
diff -r 4a15873c4ec9 -r 41ee3bfccb4d src/HOL/BNF/Examples/Infinite_Derivation_Trees/Parallel.thy
--- a/src/HOL/BNF/Examples/Infinite_Derivation_Trees/Parallel.thy	Mon Oct 15 19:03:02 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,152 +0,0 @@
-(*  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
-
-no_notation plus_class.plus (infixl "+" 65)
-no_notation Sublist.parallel (infixl "\<parallel>" 50)
-    
-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
\ No newline at end of file




