--- a/src/CCL/Trancl.thy Tue Nov 11 13:50:56 2014 +0100
+++ b/src/CCL/Trancl.thy Tue Nov 11 15:55:31 2014 +0100
@@ -9,29 +9,28 @@
imports CCL
begin
-definition trans :: "i set => o" (*transitivity predicate*)
- where "trans(r) == (ALL x y z. <x,y>:r --> <y,z>:r --> <x,z>:r)"
+definition trans :: "i set \<Rightarrow> o" (*transitivity predicate*)
+ where "trans(r) == (ALL x y z. <x,y>:r \<longrightarrow> <y,z>:r \<longrightarrow> <x,z>:r)"
definition id :: "i set" (*the identity relation*)
where "id == {p. EX x. p = <x,x>}"
-definition relcomp :: "[i set,i set] => i set" (infixr "O" 60) (*composition of relations*)
- where "r O s == {xz. EX x y z. xz = <x,z> & <x,y>:s & <y,z>:r}"
+definition relcomp :: "[i set,i set] \<Rightarrow> i set" (infixr "O" 60) (*composition of relations*)
+ where "r O s == {xz. EX x y z. xz = <x,z> \<and> <x,y>:s \<and> <y,z>:r}"
-definition rtrancl :: "i set => i set" ("(_^*)" [100] 100)
- where "r^* == lfp(%s. id Un (r O s))"
+definition rtrancl :: "i set \<Rightarrow> i set" ("(_^*)" [100] 100)
+ where "r^* == lfp(\<lambda>s. id Un (r O s))"
-definition trancl :: "i set => i set" ("(_^+)" [100] 100)
+definition trancl :: "i set \<Rightarrow> i set" ("(_^+)" [100] 100)
where "r^+ == r O rtrancl(r)"
subsection {* Natural deduction for @{text "trans(r)"} *}
-lemma transI:
- "(!! x y z. [| <x,y>:r; <y,z>:r |] ==> <x,z>:r) ==> trans(r)"
+lemma transI: "(\<And>x y z. \<lbrakk><x,y>:r; <y,z>:r\<rbrakk> \<Longrightarrow> <x,z>:r) \<Longrightarrow> trans(r)"
unfolding trans_def by blast
-lemma transD: "[| trans(r); <a,b>:r; <b,c>:r |] ==> <a,c>:r"
+lemma transD: "\<lbrakk>trans(r); <a,b>:r; <b,c>:r\<rbrakk> \<Longrightarrow> <a,c>:r"
unfolding trans_def by blast
@@ -44,8 +43,7 @@
apply (rule refl)
done
-lemma idE:
- "[| p: id; !!x.[| p = <x,x> |] ==> P |] ==> P"
+lemma idE: "\<lbrakk>p: id; \<And>x. p = <x,x> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
apply (unfold id_def)
apply (erule CollectE)
apply blast
@@ -54,20 +52,14 @@
subsection {* Composition of two relations *}
-lemma compI: "[| <a,b>:s; <b,c>:r |] ==> <a,c> : r O s"
+lemma compI: "\<lbrakk><a,b>:s; <b,c>:r\<rbrakk> \<Longrightarrow> <a,c> : r O s"
unfolding relcomp_def by blast
(*proof requires higher-level assumptions or a delaying of hyp_subst_tac*)
-lemma compE:
- "[| xz : r O s;
- !!x y z. [| xz = <x,z>; <x,y>:s; <y,z>:r |] ==> P
- |] ==> P"
+lemma compE: "\<lbrakk>xz : r O s; \<And>x y z. \<lbrakk>xz = <x,z>; <x,y>:s; <y,z>:r\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
unfolding relcomp_def by blast
-lemma compEpair:
- "[| <a,c> : r O s;
- !!y. [| <a,y>:s; <y,c>:r |] ==> P
- |] ==> P"
+lemma compEpair: "\<lbrakk><a,c> : r O s; \<And>y. \<lbrakk><a,y>:s; <y,c>:r\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
apply (erule compE)
apply (simp add: pair_inject)
done
@@ -76,13 +68,13 @@
and [elim] = compE idE
and [elim!] = pair_inject
-lemma comp_mono: "[| r'<=r; s'<=s |] ==> (r' O s') <= (r O s)"
+lemma comp_mono: "\<lbrakk>r'<=r; s'<=s\<rbrakk> \<Longrightarrow> (r' O s') <= (r O s)"
by blast
subsection {* The relation rtrancl *}
-lemma rtrancl_fun_mono: "mono(%s. id Un (r O s))"
+lemma rtrancl_fun_mono: "mono(\<lambda>s. id Un (r O s))"
apply (rule monoI)
apply (rule monoI subset_refl comp_mono Un_mono)+
apply assumption
@@ -98,13 +90,13 @@
done
(*Closure under composition with r*)
-lemma rtrancl_into_rtrancl: "[| <a,b> : r^*; <b,c> : r |] ==> <a,c> : r^*"
+lemma rtrancl_into_rtrancl: "\<lbrakk><a,b> : r^*; <b,c> : r\<rbrakk> \<Longrightarrow> <a,c> : r^*"
apply (subst rtrancl_unfold)
apply blast
done
(*rtrancl of r contains r*)
-lemma r_into_rtrancl: "[| <a,b> : r |] ==> <a,b> : r^*"
+lemma r_into_rtrancl: "<a,b> : r \<Longrightarrow> <a,b> : r^*"
apply (rule rtrancl_refl [THEN rtrancl_into_rtrancl])
apply assumption
done
@@ -113,10 +105,10 @@
subsection {* standard induction rule *}
lemma rtrancl_full_induct:
- "[| <a,b> : r^*;
- !!x. P(<x,x>);
- !!x y z.[| P(<x,y>); <x,y>: r^*; <y,z>: r |] ==> P(<x,z>) |]
- ==> P(<a,b>)"
+ "\<lbrakk><a,b> : r^*;
+ \<And>x. P(<x,x>);
+ \<And>x y z. \<lbrakk>P(<x,y>); <x,y>: r^*; <y,z>: r\<rbrakk> \<Longrightarrow> P(<x,z>)\<rbrakk>
+ \<Longrightarrow> P(<a,b>)"
apply (erule def_induct [OF rtrancl_def])
apply (rule rtrancl_fun_mono)
apply blast
@@ -124,12 +116,12 @@
(*nice induction rule*)
lemma rtrancl_induct:
- "[| <a,b> : r^*;
+ "\<lbrakk><a,b> : r^*;
P(a);
- !!y z.[| <a,y> : r^*; <y,z> : r; P(y) |] ==> P(z) |]
- ==> P(b)"
+ \<And>y z. \<lbrakk><a,y> : r^*; <y,z> : r; P(y)\<rbrakk> \<Longrightarrow> P(z) \<rbrakk>
+ \<Longrightarrow> P(b)"
(*by induction on this formula*)
- apply (subgoal_tac "ALL y. <a,b> = <a,y> --> P(y)")
+ apply (subgoal_tac "ALL y. <a,b> = <a,y> \<longrightarrow> P(y)")
(*now solve first subgoal: this formula is sufficient*)
apply blast
(*now do the induction*)
@@ -147,10 +139,8 @@
(*elimination of rtrancl -- by induction on a special formula*)
lemma rtranclE:
- "[| <a,b> : r^*; (a = b) ==> P;
- !!y.[| <a,y> : r^*; <y,b> : r |] ==> P |]
- ==> P"
- apply (subgoal_tac "a = b | (EX y. <a,y> : r^* & <y,b> : r)")
+ "\<lbrakk><a,b> : r^*; a = b \<Longrightarrow> P; \<And>y. \<lbrakk><a,y> : r^*; <y,b> : r\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
+ apply (subgoal_tac "a = b | (EX y. <a,y> : r^* \<and> <y,b> : r)")
prefer 2
apply (erule rtrancl_induct)
apply blast
@@ -163,7 +153,7 @@
subsubsection {* Conversions between trancl and rtrancl *}
-lemma trancl_into_rtrancl: "[| <a,b> : r^+ |] ==> <a,b> : r^*"
+lemma trancl_into_rtrancl: "<a,b> : r^+ \<Longrightarrow> <a,b> : r^*"
apply (unfold trancl_def)
apply (erule compEpair)
apply (erule rtrancl_into_rtrancl)
@@ -171,15 +161,15 @@
done
(*r^+ contains r*)
-lemma r_into_trancl: "[| <a,b> : r |] ==> <a,b> : r^+"
+lemma r_into_trancl: "<a,b> : r \<Longrightarrow> <a,b> : r^+"
unfolding trancl_def by (blast intro: rtrancl_refl)
(*intro rule by definition: from rtrancl and r*)
-lemma rtrancl_into_trancl1: "[| <a,b> : r^*; <b,c> : r |] ==> <a,c> : r^+"
+lemma rtrancl_into_trancl1: "\<lbrakk><a,b> : r^*; <b,c> : r\<rbrakk> \<Longrightarrow> <a,c> : r^+"
unfolding trancl_def by blast
(*intro rule from r and rtrancl*)
-lemma rtrancl_into_trancl2: "[| <a,b> : r; <b,c> : r^* |] ==> <a,c> : r^+"
+lemma rtrancl_into_trancl2: "\<lbrakk><a,b> : r; <b,c> : r^*\<rbrakk> \<Longrightarrow> <a,c> : r^+"
apply (erule rtranclE)
apply (erule subst)
apply (erule r_into_trancl)
@@ -189,11 +179,10 @@
(*elimination of r^+ -- NOT an induction rule*)
lemma tranclE:
- "[| <a,b> : r^+;
- <a,b> : r ==> P;
- !!y.[| <a,y> : r^+; <y,b> : r |] ==> P
- |] ==> P"
- apply (subgoal_tac "<a,b> : r | (EX y. <a,y> : r^+ & <y,b> : r)")
+ "\<lbrakk><a,b> : r^+;
+ <a,b> : r \<Longrightarrow> P;
+ \<And>y. \<lbrakk><a,y> : r^+; <y,b> : r\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
+ apply (subgoal_tac "<a,b> : r | (EX y. <a,y> : r^+ \<and> <y,b> : r)")
apply blast
apply (unfold trancl_def)
apply (erule compEpair)
@@ -212,7 +201,7 @@
apply assumption+
done
-lemma trancl_into_trancl2: "[| <a,b> : r; <b,c> : r^+ |] ==> <a,c> : r^+"
+lemma trancl_into_trancl2: "\<lbrakk><a,b> : r; <b,c> : r^+\<rbrakk> \<Longrightarrow> <a,c> : r^+"
apply (rule r_into_trancl [THEN trans_trancl [THEN transD]])
apply assumption+
done