(* 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 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 ∪ 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 next 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 qed also have "a ∪ (t ∪ u) = (a ∪ t) ∪ 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) ∪ (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" 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} ∪ {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 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 ∪ ?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 qed finally show ?case . qed 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 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 ∪ 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: "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) qed 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) qed finally show ?thesis . qed 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 . 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