--- a/src/HOL/Isar_Examples/Mutilated_Checkerboard.thy Fri Feb 21 17:06:48 2014 +0100
+++ b/src/HOL/Isar_Examples/Mutilated_Checkerboard.thy Fri Feb 21 18:23:11 2014 +0100
@@ -14,40 +14,40 @@
subsection {* Tilings *}
-inductive_set tiling :: "'a set set => 'a set set"
+inductive_set tiling :: "'a set set \<Rightarrow> '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"
+ empty: "{} \<in> tiling A"
+| Un: "a \<in> A \<Longrightarrow> t \<in> tiling A \<Longrightarrow> a \<subseteq> - t \<Longrightarrow> a \<union> t \<in> 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"
+ assumes "t \<in> tiling A"
+ and "u \<in> tiling A"
+ and "t \<inter> u = {}"
+ shows "t \<union> u \<in> tiling A"
proof -
let ?T = "tiling A"
- from `t : ?T` and `t Int u = {}`
- show "t Un u : ?T"
+ from `t \<in> ?T` and `t \<inter> u = {}`
+ show "t \<union> u \<in> ?T"
proof (induct t)
case empty
- with `u : ?T` show "{} Un u : ?T" by simp
+ with `u \<in> ?T` show "{} \<union> u \<in> ?T" by simp
next
case (Un a t)
- show "(a Un t) Un u : ?T"
+ show "(a \<union> t) \<union> u \<in> ?T"
proof -
- have "a Un (t Un u) : ?T"
- using `a : A`
+ have "a \<union> (t \<union> u) \<in> ?T"
+ using `a \<in> 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
+ from `(a \<union> t) \<inter> u = {}` have "t \<inter> u = {}" by blast
+ then show "t \<union> u \<in> ?T" by (rule Un)
+ from `a \<subseteq> - t` and `(a \<union> t) \<inter> u = {}`
+ show "a \<subseteq> - (t \<union> u)" by blast
qed
- also have "a Un (t Un u) = (a Un t) Un u"
+ also have "a \<union> (t \<union> u) = (a \<union> t) \<union> u"
by (simp only: Un_assoc)
finally show ?thesis .
qed
@@ -57,22 +57,21 @@
subsection {* Basic properties of ``below'' *}
-definition below :: "nat => nat set"
+definition below :: "nat \<Rightarrow> nat set"
where "below n = {i. i < n}"
-lemma below_less_iff [iff]: "(i: below k) = (i < k)"
+lemma below_less_iff [iff]: "i \<in> below k \<longleftrightarrow> 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)"
+lemma Sigma_Suc1: "m = n + 1 \<Longrightarrow> below m \<times> B = ({n} \<times> B) \<union> (below n \<times> 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)"
+ "m = n + 2 \<Longrightarrow>
+ A \<times> below m = (A \<times> {n}) \<union> (A \<times> {n + 1}) \<union> (A \<times> below n)"
by (auto simp add: below_def)
lemmas Sigma_Suc = Sigma_Suc1 Sigma_Suc2
@@ -80,22 +79,22 @@
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}"
+definition evnodd :: "(nat \<times> nat) set \<Rightarrow> nat \<Rightarrow> (nat \<times> nat) set"
+ where "evnodd A b = A \<inter> {(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_iff: "(i, j) \<in> evnodd A b \<longleftrightarrow> (i, j) \<in> A \<and> (i + j) mod 2 = b"
by (simp add: evnodd_def)
-lemma evnodd_subset: "evnodd A b <= A"
+lemma evnodd_subset: "evnodd A b \<subseteq> A"
unfolding evnodd_def by (rule Int_lower1)
-lemma evnoddD: "x : evnodd A b ==> x : A"
+lemma evnoddD: "x \<in> evnodd A b \<Longrightarrow> x \<in> A"
by (rule subsetD) (rule evnodd_subset)
-lemma evnodd_finite: "finite A ==> finite (evnodd A b)"
+lemma evnodd_finite: "finite A \<Longrightarrow> 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"
+lemma evnodd_Un: "evnodd (A \<union> B) b = evnodd A b \<union> evnodd B b"
unfolding evnodd_def by blast
lemma evnodd_Diff: "evnodd (A - B) b = evnodd A b - evnodd B b"
@@ -112,60 +111,60 @@
subsection {* Dominoes *}
-inductive_set domino :: "(nat * nat) set set"
+inductive_set domino :: "(nat \<times> nat) set set"
where
- horiz: "{(i, j), (i, j + 1)} : domino"
-| vertl: "{(i, j), (i + 1, j)} : domino"
+ horiz: "{(i, j), (i, j + 1)} \<in> domino"
+| vertl: "{(i, j), (i + 1, j)} \<in> domino"
lemma dominoes_tile_row:
- "{i} <*> below (2 * n) : tiling domino"
- (is "?B n : ?T")
+ "{i} \<times> below (2 * n) \<in> tiling domino"
+ (is "?B n \<in> ?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"
+ let ?a = "{i} \<times> {2 * n + 1} \<union> {i} \<times> {2 * n}"
+ have "?B (Suc n) = ?a \<union> ?B n"
by (auto simp add: Sigma_Suc Un_assoc)
- also have "... : ?T"
+ also have "\<dots> \<in> ?T"
proof (rule tiling.Un)
- have "{(i, 2 * n), (i, 2 * n + 1)} : domino"
+ have "{(i, 2 * n), (i, 2 * n + 1)} \<in> 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 "\<dots> \<in> domino" .
+ show "?B n \<in> ?T" by (rule Suc)
+ show "?a \<subseteq> - ?B n" by blast
qed
finally show ?case .
qed
lemma dominoes_tile_matrix:
- "below m <*> below (2 * n) : tiling domino"
- (is "?B m : ?T")
+ "below m \<times> below (2 * n) \<in> tiling domino"
+ (is "?B m \<in> ?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"
+ let ?t = "{m} \<times> below (2 * n)"
+ have "?B (Suc m) = ?t \<union> ?B m" by (simp add: Sigma_Suc)
+ also have "\<dots> \<in> ?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
+ show "?t \<in> ?T" by (rule dominoes_tile_row)
+ show "?B m \<in> ?T" by (rule Suc)
+ show "?t \<inter> ?B m = {}" by blast
qed
finally show ?case .
qed
lemma domino_singleton:
- assumes "d : domino"
+ assumes "d \<in> domino"
and "b < 2"
- shows "EX i j. evnodd d b = {(i, j)}" (is "?P d")
+ shows "\<exists>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
+ from `b < 2` have b_cases: "b = 0 \<or> 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
@@ -173,7 +172,7 @@
qed
lemma domino_finite:
- assumes "d: domino"
+ assumes "d \<in> domino"
shows "finite d"
using assms
proof induct
@@ -186,18 +185,19 @@
subsection {* Tilings of dominoes *}
lemma tiling_domino_finite:
- assumes t: "t : tiling domino" (is "t : ?T")
+ assumes t: "t \<in> tiling domino" (is "t \<in> ?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)
+ assume "a \<in> domino"
+ then have "?F a" by (rule domino_finite)
+ from this and `?F t` show "?F (a \<union> t)" by (rule finite_UnI)
qed
lemma tiling_domino_01:
- assumes t: "t : tiling domino" (is "t : ?T")
+ assumes t: "t \<in> tiling domino" (is "t \<in> ?T")
shows "card (evnodd t 0) = card (evnodd t 1)"
using t
proof induct
@@ -207,33 +207,34 @@
case (Un a t)
let ?e = evnodd
note hyp = `card (?e t 0) = card (?e t 1)`
- and at = `a <= - t`
+ and at = `a \<subseteq> - t`
have card_suc:
- "!!b. b < 2 ==> card (?e (a Un t) b) = Suc (card (?e t b))"
+ "\<And>b. b < 2 \<Longrightarrow> card (?e (a \<union> 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)
+ fix b :: nat
+ assume "b < 2"
+ have "?e (a \<union> t) b = ?e a b \<union> ?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)
+ have "\<exists>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))"
+ also have "\<dots> \<union> ?e t b = insert (i, j) (?e t b)" by simp
+ also have "card \<dots> = 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)
+ from e have "(i, j) \<in> ?e a b" by simp
+ with at show "(i, j) \<notin> ?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
+ then have "card (?e (a \<union> 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)"
+ also from card_suc have "Suc \<dots> = card (?e (a \<union> t) 1)"
by simp
finally show ?case .
qed
@@ -241,23 +242,23 @@
subsection {* Main theorem *}
-definition mutilated_board :: "nat => nat => (nat * nat) set"
+definition mutilated_board :: "nat \<Rightarrow> nat \<Rightarrow> (nat \<times> nat) set"
where
"mutilated_board m n =
- below (2 * (m + 1)) <*> below (2 * (n + 1))
+ 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"
+theorem mutil_not_tiling: "mutilated_board m n \<notin> tiling domino"
proof (unfold mutilated_board_def)
let ?T = "tiling domino"
- let ?t = "below (2 * (m + 1)) <*> below (2 * (n + 1))"
+ let ?t = "below (2 * (m + 1)) \<times> below (2 * (n + 1))"
let ?t' = "?t - {(0, 0)}"
let ?t'' = "?t' - {(2 * m + 1, 2 * n + 1)}"
- show "?t'' ~: ?T"
+ show "?t'' \<notin> ?T"
proof
- have t: "?t : ?T" by (rule dominoes_tile_matrix)
- assume t'': "?t'' : ?T"
+ have t: "?t \<in> ?T" by (rule dominoes_tile_matrix)
+ assume t'': "?t'' \<in> ?T"
let ?e = evnodd
have fin: "finite (?e ?t 0)"
@@ -271,23 +272,23 @@
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
+ show "(2 * m + 1, 2 * n + 1) \<in> ?e ?t' 0" by simp
qed
then show ?thesis by simp
qed
- also have "... < card (?e ?t 0)"
+ also have "\<dots> < card (?e ?t 0)"
proof -
- have "(0, 0) : ?e ?t 0" by simp
+ have "(0, 0) \<in> ?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)"
+ also from t have "\<dots> = 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)"
+ also from t'' have "card \<dots> = card (?e ?t'' 0)"
by (rule tiling_domino_01 [symmetric])
- finally have "... < ..." . then show False ..
+ finally have "\<dots> < \<dots>" . then show False ..
qed
qed