Quellcode-Bibliothek

^© Kompilation durch diese Firma

[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]

Datei: Stream_adm.thy Sprache: Isabelle

Original von: Isabelle^©

(*  Title:      HOL/HOLCF/FOCUS/Stream_adm.thy
    Author:     David von Oheimb, TU Muenchen
*)

section \<open>Admissibility for streams\<close>

theory Stream_adm
imports "HOLCF-Library.Stream" "HOL-Library.Order_Continuity"
begin

definition
  stream_monoP  :: "(('a stream) set \ ('a stream) set) \ bool" where
  "stream_monoP F = (\Q i. \P s. enat i \ #s \
                    (s \<in> F P) = (stream_take i\<cdot>s \<in> Q \<and> iterate i\<cdot>rt\<cdot>s \<in> P))"

definition
  stream_antiP  :: "(('a stream) set \ ('a stream) set) \ bool" where
  "stream_antiP F = (\P x. \Q i.
                (#x  < enat i \<longrightarrow> (\<forall>y. x \<sqsubseteq> y \<longrightarrow> y \<in> F P \<longrightarrow> x \<in> F P)) \<and>
                (enat i <= #x \<longrightarrow> (\<forall>y. x \<sqsubseteq> y \<longrightarrow>
                (y \<in> F P) = (stream_take i\<cdot>y \<in> Q \<and> iterate i\<cdot>rt\<cdot>y \<in> P))))"

definition
  antitonP :: "'a set => bool" where
  "antitonP P = (\x y. x \ y \ y\P \ x\P)"

(* ----------------------------------------------------------------------- *)

section "admissibility"

lemma infinite_chain_adm_lemma:
  "\Porder.chain Y; \i. P (Y i);
    \<And>Y. \<lbrakk>Porder.chain Y; \<forall>i. P (Y i); \<not> finite_chain Y\<rbrakk> \<Longrightarrow> P (\<Squnion>i. Y i)\<rbrakk>
      \<Longrightarrow> P (\<Squnion>i. Y i)"
apply (case_tac "finite_chain Y")
prefer 2 apply fast
apply (unfold finite_chain_def)
apply safe
apply (erule lub_finch1 [THEN lub_eqI, THEN ssubst])
apply assumption
apply (erule spec)
done

lemma increasing_chain_adm_lemma:
  "\Porder.chain Y; \i. P (Y i); \Y. \Porder.chain Y; \i. P (Y i);
    \<forall>i. \<exists>j>i. Y i \<noteq> Y j \<and> Y i \<sqsubseteq> Y j\<rbrakk> \<Longrightarrow> P (\<Squnion>i. Y i)\<rbrakk>
      \<Longrightarrow> P (\<Squnion>i. Y i)"
apply (erule infinite_chain_adm_lemma)
apply assumption
apply (erule thin_rl)
apply (unfold finite_chain_def)
apply (unfold max_in_chain_def)
apply (fast dest: le_imp_less_or_eq elim: chain_mono_less)
done

lemma flatstream_adm_lemma:
  assumes 1: "Porder.chain Y"
  assumes 2: "\i. P (Y i)"
  assumes 3: "(\Y. [| Porder.chain Y; \i. P (Y i); \k. \j. enat k < #((Y j)::'a::flat stream)|]
  ==> P(LUB i. Y i))"
  shows "P(LUB i. Y i)"
apply (rule increasing_chain_adm_lemma [OF 1 2])
apply (erule 3, assumption)
apply (erule thin_rl)
apply (rule allI)
apply (case_tac "\j. stream_finite (Y j)")
apply ( rule chain_incr)
apply ( rule allI)
apply ( drule spec)
apply ( safe)
apply ( rule exI)
apply ( rule slen_strict_mono)
apply (   erule spec)
apply (  assumption)
apply ( assumption)
apply (metis enat_ord_code(4) slen_infinite)
done

(* should be without reference to stream length? *)
lemma flatstream_admI: "[|(\Y. [| Porder.chain Y; \i. P (Y i);
\<forall>k. \<exists>j. enat k < #((Y j)::'a::flat stream)|] ==> P(LUB i. Y i))|]==> adm P"
apply (unfold adm_def)
apply (intro strip)
apply (erule (1) flatstream_adm_lemma)
apply (fast)
done

(* context (theory "Extended_Nat");*)
lemma ile_lemma: "enat (i + j) <= x ==> enat i <= x"
  by (rule order_trans) auto

lemma stream_monoP2I:
"\X. stream_monoP F \ \i. \l. \x y.
  enat l \<le> #x \<longrightarrow> (x::'a::flat stream) << y --> x \<in> (F ^^ i) top \<longrightarrow> y \<in> (F ^^ i) top"
apply (unfold stream_monoP_def)
apply (safe)
apply (rule_tac x="i*ia" in exI)
apply (induct_tac "ia")
apply ( simp)
apply (simp)
apply (intro strip)
apply (erule allE, erule all_dupE, drule mp, erule ile_lemma)
apply (drule_tac P="%x. x" in subst, assumption)
apply (erule allE, drule mp, rule ile_lemma) back
apply ( erule order_trans)
apply ( erule slen_mono)
apply (erule ssubst)
apply (safe)
apply ( erule (2) ile_lemma [THEN slen_take_lemma3, THEN subst])
apply (erule allE)
apply (drule mp)
apply ( erule slen_rt_mult)
apply (erule allE)
apply (drule mp)
apply (erule monofun_rt_mult)
apply (drule (1) mp)
apply (assumption)
done

lemma stream_monoP2_gfp_admI: "[| \i. \l. \x y.
enat l \<le> #x \<longrightarrow> (x::'a::flat stream) << y \<longrightarrow> x \<in> (F ^^ i) top \<longrightarrow> y \<in> (F ^^ i) top;
    inf_continuous F |] ==> adm (\<lambda>x. x \<in> gfp F)"
apply (erule inf_continuous_gfp[of F, THEN ssubst])
apply (simp (no_asm))
apply (rule adm_lemmas)
apply (rule flatstream_admI)
apply (erule allE)
apply (erule exE)
apply (erule allE, erule exE)
apply (erule allE, erule allE, drule mp) (* stream_monoP *)
apply ( drule ileI1)
apply ( drule order_trans)
apply (  rule ile_eSuc)
apply ( drule eSuc_ile_mono [THEN iffD1])
apply ( assumption)
apply (drule mp)
apply ( erule is_ub_thelub)
apply (fast)
done

lemmas fstream_gfp_admI = stream_monoP2I [THEN stream_monoP2_gfp_admI]

lemma stream_antiP2I:
"\X. [|stream_antiP (F::(('a::flat stream)set => ('a stream set)))|]
  ==> \<forall>i x y. x << y \<longrightarrow> y \<in> (F ^^ i) top \<longrightarrow> x \<in> (F ^^ i) top"
apply (unfold stream_antiP_def)
apply (rule allI)
apply (induct_tac "i")
apply ( simp)
apply (simp)
apply (intro strip)
apply (erule allE, erule all_dupE, erule exE, erule exE)
apply (erule conjE)
apply (case_tac "#x < enat i")
apply ( fast)
apply (unfold linorder_not_less)
apply (drule (1) mp)
apply (erule all_dupE, drule mp, rule below_refl)
apply (erule ssubst)
apply (erule allE, drule (1) mp)
apply (drule_tac P="%x. x" in subst, assumption)
apply (erule conjE, rule conjI)
apply ( erule slen_take_lemma3 [THEN ssubst], assumption)
apply ( assumption)
apply (erule allE, erule allE, drule mp, erule monofun_rt_mult)
apply (drule (1) mp)
apply (assumption)
done

lemma stream_antiP2_non_gfp_admI:
"\X. [|\i x y. x << y \ y \ (F ^^ i) top \ x \ (F ^^ i) top; inf_continuous F |]
  ==> adm (\<lambda>u. \<not> u \<in> gfp F)"
apply (unfold adm_def)
apply (simp add: inf_continuous_gfp)
apply (fast dest!: is_ub_thelub)
done

lemmas fstream_non_gfp_admI = stream_antiP2I [THEN stream_antiP2_non_gfp_admI]

(**new approach for adm********************************************************)

section "antitonP"

lemma antitonPD: "[| antitonP P; y \ P; x< x \ P"
apply (unfold antitonP_def)
apply auto
done

lemma antitonPI: "\x y. y \ P \ x< x \ P \ antitonP P"
apply (unfold antitonP_def)
apply (fast)
done

lemma antitonP_adm_non_P: "antitonP P \ adm (\u. u \ P)"
apply (unfold adm_def)
apply (auto dest: antitonPD elim: is_ub_thelub)
done

lemma def_gfp_adm_nonP: "P \ gfp F \ {y. \x::'a::pcpo. y \ x \ x \ P} \ F {y. \x. y \ x \ x \ P} \
  adm (\<lambda>u. u\<notin>P)"
apply (simp)
apply (rule antitonP_adm_non_P)
apply (rule antitonPI)
apply (drule gfp_upperbound)
apply (fast)
done

lemma adm_set:
"{\i. Y i |Y. Porder.chain Y \ (\i. Y i \ P)} \ P \ adm (\x. x\P)"
apply (unfold adm_def)
apply (fast)
done

lemma def_gfp_admI: "P \ gfp F \ {\i. Y i |Y. Porder.chain Y \ (\i. Y i \ P)} \
  F {\<Squnion>i. Y i |Y. Porder.chain Y \<and> (\<forall>i. Y i \<in> P)} \<Longrightarrow> adm (\<lambda>x. x\<in>P)"
apply (simp)
apply (rule adm_set)
apply (erule gfp_upperbound)
done

end

¤ Dauer der Verarbeitung: 0.1 Sekunden (vorverarbeitet) ¤

Download des Quellennavigators
Download des sprechenden Kalenders
in der Quellcodebibliothek suchen

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.