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(* Title: ZF/ex/Commutation.thy
Author: Tobias Nipkow & Sidi Ould Ehmety
Copyright 1995 TU Muenchen
Commutation theory for proving the Church Rosser theorem.
*)
theory Commutation imports ZF begin
definition
square :: "[i, i, i, i] => o" where
"square(r,s,t,u) ==
(\<forall>a b. <a,b> \<in> r \<longrightarrow> (\<forall>c. <a, c> \<in> s \<longrightarrow> (\<exists>x. <b,x> \<in> t & <c,x> \<in> u)))"
definition
commute :: "[i, i] => o" where
"commute(r,s) == square(r,s,s,r)"
definition
diamond :: "i=>o" where
"diamond(r) == commute(r, r)"
definition
strip :: "i=>o" where
"strip(r) == commute(r^*, r)"
definition
Church_Rosser :: "i => o" where
"Church_Rosser(r) == (\x y. \ (r \ converse(r))^* \
(\<exists>z. <x,z> \<in> r^* & <y,z> \<in> r^*))"
definition
confluent :: "i=>o" where
"confluent(r) == diamond(r^*)"
lemma square_sym: "square(r,s,t,u) ==> square(s,r,u,t)"
unfolding square_def by blast
lemma square_subset: "[| square(r,s,t,u); t \ t' |] ==> square(r,s,t',u)"
unfolding square_def by blast
lemma square_rtrancl:
"square(r,s,s,t) ==> field(s)<=field(t) ==> square(r^*,s,s,t^*)"
apply (unfold square_def, clarify)
apply (erule rtrancl_induct)
apply (blast intro: rtrancl_refl)
apply (blast intro: rtrancl_into_rtrancl)
done
(* A special case of square_rtrancl_on *)
lemma diamond_strip:
"diamond(r) ==> strip(r)"
apply (unfold diamond_def commute_def strip_def)
apply (rule square_rtrancl, simp_all)
done
(*** commute ***)
lemma commute_sym: "commute(r,s) ==> commute(s,r)"
unfolding commute_def by (blast intro: square_sym)
lemma commute_rtrancl:
"commute(r,s) ==> field(r)=field(s) ==> commute(r^*,s^*)"
apply (unfold commute_def)
apply (rule square_rtrancl)
apply (rule square_sym [THEN square_rtrancl, THEN square_sym])
apply (simp_all add: rtrancl_field)
done
lemma confluentD: "confluent(r) ==> diamond(r^*)"
by (simp add: confluent_def)
lemma strip_confluent: "strip(r) ==> confluent(r)"
apply (unfold strip_def confluent_def diamond_def)
apply (drule commute_rtrancl)
apply (simp_all add: rtrancl_field)
done
lemma commute_Un: "[| commute(r,t); commute(s,t) |] ==> commute(r \ s, t)"
unfolding commute_def square_def by blast
lemma diamond_Un:
"[| diamond(r); diamond(s); commute(r, s) |] ==> diamond(r \ s)"
unfolding diamond_def by (blast intro: commute_Un commute_sym)
lemma diamond_confluent:
"diamond(r) ==> confluent(r)"
apply (unfold diamond_def confluent_def)
apply (erule commute_rtrancl, simp)
done
lemma confluent_Un:
"[| confluent(r); confluent(s); commute(r^*, s^*);
relation(r); relation(s) |] ==> confluent(r \<union> s)"
apply (unfold confluent_def)
apply (rule rtrancl_Un_rtrancl [THEN subst], auto)
apply (blast dest: diamond_Un intro: diamond_confluent [THEN confluentD])
done
lemma diamond_to_confluence:
"[| diamond(r); s \ r; r<= s^* |] ==> confluent(s)"
apply (drule rtrancl_subset [symmetric], assumption)
apply (simp_all add: confluent_def)
apply (blast intro: diamond_confluent [THEN confluentD])
done
(*** Church_Rosser ***)
lemma Church_Rosser1:
"Church_Rosser(r) ==> confluent(r)"
apply (unfold confluent_def Church_Rosser_def square_def
commute_def diamond_def, auto)
apply (drule converseI)
apply (simp (no_asm_use) add: rtrancl_converse [symmetric])
apply (drule_tac x = b in spec)
apply (drule_tac x1 = c in spec [THEN mp])
apply (rule_tac b = a in rtrancl_trans)
apply (blast intro: rtrancl_mono [THEN subsetD])+
done
lemma Church_Rosser2:
"confluent(r) ==> Church_Rosser(r)"
apply (unfold confluent_def Church_Rosser_def square_def
commute_def diamond_def, auto)
apply (frule fieldI1)
apply (simp add: rtrancl_field)
apply (erule rtrancl_induct, auto)
apply (blast intro: rtrancl_refl)
apply (blast del: rtrancl_refl intro: r_into_rtrancl rtrancl_trans)+
done
lemma Church_Rosser: "Church_Rosser(r) \ confluent(r)"
by (blast intro: Church_Rosser1 Church_Rosser2)
end
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