--- a/src/HOL/Quotient.thy Sun Mar 29 15:44:54 2020 +0100
+++ b/src/HOL/Quotient.thy Sun Mar 29 21:30:52 2020 +0100
@@ -253,27 +253,29 @@
fixes P::"'a \<Rightarrow> bool"
and x::"'a"
assumes a: "equivp R2"
- shows "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
- apply(rule iffI)
- apply(rule allI)
- apply(drule_tac x="\<lambda>y. f x" in bspec)
- apply(simp add: in_respects rel_fun_def)
- apply(rule impI)
- using a equivp_reflp_symp_transp[of "R2"]
- apply (auto elim: equivpE reflpE)
- done
+ shows "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
+proof (intro allI iffI)
+ fix f
+ assume "\<forall>f \<in> Respects (R1 ===> R2). P (f x)"
+ moreover have "(\<lambda>y. f x) \<in> Respects (R1 ===> R2)"
+ using a equivp_reflp_symp_transp[of "R2"]
+ by(auto simp add: in_respects rel_fun_def elim: equivpE reflpE)
+ ultimately show "P (f x)"
+ by auto
+qed auto
lemma bex_reg_eqv_range:
assumes a: "equivp R2"
shows "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
- apply(auto)
- apply(rule_tac x="\<lambda>y. f x" in bexI)
- apply(simp)
- apply(simp add: Respects_def in_respects rel_fun_def)
- apply(rule impI)
- using a equivp_reflp_symp_transp[of "R2"]
- apply (auto elim: equivpE reflpE)
- done
+proof -
+ { fix f
+ assume "P (f x)"
+ have "(\<lambda>y. f x) \<in> Respects (R1 ===> R2)"
+ using a equivp_reflp_symp_transp[of "R2"]
+ by (auto simp add: Respects_def in_respects rel_fun_def elim: equivpE reflpE) }
+ then show ?thesis
+ by auto
+qed
(* Next four lemmas are unused *)
lemma all_reg:
@@ -322,35 +324,42 @@
assumes q: "Quotient3 R1 Abs1 Rep1"
and a: "(R1 ===> R2) f g"
shows "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
- apply (auto simp add: Babs_def in_respects rel_fun_def)
- apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
- using a apply (simp add: Babs_def rel_fun_def)
- apply (simp add: in_respects rel_fun_def)
- using Quotient3_rel[OF q]
- by metis
+proof (clarsimp simp add: Babs_def in_respects rel_fun_def)
+ fix x y
+ assume "R1 x y"
+ then have "x \<in> Respects R1 \<and> y \<in> Respects R1"
+ unfolding in_respects rel_fun_def using Quotient3_rel[OF q]by metis
+ then show "R2 (Babs (Respects R1) f x) (Babs (Respects R1) g y)"
+ using \<open>R1 x y\<close> a by (simp add: Babs_def rel_fun_def)
+qed
lemma babs_prs:
assumes q1: "Quotient3 R1 Abs1 Rep1"
and q2: "Quotient3 R2 Abs2 Rep2"
- shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
- apply (rule ext)
- apply (simp add:)
- apply (subgoal_tac "Rep1 x \<in> Respects R1")
- apply (simp add: Babs_def Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
- apply (simp add: in_respects Quotient3_rel_rep[OF q1])
- done
+shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
+proof -
+ { fix x
+ have "Rep1 x \<in> Respects R1"
+ by (simp add: in_respects Quotient3_rel_rep[OF q1])
+ then have "Abs2 (Babs (Respects R1) ((Abs1 ---> Rep2) f) (Rep1 x)) = f x"
+ by (simp add: Babs_def Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
+ }
+ then show ?thesis
+ by force
+qed
lemma babs_simp:
assumes q: "Quotient3 R1 Abs Rep"
shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
- apply(rule iffI)
- apply(simp_all only: babs_rsp[OF q])
- apply(auto simp add: Babs_def rel_fun_def)
- apply(metis Babs_def in_respects Quotient3_rel[OF q])
- done
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ unfolding rel_fun_def by (metis Babs_def in_respects Quotient3_rel[OF q])
+qed (simp add: babs_rsp[OF q])
-(* If a user proves that a particular functional relation
- is an equivalence this may be useful in regularising *)
+text \<open>If a user proves that a particular functional relation
+ is an equivalence, this may be useful in regularising\<close>
lemma babs_reg_eqv:
shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)
@@ -372,7 +381,8 @@
shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
using a by (auto elim: rel_funE simp add: Ex1_def in_respects)
-(* 2 lemmas needed for cleaning of quantifiers *)
+text \<open>Two lemmas needed for cleaning of quantifiers\<close>
+
lemma all_prs:
assumes a: "Quotient3 R absf repf"
shows "Ball (Respects R) ((absf ---> id) f) = All f"
@@ -410,10 +420,10 @@
lemma ex1_prs:
assumes "Quotient3 R absf repf"
shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
- apply (auto simp: Bex1_rel_def Respects_def)
- apply (metis Quotient3_def assms)
- apply (metis (full_types) Quotient3_def assms)
- by (meson Quotient3_rel assms)
+ (is "?lhs = ?rhs")
+ using assms
+ apply (auto simp add: Bex1_rel_def Respects_def)
+ by (metis (full_types) Quotient3_def)+
lemma bex1_bexeq_reg:
shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
@@ -543,7 +553,6 @@
subsection \<open>Quotient composition\<close>
-
lemma OOO_quotient3:
fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
@@ -558,29 +567,35 @@
proof -
have *: "(R1 OOO R2') r r \<and> (R1 OOO R2') s s \<and> (Abs2 \<circ> Abs1) r = (Abs2 \<circ> Abs1) s
\<longleftrightarrow> (R1 OOO R2') r s" for r s
- apply safe
- subgoal for a b c d
- apply simp
- apply (rule_tac b="Rep1 (Abs1 r)" in relcomppI)
- using Quotient3_refl1 R1 rep_abs_rsp apply fastforce
- apply (rule_tac b="Rep1 (Abs1 s)" in relcomppI)
- apply (metis (full_types) Rep1 Abs1 Quotient3_rel R2 Quotient3_refl1 [OF R1] Quotient3_refl2 [OF R1] Quotient3_rel_abs [OF R1])
- by (metis Quotient3_rel R1 rep_abs_rsp_left)
- subgoal for x y
- apply (drule Abs1)
- apply (erule Quotient3_refl2 [OF R1])
- apply (erule Quotient3_refl1 [OF R1])
- apply (drule Quotient3_refl1 [OF R2], drule Rep1)
- by (metis (full_types) Quotient3_def R1 relcompp.relcompI)
- subgoal for x y
- apply (drule Abs1)
- apply (erule Quotient3_refl2 [OF R1])
- apply (erule Quotient3_refl1 [OF R1])
- apply (drule Quotient3_refl2 [OF R2], drule Rep1)
- by (metis (full_types) Quotient3_def R1 relcompp.relcompI)
- subgoal for x y
- by simp (metis (full_types) Abs1 Quotient3_rel R1 R2)
- done
+ proof (intro iffI conjI; clarify)
+ show "(R1 OOO R2') r s"
+ if r: "R1 r a" "R2' a b" "R1 b r" and eq: "(Abs2 \<circ> Abs1) r = (Abs2 \<circ> Abs1) s"
+ and s: "R1 s c" "R2' c d" "R1 d s" for a b c d
+ proof -
+ have "R1 r (Rep1 (Abs1 r))"
+ using r Quotient3_refl1 R1 rep_abs_rsp by fastforce
+ moreover have "R2' (Rep1 (Abs1 r)) (Rep1 (Abs1 s))"
+ using that
+ apply (simp add: )
+ apply (metis (full_types) Rep1 Abs1 Quotient3_rel R2 Quotient3_refl1 [OF R1] Quotient3_refl2 [OF R1] Quotient3_rel_abs [OF R1])
+ done
+ moreover have "R1 (Rep1 (Abs1 s)) s"
+ by (metis s Quotient3_rel R1 rep_abs_rsp_left)
+ ultimately show ?thesis
+ by (metis relcomppI)
+ qed
+ next
+ fix x y
+ assume xy: "R1 r x" "R2' x y" "R1 y s"
+ then have "R2 (Abs1 x) (Abs1 y)"
+ by (iprover dest: Abs1 elim: Quotient3_refl1 [OF R1] Quotient3_refl2 [OF R1])
+ then have "R2' (Rep1 (Abs1 x)) (Rep1 (Abs1 x))" "R2' (Rep1 (Abs1 y)) (Rep1 (Abs1 y))"
+ by (simp_all add: Quotient3_refl1 [OF R2] Quotient3_refl2 [OF R2] Rep1)
+ with \<open>R1 r x\<close> \<open>R1 y s\<close> show "(R1 OOO R2') r r" "(R1 OOO R2') s s"
+ by (metis (full_types) Quotient3_def R1 relcompp.relcompI)+
+ show "(Abs2 \<circ> Abs1) r = (Abs2 \<circ> Abs1) s"
+ using xy by simp (metis (full_types) Abs1 Quotient3_rel R1 R2)
+ qed
show ?thesis
apply (rule Quotient3I)
using * apply (simp_all add: o_def Quotient3_abs_rep [OF R2] Quotient3_abs_rep [OF R1])
--- a/src/HOL/Transitive_Closure.thy Sun Mar 29 15:44:54 2020 +0100
+++ b/src/HOL/Transitive_Closure.thy Sun Mar 29 21:30:52 2020 +0100
@@ -87,9 +87,7 @@
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>
- apply (simp only: split_tupled_all)
- apply (erule rtrancl_refl [THEN rtrancl_into_rtrancl])
- done
+ by (simp add: split_tupled_all rtrancl_refl [THEN rtrancl_into_rtrancl])
lemma r_into_rtranclp [intro]: "r x y \<Longrightarrow> r\<^sup>*\<^sup>* x y"
\<comment> \<open>\<open>rtrancl\<close> of \<open>r\<close> contains \<open>r\<close>\<close>
@@ -97,10 +95,11 @@
lemma rtranclp_mono: "r \<le> s \<Longrightarrow> r\<^sup>*\<^sup>* \<le> s\<^sup>*\<^sup>*"
\<comment> \<open>monotonicity of \<open>rtrancl\<close>\<close>
- apply (rule predicate2I)
- apply (erule rtranclp.induct)
- apply (rule_tac [2] rtranclp.rtrancl_into_rtrancl, blast+)
- done
+proof (rule predicate2I)
+ show "s\<^sup>*\<^sup>* x y" if "r \<le> s" "r\<^sup>*\<^sup>* x y" for x y
+ using \<open>r\<^sup>*\<^sup>* x y\<close> \<open>r \<le> s\<close>
+ by (induction rule: rtranclp.induct) (blast intro: rtranclp.rtrancl_into_rtrancl)+
+qed
lemma mono_rtranclp[mono]: "(\<And>a b. x a b \<longrightarrow> y a b) \<Longrightarrow> x\<^sup>*\<^sup>* a b \<longrightarrow> y\<^sup>*\<^sup>* a b"
using rtranclp_mono[of x y] by auto
@@ -164,9 +163,7 @@
qed
lemma rtrancl_Int_subset: "Id \<subseteq> s \<Longrightarrow> (r\<^sup>* \<inter> s) O r \<subseteq> s \<Longrightarrow> r\<^sup>* \<subseteq> s"
- apply clarify
- apply (erule rtrancl_induct, auto)
- done
+ by (fastforce elim: rtrancl_induct)
lemma converse_rtranclp_into_rtranclp: "r a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>*\<^sup>* a c"
by (rule rtranclp_trans) iprover+
@@ -176,27 +173,23 @@
text \<open>\<^medskip> More \<^term>\<open>r\<^sup>*\<close> equations and inclusions.\<close>
lemma rtranclp_idemp [simp]: "(r\<^sup>*\<^sup>*)\<^sup>*\<^sup>* = r\<^sup>*\<^sup>*"
- apply (auto intro!: order_antisym)
- apply (erule rtranclp_induct)
- apply (rule rtranclp.rtrancl_refl)
- apply (blast intro: rtranclp_trans)
- done
+proof -
+ have "r\<^sup>*\<^sup>*\<^sup>*\<^sup>* x y \<Longrightarrow> r\<^sup>*\<^sup>* x y" for x y
+ by (induction rule: rtranclp_induct) (blast intro: rtranclp_trans)+
+ then show ?thesis
+ by (auto intro!: order_antisym)
+qed
lemmas rtrancl_idemp [simp] = rtranclp_idemp [to_set]
lemma rtrancl_idemp_self_comp [simp]: "R\<^sup>* O R\<^sup>* = R\<^sup>*"
- apply (rule set_eqI)
- apply (simp only: split_tupled_all)
- apply (blast intro: rtrancl_trans)
- done
+ by (force intro: rtrancl_trans)
lemma rtrancl_subset_rtrancl: "r \<subseteq> s\<^sup>* \<Longrightarrow> r\<^sup>* \<subseteq> s\<^sup>*"
-by (drule rtrancl_mono, simp)
+ 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, simp)
- done
+ by (fastforce dest: rtranclp_mono)
lemmas rtrancl_subset = rtranclp_subset [to_set]
@@ -319,11 +312,15 @@
subsection \<open>Transitive closure\<close>
-lemma trancl_mono: "\<And>p. p \<in> r\<^sup>+ \<Longrightarrow> r \<subseteq> s \<Longrightarrow> p \<in> s\<^sup>+"
- apply (simp add: split_tupled_all)
- apply (erule trancl.induct)
- apply (iprover dest: subsetD)+
- done
+lemma trancl_mono:
+ assumes "p \<in> r\<^sup>+" "r \<subseteq> s"
+ shows "p \<in> s\<^sup>+"
+proof -
+ have "\<lbrakk>(a, b) \<in> r\<^sup>+; r \<subseteq> s\<rbrakk> \<Longrightarrow> (a, b) \<in> s\<^sup>+" for a b
+ by (induction rule: trancl.induct) (iprover dest: subsetD)+
+ with assms show ?thesis
+ by (cases p) force
+qed
lemma r_into_trancl': "\<And>p. p \<in> r \<Longrightarrow> p \<in> r\<^sup>+"
by (simp only: split_tupled_all) (erule r_into_trancl)
@@ -342,12 +339,19 @@
lemmas rtrancl_into_trancl1 = rtranclp_into_tranclp1 [to_set]
-lemma rtranclp_into_tranclp2: "r a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>+\<^sup>+ a c"
+lemma rtranclp_into_tranclp2:
+ assumes "r a b" "r\<^sup>*\<^sup>* b c" shows "r\<^sup>+\<^sup>+ a c"
\<comment> \<open>intro rule from \<open>r\<close> and \<open>rtrancl\<close>\<close>
- apply (erule rtranclp.cases, iprover)
- apply (rule rtranclp_trans [THEN rtranclp_into_tranclp1])
- apply (simp | rule r_into_rtranclp)+
- done
+ using \<open>r\<^sup>*\<^sup>* b c\<close>
+proof (cases rule: rtranclp.cases)
+ case rtrancl_refl
+ with assms show ?thesis
+ by iprover
+next
+ case rtrancl_into_rtrancl
+ with assms show ?thesis
+ by (auto intro: rtranclp_trans [THEN rtranclp_into_tranclp1])
+qed
lemmas rtrancl_into_trancl2 = rtranclp_into_tranclp2 [to_set]
@@ -384,9 +388,7 @@
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 clarify
- apply (erule trancl_induct, auto)
- done
+ by (fastforce simp add: elim: trancl_induct)
lemma trancl_unfold: "r\<^sup>+ = r \<union> r\<^sup>+ O r"
by (auto intro: trancl_into_trancl elim: tranclE)
@@ -418,10 +420,7 @@
using assms(2,1) by induct iprover+
lemma trancl_id [simp]: "trans r \<Longrightarrow> r\<^sup>+ = r"
- apply auto
- apply (erule trancl_induct, assumption)
- apply (unfold trans_def, blast)
- done
+ unfolding trans_def by (fastforce simp add: elim: trancl_induct)
lemma rtranclp_tranclp_tranclp:
assumes "r\<^sup>*\<^sup>* x y"
@@ -435,19 +434,30 @@
lemmas trancl_into_trancl2 = tranclp_into_tranclp2 [to_set]
-lemma tranclp_converseI: "(r\<^sup>+\<^sup>+)\<inverse>\<inverse> x y \<Longrightarrow> (r\<inverse>\<inverse>)\<^sup>+\<^sup>+ x y"
- apply (drule conversepD)
- apply (erule tranclp_induct)
- apply (iprover intro: conversepI tranclp_trans)+
- done
+lemma tranclp_converseI:
+ assumes "(r\<^sup>+\<^sup>+)\<inverse>\<inverse> x y" shows "(r\<inverse>\<inverse>)\<^sup>+\<^sup>+ x y"
+ using conversepD [OF assms]
+proof (induction rule: tranclp_induct)
+ case (base y)
+ then show ?case
+ by (iprover intro: conversepI)
+next
+ case (step y z)
+ then show ?case
+ by (iprover intro: conversepI tranclp_trans)
+qed
lemmas trancl_converseI = tranclp_converseI [to_set]
-lemma tranclp_converseD: "(r\<inverse>\<inverse>)\<^sup>+\<^sup>+ x y \<Longrightarrow> (r\<^sup>+\<^sup>+)\<inverse>\<inverse> x y"
- apply (rule conversepI)
- apply (erule tranclp_induct)
- apply (iprover dest: conversepD intro: tranclp_trans)+
- done
+lemma tranclp_converseD:
+ assumes "(r\<inverse>\<inverse>)\<^sup>+\<^sup>+ x y" shows "(r\<^sup>+\<^sup>+)\<inverse>\<inverse> x y"
+proof -
+ have "r\<^sup>+\<^sup>+ y x"
+ using assms
+ by (induction rule: tranclp_induct) (iprover dest: conversepD intro: tranclp_trans)+
+ then show ?thesis
+ by (rule conversepI)
+qed
lemmas trancl_converseD = tranclp_converseD [to_set]
@@ -463,17 +473,21 @@
assumes major: "r\<^sup>+\<^sup>+ a b"
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 (blast intro: cases)
- apply (blast intro: assms dest!: tranclp_converseD)
- done
+proof -
+ have "r\<inverse>\<inverse>\<^sup>+\<^sup>+ b a"
+ by (intro tranclp_converseI conversepI major)
+ then show ?thesis
+ by (induction rule: tranclp_induct) (blast intro: cases dest: tranclp_converseD)+
+qed
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, auto)
- apply (blast intro: rtranclp_trans)
- done
+proof (induction rule: converse_tranclp_induct)
+ case (step u v)
+ then show ?case
+ by (blast intro: rtranclp_trans)
+qed auto
lemmas tranclD = tranclpD [to_set]
@@ -492,7 +506,7 @@
by iprover
next
case rtrancl_into_rtrancl
- from this have "tranclp r y z"
+ then have "tranclp r y z"
by (iprover intro: rtranclp_into_tranclp1)
with \<open>r x y\<close> step show P
by iprover
@@ -513,17 +527,15 @@
lemma trancl_subset_Sigma_aux: "(a, b) \<in> r\<^sup>* \<Longrightarrow> r \<subseteq> A \<times> A \<Longrightarrow> a = b \<or> a \<in> A"
by (induct rule: rtrancl_induct) auto
-lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A \<Longrightarrow> r\<^sup>+ \<subseteq> A \<times> A"
- apply (clarsimp simp:)
- apply (erule tranclE)
- apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+
- done
+lemma trancl_subset_Sigma:
+ assumes "r \<subseteq> A \<times> A" shows "r\<^sup>+ \<subseteq> A \<times> A"
+proof (rule trancl_Int_subset [OF assms])
+ show "(r\<^sup>+ \<inter> A \<times> A) O r \<subseteq> A \<times> A"
+ using assms by auto
+qed
lemma reflclp_tranclp [simp]: "(r\<^sup>+\<^sup>+)\<^sup>=\<^sup>= = r\<^sup>*\<^sup>*"
- apply (safe intro!: order_antisym)
- apply (erule tranclp_into_rtranclp)
- apply (blast elim: rtranclp.cases dest: rtranclp_into_tranclp1)
- done
+ by (fast elim: rtranclp.cases tranclp_into_rtranclp dest: rtranclp_into_tranclp1)
lemmas reflcl_trancl [simp] = reflclp_tranclp [to_set]
@@ -629,19 +641,14 @@
qed
lemma trancl_subset_Field2: "r\<^sup>+ \<subseteq> Field r \<times> Field r"
- apply clarify
- apply (erule trancl_induct)
- apply (auto simp: Field_def)
- done
+ by (rule trancl_Int_subset) (auto simp: Field_def)
lemma finite_trancl[simp]: "finite (r\<^sup>+) = finite r"
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 (auto simp: finite_Field)
- done
+ by (auto simp: finite_Field trancl_subset_Field2 [THEN finite_subset])
qed
lemma finite_rtrancl_Image[simp]: assumes "finite R" "finite A" shows "finite (R\<^sup>* `` A)"
@@ -665,9 +672,7 @@
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
+ by (auto elim: converse_rtranclE dest: single_valuedD intro: rtrancl_trans)
qed
lemma r_r_into_trancl: "(a, b) \<in> R \<Longrightarrow> (b, c) \<in> R \<Longrightarrow> (a, c) \<in> R\<^sup>+"
@@ -676,12 +681,14 @@
lemma trancl_into_trancl: "(a, b) \<in> r\<^sup>+ \<Longrightarrow> (b, c) \<in> r \<Longrightarrow> (a, c) \<in> r\<^sup>+"
by (induct rule: trancl_induct) (fast intro: r_r_into_trancl trancl_trans)+
-lemma tranclp_rtranclp_tranclp: "r\<^sup>+\<^sup>+ a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>+\<^sup>+ a c"
- apply (drule tranclpD)
- apply (elim exE conjE)
- apply (drule rtranclp_trans, assumption)
- apply (drule (2) rtranclp_into_tranclp2)
- done
+lemma tranclp_rtranclp_tranclp:
+ assumes "r\<^sup>+\<^sup>+ a b" "r\<^sup>*\<^sup>* b c" shows "r\<^sup>+\<^sup>+ a c"
+proof -
+ obtain z where "r a z" "r\<^sup>*\<^sup>* z c"
+ using assms by (iprover dest: tranclpD rtranclp_trans)
+ then show ?thesis
+ by (blast dest: rtranclp_into_tranclp2)
+qed
lemma rtranclp_conversep: "r\<inverse>\<inverse>\<^sup>*\<^sup>* = r\<^sup>*\<^sup>*\<inverse>\<inverse>"
by(auto simp add: fun_eq_iff intro: rtranclp_converseI rtranclp_converseD)
@@ -717,13 +724,13 @@
lemma symclpI [simp, intro?]:
shows symclpI1: "r x y \<Longrightarrow> symclp r x y"
- and symclpI2: "r y x \<Longrightarrow> symclp r x y"
-by(simp_all add: symclp_def)
+ and symclpI2: "r y x \<Longrightarrow> symclp r x y"
+ by(simp_all add: symclp_def)
lemma symclpE [consumes 1, cases pred]:
assumes "symclp r x y"
obtains (base) "r x y" | (sym) "r y x"
-using assms by(auto simp add: symclp_def)
+ using assms by(auto simp add: symclp_def)
lemma symclp_pointfree: "symclp r = sup r r\<inverse>\<inverse>"
by(auto simp add: symclp_def fun_eq_iff)
@@ -962,12 +969,10 @@
lemma trancl_power: "p \<in> R\<^sup>+ \<longleftrightarrow> (\<exists>n > 0. p \<in> R ^^ n)"
proof -
- have "((a, b) \<in> R\<^sup>+) = (\<exists>n>0. (a, b) \<in> R ^^ n)" for a b
+ have "(a, b) \<in> R\<^sup>+ \<longleftrightarrow> (\<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
+ by (fastforce simp: rtrancl_is_UN_relpow relcomp_unfold dest: tranclD2)
show "(a, b) \<in> R\<^sup>+" if "n > 0" "(a, b) \<in> R ^^ n" for n
proof (cases n)
case (Suc m)
@@ -1117,17 +1122,23 @@
fixes R :: "('a \<times> 'a) set"
assumes "finite R"
shows "R^^k \<subseteq> (\<Union>n\<in>{n. n \<le> card R}. R^^n)"
- apply (cases k, force)
- apply (use relpow_finite_bounded1[OF assms, of k] in auto)
- done
+proof (cases k)
+ case (Suc k')
+ then show ?thesis
+ using relpow_finite_bounded1[OF assms, of k] by auto
+qed force
lemma rtrancl_finite_eq_relpow: "finite R \<Longrightarrow> R\<^sup>* = (\<Union>n\<in>{n. n \<le> card R}. R^^n)"
by (fastforce simp: rtrancl_power dest: relpow_finite_bounded)
-lemma trancl_finite_eq_relpow: "finite R \<Longrightarrow> R\<^sup>+ = (\<Union>n\<in>{n. 0 < n \<and> n \<le> card R}. R^^n)"
- apply (auto simp: trancl_power)
- apply (auto dest: relpow_finite_bounded1)
- done
+lemma trancl_finite_eq_relpow:
+ assumes "finite R" shows "R\<^sup>+ = (\<Union>n\<in>{n. 0 < n \<and> n \<le> card R}. R^^n)"
+proof -
+ have "\<And>a b n. \<lbrakk>0 < n; (a, b) \<in> R ^^ n\<rbrakk> \<Longrightarrow> \<exists>x>0. x \<le> card R \<and> (a, b) \<in> R ^^ x"
+ using assms by (auto dest: relpow_finite_bounded1)
+ then show ?thesis
+ by (auto simp: trancl_power)
+qed
lemma finite_relcomp[simp,intro]:
assumes "finite R" and "finite S"
@@ -1189,7 +1200,7 @@
show ?thesis
proof (cases "i = 1")
case True
- from this \<open>(a, b) \<in> R ^^ i\<close> show ?thesis
+ with \<open>(a, b) \<in> R ^^ i\<close> show ?thesis
by (auto simp: ntrancl_def)
next
case False