# Theory Trancl

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theory Trancl
imports CCL
`(*  Title:      CCL/Trancl.thy    Author:     Martin Coen, Cambridge University Computer Laboratory    Copyright   1993  University of Cambridge*)header {* Transitive closure of a relation *}theory Tranclimports CCLbegindefinition trans :: "i set => o"  (*transitivity predicate*)  where "trans(r) == (ALL x y z. <x,y>:r --> <y,z>:r --> <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 rtrancl :: "i set => i set"  ("(_^*)" [100] 100)  where "r^* == lfp(%s. id Un (r O s))"definition trancl :: "i set => 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)"  unfolding trans_def by blastlemma transD: "[| trans(r);  <a,b>:r;  <b,c>:r |] ==> <a,c>:r"  unfolding trans_def by blastsubsection {* Identity relation *}lemma idI: "<a,a> : id"  apply (unfold id_def)  apply (rule CollectI)  apply (rule exI)  apply (rule refl)  donelemma idE:    "[| p: id;  !!x.[| p = <x,x> |] ==> P |] ==>  P"  apply (unfold id_def)  apply (erule CollectE)  apply blast  donesubsection {* Composition of two relations *}lemma compI: "[| <a,b>:s; <b,c>:r |] ==> <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"  unfolding relcomp_def by blastlemma compEpair:  "[| <a,c> : r O s;      !!y. [| <a,y>:s;  <y,c>:r |] ==> P   |] ==> P"  apply (erule compE)  apply (simp add: pair_inject)  donelemmas [intro] = compI idI  and [elim] = compE idE  and [elim!] = pair_injectlemma comp_mono: "[| r'<=r; s'<=s |] ==> (r' O s') <= (r O s)"  by blastsubsection {* The relation rtrancl *}lemma rtrancl_fun_mono: "mono(%s. id Un (r O s))"  apply (rule monoI)  apply (rule monoI subset_refl comp_mono Un_mono)+  apply assumption  donelemma rtrancl_unfold: "r^* = id Un (r O r^*)"  by (rule rtrancl_fun_mono [THEN rtrancl_def [THEN def_lfp_Tarski]])(*Reflexivity of rtrancl*)lemma rtrancl_refl: "<a,a> : r^*"  apply (subst rtrancl_unfold)  apply blast  done(*Closure under composition with r*)lemma rtrancl_into_rtrancl: "[| <a,b> : r^*;  <b,c> : r |] ==> <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^*"  apply (rule rtrancl_refl [THEN rtrancl_into_rtrancl])  apply assumption  donesubsection {* 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>)"  apply (erule def_induct [OF rtrancl_def])   apply (rule rtrancl_fun_mono)  apply blast  done(*nice induction rule*)lemma rtrancl_induct:  "[| <a,b> : r^*;      P(a);      !!y z.[| <a,y> : r^*;  <y,z> : r;  P(y) |] ==> P(z) |]    ==> P(b)"(*by induction on this formula*)  apply (subgoal_tac "ALL y. <a,b> = <a,y> --> P(y)")(*now solve first subgoal: this formula is sufficient*)  apply blast(*now do the induction*)  apply (erule rtrancl_full_induct)   apply blast  apply blast  done(*transitivity of transitive closure!! -- by induction.*)lemma trans_rtrancl: "trans(r^*)"  apply (rule transI)  apply (rule_tac b = z in rtrancl_induct)    apply (fast elim: rtrancl_into_rtrancl)+  done(*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)")   prefer 2   apply (erule rtrancl_induct)    apply blast   apply blast  apply blast  donesubsection {* The relation trancl *}subsubsection {* Conversions between trancl and rtrancl *}lemma trancl_into_rtrancl: "[| <a,b> : r^+ |] ==> <a,b> : r^*"  apply (unfold trancl_def)  apply (erule compEpair)  apply (erule rtrancl_into_rtrancl)  apply assumption  done(*r^+ contains r*)lemma r_into_trancl: "[| <a,b> : r |] ==> <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^+"  unfolding trancl_def by blast(*intro rule from r and rtrancl*)lemma rtrancl_into_trancl2: "[| <a,b> : r;  <b,c> : r^* |]   ==>  <a,c> : r^+"  apply (erule rtranclE)   apply (erule subst)   apply (erule r_into_trancl)  apply (rule trans_rtrancl [THEN transD, THEN rtrancl_into_trancl1])    apply (assumption | rule r_into_rtrancl)+  done(*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)")   apply blast  apply (unfold trancl_def)  apply (erule compEpair)  apply (erule rtranclE)   apply blast  apply (blast intro!: rtrancl_into_trancl1)  done(*Transitivity of r^+.  Proved by unfolding since it uses transitivity of rtrancl. *)lemma trans_trancl: "trans(r^+)"  apply (unfold trancl_def)  apply (rule transI)  apply (erule compEpair)+  apply (erule rtrancl_into_rtrancl [THEN trans_rtrancl [THEN transD, THEN compI]])    apply assumption+  donelemma trancl_into_trancl2: "[| <a,b> : r;  <b,c> : r^+ |]   ==>  <a,c> : r^+"  apply (rule r_into_trancl [THEN trans_trancl [THEN transD]])   apply assumption+  doneend`