src/HOL/HOLCF/FOCUS/Fstreams.thy
author wenzelm
Sat, 07 Apr 2012 16:41:59 +0200
changeset 47389 e8552cba702d
parent 44019 ee784502aed5
child 49521 06cb12198b92
permissions -rw-r--r--
explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions; tuned;
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(*  Title:      HOL/HOLCF/FOCUS/Fstreams.thy
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    Author:     Borislav Gajanovic
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FOCUS flat streams (with lifted elements).
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TODO: integrate this with Fstream.
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*)
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theory Fstreams
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imports "~~/src/HOL/HOLCF/Library/Stream"
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begin
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default_sort type
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type_synonym 'a fstream = "('a lift) stream"
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definition
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  fsingleton    :: "'a => 'a fstream"  ("<_>" [1000] 999) where
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  fsingleton_def2: "fsingleton = (%a. Def a && UU)"
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definition
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  fsfilter      :: "'a set \<Rightarrow> 'a fstream \<rightarrow> 'a fstream" where
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  "fsfilter A = sfilter\<cdot>(flift2 (\<lambda>x. x\<in>A))"
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definition
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  fsmap         :: "('a => 'b) => 'a fstream -> 'b fstream" where
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  "fsmap f = smap$(flift2 f)"
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definition
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  jth           :: "nat => 'a fstream => 'a" where
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  "jth = (%n s. if enat n < #s then THE a. i_th n s = Def a else undefined)"
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definition
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  first         :: "'a fstream => 'a" where
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  "first = (%s. jth 0 s)"
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definition
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  last          :: "'a fstream => 'a" where
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  "last = (%s. case #s of enat n => (if n~=0 then jth (THE k. Suc k = n) s else undefined))"
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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abbreviation
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  emptystream :: "'a fstream"  ("<>") where
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  "<> == \<bottom>"
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abbreviation
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  fsfilter' :: "'a set \<Rightarrow> 'a fstream \<Rightarrow> 'a fstream"       ("(_'(C')_)" [64,63] 63) where
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  "A(C)s == fsfilter A\<cdot>s"
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notation (xsymbols)
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  fsfilter'  ("(_\<copyright>_)" [64,63] 63)
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma ft_fsingleton[simp]: "ft$(<a>) = Def a"
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by (simp add: fsingleton_def2)
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lemma slen_fsingleton[simp]: "#(<a>) = enat 1"
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by (simp add: fsingleton_def2 enat_defs)
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lemma slen_fstreams[simp]: "#(<a> ooo s) = eSuc (#s)"
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by (simp add: fsingleton_def2)
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lemma slen_fstreams2[simp]: "#(s ooo <a>) = eSuc (#s)"
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apply (cases "#s")
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apply (auto simp add: eSuc_enat)
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apply (insert slen_sconc [of _ s "Suc 0" "<a>"], auto)
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by (simp add: sconc_def)
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lemma j_th_0_fsingleton[simp]:"jth 0 (<a>) = a"
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apply (simp add: fsingleton_def2 jth_def)
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by (simp add: i_th_def enat_0)
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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)
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by (simp add: i_th_def enat_0)
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lemma first_sconc[simp]: "first (<a> ooo s) = a"
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by (simp add: first_def)
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lemma first_fsingleton[simp]: "first (<a>) = a"
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by (simp add: first_def)
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lemma jth_n[simp]: "enat n = #s ==> jth n (s ooo <a>) = a"
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apply (simp add: jth_def, auto)
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apply (simp add: i_th_def rt_sconc1)
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by (simp add: enat_defs split: enat.splits)
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lemma last_sconc[simp]: "enat n = #s ==> last (s ooo <a>) = a"
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apply (simp add: last_def)
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apply (simp add: enat_defs split:enat.splits)
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by (drule sym, auto)
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lemma last_fsingleton[simp]: "last (<a>) = a"
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by (simp add: last_def)
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lemma first_UU[simp]: "first UU = undefined"
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by (simp add: first_def jth_def)
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lemma last_UU[simp]:"last UU = undefined"
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by (simp add: last_def jth_def enat_defs)
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lemma last_infinite[simp]:"#s = \<infinity> ==> last s = undefined"
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by (simp add: last_def)
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lemma jth_slen_lemma1:"n <= k & enat n = #s ==> jth k s = undefined"
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by (simp add: jth_def enat_defs split:enat.splits, auto)
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lemma jth_UU[simp]:"jth n UU = undefined" 
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by (simp add: jth_def)
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lemma ext_last:"[|s ~= UU; enat (Suc n) = #s|] ==> (stream_take n$s) ooo <(last s)> = s" 
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apply (simp add: last_def)
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apply (case_tac "#s", auto)
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apply (simp add: fsingleton_def2)
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apply (subgoal_tac "Def (jth n s) = i_th n s")
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apply (auto simp add: i_th_last)
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apply (drule slen_take_lemma1, auto)
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apply (simp add: jth_def)
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apply (case_tac "i_th n s = UU")
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apply auto
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apply (simp add: i_th_def)
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apply (case_tac "i_rt n s = UU", auto)
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apply (drule i_rt_slen [THEN iffD1])
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apply (drule slen_take_eq_rev [rule_format, THEN iffD2],auto)
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by (drule not_Undef_is_Def [THEN iffD1], auto)
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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)"
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by (simp add: fsingleton_def2)
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9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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lemma fsingleton_lemma2[simp]: "<a> ~= <>"
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by (simp add: fsingleton_def2)
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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"
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by (simp add: fsingleton_def2)
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lemma fstreams_ind: 
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  "[| adm P; P <>; !!a s. P s ==> P (<a> ooo s) |] ==> P x"
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apply (simp add: fsingleton_def2)
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apply (rule stream.induct, auto)
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   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)
30807
a167ed35ec0d domain package declares more simp rules
huffman
parents: 28524
diff changeset
   178
done
15188
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)
25920
8df5eabda5f6 change class axiom ax_flat to rule_format
huffman
parents: 24107
diff changeset
   184
apply (drule ax_flat, simp)
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   185
by (erule sconc_mono)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   186
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   187
lemma ft_fstreams[simp]: "ft$(<a> ooo s) = Def a"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   188
by (simp add: fsingleton_def2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   189
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   190
lemma rt_fstreams[simp]: "rt$(<a> ooo s) = s"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   191
by (simp add: fsingleton_def2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   192
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   193
lemma ft_eq[simp]: "(ft$s = Def a) = (EX t. s = <a> ooo t)"
35781
b7738ab762b1 renamed some lemmas generated by the domain package
huffman
parents: 35572
diff changeset
   194
apply (cases s, auto)
16746
3f3fbfd8ce04 fixes to work with UU_reorient_simproc
huffman
parents: 16417
diff changeset
   195
by ((*drule sym,*) auto simp add: fsingleton_def2)
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   196
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   197
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
   198
by auto
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   199
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   200
lemma fstreams_mono: "<a> ooo b << <a> ooo c ==> b << c"
30807
a167ed35ec0d domain package declares more simp rules
huffman
parents: 28524
diff changeset
   201
by (simp add: fsingleton_def2)
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   202
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   203
lemma fsmap_UU[simp]: "fsmap f$UU = UU"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   204
by (simp add: fsmap_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   205
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   206
lemma fsmap_fsingleton_sconc: "fsmap f$(<x> ooo xs) = <(f x)> ooo (fsmap f$xs)"
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   207
by (simp add: fsmap_def fsingleton_def2 flift2_def)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   208
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   209
lemma fsmap_fsingleton[simp]: "fsmap f$(<x>) = <(f x)>"
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
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   213
lemma fstreams_chain_lemma[rule_format]:
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   214
  "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
   215
apply (induct_tac n, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   216
apply (case_tac "s=UU", auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   217
apply (drule stream_exhaust_eq [THEN iffD1], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   218
apply (case_tac "y=UU", auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   219
apply (drule stream_exhaust_eq [THEN iffD1], auto)
40431
682d6c455670 discontinue a bunch of legacy theorem names
huffman
parents: 37110
diff changeset
   220
apply (simp add: flat_below_iff)
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   221
apply (case_tac "s=UU", auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   222
apply (drule stream_exhaust_eq [THEN iffD1], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   223
apply (erule_tac x="ya" in allE)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   224
apply (drule stream_prefix, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   225
apply (case_tac "y=UU",auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   226
apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)
35215
a03462cbf86f get rid of warnings about duplicate simp rules in all HOLCF theories
huffman
parents: 32960
diff changeset
   227
apply auto
40431
682d6c455670 discontinue a bunch of legacy theorem names
huffman
parents: 37110
diff changeset
   228
apply (simp add: flat_below_iff)
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   229
apply (erule_tac x="tt" in allE)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   230
apply (erule_tac x="yb" in allE, auto)
40431
682d6c455670 discontinue a bunch of legacy theorem names
huffman
parents: 37110
diff changeset
   231
apply (simp add: flat_below_iff)
682d6c455670 discontinue a bunch of legacy theorem names
huffman
parents: 37110
diff changeset
   232
by (simp add: flat_below_iff)
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   233
27413
3154f3765cc7 replace lub (range Y) with (LUB i. Y i)
huffman
parents: 27291
diff changeset
   234
lemma fstreams_lub_lemma1: "[| chain Y; (LUB i. Y i) = <a> ooo s |] ==> EX j t. Y j = <a> ooo t"
3154f3765cc7 replace lub (range Y) with (LUB i. Y i)
huffman
parents: 27291
diff changeset
   235
apply (subgoal_tac "(LUB i. Y i) ~= UU")
41431
138f414f14cb remove various lemmas redundant with lub_eq_bottom_iff
huffman
parents: 41413
diff changeset
   236
apply (frule lub_eq_bottom_iff, auto)
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   237
apply (drule_tac x="i" in is_ub_thelub, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   238
by (drule fstreams_prefix' [THEN iffD1], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   239
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   240
lemma fstreams_lub1: 
27413
3154f3765cc7 replace lub (range Y) with (LUB i. Y i)
huffman
parents: 27291
diff changeset
   241
 "[| chain Y; (LUB i. Y i) = <a> ooo s |]
3154f3765cc7 replace lub (range Y) with (LUB i. Y i)
huffman
parents: 27291
diff changeset
   242
     ==> (EX j t. Y j = <a> ooo t) & (EX X. chain X & (ALL i. EX j. <a> ooo X i << Y j) & (LUB i. X i) = s)"
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   243
apply (auto simp add: fstreams_lub_lemma1)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   244
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
   245
apply (induct_tac i, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   246
apply (drule fstreams_lub_lemma1, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   247
apply (rule_tac x="j" in exI, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   248
apply (case_tac "max_in_chain j Y")
40771
1c6f7d4b110e renamed several HOLCF theorems (listed in NEWS)
huffman
parents: 40431
diff changeset
   249
apply (frule lub_finch1 [THEN lub_eqI], auto)
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   250
apply (rule_tac x="j" in exI)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   251
apply (erule subst) back back
40431
682d6c455670 discontinue a bunch of legacy theorem names
huffman
parents: 37110
diff changeset
   252
apply (simp add: below_prod_def sconc_mono)
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   253
apply (simp add: max_in_chain_def, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   254
apply (rule_tac x="ja" in exI)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   255
apply (subgoal_tac "Y j << Y ja")
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   256
apply (drule fstreams_prefix, auto)+
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   257
apply (rule sconc_mono)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   258
apply (rule fstreams_chain_lemma, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   259
apply (subgoal_tac "Y ja << (LUB i. (Y i))", clarsimp)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   260
apply (drule fstreams_mono, simp)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   261
apply (rule is_ub_thelub, simp)
25922
cb04d05e95fb rename lemma chain_mono3 -> chain_mono, chain_mono -> chain_mono_less
huffman
parents: 25920
diff changeset
   262
apply (blast intro: chain_mono)
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   263
by (rule stream_reach2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   264
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   265
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   266
lemma lub_Pair_not_UU_lemma: 
27413
3154f3765cc7 replace lub (range Y) with (LUB i. Y i)
huffman
parents: 27291
diff changeset
   267
  "[| chain Y; (LUB i. Y i) = ((a::'a::flat), b); a ~= UU; b ~= UU |] 
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   268
      ==> EX j c d. Y j = (c, d) & c ~= UU & d ~= UU"
40771
1c6f7d4b110e renamed several HOLCF theorems (listed in NEWS)
huffman
parents: 40431
diff changeset
   269
apply (frule lub_prod, clarsimp)
41431
138f414f14cb remove various lemmas redundant with lub_eq_bottom_iff
huffman
parents: 41413
diff changeset
   270
apply (clarsimp simp add: lub_eq_bottom_iff [where Y="\<lambda>i. fst (Y i)"])
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   271
apply (case_tac "Y i", clarsimp)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   272
apply (case_tac "max_in_chain i Y")
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   273
apply (drule maxinch_is_thelub, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   274
apply (rule_tac x="i" in exI, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   275
apply (simp add: max_in_chain_def, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   276
apply (subgoal_tac "Y i << Y j",auto)
40431
682d6c455670 discontinue a bunch of legacy theorem names
huffman
parents: 37110
diff changeset
   277
apply (simp add: below_prod_def, clarsimp)
25920
8df5eabda5f6 change class axiom ax_flat to rule_format
huffman
parents: 24107
diff changeset
   278
apply (drule ax_flat, auto)
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   279
apply (case_tac "snd (Y j) = UU",auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   280
apply (case_tac "Y j", auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   281
apply (rule_tac x="j" in exI)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   282
apply (case_tac "Y j",auto)
25922
cb04d05e95fb rename lemma chain_mono3 -> chain_mono, chain_mono -> chain_mono_less
huffman
parents: 25920
diff changeset
   283
by (drule chain_mono, auto)
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   284
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   285
lemma fstreams_lub_lemma2: 
27413
3154f3765cc7 replace lub (range Y) with (LUB i. Y i)
huffman
parents: 27291
diff changeset
   286
  "[| chain Y; (LUB i. Y i) = (a, <m> ooo ms); (a::'a::flat) ~= UU |] ==> EX j t. Y j = (a, <m> ooo t)"
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   287
apply (frule lub_Pair_not_UU_lemma, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   288
apply (drule_tac x="j" in is_ub_thelub, auto)
25920
8df5eabda5f6 change class axiom ax_flat to rule_format
huffman
parents: 24107
diff changeset
   289
apply (drule ax_flat, clarsimp)
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   290
by (drule fstreams_prefix' [THEN iffD1], auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   291
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   292
lemma fstreams_lub2:
27413
3154f3765cc7 replace lub (range Y) with (LUB i. Y i)
huffman
parents: 27291
diff changeset
   293
  "[| chain Y; (LUB i. Y i) = (a, <m> ooo ms); (a::'a::flat) ~= UU |] 
3154f3765cc7 replace lub (range Y) with (LUB i. Y i)
huffman
parents: 27291
diff changeset
   294
      ==> (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 i. X i) = ms)"
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   295
apply (auto simp add: fstreams_lub_lemma2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   296
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
   297
apply (induct_tac i, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   298
apply (drule fstreams_lub_lemma2, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   299
apply (rule_tac x="j" in exI, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   300
apply (case_tac "max_in_chain j Y")
40771
1c6f7d4b110e renamed several HOLCF theorems (listed in NEWS)
huffman
parents: 40431
diff changeset
   301
apply (frule lub_finch1 [THEN lub_eqI], auto)
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   302
apply (rule_tac x="j" in exI)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   303
apply (erule subst) back back
26029
46e84ca065f1 add lemmas prod_lessI and Pair_less_iff [simp]
huffman
parents: 25922
diff changeset
   304
apply (simp add: sconc_mono)
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   305
apply (simp add: max_in_chain_def, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   306
apply (rule_tac x="ja" in exI)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   307
apply (subgoal_tac "Y j << Y ja")
40431
682d6c455670 discontinue a bunch of legacy theorem names
huffman
parents: 37110
diff changeset
   308
apply (simp add: below_prod_def, auto)
682d6c455670 discontinue a bunch of legacy theorem names
huffman
parents: 37110
diff changeset
   309
apply (drule below_trans)
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   310
apply (simp add: ax_flat, auto)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   311
apply (drule fstreams_prefix, auto)+
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   312
apply (rule sconc_mono)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   313
apply (subgoal_tac "tt ~= tta" "tta << ms")
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   314
apply (blast intro: fstreams_chain_lemma)
40771
1c6f7d4b110e renamed several HOLCF theorems (listed in NEWS)
huffman
parents: 40431
diff changeset
   315
apply (frule lub_prod, auto)
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   316
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
   317
apply (drule fstreams_mono, simp)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   318
apply (rule is_ub_thelub chainI)
40431
682d6c455670 discontinue a bunch of legacy theorem names
huffman
parents: 37110
diff changeset
   319
apply (simp add: chain_def below_prod_def)
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   320
apply (subgoal_tac "fst (Y j) ~= fst (Y ja) | snd (Y j) ~= snd (Y ja)", simp)
25920
8df5eabda5f6 change class axiom ax_flat to rule_format
huffman
parents: 24107
diff changeset
   321
apply (drule ax_flat, simp)+
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   322
apply (drule prod_eqI, auto)
25922
cb04d05e95fb rename lemma chain_mono3 -> chain_mono, chain_mono -> chain_mono_less
huffman
parents: 25920
diff changeset
   323
apply (simp add: chain_mono)
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   324
by (rule stream_reach2)
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   325
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   326
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   327
lemma cpo_cont_lemma:
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   328
  "[| monofun (f::'a::cpo => 'b::cpo); (!Y. chain Y --> f (lub(range Y)) << (LUB i. f (Y i))) |] ==> cont f"
35914
91a7311177c4 remove contlub predicate
huffman
parents: 35781
diff changeset
   329
by (erule contI2, simp)
15188
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   330
9d57263faf9e integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
diff changeset
   331
end