turned translation for 1::nat into def.
introduced 1' and replaced most occurrences of 1 by 1'.
(* Title: HOL/Induct/Mutil.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
*)
header {* The Mutilated Chess Board Problem *}
theory Mutil = Main:
text {*
The Mutilated Chess Board Problem, formalized inductively.
Originator is Max Black, according to J A Robinson. Popularized as
the Mutilated Checkerboard Problem by J McCarthy.
*}
consts tiling :: "'a set set => 'a set set"
inductive "tiling A"
intros [simp, intro]
empty: "{} \<in> tiling A"
Un: "a \<in> A ==> t \<in> tiling A ==> a \<inter> t = {} ==> a \<union> t \<in> tiling A"
consts domino :: "(nat \<times> nat) set set"
inductive domino
intros [simp]
horiz: "{(i, j), (i, Suc j)} \<in> domino"
vertl: "{(i, j), (Suc i, j)} \<in> domino"
constdefs
coloured :: "nat => (nat \<times> nat) set"
"coloured b == {(i, j). (i + j) mod #2 = b}"
text {* \medskip The union of two disjoint tilings is a tiling *}
lemma tiling_UnI [rule_format, intro]:
"t \<in> tiling A ==> u \<in> tiling A --> t \<inter> u = {} --> t \<union> u \<in> tiling A"
apply (erule tiling.induct)
prefer 2
apply (simp add: Un_assoc)
apply auto
done
text {* \medskip Chess boards *}
lemma Sigma_Suc1 [simp]:
"lessThan (Suc n) \<times> B = ({n} \<times> B) \<union> ((lessThan n) \<times> B)"
apply (unfold lessThan_def)
apply auto
done
lemma Sigma_Suc2 [simp]:
"A \<times> lessThan (Suc n) = (A \<times> {n}) \<union> (A \<times> (lessThan n))"
apply (unfold lessThan_def)
apply auto
done
lemma sing_Times_lemma: "({i} \<times> {n}) \<union> ({i} \<times> {m}) = {(i, m), (i, n)}"
apply auto
done
lemma dominoes_tile_row [intro!]: "{i} \<times> lessThan (#2 * n) \<in> tiling domino"
apply (induct n)
apply (simp_all add: Un_assoc [symmetric])
apply (rule tiling.Un)
apply (auto simp add: sing_Times_lemma)
done
lemma dominoes_tile_matrix: "(lessThan m) \<times> lessThan (#2 * n) \<in> tiling domino"
apply (induct m)
apply auto
done
text {* \medskip @{term coloured} and Dominoes *}
lemma coloured_insert [simp]:
"coloured b \<inter> (insert (i, j) t) =
(if (i + j) mod #2 = b then insert (i, j) (coloured b \<inter> t)
else coloured b \<inter> t)"
apply (unfold coloured_def)
apply auto
done
lemma domino_singletons:
"d \<in> domino ==>
(\<exists>i j. coloured 0 \<inter> d = {(i, j)}) \<and>
(\<exists>m n. coloured 1 \<inter> d = {(m, n)})"
apply (erule domino.cases)
apply (auto simp add: mod_Suc)
done
lemma domino_finite [simp]: "d \<in> domino ==> finite d"
apply (erule domino.cases)
apply auto
done
text {* \medskip Tilings of dominoes *}
lemma tiling_domino_finite [simp]: "t \<in> tiling domino ==> finite t"
apply (erule tiling.induct)
apply auto
done
declare
Int_Un_distrib [simp]
Diff_Int_distrib [simp]
lemma tiling_domino_0_1:
"t \<in> tiling domino ==> card (coloured 0 \<inter> t) = card (coloured 1' \<inter> t)"
apply (erule tiling.induct)
apply (drule_tac [2] domino_singletons)
apply auto
apply (subgoal_tac "\<forall>p C. C \<inter> a = {p} --> p \<notin> t")
-- {* this lemma tells us that both ``inserts'' are non-trivial *}
apply (simp (no_asm_simp))
apply blast
done
text {* \medskip Final argument is surprisingly complex *}
theorem gen_mutil_not_tiling:
"t \<in> tiling domino ==>
(i + j) mod #2 = 0 ==> (m + n) mod #2 = 0 ==>
{(i, j), (m, n)} \<subseteq> t
==> (t - {(i, j)} - {(m, n)}) \<notin> tiling domino"
apply (rule notI)
apply (subgoal_tac
"card (coloured 0 \<inter> (t - {(i, j)} - {(m, n)})) <
card (coloured 1' \<inter> (t - {(i, j)} - {(m, n)}))")
apply (force simp only: tiling_domino_0_1)
apply (simp add: tiling_domino_0_1 [symmetric])
apply (simp add: coloured_def card_Diff2_less)
done
text {* Apply the general theorem to the well-known case *}
theorem mutil_not_tiling:
"t = lessThan (#2 * Suc m) \<times> lessThan (#2 * Suc n)
==> t - {(0, 0)} - {(Suc (#2 * m), Suc (#2 * n))} \<notin> tiling domino"
apply (rule gen_mutil_not_tiling)
apply (blast intro!: dominoes_tile_matrix)
apply auto
done
end