author | wenzelm |
Wed, 23 Nov 2005 22:26:13 +0100 | |
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parent 16417 | 9bc16273c2d4 |
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(* Title: HOL/Lambda/ParRed.thy |
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ID: $Id$ |
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Author: Tobias Nipkow |
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Copyright 1995 TU Muenchen |
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Properties of => and "cd", in particular the diamond property of => and |
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confluence of beta. |
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*) |
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header {* Parallel reduction and a complete developments *} |
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theory ParRed imports Lambda Commutation begin |
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subsection {* Parallel reduction *} |
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consts |
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par_beta :: "(dB \<times> dB) set" |
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syntax |
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par_beta :: "[dB, dB] => bool" (infixl "=>" 50) |
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translations |
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"s => t" == "(s, t) \<in> par_beta" |
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inductive par_beta |
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intros |
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var [simp, intro!]: "Var n => Var n" |
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abs [simp, intro!]: "s => t ==> Abs s => Abs t" |
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app [simp, intro!]: "[| s => s'; t => t' |] ==> s \<degree> t => s' \<degree> t'" |
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beta [simp, intro!]: "[| s => s'; t => t' |] ==> (Abs s) \<degree> t => s'[t'/0]" |
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inductive_cases par_beta_cases [elim!]: |
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"Var n => t" |
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"Abs s => Abs t" |
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"(Abs s) \<degree> t => u" |
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"s \<degree> t => u" |
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"Abs s => t" |
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subsection {* Inclusions *} |
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text {* @{text "beta \<subseteq> par_beta \<subseteq> beta^*"} \medskip *} |
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lemma par_beta_varL [simp]: |
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"(Var n => t) = (t = Var n)" |
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by blast |
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lemma par_beta_refl [simp]: "t => t" (* par_beta_refl [intro!] causes search to blow up *) |
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by (induct t) simp_all |
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lemma beta_subset_par_beta: "beta <= par_beta" |
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apply (rule subsetI) |
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apply clarify |
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apply (erule beta.induct) |
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apply (blast intro!: par_beta_refl)+ |
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done |
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lemma par_beta_subset_beta: "par_beta <= beta^*" |
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apply (rule subsetI) |
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apply clarify |
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apply (erule par_beta.induct) |
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apply blast |
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apply (blast del: rtrancl_refl intro: rtrancl_into_rtrancl)+ |
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-- {* @{thm[source] rtrancl_refl} complicates the proof by increasing the branching factor *} |
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done |
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subsection {* Misc properties of par-beta *} |
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lemma par_beta_lift [simp]: |
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"t => t' \<Longrightarrow> lift t n => lift t' n" |
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by (induct t fixing: t' n) fastsimp+ |
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lemma par_beta_subst: |
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"s => s' \<Longrightarrow> t => t' \<Longrightarrow> t[s/n] => t'[s'/n]" |
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apply (induct t fixing: s s' t' n) |
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apply (simp add: subst_Var) |
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apply (erule par_beta_cases) |
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apply simp |
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apply (simp add: subst_subst [symmetric]) |
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apply (fastsimp intro!: par_beta_lift) |
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apply fastsimp |
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done |
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subsection {* Confluence (directly) *} |
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lemma diamond_par_beta: "diamond par_beta" |
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apply (unfold diamond_def commute_def square_def) |
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apply (rule impI [THEN allI [THEN allI]]) |
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apply (erule par_beta.induct) |
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apply (blast intro!: par_beta_subst)+ |
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done |
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subsection {* Complete developments *} |
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consts |
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"cd" :: "dB => dB" |
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recdef "cd" "measure size" |
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"cd (Var n) = Var n" |
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"cd (Var n \<degree> t) = Var n \<degree> cd t" |
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"cd ((s1 \<degree> s2) \<degree> t) = cd (s1 \<degree> s2) \<degree> cd t" |
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"cd (Abs u \<degree> t) = (cd u)[cd t/0]" |
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"cd (Abs s) = Abs (cd s)" |
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lemma par_beta_cd: "s => t \<Longrightarrow> t => cd s" |
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apply (induct s fixing: t rule: cd.induct) |
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apply auto |
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apply (fast intro!: par_beta_subst) |
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done |
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subsection {* Confluence (via complete developments) *} |
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lemma diamond_par_beta2: "diamond par_beta" |
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apply (unfold diamond_def commute_def square_def) |
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apply (blast intro: par_beta_cd) |
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done |
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theorem beta_confluent: "confluent beta" |
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apply (rule diamond_par_beta2 diamond_to_confluence |
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par_beta_subset_beta beta_subset_par_beta)+ |
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done |
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end |