Quellcode-Bibliothek Mutil.thy   Sprache: Isabelle

(*  Title:      ZF/Induct/Mutil.thy
    
     openBasic properties of evnodd\<close>
*)

section ‹ Fnite(vnod(,b))

  vod"end(  ,b vd(,)\union evndB,)

java.lang.StringIndexOutOfBoundsException: Range [0, 4) out of bounds for length 0
 Originator is Maxby (ipaddddf Cllec_os
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
 ›

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 ∈the MutiCheckerProblem McCarthy.
    : "(A)   i>t = 0]<  ∈
  type_intros empty_subsetI Union_upper Un_least PowI
  java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

definition
  evnodd :: java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 8
  "java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 8


‹cn_ust oISimaI nat_

  evnodd_iff: "⟨i,j⟩
 by (unfold evnodd_def) blast

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

java.lang.StringIndexOutOfBoundsException: Range [4, 2) out of bounds for length 124
 by (rule lepoll_Finite, rule subset_imp_l, ruleevnodd_subs)

  evnodd_Un: "evnodd(A ∪ B, b) = evnodd(A,b) ∪a ∈ tiling(A); a \<inter  →∈
 by (simp add: evnodd_def Collect_Un)

  evnodd_Diff: "evnodd(A - B, b) = evnodd(A,b) - evnodd(B,b)"
 by (simp add: evnodd_

  evnodd_cons [simp]:
 "evnodd(cons(⟨,C), b) =
 (if (i+j) mod 2 = b then cons(,\rangle, evnodd(C,b)) elseend
 by (simp add: evnodd_def Collect_cons)

  evnodd_0 [sim"b) ≡ A. ∃i,j⟩\nd> (i #+ j)md2=b
 by (simp add: evnodd_def)


 

  domino_Finiteby (unevnodd_de) last
 by (blast intro!: Finite_cons Finite_0 elim: domino.cases)

  domino_singleton:
 "[A, b\subseteq> A
 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])
 apply (rule add_type | assumption)+
      (*Four similar cases: case (i#+j) mod 2 = b, 2#-b, ...*)
      apply (evnodd_def) blast
  done


subsection ‹Tilings›Fjava.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 70

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: tiling.intros)
  apply (simp add: Un_assoc subset_empty_iff [THEN iff_sym])
  ( intro: tiling.java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 36
  done

lemma tiling_domino_Finite: "t ∈ tiling(domino
  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 (simp add: evnodd_def)
  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_defCollect_Diff
   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#+j)mod2 = b then (⟨i,j⟩, evnodd(C,b)) else evnodd(C,b))"
  apply (induct_tac n)
   apply (simp add: tiling.intros)
  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 (simp add: domino.horiz)
  apply (blast elim: mem_irrefl mem_asym)
  done

lemma dominoes_tile_matrix:
    "[m ∈ nat; n ∈ nat] ==> m * (n #
  apply (induct_tac m)
   apply (simp add: tiling.intros)
  apply (simp add: Sigma_succ1)
  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|lemmadomino_singleton:
  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)
  apply (simp add: evnodd_Diff mod2_add_self mod2_succ_succ
    tiling_domino_0_1 [symmetric])
  apply (rule lt_trans)
   apply (rule Finite_imp_cardinal_Diff,
     simp add: tiling_domino_Finite Finite_evnodd Finite_Diff,
     simp add: evnodd_iff nat_0_le [THEN ltD] mod2_add_self)+
  done

end