B (Suc n) = ?a Un ?B n"
by (auto simp add: Sigma_Suc Un_assoc)
moreover 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
ultimately show ?case by simp
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 Un ?B m" by (simp add: Sigma_Suc)
moreover have "... : ?T"
proof (rule tiling_Un)
show "?t : ?T" by (rule dominoes_tile_row)
show "?B m : ?T" by (rule Suc)
show "?t Int ?B m = {}" by blast
qed
ultimately show ?case by simp
qed
lemma domino_singleton:
assumes d: "d : domino" and "b < 2"
shows "EX i j. evnodd d b = {(i, j)}" (is "?P d")
using d
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: "d: domino"
shows "finite d"
using d
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 Un 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:
"!!b. b < 2 ==> card (?e (a Un t) b) = Suc (card (?e t b))"
proof -
fix b :: nat assume "b < 2"
have "?e (a Un t) b = ?e a b Un ?e t b" by (rule evnodd_Un)
also obtain i j where e: "?e a b = {(i, j)}"
proof -
from `a \<in> domino` and `b < 2`
have "EX i j. ?e a b = {(i, j)}" by (rule domino_singleton)
then show ?thesis by (blast intro: that)
qed
moreover have "... Un ?e t b = insert (i, j) (?e t b)" by simp
moreover have "card ... = Suc (card (?e t b))"
proof (rule card_insert_disjoint)
from `t \<in> 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
ultimately show "?thesis b" by simp
qed
then have "card (?e (a Un 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 Un t) 1)"
by simp
finally show ?case .
qed
subsection {* Main theorem *}
constdefs
mutilated_board :: "nat => nat => (nat * nat) set"
"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