(*  Title:      HOL/HOLCF/FOCUS/Fstreams.thy
    Author:     Borislav Gajanovic

FOCUS flat streams (with lifted elements).

TODO: integrate this with Fstream.
*)

theory Fstreams
imports "HOLCF-Library.Stream"
begin

default_sort type

type_synonym 'a fstream = "('a lift) stream"

definition
  fsingleton    :: "'a => 'a fstream"  ("<_>" [1000] 999) where
  fsingleton_def2: "fsingleton = (%a. Def a && UU)"

definition
  fsfilter      :: "'a set \<Rightarrow> 'a fstream \<rightarrow> 'a fstream" where
  "fsfilter A = sfilter\<cdot>(flift2 (\<lambda>x. x\<in>A))"

definition
  fsmap         :: "('a => 'b) => 'a fstream -> 'b fstream" where
  "fsmap f = smap\<cdot>(flift2 f)"

definition
  jth           :: "nat => 'a fstream => 'a" where
  "jth = (%n s. if enat n < #s then THE a. i_th n s = Def a else undefined)"

definition
  first         :: "'a fstream => 'a" where
  "first = (%s. jth 0 s)"

definition
  last          :: "'a fstream => 'a" where
  "last = (%s. case #s of enat n => (if n~=0 then jth (THE k. Suc k = n) s else undefined))"


abbreviation
  emptystream :: "'a fstream"  ("<>") where
  "<> == \<bottom>"

abbreviation
  fsfilter' :: "'a set \<Rightarrow> 'a fstream \<Rightarrow> 'a fstream"       ("(_\<copyright>_)" [64,63] 63) where
  "A\<copyright>s == fsfilter A\<cdot>s"

notation (ASCII)
  fsfilter'  ("(_'(C')_)" [64,63] 63)


lemma ft_fsingleton[simp]: "ft\<cdot>(<a>) = Def a"
by (simp add: fsingleton_def2)

lemma slen_fsingleton[simp]: "#(<a>) = enat 1"
by (simp add: fsingleton_def2 enat_defs)

lemma slen_fstreams[simp]: "#(<a> ooo s) = eSuc (#s)"
by (simp add: fsingleton_def2)

lemma slen_fstreams2[simp]: "#(s ooo <a>) = eSuc (#s)"
apply (cases "#s")
apply (auto simp add: eSuc_enat)
apply (insert slen_sconc [of _ s "Suc 0" "<a>"], auto)
apply (simp add: sconc_def)
done

lemma j_th_0_fsingleton[simp]:"jth 0 (<a>) = a"
apply (simp add: fsingleton_def2 jth_def)
apply (simp add: i_th_def enat_0)
done

lemma jth_0[simp]: "jth 0 (<a> ooo s) = a"  
apply (simp add: fsingleton_def2 jth_def)
apply (simp add: i_th_def enat_0)
done

lemma first_sconc[simp]: "first (<a> ooo s) = a"
by (simp add: first_def)

lemma first_fsingleton[simp]: "first (<a>) = a"
by (simp add: first_def)

lemma jth_n[simp]: "enat n = #s ==> jth n (s ooo <a>) = a"
apply (simp add: jth_def, auto)
apply (simp add: i_th_def rt_sconc1)
apply (simp add: enat_defs split: enat.splits)
done

lemma last_sconc[simp]: "enat n = #s ==> last (s ooo <a>) = a"
apply (simp add: last_def)
apply (simp add: enat_defs split:enat.splits)
apply (drule sym, auto)
done

lemma last_fsingleton[simp]: "last (<a>) = a"
by (simp add: last_def)

lemma first_UU[simp]: "first UU = undefined"
by (simp add: first_def jth_def)

lemma last_UU[simp]:"last UU = undefined"
by (simp add: last_def jth_def enat_defs)

lemma last_infinite[simp]:"#s = \<infinity> \<Longrightarrow> last s = undefined"
by (simp add: last_def)

lemma jth_slen_lemma1:"n \<le> k \<and> enat n = #s \<Longrightarrow> jth k s = undefined"
by (simp add: jth_def enat_defs split:enat.splits, auto)

lemma jth_UU[simp]:"jth n UU = undefined" 
by (simp add: jth_def)

lemma ext_last:"\<lbrakk>s \<noteq> UU; enat (Suc n) = #s\<rbrakk> \<Longrightarrow> (stream_take n\<cdot>s) ooo <(last s)> = s"
apply (simp add: last_def)
apply (case_tac "#s", auto)
apply (simp add: fsingleton_def2)
apply (subgoal_tac "Def (jth n s) = i_th n s")
apply (auto simp add: i_th_last)
apply (drule slen_take_lemma1, auto)
apply (simp add: jth_def)
apply (case_tac "i_th n s = UU")
apply auto
apply (simp add: i_th_def)
apply (case_tac "i_rt n s = UU", auto)
apply (drule i_rt_slen [THEN iffD1])
apply (drule slen_take_eq_rev [rule_format, THEN iffD2],auto)
apply (drule not_Undef_is_Def [THEN iffD1], auto)
done


lemma fsingleton_lemma1[simp]: "(<a> = <b>) = (a=b)"
by (simp add: fsingleton_def2)

lemma fsingleton_lemma2[simp]: "<a> ~= <>"
by (simp add: fsingleton_def2)

lemma fsingleton_sconc:"<a> ooo s = Def a && s"
by (simp add: fsingleton_def2)

lemma fstreams_ind: 
  "[| adm P; P <>; !!a s. P s ==> P (<a> ooo s) |] ==> P x"
apply (simp add: fsingleton_def2)
apply (rule stream.induct, auto)
apply (drule not_Undef_is_Def [THEN iffD1], auto)
done

lemma fstreams_ind2:
  "[| adm P; P <>; !!a. P (<a>); !!a b s. P s ==> P (<a> ooo <b> ooo s) |] ==> P x"
apply (simp add: fsingleton_def2)
apply (rule stream_ind2, auto)
apply (drule not_Undef_is_Def [THEN iffD1], auto)+
done

lemma fstreams_take_Suc[simp]: "stream_take (Suc n)\<cdot>(<a> ooo s) = <a> ooo stream_take n\<cdot>s"
by (simp add: fsingleton_def2)

lemma fstreams_not_empty[simp]: "<a> ooo s \<noteq> <>"
by (simp add: fsingleton_def2)

lemma fstreams_not_empty2[simp]: "s ooo <a> \<noteq> <>"
by (case_tac "s=UU", auto)

lemma fstreams_exhaust: "x = UU \<or> (\<exists>a s. x = <a> ooo s)"
apply (simp add: fsingleton_def2, auto)
apply (erule contrapos_pp, auto)
apply (drule stream_exhaust_eq [THEN iffD1], auto)
apply (drule not_Undef_is_Def [THEN iffD1], auto)
done

lemma fstreams_cases: "\<lbrakk>x = UU \<Longrightarrow> P; \<And>a y. x = <a> ooo y \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
by (insert fstreams_exhaust [of x], auto)

lemma fstreams_exhaust_eq: "x \<noteq> UU \<longleftrightarrow> (\<exists>a y. x = <a> ooo y)"
apply (simp add: fsingleton_def2, auto)
apply (drule stream_exhaust_eq [THEN iffD1], auto)
apply (drule not_Undef_is_Def [THEN iffD1], auto)
done

lemma fstreams_inject: "<a> ooo s = <b> ooo t \<longleftrightarrow> a = b \<and> s = t"
by (simp add: fsingleton_def2)

lemma fstreams_prefix: "<a> ooo s << t \<Longrightarrow> \<exists>tt. t = <a> ooo tt \<and> s << tt"
apply (simp add: fsingleton_def2)
apply (insert stream_prefix [of "Def a" s t], auto)
done

lemma fstreams_prefix': "x << <a> ooo z \<longleftrightarrow> x = <> \<or> (\<exists>y. x = <a> ooo y \<and> y << z)"
apply (auto, case_tac "x=UU", auto)
apply (drule stream_exhaust_eq [THEN iffD1], auto)
apply (simp add: fsingleton_def2, auto)
apply (drule ax_flat, simp)
apply (erule sconc_mono)
done

lemma ft_fstreams[simp]: "ft\<cdot>(<a> ooo s) = Def a"
by (simp add: fsingleton_def2)

lemma rt_fstreams[simp]: "rt\<cdot>(<a> ooo s) = s"
by (simp add: fsingleton_def2)

lemma ft_eq[simp]: "ft\<cdot>s = Def a \<longleftrightarrow> (\<exists>t. s = <a> ooo t)"
apply (cases s, auto)
apply (auto simp add: fsingleton_def2)
done

lemma surjective_fstreams: "<d> ooo y = x \<longleftrightarrow> (ft\<cdot>x = Def d \<and> rt\<cdot>x = y)"
by auto

lemma fstreams_mono: "<a> ooo b << <a> ooo c \<Longrightarrow> b << c"
by (simp add: fsingleton_def2)

lemma fsmap_UU[simp]: "fsmap f\<cdot>UU = UU"
by (simp add: fsmap_def)

lemma fsmap_fsingleton_sconc: "fsmap f\<cdot>(<x> ooo xs) = <(f x)> ooo (fsmap f\<cdot>xs)"
by (simp add: fsmap_def fsingleton_def2 flift2_def)

lemma fsmap_fsingleton[simp]: "fsmap f\<cdot>(<x>) = <(f x)>"
by (simp add: fsmap_def fsingleton_def2 flift2_def)


lemma fstreams_chain_lemma[rule_format]:
  "\<forall>s x y. stream_take n\<cdot>(s::'a fstream) << x \<and> x << y \<and> y << s \<and> x \<noteq> y \<longrightarrow> stream_take (Suc n)\<cdot>s << y"
apply (induct_tac n, auto)
apply (case_tac "s=UU", auto)
apply (drule stream_exhaust_eq [THEN iffD1], auto)
apply (case_tac "y=UU", auto)
apply (drule stream_exhaust_eq [THEN iffD1], auto)
apply (simp add: flat_below_iff)
apply (case_tac "s=UU", auto)
apply (drule stream_exhaust_eq [THEN iffD1], auto)
apply (erule_tac x="ya" in allE)
apply (drule stream_prefix, auto)
apply (case_tac "y=UU",auto)
apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)
apply auto
apply (simp add: flat_below_iff)
apply (erule_tac x="tt" in allE)
apply (erule_tac x="yb" in allE, auto)
apply (simp add: flat_below_iff)
apply (simp add: flat_below_iff)
done

lemma fstreams_lub_lemma1: "\<lbrakk>chain Y; (LUB i. Y i) = <a> ooo s\<rbrakk> \<Longrightarrow> \<exists>j t. Y j = <a> ooo t"
apply (subgoal_tac "(LUB i. Y i) ~= UU")
apply (frule lub_eq_bottom_iff, auto)
apply (drule_tac x="i" in is_ub_thelub, auto)
apply (drule fstreams_prefix' [THEN iffD1], auto)
done

lemma fstreams_lub1: 
 "\<lbrakk>chain Y; (LUB i. Y i) = <a> ooo s\<rbrakk>
     \<Longrightarrow> (\<exists>j t. Y j = <a> ooo t) \<and> (\<exists>X. chain X \<and> (\<forall>i. \<exists>j. <a> ooo X i << Y j) \<and> (LUB i. X i) = s)"
apply (auto simp add: fstreams_lub_lemma1)
apply (rule_tac x="\<lambda>n. stream_take n\<cdot>s" in exI, auto)
apply (induct_tac i, auto)
apply (drule fstreams_lub_lemma1, auto)
apply (rule_tac x="j" in exI, auto)
apply (case_tac "max_in_chain j Y")
apply (frule lub_finch1 [THEN lub_eqI], auto)
apply (rule_tac x="j" in exI)
apply (erule subst) back back
apply (simp add: below_prod_def sconc_mono)
apply (simp add: max_in_chain_def, auto)
apply (rule_tac x="ja" in exI)
apply (subgoal_tac "Y j << Y ja")
apply (drule fstreams_prefix, auto)+
apply (rule sconc_mono)
apply (rule fstreams_chain_lemma, auto)
apply (subgoal_tac "Y ja << (LUB i. (Y i))", clarsimp)
apply (drule fstreams_mono, simp)
apply (rule is_ub_thelub, simp)
apply (blast intro: chain_mono)
by (rule stream_reach2)


lemma lub_Pair_not_UU_lemma: 
  "\<lbrakk>chain Y; (LUB i. Y i) = ((a::'a::flat), b); a \<noteq> UU; b \<noteq> UU\<rbrakk>
      \<Longrightarrow> \<exists>j c d. Y j = (c, d) \<and> c \<noteq> UU \<and> d \<noteq> UU"
apply (frule lub_prod, clarsimp)
apply (clarsimp simp add: lub_eq_bottom_iff [where Y="\<lambda>i. fst (Y i)"])
apply (case_tac "Y i", clarsimp)
apply (case_tac "max_in_chain i Y")
apply (drule maxinch_is_thelub, auto)
apply (rule_tac x="i" in exI, auto)
apply (simp add: max_in_chain_def, auto)
apply (subgoal_tac "Y i << Y j",auto)
apply (simp add: below_prod_def, clarsimp)
apply (drule ax_flat, auto)
apply (case_tac "snd (Y j) = UU",auto)
apply (case_tac "Y j", auto)
apply (rule_tac x="j" in exI)
apply (case_tac "Y j",auto)
apply (drule chain_mono, auto)
done

lemma fstreams_lub_lemma2: 
  "\<lbrakk>chain Y; (LUB i. Y i) = (a, <m> ooo ms); (a::'a::flat) \<noteq> UU\<rbrakk> \<Longrightarrow> \<exists>j t. Y j = (a, <m> ooo t)"
apply (frule lub_Pair_not_UU_lemma, auto)
apply (drule_tac x="j" in is_ub_thelub, auto)
apply (drule ax_flat, clarsimp)
apply (drule fstreams_prefix' [THEN iffD1], auto)
done

lemma fstreams_lub2:
  "\<lbrakk>chain Y; (LUB i. Y i) = (a, <m> ooo ms); (a::'a::flat) \<noteq> UU\<rbrakk>
      \<Longrightarrow> (\<exists>j t. Y j = (a, <m> ooo t)) \<and> (\<exists>X. chain X \<and> (\<forall>i. \<exists>j. (a, <m> ooo X i) << Y j) \<and> (LUB i. X i) = ms)"
apply (auto simp add: fstreams_lub_lemma2)
apply (rule_tac x="\<lambda>n. stream_take n\<cdot>ms" in exI, auto)
apply (induct_tac i, auto)
apply (drule fstreams_lub_lemma2, auto)
apply (rule_tac x="j" in exI, auto)
apply (case_tac "max_in_chain j Y")
apply (frule lub_finch1 [THEN lub_eqI], auto)
apply (rule_tac x="j" in exI)
apply (erule subst) back back
apply (simp add: sconc_mono)
apply (simp add: max_in_chain_def, auto)
apply (rule_tac x="ja" in exI)
apply (subgoal_tac "Y j << Y ja")
apply (simp add: below_prod_def, auto)
apply (drule below_trans)
apply (simp add: ax_flat, auto)
apply (drule fstreams_prefix, auto)+
apply (rule sconc_mono)
apply (subgoal_tac "tt \<noteq> tta" "tta << ms")
apply (blast intro: fstreams_chain_lemma)
apply (frule lub_prod, auto)
apply (subgoal_tac "snd (Y ja) << (LUB i. snd (Y i))", clarsimp)
apply (drule fstreams_mono, simp)
apply (rule is_ub_thelub chainI)
apply (simp add: chain_def below_prod_def)
apply (subgoal_tac "fst (Y j) \<noteq> fst (Y ja) \<or> snd (Y j) \<noteq> snd (Y ja)", simp)
apply (drule ax_flat, simp)+
apply (drule prod_eqI, auto)
apply (simp add: chain_mono)
apply (rule stream_reach2)
done


lemma cpo_cont_lemma:
  "\<lbrakk>monofun (f::'a::cpo \<Rightarrow> 'b::cpo); (\<forall>Y. chain Y \<longrightarrow> f (lub(range Y)) << (LUB i. f (Y i)))\<rbrakk> \<Longrightarrow> cont f"
apply (erule contI2)
apply simp
done

end