(* Title: HOL/Datatype_Examples/Derivation_Trees/Parallel_Composition.thy
Author: Andrei Popescu, TU Muenchen
Copyright 2012
Parallel composition.
*)
section \<open>Parallel Composition\<close>
theory Parallel_Composition
imports DTree
begin
no_notation plus_class.plus (infixl "+" 65)
consts Nplus :: "N \<Rightarrow> N \<Rightarrow> N" (infixl "+" 60)
axiomatization where
Nplus_comm: "(a::N) + b = b + (a::N)"
and Nplus_assoc: "((a::N) + b) + c = a + (b + c)"
subsection\<open>Corecursive Definition of Parallel Composition\<close>
fun par_r where "par_r (tr1,tr2) = root tr1 + root tr2"
fun par_c where
"par_c (tr1,tr2) =
Inl ` (Inl -` (cont tr1 \<union> cont tr2)) \<union>
Inr ` (Inr -` cont tr1 \<times> Inr -` cont tr2)"
declare par_r.simps[simp del] declare par_c.simps[simp del]
definition par :: "dtree \<times> dtree \<Rightarrow> dtree" where
"par \<equiv> unfold par_r par_c"
abbreviation par_abbr (infixr "\<parallel>" 80) where "tr1 \<parallel> tr2 \<equiv> par (tr1, tr2)"
lemma finite_par_c: "finite (par_c (tr1, tr2))"
unfolding par_c.simps apply(rule finite_UnI)
apply (metis finite_Un finite_cont finite_imageI finite_vimageI inj_Inl)
apply(intro finite_imageI finite_cartesian_product finite_vimageI)
using finite_cont by auto
lemma root_par: "root (tr1 \<parallel> tr2) = root tr1 + root tr2"
using unfold(1)[of par_r par_c "(tr1,tr2)"] unfolding par_def par_r.simps by simp
lemma cont_par:
"cont (tr1 \<parallel> tr2) = (id \<oplus> par) ` par_c (tr1,tr2)"
using unfold(2)[of par_c "(tr1,tr2)" par_r, OF finite_par_c]
unfolding par_def ..
lemma Inl_cont_par[simp]:
"Inl -` (cont (tr1 \<parallel> tr2)) = Inl -` (cont tr1 \<union> cont tr2)"
unfolding cont_par par_c.simps by auto
lemma Inr_cont_par[simp]:
"Inr -` (cont (tr1 \<parallel> tr2)) = par ` (Inr -` cont tr1 \<times> Inr -` cont tr2)"
unfolding cont_par par_c.simps by auto
lemma Inl_in_cont_par:
"Inl t \<in> cont (tr1 \<parallel> tr2) \<longleftrightarrow> (Inl t \<in> cont tr1 \<or> Inl t \<in> cont tr2)"
using Inl_cont_par[of tr1 tr2] unfolding vimage_def by auto
lemma Inr_in_cont_par:
"Inr t \<in> cont (tr1 \<parallel> tr2) \<longleftrightarrow> (t \<in> par ` (Inr -` cont tr1 \<times> Inr -` cont tr2))"
using Inr_cont_par[of tr1 tr2] unfolding vimage_def by auto
subsection\<open>Structural Coinduction Proofs\<close>
lemma rel_set_rel_sum_eq[simp]:
"rel_set (rel_sum (=) \<phi>) A1 A2 \<longleftrightarrow>
Inl -` A1 = Inl -` A2 \<and> rel_set \<phi> (Inr -` A1) (Inr -` A2)"
unfolding rel_set_rel_sum rel_set_eq ..
(* Detailed proofs of commutativity and associativity: *)
theorem par_com: "tr1 \<parallel> tr2 = tr2 \<parallel> tr1"
proof-
let ?\<theta> = "\<lambda> trA trB. \<exists> tr1 tr2. trA = tr1 \<parallel> tr2 \<and> trB = tr2 \<parallel> tr1"
{fix trA trB
assume "?\<theta> trA trB" hence "trA = trB"
apply (induct rule: dtree_coinduct)
unfolding rel_set_rel_sum rel_set_eq unfolding rel_set_def proof safe
fix tr1 tr2 show "root (tr1 \<parallel> tr2) = root (tr2 \<parallel> tr1)"
unfolding root_par by (rule Nplus_comm)
next
fix n tr1 tr2 assume "Inl n \<in> cont (tr1 \<parallel> tr2)" thus "n \<in> Inl -` (cont (tr2 \<parallel> tr1))"
unfolding Inl_in_cont_par by auto
next
fix n tr1 tr2 assume "Inl n \<in> cont (tr2 \<parallel> tr1)" thus "n \<in> Inl -` (cont (tr1 \<parallel> tr2))"
unfolding Inl_in_cont_par by auto
next
fix tr1 tr2 trA' assume "Inr trA' \<in> cont (tr1 \<parallel> tr2)"
then obtain tr1' tr2' where "trA' = tr1' \<parallel> tr2'"
and "Inr tr1' \<in> cont tr1" and "Inr tr2' \<in> cont tr2"
unfolding Inr_in_cont_par by auto
thus "\<exists> trB' \<in> Inr -` (cont (tr2 \<parallel> tr1)). ?\<theta> trA' trB'"
apply(intro bexI[of _ "tr2' \<parallel> tr1'"]) unfolding Inr_in_cont_par by auto
next
fix tr1 tr2 trB' assume "Inr trB' \<in> cont (tr2 \<parallel> tr1)"
then obtain tr1' tr2' where "trB' = tr2' \<parallel> tr1'"
and "Inr tr1' \<in> cont tr1" and "Inr tr2' \<in> cont tr2"
unfolding Inr_in_cont_par by auto
thus "\<exists> trA' \<in> Inr -` (cont (tr1 \<parallel> tr2)). ?\<theta> trA' trB'"
apply(intro bexI[of _ "tr1' \<parallel> tr2'"]) unfolding Inr_in_cont_par by auto
qed
}
thus ?thesis by blast
qed
lemma par_assoc: "(tr1 \<parallel> tr2) \<parallel> tr3 = tr1 \<parallel> (tr2 \<parallel> tr3)"
proof-
let ?\<theta> =
"\<lambda> trA trB. \<exists> tr1 tr2 tr3. trA = (tr1 \<parallel> tr2) \<parallel> tr3 \<and> trB = tr1 \<parallel> (tr2 \<parallel> tr3)"
{fix trA trB
assume "?\<theta> trA trB" hence "trA = trB"
apply (induct rule: dtree_coinduct)
unfolding rel_set_rel_sum rel_set_eq unfolding rel_set_def proof safe
fix tr1 tr2 tr3 show "root ((tr1 \<parallel> tr2) \<parallel> tr3) = root (tr1 \<parallel> (tr2 \<parallel> tr3))"
unfolding root_par by (rule Nplus_assoc)
next
fix n tr1 tr2 tr3 assume "Inl n \<in> (cont ((tr1 \<parallel> tr2) \<parallel> tr3))"
thus "n \<in> Inl -` (cont (tr1 \<parallel> tr2 \<parallel> tr3))" unfolding Inl_in_cont_par by simp
next
fix n tr1 tr2 tr3 assume "Inl n \<in> (cont (tr1 \<parallel> tr2 \<parallel> tr3))"
thus "n \<in> Inl -` (cont ((tr1 \<parallel> tr2) \<parallel> tr3))" unfolding Inl_in_cont_par by simp
next
fix trA' tr1 tr2 tr3 assume "Inr trA' \<in> cont ((tr1 \<parallel> tr2) \<parallel> tr3)"
then obtain tr1' tr2' tr3' where "trA' = (tr1' \<parallel> tr2') \<parallel> tr3'"
and "Inr tr1' \<in> cont tr1" and "Inr tr2' \<in> cont tr2"
and "Inr tr3' \<in> cont tr3" unfolding Inr_in_cont_par by auto
thus "\<exists> trB' \<in> Inr -` (cont (tr1 \<parallel> tr2 \<parallel> tr3)). ?\<theta> trA' trB'"
apply(intro bexI[of _ "tr1' \<parallel> tr2' \<parallel> tr3'"])
unfolding Inr_in_cont_par by auto
next
fix trB' tr1 tr2 tr3 assume "Inr trB' \<in> cont (tr1 \<parallel> tr2 \<parallel> tr3)"
then obtain tr1' tr2' tr3' where "trB' = tr1' \<parallel> (tr2' \<parallel> tr3')"
and "Inr tr1' \<in> cont tr1" and "Inr tr2' \<in> cont tr2"
and "Inr tr3' \<in> cont tr3" unfolding Inr_in_cont_par by auto
thus "\<exists> trA' \<in> Inr -` cont ((tr1 \<parallel> tr2) \<parallel> tr3). ?\<theta> trA' trB'"
apply(intro bexI[of _ "(tr1' \<parallel> tr2') \<parallel> tr3'"])
unfolding Inr_in_cont_par by auto
qed
}
thus ?thesis by blast
qed
end