(* Title: HOL/Isar_examples/MutilatedCheckerboard.thy
ID: $Id$
Author: Markus Wenzel, TU Muenchen (Isar document)
Lawrence C Paulson, Cambridge University Computer Laboratory (original scripts)
The Mutilated Checker Board Problem, formalized inductively.
Originator is Max Black, according to J A Robinson.
Popularized as the Mutilated Checkerboard Problem by J McCarthy.
See also HOL/Induct/Mutil for the original Isabelle tactic scripts.
*)
theory MutilatedCheckerboard = Main:;
section {* Tilings *};
consts
tiling :: "'a set set => 'a set set";
inductive "tiling A"
intrs
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: "t : tiling A --> u : tiling A --> t Int u = {} --> t Un u : tiling A";
proof;
assume "t : tiling A" (is "_ : ??T");
thus "u : ??T --> t Int u = {} --> t Un u : ??T" (is "??P t");
proof (induct t set: tiling);
show "??P {}"; by simp;
fix a t;
assume "a : A" "t : ??T" "??P t" "a <= - t";
show "??P (a Un t)";
proof (intro impI);
assume "u : ??T" "(a Un t) Int u = {}";
have hyp: "t Un u: ??T"; by blast;
have "a <= - (t Un u)"; by blast;
with _ hyp; have "a Un (t Un u) : ??T"; by (rule tiling.Un);
also; have "a Un (t Un u) = (a Un t) Un u"; by (simp only: Un_assoc);
finally; show "... : ??T"; .;
qed;
qed;
qed;
section {* 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: "below (Suc n) Times B = ({n} Times B) Un (below n Times B)";
by (simp add: below_def less_Suc_eq) blast;
lemma Sigma_Suc2: "A Times below (Suc n) = (A Times {n}) Un (A Times (below n))";
by (simp add: below_def less_Suc_eq) blast;
lemmas Sigma_Suc = Sigma_Suc1 Sigma_Suc2;
section {* 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;
section {* Dominoes *};
consts
domino :: "(nat * nat) set set";
inductive domino
intrs
horiz: "{(i, j), (i, j + 1)} : domino"
vertl: "{(i, j), (i + 1, j)} : domino";
lemma dominoes_tile_row: "{i} Times below (2 * n) : tiling domino"
(is "??P n" is "??B n : ??T");
proof (induct n);
show "??P 0"; by (simp add: below_0 tiling.empty);
fix n; assume hyp: "??P n";
let ??a = "{i} Times {2 * n + 1} Un {i} Times {2 * n}";
have "??B (Suc n) = ??a Un ??B n"; by (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"; .;
from hyp; show "??B n : ??T"; .;
show "??a <= - ??B n"; by force;
qed;
finally; show "??P (Suc n)"; .;
qed;
lemma dominoes_tile_matrix: "below m Times below (2 * n) : tiling domino"
(is "??P m" is "??B m : ??T");
proof (induct m);
show "??P 0"; by (simp add: below_0 tiling.empty);
fix m; assume hyp: "??P m";
let ??t = "{m} Times below (2 * n)";
have "??B (Suc m) = ??t Un ??B m"; by (simp add: Sigma_Suc);
also; have "... : ??T";
proof (rule tiling_Un [rulify]);
show "??t : ??T"; by (rule dominoes_tile_row);
from hyp; show "??B m : ??T"; .;
show "??t Int ??B m = {}"; by blast;
qed;
finally; show "??P (Suc m)"; .;
qed;
lemma domino_singleton: "[| d : domino; b < 2 |] ==> EX i j. evnodd d b = {(i, j)}";
proof -;
assume "b < 2";
assume "d : domino";
thus ??thesis (is "??P d");
proof (induct d set: domino);
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;
qed;
lemma domino_finite: "d: domino ==> finite d";
proof (induct set: domino);
fix i j :: nat;
show "finite {(i, j), (i, j + 1)}"; by (intro Finites.intrs);
show "finite {(i, j), (i + 1, j)}"; by (intro Finites.intrs);
qed;
section {* Tilings of dominoes *};
lemma tiling_domino_finite: "t : tiling domino ==> finite t" (is "t : ??T ==> ??F t");
proof -;
assume "t : ??T";
thus "??F t";
proof (induct t set: tiling);
show "??F {}"; by (rule Finites.emptyI);
fix a t; assume "??F t";
assume "a : domino"; hence "??F a"; by (rule domino_finite);
thus "??F (a Un t)"; by (rule finite_UnI);
qed;
qed;
lemma tiling_domino_01: "t : tiling domino ==> card (evnodd t 0) = card (evnodd t 1)"
(is "t : ??T ==> ??P t");
proof -;
assume "t : ??T";
thus "??P t";
proof (induct t set: tiling);
show "??P {}"; by (simp add: evnodd_def);
fix a t;
let ??e = evnodd;
assume "a : domino" "t : ??T"
and hyp: "card (??e t 0) = card (??e t 1)"
and "a <= - t";
have card_suc: "!!b. b < 2 ==> card (??e (a Un t) b) = Suc (card (??e t b))";
proof -;
fix b; assume "b < 2";
have "EX i j. ??e a b = {(i, j)}"; by (rule domino_singleton);
thus "??thesis b";
proof (elim exE);
have "??e (a Un t) b = ??e a b Un ??e t b"; by (rule evnodd_Un);
also; fix i j; assume "??e a b = {(i, j)}";
also; have "... Un ??e t b = insert (i, j) (??e t b)"; by simp;
also; have "card ... = Suc (card (??e t b))";
proof (rule card_insert_disjoint);
show "finite (??e t b)"; by (rule evnodd_finite, rule tiling_domino_finite);
have "(i, j) : ??e a b"; by asm_simp;
thus "(i, j) ~: ??e t b"; by (force dest: evnoddD);
qed;
finally; show ??thesis; .;
qed;
qed;
hence "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 "??P (a Un t)"; .;
qed;
qed;
section {* Main theorem *};
constdefs
mutilated_board :: "nat => nat => (nat * nat) set"
"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 ~: 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'' ~: ??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);
show "finite (??e ??t' 0)"; by (rule finite_subset, rule fin) force;
show "(2 * m + 1, 2 * n + 1) : ??e ??t' 0"; by simp;
qed;
thus ??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);
thus ??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 [RS sym]);
finally; show False; ..;
qed;
qed;
end;