# Theory MutilatedCheckerboard

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theory MutilatedCheckerboard
imports Main
begin

    (*  Title:      HOL/Isar_examples/MutilatedCheckerboard.thy
        ID:         $Id: MutilatedCheckerboard.thy,v 1.30 2005/06/17 14:13:08 haftmann Exp $
        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 *}

    consts
      tiling :: "'a set set => 'a set set"

    inductive "tiling A"
      intros
        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 -
      let ?T = "tiling A"
      assume u: "u : ?T"
      assume "t : ?T"
      thus "t Int u = {} ==> t Un u : ?T" (is "PROP ?P t")
      proof (induct t)
        case empty
        with u 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"
          proof (rule tiling.Un)
            show "a : A" .
            have atu: "(a Un t) Int u = {}" .
            hence "t Int u = {}" by blast
            thus "t Un u: ?T" by (rule Un)
            have "a <= - t" .
            with atu 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 *}

    consts
      domino :: "(nat * nat) set set"

    inductive domino
      intros
        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)
      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" .
        show "?B n : ?T" by (rule Suc)
        show "?a <= - ?B n" by blast
      qed
      finally show ?case .
    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)
      also 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
      finally show ?case .
    qed

    lemma domino_singleton:
      "d : domino ==> b < 2 ==> EX i j. evnodd d b = {(i, j)}"
    proof -
      assume b: "b < 2"
      assume "d : domino"
      thus ?thesis (is "?P d")
      proof induct
        from b 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 -
      assume "d: domino"
      thus ?thesis
      proof induct
        fix i j :: nat
        show "finite {(i, j), (i, j + 1)}" by (intro Finites.intros)
        show "finite {(i, j), (i + 1, j)}" by (intro Finites.intros)
      qed
    qed

    subsection {* 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
        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 ==> _")
    proof -
      assume "t : ?T"
      thus ?thesis
      proof induct
        case empty
        show ?case by (simp add: evnodd_def)
      next
        case (Un a t)
        let ?e = evnodd
        have hyp: "card (?e t 0) = card (?e t 1)" .
        have 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 -
            have "EX i j. ?e a b = {(i, j)}" by (rule domino_singleton)
            thus ?thesis by (blast intro: that)
          qed
          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)
            from e have "(i, j) : ?e a b" by simp
            with at show "(i, j) ~: ?e t b" by (blast dest: evnoddD)
          qed
          finally show "?thesis b" .
        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 ?case .
      qed
    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
          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 [symmetric])
        finally have "... < ..." . thus False ..
      qed
    qed

    end

### Tilings

lemma tiling_Un:

      [| t ∈ tiling A; u ∈ tiling A; t ∩ u = {} |] ==> t ∪ u ∈ tiling A

### Basic properties of ``below''

lemma below_less_iff:

      (i ∈ below k) = (i < k)

lemma below_0:

      below 0 = {}

lemma Sigma_Suc1:

      m = n + 1 ==> below m × B = {n} × B ∪ below n × B

lemma Sigma_Suc2:

      m = n + 2 ==> A × below m = A × {n} ∪ A × {n + 1} ∪ A × below n

lemmas Sigma_Suc:

      m = n + 1 ==> below m × B = {n} × B ∪ below n × B

      m = n + 2 ==> A × below m = A × {n} ∪ A × {n + 1} ∪ A × below n

lemmas Sigma_Suc:

      m = n + 1 ==> below m × B = {n} × B ∪ below n × B

      m = n + 2 ==> A × below m = A × {n} ∪ A × {n + 1} ∪ A × below n

### Basic properties of ``evnodd''

lemma evnodd_iff:

      ((i, j) ∈ evnodd A b) = ((i, j) ∈ A ∧ (i + j) mod 2 = b)

lemma evnodd_subset:

      evnodd A b ⊆ A

lemma evnoddD:

      x ∈ evnodd A b ==> x ∈ A

lemma evnodd_finite:

      finite A ==> finite (evnodd A b)

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_empty:

      evnodd {} b = {}

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)

### Dominoes

lemma dominoes_tile_row:

      {i} × below (2 * n) ∈ tiling domino

lemma dominoes_tile_matrix:

      below m × below (2 * n) ∈ tiling domino

lemma domino_singleton:

      [| d ∈ domino; b < 2 |] ==> ∃i j. evnodd d b = {(i, j)}

lemma domino_finite:

      d ∈ domino ==> finite d

### Tilings of dominoes

lemma tiling_domino_finite:

      t ∈ tiling domino ==> finite t

lemma tiling_domino_01:

      t ∈ tiling domino ==> card (evnodd t 0) = card (evnodd t 1)

### Main theorem

theorem mutil_not_tiling:

      mutilated_board m n ∉ tiling domino