(* Title: HOL/BNF/Examples/Infinite_Derivation_Trees/Parallel.thy
Author: Andrei Popescu, TU Muenchen
Copyright 2012
Parallel composition.
*)
header {* Parallel Composition *}
theory Parallel
imports Tree
begin
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)"
section{* Parallel composition *}
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 :: "Tree \<times> Tree \<Rightarrow> Tree" 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
section{* =-coinductive proofs *}
(* Detailed proofs of commutativity and associativity: *)
theorem par_com: "tr1 \<parallel> tr2 = tr2 \<parallel> tr1"
proof-
let ?\<phi> = "\<lambda> trA trB. \<exists> tr1 tr2. trA = tr1 \<parallel> tr2 \<and> trB = tr2 \<parallel> tr1"
{fix trA trB
assume "?\<phi> trA trB" hence "trA = trB"
proof (induct rule: Tree_coind, safe)
fix tr1 tr2
show "root (tr1 \<parallel> tr2) = root (tr2 \<parallel> tr1)"
unfolding root_par by (rule Nplus_comm)
next
fix tr1 tr2 :: Tree
let ?trA = "tr1 \<parallel> tr2" let ?trB = "tr2 \<parallel> tr1"
show "(?\<phi> ^#2) (cont ?trA) (cont ?trB)"
unfolding lift2_def proof(intro conjI allI impI)
fix n show "Inl n \<in> cont (tr1 \<parallel> tr2) \<longleftrightarrow> Inl n \<in> cont (tr2 \<parallel> tr1)"
unfolding Inl_in_cont_par by auto
next
fix trA' assume "Inr trA' \<in> cont ?trA"
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'. Inr trB' \<in> cont ?trB \<and> ?\<phi> trA' trB'"
apply(intro exI[of _ "tr2' \<parallel> tr1'"]) unfolding Inr_in_cont_par by auto
next
fix trB' assume "Inr trB' \<in> cont ?trB"
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'. Inr trA' \<in> cont ?trA \<and> ?\<phi> trA' trB'"
apply(intro exI[of _ "tr1' \<parallel> tr2'"]) unfolding Inr_in_cont_par by auto
qed
qed
}
thus ?thesis by blast
qed
theorem par_assoc: "(tr1 \<parallel> tr2) \<parallel> tr3 = tr1 \<parallel> (tr2 \<parallel> tr3)"
proof-
let ?\<phi> =
"\<lambda> trA trB. \<exists> tr1 tr2 tr3. trA = (tr1 \<parallel> tr2) \<parallel> tr3 \<and>
trB = tr1 \<parallel> (tr2 \<parallel> tr3)"
{fix trA trB
assume "?\<phi> trA trB" hence "trA = trB"
proof (induct rule: Tree_coind, 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 tr1 tr2 tr3
let ?trA = "(tr1 \<parallel> tr2) \<parallel> tr3" let ?trB = "tr1 \<parallel> (tr2 \<parallel> tr3)"
show "(?\<phi> ^#2) (cont ?trA) (cont ?trB)"
unfolding lift2_def proof(intro conjI allI impI)
fix n show "Inl n \<in> (cont ?trA) \<longleftrightarrow> Inl n \<in> (cont ?trB)"
unfolding Inl_in_cont_par by simp
next
fix trA' assume "Inr trA' \<in> cont ?trA"
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'. Inr trB' \<in> cont ?trB \<and> ?\<phi> trA' trB'"
apply(intro exI[of _ "tr1' \<parallel> (tr2' \<parallel> tr3')"])
unfolding Inr_in_cont_par by auto
next
fix trB' assume "Inr trB' \<in> cont ?trB"
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'. Inr trA' \<in> cont ?trA \<and> ?\<phi> trA' trB'"
apply(intro exI[of _ "(tr1' \<parallel> tr2') \<parallel> tr3'"])
unfolding Inr_in_cont_par by auto
qed
qed
}
thus ?thesis by blast
qed
end