author wenzelm Mon, 30 Aug 1999 11:13:27 +0200 changeset 7385 1d486a5b6176 parent 7384 33c976216121 child 7386 1048bc161c05
tuned;
```--- a/src/HOL/Isar_examples/MutilatedCheckerboard.thy	Sun Aug 29 17:53:03 1999 +0200
+++ b/src/HOL/Isar_examples/MutilatedCheckerboard.thy	Mon Aug 30 11:13:27 1999 +0200
@@ -1,11 +1,13 @@
(*  Title:      HOL/Isar_examples/MutilatedCheckerboard.thy
ID:         \$Id\$
-    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory (original script)
-                Markus Wenzel, TU Muenchen (Isar document)
+    Author:     Markus Wenzel, TU Muenchen (Isar document)
+                Lawrence C Paulson, Cambridge University Computer Laboratory (original scripts)

-The Mutilated Chess Board Problem, formalized inductively.
+The Mutilated Checker Board Problem, formalized inductively.
Originator is Max Black, according to J A Robinson.
Popularized as the Mutilated Checkerboard Problem by J McCarthy.
+
*)

theory MutilatedCheckerboard = Main:;
@@ -32,7 +34,7 @@
show "??P {}"; by simp;

fix a t;
-    assume "a:A" "t : ??T" "??P t" "a <= - t";
+    assume "a : A" "t : ??T" "??P t" "a <= - t";
show "??P (a Un t)";
proof (intro impI);
assume "u : ??T" "(a Un t) Int u = {}";
@@ -45,9 +47,6 @@
qed;
qed;

-lemma tiling_UnI: "[| t : tiling A; u : tiling A; t Int u = {} |] ==> t Un u : tiling A";
-  by (rule tiling_Un [rulify]);
-

section {* Basic properties of below *};

@@ -58,7 +57,7 @@
lemma below_less_iff [iff]: "(i: below k) = (i < k)";

-lemma below_0 [simp]: "below 0 = {}";
+lemma below_0: "below 0 = {}";

lemma Sigma_Suc1: "below (Suc n) Times B = ({n} Times B) Un (below n Times B)";
@@ -73,33 +72,31 @@
section {* Basic properties of evnodd *};

constdefs
-  evnodd :: "[(nat * nat) set, nat] => (nat * nat) set"
+  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)";

lemma evnodd_subset: "evnodd A b <= A";
-proof (unfold evnodd_def);
-  show "!!B. A Int B <= A"; by (rule Int_lower1);
-qed;
+  by (unfold evnodd_def, rule Int_lower1);

lemma evnoddD: "x : evnodd A b ==> x : A";
by (rule subsetD, rule evnodd_subset);

-lemma evnodd_finite [simp]: "finite A ==> finite (evnodd A b)";
+lemma evnodd_finite: "finite A ==> finite (evnodd A b)";
by (rule finite_subset, rule evnodd_subset);

-lemma evnodd_Un [simp]: "evnodd (A Un B) b = evnodd A b Un evnodd B b";
+lemma evnodd_Un: "evnodd (A Un B) b = evnodd A b Un evnodd B b";
by (unfold evnodd_def) blast;

-lemma evnodd_Diff [simp]: "evnodd (A - B) b = evnodd A b - evnodd B b";
+lemma evnodd_Diff: "evnodd (A - B) b = evnodd A b - evnodd B b";
by (unfold evnodd_def) blast;

-lemma evnodd_empty [simp]: "evnodd {} b = {}";
+lemma evnodd_empty: "evnodd {} b = {}";

-lemma evnodd_insert [simp]: "evnodd (insert (i, j) C) 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)";

@@ -111,43 +108,41 @@

inductive domino
intrs
-    horiz:  "{(i, j), (i, Suc j)} : domino"
-    vertl:  "{(i, j), (Suc i, j)} : domino";
+    horiz:  "{(i, j), (i, j + 1)} : domino"
+    vertl:  "{(i, j), (i + 1, j)} : domino";

-lemma dominoes_tile_row: "{i} Times below (n + n) : tiling domino"
+lemma dominoes_tile_row: "{i} Times below (2 * n) : tiling domino"
(is "??P n" is "??B n : ??T");
proof (induct n);
-  have "??B 0 = {}"; by simp;
-  also; have "... : ??T"; by (rule tiling.empty);
-  finally; show "??P 0"; .;
+  show "??P 0"; by (simp add: below_0 tiling.empty);

fix n; assume hyp: "??P n";
-  let ??a = "{i} Times {Suc (n + n)} Un {i} Times {n + n}";
+  let ??a = "{i} Times {2 * n + 1} Un {i} Times {2 * n}";

have "??B (Suc n) = ??a Un ??B n"; by (simp add: Sigma_Suc Un_assoc);
also; have "... : ??T";
proof (rule tiling.Un);
-    have "{(i, n + n), (i, Suc (n + n))} : domino"; by (rule domino.horiz);
-    also; have "{(i, n + n), (i, Suc (n + n))} = ??a"; by blast;
-    finally; show "??a : domino"; .;
+    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 hyp);
show "??a <= - ??B n"; by force;
qed;
finally; show "??P (Suc n)"; .;
qed;

-lemma dominoes_tile_matrix: "below m Times below (n + n) : tiling domino"
+lemma dominoes_tile_matrix: "below m Times below (2 * n) : tiling domino"
(is "??P m" is "??B m : ??T");
proof (induct m);
-  show "??P 0"; by (simp add: tiling.empty) -- {* same as above *};
+  show "??P 0"; by (simp add: below_0 tiling.empty);

fix m; assume hyp: "??P m";
-  let ??t = "{m} Times below (n + n)";
+  let ??t = "{m} Times below (2 * n)";

have "??B (Suc m) = ??t Un ??B m"; by (simp add: Sigma_Suc);
also; have "... : ??T";
-  proof (rule tiling_UnI);
+  proof (rule tiling_Un [rulify]);
show "??t : ??T"; by (rule dominoes_tile_row);
show "??B m : ??T"; by (rule hyp);
show "??t Int ??B m = {}"; by blast;
@@ -162,19 +157,19 @@
assume "d : domino";
thus ??thesis (is "??P d");
proof (induct d set: domino);
+    have b_cases: "b = 0 | b = 1"; by arith;
fix i j;
-    have b_cases: "b = 0 | b = 1"; by arith;
-    note [simp] = less_Suc_eq mod_Suc;
-    from b_cases; show "??P {(i, j), (i, Suc j)}"; by rule auto;
-    from b_cases; show "??P {(i, j), (Suc i, j)}"; by rule auto;
+    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 (induct set: domino);
fix i j;
-  show "finite {(i, j), (i, Suc j)}"; by (intro Finites.intrs);
-  show "finite {(i, j), (Suc i, j)}"; by (intro Finites.intrs);
+  show "finite {(i, j), (i, j + 1)}"; by (intro Finites.intrs);
+  show "finite {(i, j), (i + 1, j)}"; by (intro Finites.intrs);
qed;

@@ -184,7 +179,7 @@
proof -;
assume "t : ??T";
thus "??F t";
-  proof (induct set: tiling);
+  proof (induct t set: tiling);
show "??F {}"; by (rule Finites.emptyI);
fix a t; assume "??F t";
assume "a : domino"; hence "??F a"; by (rule domino_finite);
@@ -197,7 +192,7 @@
proof -;
assume "t : ??T";
thus "??P t";
-  proof (induct set: tiling);
+  proof (induct t set: tiling);
show "??P {}"; by (simp add: evnodd_def);

fix a t;
@@ -237,29 +232,30 @@

constdefs
mutilated_board :: "nat => nat => (nat * nat) set"
-  "mutilated_board m n == below (Suc m + Suc m) Times below (Suc n + Suc n)
-    - {(0, 0)} - {(Suc (m + m), Suc (n + n))}";
+  "mutilated_board m n == below (2 * (m + 1)) Times below (2 * (n + 1))
+    - {(0, 0)} - {(2 * m + 1, 2 * n + 1)}";

-theorem mutil_not_tiling: "mutilated_board m n ~: tiling domino" (is "_ ~: ??T");
+theorem mutil_not_tiling: "mutilated_board m n ~: tiling domino";
proof (unfold mutilated_board_def);
-  let ??t = "below (Suc m + Suc m) Times below (Suc n + Suc n)";
+  let ??T = "tiling domino";
+  let ??t = "below (2 * (m + 1)) Times below (2 * (n + 1))";
let ??t' = "??t - {(0, 0)}";
-  let ??t'' = "??t' - {(Suc (m + m), Suc (n + n))}";
+  let ??t'' = "??t' - {(2 * m + 1, 2 * n + 1)}";
show "??t'' ~: ??T";
proof;
-    let ??e = evnodd;
-    note [simp] = evnodd_iff;
+    have t: "??t : ??T"; by (rule dominoes_tile_matrix);
assume t'': "??t'' : ??T";

-    have t: "??t : ??T"; by (rule dominoes_tile_matrix);
+    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 - {(Suc (m + m), Suc (n + n))}) < card (??e ??t' 0)";
+      have "card (??e ??t' 0 - {(2 * m + 1, 2 * n + 1)}) < card (??e ??t' 0)";
proof (rule card_Diff1_less);
show "finite (??e ??t' 0)"; by (rule finite_subset, rule fin) force;
-	show "(Suc (m + m), Suc (n + n)) : ??e ??t' 0"; by simp;
+	show "(2 * m + 1, 2 * n + 1) : ??e ??t' 0"; by simp;
qed;
thus ??thesis; by simp;
qed;
@@ -271,7 +267,7 @@
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; have "card ... = card (??e ??t'' 0)"; by (rule sym, rule tiling_domino_01);
+    also; from t''; have "card ... = card (??e ??t'' 0)"; by (rule tiling_domino_01 [RS sym]);
finally; show False; ..;
qed;
qed;```