src/HOLCF/FOCUS/Fstreams.thy
author oheimb
Tue, 07 Sep 2004 16:02:42 +0200
changeset 15188 9d57263faf9e
child 16219 af5ed1a10cd7
permissions -rw-r--r--
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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(*  Title: 	HOLCF/FOCUS/Fstreams.thy
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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    ID:         $Id$
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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    Author: 	Borislav Gajanovic
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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FOCUS flat streams (with lifted elements)
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###TODO: integrate this with Fstream.*
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*)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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theory Fstreams = Stream:
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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defaultsort type
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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types 'a fstream = "('a lift) stream"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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consts
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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  fsingleton    :: "'a => 'a fstream"  ("<_>" [1000] 999)
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  fsfilter      :: "'a set \<Rightarrow> 'a fstream \<rightarrow> 'a fstream"
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  fsmap		:: "('a => 'b) => 'a fstream -> 'b fstream"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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  jth           :: "nat => 'a fstream => 'a"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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  first         :: "'a fstream => 'a"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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  last          :: "'a fstream => 'a"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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syntax
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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  "@emptystream":: "'a fstream" 			("<>")
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  "@fsfilter"	:: "'a set \<Rightarrow> 'a fstream \<Rightarrow> 'a fstream"	("(_'(C')_)" [64,63] 63)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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syntax (xsymbols)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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  "@fsfilter"	:: "'a set \<Rightarrow> 'a fstream \<Rightarrow> 'a fstream" ("(_\<copyright>_)"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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								     [64,63] 63)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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translations
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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  "<>"    => "\<bottom>"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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  "A(C)s" == "fsfilter A\<cdot>s"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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defs
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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  fsingleton_def2:    "fsingleton  == %a. Def a && UU"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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  jth_def:            "jth == %n s. if Fin n < #s then THE a. i_th n s = Def a else arbitrary" 
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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  first_def:          "first == %s. jth 0 s"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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  last_def:           "last == %s. case #s of Fin n => (if n~=0 then jth (THE k. Suc k = n) s else arbitrary)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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                              | Infty => arbitrary"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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  fsfilter_def:       "fsfilter A \<equiv> sfilter\<cdot>(flift2 (\<lambda>x. x\<in>A))"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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  fsmap_def:   	      "fsmap f  == smap$(flift2 f)"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma ft_fsingleton[simp]: "ft$(<a>) = Def a"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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by (simp add: fsingleton_def2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma slen_fsingleton[simp]: "#(<a>) = Fin 1"
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by (simp add: fsingleton_def2 inat_defs)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma slen_fstreams[simp]: "#(<a> ooo s) = iSuc (#s)"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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by (simp add: fsingleton_def2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma slen_fstreams2[simp]: "#(s ooo <a>) = iSuc (#s)"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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apply (case_tac "#s", auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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apply (insert slen_sconc [of _ s "Suc 0" "<a>"], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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by (simp add: sconc_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma j_th_0_fsingleton[simp]:"jth 0 (<a>) = a"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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apply (simp add: fsingleton_def2 jth_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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by (simp add: i_th_def Fin_0)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma jth_0[simp]: "jth 0 (<a> ooo s) = a"  
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apply (simp add: fsingleton_def2 jth_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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by (simp add: i_th_def Fin_0)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma first_sconc[simp]: "first (<a> ooo s) = a"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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by (simp add: first_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma first_fsingleton[simp]: "first (<a>) = a"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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by (simp add: first_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma jth_n[simp]: "Fin n = #s ==> jth n (s ooo <a>) = a"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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apply (simp add: jth_def, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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apply (simp add: i_th_def rt_sconc1)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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by (simp add: inat_defs split: inat_splits)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma last_sconc[simp]: "Fin n = #s ==> last (s ooo <a>) = a"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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apply (simp add: last_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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apply (simp add: inat_defs split:inat_splits)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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by (drule sym, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma last_fsingleton[simp]: "last (<a>) = a"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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by (simp add: last_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma first_UU[simp]: "first UU = arbitrary"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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by (simp add: first_def jth_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma last_UU[simp]:"last UU = arbitrary"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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by (simp add: last_def jth_def inat_defs)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma last_infinite[simp]:"#s = Infty ==> last s = arbitrary"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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by (simp add: last_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma jth_slen_lemma1:"n <= k & Fin n = #s ==> jth k s = arbitrary"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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by (simp add: jth_def inat_defs split:inat_splits, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma jth_UU[simp]:"jth n UU = arbitrary" 
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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by (simp add: jth_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma ext_last:"[|s ~= UU; Fin (Suc n) = #s|] ==> (stream_take n$s) ooo <(last s)> = s" 
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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apply (simp add: last_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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apply (case_tac "#s", auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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apply (simp add: fsingleton_def2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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apply (subgoal_tac "Def (jth n s) = i_th n s")
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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apply (auto simp add: i_th_last)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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apply (drule slen_take_lemma1, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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apply (simp add: jth_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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apply (case_tac "i_th n s = UU")
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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   120
apply auto
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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apply (simp add: i_th_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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   122
apply (case_tac "i_rt n s = UU", auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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apply (drule i_rt_slen [THEN iffD1])
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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apply (drule slen_take_eq_rev [rule_format, THEN iffD2],auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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by (drule not_Undef_is_Def [THEN iffD1], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma fsingleton_lemma1[simp]: "(<a> = <b>) = (a=b)"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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by (simp add: fsingleton_def2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma fsingleton_lemma2[simp]: "<a> ~= <>"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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by (simp add: fsingleton_def2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma fsingleton_sconc:"<a> ooo s = Def a && s"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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by (simp add: fsingleton_def2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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   137
lemma fstreams_ind: 
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   138
  "[| adm P; P <>; !!a s. P s ==> P (<a> ooo s) |] ==> P x"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   139
apply (simp add: fsingleton_def2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   140
apply (rule stream.ind, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   141
by (drule not_Undef_is_Def [THEN iffD1], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   142
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   143
lemma fstreams_ind2:
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   144
  "[| adm P; P <>; !!a. P (<a>); !!a b s. P s ==> P (<a> ooo <b> ooo s) |] ==> P x"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   145
apply (simp add: fsingleton_def2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   146
apply (rule stream_ind2, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   147
by (drule not_Undef_is_Def [THEN iffD1], auto)+
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   148
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   149
lemma fstreams_take_Suc[simp]: "stream_take (Suc n)$(<a> ooo s) = <a> ooo stream_take n$s"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   150
by (simp add: fsingleton_def2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   151
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   152
lemma fstreams_not_empty[simp]: "<a> ooo s ~= <>"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   153
by (simp add: fsingleton_def2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   154
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   155
lemma fstreams_not_empty2[simp]: "s ooo <a> ~= <>"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   156
by (case_tac "s=UU", auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   157
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   158
lemma fstreams_exhaust: "x = UU | (EX a s. x = <a> ooo s)"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   159
apply (simp add: fsingleton_def2, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   160
apply (erule contrapos_pp, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   161
apply (drule stream_exhaust_eq [THEN iffD1], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   162
by (drule not_Undef_is_Def [THEN iffD1], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   163
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   164
lemma fstreams_cases: "[| x = UU ==> P; !!a y. x = <a> ooo y ==> P |] ==> P"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   165
by (insert fstreams_exhaust [of x], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   166
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   167
lemma fstreams_exhaust_eq: "(x ~= UU) = (? a y. x = <a> ooo y)"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   168
apply (simp add: fsingleton_def2, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   169
apply (drule stream_exhaust_eq [THEN iffD1], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   170
by (drule not_Undef_is_Def [THEN iffD1], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   171
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   172
lemma fstreams_inject: "(<a> ooo s = <b> ooo t) = (a=b & s=t)"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   173
by (simp add: fsingleton_def2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   174
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   175
lemma fstreams_prefix: "<a> ooo s << t ==> EX tt. t = <a> ooo tt &  s << tt"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   176
apply (simp add: fsingleton_def2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   177
apply (insert stream_prefix [of "Def a" s t], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   178
by (drule stream.inverts, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   179
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   180
lemma fstreams_prefix': "x << <a> ooo z = (x = <> |  (EX y. x = <a> ooo y &  y << z))"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   181
apply (auto, case_tac "x=UU", auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   182
apply (drule stream_exhaust_eq [THEN iffD1], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   183
apply (simp add: fsingleton_def2, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   184
apply (drule stream.inverts, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   185
apply (drule ax_flat [rule_format], simp)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   186
apply (drule stream.inverts, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   187
by (erule sconc_mono)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   189
lemma ft_fstreams[simp]: "ft$(<a> ooo s) = Def a"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   190
by (simp add: fsingleton_def2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   191
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   192
lemma rt_fstreams[simp]: "rt$(<a> ooo s) = s"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   193
by (simp add: fsingleton_def2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   194
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   195
lemma ft_eq[simp]: "(ft$s = Def a) = (EX t. s = <a> ooo t)"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   196
apply (rule stream.casedist [of s],auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   197
by (drule sym, auto simp add: fsingleton_def2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   198
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   199
lemma surjective_fstreams: "(<d> ooo y = x) = (ft$x = Def d & rt$x = y)"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   200
by auto
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   201
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   202
lemma fstreams_mono: "<a> ooo b << <a> ooo c ==> b << c"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   203
apply (simp add: fsingleton_def2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   204
by (drule stream.inverts, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   205
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   206
lemma fsmap_UU[simp]: "fsmap f$UU = UU"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   207
by (simp add: fsmap_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   208
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   209
lemma fsmap_fsingleton_sconc: "fsmap f$(<x> ooo xs) = <(f x)> ooo (fsmap f$xs)"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   210
by (simp add: fsmap_def fsingleton_def2 flift2_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   211
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   212
lemma fsmap_fsingleton[simp]: "fsmap f$(<x>) = <(f x)>"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   213
by (simp add: fsmap_def fsingleton_def2 flift2_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   214
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   215
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   216
declare range_composition[simp del]
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   217
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   218
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   219
lemma fstreams_chain_lemma[rule_format]:
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   220
  "ALL s x y. stream_take n$(s::'a fstream) << x & x << y & y << s & x ~= y --> stream_take (Suc n)$s << y"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   221
apply (induct_tac n, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   222
apply (case_tac "s=UU", auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   223
apply (drule stream_exhaust_eq [THEN iffD1], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   224
apply (case_tac "y=UU", auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   225
apply (drule stream_exhaust_eq [THEN iffD1], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   226
apply (drule stream.inverts, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   227
apply (simp add: less_lift_def, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   228
apply (rule monofun_cfun, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   229
apply (case_tac "s=UU", auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   230
apply (drule stream_exhaust_eq [THEN iffD1], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   231
apply (erule_tac x="ya" in allE)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   232
apply (drule stream_prefix, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   233
apply (case_tac "y=UU",auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   234
apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   235
apply (drule stream.inverts, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   236
apply (simp add: less_lift_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   237
apply (drule stream.inverts, auto)+
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   238
apply (erule_tac x="tt" in allE)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   239
apply (erule_tac x="yb" in allE, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   240
apply (simp add: less_lift_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   241
by (rule monofun_cfun, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   242
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   243
lemma fstreams_lub_lemma1: "[| chain Y; lub (range Y) = <a> ooo s |] ==> EX j t. Y j = <a> ooo t"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   244
apply (subgoal_tac "lub(range Y) ~= UU")
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   245
apply (drule chain_UU_I_inverse2, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   246
apply (drule_tac x="i" in is_ub_thelub, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   247
by (drule fstreams_prefix' [THEN iffD1], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   248
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   249
lemma fstreams_lub1: 
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   250
 "[| chain Y; lub (range Y) = <a> ooo s |]
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   251
     ==> (EX j t. Y j = <a> ooo t) & (EX X. chain X & (ALL i. EX j. <a> ooo X i << Y j) & lub (range X) = s)"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   252
apply (auto simp add: fstreams_lub_lemma1)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   253
apply (rule_tac x="%n. stream_take n$s" in exI, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   254
apply (simp add: chain_stream_take)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   255
apply (induct_tac i, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   256
apply (drule fstreams_lub_lemma1, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   257
apply (rule_tac x="j" in exI, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   258
apply (case_tac "max_in_chain j Y")
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   259
apply (frule lub_finch1 [THEN thelubI], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   260
apply (rule_tac x="j" in exI)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   261
apply (erule subst) back back
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   262
apply (simp add: less_cprod_def sconc_mono)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   263
apply (simp add: max_in_chain_def, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   264
apply (rule_tac x="ja" in exI)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   265
apply (subgoal_tac "Y j << Y ja")
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   266
apply (drule fstreams_prefix, auto)+
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   267
apply (rule sconc_mono)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   268
apply (rule fstreams_chain_lemma, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   269
apply (subgoal_tac "Y ja << (LUB i. (Y i))", clarsimp)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   270
apply (drule fstreams_mono, simp)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   271
apply (rule is_ub_thelub, simp)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   272
apply (blast intro: chain_mono3)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   273
by (rule stream_reach2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   274
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   275
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   276
lemma lub_Pair_not_UU_lemma: 
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   277
  "[| chain Y; lub (range Y) = ((a::'a::flat), b); a ~= UU; b ~= UU |] 
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   278
      ==> EX j c d. Y j = (c, d) & c ~= UU & d ~= UU"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   279
apply (frule thelub_cprod, clarsimp)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   280
apply (drule chain_UU_I_inverse2, clarsimp)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   281
apply (case_tac "Y i", clarsimp)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   282
apply (case_tac "max_in_chain i Y")
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   283
apply (drule maxinch_is_thelub, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   284
apply (rule_tac x="i" in exI, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   285
apply (simp add: max_in_chain_def, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   286
apply (subgoal_tac "Y i << Y j",auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   287
apply (simp add: less_cprod_def, clarsimp)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   288
apply (drule ax_flat [rule_format], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   289
apply (case_tac "snd (Y j) = UU",auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   290
apply (case_tac "Y j", auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   291
apply (rule_tac x="j" in exI)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   292
apply (case_tac "Y j",auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   293
by (drule chain_mono3, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   294
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   295
lemma fstreams_lub_lemma2: 
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   296
  "[| chain Y; lub (range Y) = (a, <m> ooo ms); (a::'a::flat) ~= UU |] ==> EX j t. Y j = (a, <m> ooo t)"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   297
apply (frule lub_Pair_not_UU_lemma, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   298
apply (drule_tac x="j" in is_ub_thelub, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   299
apply (simp add: less_cprod_def, clarsimp)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   300
apply (drule ax_flat [rule_format], clarsimp)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   301
by (drule fstreams_prefix' [THEN iffD1], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   302
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   303
lemma fstreams_lub2:
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   304
  "[| chain Y; lub (range Y) = (a, <m> ooo ms); (a::'a::flat) ~= UU |] 
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   305
      ==> (EX j t. Y j = (a, <m> ooo t)) & (EX X. chain X & (ALL i. EX j. (a, <m> ooo X i) << Y j) & lub (range X) = ms)"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   306
apply (auto simp add: fstreams_lub_lemma2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   307
apply (rule_tac x="%n. stream_take n$ms" in exI, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   308
apply (simp add: chain_stream_take)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   309
apply (induct_tac i, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   310
apply (drule fstreams_lub_lemma2, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   311
apply (rule_tac x="j" in exI, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   312
apply (simp add: less_cprod_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   313
apply (case_tac "max_in_chain j Y")
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   314
apply (frule lub_finch1 [THEN thelubI], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   315
apply (rule_tac x="j" in exI)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   316
apply (erule subst) back back
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   317
apply (simp add: less_cprod_def sconc_mono)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   318
apply (simp add: max_in_chain_def, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   319
apply (rule_tac x="ja" in exI)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   320
apply (subgoal_tac "Y j << Y ja")
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   321
apply (simp add: less_cprod_def, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   322
apply (drule trans_less)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   323
apply (simp add: ax_flat, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   324
apply (drule fstreams_prefix, auto)+
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   325
apply (rule sconc_mono)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   326
apply (subgoal_tac "tt ~= tta" "tta << ms")
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   327
apply (blast intro: fstreams_chain_lemma)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   328
apply (frule lub_cprod [THEN thelubI], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   329
apply (subgoal_tac "snd (Y ja) << (LUB i. snd (Y i))", clarsimp)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   330
apply (drule fstreams_mono, simp)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   331
apply (rule is_ub_thelub chainI)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   332
apply (simp add: chain_def less_cprod_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   333
apply (subgoal_tac "fst (Y j) ~= fst (Y ja) | snd (Y j) ~= snd (Y ja)", simp)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   334
apply (drule ax_flat[rule_format], simp)+
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   335
apply (drule prod_eqI, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   336
apply (simp add: chain_mono3)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   337
by (rule stream_reach2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   338
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   339
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   340
lemma cpo_cont_lemma:
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   341
  "[| monofun (f::'a::cpo => 'b::cpo); (!Y. chain Y --> f (lub(range Y)) << (LUB i. f (Y i))) |] ==> cont f"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   342
apply (rule monocontlub2cont, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   343
apply (simp add: contlub, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   344
apply (erule_tac x="Y" in allE, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   345
apply (simp add: po_eq_conv)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   346
apply (frule cpo,auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   347
apply (frule is_lubD1)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   348
apply (frule ub2ub_monofun, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   349
apply (drule thelubI, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   350
apply (rule is_lub_thelub, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   351
by (erule ch2ch_monofun, simp)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   352
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   353
end
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   354
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   355
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   356
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   357