# Theory Mutil

theory Mutil
imports ZF
```(*  Title:      ZF/Induct/Mutil.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

section ‹The Mutilated Chess Board Problem, formalized inductively›

theory Mutil imports ZF begin

text ‹
Originator is Max Black, according to J A Robinson.  Popularized as
the Mutilated Checkerboard Problem by J McCarthy.
›

consts
domino :: i
tiling :: "i => i"

inductive
domains "domino" ⊆ "Pow(nat × nat)"
intros
horiz: "[| i ∈ nat;  j ∈ nat |] ==> {<i,j>, <i,succ(j)>} ∈ domino"
vertl: "[| i ∈ nat;  j ∈ nat |] ==> {<i,j>, <succ(i),j>} ∈ domino"
type_intros empty_subsetI cons_subsetI PowI SigmaI nat_succI

inductive
domains "tiling(A)" ⊆ "Pow(⋃(A))"
intros
empty: "0 ∈ tiling(A)"
Un: "[| a ∈ A;  t ∈ tiling(A);  a ∩ t = 0 |] ==> a ∪ t ∈ tiling(A)"
type_intros empty_subsetI Union_upper Un_least PowI
type_elims PowD [elim_format]

definition
evnodd :: "[i, i] => i"  where
"evnodd(A,b) == {z ∈ A. ∃i j. z = <i,j> ∧ (i #+ j) mod 2 = b}"

subsection ‹Basic properties of evnodd›

lemma evnodd_iff: "<i,j>: evnodd(A,b) ⟷ <i,j>: A & (i#+j) mod 2 = b"
by (unfold evnodd_def) blast

lemma evnodd_subset: "evnodd(A, b) ⊆ A"
by (unfold evnodd_def) blast

lemma Finite_evnodd: "Finite(X) ==> Finite(evnodd(X,b))"
by (rule lepoll_Finite, rule subset_imp_lepoll, rule evnodd_subset)

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

lemma evnodd_Diff: "evnodd(A - B, b) = evnodd(A,b) - evnodd(B,b)"

lemma evnodd_cons [simp]:
"evnodd(cons(<i,j>,C), b) =
(if (i#+j) mod 2 = b then cons(<i,j>, evnodd(C,b)) else evnodd(C,b))"

lemma evnodd_0 [simp]: "evnodd(0, b) = 0"

subsection ‹Dominoes›

lemma domino_Finite: "d ∈ domino ==> Finite(d)"
by (blast intro!: Finite_cons Finite_0 elim: domino.cases)

lemma domino_singleton:
"[| d ∈ domino; b<2 |] ==> ∃i' j'. evnodd(d,b) = {<i',j'>}"
apply (erule domino.cases)
apply (rule_tac [2] k1 = "i#+j" in mod2_cases [THEN disjE])
apply (rule_tac k1 = "i#+j" in mod2_cases [THEN disjE])
(*Four similar cases: case (i#+j) mod 2 = b, 2#-b, ...*)
apply (auto simp add: mod_succ succ_neq_self dest: ltD)
done

subsection ‹Tilings›

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

lemma tiling_UnI:
"t ∈ tiling(A) ==> u ∈ tiling(A) ==> t ∩ u = 0 ==> t ∪ u ∈ tiling(A)"
apply (induct set: tiling)
apply (simp add: Un_assoc subset_empty_iff [THEN iff_sym])
apply (blast intro: tiling.intros)
done

lemma tiling_domino_Finite: "t ∈ tiling(domino) ==> Finite(t)"
apply (induct set: tiling)
apply (rule Finite_0)
apply (blast intro!: Finite_Un intro: domino_Finite)
done

lemma tiling_domino_0_1: "t ∈ tiling(domino) ==> |evnodd(t,0)| = |evnodd(t,1)|"
apply (induct set: tiling)
apply (rule_tac b1 = 0 in domino_singleton [THEN exE])
prefer 2
apply simp
apply assumption
apply (rule_tac b1 = 1 in domino_singleton [THEN exE])
prefer 2
apply simp
apply assumption
apply safe
apply (subgoal_tac "∀p b. p ∈ evnodd (a,b) ⟶ p∉evnodd (t,b)")
apply (simp add: evnodd_Un Un_cons tiling_domino_Finite
evnodd_subset [THEN subset_Finite] Finite_imp_cardinal_cons)
apply (blast dest!: evnodd_subset [THEN subsetD] elim: equalityE)
done

lemma dominoes_tile_row:
"[| i ∈ nat;  n ∈ nat |] ==> {i} * (n #+ n) ∈ tiling(domino)"
apply (induct_tac n)
apply (simp add: Un_assoc [symmetric] Sigma_succ2)
apply (rule tiling.intros)
prefer 2 apply assumption
apply (rename_tac n')
apply (subgoal_tac (*seems the easiest way of turning one to the other*)
"{i}*{succ (n'#+n') } ∪ {i}*{n'#+n'} =
{<i,n'#+n'>, <i,succ (n'#+n') >}")
prefer 2 apply blast
apply (blast elim: mem_irrefl mem_asym)
done

lemma dominoes_tile_matrix:
"[| m ∈ nat;  n ∈ nat |] ==> m * (n #+ n) ∈ tiling(domino)"
apply (induct_tac m)
apply (blast intro: tiling_UnI dominoes_tile_row elim: mem_irrefl)
done

lemma eq_lt_E: "[| x=y; x<y |] ==> P"
by auto

theorem mutil_not_tiling: "[| m ∈ nat;  n ∈ nat;
t = (succ(m)#+succ(m))*(succ(n)#+succ(n));
t' = t - {<0,0>} - {<succ(m#+m), succ(n#+n)>} |]
==> t' ∉ tiling(domino)"
apply (rule notI)
apply (drule tiling_domino_0_1)
apply (erule_tac x = "|A|" for A in eq_lt_E)
apply (subgoal_tac "t ∈ tiling (domino)")
prefer 2 (*Requires a small simpset that won't move the succ applications*)
apply (simp only: nat_succI add_type dominoes_tile_matrix)
tiling_domino_0_1 [symmetric])
apply (rule lt_trans)
apply (rule Finite_imp_cardinal_Diff,
done

end
```