--- a/src/HOL/Transitive_Closure.thy Wed Feb 07 17:26:49 2007 +0100
+++ b/src/HOL/Transitive_Closure.thy Wed Feb 07 17:28:09 2007 +0100
@@ -7,7 +7,7 @@
header {* Reflexive and Transitive closure of a relation *}
theory Transitive_Closure
-imports Inductive
+imports Predicate
uses "~~/src/Provers/trancl.ML"
begin
@@ -20,56 +20,85 @@
operands to be atomic.
*}
-consts
- rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^*)" [1000] 999)
-
-inductive "r^*"
- intros
- rtrancl_refl [intro!, Pure.intro!, simp]: "(a, a) : r^*"
- rtrancl_into_rtrancl [Pure.intro]: "(a, b) : r^* ==> (b, c) : r ==> (a, c) : r^*"
+inductive2
+ rtrancl :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" ("(_^**)" [1000] 1000)
+ for r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
+where
+ rtrancl_refl [intro!, Pure.intro!, simp]: "r^** a a"
+ | rtrancl_into_rtrancl [Pure.intro]: "r^** a b ==> r b c ==> r^** a c"
-consts
- trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^+)" [1000] 999)
+inductive2
+ trancl :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" ("(_^++)" [1000] 1000)
+ for r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
+where
+ r_into_trancl [intro, Pure.intro]: "r a b ==> r^++ a b"
+ | trancl_into_trancl [Pure.intro]: "r^++ a b ==> r b c ==> r^++ a c"
-inductive "r^+"
- intros
- r_into_trancl [intro, Pure.intro]: "(a, b) : r ==> (a, b) : r^+"
- trancl_into_trancl [Pure.intro]: "(a, b) : r^+ ==> (b, c) : r ==> (a,c) : r^+"
+constdefs
+ rtrancl_set :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^*)" [1000] 999)
+ "r^* == Collect2 (member2 r)^**"
+
+ trancl_set :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^+)" [1000] 999)
+ "r^+ == Collect2 (member2 r)^++"
abbreviation
- reflcl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^=)" [1000] 999) where
+ reflcl :: "('a => 'a => bool) => 'a => 'a => bool" ("(_^==)" [1000] 1000) where
+ "r^== == join r op ="
+
+abbreviation
+ reflcl_set :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^=)" [1000] 999) where
"r^= == r \<union> Id"
notation (xsymbols)
- rtrancl ("(_\<^sup>*)" [1000] 999) and
- trancl ("(_\<^sup>+)" [1000] 999) and
- reflcl ("(_\<^sup>=)" [1000] 999)
+ rtrancl ("(_\<^sup>*\<^sup>*)" [1000] 1000) and
+ trancl ("(_\<^sup>+\<^sup>+)" [1000] 1000) and
+ reflcl ("(_\<^sup>=\<^sup>=)" [1000] 1000) and
+ rtrancl_set ("(_\<^sup>*)" [1000] 999) and
+ trancl_set ("(_\<^sup>+)" [1000] 999) and
+ reflcl_set ("(_\<^sup>=)" [1000] 999)
notation (HTML output)
- rtrancl ("(_\<^sup>*)" [1000] 999) and
- trancl ("(_\<^sup>+)" [1000] 999) and
- reflcl ("(_\<^sup>=)" [1000] 999)
+ rtrancl ("(_\<^sup>*\<^sup>*)" [1000] 1000) and
+ trancl ("(_\<^sup>+\<^sup>+)" [1000] 1000) and
+ reflcl ("(_\<^sup>=\<^sup>=)" [1000] 1000) and
+ rtrancl_set ("(_\<^sup>*)" [1000] 999) and
+ trancl_set ("(_\<^sup>+)" [1000] 999) and
+ reflcl_set ("(_\<^sup>=)" [1000] 999)
subsection {* Reflexive-transitive closure *}
+lemma rtrancl_set_eq [pred_set_conv]: "(member2 r)^** = member2 (r^*)"
+ by (simp add: rtrancl_set_def)
+
+lemma reflcl_set_eq [pred_set_conv]: "(join (member2 r) op =) = member2 (r Un Id)"
+ by (simp add: expand_fun_eq)
+
+lemmas rtrancl_refl [intro!, Pure.intro!, simp] = rtrancl_refl [to_set]
+lemmas rtrancl_into_rtrancl [Pure.intro] = rtrancl_into_rtrancl [to_set]
+
lemma r_into_rtrancl [intro]: "!!p. p \<in> r ==> p \<in> r^*"
-- {* @{text rtrancl} of @{text r} contains @{text r} *}
apply (simp only: split_tupled_all)
apply (erule rtrancl_refl [THEN rtrancl_into_rtrancl])
done
-lemma rtrancl_mono: "r \<subseteq> s ==> r^* \<subseteq> s^*"
+lemma r_into_rtrancl' [intro]: "r x y ==> r^** x y"
+ -- {* @{text rtrancl} of @{text r} contains @{text r} *}
+ by (erule rtrancl.rtrancl_refl [THEN rtrancl.rtrancl_into_rtrancl])
+
+lemma rtrancl_mono': "r \<le> s ==> r^** \<le> s^**"
-- {* monotonicity of @{text rtrancl} *}
- apply (rule subsetI)
- apply (simp only: split_tupled_all)
+ apply (rule predicate2I)
apply (erule rtrancl.induct)
- apply (rule_tac [2] rtrancl_into_rtrancl, blast+)
+ apply (rule_tac [2] rtrancl.rtrancl_into_rtrancl, blast+)
done
-theorem rtrancl_induct [consumes 1, induct set: rtrancl]:
- assumes a: "(a, b) : r^*"
- and cases: "P a" "!!y z. [| (a, y) : r^*; (y, z) : r; P y |] ==> P z"
+lemmas rtrancl_mono = rtrancl_mono' [to_set]
+
+theorem rtrancl_induct' [consumes 1, induct set: rtrancl]:
+ assumes a: "r^** a b"
+ and cases: "P a" "!!y z. [| r^** a y; r y z; P y |] ==> P z"
shows "P b"
proof -
from a have "a = a --> P b"
@@ -77,6 +106,12 @@
thus ?thesis by iprover
qed
+lemmas rtrancl_induct [consumes 1, induct set: rtrancl_set] = rtrancl_induct' [to_set]
+
+lemmas rtrancl_induct2' =
+ rtrancl_induct'[of _ "(ax,ay)" "(bx,by)", split_rule,
+ consumes 1, case_names refl step]
+
lemmas rtrancl_induct2 =
rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete),
consumes 1, case_names refl step]
@@ -95,6 +130,12 @@
lemmas rtrancl_trans = trans_rtrancl [THEN transD, standard]
+lemma rtrancl_trans':
+ assumes xy: "r^** x y"
+ and yz: "r^** y z"
+ shows "r^** x z" using yz xy
+ by induct iprover+
+
lemma rtranclE:
assumes major: "(a::'a,b) : r^*"
and cases: "(a = b) ==> P"
@@ -114,21 +155,25 @@
apply (erule rtrancl_induct, auto)
done
-lemma converse_rtrancl_into_rtrancl:
- "(a, b) \<in> r \<Longrightarrow> (b, c) \<in> r\<^sup>* \<Longrightarrow> (a, c) \<in> r\<^sup>*"
- by (rule rtrancl_trans) iprover+
+lemma converse_rtrancl_into_rtrancl':
+ "r a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>*\<^sup>* a c"
+ by (rule rtrancl_trans') iprover+
+
+lemmas converse_rtrancl_into_rtrancl = converse_rtrancl_into_rtrancl' [to_set]
text {*
\medskip More @{term "r^*"} equations and inclusions.
*}
-lemma rtrancl_idemp [simp]: "(r^*)^* = r^*"
- apply auto
- apply (erule rtrancl_induct)
- apply (rule rtrancl_refl)
- apply (blast intro: rtrancl_trans)
+lemma rtrancl_idemp' [simp]: "(r^**)^** = r^**"
+ apply (auto intro!: order_antisym)
+ apply (erule rtrancl_induct')
+ apply (rule rtrancl.rtrancl_refl)
+ apply (blast intro: rtrancl_trans')
done
+lemmas rtrancl_idemp [simp] = rtrancl_idemp' [to_set]
+
lemma rtrancl_idemp_self_comp [simp]: "R^* O R^* = R^*"
apply (rule set_ext)
apply (simp only: split_tupled_all)
@@ -138,16 +183,22 @@
lemma rtrancl_subset_rtrancl: "r \<subseteq> s^* ==> r^* \<subseteq> s^*"
by (drule rtrancl_mono, simp)
-lemma rtrancl_subset: "R \<subseteq> S ==> S \<subseteq> R^* ==> S^* = R^*"
- apply (drule rtrancl_mono)
- apply (drule rtrancl_mono, simp)
+lemma rtrancl_subset': "R \<le> S ==> S \<le> R^** ==> S^** = R^**"
+ apply (drule rtrancl_mono')
+ apply (drule rtrancl_mono', simp)
done
-lemma rtrancl_Un_rtrancl: "(R^* \<union> S^*)^* = (R \<union> S)^*"
- by (blast intro!: rtrancl_subset intro: r_into_rtrancl rtrancl_mono [THEN subsetD])
+lemmas rtrancl_subset = rtrancl_subset' [to_set]
+
+lemma rtrancl_Un_rtrancl': "(join (R^**) (S^**))^** = (join R S)^**"
+ by (blast intro!: rtrancl_subset' intro: rtrancl_mono' [THEN predicate2D])
-lemma rtrancl_reflcl [simp]: "(R^=)^* = R^*"
- by (blast intro!: rtrancl_subset intro: r_into_rtrancl)
+lemmas rtrancl_Un_rtrancl = rtrancl_Un_rtrancl' [to_set]
+
+lemma rtrancl_reflcl' [simp]: "(R^==)^** = R^**"
+ by (blast intro!: rtrancl_subset')
+
+lemmas rtrancl_reflcl [simp] = rtrancl_reflcl' [to_set]
lemma rtrancl_r_diff_Id: "(r - Id)^* = r^*"
apply (rule sym)
@@ -157,58 +208,75 @@
apply (blast intro!: r_into_rtrancl)
done
-theorem rtrancl_converseD:
- assumes r: "(x, y) \<in> (r^-1)^*"
- shows "(y, x) \<in> r^*"
+lemma rtrancl_r_diff_Id': "(meet r op ~=)^** = r^**"
+ apply (rule sym)
+ apply (rule rtrancl_subset')
+ apply blast+
+ done
+
+theorem rtrancl_converseD':
+ assumes r: "(r^--1)^** x y"
+ shows "r^** y x"
proof -
from r show ?thesis
- by induct (iprover intro: rtrancl_trans dest!: converseD)+
+ by induct (iprover intro: rtrancl_trans' dest!: conversepD)+
qed
-theorem rtrancl_converseI:
- assumes r: "(y, x) \<in> r^*"
- shows "(x, y) \<in> (r^-1)^*"
+lemmas rtrancl_converseD = rtrancl_converseD' [to_set]
+
+theorem rtrancl_converseI':
+ assumes r: "r^** y x"
+ shows "(r^--1)^** x y"
proof -
from r show ?thesis
- by induct (iprover intro: rtrancl_trans converseI)+
+ by induct (iprover intro: rtrancl_trans' conversepI)+
qed
+lemmas rtrancl_converseI = rtrancl_converseI' [to_set]
+
lemma rtrancl_converse: "(r^-1)^* = (r^*)^-1"
by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI)
lemma sym_rtrancl: "sym r ==> sym (r^*)"
by (simp only: sym_conv_converse_eq rtrancl_converse [symmetric])
-theorem converse_rtrancl_induct[consumes 1]:
- assumes major: "(a, b) : r^*"
- and cases: "P b" "!!y z. [| (y, z) : r; (z, b) : r^*; P z |] ==> P y"
+theorem converse_rtrancl_induct'[consumes 1]:
+ assumes major: "r^** a b"
+ and cases: "P b" "!!y z. [| r y z; r^** z b; P z |] ==> P y"
shows "P a"
proof -
- from rtrancl_converseI [OF major]
+ from rtrancl_converseI' [OF major]
show ?thesis
- by induct (iprover intro: cases dest!: converseD rtrancl_converseD)+
+ by induct (iprover intro: cases dest!: conversepD rtrancl_converseD')+
qed
+lemmas converse_rtrancl_induct[consumes 1] = converse_rtrancl_induct' [to_set]
+
+lemmas converse_rtrancl_induct2' =
+ converse_rtrancl_induct'[of _ "(ax,ay)" "(bx,by)", split_rule,
+ consumes 1, case_names refl step]
+
lemmas converse_rtrancl_induct2 =
converse_rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete),
consumes 1, case_names refl step]
-lemma converse_rtranclE:
- assumes major: "(x,z):r^*"
+lemma converse_rtranclE':
+ assumes major: "r^** x z"
and cases: "x=z ==> P"
- "!!y. [| (x,y):r; (y,z):r^* |] ==> P"
+ "!!y. [| r x y; r^** y z |] ==> P"
shows P
- apply (subgoal_tac "x = z | (EX y. (x,y) : r & (y,z) : r^*)")
- apply (rule_tac [2] major [THEN converse_rtrancl_induct])
+ apply (subgoal_tac "x = z | (EX y. r x y & r^** y z)")
+ apply (rule_tac [2] major [THEN converse_rtrancl_induct'])
prefer 2 apply iprover
prefer 2 apply iprover
apply (erule asm_rl exE disjE conjE cases)+
done
-ML_setup {*
- bind_thm ("converse_rtranclE2", split_rule
- (read_instantiate [("x","(xa,xb)"), ("z","(za,zb)")] (thm "converse_rtranclE")));
-*}
+lemmas converse_rtranclE = converse_rtranclE' [to_set]
+
+lemmas converse_rtranclE2' = converse_rtranclE' [of _ "(xa,xb)" "(za,zb)", split_rule]
+
+lemmas converse_rtranclE2 = converse_rtranclE [of "(xa,xb)" "(za,zb)", split_rule]
lemma r_comp_rtrancl_eq: "r O r^* = r^* O r"
by (blast elim: rtranclE converse_rtranclE
@@ -220,8 +288,14 @@
subsection {* Transitive closure *}
+lemma trancl_set_eq [pred_set_conv]: "(member2 r)^++ = member2 (r^+)"
+ by (simp add: trancl_set_def)
+
+lemmas r_into_trancl [intro, Pure.intro] = r_into_trancl [to_set]
+lemmas trancl_into_trancl [Pure.intro] = trancl_into_trancl [to_set]
+
lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> s^+"
- apply (simp only: split_tupled_all)
+ apply (simp add: split_tupled_all trancl_set_def)
apply (erule trancl.induct)
apply (iprover dest: subsetD)+
done
@@ -233,24 +307,30 @@
\medskip Conversions between @{text trancl} and @{text rtrancl}.
*}
-lemma trancl_into_rtrancl: "(a, b) \<in> r^+ ==> (a, b) \<in> r^*"
+lemma trancl_into_rtrancl': "r^++ a b ==> r^** a b"
by (erule trancl.induct) iprover+
-lemma rtrancl_into_trancl1: assumes r: "(a, b) \<in> r^*"
- shows "!!c. (b, c) \<in> r ==> (a, c) \<in> r^+" using r
+lemmas trancl_into_rtrancl = trancl_into_rtrancl' [to_set]
+
+lemma rtrancl_into_trancl1': assumes r: "r^** a b"
+ shows "!!c. r b c ==> r^++ a c" using r
by induct iprover+
-lemma rtrancl_into_trancl2: "[| (a,b) : r; (b,c) : r^* |] ==> (a,c) : r^+"
+lemmas rtrancl_into_trancl1 = rtrancl_into_trancl1' [to_set]
+
+lemma rtrancl_into_trancl2': "[| r a b; r^** b c |] ==> r^++ a c"
-- {* intro rule from @{text r} and @{text rtrancl} *}
- apply (erule rtranclE, iprover)
- apply (rule rtrancl_trans [THEN rtrancl_into_trancl1])
- apply (assumption | rule r_into_rtrancl)+
+ apply (erule rtrancl.cases, iprover)
+ apply (rule rtrancl_trans' [THEN rtrancl_into_trancl1'])
+ apply (simp | rule r_into_rtrancl')+
done
-lemma trancl_induct [consumes 1, induct set: trancl]:
- assumes a: "(a,b) : r^+"
- and cases: "!!y. (a, y) : r ==> P y"
- "!!y z. (a,y) : r^+ ==> (y, z) : r ==> P y ==> P z"
+lemmas rtrancl_into_trancl2 = rtrancl_into_trancl2' [to_set]
+
+lemma trancl_induct' [consumes 1, induct set: trancl]:
+ assumes a: "r^++ a b"
+ and cases: "!!y. r a y ==> P y"
+ "!!y z. r^++ a y ==> r y z ==> P y ==> P z"
shows "P b"
-- {* Nice induction rule for @{text trancl} *}
proof -
@@ -259,19 +339,32 @@
thus ?thesis by iprover
qed
+lemmas trancl_induct [consumes 1, induct set: trancl_set] = trancl_induct' [to_set]
+
+lemmas trancl_induct2' =
+ trancl_induct'[of _ "(ax,ay)" "(bx,by)", split_rule,
+ consumes 1, case_names base step]
+
lemmas trancl_induct2 =
trancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete),
consumes 1, case_names base step]
-lemma trancl_trans_induct:
- assumes major: "(x,y) : r^+"
- and cases: "!!x y. (x,y) : r ==> P x y"
- "!!x y z. [| (x,y) : r^+; P x y; (y,z) : r^+; P y z |] ==> P x z"
+lemma trancl_trans_induct':
+ assumes major: "r^++ x y"
+ and cases: "!!x y. r x y ==> P x y"
+ "!!x y z. [| r^++ x y; P x y; r^++ y z; P y z |] ==> P x z"
shows "P x y"
-- {* Another induction rule for trancl, incorporating transitivity *}
- by (iprover intro: r_into_trancl major [THEN trancl_induct] cases)
+ by (iprover intro: major [THEN trancl_induct'] cases)
+
+lemmas trancl_trans_induct = trancl_trans_induct' [to_set]
-inductive_cases tranclE: "(a, b) : r^+"
+lemma tranclE:
+ assumes H: "(a, b) : r^+"
+ and cases: "(a, b) : r ==> P" "\<And>c. (a, c) : r^+ ==> (c, b) : r ==> P"
+ shows P
+ using H [simplified trancl_set_def, simplified]
+ by cases (auto intro: cases [simplified trancl_set_def, simplified])
lemma trancl_Int_subset: "[| r \<subseteq> s; r O (r^+ \<inter> s) \<subseteq> s|] ==> r^+ \<subseteq> s"
apply (rule subsetI)
@@ -293,6 +386,12 @@
lemmas trancl_trans = trans_trancl [THEN transD, standard]
+lemma trancl_trans':
+ assumes xy: "r^++ x y"
+ and yz: "r^++ y z"
+ shows "r^++ x z" using yz xy
+ by induct iprover+
+
lemma trancl_id[simp]: "trans r \<Longrightarrow> r^+ = r"
apply(auto)
apply(erule trancl_induct)
@@ -301,12 +400,16 @@
apply(blast)
done
-lemma rtrancl_trancl_trancl: assumes r: "(x, y) \<in> r^*"
- shows "!!z. (y, z) \<in> r^+ ==> (x, z) \<in> r^+" using r
- by induct (iprover intro: trancl_trans)+
+lemma rtrancl_trancl_trancl': assumes r: "r^** x y"
+ shows "!!z. r^++ y z ==> r^++ x z" using r
+ by induct (iprover intro: trancl_trans')+
-lemma trancl_into_trancl2: "(a, b) \<in> r ==> (b, c) \<in> r^+ ==> (a, c) \<in> r^+"
- by (erule transD [OF trans_trancl r_into_trancl])
+lemmas rtrancl_trancl_trancl = rtrancl_trancl_trancl' [to_set]
+
+lemma trancl_into_trancl2': "r a b ==> r^++ b c ==> r^++ a c"
+ by (erule trancl_trans' [OF trancl.r_into_trancl])
+
+lemmas trancl_into_trancl2 = trancl_into_trancl2' [to_set]
lemma trancl_insert:
"(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}"
@@ -321,41 +424,51 @@
[THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
done
-lemma trancl_converseI: "(x, y) \<in> (r^+)^-1 ==> (x, y) \<in> (r^-1)^+"
- apply (drule converseD)
- apply (erule trancl.induct)
- apply (iprover intro: converseI trancl_trans)+
+lemma trancl_converseI': "(r^++)^--1 x y ==> (r^--1)^++ x y"
+ apply (drule conversepD)
+ apply (erule trancl_induct')
+ apply (iprover intro: conversepI trancl_trans')+
done
-lemma trancl_converseD: "(x, y) \<in> (r^-1)^+ ==> (x, y) \<in> (r^+)^-1"
- apply (rule converseI)
- apply (erule trancl.induct)
- apply (iprover dest: converseD intro: trancl_trans)+
+lemmas trancl_converseI = trancl_converseI' [to_set]
+
+lemma trancl_converseD': "(r^--1)^++ x y ==> (r^++)^--1 x y"
+ apply (rule conversepI)
+ apply (erule trancl_induct')
+ apply (iprover dest: conversepD intro: trancl_trans')+
done
-lemma trancl_converse: "(r^-1)^+ = (r^+)^-1"
- by (fastsimp simp add: split_tupled_all
- intro!: trancl_converseI trancl_converseD)
+lemmas trancl_converseD = trancl_converseD' [to_set]
+
+lemma trancl_converse': "(r^--1)^++ = (r^++)^--1"
+ by (fastsimp simp add: expand_fun_eq
+ intro!: trancl_converseI' dest!: trancl_converseD')
+
+lemmas trancl_converse = trancl_converse' [to_set]
lemma sym_trancl: "sym r ==> sym (r^+)"
by (simp only: sym_conv_converse_eq trancl_converse [symmetric])
-lemma converse_trancl_induct:
- assumes major: "(a,b) : r^+"
- and cases: "!!y. (y,b) : r ==> P(y)"
- "!!y z.[| (y,z) : r; (z,b) : r^+; P(z) |] ==> P(y)"
+lemma converse_trancl_induct':
+ assumes major: "r^++ a b"
+ and cases: "!!y. r y b ==> P(y)"
+ "!!y z.[| r y z; r^++ z b; P(z) |] ==> P(y)"
shows "P a"
- apply (rule major [THEN converseI, THEN trancl_converseI [THEN trancl_induct]])
+ apply (rule trancl_induct' [OF trancl_converseI', OF conversepI, OF major])
apply (rule cases)
- apply (erule converseD)
- apply (blast intro: prems dest!: trancl_converseD)
+ apply (erule conversepD)
+ apply (blast intro: prems dest!: trancl_converseD' conversepD)
done
-lemma tranclD: "(x, y) \<in> R^+ ==> EX z. (x, z) \<in> R \<and> (z, y) \<in> R^*"
- apply (erule converse_trancl_induct, auto)
- apply (blast intro: rtrancl_trans)
+lemmas converse_trancl_induct = converse_trancl_induct' [to_set]
+
+lemma tranclD': "R^++ x y ==> EX z. R x z \<and> R^** z y"
+ apply (erule converse_trancl_induct', auto)
+ apply (blast intro: rtrancl_trans')
done
+lemmas tranclD = tranclD' [to_set]
+
lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+"
by (blast elim: tranclE dest: trancl_into_rtrancl)
@@ -373,12 +486,14 @@
apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+
done
-lemma reflcl_trancl [simp]: "(r^+)^= = r^*"
- apply safe
- apply (erule trancl_into_rtrancl)
- apply (blast elim: rtranclE dest: rtrancl_into_trancl1)
+lemma reflcl_trancl' [simp]: "(r^++)^== = r^**"
+ apply (safe intro!: order_antisym)
+ apply (erule trancl_into_rtrancl')
+ apply (blast elim: rtrancl.cases dest: rtrancl_into_trancl1')
done
+lemmas reflcl_trancl [simp] = reflcl_trancl' [to_set]
+
lemma trancl_reflcl [simp]: "(r^=)^+ = r^*"
apply safe
apply (drule trancl_into_rtrancl, simp)
@@ -394,8 +509,10 @@
lemma rtrancl_empty [simp]: "{}^* = Id"
by (rule subst [OF reflcl_trancl]) simp
-lemma rtranclD: "(a, b) \<in> R^* ==> a = b \<or> a \<noteq> b \<and> (a, b) \<in> R^+"
- by (force simp add: reflcl_trancl [symmetric] simp del: reflcl_trancl)
+lemma rtranclD': "R^** a b ==> a = b \<or> a \<noteq> b \<and> R^++ a b"
+ by (force simp add: reflcl_trancl' [symmetric] simp del: reflcl_trancl')
+
+lemmas rtranclD = rtranclD' [to_set]
lemma rtrancl_eq_or_trancl:
"(x,y) \<in> R\<^sup>* = (x=y \<or> x\<noteq>y \<and> (x,y) \<in> R\<^sup>+)"
@@ -450,24 +567,32 @@
apply (fast intro: r_r_into_trancl trancl_trans)
done
-lemma trancl_rtrancl_trancl:
- "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r\<^sup>* ==> (a, c) \<in> r\<^sup>+"
- apply (drule tranclD)
+lemma trancl_rtrancl_trancl':
+ "r\<^sup>+\<^sup>+ a b ==> r\<^sup>*\<^sup>* b c ==> r\<^sup>+\<^sup>+ a c"
+ apply (drule tranclD')
apply (erule exE, erule conjE)
- apply (drule rtrancl_trans, assumption)
- apply (drule rtrancl_into_trancl2, assumption, assumption)
+ apply (drule rtrancl_trans', assumption)
+ apply (drule rtrancl_into_trancl2', assumption, assumption)
done
+lemmas trancl_rtrancl_trancl = trancl_rtrancl_trancl' [to_set]
+
lemmas transitive_closure_trans [trans] =
r_r_into_trancl trancl_trans rtrancl_trans
trancl_into_trancl trancl_into_trancl2
rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
rtrancl_trancl_trancl trancl_rtrancl_trancl
+lemmas transitive_closure_trans' [trans] =
+ trancl_trans' rtrancl_trans'
+ trancl.trancl_into_trancl trancl_into_trancl2'
+ rtrancl.rtrancl_into_rtrancl converse_rtrancl_into_rtrancl'
+ rtrancl_trancl_trancl' trancl_rtrancl_trancl'
+
declare trancl_into_rtrancl [elim]
-declare rtranclE [cases set: rtrancl]
-declare tranclE [cases set: trancl]
+declare rtranclE [cases set: rtrancl_set]
+declare tranclE [cases set: trancl_set]
@@ -490,8 +615,8 @@
fun decomp (Trueprop $ t) =
let fun dec (Const ("op :", _) $ (Const ("Pair", _) $ a $ b) $ rel ) =
- let fun decr (Const ("Transitive_Closure.rtrancl", _ ) $ r) = (r,"r*")
- | decr (Const ("Transitive_Closure.trancl", _ ) $ r) = (r,"r+")
+ let fun decr (Const ("Transitive_Closure.rtrancl_set", _ ) $ r) = (r,"r*")
+ | decr (Const ("Transitive_Closure.trancl_set", _ ) $ r) = (r,"r+")
| decr r = (r,"r");
val (rel,r) = decr rel;
in SOME (a,b,rel,r) end
@@ -500,9 +625,34 @@
end);
+structure Tranclp_Tac = Trancl_Tac_Fun (
+ struct
+ val r_into_trancl = thm "trancl.r_into_trancl";
+ val trancl_trans = thm "trancl_trans'";
+ val rtrancl_refl = thm "rtrancl.rtrancl_refl";
+ val r_into_rtrancl = thm "r_into_rtrancl'";
+ val trancl_into_rtrancl = thm "trancl_into_rtrancl'";
+ val rtrancl_trancl_trancl = thm "rtrancl_trancl_trancl'";
+ val trancl_rtrancl_trancl = thm "trancl_rtrancl_trancl'";
+ val rtrancl_trans = thm "rtrancl_trans'";
+
+ fun decomp (Trueprop $ t) =
+ let fun dec (rel $ a $ b) =
+ let fun decr (Const ("Transitive_Closure.rtrancl", _ ) $ r) = (r,"r*")
+ | decr (Const ("Transitive_Closure.trancl", _ ) $ r) = (r,"r+")
+ | decr r = (r,"r");
+ val (rel,r) = decr rel;
+ in SOME (a, b, Envir.beta_eta_contract rel, r) end
+ | dec _ = NONE
+ in dec t end;
+
+ end);
+
change_simpset (fn ss => ss
addSolver (mk_solver "Trancl" (fn _ => Trancl_Tac.trancl_tac))
- addSolver (mk_solver "Rtrancl" (fn _ => Trancl_Tac.rtrancl_tac)));
+ addSolver (mk_solver "Rtrancl" (fn _ => Trancl_Tac.rtrancl_tac))
+ addSolver (mk_solver "Tranclp" (fn _ => Tranclp_Tac.trancl_tac))
+ addSolver (mk_solver "Rtranclp" (fn _ => Tranclp_Tac.rtrancl_tac)));
*}
@@ -514,5 +664,11 @@
method_setup rtrancl =
{* Method.no_args (Method.SIMPLE_METHOD' Trancl_Tac.rtrancl_tac) *}
{* simple transitivity reasoner *}
+method_setup tranclp =
+ {* Method.no_args (Method.SIMPLE_METHOD' Tranclp_Tac.trancl_tac) *}
+ {* simple transitivity reasoner (predicate version) *}
+method_setup rtranclp =
+ {* Method.no_args (Method.SIMPLE_METHOD' Tranclp_Tac.rtrancl_tac) *}
+ {* simple transitivity reasoner (predicate version) *}
end