Theory Mutilated_Checkerboard

(*  Title:      HOL/Isar_Examples/Mutilated_Checkerboard.thy
    Author:     Markus Wenzel, TU Muenchen (Isar document)
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory (original scripts)

section ‹The Mutilated Checker Board Problem›

theory Mutilated_Checkerboard
  imports Main

text ‹
  The Mutilated Checker Board Problem, formalized inductively. See cite"paulson-mutilated-board" for the original tactic script version.

subsection ‹Tilings›

inductive_set tiling :: "'a set set  'a set set" for A :: "'a set set"
    empty: "{}  tiling A"
  | Un: "a  t  tiling A" if "a  A" and "t  tiling A" and "a  - t"

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

lemma tiling_Un:
  assumes "t  tiling A"
    and "u  tiling A"
    and "t  u = {}"
  shows "t  u  tiling A"
proof -
  let ?T = "tiling A"
  from t  ?T and t  u = {}
  show "t  u  ?T"
  proof (induct t)
    case empty
    with u  ?T show "{}  u  ?T" by simp
    case (Un a t)
    show "(a  t)  u  ?T"
    proof -
      have "a  (t  u)  ?T"
        using a  A
      proof (rule tiling.Un)
        from (a  t)  u = {} have "t  u = {}" by blast
        then show "t  u  ?T" by (rule Un)
        from a  - t and (a  t)  u = {}
        show "a  - (t  u)" by blast
      also have "a  (t  u) = (a  t)  u"
        by (simp only: Un_assoc)
      finally show ?thesis .

subsection ‹Basic properties of ``below''›

definition below :: "nat  nat set"
  where "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)  (below n × B)"
  by (simp add: below_def less_Suc_eq) blast

lemma Sigma_Suc2:
  "m = n + 2 
    A × below m = (A × {n})  (A × {n + 1})  (A × below n)"
  by (auto simp add: below_def)

lemmas Sigma_Suc = Sigma_Suc1 Sigma_Suc2

subsection ‹Basic properties of ``evnodd''›

definition evnodd :: "(nat × nat) set  nat  (nat × nat) set"
  where "evnodd A b = A  {(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"
  unfolding evnodd_def by (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  B) b = evnodd A b  evnodd B b"
  unfolding evnodd_def by blast

lemma evnodd_Diff: "evnodd (A - B) b = evnodd A b - evnodd B b"
  unfolding evnodd_def by 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)

subsection ‹Dominoes›

inductive_set domino :: "(nat × nat) set set"
    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)
  case (Suc n)
  let ?a = "{i} × {2 * n + 1}  {i} × {2 * n}"
  have "?B (Suc n) = ?a  ?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
  finally show ?case .

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)
  case (Suc m)
  let ?t = "{m} × below (2 * n)"
  have "?B (Suc m) = ?t  ?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  ?B m = {}" by blast
  finally show ?case .

lemma domino_singleton:
  assumes "d  domino"
    and "b < 2"
  shows "i j. evnodd d b = {(i, j)}"  (is "?P d")
  using assms
proof induct
  from b < 2 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

lemma domino_finite:
  assumes "d  domino"
  shows "finite d"
  using assms
proof induct
  fix i j :: nat
  show "finite {(i, j), (i, j + 1)}" by (intro finite.intros)
  show "finite {(i, j), (i + 1, j)}" by (intro finite.intros)

subsection ‹Tilings of dominoes›

lemma tiling_domino_finite:
  assumes t: "t  tiling domino"  (is "t  ?T")
  shows "finite t"  (is "?F t")
  using t
proof induct
  show "?F {}" by (rule finite.emptyI)
  fix a t assume "?F t"
  assume "a  domino"
  then have "?F a" by (rule domino_finite)
  from this and ?F t show "?F (a  t)" by (rule finite_UnI)

lemma tiling_domino_01:
  assumes t: "t  tiling domino"  (is "t  ?T")
  shows "card (evnodd t 0) = card (evnodd t 1)"
  using t
proof induct
  case empty
  show ?case by (simp add: evnodd_def)
  case (Un a t)
  let ?e = evnodd
  note hyp = card (?e t 0) = card (?e t 1)
    and at = a  - t
  have card_suc: "card (?e (a  t) b) = Suc (card (?e t b))" if "b < 2" for b :: nat
  proof -
    have "?e (a  t) b = ?e a b  ?e t b" by (rule evnodd_Un)
    also obtain i j where e: "?e a b = {(i, j)}"
    proof -
      from a  domino and b < 2
      have "i j. ?e a b = {(i, j)}" by (rule domino_singleton)
      then show ?thesis by (blast intro: that)
    also have "  ?e t b = insert (i, j) (?e t b)" by simp
    also have "card  = Suc (card (?e t b))"
    proof (rule card_insert_disjoint)
      from t  tiling domino have "finite t"
        by (rule tiling_domino_finite)
      then show "finite (?e t b)"
        by (rule evnodd_finite)
      from e have "(i, j)  ?e a b" by simp
      with at show "(i, j)  ?e t b" by (blast dest: evnoddD)
    finally show ?thesis .
  then have "card (?e (a  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  t) 1)"
    by simp
  finally show ?case .

subsection ‹Main theorem›

definition mutilated_board :: "nat  nat  (nat × nat) set"
  where "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"
    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
      then show ?thesis by simp
    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)
      then show ?thesis by simp
    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 " < " . then show False ..