
nipkow@10213: (*  Title:      HOL/Transitive_Closure.thy
nipkow@10213:     ID:         $Id$
nipkow@10213:     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
nipkow@10213:     Copyright   1992  University of Cambridge
nipkow@10213: 
nipkow@10213: Relfexive and Transitive closure of a relation
nipkow@10213: 
nipkow@10213: rtrancl is reflexive/transitive closure;
nipkow@10213: trancl  is transitive closure
nipkow@10213: reflcl  is reflexive closure
nipkow@10213: 
wenzelm@10331: These postfix operators have MAXIMUM PRIORITY, forcing their operands
wenzelm@10331: to be atomic.
nipkow@10213: *)
nipkow@10213: 
wenzelm@10980: theory Transitive_Closure = Lfp + Relation
wenzelm@10980: files ("Transitive_Closure_lemmas.ML"):
nipkow@10213: 
nipkow@10213: constdefs
wenzelm@10331:   rtrancl :: "('a * 'a) set => ('a * 'a) set"    ("(_^*)" [1000] 999)
wenzelm@10331:   "r^* == lfp (%s. Id Un (r O s))"
nipkow@10213: 
wenzelm@10331:   trancl :: "('a * 'a) set => ('a * 'a) set"    ("(_^+)" [1000] 999)
wenzelm@10331:   "r^+ ==  r O rtrancl r"
nipkow@10213: 
nipkow@10213: syntax
wenzelm@10331:   "_reflcl" :: "('a * 'a) set => ('a * 'a) set"    ("(_^=)" [1000] 999)
nipkow@10213: translations
nipkow@10213:   "r^=" == "r Un Id"
nipkow@10213: 
wenzelm@10827: syntax (xsymbols)
wenzelm@10331:   rtrancl :: "('a * 'a) set => ('a * 'a) set"    ("(_\\<^sup>*)" [1000] 999)
wenzelm@10331:   trancl :: "('a * 'a) set => ('a * 'a) set"    ("(_\\<^sup>+)" [1000] 999)
wenzelm@10331:   "_reflcl" :: "('a * 'a) set => ('a * 'a) set"    ("(_\\<^sup>=)" [1000] 999)
wenzelm@10331: 
wenzelm@10980: use "Transitive_Closure_lemmas.ML"
wenzelm@10980: 
nipkow@10996: 
nipkow@10996: lemma reflcl_trancl[simp]: "(r\<^sup>+)\<^sup>= = r\<^sup>*"
nipkow@10996: apply safe;
nipkow@10996: apply (erule trancl_into_rtrancl);
nipkow@10996: by (blast elim:rtranclE dest:rtrancl_into_trancl1)
nipkow@10996: 
nipkow@10996: lemma trancl_reflcl[simp]: "(r\<^sup>=)\<^sup>+ = r\<^sup>*"
nipkow@10996: apply safe
nipkow@10996:  apply (drule trancl_into_rtrancl)
nipkow@10996:  apply simp;
nipkow@10996: apply (erule rtranclE)
nipkow@10996:  apply safe
nipkow@10996:  apply(rule r_into_trancl)
nipkow@10996:  apply simp
nipkow@10996: apply(rule rtrancl_into_trancl1)
nipkow@10996:  apply(erule rtrancl_reflcl[THEN equalityD2, THEN subsetD])
nipkow@10996: apply fast
nipkow@10996: done
nipkow@10996: 
nipkow@10996: lemma trancl_empty[simp]: "{}\<^sup>+ = {}"
nipkow@10996: by (auto elim:trancl_induct)
nipkow@10996: 
nipkow@10996: lemma rtrancl_empty[simp]: "{}\<^sup>* = Id"
nipkow@10996: by(rule subst[OF reflcl_trancl], simp)
nipkow@10996: 
nipkow@10996: lemma rtranclD: "(a,b) \<in> R\<^sup>* \<Longrightarrow> a=b \<or> a\<noteq>b \<and> (a,b) \<in> R\<^sup>+"
nipkow@10996: by(force simp add: reflcl_trancl[THEN sym] simp del: reflcl_trancl)
nipkow@10996: 
nipkow@10996: (* should be merged with the main body of lemmas: *)
nipkow@10996: 
nipkow@10996: lemma Domain_rtrancl[simp]: "Domain(R\<^sup>*) = UNIV"
nipkow@10996: by blast
nipkow@10996: 
nipkow@10996: lemma Range_rtrancl[simp]: "Range(R\<^sup>*) = UNIV"
nipkow@10996: by blast
nipkow@10996: 
nipkow@10996: lemma rtrancl_Un_subset: "(R\<^sup>* \<union> S\<^sup>*) \<subseteq> (R Un S)\<^sup>*"
nipkow@10996: by(rule rtrancl_Un_rtrancl[THEN subst], fast)
nipkow@10996: 
nipkow@10996: lemma in_rtrancl_UnI: "x \<in> R\<^sup>* \<or> x \<in> S\<^sup>* \<Longrightarrow> x \<in> (R \<union> S)\<^sup>*"
nipkow@10996: by (blast intro: subsetD[OF rtrancl_Un_subset])
nipkow@10996: 
nipkow@10996: lemma trancl_domain [simp]: "Domain (r\<^sup>+) = Domain r"
nipkow@10996: by (unfold Domain_def, blast dest:tranclD)
nipkow@10996: 
nipkow@10996: lemma trancl_range [simp]: "Range (r\<^sup>+) = Range r"
nipkow@10996: by (simp add:Range_def trancl_converse[THEN sym])
nipkow@10996: 
nipkow@10213: end




