--- a/src/HOL/Finite_Set.thy Wed Jul 06 14:09:13 2016 +0200
+++ b/src/HOL/Finite_Set.thy Wed Jul 06 20:19:51 2016 +0200
@@ -16,9 +16,9 @@
begin
inductive finite :: "'a set \<Rightarrow> bool"
- where
- emptyI [simp, intro!]: "finite {}"
- | insertI [simp, intro!]: "finite A \<Longrightarrow> finite (insert a A)"
+where
+ emptyI [simp, intro!]: "finite {}"
+| insertI [simp, intro!]: "finite A \<Longrightarrow> finite (insert a A)"
end
@@ -32,14 +32,16 @@
assumes "P {}"
and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
shows "P F"
-using \<open>finite F\<close>
+ using \<open>finite F\<close>
proof induct
show "P {}" by fact
- fix x F assume F: "finite F" and P: "P F"
+next
+ fix x F
+ assume F: "finite F" and P: "P F"
show "P (insert x F)"
proof cases
assume "x \<in> F"
- hence "insert x F = F" by (rule insert_absorb)
+ then have "insert x F = F" by (rule insert_absorb)
with P show ?thesis by (simp only:)
next
assume "x \<notin> F"
@@ -49,13 +51,15 @@
lemma infinite_finite_induct [case_names infinite empty insert]:
assumes infinite: "\<And>A. \<not> finite A \<Longrightarrow> P A"
- assumes empty: "P {}"
- assumes insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
+ and empty: "P {}"
+ and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
shows "P A"
proof (cases "finite A")
- case False with infinite show ?thesis .
+ case False
+ with infinite show ?thesis .
next
- case True then show ?thesis by (induct A) (fact empty insert)+
+ case True
+ then show ?thesis by (induct A) (fact empty insert)+
qed
@@ -71,16 +75,18 @@
text \<open>A finite choice principle. Does not need the SOME choice operator.\<close>
-lemma finite_set_choice:
- "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
+lemma finite_set_choice: "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
proof (induct rule: finite_induct)
- case empty then show ?case by simp
+ case empty
+ then show ?case by simp
next
case (insert a A)
- then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto
- show ?case (is "EX f. ?P f")
+ then obtain f b where f: "\<forall>x\<in>A. P x (f x)" and ab: "P a b"
+ by auto
+ show ?case (is "\<exists>f. ?P f")
proof
- show "?P(%x. if x = a then b else f x)" using f ab by auto
+ show "?P (\<lambda>x. if x = a then b else f x)"
+ using f ab by auto
qed
qed
@@ -88,100 +94,101 @@
subsubsection \<open>Finite sets are the images of initial segments of natural numbers\<close>
lemma finite_imp_nat_seg_image_inj_on:
- assumes "finite A"
+ assumes "finite A"
shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
-using assms
+ using assms
proof induct
case empty
show ?case
proof
- show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}" by simp
+ show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}"
+ by simp
qed
next
case (insert a A)
have notinA: "a \<notin> A" by fact
- from insert.hyps obtain n f
- where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
- hence "insert a A = f(n:=a) ` {i. i < Suc n}"
- "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
- by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
- thus ?case by blast
+ from insert.hyps obtain n f where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}"
+ by blast
+ then have "insert a A = f(n:=a) ` {i. i < Suc n}" and "inj_on (f(n:=a)) {i. i < Suc n}"
+ using notinA by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
+ then show ?case by blast
qed
-lemma nat_seg_image_imp_finite:
- "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"
+lemma nat_seg_image_imp_finite: "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"
proof (induct n arbitrary: A)
- case 0 thus ?case by simp
+ case 0
+ then show ?case by simp
next
case (Suc n)
let ?B = "f ` {i. i < n}"
- have finB: "finite ?B" by(rule Suc.hyps[OF refl])
+ have finB: "finite ?B" by (rule Suc.hyps[OF refl])
show ?case
- proof cases
- assume "\<exists>k<n. f n = f k"
- hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
- thus ?thesis using finB by simp
+ proof (cases "\<exists>k<n. f n = f k")
+ case True
+ then have "A = ?B"
+ using Suc.prems by (auto simp:less_Suc_eq)
+ then show ?thesis
+ using finB by simp
next
- assume "\<not>(\<exists> k<n. f n = f k)"
- hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
- thus ?thesis using finB by simp
+ case False
+ then have "A = insert (f n) ?B"
+ using Suc.prems by (auto simp:less_Suc_eq)
+ then show ?thesis using finB by simp
qed
qed
-lemma finite_conv_nat_seg_image:
- "finite A \<longleftrightarrow> (\<exists>(n::nat) f. A = f ` {i::nat. i < n})"
+lemma finite_conv_nat_seg_image: "finite A \<longleftrightarrow> (\<exists>(n::nat) f. A = f ` {i::nat. i < n})"
by (blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
lemma finite_imp_inj_to_nat_seg:
assumes "finite A"
shows "\<exists>f n::nat. f ` A = {i. i < n} \<and> inj_on f A"
proof -
- from finite_imp_nat_seg_image_inj_on[OF \<open>finite A\<close>]
+ from finite_imp_nat_seg_image_inj_on [OF \<open>finite A\<close>]
obtain f and n::nat where bij: "bij_betw f {i. i<n} A"
- by (auto simp:bij_betw_def)
+ by (auto simp: bij_betw_def)
let ?f = "the_inv_into {i. i<n} f"
- have "inj_on ?f A & ?f ` A = {i. i<n}"
+ have "inj_on ?f A \<and> ?f ` A = {i. i<n}"
by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
- thus ?thesis by blast
+ then show ?thesis by blast
qed
-lemma finite_Collect_less_nat [iff]:
- "finite {n::nat. n < k}"
+lemma finite_Collect_less_nat [iff]: "finite {n::nat. n < k}"
by (fastforce simp: finite_conv_nat_seg_image)
-lemma finite_Collect_le_nat [iff]:
- "finite {n::nat. n \<le> k}"
+lemma finite_Collect_le_nat [iff]: "finite {n::nat. n \<le> k}"
by (simp add: le_eq_less_or_eq Collect_disj_eq)
subsubsection \<open>Finiteness and common set operations\<close>
-lemma rev_finite_subset:
- "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"
+lemma rev_finite_subset: "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"
proof (induct arbitrary: A rule: finite_induct)
case empty
then show ?case by simp
next
case (insert x F A)
- have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})" by fact+
+ have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})"
+ by fact+
show "finite A"
proof cases
assume x: "x \<in> A"
with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
with r have "finite (A - {x})" .
- hence "finite (insert x (A - {x}))" ..
- also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
+ then have "finite (insert x (A - {x}))" ..
+ also have "insert x (A - {x}) = A"
+ using x by (rule insert_Diff)
finally show ?thesis .
next
show ?thesis when "A \<subseteq> F"
using that by fact
assume "x \<notin> A"
- with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
+ with A show "A \<subseteq> F"
+ by (simp add: subset_insert_iff)
qed
qed
-lemma finite_subset:
- "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
+lemma finite_subset: "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
by (rule rev_finite_subset)
lemma finite_UnI:
@@ -189,8 +196,7 @@
shows "finite (F \<union> G)"
using assms by induct simp_all
-lemma finite_Un [iff]:
- "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
+lemma finite_Un [iff]: "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
by (blast intro: finite_UnI finite_subset [of _ "F \<union> G"])
lemma finite_insert [simp]: "finite (insert a A) \<longleftrightarrow> finite A"
@@ -200,8 +206,7 @@
then show ?thesis by simp
qed
-lemma finite_Int [simp, intro]:
- "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
+lemma finite_Int [simp, intro]: "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
by (blast intro: finite_subset)
lemma finite_Collect_conjI [simp, intro]:
@@ -212,112 +217,110 @@
"finite {x. P x \<or> Q x} \<longleftrightarrow> finite {x. P x} \<and> finite {x. Q x}"
by (simp add: Collect_disj_eq)
-lemma finite_Diff [simp, intro]:
- "finite A \<Longrightarrow> finite (A - B)"
+lemma finite_Diff [simp, intro]: "finite A \<Longrightarrow> finite (A - B)"
by (rule finite_subset, rule Diff_subset)
lemma finite_Diff2 [simp]:
assumes "finite B"
shows "finite (A - B) \<longleftrightarrow> finite A"
proof -
- have "finite A \<longleftrightarrow> finite((A - B) \<union> (A \<inter> B))" by (simp add: Un_Diff_Int)
- also have "\<dots> \<longleftrightarrow> finite (A - B)" using \<open>finite B\<close> by simp
+ have "finite A \<longleftrightarrow> finite ((A - B) \<union> (A \<inter> B))"
+ by (simp add: Un_Diff_Int)
+ also have "\<dots> \<longleftrightarrow> finite (A - B)"
+ using \<open>finite B\<close> by simp
finally show ?thesis ..
qed
-lemma finite_Diff_insert [iff]:
- "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
+lemma finite_Diff_insert [iff]: "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
proof -
have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp
moreover have "A - insert a B = A - B - {a}" by auto
ultimately show ?thesis by simp
qed
-lemma finite_compl[simp]:
+lemma finite_compl [simp]:
"finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)"
by (simp add: Compl_eq_Diff_UNIV)
-lemma finite_Collect_not[simp]:
+lemma finite_Collect_not [simp]:
"finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)"
by (simp add: Collect_neg_eq)
lemma finite_Union [simp, intro]:
- "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite(\<Union>A)"
+ "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite (\<Union>A)"
by (induct rule: finite_induct) simp_all
lemma finite_UN_I [intro]:
"finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (\<Union>a\<in>A. B a)"
by (induct rule: finite_induct) simp_all
-lemma finite_UN [simp]:
- "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
+lemma finite_UN [simp]: "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
by (blast intro: finite_subset)
-lemma finite_Inter [intro]:
- "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
+lemma finite_Inter [intro]: "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
by (blast intro: Inter_lower finite_subset)
-lemma finite_INT [intro]:
- "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
+lemma finite_INT [intro]: "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
by (blast intro: INT_lower finite_subset)
-lemma finite_imageI [simp, intro]:
- "finite F \<Longrightarrow> finite (h ` F)"
+lemma finite_imageI [simp, intro]: "finite F \<Longrightarrow> finite (h ` F)"
by (induct rule: finite_induct) simp_all
-lemma finite_image_set [simp]:
- "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
+lemma finite_image_set [simp]: "finite {x. P x} \<Longrightarrow> finite {f x |x. P x}"
by (simp add: image_Collect [symmetric])
lemma finite_image_set2:
- "finite {x. P x} \<Longrightarrow> finite {y. Q y} \<Longrightarrow> finite {f x y | x y. P x \<and> Q y}"
+ "finite {x. P x} \<Longrightarrow> finite {y. Q y} \<Longrightarrow> finite {f x y |x y. P x \<and> Q y}"
by (rule finite_subset [where B = "\<Union>x \<in> {x. P x}. \<Union>y \<in> {y. Q y}. {f x y}"]) auto
lemma finite_imageD:
assumes "finite (f ` A)" and "inj_on f A"
shows "finite A"
-using assms
+ using assms
proof (induct "f ` A" arbitrary: A)
- case empty then show ?case by simp
+ case empty
+ then show ?case by simp
next
case (insert x B)
- then have B_A: "insert x B = f ` A" by simp
- then obtain y where "x = f y" and "y \<in> A" by blast
- from B_A \<open>x \<notin> B\<close> have "B = f ` A - {x}" by blast
- with B_A \<open>x \<notin> B\<close> \<open>x = f y\<close> \<open>inj_on f A\<close> \<open>y \<in> A\<close> have "B = f ` (A - {y})"
+ then have B_A: "insert x B = f ` A"
+ by simp
+ then obtain y where "x = f y" and "y \<in> A"
+ by blast
+ from B_A \<open>x \<notin> B\<close> have "B = f ` A - {x}"
+ by blast
+ with B_A \<open>x \<notin> B\<close> \<open>x = f y\<close> \<open>inj_on f A\<close> \<open>y \<in> A\<close> have "B = f ` (A - {y})"
by (simp add: inj_on_image_set_diff Set.Diff_subset)
- moreover from \<open>inj_on f A\<close> have "inj_on f (A - {y})" by (rule inj_on_diff)
- ultimately have "finite (A - {y})" by (rule insert.hyps)
- then show "finite A" by simp
+ moreover from \<open>inj_on f A\<close> have "inj_on f (A - {y})"
+ by (rule inj_on_diff)
+ ultimately have "finite (A - {y})"
+ by (rule insert.hyps)
+ then show "finite A"
+ by simp
qed
-lemma finite_image_iff:
- assumes "inj_on f A"
- shows "finite (f ` A) \<longleftrightarrow> finite A"
-using assms finite_imageD by blast
+lemma finite_image_iff: "inj_on f A \<Longrightarrow> finite (f ` A) \<longleftrightarrow> finite A"
+ using finite_imageD by blast
-lemma finite_surj:
- "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
+lemma finite_surj: "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
by (erule finite_subset) (rule finite_imageI)
-lemma finite_range_imageI:
- "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
+lemma finite_range_imageI: "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
by (drule finite_imageI) (simp add: range_composition)
lemma finite_subset_image:
assumes "finite B"
shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
-using assms
+ using assms
proof induct
- case empty then show ?case by simp
+ case empty
+ then show ?case by simp
next
- case insert then show ?case
- by (clarsimp simp del: image_insert simp add: image_insert [symmetric])
- blast
+ case insert
+ then show ?case
+ by (clarsimp simp del: image_insert simp add: image_insert [symmetric]) blast
qed
-lemma finite_vimage_IntI:
- "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
+lemma finite_vimage_IntI: "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
apply (induct rule: finite_induct)
apply simp_all
apply (subst vimage_insert)
@@ -334,15 +337,14 @@
by (simp only: * assms finite_UN_I)
qed
-lemma finite_vimageI:
- "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
+lemma finite_vimageI: "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
using finite_vimage_IntI[of F h UNIV] by auto
-lemma finite_vimageD': "\<lbrakk> finite (f -` A); A \<subseteq> range f \<rbrakk> \<Longrightarrow> finite A"
-by(auto simp add: subset_image_iff intro: finite_subset[rotated])
+lemma finite_vimageD': "finite (f -` A) \<Longrightarrow> A \<subseteq> range f \<Longrightarrow> finite A"
+ by (auto simp add: subset_image_iff intro: finite_subset[rotated])
-lemma finite_vimageD: "\<lbrakk> finite (h -` F); surj h \<rbrakk> \<Longrightarrow> finite F"
-by(auto dest: finite_vimageD')
+lemma finite_vimageD: "finite (h -` F) \<Longrightarrow> surj h \<Longrightarrow> finite F"
+ by (auto dest: finite_vimageD')
lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
@@ -359,30 +361,36 @@
assumes "finite {y. P y}"
shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
proof -
- have "{x. EX y. P y & Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto
- with assms show ?thesis by simp
+ have "{x. \<exists>y. P y \<and> Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})"
+ by auto
+ with assms show ?thesis
+ by simp
qed
-lemma finite_Plus:
- "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
+lemma finite_Plus: "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
by (simp add: Plus_def)
-lemma finite_PlusD:
+lemma finite_PlusD:
fixes A :: "'a set" and B :: "'b set"
assumes fin: "finite (A <+> B)"
shows "finite A" "finite B"
proof -
- have "Inl ` A \<subseteq> A <+> B" by auto
- then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)
- then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)
+ have "Inl ` A \<subseteq> A <+> B"
+ by auto
+ then have "finite (Inl ` A :: ('a + 'b) set)"
+ using fin by (rule finite_subset)
+ then show "finite A"
+ by (rule finite_imageD) (auto intro: inj_onI)
next
- have "Inr ` B \<subseteq> A <+> B" by auto
- then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)
- then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)
+ have "Inr ` B \<subseteq> A <+> B"
+ by auto
+ then have "finite (Inr ` B :: ('a + 'b) set)"
+ using fin by (rule finite_subset)
+ then show "finite B"
+ by (rule finite_imageD) (auto intro: inj_onI)
qed
-lemma finite_Plus_iff [simp]:
- "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
+lemma finite_Plus_iff [simp]: "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
by (auto intro: finite_PlusD finite_Plus)
lemma finite_Plus_UNIV_iff [simp]:
@@ -390,21 +398,22 @@
by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
lemma finite_SigmaI [simp, intro]:
- "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) ==> finite (SIGMA a:A. B a)"
- by (unfold Sigma_def) blast
+ "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (SIGMA a:A. B a)"
+ unfolding Sigma_def by blast
lemma finite_SigmaI2:
assumes "finite {x\<in>A. B x \<noteq> {}}"
and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)"
shows "finite (Sigma A B)"
proof -
- from assms have "finite (Sigma {x\<in>A. B x \<noteq> {}} B)" by auto
- also have "Sigma {x:A. B x \<noteq> {}} B = Sigma A B" by auto
+ from assms have "finite (Sigma {x\<in>A. B x \<noteq> {}} B)"
+ by auto
+ also have "Sigma {x:A. B x \<noteq> {}} B = Sigma A B"
+ by auto
finally show ?thesis .
qed
-lemma finite_cartesian_product:
- "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
+lemma finite_cartesian_product: "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
by (rule finite_SigmaI)
lemma finite_Prod_UNIV:
@@ -417,10 +426,12 @@
proof -
from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
by (auto simp add: finite_conv_nat_seg_image)
- then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp
+ then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}"
+ by simp
with \<open>B \<noteq> {}\<close> have "A = (fst \<circ> f) ` {i::nat. i < n}"
by (simp add: image_comp)
- then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast
+ then have "\<exists>n f. A = f ` {i::nat. i < n}"
+ by blast
then show ?thesis
by (auto simp add: finite_conv_nat_seg_image)
qed
@@ -431,10 +442,12 @@
proof -
from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
by (auto simp add: finite_conv_nat_seg_image)
- then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp
+ then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}"
+ by simp
with \<open>A \<noteq> {}\<close> have "B = (snd \<circ> f) ` {i::nat. i < n}"
by (simp add: image_comp)
- then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast
+ then have "\<exists>n f. B = f ` {i::nat. i < n}"
+ by blast
then show ?thesis
by (auto simp add: finite_conv_nat_seg_image)
qed
@@ -443,48 +456,52 @@
"finite (A \<times> B) \<longleftrightarrow> (A = {} \<or> B = {} \<or> (finite A \<and> finite B))"
by (auto dest: finite_cartesian_productD1 finite_cartesian_productD2 finite_cartesian_product)
-lemma finite_prod:
+lemma finite_prod:
"finite (UNIV :: ('a \<times> 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
using finite_cartesian_product_iff[of UNIV UNIV] by simp
-lemma finite_Pow_iff [iff]:
- "finite (Pow A) \<longleftrightarrow> finite A"
+lemma finite_Pow_iff [iff]: "finite (Pow A) \<longleftrightarrow> finite A"
proof
assume "finite (Pow A)"
- then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
- then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
+ then have "finite ((\<lambda>x. {x}) ` A)"
+ by (blast intro: finite_subset)
+ then show "finite A"
+ by (rule finite_imageD [unfolded inj_on_def]) simp
next
assume "finite A"
then show "finite (Pow A)"
by induct (simp_all add: Pow_insert)
qed
-corollary finite_Collect_subsets [simp, intro]:
- "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
+corollary finite_Collect_subsets [simp, intro]: "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
by (simp add: Pow_def [symmetric])
lemma finite_set: "finite (UNIV :: 'a set set) \<longleftrightarrow> finite (UNIV :: 'a set)"
-by(simp only: finite_Pow_iff Pow_UNIV[symmetric])
+ by (simp only: finite_Pow_iff Pow_UNIV[symmetric])
-lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
+lemma finite_UnionD: "finite (\<Union>A) \<Longrightarrow> finite A"
by (blast intro: finite_subset [OF subset_Pow_Union])
-lemma finite_set_of_finite_funs: assumes "finite A" "finite B"
-shows "finite{f. \<forall>x. (x \<in> A \<longrightarrow> f x \<in> B) \<and> (x \<notin> A \<longrightarrow> f x = d)}" (is "finite ?S")
-proof-
+lemma finite_set_of_finite_funs:
+ assumes "finite A" "finite B"
+ shows "finite {f. \<forall>x. (x \<in> A \<longrightarrow> f x \<in> B) \<and> (x \<notin> A \<longrightarrow> f x = d)}" (is "finite ?S")
+proof -
let ?F = "\<lambda>f. {(a,b). a \<in> A \<and> b = f a}"
- have "?F ` ?S \<subseteq> Pow(A \<times> B)" by auto
- from finite_subset[OF this] assms have 1: "finite (?F ` ?S)" by simp
+ have "?F ` ?S \<subseteq> Pow(A \<times> B)"
+ by auto
+ from finite_subset[OF this] assms have 1: "finite (?F ` ?S)"
+ by simp
have 2: "inj_on ?F ?S"
- by(fastforce simp add: inj_on_def set_eq_iff fun_eq_iff)
- show ?thesis by(rule finite_imageD[OF 1 2])
+ by (fastforce simp add: inj_on_def set_eq_iff fun_eq_iff)
+ show ?thesis
+ by (rule finite_imageD [OF 1 2])
qed
lemma not_finite_existsD:
assumes "\<not> finite {a. P a}"
shows "\<exists>a. P a"
proof (rule classical)
- assume "\<not> (\<exists>a. P a)"
+ assume "\<not> ?thesis"
with assms show ?thesis by auto
qed
@@ -496,11 +513,13 @@
assumes "\<And>x. P {x}"
and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
shows "P F"
-using assms
+ using assms
proof induct
- case empty then show ?case by simp
+ case empty
+ then show ?case by simp
next
- case (insert x F) then show ?case by cases auto
+ case (insert x F)
+ then show ?case by cases auto
qed
lemma finite_subset_induct [consumes 2, case_names empty insert]:
@@ -508,13 +527,12 @@
assumes empty: "P {}"
and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"
shows "P F"
-using \<open>finite F\<close> \<open>F \<subseteq> A\<close>
+ using \<open>finite F\<close> \<open>F \<subseteq> A\<close>
proof induct
show "P {}" by fact
next
fix x F
- assume "finite F" and "x \<notin> F" and
- P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
+ assume "finite F" and "x \<notin> F" and P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
show "P (insert x F)"
proof (rule insert)
from i show "x \<in> A" by blast
@@ -531,11 +549,10 @@
and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
shows "P {}"
proof -
- have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
+ have "P (A - B)" if "B \<subseteq> A" for B :: "'a set"
proof -
- fix B :: "'a set"
- assume "B \<subseteq> A"
- with \<open>finite A\<close> have "finite B" by (rule rev_finite_subset)
+ from \<open>finite A\<close> that have "finite B"
+ by (rule rev_finite_subset)
from this \<open>B \<subseteq> A\<close> show "P (A - B)"
proof induct
case empty
@@ -544,11 +561,15 @@
case (insert b B)
have "P (A - B - {b})"
proof (rule remove)
- from \<open>finite A\<close> show "finite (A - B)" by induct auto
- from insert show "b \<in> A - B" by simp
- from insert show "P (A - B)" by simp
+ from \<open>finite A\<close> show "finite (A - B)"
+ by induct auto
+ from insert show "b \<in> A - B"
+ by simp
+ from insert show "P (A - B)"
+ by simp
qed
- also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
+ also have "A - B - {b} = A - insert b B"
+ by (rule Diff_insert [symmetric])
finally show ?case .
qed
qed
@@ -558,11 +579,13 @@
lemma finite_update_induct [consumes 1, case_names const update]:
assumes finite: "finite {a. f a \<noteq> c}"
- assumes const: "P (\<lambda>a. c)"
- assumes update: "\<And>a b f. finite {a. f a \<noteq> c} \<Longrightarrow> f a = c \<Longrightarrow> b \<noteq> c \<Longrightarrow> P f \<Longrightarrow> P (f(a := b))"
+ and const: "P (\<lambda>a. c)"
+ and update: "\<And>a b f. finite {a. f a \<noteq> c} \<Longrightarrow> f a = c \<Longrightarrow> b \<noteq> c \<Longrightarrow> P f \<Longrightarrow> P (f(a := b))"
shows "P f"
-using finite proof (induct "{a. f a \<noteq> c}" arbitrary: f)
- case empty with const show ?case by simp
+ using finite
+proof (induct "{a. f a \<noteq> c}" arbitrary: f)
+ case empty
+ with const show ?case by simp
next
case (insert a A)
then have "A = {a'. (f(a := c)) a' \<noteq> c}" and "f a \<noteq> c"
@@ -573,7 +596,8 @@
by simp
from insert \<open>A = {a'. (f(a := c)) a' \<noteq> c}\<close> have "P (f(a := c))"
by simp
- with \<open>finite {a'. (f(a := c)) a' \<noteq> c}\<close> \<open>(f(a := c)) a = c\<close> \<open>f a \<noteq> c\<close> have "P ((f(a := c))(a := f a))"
+ with \<open>finite {a'. (f(a := c)) a' \<noteq> c}\<close> \<open>(f(a := c)) a = c\<close> \<open>f a \<noteq> c\<close>
+ have "P ((f(a := c))(a := f a))"
by (rule update)
then show ?case by simp
qed
@@ -581,8 +605,7 @@
subsection \<open>Class \<open>finite\<close>\<close>
-class finite =
- assumes finite_UNIV: "finite (UNIV :: 'a set)"
+class finite = assumes finite_UNIV: "finite (UNIV :: 'a set)"
begin
lemma finite [simp]: "finite (A :: 'a set)"
@@ -596,20 +619,22 @@
instance prod :: (finite, finite) finite
by standard (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
-lemma inj_graph: "inj (%f. {(x, y). y = f x})"
- by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
+lemma inj_graph: "inj (\<lambda>f. {(x, y). y = f x})"
+ by (rule inj_onI) (auto simp add: set_eq_iff fun_eq_iff)
instance "fun" :: (finite, finite) finite
proof
- show "finite (UNIV :: ('a => 'b) set)"
+ show "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
proof (rule finite_imageD)
- let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
- have "range ?graph \<subseteq> Pow UNIV" by simp
+ let ?graph = "\<lambda>f::'a \<Rightarrow> 'b. {(x, y). y = f x}"
+ have "range ?graph \<subseteq> Pow UNIV"
+ by simp
moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
by (simp only: finite_Pow_iff finite)
ultimately show "finite (range ?graph)"
by (rule finite_subset)
- show "inj ?graph" by (rule inj_graph)
+ show "inj ?graph"
+ by (rule inj_graph)
qed
qed
@@ -629,8 +654,8 @@
subsection \<open>A basic fold functional for finite sets\<close>
text \<open>The intended behaviour is
-\<open>fold f z {x\<^sub>1, ..., x\<^sub>n} = f x\<^sub>1 (\<dots> (f x\<^sub>n z)\<dots>)\<close>
-if \<open>f\<close> is ``left-commutative'':
+ \<open>fold f z {x\<^sub>1, \<dots>, x\<^sub>n} = f x\<^sub>1 (\<dots> (f x\<^sub>n z)\<dots>)\<close>
+ if \<open>f\<close> is ``left-commutative'':
\<close>
locale comp_fun_commute =
@@ -641,34 +666,35 @@
lemma fun_left_comm: "f y (f x z) = f x (f y z)"
using comp_fun_commute by (simp add: fun_eq_iff)
-lemma commute_left_comp:
- "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
+lemma commute_left_comp: "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
by (simp add: o_assoc comp_fun_commute)
end
inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
-for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
- emptyI [intro]: "fold_graph f z {} z" |
- insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
- \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
+ for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b
+where
+ emptyI [intro]: "fold_graph f z {} z"
+| insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
-definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
- "fold f z A = (if finite A then (THE y. fold_graph f z A y) else z)"
+definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
+ where "fold f z A = (if finite A then (THE y. fold_graph f z A y) else z)"
-text\<open>A tempting alternative for the definiens is
-@{term "if finite A then THE y. fold_graph f z A y else e"}.
-It allows the removal of finiteness assumptions from the theorems
-\<open>fold_comm\<close>, \<open>fold_reindex\<close> and \<open>fold_distrib\<close>.
-The proofs become ugly. It is not worth the effort. (???)\<close>
+text \<open>
+ A tempting alternative for the definiens is
+ @{term "if finite A then THE y. fold_graph f z A y else e"}.
+ It allows the removal of finiteness assumptions from the theorems
+ \<open>fold_comm\<close>, \<open>fold_reindex\<close> and \<open>fold_distrib\<close>.
+ The proofs become ugly. It is not worth the effort. (???)
+\<close>
lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
-by (induct rule: finite_induct) auto
+ by (induct rule: finite_induct) auto
-subsubsection\<open>From @{const fold_graph} to @{term fold}\<close>
+subsubsection \<open>From @{const fold_graph} to @{term fold}\<close>
context comp_fun_commute
begin
@@ -681,11 +707,16 @@
lemma fold_graph_insertE_aux:
"fold_graph f z A y \<Longrightarrow> a \<in> A \<Longrightarrow> \<exists>y'. y = f a y' \<and> fold_graph f z (A - {a}) y'"
proof (induct set: fold_graph)
- case (insertI x A y) show ?case
+ case emptyI
+ then show ?case by simp
+next
+ case (insertI x A y)
+ show ?case
proof (cases "x = a")
- assume "x = a" with insertI show ?case by auto
+ case True
+ with insertI show ?thesis by auto
next
- assume "x \<noteq> a"
+ case False
then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
using insertI by auto
have "f x y = f a (f x y')"
@@ -693,86 +724,96 @@
moreover have "fold_graph f z (insert x A - {a}) (f x y')"
using y' and \<open>x \<noteq> a\<close> and \<open>x \<notin> A\<close>
by (simp add: insert_Diff_if fold_graph.insertI)
- ultimately show ?case by fast
+ ultimately show ?thesis
+ by fast
qed
-qed simp
+qed
lemma fold_graph_insertE:
assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
obtains y where "v = f x y" and "fold_graph f z A y"
-using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
+ using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
-lemma fold_graph_determ:
- "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
+lemma fold_graph_determ: "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
proof (induct arbitrary: y set: fold_graph)
+ case emptyI
+ then show ?case by fast
+next
case (insertI x A y v)
from \<open>fold_graph f z (insert x A) v\<close> and \<open>x \<notin> A\<close>
obtain y' where "v = f x y'" and "fold_graph f z A y'"
by (rule fold_graph_insertE)
- from \<open>fold_graph f z A y'\<close> have "y' = y" by (rule insertI)
- with \<open>v = f x y'\<close> show "v = f x y" by simp
-qed fast
+ from \<open>fold_graph f z A y'\<close> have "y' = y"
+ by (rule insertI)
+ with \<open>v = f x y'\<close> show "v = f x y"
+ by simp
+qed
-lemma fold_equality:
- "fold_graph f z A y \<Longrightarrow> fold f z A = y"
+lemma fold_equality: "fold_graph f z A y \<Longrightarrow> fold f z A = y"
by (cases "finite A") (auto simp add: fold_def intro: fold_graph_determ dest: fold_graph_finite)
lemma fold_graph_fold:
assumes "finite A"
shows "fold_graph f z A (fold f z A)"
proof -
- from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph)
+ from assms have "\<exists>x. fold_graph f z A x"
+ by (rule finite_imp_fold_graph)
moreover note fold_graph_determ
- ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)
- then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')
- with assms show ?thesis by (simp add: fold_def)
+ ultimately have "\<exists>!x. fold_graph f z A x"
+ by (rule ex_ex1I)
+ then have "fold_graph f z A (The (fold_graph f z A))"
+ by (rule theI')
+ with assms show ?thesis
+ by (simp add: fold_def)
qed
text \<open>The base case for \<open>fold\<close>:\<close>
-lemma (in -) fold_infinite [simp]:
- assumes "\<not> finite A"
- shows "fold f z A = z"
- using assms by (auto simp add: fold_def)
+lemma (in -) fold_infinite [simp]: "\<not> finite A \<Longrightarrow> fold f z A = z"
+ by (auto simp: fold_def)
-lemma (in -) fold_empty [simp]:
- "fold f z {} = z"
- by (auto simp add: fold_def)
+lemma (in -) fold_empty [simp]: "fold f z {} = z"
+ by (auto simp: fold_def)
-text\<open>The various recursion equations for @{const fold}:\<close>
+text \<open>The various recursion equations for @{const fold}:\<close>
lemma fold_insert [simp]:
assumes "finite A" and "x \<notin> A"
shows "fold f z (insert x A) = f x (fold f z A)"
proof (rule fold_equality)
fix z
- from \<open>finite A\<close> have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
- with \<open>x \<notin> A\<close> have "fold_graph f z (insert x A) (f x (fold f z A))" by (rule fold_graph.insertI)
- then show "fold_graph f z (insert x A) (f x (fold f z A))" by simp
+ from \<open>finite A\<close> have "fold_graph f z A (fold f z A)"
+ by (rule fold_graph_fold)
+ with \<open>x \<notin> A\<close> have "fold_graph f z (insert x A) (f x (fold f z A))"
+ by (rule fold_graph.insertI)
+ then show "fold_graph f z (insert x A) (f x (fold f z A))"
+ by simp
qed
declare (in -) empty_fold_graphE [rule del] fold_graph.intros [rule del]
\<comment> \<open>No more proofs involve these.\<close>
-lemma fold_fun_left_comm:
- "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
+lemma fold_fun_left_comm: "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
proof (induct rule: finite_induct)
- case empty then show ?case by simp
+ case empty
+ then show ?case by simp
next
- case (insert y A) then show ?case
+ case insert
+ then show ?case
by (simp add: fun_left_comm [of x])
qed
-lemma fold_insert2:
- "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
+lemma fold_insert2: "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
by (simp add: fold_fun_left_comm)
lemma fold_rec:
assumes "finite A" and "x \<in> A"
shows "fold f z A = f x (fold f z (A - {x}))"
proof -
- have A: "A = insert x (A - {x})" using \<open>x \<in> A\<close> by blast
- then have "fold f z A = fold f z (insert x (A - {x}))" by simp
+ have A: "A = insert x (A - {x})"
+ using \<open>x \<in> A\<close> by blast
+ then have "fold f z A = fold f z (insert x (A - {x}))"
+ by simp
also have "\<dots> = f x (fold f z (A - {x}))"
by (rule fold_insert) (simp add: \<open>finite A\<close>)+
finally show ?thesis .
@@ -782,27 +823,32 @@
assumes "finite A"
shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
proof -
- from \<open>finite A\<close> have "finite (insert x A)" by auto
- moreover have "x \<in> insert x A" by auto
+ from \<open>finite A\<close> have "finite (insert x A)"
+ by auto
+ moreover have "x \<in> insert x A"
+ by auto
ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
by (rule fold_rec)
- then show ?thesis by simp
+ then show ?thesis
+ by simp
qed
lemma fold_set_union_disj:
assumes "finite A" "finite B" "A \<inter> B = {}"
shows "Finite_Set.fold f z (A \<union> B) = Finite_Set.fold f (Finite_Set.fold f z A) B"
-using assms(2,1,3) by induction simp_all
+ using assms(2,1,3) by induct simp_all
end
-text\<open>Other properties of @{const fold}:\<close>
+text \<open>Other properties of @{const fold}:\<close>
lemma fold_image:
assumes "inj_on g A"
shows "fold f z (g ` A) = fold (f \<circ> g) z A"
proof (cases "finite A")
- case False with assms show ?thesis by (auto dest: finite_imageD simp add: fold_def)
+ case False
+ with assms show ?thesis
+ by (auto dest: finite_imageD simp add: fold_def)
next
case True
have "fold_graph f z (g ` A) = fold_graph (f \<circ> g) z A"
@@ -810,48 +856,63 @@
fix w
show "fold_graph f z (g ` A) w \<longleftrightarrow> fold_graph (f \<circ> g) z A w" (is "?P \<longleftrightarrow> ?Q")
proof
- assume ?P then show ?Q using assms
+ assume ?P
+ then show ?Q
+ using assms
proof (induct "g ` A" w arbitrary: A)
- case emptyI then show ?case by (auto intro: fold_graph.emptyI)
+ case emptyI
+ then show ?case by (auto intro: fold_graph.emptyI)
next
case (insertI x A r B)
- from \<open>inj_on g B\<close> \<open>x \<notin> A\<close> \<open>insert x A = image g B\<close> obtain x' A' where
- "x' \<notin> A'" and [simp]: "B = insert x' A'" "x = g x'" "A = g ` A'"
+ from \<open>inj_on g B\<close> \<open>x \<notin> A\<close> \<open>insert x A = image g B\<close> obtain x' A'
+ where "x' \<notin> A'" and [simp]: "B = insert x' A'" "x = g x'" "A = g ` A'"
by (rule inj_img_insertE)
- from insertI.prems have "fold_graph (f o g) z A' r"
+ from insertI.prems have "fold_graph (f \<circ> g) z A' r"
by (auto intro: insertI.hyps)
with \<open>x' \<notin> A'\<close> have "fold_graph (f \<circ> g) z (insert x' A') ((f \<circ> g) x' r)"
by (rule fold_graph.insertI)
- then show ?case by simp
+ then show ?case
+ by simp
qed
next
- assume ?Q then show ?P using assms
+ assume ?Q
+ then show ?P
+ using assms
proof induct
- case emptyI thus ?case by (auto intro: fold_graph.emptyI)
+ case emptyI
+ then show ?case
+ by (auto intro: fold_graph.emptyI)
next
case (insertI x A r)
- from \<open>x \<notin> A\<close> insertI.prems have "g x \<notin> g ` A" by auto
- moreover from insertI have "fold_graph f z (g ` A) r" by simp
+ from \<open>x \<notin> A\<close> insertI.prems have "g x \<notin> g ` A"
+ by auto
+ moreover from insertI have "fold_graph f z (g ` A) r"
+ by simp
ultimately have "fold_graph f z (insert (g x) (g ` A)) (f (g x) r)"
by (rule fold_graph.insertI)
- then show ?case by simp
+ then show ?case
+ by simp
qed
qed
qed
- with True assms show ?thesis by (auto simp add: fold_def)
+ with True assms show ?thesis
+ by (auto simp add: fold_def)
qed
lemma fold_cong:
assumes "comp_fun_commute f" "comp_fun_commute g"
- assumes "finite A" and cong: "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
+ and "finite A"
+ and cong: "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
and "s = t" and "A = B"
shows "fold f s A = fold g t B"
proof -
- have "fold f s A = fold g s A"
- using \<open>finite A\<close> cong proof (induct A)
- case empty then show ?case by simp
+ have "fold f s A = fold g s A"
+ using \<open>finite A\<close> cong
+ proof (induct A)
+ case empty
+ then show ?case by simp
next
- case (insert x A)
+ case insert
interpret f: comp_fun_commute f by (fact \<open>comp_fun_commute f\<close>)
interpret g: comp_fun_commute g by (fact \<open>comp_fun_commute g\<close>)
from insert show ?case by simp
@@ -874,16 +935,19 @@
shows "fold f z (insert x A) = f x (fold f z A)"
proof cases
assume "x \<in> A"
- then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
- then show ?thesis using assms by (simp add: comp_fun_idem fun_left_idem)
+ then obtain B where "A = insert x B" and "x \<notin> B"
+ by (rule set_insert)
+ then show ?thesis
+ using assms by (simp add: comp_fun_idem fun_left_idem)
next
- assume "x \<notin> A" then show ?thesis using assms by simp
+ assume "x \<notin> A"
+ then show ?thesis
+ using assms by simp
qed
declare fold_insert [simp del] fold_insert_idem [simp]
-lemma fold_insert_idem2:
- "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
+lemma fold_insert_idem2: "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
by (simp add: fold_fun_left_comm)
end
@@ -891,50 +955,54 @@
subsubsection \<open>Liftings to \<open>comp_fun_commute\<close> etc.\<close>
-lemma (in comp_fun_commute) comp_comp_fun_commute:
- "comp_fun_commute (f \<circ> g)"
-proof
-qed (simp_all add: comp_fun_commute)
+lemma (in comp_fun_commute) comp_comp_fun_commute: "comp_fun_commute (f \<circ> g)"
+ by standard (simp_all add: comp_fun_commute)
-lemma (in comp_fun_idem) comp_comp_fun_idem:
- "comp_fun_idem (f \<circ> g)"
+lemma (in comp_fun_idem) comp_comp_fun_idem: "comp_fun_idem (f \<circ> g)"
by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)
(simp_all add: comp_fun_idem)
-lemma (in comp_fun_commute) comp_fun_commute_funpow:
- "comp_fun_commute (\<lambda>x. f x ^^ g x)"
+lemma (in comp_fun_commute) comp_fun_commute_funpow: "comp_fun_commute (\<lambda>x. f x ^^ g x)"
proof
- fix y x
- show "f y ^^ g y \<circ> f x ^^ g x = f x ^^ g x \<circ> f y ^^ g y"
+ show "f y ^^ g y \<circ> f x ^^ g x = f x ^^ g x \<circ> f y ^^ g y" for x y
proof (cases "x = y")
- case True then show ?thesis by simp
+ case True
+ then show ?thesis by simp
next
- case False show ?thesis
+ case False
+ show ?thesis
proof (induct "g x" arbitrary: g)
- case 0 then show ?case by simp
+ case 0
+ then show ?case by simp
next
case (Suc n g)
have hyp1: "f y ^^ g y \<circ> f x = f x \<circ> f y ^^ g y"
proof (induct "g y" arbitrary: g)
- case 0 then show ?case by simp
+ case 0
+ then show ?case by simp
next
case (Suc n g)
define h where "h z = g z - 1" for z
- with Suc have "n = h y" by simp
+ with Suc have "n = h y"
+ by simp
with Suc have hyp: "f y ^^ h y \<circ> f x = f x \<circ> f y ^^ h y"
by auto
- from Suc h_def have "g y = Suc (h y)" by simp
- then show ?case by (simp add: comp_assoc hyp)
- (simp add: o_assoc comp_fun_commute)
+ from Suc h_def have "g y = Suc (h y)"
+ by simp
+ then show ?case
+ by (simp add: comp_assoc hyp) (simp add: o_assoc comp_fun_commute)
qed
define h where "h z = (if z = x then g x - 1 else g z)" for z
- with Suc have "n = h x" by simp
+ with Suc have "n = h x"
+ by simp
with Suc have "f y ^^ h y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ h y"
by auto
- with False h_def have hyp2: "f y ^^ g y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ g y" by simp
- from Suc h_def have "g x = Suc (h x)" by simp
- then show ?case by (simp del: funpow.simps add: funpow_Suc_right o_assoc hyp2)
- (simp add: comp_assoc hyp1)
+ with False h_def have hyp2: "f y ^^ g y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ g y"
+ by simp
+ from Suc h_def have "g x = Suc (h x)"
+ by simp
+ then show ?case
+ by (simp del: funpow.simps add: funpow_Suc_right o_assoc hyp2) (simp add: comp_assoc hyp1)
qed
qed
qed
@@ -942,51 +1010,45 @@
subsubsection \<open>Expressing set operations via @{const fold}\<close>
-lemma comp_fun_commute_const:
- "comp_fun_commute (\<lambda>_. f)"
-proof
-qed rule
+lemma comp_fun_commute_const: "comp_fun_commute (\<lambda>_. f)"
+ by standard rule
-lemma comp_fun_idem_insert:
- "comp_fun_idem insert"
-proof
-qed auto
+lemma comp_fun_idem_insert: "comp_fun_idem insert"
+ by standard auto
-lemma comp_fun_idem_remove:
- "comp_fun_idem Set.remove"
-proof
-qed auto
+lemma comp_fun_idem_remove: "comp_fun_idem Set.remove"
+ by standard auto
-lemma (in semilattice_inf) comp_fun_idem_inf:
- "comp_fun_idem inf"
-proof
-qed (auto simp add: inf_left_commute)
+lemma (in semilattice_inf) comp_fun_idem_inf: "comp_fun_idem inf"
+ by standard (auto simp add: inf_left_commute)
-lemma (in semilattice_sup) comp_fun_idem_sup:
- "comp_fun_idem sup"
-proof
-qed (auto simp add: sup_left_commute)
+lemma (in semilattice_sup) comp_fun_idem_sup: "comp_fun_idem sup"
+ by standard (auto simp add: sup_left_commute)
lemma union_fold_insert:
assumes "finite A"
shows "A \<union> B = fold insert B A"
proof -
- interpret comp_fun_idem insert by (fact comp_fun_idem_insert)
- from \<open>finite A\<close> show ?thesis by (induct A arbitrary: B) simp_all
+ interpret comp_fun_idem insert
+ by (fact comp_fun_idem_insert)
+ from \<open>finite A\<close> show ?thesis
+ by (induct A arbitrary: B) simp_all
qed
lemma minus_fold_remove:
assumes "finite A"
shows "B - A = fold Set.remove B A"
proof -
- interpret comp_fun_idem Set.remove by (fact comp_fun_idem_remove)
- from \<open>finite A\<close> have "fold Set.remove B A = B - A" by (induct A arbitrary: B) auto
+ interpret comp_fun_idem Set.remove
+ by (fact comp_fun_idem_remove)
+ from \<open>finite A\<close> have "fold Set.remove B A = B - A"
+ by (induct A arbitrary: B) auto
then show ?thesis ..
qed
lemma comp_fun_commute_filter_fold:
"comp_fun_commute (\<lambda>x A'. if P x then Set.insert x A' else A')"
-proof -
+proof -
interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert)
show ?thesis by standard (auto simp: fun_eq_iff)
qed
@@ -994,77 +1056,79 @@
lemma Set_filter_fold:
assumes "finite A"
shows "Set.filter P A = fold (\<lambda>x A'. if P x then Set.insert x A' else A') {} A"
-using assms
-by (induct A)
- (auto simp add: Set.filter_def comp_fun_commute.fold_insert[OF comp_fun_commute_filter_fold])
+ using assms
+ by induct
+ (auto simp add: Set.filter_def comp_fun_commute.fold_insert[OF comp_fun_commute_filter_fold])
-lemma inter_Set_filter:
+lemma inter_Set_filter:
assumes "finite B"
shows "A \<inter> B = Set.filter (\<lambda>x. x \<in> A) B"
-using assms
-by (induct B) (auto simp: Set.filter_def)
+ using assms
+ by induct (auto simp: Set.filter_def)
lemma image_fold_insert:
assumes "finite A"
shows "image f A = fold (\<lambda>k A. Set.insert (f k) A) {} A"
-using assms
proof -
- interpret comp_fun_commute "\<lambda>k A. Set.insert (f k) A" by standard auto
- show ?thesis using assms by (induct A) auto
+ interpret comp_fun_commute "\<lambda>k A. Set.insert (f k) A"
+ by standard auto
+ show ?thesis
+ using assms by (induct A) auto
qed
lemma Ball_fold:
assumes "finite A"
shows "Ball A P = fold (\<lambda>k s. s \<and> P k) True A"
-using assms
proof -
- interpret comp_fun_commute "\<lambda>k s. s \<and> P k" by standard auto
- show ?thesis using assms by (induct A) auto
+ interpret comp_fun_commute "\<lambda>k s. s \<and> P k"
+ by standard auto
+ show ?thesis
+ using assms by (induct A) auto
qed
lemma Bex_fold:
assumes "finite A"
shows "Bex A P = fold (\<lambda>k s. s \<or> P k) False A"
-using assms
proof -
- interpret comp_fun_commute "\<lambda>k s. s \<or> P k" by standard auto
- show ?thesis using assms by (induct A) auto
+ interpret comp_fun_commute "\<lambda>k s. s \<or> P k"
+ by standard auto
+ show ?thesis
+ using assms by (induct A) auto
qed
-lemma comp_fun_commute_Pow_fold:
- "comp_fun_commute (\<lambda>x A. A \<union> Set.insert x ` A)"
+lemma comp_fun_commute_Pow_fold: "comp_fun_commute (\<lambda>x A. A \<union> Set.insert x ` A)"
by (clarsimp simp: fun_eq_iff comp_fun_commute_def) blast
lemma Pow_fold:
assumes "finite A"
shows "Pow A = fold (\<lambda>x A. A \<union> Set.insert x ` A) {{}} A"
-using assms
proof -
- interpret comp_fun_commute "\<lambda>x A. A \<union> Set.insert x ` A" by (rule comp_fun_commute_Pow_fold)
- show ?thesis using assms by (induct A) (auto simp: Pow_insert)
+ interpret comp_fun_commute "\<lambda>x A. A \<union> Set.insert x ` A"
+ by (rule comp_fun_commute_Pow_fold)
+ show ?thesis
+ using assms by (induct A) (auto simp: Pow_insert)
qed
lemma fold_union_pair:
assumes "finite B"
shows "(\<Union>y\<in>B. {(x, y)}) \<union> A = fold (\<lambda>y. Set.insert (x, y)) A B"
proof -
- interpret comp_fun_commute "\<lambda>y. Set.insert (x, y)" by standard auto
- show ?thesis using assms by (induct B arbitrary: A) simp_all
+ interpret comp_fun_commute "\<lambda>y. Set.insert (x, y)"
+ by standard auto
+ show ?thesis
+ using assms by (induct arbitrary: A) simp_all
qed
-lemma comp_fun_commute_product_fold:
- assumes "finite B"
- shows "comp_fun_commute (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B)"
- by standard (auto simp: fold_union_pair[symmetric] assms)
+lemma comp_fun_commute_product_fold:
+ "finite B \<Longrightarrow> comp_fun_commute (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B)"
+ by standard (auto simp: fold_union_pair [symmetric])
lemma product_fold:
- assumes "finite A"
- assumes "finite B"
+ assumes "finite A" "finite B"
shows "A \<times> B = fold (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B) {} A"
-using assms unfolding Sigma_def
-by (induct A)
- (simp_all add: comp_fun_commute.fold_insert[OF comp_fun_commute_product_fold] fold_union_pair)
-
+ using assms unfolding Sigma_def
+ by (induct A)
+ (simp_all add: comp_fun_commute.fold_insert[OF comp_fun_commute_product_fold] fold_union_pair)
context complete_lattice
begin
@@ -1073,61 +1137,55 @@
assumes "finite A"
shows "inf (Inf A) B = fold inf B A"
proof -
- interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
- from \<open>finite A\<close> fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
- (simp_all add: inf_commute fun_eq_iff)
+ interpret comp_fun_idem inf
+ by (fact comp_fun_idem_inf)
+ from \<open>finite A\<close> fold_fun_left_comm show ?thesis
+ by (induct A arbitrary: B) (simp_all add: inf_commute fun_eq_iff)
qed
lemma sup_Sup_fold_sup:
assumes "finite A"
shows "sup (Sup A) B = fold sup B A"
proof -
- interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
- from \<open>finite A\<close> fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
- (simp_all add: sup_commute fun_eq_iff)
+ interpret comp_fun_idem sup
+ by (fact comp_fun_idem_sup)
+ from \<open>finite A\<close> fold_fun_left_comm show ?thesis
+ by (induct A arbitrary: B) (simp_all add: sup_commute fun_eq_iff)
qed
-lemma Inf_fold_inf:
- assumes "finite A"
- shows "Inf A = fold inf top A"
- using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
+lemma Inf_fold_inf: "finite A \<Longrightarrow> Inf A = fold inf top A"
+ using inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
-lemma Sup_fold_sup:
- assumes "finite A"
- shows "Sup A = fold sup bot A"
- using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
+lemma Sup_fold_sup: "finite A \<Longrightarrow> Sup A = fold sup bot A"
+ using sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
lemma inf_INF_fold_inf:
assumes "finite A"
- shows "inf B (INFIMUM A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold")
-proof (rule sym)
+ shows "inf B (INFIMUM A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold")
+proof -
interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
- from \<open>finite A\<close> show "?fold = ?inf"
- by (induct A arbitrary: B)
- (simp_all add: inf_left_commute)
+ from \<open>finite A\<close> have "?fold = ?inf"
+ by (induct A arbitrary: B) (simp_all add: inf_left_commute)
+ then show ?thesis ..
qed
lemma sup_SUP_fold_sup:
assumes "finite A"
- shows "sup B (SUPREMUM A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold")
-proof (rule sym)
+ shows "sup B (SUPREMUM A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold")
+proof -
interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
- from \<open>finite A\<close> show "?fold = ?sup"
- by (induct A arbitrary: B)
- (simp_all add: sup_left_commute)
+ from \<open>finite A\<close> have "?fold = ?sup"
+ by (induct A arbitrary: B) (simp_all add: sup_left_commute)
+ then show ?thesis ..
qed
-lemma INF_fold_inf:
- assumes "finite A"
- shows "INFIMUM A f = fold (inf \<circ> f) top A"
- using assms inf_INF_fold_inf [of A top] by simp
+lemma INF_fold_inf: "finite A \<Longrightarrow> INFIMUM A f = fold (inf \<circ> f) top A"
+ using inf_INF_fold_inf [of A top] by simp
-lemma SUP_fold_sup:
- assumes "finite A"
- shows "SUPREMUM A f = fold (sup \<circ> f) bot A"
- using assms sup_SUP_fold_sup [of A bot] by simp
+lemma SUP_fold_sup: "finite A \<Longrightarrow> SUPREMUM A f = fold (sup \<circ> f) bot A"
+ using sup_SUP_fold_sup [of A bot] by simp
end
@@ -1146,15 +1204,14 @@
by standard (insert comp_fun_commute, simp add: fun_eq_iff)
definition F :: "'a set \<Rightarrow> 'b"
-where
- eq_fold: "F A = fold f z A"
+ where eq_fold: "F A = fold f z A"
lemma empty [simp]:"F {} = z"
by (simp add: eq_fold)
lemma infinite [simp]: "\<not> finite A \<Longrightarrow> F A = z"
by (simp add: eq_fold)
-
+
lemma insert [simp]:
assumes "finite A" and "x \<notin> A"
shows "F (insert x A) = f x (F A)"
@@ -1163,7 +1220,7 @@
have "fold f z (insert x A) = f x (fold f z A)" by simp
with \<open>finite A\<close> show ?thesis by (simp add: eq_fold fun_eq_iff)
qed
-
+
lemma remove:
assumes "finite A" and "x \<in> A"
shows "F A = f x (F (A - {x}))"
@@ -1174,10 +1231,8 @@
ultimately show ?thesis by simp
qed
-lemma insert_remove:
- assumes "finite A"
- shows "F (insert x A) = f x (F (A - {x}))"
- using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
+lemma insert_remove: "finite A \<Longrightarrow> F (insert x A) = f x (F (A - {x}))"
+ by (cases "x \<in> A") (simp_all add: remove insert_absorb)
end
@@ -1209,7 +1264,7 @@
text \<open>
The traditional definition
- @{prop "card A \<equiv> LEAST n. EX f. A = {f i | i. i < n}"}
+ @{prop "card A \<equiv> LEAST n. \<exists>f. A = {f i |i. i < n}"}
is ugly to work with.
But now that we have @{const fold} things are easy:
\<close>
@@ -1218,60 +1273,49 @@
defines card = "folding.F (\<lambda>_. Suc) 0"
by standard rule
-lemma card_infinite:
- "\<not> finite A \<Longrightarrow> card A = 0"
+lemma card_infinite: "\<not> finite A \<Longrightarrow> card A = 0"
by (fact card.infinite)
-lemma card_empty:
- "card {} = 0"
+lemma card_empty: "card {} = 0"
by (fact card.empty)
-lemma card_insert_disjoint:
- "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> card (insert x A) = Suc (card A)"
+lemma card_insert_disjoint: "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> card (insert x A) = Suc (card A)"
by (fact card.insert)
-lemma card_insert_if:
- "finite A \<Longrightarrow> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
+lemma card_insert_if: "finite A \<Longrightarrow> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
by auto (simp add: card.insert_remove card.remove)
-lemma card_ge_0_finite:
- "card A > 0 \<Longrightarrow> finite A"
+lemma card_ge_0_finite: "card A > 0 \<Longrightarrow> finite A"
by (rule ccontr) simp
-lemma card_0_eq [simp]:
- "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
+lemma card_0_eq [simp]: "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
by (auto dest: mk_disjoint_insert)
-lemma finite_UNIV_card_ge_0:
- "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
+lemma finite_UNIV_card_ge_0: "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
by (rule ccontr) simp
-lemma card_eq_0_iff:
- "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
+lemma card_eq_0_iff: "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
by auto
-lemma card_range_greater_zero:
- "finite (range f) \<Longrightarrow> card (range f) > 0"
+lemma card_range_greater_zero: "finite (range f) \<Longrightarrow> card (range f) > 0"
by (rule ccontr) (simp add: card_eq_0_iff)
-lemma card_gt_0_iff:
- "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
- by (simp add: neq0_conv [symmetric] card_eq_0_iff)
+lemma card_gt_0_iff: "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
+ by (simp add: neq0_conv [symmetric] card_eq_0_iff)
-lemma card_Suc_Diff1:
- "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> Suc (card (A - {x})) = card A"
-apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
-apply(simp del:insert_Diff_single)
-done
+lemma card_Suc_Diff1: "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> Suc (card (A - {x})) = card A"
+ apply (rule insert_Diff [THEN subst, where t = A])
+ apply assumption
+ apply (simp del: insert_Diff_single)
+ done
-lemma card_insert_le_m1: "n>0 \<Longrightarrow> card y \<le> n-1 \<Longrightarrow> card (insert x y) \<le> n"
+lemma card_insert_le_m1: "n > 0 \<Longrightarrow> card y \<le> n - 1 \<Longrightarrow> card (insert x y) \<le> n"
apply (cases "finite y")
apply (cases "x \<in> y")
apply (auto simp: insert_absorb)
done
-lemma card_Diff_singleton:
- "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> card (A - {x}) = card A - 1"
+lemma card_Diff_singleton: "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> card (A - {x}) = card A - 1"
by (simp add: card_Suc_Diff1 [symmetric])
lemma card_Diff_singleton_if:
@@ -1282,124 +1326,137 @@
assumes "finite A" and "a \<in> A" and "a \<notin> B"
shows "card (A - insert a B) = card (A - B) - 1"
proof -
- have "A - insert a B = (A - B) - {a}" using assms by blast
- then show ?thesis using assms by(simp add: card_Diff_singleton)
+ have "A - insert a B = (A - B) - {a}"
+ using assms by blast
+ then show ?thesis
+ using assms by (simp add: card_Diff_singleton)
qed
-lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
+lemma card_insert: "finite A \<Longrightarrow> card (insert x A) = Suc (card (A - {x}))"
by (fact card.insert_remove)
-lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
-by (simp add: card_insert_if)
+lemma card_insert_le: "finite A \<Longrightarrow> card A \<le> card (insert x A)"
+ by (simp add: card_insert_if)
-lemma card_Collect_less_nat[simp]: "card{i::nat. i < n} = n"
-by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
+lemma card_Collect_less_nat[simp]: "card {i::nat. i < n} = n"
+ by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
-lemma card_Collect_le_nat[simp]: "card{i::nat. i <= n} = Suc n"
-using card_Collect_less_nat[of "Suc n"] by(simp add: less_Suc_eq_le)
+lemma card_Collect_le_nat[simp]: "card {i::nat. i \<le> n} = Suc n"
+ using card_Collect_less_nat[of "Suc n"] by (simp add: less_Suc_eq_le)
lemma card_mono:
assumes "finite B" and "A \<subseteq> B"
shows "card A \<le> card B"
proof -
- from assms have "finite A" by (auto intro: finite_subset)
- then show ?thesis using assms proof (induct A arbitrary: B)
- case empty then show ?case by simp
+ from assms have "finite A"
+ by (auto intro: finite_subset)
+ then show ?thesis
+ using assms
+ proof (induct A arbitrary: B)
+ case empty
+ then show ?case by simp
next
case (insert x A)
- then have "x \<in> B" by simp
- from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
- with insert.hyps have "card A \<le> card (B - {x})" by auto
- with \<open>finite A\<close> \<open>x \<notin> A\<close> \<open>finite B\<close> \<open>x \<in> B\<close> show ?case by simp (simp only: card.remove)
+ then have "x \<in> B"
+ by simp
+ from insert have "A \<subseteq> B - {x}" and "finite (B - {x})"
+ by auto
+ with insert.hyps have "card A \<le> card (B - {x})"
+ by auto
+ with \<open>finite A\<close> \<open>x \<notin> A\<close> \<open>finite B\<close> \<open>x \<in> B\<close> show ?case
+ by simp (simp only: card.remove)
qed
qed
-lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
-apply (induct rule: finite_induct)
-apply simp
-apply clarify
-apply (subgoal_tac "finite A & A - {x} <= F")
- prefer 2 apply (blast intro: finite_subset, atomize)
-apply (drule_tac x = "A - {x}" in spec)
-apply (simp add: card_Diff_singleton_if split add: if_split_asm)
-apply (case_tac "card A", auto)
-done
+lemma card_seteq: "finite B \<Longrightarrow> (\<And>A. A \<subseteq> B \<Longrightarrow> card B \<le> card A \<Longrightarrow> A = B)"
+ apply (induct rule: finite_induct)
+ apply simp
+ apply clarify
+ apply (subgoal_tac "finite A \<and> A - {x} \<subseteq> F")
+ prefer 2 apply (blast intro: finite_subset, atomize)
+ apply (drule_tac x = "A - {x}" in spec)
+ apply (simp add: card_Diff_singleton_if split add: if_split_asm)
+ apply (case_tac "card A", auto)
+ done
-lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
-apply (simp add: psubset_eq linorder_not_le [symmetric])
-apply (blast dest: card_seteq)
-done
+lemma psubset_card_mono: "finite B \<Longrightarrow> A < B \<Longrightarrow> card A < card B"
+ apply (simp add: psubset_eq linorder_not_le [symmetric])
+ apply (blast dest: card_seteq)
+ done
lemma card_Un_Int:
- assumes "finite A" and "finite B"
+ assumes "finite A" "finite B"
shows "card A + card B = card (A \<union> B) + card (A \<inter> B)"
-using assms proof (induct A)
- case empty then show ?case by simp
+ using assms
+proof (induct A)
+ case empty
+ then show ?case by simp
next
- case (insert x A) then show ?case
+ case insert
+ then show ?case
by (auto simp add: insert_absorb Int_insert_left)
qed
-lemma card_Un_disjoint:
- assumes "finite A" and "finite B"
- assumes "A \<inter> B = {}"
- shows "card (A \<union> B) = card A + card B"
-using assms card_Un_Int [of A B] by simp
+lemma card_Un_disjoint: "finite A \<Longrightarrow> finite B \<Longrightarrow> A \<inter> B = {} \<Longrightarrow> card (A \<union> B) = card A + card B"
+ using card_Un_Int [of A B] by simp
lemma card_Un_le: "card (A \<union> B) \<le> card A + card B"
-apply(cases "finite A")
- apply(cases "finite B")
- using le_iff_add card_Un_Int apply blast
- apply simp
-apply simp
-done
+ apply (cases "finite A")
+ apply (cases "finite B")
+ using le_iff_add card_Un_Int apply blast
+ apply simp
+ apply simp
+ done
lemma card_Diff_subset:
- assumes "finite B" and "B \<subseteq> A"
+ assumes "finite B"
+ and "B \<subseteq> A"
shows "card (A - B) = card A - card B"
proof (cases "finite A")
- case False with assms show ?thesis by simp
+ case False
+ with assms show ?thesis
+ by simp
next
- case True with assms show ?thesis by (induct B arbitrary: A) simp_all
+ case True
+ with assms show ?thesis
+ by (induct B arbitrary: A) simp_all
qed
lemma card_Diff_subset_Int:
- assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
+ assumes "finite (A \<inter> B)"
+ shows "card (A - B) = card A - card (A \<inter> B)"
proof -
have "A - B = A - A \<inter> B" by auto
- thus ?thesis
- by (simp add: card_Diff_subset AB)
+ with assms show ?thesis
+ by (simp add: card_Diff_subset)
qed
lemma diff_card_le_card_Diff:
-assumes "finite B" shows "card A - card B \<le> card(A - B)"
-proof-
+ assumes "finite B"
+ shows "card A - card B \<le> card (A - B)"
+proof -
have "card A - card B \<le> card A - card (A \<inter> B)"
using card_mono[OF assms Int_lower2, of A] by arith
- also have "\<dots> = card(A-B)" using assms by(simp add: card_Diff_subset_Int)
+ also have "\<dots> = card (A - B)"
+ using assms by (simp add: card_Diff_subset_Int)
finally show ?thesis .
qed
-lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
-apply (rule Suc_less_SucD)
-apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
-done
+lemma card_Diff1_less: "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> card (A - {x}) < card A"
+ by (rule Suc_less_SucD) (simp add: card_Suc_Diff1 del: card_Diff_insert)
-lemma card_Diff2_less:
- "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
-apply (case_tac "x = y")
- apply (simp add: card_Diff1_less del:card_Diff_insert)
-apply (rule less_trans)
- prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
-done
+lemma card_Diff2_less: "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> card (A - {x} - {y}) < card A"
+ apply (cases "x = y")
+ apply (simp add: card_Diff1_less del:card_Diff_insert)
+ apply (rule less_trans)
+ prefer 2 apply (auto intro!: card_Diff1_less simp del: card_Diff_insert)
+ done
-lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
-apply (case_tac "x : A")
- apply (simp_all add: card_Diff1_less less_imp_le)
-done
+lemma card_Diff1_le: "finite A \<Longrightarrow> card (A - {x}) \<le> card A"
+ by (cases "x \<in> A") (simp_all add: card_Diff1_less less_imp_le)
-lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
-by (erule psubsetI, blast)
+lemma card_psubset: "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> card A < card B \<Longrightarrow> A < B"
+ by (erule psubsetI) blast
lemma card_le_inj:
assumes fA: "finite A"
@@ -1413,7 +1470,7 @@
next
case (insert x s t)
then show ?case
- proof (induct rule: finite_induct[OF "insert.prems"(1)])
+ proof (induct rule: finite_induct [OF insert.prems(1)])
case 1
then show ?case by simp
next
@@ -1454,41 +1511,43 @@
qed
lemma insert_partition:
- "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
- \<Longrightarrow> x \<inter> \<Union>F = {}"
-by auto
+ "x \<notin> F \<Longrightarrow> \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<Longrightarrow> x \<inter> \<Union>F = {}"
+ by auto
-lemma finite_psubset_induct[consumes 1, case_names psubset]:
- assumes fin: "finite A"
- and major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A"
+lemma finite_psubset_induct [consumes 1, case_names psubset]:
+ assumes finite: "finite A"
+ and major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A"
shows "P A"
-using fin
+ using finite
proof (induct A taking: card rule: measure_induct_rule)
case (less A)
have fin: "finite A" by fact
- have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
- { fix B
- assume asm: "B \<subset> A"
- from asm have "card B < card A" using psubset_card_mono fin by blast
+ have ih: "card B < card A \<Longrightarrow> finite B \<Longrightarrow> P B" for B by fact
+ have "P B" if "B \<subset> A" for B
+ proof -
+ from that have "card B < card A"
+ using psubset_card_mono fin by blast
moreover
- from asm have "B \<subseteq> A" by auto
- then have "finite B" using fin finite_subset by blast
- ultimately
- have "P B" using ih by simp
- }
+ from that have "B \<subseteq> A"
+ by auto
+ then have "finite B"
+ using fin finite_subset by blast
+ ultimately show ?thesis using ih by simp
+ qed
with fin show "P A" using major by blast
qed
-lemma finite_induct_select[consumes 1, case_names empty select]:
+lemma finite_induct_select [consumes 1, case_names empty select]:
assumes "finite S"
- assumes "P {}"
- assumes select: "\<And>T. T \<subset> S \<Longrightarrow> P T \<Longrightarrow> \<exists>s\<in>S - T. P (insert s T)"
+ and "P {}"
+ and select: "\<And>T. T \<subset> S \<Longrightarrow> P T \<Longrightarrow> \<exists>s\<in>S - T. P (insert s T)"
shows "P S"
proof -
have "0 \<le> card S" by simp
then have "\<exists>T \<subseteq> S. card T = card S \<and> P T"
proof (induct rule: dec_induct)
- case base with \<open>P {}\<close> show ?case
+ case base with \<open>P {}\<close>
+ show ?case
by (intro exI[of _ "{}"]) auto
next
case (step n)
@@ -1506,24 +1565,27 @@
qed
lemma remove_induct [case_names empty infinite remove]:
- assumes empty: "P ({} :: 'a set)" and infinite: "\<not>finite B \<Longrightarrow> P B"
- and remove: "\<And>A. finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P (A - {x})) \<Longrightarrow> P A"
+ assumes empty: "P ({} :: 'a set)"
+ and infinite: "\<not> finite B \<Longrightarrow> P B"
+ and remove: "\<And>A. finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P (A - {x})) \<Longrightarrow> P A"
shows "P B"
proof (cases "finite B")
assume "\<not>finite B"
- thus ?thesis by (rule infinite)
+ then show ?thesis by (rule infinite)
next
define A where "A = B"
assume "finite B"
- hence "finite A" "A \<subseteq> B" by (simp_all add: A_def)
- thus "P A"
- proof (induction "card A" arbitrary: A)
+ then have "finite A" "A \<subseteq> B"
+ by (simp_all add: A_def)
+ then show "P A"
+ proof (induct "card A" arbitrary: A)
case 0
- hence "A = {}" by auto
+ then have "A = {}" by auto
with empty show ?case by simp
next
case (Suc n A)
- from \<open>A \<subseteq> B\<close> and \<open>finite B\<close> have "finite A" by (rule finite_subset)
+ from \<open>A \<subseteq> B\<close> and \<open>finite B\<close> have "finite A"
+ by (rule finite_subset)
moreover from Suc.hyps have "A \<noteq> {}" by auto
moreover note \<open>A \<subseteq> B\<close>
moreover have "P (A - {x})" if x: "x \<in> A" for x
@@ -1533,92 +1595,99 @@
qed
lemma finite_remove_induct [consumes 1, case_names empty remove]:
- assumes finite: "finite B" and empty: "P ({} :: 'a set)"
- and rm: "\<And>A. finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P (A - {x})) \<Longrightarrow> P A"
+ fixes P :: "'a set \<Rightarrow> bool"
+ assumes finite: "finite B"
+ and empty: "P {}"
+ and rm: "\<And>A. finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P (A - {x})) \<Longrightarrow> P A"
defines "B' \<equiv> B"
- shows "P B'"
- by (induction B' rule: remove_induct) (simp_all add: assms)
+ shows "P B'"
+ by (induct B' rule: remove_induct) (simp_all add: assms)
-text\<open>main cardinality theorem\<close>
+text \<open>Main cardinality theorem.\<close>
lemma card_partition [rule_format]:
- "finite C ==>
- finite (\<Union>C) -->
- (\<forall>c\<in>C. card c = k) -->
- (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
- k * card(C) = card (\<Union>C)"
-apply (erule finite_induct, simp)
-apply (simp add: card_Un_disjoint insert_partition
- finite_subset [of _ "\<Union>(insert x F)"])
-done
+ "finite C \<Longrightarrow> finite (\<Union>C) \<Longrightarrow> (\<forall>c\<in>C. card c = k) \<Longrightarrow>
+ (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {}) \<Longrightarrow>
+ k * card C = card (\<Union>C)"
+ apply (induct rule: finite_induct)
+ apply simp
+ apply (simp add: card_Un_disjoint insert_partition finite_subset [of _ "\<Union>(insert x F)"])
+ done
lemma card_eq_UNIV_imp_eq_UNIV:
assumes fin: "finite (UNIV :: 'a set)"
- and card: "card A = card (UNIV :: 'a set)"
+ and card: "card A = card (UNIV :: 'a set)"
shows "A = (UNIV :: 'a set)"
proof
show "A \<subseteq> UNIV" by simp
show "UNIV \<subseteq> A"
proof
- fix x
- show "x \<in> A"
+ show "x \<in> A" for x
proof (rule ccontr)
assume "x \<notin> A"
then have "A \<subset> UNIV" by auto
- with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
+ with fin have "card A < card (UNIV :: 'a set)"
+ by (fact psubset_card_mono)
with card show False by simp
qed
qed
qed
-text\<open>The form of a finite set of given cardinality\<close>
+text \<open>The form of a finite set of given cardinality\<close>
lemma card_eq_SucD:
-assumes "card A = Suc k"
-shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
+ assumes "card A = Suc k"
+ shows "\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {})"
proof -
- have fin: "finite A" using assms by (auto intro: ccontr)
- moreover have "card A \<noteq> 0" using assms by auto
- ultimately obtain b where b: "b \<in> A" by auto
+ have fin: "finite A"
+ using assms by (auto intro: ccontr)
+ moreover have "card A \<noteq> 0"
+ using assms by auto
+ ultimately obtain b where b: "b \<in> A"
+ by auto
show ?thesis
proof (intro exI conjI)
- show "A = insert b (A-{b})" using b by blast
- show "b \<notin> A - {b}" by blast
+ show "A = insert b (A - {b})"
+ using b by blast
+ show "b \<notin> A - {b}"
+ by blast
show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
using assms b fin by(fastforce dest:mk_disjoint_insert)+
qed
qed
lemma card_Suc_eq:
- "(card A = Suc k) =
- (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
- apply(auto elim!: card_eq_SucD)
- apply(subst card.insert)
- apply(auto simp add: intro:ccontr)
- done
+ "card A = Suc k \<longleftrightarrow>
+ (\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {}))"
+ apply (auto elim!: card_eq_SucD)
+ apply (subst card.insert)
+ apply (auto simp add: intro:ccontr)
+ done
lemma card_1_singletonE:
- assumes "card A = 1" obtains x where "A = {x}"
+ assumes "card A = 1"
+ obtains x where "A = {x}"
using assms by (auto simp: card_Suc_eq)
lemma is_singleton_altdef: "is_singleton A \<longleftrightarrow> card A = 1"
unfolding is_singleton_def
by (auto elim!: card_1_singletonE is_singletonE simp del: One_nat_def)
-lemma card_le_Suc_iff: "finite A \<Longrightarrow>
- Suc n \<le> card A = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
-by (fastforce simp: card_Suc_eq less_eq_nat.simps(2) insert_eq_iff
- dest: subset_singletonD split: nat.splits if_splits)
+lemma card_le_Suc_iff:
+ "finite A \<Longrightarrow> Suc n \<le> card A = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
+ by (fastforce simp: card_Suc_eq less_eq_nat.simps(2) insert_eq_iff
+ dest: subset_singletonD split: nat.splits if_splits)
lemma finite_fun_UNIVD2:
assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
shows "finite (UNIV :: 'b set)"
proof -
- from fin have "\<And>arbitrary. finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
+ from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))" for arbitrary
by (rule finite_imageI)
- moreover have "\<And>arbitrary. UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
+ moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)" for arbitrary
by (rule UNIV_eq_I) auto
- ultimately show "finite (UNIV :: 'b set)" by simp
+ ultimately show "finite (UNIV :: 'b set)"
+ by simp
qed
lemma card_UNIV_unit [simp]: "card (UNIV :: unit set) = 1"
@@ -1628,164 +1697,176 @@
assumes "\<not> finite A"
shows "\<exists>B. finite B \<and> card B = n \<and> B \<subseteq> A"
proof (induction n)
- case 0 show ?case by (intro exI[of _ "{}"]) auto
-next
+ case 0
+ show ?case by (intro exI[of _ "{}"]) auto
+next
case (Suc n)
- then guess B .. note B = this
+ then obtain B where B: "finite B \<and> card B = n \<and> B \<subseteq> A" ..
with \<open>\<not> finite A\<close> have "A \<noteq> B" by auto
with B have "B \<subset> A" by auto
- hence "\<exists>x. x \<in> A - B" by (elim psubset_imp_ex_mem)
- then guess x .. note x = this
+ then have "\<exists>x. x \<in> A - B"
+ by (elim psubset_imp_ex_mem)
+ then obtain x where x: "x \<in> A - B" ..
with B have "finite (insert x B) \<and> card (insert x B) = Suc n \<and> insert x B \<subseteq> A"
by auto
- thus "\<exists>B. finite B \<and> card B = Suc n \<and> B \<subseteq> A" ..
+ then show "\<exists>B. finite B \<and> card B = Suc n \<and> B \<subseteq> A" ..
qed
+
subsubsection \<open>Cardinality of image\<close>
-lemma card_image_le: "finite A ==> card (f ` A) \<le> card A"
+lemma card_image_le: "finite A \<Longrightarrow> card (f ` A) \<le> card A"
by (induct rule: finite_induct) (simp_all add: le_SucI card_insert_if)
lemma card_image:
assumes "inj_on f A"
shows "card (f ` A) = card A"
proof (cases "finite A")
- case True then show ?thesis using assms by (induct A) simp_all
+ case True
+ then show ?thesis
+ using assms by (induct A) simp_all
next
- case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
- with False show ?thesis by simp
+ case False
+ then have "\<not> finite (f ` A)"
+ using assms by (auto dest: finite_imageD)
+ with False show ?thesis
+ by simp
qed
lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
-by(auto simp: card_image bij_betw_def)
+ by(auto simp: card_image bij_betw_def)
-lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
-by (simp add: card_seteq card_image)
+lemma endo_inj_surj: "finite A \<Longrightarrow> f ` A \<subseteq> A \<Longrightarrow> inj_on f A \<Longrightarrow> f ` A = A"
+ by (simp add: card_seteq card_image)
lemma eq_card_imp_inj_on:
- assumes "finite A" "card(f ` A) = card A" shows "inj_on f A"
-using assms
+ assumes "finite A" "card(f ` A) = card A"
+ shows "inj_on f A"
+ using assms
proof (induct rule:finite_induct)
- case empty show ?case by simp
+ case empty
+ show ?case by simp
next
case (insert x A)
- then show ?case using card_image_le [of A f]
- by (simp add: card_insert_if split: if_splits)
+ then show ?case
+ using card_image_le [of A f] by (simp add: card_insert_if split: if_splits)
qed
-lemma inj_on_iff_eq_card: "finite A \<Longrightarrow> inj_on f A \<longleftrightarrow> card(f ` A) = card A"
+lemma inj_on_iff_eq_card: "finite A \<Longrightarrow> inj_on f A \<longleftrightarrow> card (f ` A) = card A"
by (blast intro: card_image eq_card_imp_inj_on)
lemma card_inj_on_le:
- assumes "inj_on f A" "f ` A \<subseteq> B" "finite B" shows "card A \<le> card B"
+ assumes "inj_on f A" "f ` A \<subseteq> B" "finite B"
+ shows "card A \<le> card B"
proof -
- have "finite A" using assms
- by (blast intro: finite_imageD dest: finite_subset)
- then show ?thesis using assms
- by (force intro: card_mono simp: card_image [symmetric])
+ have "finite A"
+ using assms by (blast intro: finite_imageD dest: finite_subset)
+ then show ?thesis
+ using assms by (force intro: card_mono simp: card_image [symmetric])
qed
lemma surj_card_le: "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> card B \<le> card A"
by (blast intro: card_image_le card_mono le_trans)
lemma card_bij_eq:
- "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
- finite A; finite B |] ==> card A = card B"
-by (auto intro: le_antisym card_inj_on_le)
+ "inj_on f A \<Longrightarrow> f ` A \<subseteq> B \<Longrightarrow> inj_on g B \<Longrightarrow> g ` B \<subseteq> A \<Longrightarrow> finite A \<Longrightarrow> finite B
+ \<Longrightarrow> card A = card B"
+ by (auto intro: le_antisym card_inj_on_le)
+
+lemma bij_betw_finite: "bij_betw f A B \<Longrightarrow> finite A \<longleftrightarrow> finite B"
+ unfolding bij_betw_def using finite_imageD [of f A] by auto
-lemma bij_betw_finite:
- assumes "bij_betw f A B"
- shows "finite A \<longleftrightarrow> finite B"
-using assms unfolding bij_betw_def
-using finite_imageD[of f A] by auto
+lemma inj_on_finite: "inj_on f A \<Longrightarrow> f ` A \<le> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
+ using finite_imageD finite_subset by blast
-lemma inj_on_finite:
-assumes "inj_on f A" "f ` A \<le> B" "finite B"
-shows "finite A"
-using assms finite_imageD finite_subset by blast
+lemma card_vimage_inj: "inj f \<Longrightarrow> A \<subseteq> range f \<Longrightarrow> card (f -` A) = card A"
+ by (auto 4 3 simp: subset_image_iff inj_vimage_image_eq
+ intro: card_image[symmetric, OF subset_inj_on])
-lemma card_vimage_inj: "\<lbrakk> inj f; A \<subseteq> range f \<rbrakk> \<Longrightarrow> card (f -` A) = card A"
-by(auto 4 3 simp add: subset_image_iff inj_vimage_image_eq intro: card_image[symmetric, OF subset_inj_on])
subsubsection \<open>Pigeonhole Principles\<close>
-lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
-by (auto dest: card_image less_irrefl_nat)
+lemma pigeonhole: "card A > card (f ` A) \<Longrightarrow> \<not> inj_on f A "
+ by (auto dest: card_image less_irrefl_nat)
lemma pigeonhole_infinite:
-assumes "~ finite A" and "finite(f`A)"
-shows "EX a0:A. ~finite{a:A. f a = f a0}"
-proof -
- have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
- proof(induct "f`A" arbitrary: A rule: finite_induct)
- case empty thus ?case by simp
+ assumes "\<not> finite A" and "finite (f`A)"
+ shows "\<exists>a0\<in>A. \<not> finite {a\<in>A. f a = f a0}"
+ using assms(2,1)
+proof (induct "f`A" arbitrary: A rule: finite_induct)
+ case empty
+ then show ?case by simp
+next
+ case (insert b F)
+ show ?case
+ proof (cases "finite {a\<in>A. f a = b}")
+ case True
+ with \<open>\<not> finite A\<close> have "\<not> finite (A - {a\<in>A. f a = b})"
+ by simp
+ also have "A - {a\<in>A. f a = b} = {a\<in>A. f a \<noteq> b}"
+ by blast
+ finally have "\<not> finite {a\<in>A. f a \<noteq> b}" .
+ from insert(3)[OF _ this] insert(2,4) show ?thesis
+ by simp (blast intro: rev_finite_subset)
next
- case (insert b F)
- show ?case
- proof cases
- assume "finite{a:A. f a = b}"
- hence "~ finite(A - {a:A. f a = b})" using \<open>\<not> finite A\<close> by simp
- also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
- finally have "~ finite({a:A. f a \<noteq> b})" .
- from insert(3)[OF _ this]
- show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
- next
- assume 1: "~finite{a:A. f a = b}"
- hence "{a \<in> A. f a = b} \<noteq> {}" by force
- thus ?thesis using 1 by blast
- qed
+ case False
+ then have "{a \<in> A. f a = b} \<noteq> {}" by force
+ with False show ?thesis by blast
qed
- from this[OF assms(2,1)] show ?thesis .
qed
lemma pigeonhole_infinite_rel:
-assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
-shows "EX b:B. ~finite{a:A. R a b}"
+ assumes "\<not> finite A"
+ and "finite B"
+ and "\<forall>a\<in>A. \<exists>b\<in>B. R a b"
+ shows "\<exists>b\<in>B. \<not> finite {a:A. R a b}"
proof -
- let ?F = "%a. {b:B. R a b}"
- from finite_Pow_iff[THEN iffD2, OF \<open>finite B\<close>]
- have "finite(?F ` A)" by(blast intro: rev_finite_subset)
- from pigeonhole_infinite[where f = ?F, OF assms(1) this]
- obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
- obtain b0 where "b0 : B" and "R a0 b0" using \<open>a0:A\<close> assms(3) by blast
- { assume "finite{a:A. R a b0}"
- then have "finite {a\<in>A. ?F a = ?F a0}"
- using \<open>b0 : B\<close> \<open>R a0 b0\<close> by(blast intro: rev_finite_subset)
- }
- with 1 \<open>b0 : B\<close> show ?thesis by blast
+ let ?F = "\<lambda>a. {b\<in>B. R a b}"
+ from finite_Pow_iff[THEN iffD2, OF \<open>finite B\<close>] have "finite (?F ` A)"
+ by (blast intro: rev_finite_subset)
+ from pigeonhole_infinite [where f = ?F, OF assms(1) this]
+ obtain a0 where "a0 \<in> A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
+ obtain b0 where "b0 \<in> B" and "R a0 b0"
+ using \<open>a0 \<in> A\<close> assms(3) by blast
+ have "finite {a\<in>A. ?F a = ?F a0}" if "finite{a:A. R a b0}"
+ using \<open>b0 \<in> B\<close> \<open>R a0 b0\<close> that by (blast intro: rev_finite_subset)
+ with 1 \<open>b0 : B\<close> show ?thesis
+ by blast
qed
subsubsection \<open>Cardinality of sums\<close>
lemma card_Plus:
- assumes "finite A" and "finite B"
+ assumes "finite A" "finite B"
shows "card (A <+> B) = card A + card B"
proof -
have "Inl`A \<inter> Inr`B = {}" by fast
with assms show ?thesis
- unfolding Plus_def
- by (simp add: card_Un_disjoint card_image)
+ by (simp add: Plus_def card_Un_disjoint card_image)
qed
lemma card_Plus_conv_if:
"card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
by (auto simp add: card_Plus)
-text \<open>Relates to equivalence classes. Based on a theorem of F. Kamm\"uller.\<close>
+text \<open>Relates to equivalence classes. Based on a theorem of F. Kammüller.\<close>
lemma dvd_partition:
- assumes f: "finite (\<Union>C)" and "\<forall>c\<in>C. k dvd card c" "\<forall>c1\<in>C. \<forall>c2\<in>C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {}"
- shows "k dvd card (\<Union>C)"
+ assumes f: "finite (\<Union>C)"
+ and "\<forall>c\<in>C. k dvd card c" "\<forall>c1\<in>C. \<forall>c2\<in>C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {}"
+ shows "k dvd card (\<Union>C)"
proof -
- have "finite C"
+ have "finite C"
by (rule finite_UnionD [OF f])
- then show ?thesis using assms
+ then show ?thesis
+ using assms
proof (induct rule: finite_induct)
- case empty show ?case by simp
+ case empty
+ show ?case by simp
next
- case (insert c C)
- then show ?case
+ case insert
+ then show ?case
apply simp
apply (subst card_Un_disjoint)
apply (auto simp add: disjoint_eq_subset_Compl)
@@ -1793,34 +1874,33 @@
qed
qed
+
subsubsection \<open>Relating injectivity and surjectivity\<close>
-lemma finite_surj_inj: assumes "finite A" "A \<subseteq> f ` A" shows "inj_on f A"
+lemma finite_surj_inj:
+ assumes "finite A" "A \<subseteq> f ` A"
+ shows "inj_on f A"
proof -
- have "f ` A = A"
+ have "f ` A = A"
by (rule card_seteq [THEN sym]) (auto simp add: assms card_image_le)
then show ?thesis using assms
by (simp add: eq_card_imp_inj_on)
qed
-lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
-shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
-by (blast intro: finite_surj_inj subset_UNIV)
+lemma finite_UNIV_surj_inj: "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f" for f :: "'a \<Rightarrow> 'a"
+ by (blast intro: finite_surj_inj subset_UNIV)
-lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
-shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
-by(fastforce simp:surj_def dest!: endo_inj_surj)
+lemma finite_UNIV_inj_surj: "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f" for f :: "'a \<Rightarrow> 'a"
+ by (fastforce simp:surj_def dest!: endo_inj_surj)
-corollary infinite_UNIV_nat [iff]:
- "\<not> finite (UNIV :: nat set)"
+corollary infinite_UNIV_nat [iff]: "\<not> finite (UNIV :: nat set)"
proof
assume "finite (UNIV :: nat set)"
- with finite_UNIV_inj_surj [of Suc]
- show False by simp (blast dest: Suc_neq_Zero surjD)
+ with finite_UNIV_inj_surj [of Suc] show False
+ by simp (blast dest: Suc_neq_Zero surjD)
qed
-lemma infinite_UNIV_char_0:
- "\<not> finite (UNIV :: 'a::semiring_char_0 set)"
+lemma infinite_UNIV_char_0: "\<not> finite (UNIV :: 'a::semiring_char_0 set)"
proof
assume "finite (UNIV :: 'a set)"
with subset_UNIV have "finite (range of_nat :: 'a set)"
@@ -1836,7 +1916,7 @@
hide_const (open) Finite_Set.fold
-subsection "Infinite Sets"
+subsection \<open>Infinite Sets\<close>
text \<open>
Some elementary facts about infinite sets, mostly by Stephan Merz.
@@ -1859,19 +1939,18 @@
by simp
lemma Diff_infinite_finite:
- assumes T: "finite T" and S: "infinite S"
+ assumes "finite T" "infinite S"
shows "infinite (S - T)"
- using T
+ using \<open>finite T\<close>
proof induct
- from S
- show "infinite (S - {})" by auto
+ from \<open>infinite S\<close> show "infinite (S - {})"
+ by auto
next
fix T x
assume ih: "infinite (S - T)"
have "S - (insert x T) = (S - T) - {x}"
by (rule Diff_insert)
- with ih
- show "infinite (S - (insert x T))"
+ with ih show "infinite (S - (insert x T))"
by (simp add: infinite_remove)
qed
@@ -1882,21 +1961,23 @@
by simp
lemma infinite_super:
- assumes T: "S \<subseteq> T" and S: "infinite S"
+ assumes "S \<subseteq> T"
+ and "infinite S"
shows "infinite T"
proof
assume "finite T"
- with T have "finite S" by (simp add: finite_subset)
- with S show False by simp
+ with \<open>S \<subseteq> T\<close> have "finite S" by (simp add: finite_subset)
+ with \<open>infinite S\<close> show False by simp
qed
proposition infinite_coinduct [consumes 1, case_names infinite]:
assumes "X A"
- and step: "\<And>A. X A \<Longrightarrow> \<exists>x\<in>A. X (A - {x}) \<or> infinite (A - {x})"
+ and step: "\<And>A. X A \<Longrightarrow> \<exists>x\<in>A. X (A - {x}) \<or> infinite (A - {x})"
shows "infinite A"
proof
assume "finite A"
- then show False using \<open>X A\<close>
+ then show False
+ using \<open>X A\<close>
proof (induction rule: finite_psubset_induct)
case (psubset A)
then obtain x where "x \<in> A" "X (A - {x}) \<or> infinite (A - {x})"
@@ -1906,7 +1987,8 @@
show False
apply (rule psubset.IH [where B = "A - {x}"])
using \<open>x \<in> A\<close> apply blast
- by (simp add: \<open>X (A - {x})\<close>)
+ apply (simp add: \<open>X (A - {x})\<close>)
+ done
qed
qed
@@ -1918,14 +2000,14 @@
\<close>
lemma inf_img_fin_dom':
- assumes img: "finite (f ` A)" and dom: "infinite A"
+ assumes img: "finite (f ` A)"
+ and dom: "infinite A"
shows "\<exists>y \<in> f ` A. infinite (f -` {y} \<inter> A)"
proof (rule ccontr)
have "A \<subseteq> (\<Union>y\<in>f ` A. f -` {y} \<inter> A)" by auto
- moreover
- assume "\<not> ?thesis"
+ moreover assume "\<not> ?thesis"
with img have "finite (\<Union>y\<in>f ` A. f -` {y} \<inter> A)" by blast
- ultimately have "finite A" by(rule finite_subset)
+ ultimately have "finite A" by (rule finite_subset)
with dom show False by contradiction
qed
@@ -1937,16 +2019,15 @@
lemma inf_img_fin_dom:
assumes img: "finite (f`A)" and dom: "infinite A"
shows "\<exists>y \<in> f`A. infinite (f -` {y})"
-using inf_img_fin_dom'[OF assms] by auto
+ using inf_img_fin_dom'[OF assms] by auto
lemma inf_img_fin_domE:
assumes "finite (f`A)" and "infinite A"
obtains y where "y \<in> f`A" and "infinite (f -` {y})"
using assms by (blast dest: inf_img_fin_dom)
-proposition finite_image_absD:
- fixes S :: "'a::linordered_ring set"
- shows "finite (abs ` S) \<Longrightarrow> finite S"
+proposition finite_image_absD: "finite (abs ` S) \<Longrightarrow> finite S"
+ for S :: "'a::linordered_ring set"
by (rule ccontr) (auto simp: abs_eq_iff vimage_def dest: inf_img_fin_dom)
end