--- a/src/HOL/Transitive_Closure.thy Wed Jul 11 23:24:25 2018 +0100
+++ b/src/HOL/Transitive_Closure.thy Thu Jul 12 17:22:39 2018 +0100
@@ -83,7 +83,7 @@
subsection \<open>Reflexive-transitive closure\<close>
lemma reflcl_set_eq [pred_set_conv]: "(sup (\<lambda>x y. (x, y) \<in> r) (=)) = (\<lambda>x y. (x, y) \<in> r \<union> Id)"
- by (auto simp add: fun_eq_iff)
+ by (auto simp: fun_eq_iff)
lemma r_into_rtrancl [intro]: "\<And>p. p \<in> r \<Longrightarrow> p \<in> r\<^sup>*"
\<comment> \<open>\<open>rtrancl\<close> of \<open>r\<close> contains \<open>r\<close>\<close>
@@ -156,18 +156,16 @@
(base) "a = b"
| (step) y where "(a, y) \<in> r\<^sup>*" and "(y, b) \<in> r"
\<comment> \<open>elimination of \<open>rtrancl\<close> -- by induction on a special formula\<close>
- apply (subgoal_tac "a = b \<or> (\<exists>y. (a, y) \<in> r\<^sup>* \<and> (y, b) \<in> r)")
- apply (rule_tac [2] major [THEN rtrancl_induct])
- prefer 2 apply blast
- prefer 2 apply blast
- apply (erule asm_rl exE disjE conjE base step)+
- done
+proof -
+ have "a = b \<or> (\<exists>y. (a, y) \<in> r\<^sup>* \<and> (y, b) \<in> r)"
+ by (rule major [THEN rtrancl_induct]) blast+
+ then show ?thesis
+ by (auto intro: base step)
+qed
lemma rtrancl_Int_subset: "Id \<subseteq> s \<Longrightarrow> (r\<^sup>* \<inter> s) O r \<subseteq> s \<Longrightarrow> r\<^sup>* \<subseteq> s"
- apply (rule subsetI)
- apply auto
- apply (erule rtrancl_induct)
- apply auto
+ apply clarify
+ apply (erule rtrancl_induct, auto)
done
lemma converse_rtranclp_into_rtranclp: "r a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>*\<^sup>* a c"
@@ -193,14 +191,11 @@
done
lemma rtrancl_subset_rtrancl: "r \<subseteq> s\<^sup>* \<Longrightarrow> r\<^sup>* \<subseteq> s\<^sup>*"
- apply (drule rtrancl_mono)
- apply simp
- done
+by (drule rtrancl_mono, simp)
lemma rtranclp_subset: "R \<le> S \<Longrightarrow> S \<le> R\<^sup>*\<^sup>* \<Longrightarrow> S\<^sup>*\<^sup>* = R\<^sup>*\<^sup>*"
apply (drule rtranclp_mono)
- apply (drule rtranclp_mono)
- apply simp
+ apply (drule rtranclp_mono, simp)
done
lemmas rtrancl_subset = rtranclp_subset [to_set]
@@ -216,21 +211,10 @@
lemmas rtrancl_reflcl [simp] = rtranclp_reflclp [to_set]
lemma rtrancl_r_diff_Id: "(r - Id)\<^sup>* = r\<^sup>*"
- apply (rule sym)
- apply (rule rtrancl_subset)
- apply blast
- apply clarify
- apply (rename_tac a b)
- apply (case_tac "a = b")
- apply blast
- apply blast
- done
+ by (rule rtrancl_subset [symmetric]) auto
lemma rtranclp_r_diff_Id: "(inf r (\<noteq>))\<^sup>*\<^sup>* = r\<^sup>*\<^sup>*"
- apply (rule sym)
- apply (rule rtranclp_subset)
- apply blast+
- done
+ by (rule rtranclp_subset [symmetric]) auto
theorem rtranclp_converseD:
assumes "(r\<inverse>\<inverse>)\<^sup>*\<^sup>* x y"
@@ -272,12 +256,12 @@
assumes major: "r\<^sup>*\<^sup>* x z"
and cases: "x = z \<Longrightarrow> P" "\<And>y. r x y \<Longrightarrow> r\<^sup>*\<^sup>* y z \<Longrightarrow> P"
shows P
- apply (subgoal_tac "x = z \<or> (\<exists>y. r x y \<and> r\<^sup>*\<^sup>* y z)")
- apply (rule_tac [2] major [THEN converse_rtranclp_induct])
- prefer 2 apply iprover
- prefer 2 apply iprover
- apply (erule asm_rl exE disjE conjE cases)+
- done
+proof -
+ have "x = z \<or> (\<exists>y. r x y \<and> r\<^sup>*\<^sup>* y z)"
+ by (rule_tac major [THEN converse_rtranclp_induct]) iprover+
+ then show ?thesis
+ by (auto intro: cases)
+qed
lemmas converse_rtranclE = converse_rtranclpE [to_set]
@@ -360,8 +344,7 @@
lemma rtranclp_into_tranclp2: "r a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>+\<^sup>+ a c"
\<comment> \<open>intro rule from \<open>r\<close> and \<open>rtrancl\<close>\<close>
- apply (erule rtranclp.cases)
- apply iprover
+ apply (erule rtranclp.cases, iprover)
apply (rule rtranclp_trans [THEN rtranclp_into_tranclp1])
apply (simp | rule r_into_rtranclp)+
done
@@ -401,10 +384,8 @@
using assms by cases simp_all
lemma trancl_Int_subset: "r \<subseteq> s \<Longrightarrow> (r\<^sup>+ \<inter> s) O r \<subseteq> s \<Longrightarrow> r\<^sup>+ \<subseteq> s"
- apply (rule subsetI)
- apply auto
- apply (erule trancl_induct)
- apply auto
+ apply clarify
+ apply (erule trancl_induct, auto)
done
lemma trancl_unfold: "r\<^sup>+ = r \<union> r\<^sup>+ O r"
@@ -438,10 +419,8 @@
lemma trancl_id [simp]: "trans r \<Longrightarrow> r\<^sup>+ = r"
apply auto
- apply (erule trancl_induct)
- apply assumption
- apply (unfold trans_def)
- apply blast
+ apply (erule trancl_induct, assumption)
+ apply (unfold trans_def, blast)
done
lemma rtranclp_tranclp_tranclp:
@@ -485,16 +464,14 @@
and cases: "\<And>y. r y b \<Longrightarrow> P y" "\<And>y z. r y z \<Longrightarrow> r\<^sup>+\<^sup>+ z b \<Longrightarrow> P z \<Longrightarrow> P y"
shows "P a"
apply (rule tranclp_induct [OF tranclp_converseI, OF conversepI, OF major])
- apply (rule cases)
- apply (erule conversepD)
+ apply (blast intro: cases)
apply (blast intro: assms dest!: tranclp_converseD)
done
lemmas converse_trancl_induct = converse_tranclp_induct [to_set]
lemma tranclpD: "R\<^sup>+\<^sup>+ x y \<Longrightarrow> \<exists>z. R x z \<and> R\<^sup>*\<^sup>* z y"
- apply (erule converse_tranclp_induct)
- apply auto
+ apply (erule converse_tranclp_induct, auto)
apply (blast intro: rtranclp_trans)
done
@@ -537,8 +514,7 @@
by (induct rule: rtrancl_induct) auto
lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A \<Longrightarrow> r\<^sup>+ \<subseteq> A \<times> A"
- apply (rule subsetI)
- apply (simp only: split_tupled_all)
+ apply (clarsimp simp:)
apply (erule tranclE)
apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+
done
@@ -552,13 +528,19 @@
lemmas reflcl_trancl [simp] = reflclp_tranclp [to_set]
lemma trancl_reflcl [simp]: "(r\<^sup>=)\<^sup>+ = r\<^sup>*"
- apply safe
- apply (drule trancl_into_rtrancl, simp)
- apply (erule rtranclE, safe)
- apply (rule r_into_trancl, simp)
- apply (rule rtrancl_into_trancl1)
- apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)
- done
+proof -
+ have "(a, b) \<in> (r\<^sup>=)\<^sup>+ \<Longrightarrow> (a, b) \<in> r\<^sup>*" for a b
+ by (force dest: trancl_into_rtrancl)
+ moreover have "(a, b) \<in> (r\<^sup>=)\<^sup>+" if "(a, b) \<in> r\<^sup>*" for a b
+ using that
+ proof (cases a b rule: rtranclE)
+ case step
+ show ?thesis
+ by (rule rtrancl_into_trancl1) (use step in auto)
+ qed auto
+ ultimately show ?thesis
+ by auto
+qed
lemma rtrancl_trancl_reflcl [code]: "r\<^sup>* = (r\<^sup>+)\<^sup>="
by simp
@@ -570,7 +552,7 @@
by (rule subst [OF reflcl_trancl]) simp
lemma rtranclpD: "R\<^sup>*\<^sup>* a b \<Longrightarrow> a = b \<or> a \<noteq> b \<and> R\<^sup>+\<^sup>+ a b"
- by (force simp add: reflclp_tranclp [symmetric] simp del: reflclp_tranclp)
+ by (force simp: reflclp_tranclp [symmetric] simp del: reflclp_tranclp)
lemmas rtranclD = rtranclpD [to_set]
@@ -585,19 +567,20 @@
lemma trancl_insert: "(insert (y, x) r)\<^sup>+ = r\<^sup>+ \<union> {(a, b). (a, y) \<in> r\<^sup>* \<and> (x, b) \<in> r\<^sup>*}"
\<comment> \<open>primitive recursion for \<open>trancl\<close> over finite relations\<close>
- apply (rule equalityI)
- apply (rule subsetI)
- apply (simp only: split_tupled_all)
- apply (erule trancl_induct, blast)
- apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl trancl_trans)
- apply (rule subsetI)
- apply (blast intro: trancl_mono rtrancl_mono
- [THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
- done
+proof -
+ have "\<And>a b. (a, b) \<in> (insert (y, x) r)\<^sup>+ \<Longrightarrow>
+ (a, b) \<in> r\<^sup>+ \<union> {(a, b). (a, y) \<in> r\<^sup>* \<and> (x, b) \<in> r\<^sup>*}"
+ by (erule trancl_induct) (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl trancl_trans)+
+ moreover have "r\<^sup>+ \<union> {(a, b). (a, y) \<in> r\<^sup>* \<and> (x, b) \<in> r\<^sup>*} \<subseteq> (insert (y, x) r)\<^sup>+"
+ by (blast intro: trancl_mono rtrancl_mono [THEN [2] rev_subsetD]
+ rtrancl_trancl_trancl rtrancl_into_trancl2)
+ ultimately show ?thesis
+ by auto
+qed
lemma trancl_insert2:
"(insert (a, b) r)\<^sup>+ = r\<^sup>+ \<union> {(x, y). ((x, a) \<in> r\<^sup>+ \<or> x = a) \<and> ((b, y) \<in> r\<^sup>+ \<or> y = b)}"
- by (auto simp add: trancl_insert rtrancl_eq_or_trancl)
+ by (auto simp: trancl_insert rtrancl_eq_or_trancl)
lemma rtrancl_insert: "(insert (a,b) r)\<^sup>* = r\<^sup>* \<union> {(x, y). (x, a) \<in> r\<^sup>* \<and> (b, y) \<in> r\<^sup>*}"
using trancl_insert[of a b r]
@@ -636,28 +619,30 @@
lemma trancl_range [simp]: "Range (r\<^sup>+) = Range r"
unfolding Domain_converse [symmetric] by (simp add: trancl_converse [symmetric])
-lemma Not_Domain_rtrancl: "x \<notin> Domain R \<Longrightarrow> (x, y) \<in> R\<^sup>* \<longleftrightarrow> x = y"
- apply auto
- apply (erule rev_mp)
- apply (erule rtrancl_induct)
- apply auto
- done
+lemma Not_Domain_rtrancl:
+ assumes "x \<notin> Domain R" shows "(x, y) \<in> R\<^sup>* \<longleftrightarrow> x = y"
+proof -
+have "(x, y) \<in> R\<^sup>* \<Longrightarrow> x = y"
+ by (erule rtrancl_induct) (use assms in auto)
+ then show ?thesis
+ by auto
+qed
lemma trancl_subset_Field2: "r\<^sup>+ \<subseteq> Field r \<times> Field r"
apply clarify
apply (erule trancl_induct)
- apply (auto simp add: Field_def)
+ apply (auto simp: Field_def)
done
lemma finite_trancl[simp]: "finite (r\<^sup>+) = finite r"
- apply auto
- prefer 2
+proof
+ show "finite (r\<^sup>+) \<Longrightarrow> finite r"
+ by (blast intro: r_into_trancl' finite_subset)
+ show "finite r \<Longrightarrow> finite (r\<^sup>+)"
apply (rule trancl_subset_Field2 [THEN finite_subset])
- apply (rule finite_SigmaI)
- prefer 3
- apply (blast intro: r_into_trancl' finite_subset)
- apply (auto simp add: finite_Field)
+ apply (auto simp: finite_Field)
done
+qed
lemma finite_rtrancl_Image[simp]: assumes "finite R" "finite A" shows "finite (R\<^sup>* `` A)"
proof (rule ccontr)
@@ -670,13 +655,20 @@
be merged with main body.\<close>
lemma single_valued_confluent:
- "single_valued r \<Longrightarrow> (x, y) \<in> r\<^sup>* \<Longrightarrow> (x, z) \<in> r\<^sup>* \<Longrightarrow> (y, z) \<in> r\<^sup>* \<or> (z, y) \<in> r\<^sup>*"
- apply (erule rtrancl_induct)
- apply simp
- apply (erule disjE)
- apply (blast elim:converse_rtranclE dest:single_valuedD)
- apply (blast intro:rtrancl_trans)
- done
+ assumes "single_valued r" and xy: "(x, y) \<in> r\<^sup>*" and xz: "(x, z) \<in> r\<^sup>*"
+ shows "(y, z) \<in> r\<^sup>* \<or> (z, y) \<in> r\<^sup>*"
+ using xy
+proof (induction rule: rtrancl_induct)
+ case base
+ show ?case
+ by (simp add: assms)
+next
+ case (step y z)
+ with xz \<open>single_valued r\<close> show ?case
+ apply (auto simp: elim: converse_rtranclE dest: single_valuedD)
+ apply (blast intro: rtrancl_trans)
+ done
+qed
lemma r_r_into_trancl: "(a, b) \<in> R \<Longrightarrow> (b, c) \<in> R \<Longrightarrow> (a, c) \<in> R\<^sup>+"
by (fast intro: trancl_trans)
@@ -824,16 +816,18 @@
by (fact relpow_Suc_D2' [to_pred])
lemma relpow_E2:
- "(x, z) \<in> R ^^ n \<Longrightarrow>
- (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P) \<Longrightarrow>
- (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ m \<Longrightarrow> P) \<Longrightarrow> P"
- apply (cases n)
- apply simp
- apply (rename_tac nat)
- apply (cut_tac n=nat and R=R in relpow_Suc_D2')
- apply simp
- apply blast
- done
+ assumes "(x, z) \<in> R ^^ n" "n = 0 \<Longrightarrow> x = z \<Longrightarrow> P"
+ "\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ m \<Longrightarrow> P"
+ shows "P"
+proof (cases n)
+ case 0
+ with assms show ?thesis
+ by simp
+next
+ case (Suc m)
+ with assms relpow_Suc_D2' [of m R] show ?thesis
+ by force
+qed
lemma relpowp_E2:
"(P ^^ n) x z \<Longrightarrow>
@@ -915,22 +909,23 @@
by (simp add: rtranclp_is_Sup_relpowp)
lemma trancl_power: "p \<in> R\<^sup>+ \<longleftrightarrow> (\<exists>n > 0. p \<in> R ^^ n)"
- apply (cases p)
- apply simp
- apply (rule iffI)
- apply (drule tranclD2)
- apply (clarsimp simp: rtrancl_is_UN_relpow)
- apply (rule_tac x="Suc x" in exI)
- apply (clarsimp simp: relcomp_unfold)
- apply fastforce
- apply clarsimp
- apply (case_tac n)
- apply simp
- apply clarsimp
- apply (drule relpow_imp_rtrancl)
- apply (drule rtrancl_into_trancl1)
- apply auto
- done
+proof -
+ have "((a, b) \<in> R\<^sup>+) = (\<exists>n>0. (a, b) \<in> R ^^ n)" for a b
+ proof safe
+ show "(a, b) \<in> R\<^sup>+ \<Longrightarrow> \<exists>n>0. (a, b) \<in> R ^^ n"
+ apply (drule tranclD2)
+ apply (fastforce simp: rtrancl_is_UN_relpow relcomp_unfold)
+ done
+ show "(a, b) \<in> R\<^sup>+" if "n > 0" "(a, b) \<in> R ^^ n" for n
+ proof (cases n)
+ case (Suc m)
+ with that show ?thesis
+ by (auto simp: dest: relpow_imp_rtrancl rtrancl_into_trancl1)
+ qed (use that in auto)
+ qed
+ then show ?thesis
+ by (cases p) auto
+qed
lemma tranclp_power: "(P\<^sup>+\<^sup>+) x y \<longleftrightarrow> (\<exists>n > 0. (P ^^ n) x y)"
using trancl_power [to_pred, of P "(x, y)"] by simp
@@ -1070,8 +1065,7 @@
fixes R :: "('a \<times> 'a) set"
assumes "finite R"
shows "R^^k \<subseteq> (UN n:{n. n \<le> card R}. R^^n)"
- apply (cases k)
- apply force
+ apply (cases k, force)
apply (use relpow_finite_bounded1[OF assms, of k] in auto)
done
@@ -1088,7 +1082,7 @@
shows "finite (R O S)"
proof-
have "R O S = (\<Union>(x, y)\<in>R. \<Union>(u, v)\<in>S. if u = y then {(x, v)} else {})"
- by (force simp add: split_def image_constant_conv split: if_splits)
+ by (force simp: split_def image_constant_conv split: if_splits)
then show ?thesis
using assms by clarsimp
qed
@@ -1166,7 +1160,7 @@
by (auto simp: Let_def)
lemma finite_trancl_ntranl: "finite R \<Longrightarrow> trancl R = ntrancl (card R - 1) R"
- by (cases "card R") (auto simp add: trancl_finite_eq_relpow relpow_empty ntrancl_def)
+ by (cases "card R") (auto simp: trancl_finite_eq_relpow relpow_empty ntrancl_def)
subsection \<open>Acyclic relations\<close>