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src/HOL/Isar_Examples/Mutilated_Checkerboard.thy

author | wenzelm |

Sat, 07 Apr 2012 16:41:59 +0200 | |

changeset 47389 | e8552cba702d |

parent 46582 | dcc312f22ee8 |

child 55656 | eb07b0acbebc |

permissions | -rw-r--r-- |

explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions;
tuned;

(* Title: HOL/Isar_Examples/Mutilated_Checkerboard.thy Author: Markus Wenzel, TU Muenchen (Isar document) Author: Lawrence C Paulson, Cambridge University Computer Laboratory (original scripts) *) header {* The Mutilated Checker Board Problem *} theory Mutilated_Checkerboard imports Main begin 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" where 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: assumes "t : tiling A" and "u : tiling A" and "t Int u = {}" shows "t Un u : tiling A" proof - let ?T = "tiling A" from `t : ?T` and `t Int u = {}` show "t Un u : ?T" proof (induct t) case empty with `u : ?T` 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" using `a : A` proof (rule tiling.Un) from `(a Un t) Int u = {}` have "t Int u = {}" by blast then show "t Un u: ?T" by (rule Un) from `a <= - t` and `(a Un t) Int u = {}` 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'' *} 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) 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'' *} definition evnodd :: "(nat * nat) set => nat => (nat * nat) set" where "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" 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 Un B) b = evnodd A b Un 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" where 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: assumes "d : domino" and "b < 2" shows "EX 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 qed 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) qed 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 Un t)" by (rule finite_UnI) qed 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) next case (Un a t) let ?e = evnodd note hyp = `card (?e t 0) = card (?e t 1)` and 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 - from `a \<in> domino` and `b < 2` have "EX i j. ?e a b = {(i, j)}" by (rule domino_singleton) then show ?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) from `t \<in> 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) qed finally show "?thesis b" . qed then have "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 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" 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 then show ?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) then show ?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 "... < ..." . then show False .. qed qed end