(* Title: HOL/Lambda/ParRed.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 1995 TU Muenchen
Properties of => and "cd", in particular the diamond property of => and
confluence of beta.
*)
header {* Parallel reduction and a complete developments *}
theory ParRed imports Lambda Commutation begin
subsection {* Parallel reduction *}
inductive par_beta :: "[dB, dB] => bool" (infixl "=>" 50)
where
var [simp, intro!]: "Var n => Var n"
| abs [simp, intro!]: "s => t ==> Abs s => Abs t"
| app [simp, intro!]: "[| s => s'; t => t' |] ==> s \<degree> t => s' \<degree> t'"
| beta [simp, intro!]: "[| s => s'; t => t' |] ==> (Abs s) \<degree> t => s'[t'/0]"
inductive_cases par_beta_cases [elim!]:
"Var n => t"
"Abs s => Abs t"
"(Abs s) \<degree> t => u"
"s \<degree> t => u"
"Abs s => t"
subsection {* Inclusions *}
text {* @{text "beta \<subseteq> par_beta \<subseteq> beta^*"} \medskip *}
lemma par_beta_varL [simp]:
"(Var n => t) = (t = Var n)"
by blast
lemma par_beta_refl [simp]: "t => t" (* par_beta_refl [intro!] causes search to blow up *)
by (induct t) simp_all
lemma beta_subset_par_beta: "beta <= par_beta"
apply (rule predicate2I)
apply (erule beta.induct)
apply (blast intro!: par_beta_refl)+
done
lemma par_beta_subset_beta: "par_beta <= beta^**"
apply (rule predicate2I)
apply (erule par_beta.induct)
apply blast
apply (blast del: rtranclp.rtrancl_refl intro: rtranclp.rtrancl_into_rtrancl)+
-- {* @{thm[source] rtrancl_refl} complicates the proof by increasing the branching factor *}
done
subsection {* Misc properties of @{text "par_beta"} *}
lemma par_beta_lift [simp]:
"t => t' \<Longrightarrow> lift t n => lift t' n"
by (induct t arbitrary: t' n) fastsimp+
lemma par_beta_subst:
"s => s' \<Longrightarrow> t => t' \<Longrightarrow> t[s/n] => t'[s'/n]"
apply (induct t arbitrary: s s' t' n)
apply (simp add: subst_Var)
apply (erule par_beta_cases)
apply simp
apply (simp add: subst_subst [symmetric])
apply (fastsimp intro!: par_beta_lift)
apply fastsimp
done
subsection {* Confluence (directly) *}
lemma diamond_par_beta: "diamond par_beta"
apply (unfold diamond_def commute_def square_def)
apply (rule impI [THEN allI [THEN allI]])
apply (erule par_beta.induct)
apply (blast intro!: par_beta_subst)+
done
subsection {* Complete developments *}
consts
"cd" :: "dB => dB"
recdef "cd" "measure size"
"cd (Var n) = Var n"
"cd (Var n \<degree> t) = Var n \<degree> cd t"
"cd ((s1 \<degree> s2) \<degree> t) = cd (s1 \<degree> s2) \<degree> cd t"
"cd (Abs u \<degree> t) = (cd u)[cd t/0]"
"cd (Abs s) = Abs (cd s)"
lemma par_beta_cd: "s => t \<Longrightarrow> t => cd s"
apply (induct s arbitrary: t rule: cd.induct)
apply auto
apply (fast intro!: par_beta_subst)
done
subsection {* Confluence (via complete developments) *}
lemma diamond_par_beta2: "diamond par_beta"
apply (unfold diamond_def commute_def square_def)
apply (blast intro: par_beta_cd)
done
theorem beta_confluent: "confluent beta"
apply (rule diamond_par_beta2 diamond_to_confluence
par_beta_subset_beta beta_subset_par_beta)+
done
end