(*  Title:      HOLCF/ex/Stream.thy

    Author:     Franz Regensburger, David von Oheimb, Borislav Gajanovic

*)

header {* General Stream domain *}

theory Stream

imports HOLCF Nat_Infinity

begin

default_sort pcpo

domain (unsafe) 'a stream = scons (ft::'a) (lazy rt::"'a stream") (infixr "&&" 65)

definition

  smap :: "('a \<rightarrow> 'b) \<rightarrow> 'a stream \<rightarrow> 'b stream" where

  "smap = fix\<cdot>(\<Lambda> h f s. case s of x && xs \<Rightarrow> f\<cdot>x && h\<cdot>f\<cdot>xs)"

definition

  sfilter :: "('a \<rightarrow> tr) \<rightarrow> 'a stream \<rightarrow> 'a stream" where

  "sfilter = fix\<cdot>(\<Lambda> h p s. case s of x && xs \<Rightarrow>

                                     If p\<cdot>x then x && h\<cdot>p\<cdot>xs else h\<cdot>p\<cdot>xs)"

definition

  slen :: "'a stream \<Rightarrow> inat"  ("#_" [1000] 1000) where

  "#s = (if stream_finite s then Fin (LEAST n. stream_take n\<cdot>s = s) else \<infinity>)"

(* concatenation *)

definition

  i_rt :: "nat => 'a stream => 'a stream" where (* chops the first i elements *)

  "i_rt = (%i s. iterate i$rt$s)"

definition

  i_th :: "nat => 'a stream => 'a" where (* the i-th element *)

  "i_th = (%i s. ft$(i_rt i s))"

definition

  sconc :: "'a stream => 'a stream => 'a stream"  (infixr "ooo" 65) where

  "s1 ooo s2 = (case #s1 of

                  Fin n \<Rightarrow> (SOME s. (stream_take n$s=s1) & (i_rt n s = s2))

               | \<infinity>     \<Rightarrow> s1)"

primrec constr_sconc' :: "nat => 'a stream => 'a stream => 'a stream"

where

  constr_sconc'_0:   "constr_sconc' 0 s1 s2 = s2"

| constr_sconc'_Suc: "constr_sconc' (Suc n) s1 s2 = ft$s1 &&

                                                    constr_sconc' n (rt$s1) s2"

definition

  constr_sconc  :: "'a stream => 'a stream => 'a stream" where (* constructive *)

  "constr_sconc s1 s2 = (case #s1 of

                          Fin n \<Rightarrow> constr_sconc' n s1 s2

                        | \<infinity>    \<Rightarrow> s1)"

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

(* theorems about scons                                                    *)

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

section "scons"

lemma scons_eq_UU: "(a && s = UU) = (a = UU)"

by simp

lemma scons_not_empty: "[| a && x = UU; a ~= UU |] ==> R"

by simp

lemma stream_exhaust_eq: "(x ~= UU) = (EX a y. a ~= UU &  x = a && y)"

by (cases x, auto)

lemma stream_neq_UU: "x~=UU ==> EX a a_s. x=a&&a_s & a~=UU"

by (simp add: stream_exhaust_eq,auto)

lemma stream_prefix:

  "[| a && s << t; a ~= UU  |] ==> EX b tt. t = b && tt &  b ~= UU &  s << tt"

by (cases t, auto)

lemma stream_prefix':

  "b ~= UU ==> x << b && z =

   (x = UU |  (EX a y. x = a && y &  a ~= UU &  a << b &  y << z))"

by (cases x, auto)

(*

lemma stream_prefix1: "[| x<<y; xs<<ys |] ==> x&&xs << y&&ys"

by (insert stream_prefix' [of y "x&&xs" ys],force)

*)

lemma stream_flat_prefix:

  "[| x && xs << y && ys; (x::'a::flat) ~= UU|] ==> x = y & xs << ys"

apply (case_tac "y=UU",auto)

by (drule ax_flat,simp)

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

(* theorems about stream_case                                              *)

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

section "stream_case"

lemma stream_case_strictf: "stream_case$UU$s=UU"

by (cases s, auto)

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

(* theorems about ft and rt                                                *)

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

section "ft & rt"

lemma ft_defin: "s~=UU ==> ft$s~=UU"

by simp

lemma rt_strict_rev: "rt$s~=UU ==> s~=UU"

by auto

lemma surjectiv_scons: "(ft$s)&&(rt$s)=s"

by (cases s, auto)

lemma monofun_rt_mult: "x << s ==> iterate i$rt$x << iterate i$rt$s"

by (rule monofun_cfun_arg)

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

(* theorems about stream_take                                              *)

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

section "stream_take"

lemma stream_reach2: "(LUB i. stream_take i$s) = s"

by (rule stream.reach)

lemma chain_stream_take: "chain (%i. stream_take i$s)"

by simp

lemma stream_take_prefix [simp]: "stream_take n$s << s"

apply (insert stream_reach2 [of s])

apply (erule subst) back

apply (rule is_ub_thelub)

by (simp only: chain_stream_take)

lemma stream_take_more [rule_format]:

  "ALL x. stream_take n$x = x --> stream_take (Suc n)$x = x"

apply (induct_tac n,auto)

apply (case_tac "x=UU",auto)

by (drule stream_exhaust_eq [THEN iffD1],auto)

lemma stream_take_lemma3 [rule_format]:

  "ALL x xs. x~=UU --> stream_take n$(x && xs) = x && xs --> stream_take n$xs=xs"

apply (induct_tac n,clarsimp)

(*apply (drule sym, erule scons_not_empty, simp)*)

apply (clarify, rule stream_take_more)

apply (erule_tac x="x" in allE)

by (erule_tac x="xs" in allE,simp)

lemma stream_take_lemma4:

  "ALL x xs. stream_take n$xs=xs --> stream_take (Suc n)$(x && xs) = x && xs"

by auto

lemma stream_take_idempotent [rule_format, simp]:

 "ALL s. stream_take n$(stream_take n$s) = stream_take n$s"

apply (induct_tac n, auto)

apply (case_tac "s=UU", auto)

by (drule stream_exhaust_eq [THEN iffD1], auto)

lemma stream_take_take_Suc [rule_format, simp]:

  "ALL s. stream_take n$(stream_take (Suc n)$s) =

                                    stream_take n$s"

apply (induct_tac n, auto)

apply (case_tac "s=UU", auto)

by (drule stream_exhaust_eq [THEN iffD1], auto)

lemma mono_stream_take_pred:

  "stream_take (Suc n)$s1 << stream_take (Suc n)$s2 ==>

                       stream_take n$s1 << stream_take n$s2"

by (insert monofun_cfun_arg [of "stream_take (Suc n)$s1"

  "stream_take (Suc n)$s2" "stream_take n"], auto)

(*

lemma mono_stream_take_pred:

  "stream_take (Suc n)$s1 << stream_take (Suc n)$s2 ==>

                       stream_take n$s1 << stream_take n$s2"

by (drule mono_stream_take [of _ _ n],simp)

*)

lemma stream_take_lemma10 [rule_format]:

  "ALL k<=n. stream_take n$s1 << stream_take n$s2

                             --> stream_take k$s1 << stream_take k$s2"

apply (induct_tac n,simp,clarsimp)

apply (case_tac "k=Suc n",blast)

apply (erule_tac x="k" in allE)

by (drule mono_stream_take_pred,simp)

lemma stream_take_le_mono : "k<=n ==> stream_take k$s1 << stream_take n$s1"

apply (insert chain_stream_take [of s1])

by (drule chain_mono,auto)

lemma mono_stream_take: "s1 << s2 ==> stream_take n$s1 << stream_take n$s2"

by (simp add: monofun_cfun_arg)

(*

lemma stream_take_prefix [simp]: "stream_take n$s << s"

apply (subgoal_tac "s=(LUB n. stream_take n$s)")

 apply (erule ssubst, rule is_ub_thelub)

 apply (simp only: chain_stream_take)

by (simp only: stream_reach2)

*)

lemma stream_take_take_less:"stream_take k$(stream_take n$s) << stream_take k$s"

by (rule monofun_cfun_arg,auto)

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

(* special induction rules                                                   *)

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

section "induction"

lemma stream_finite_ind:

 "[| stream_finite x; P UU; !!a s. [| a ~= UU; P s |] ==> P (a && s) |] ==> P x"

apply (simp add: stream.finite_def,auto)

apply (erule subst)

by (drule stream.finite_induct [of P _ x], auto)

lemma stream_finite_ind2:

"[| P UU; !! x. x ~= UU ==> P (x && UU); !! y z s. [| y ~= UU; z ~= UU; P s |] ==> P (y && z && s )|] ==>

                                 !s. P (stream_take n$s)"

apply (rule nat_less_induct [of _ n],auto)

apply (case_tac n, auto) 

apply (case_tac nat, auto) 

apply (case_tac "s=UU",clarsimp)

apply (drule stream_exhaust_eq [THEN iffD1],clarsimp)

apply (case_tac "s=UU",clarsimp)

apply (drule stream_exhaust_eq [THEN iffD1],clarsimp)

apply (case_tac "y=UU",clarsimp)

by (drule stream_exhaust_eq [THEN iffD1],clarsimp)

lemma stream_ind2:

"[| adm P; P UU; !!a. a ~= UU ==> P (a && UU); !!a b s. [| a ~= UU; b ~= UU; P s |] ==> P (a && b && s) |] ==> P x"

apply (insert stream.reach [of x],erule subst)

apply (erule admD, rule chain_stream_take)

apply (insert stream_finite_ind2 [of P])

by simp

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

(* simplify use of coinduction                                             *)

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

section "coinduction"

lemma stream_coind_lemma2: "!s1 s2. R s1 s2 --> ft$s1 = ft$s2 &  R (rt$s1) (rt$s2) ==> stream_bisim R"

 apply (simp add: stream.bisim_def,clarsimp)

 apply (drule spec, drule spec, drule (1) mp)

 apply (case_tac "x", simp)

 apply (case_tac "y", simp)

by auto

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

(* theorems about stream_finite                                            *)

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

section "stream_finite"

lemma stream_finite_UU [simp]: "stream_finite UU"

by (simp add: stream.finite_def)

lemma stream_finite_UU_rev: "~  stream_finite s ==> s ~= UU"

by (auto simp add: stream.finite_def)

lemma stream_finite_lemma1: "stream_finite xs ==> stream_finite (x && xs)"

apply (simp add: stream.finite_def,auto)

apply (rule_tac x="Suc n" in exI)

by (simp add: stream_take_lemma4)

lemma stream_finite_lemma2: "[| x ~= UU; stream_finite (x && xs) |] ==> stream_finite xs"

apply (simp add: stream.finite_def, auto)

apply (rule_tac x="n" in exI)

by (erule stream_take_lemma3,simp)

lemma stream_finite_rt_eq: "stream_finite (rt$s) = stream_finite s"

apply (cases s, auto)

apply (rule stream_finite_lemma1, simp)

by (rule stream_finite_lemma2,simp)

lemma stream_finite_less: "stream_finite s ==> !t. t<<s --> stream_finite t"

apply (erule stream_finite_ind [of s], auto)

apply (case_tac "t=UU", auto)

apply (drule stream_exhaust_eq [THEN iffD1],auto)

apply (erule_tac x="y" in allE, simp)

by (rule stream_finite_lemma1, simp)

lemma stream_take_finite [simp]: "stream_finite (stream_take n$s)"

apply (simp add: stream.finite_def)

by (rule_tac x="n" in exI,simp)

lemma adm_not_stream_finite: "adm (%x. ~ stream_finite x)"

apply (rule adm_upward)

apply (erule contrapos_nn)

apply (erule (1) stream_finite_less [rule_format])

done

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

(* theorems about stream length                                            *)

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

section "slen"

lemma slen_empty [simp]: "#\<bottom> = 0"

by (simp add: slen_def stream.finite_def zero_inat_def Least_equality)

lemma slen_scons [simp]: "x ~= \<bottom> ==> #(x&&xs) = iSuc (#xs)"

apply (case_tac "stream_finite (x && xs)")

apply (simp add: slen_def, auto)

apply (simp add: stream.finite_def, auto simp add: iSuc_Fin)

apply (rule Least_Suc2, auto)

(*apply (drule sym)*)

(*apply (drule sym scons_eq_UU [THEN iffD1],simp)*)

apply (erule stream_finite_lemma2, simp)

apply (simp add: slen_def, auto)

by (drule stream_finite_lemma1,auto)

lemma slen_less_1_eq: "(#x < Fin (Suc 0)) = (x = \<bottom>)"

by (cases x, auto simp add: Fin_0 iSuc_Fin[THEN sym])

lemma slen_empty_eq: "(#x = 0) = (x = \<bottom>)"

by (cases x, auto)

lemma slen_scons_eq: "(Fin (Suc n) < #x) = (? a y. x = a && y &  a ~= \<bottom> &  Fin n < #y)"

apply (auto, case_tac "x=UU",auto)

apply (drule stream_exhaust_eq [THEN iffD1], auto)

apply (case_tac "#y") apply simp_all

apply (case_tac "#y") apply simp_all

done

lemma slen_iSuc: "#x = iSuc n --> (? a y. x = a&&y &  a ~= \<bottom> &  #y = n)"

by (cases x, auto)

lemma slen_stream_take_finite [simp]: "#(stream_take n$s) ~= \<infinity>"

by (simp add: slen_def)

lemma slen_scons_eq_rev: "(#x < Fin (Suc (Suc n))) = (!a y. x ~= a && y |  a = \<bottom> |  #y < Fin (Suc n))"

 apply (cases x, auto)

   apply (simp add: zero_inat_def)

  apply (case_tac "#stream") apply (simp_all add: iSuc_Fin)

 apply (case_tac "#stream") apply (simp_all add: iSuc_Fin)

done

lemma slen_take_lemma4 [rule_format]:

  "!s. stream_take n$s ~= s --> #(stream_take n$s) = Fin n"

apply (induct n, auto simp add: Fin_0)

apply (case_tac "s=UU", simp)

by (drule stream_exhaust_eq [THEN iffD1], auto simp add: iSuc_Fin)

(*

lemma stream_take_idempotent [simp]:

 "stream_take n$(stream_take n$s) = stream_take n$s"

apply (case_tac "stream_take n$s = s")

apply (auto,insert slen_take_lemma4 [of n s]);

by (auto,insert slen_take_lemma1 [of "stream_take n$s" n],simp)

lemma stream_take_take_Suc [simp]: "stream_take n$(stream_take (Suc n)$s) =

                                    stream_take n$s"

apply (simp add: po_eq_conv,auto)

 apply (simp add: stream_take_take_less)

apply (subgoal_tac "stream_take n$s = stream_take n$(stream_take n$s)")

 apply (erule ssubst)

 apply (rule_tac monofun_cfun_arg)

 apply (insert chain_stream_take [of s])

by (simp add: chain_def,simp)

*)

lemma slen_take_eq: "ALL x. (Fin n < #x) = (stream_take n\<cdot>x ~= x)"

apply (induct_tac n, auto)

apply (simp add: Fin_0, clarsimp)

apply (drule not_sym)

apply (drule slen_empty_eq [THEN iffD1], simp)

apply (case_tac "x=UU", simp)

apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)

apply (erule_tac x="y" in allE, auto)

apply (simp_all add: not_less iSuc_Fin)

apply (case_tac "#y") apply simp_all

apply (case_tac "x=UU", simp)

apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)

apply (erule_tac x="y" in allE, simp)

apply (case_tac "#y") by simp_all

lemma slen_take_eq_rev: "(#x <= Fin n) = (stream_take n\<cdot>x = x)"

by (simp add: linorder_not_less [symmetric] slen_take_eq)

lemma slen_take_lemma1: "#x = Fin n ==> stream_take n\<cdot>x = x"

by (rule slen_take_eq_rev [THEN iffD1], auto)

lemma slen_rt_mono: "#s2 <= #s1 ==> #(rt$s2) <= #(rt$s1)"

apply (cases s1)

 by (cases s2, simp+)+

lemma slen_take_lemma5: "#(stream_take n$s) <= Fin n"

apply (case_tac "stream_take n$s = s")

 apply (simp add: slen_take_eq_rev)

by (simp add: slen_take_lemma4)

lemma slen_take_lemma2: "!x. ~stream_finite x --> #(stream_take i\<cdot>x) = Fin i"

apply (simp add: stream.finite_def, auto)

by (simp add: slen_take_lemma4)

lemma slen_infinite: "stream_finite x = (#x ~= Infty)"

by (simp add: slen_def)

lemma slen_mono_lemma: "stream_finite s ==> ALL t. s << t --> #s <= #t"

apply (erule stream_finite_ind [of s], auto)

apply (case_tac "t=UU", auto)

apply (drule stream_exhaust_eq [THEN iffD1], auto)

done

lemma slen_mono: "s << t ==> #s <= #t"

apply (case_tac "stream_finite t")

apply (frule stream_finite_less)

apply (erule_tac x="s" in allE, simp)

apply (drule slen_mono_lemma, auto)

by (simp add: slen_def)

lemma iterate_lemma: "F$(iterate n$F$x) = iterate n$F$(F$x)"

by (insert iterate_Suc2 [of n F x], auto)

lemma slen_rt_mult [rule_format]: "!x. Fin (i + j) <= #x --> Fin j <= #(iterate i$rt$x)"

apply (induct i, auto)

apply (case_tac "x=UU", auto simp add: zero_inat_def)

apply (drule stream_exhaust_eq [THEN iffD1], auto)

apply (erule_tac x="y" in allE, auto)

apply (simp add: not_le) apply (case_tac "#y") apply (simp_all add: iSuc_Fin)

by (simp add: iterate_lemma)

lemma slen_take_lemma3 [rule_format]:

  "!(x::'a::flat stream) y. Fin n <= #x --> x << y --> stream_take n\<cdot>x = stream_take n\<cdot>y"

apply (induct_tac n, auto)

apply (case_tac "x=UU", auto)

apply (simp add: zero_inat_def)

apply (simp add: Suc_ile_eq)

apply (case_tac "y=UU", clarsimp)

apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)+

apply (erule_tac x="ya" in allE, simp)

by (drule ax_flat, simp)

lemma slen_strict_mono_lemma:

  "stream_finite t ==> !s. #(s::'a::flat stream) = #t &  s << t --> s = t"

apply (erule stream_finite_ind, auto)

apply (case_tac "sa=UU", auto)

apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)

by (drule ax_flat, simp)

lemma slen_strict_mono: "[|stream_finite t; s ~= t; s << (t::'a::flat stream) |] ==> #s < #t"

by (auto simp add: slen_mono less_le dest: slen_strict_mono_lemma)

lemma stream_take_Suc_neq: "stream_take (Suc n)$s ~=s ==>

                     stream_take n$s ~= stream_take (Suc n)$s"

apply auto

apply (subgoal_tac "stream_take n$s ~=s")

 apply (insert slen_take_lemma4 [of n s],auto)

apply (cases s, simp)

by (simp add: slen_take_lemma4 iSuc_Fin)

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

(* theorems about smap                                                     *)

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

section "smap"

lemma smap_unfold: "smap = (\<Lambda> f t. case t of x&&xs \<Rightarrow> f$x && smap$f$xs)"

by (insert smap_def [where 'a='a and 'b='b, THEN eq_reflection, THEN fix_eq2], auto)

lemma smap_empty [simp]: "smap\<cdot>f\<cdot>\<bottom> = \<bottom>"

by (subst smap_unfold, simp)

lemma smap_scons [simp]: "x~=\<bottom> ==> smap\<cdot>f\<cdot>(x&&xs) = (f\<cdot>x)&&(smap\<cdot>f\<cdot>xs)"

by (subst smap_unfold, force)

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

(* theorems about sfilter                                                  *)

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

section "sfilter"

lemma sfilter_unfold:

 "sfilter = (\<Lambda> p s. case s of x && xs \<Rightarrow>

  If p\<cdot>x then x && sfilter\<cdot>p\<cdot>xs else sfilter\<cdot>p\<cdot>xs)"

by (insert sfilter_def [where 'a='a, THEN eq_reflection, THEN fix_eq2], auto)

lemma strict_sfilter: "sfilter\<cdot>\<bottom> = \<bottom>"

apply (rule cfun_eqI)

apply (subst sfilter_unfold, auto)

apply (case_tac "x=UU", auto)

by (drule stream_exhaust_eq [THEN iffD1], auto)

lemma sfilter_empty [simp]: "sfilter\<cdot>f\<cdot>\<bottom> = \<bottom>"

by (subst sfilter_unfold, force)

lemma sfilter_scons [simp]:

  "x ~= \<bottom> ==> sfilter\<cdot>f\<cdot>(x && xs) =

                           If f\<cdot>x then x && sfilter\<cdot>f\<cdot>xs else sfilter\<cdot>f\<cdot>xs"

by (subst sfilter_unfold, force)

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

   section "i_rt"

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

lemma i_rt_UU [simp]: "i_rt n UU = UU"

  by (induct n) (simp_all add: i_rt_def)

lemma i_rt_0 [simp]: "i_rt 0 s = s"

by (simp add: i_rt_def)

lemma i_rt_Suc [simp]: "a ~= UU ==> i_rt (Suc n) (a&&s) = i_rt n s"

by (simp add: i_rt_def iterate_Suc2 del: iterate_Suc)

lemma i_rt_Suc_forw: "i_rt (Suc n) s = i_rt n (rt$s)"

by (simp only: i_rt_def iterate_Suc2)

lemma i_rt_Suc_back:"i_rt (Suc n) s = rt$(i_rt n s)"

by (simp only: i_rt_def,auto)

lemma i_rt_mono: "x << s ==> i_rt n x  << i_rt n s"

by (simp add: i_rt_def monofun_rt_mult)

lemma i_rt_ij_lemma: "Fin (i + j) <= #x ==> Fin j <= #(i_rt i x)"

by (simp add: i_rt_def slen_rt_mult)

lemma slen_i_rt_mono: "#s2 <= #s1 ==> #(i_rt n s2) <= #(i_rt n s1)"

apply (induct_tac n,auto)

apply (simp add: i_rt_Suc_back)

by (drule slen_rt_mono,simp)

lemma i_rt_take_lemma1 [rule_format]: "ALL s. i_rt n (stream_take n$s) = UU"

apply (induct_tac n)

 apply (simp add: i_rt_Suc_back,auto)

apply (case_tac "s=UU",auto)

by (drule stream_exhaust_eq [THEN iffD1],auto)

lemma i_rt_slen: "(i_rt n s = UU) = (stream_take n$s = s)"

apply auto

 apply (insert i_rt_ij_lemma [of n "Suc 0" s])

 apply (subgoal_tac "#(i_rt n s)=0")

  apply (case_tac "stream_take n$s = s",simp+)

  apply (insert slen_take_eq [rule_format,of n s],simp)

  apply (cases "#s") apply (simp_all add: zero_inat_def)

  apply (simp add: slen_take_eq)

  apply (cases "#s")

  using i_rt_take_lemma1 [of n s]

  apply (simp_all add: zero_inat_def)

  done

lemma i_rt_lemma_slen: "#s=Fin n ==> i_rt n s = UU"

by (simp add: i_rt_slen slen_take_lemma1)

lemma stream_finite_i_rt [simp]: "stream_finite (i_rt n s) = stream_finite s"

apply (induct_tac n, auto)

 apply (cases s, auto simp del: i_rt_Suc)

by (simp add: i_rt_Suc_back stream_finite_rt_eq)+

lemma take_i_rt_len_lemma: "ALL sl x j t. Fin sl = #x & n <= sl &

                            #(stream_take n$x) = Fin t & #(i_rt n x)= Fin j

                                              --> Fin (j + t) = #x"

apply (induct n, auto)

 apply (simp add: zero_inat_def)

apply (case_tac "x=UU",auto)

 apply (simp add: zero_inat_def)

apply (drule stream_exhaust_eq [THEN iffD1],clarsimp)

apply (subgoal_tac "EX k. Fin k = #y",clarify)

 apply (erule_tac x="k" in allE)

 apply (erule_tac x="y" in allE,auto)

 apply (erule_tac x="THE p. Suc p = t" in allE,auto)

   apply (simp add: iSuc_def split: inat.splits)

  apply (simp add: iSuc_def split: inat.splits)

  apply (simp only: the_equality)

 apply (simp add: iSuc_def split: inat.splits)

 apply force

apply (simp add: iSuc_def split: inat.splits)

done

lemma take_i_rt_len:

"[| Fin sl = #x; n <= sl; #(stream_take n$x) = Fin t; #(i_rt n x) = Fin j |] ==>

    Fin (j + t) = #x"

by (blast intro: take_i_rt_len_lemma [rule_format])

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

   section "i_th"

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

lemma i_th_i_rt_step:

"[| i_th n s1 << i_th n s2; i_rt (Suc n) s1 << i_rt (Suc n) s2 |] ==>

   i_rt n s1 << i_rt n s2"

apply (simp add: i_th_def i_rt_Suc_back)

apply (cases "i_rt n s1", simp)

apply (cases "i_rt n s2", auto)

done

lemma i_th_stream_take_Suc [rule_format]:

 "ALL s. i_th n (stream_take (Suc n)$s) = i_th n s"

apply (induct_tac n,auto)

 apply (simp add: i_th_def)

 apply (case_tac "s=UU",auto)

 apply (drule stream_exhaust_eq [THEN iffD1],auto)

apply (case_tac "s=UU",simp add: i_th_def)

apply (drule stream_exhaust_eq [THEN iffD1],auto)

by (simp add: i_th_def i_rt_Suc_forw)

lemma i_th_last: "i_th n s && UU = i_rt n (stream_take (Suc n)$s)"

apply (insert surjectiv_scons [of "i_rt n (stream_take (Suc n)$s)"])

apply (rule i_th_stream_take_Suc [THEN subst])

apply (simp add: i_th_def  i_rt_Suc_back [symmetric])

by (simp add: i_rt_take_lemma1)

lemma i_th_last_eq:

"i_th n s1 = i_th n s2 ==> i_rt n (stream_take (Suc n)$s1) = i_rt n (stream_take (Suc n)$s2)"

apply (insert i_th_last [of n s1])

apply (insert i_th_last [of n s2])

by auto

lemma i_th_prefix_lemma:

"[| k <= n; stream_take (Suc n)$s1 << stream_take (Suc n)$s2 |] ==>

    i_th k s1 << i_th k s2"

apply (insert i_th_stream_take_Suc [of k s1, THEN sym])

apply (insert i_th_stream_take_Suc [of k s2, THEN sym],auto)

apply (simp add: i_th_def)

apply (rule monofun_cfun, auto)

apply (rule i_rt_mono)

by (blast intro: stream_take_lemma10)

lemma take_i_rt_prefix_lemma1:

  "stream_take (Suc n)$s1 << stream_take (Suc n)$s2 ==>

   i_rt (Suc n) s1 << i_rt (Suc n) s2 ==>

   i_rt n s1 << i_rt n s2 & stream_take n$s1 << stream_take n$s2"

apply auto

 apply (insert i_th_prefix_lemma [of n n s1 s2])

 apply (rule i_th_i_rt_step,auto)

by (drule mono_stream_take_pred,simp)

lemma take_i_rt_prefix_lemma:

"[| stream_take n$s1 << stream_take n$s2; i_rt n s1 << i_rt n s2 |] ==> s1 << s2"

apply (case_tac "n=0",simp)

apply (auto)

apply (subgoal_tac "stream_take 0$s1 << stream_take 0$s2 &

                    i_rt 0 s1 << i_rt 0 s2")

 defer 1

 apply (rule zero_induct,blast)

 apply (blast dest: take_i_rt_prefix_lemma1)

by simp

lemma streams_prefix_lemma: "(s1 << s2) =

  (stream_take n$s1 << stream_take n$s2 & i_rt n s1 << i_rt n s2)"

apply auto

  apply (simp add: monofun_cfun_arg)

 apply (simp add: i_rt_mono)

by (erule take_i_rt_prefix_lemma,simp)

lemma streams_prefix_lemma1:

 "[| stream_take n$s1 = stream_take n$s2; i_rt n s1 = i_rt n s2 |] ==> s1 = s2"

apply (simp add: po_eq_conv,auto)

 apply (insert streams_prefix_lemma)

 by blast+

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

   section "sconc"

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

lemma UU_sconc [simp]: " UU ooo s = s "

by (simp add: sconc_def zero_inat_def)

lemma scons_neq_UU: "a~=UU ==> a && s ~=UU"

by auto

lemma singleton_sconc [rule_format, simp]: "x~=UU --> (x && UU) ooo y = x && y"

apply (simp add: sconc_def zero_inat_def iSuc_def split: inat.splits, auto)

apply (rule someI2_ex,auto)

 apply (rule_tac x="x && y" in exI,auto)

apply (simp add: i_rt_Suc_forw)

apply (case_tac "xa=UU",simp)

by (drule stream_exhaust_eq [THEN iffD1],auto)

lemma ex_sconc [rule_format]:

  "ALL k y. #x = Fin k --> (EX w. stream_take k$w = x & i_rt k w = y)"

apply (case_tac "#x")

 apply (rule stream_finite_ind [of x],auto)

  apply (simp add: stream.finite_def)

  apply (drule slen_take_lemma1,blast)

 apply (simp_all add: zero_inat_def iSuc_def split: inat.splits)

apply (erule_tac x="y" in allE,auto)

by (rule_tac x="a && w" in exI,auto)

lemma rt_sconc1: "Fin n = #x ==> i_rt n (x ooo y) = y"

apply (simp add: sconc_def split: inat.splits, arith?,auto)

apply (rule someI2_ex,auto)

by (drule ex_sconc,simp)

lemma sconc_inj2: "\<lbrakk>Fin n = #x; x ooo y = x ooo z\<rbrakk> \<Longrightarrow> y = z"

apply (frule_tac y=y in rt_sconc1)

by (auto elim: rt_sconc1)

lemma sconc_UU [simp]:"s ooo UU = s"

apply (case_tac "#s")

 apply (simp add: sconc_def)

 apply (rule someI2_ex)

  apply (rule_tac x="s" in exI)

  apply auto

   apply (drule slen_take_lemma1,auto)

  apply (simp add: i_rt_lemma_slen)

 apply (drule slen_take_lemma1,auto)

 apply (simp add: i_rt_slen)

by (simp add: sconc_def)

lemma stream_take_sconc [simp]: "Fin n = #x ==> stream_take n$(x ooo y) = x"

apply (simp add: sconc_def)

apply (cases "#x")

apply auto

apply (rule someI2_ex, auto)

by (drule ex_sconc,simp)

lemma scons_sconc [rule_format,simp]: "a~=UU --> (a && x) ooo y = a && x ooo y"

apply (cases "#x",auto)

 apply (simp add: sconc_def iSuc_Fin)

 apply (rule someI2_ex)

  apply (drule ex_sconc, simp)

 apply (rule someI2_ex, auto)

  apply (simp add: i_rt_Suc_forw)

  apply (rule_tac x="a && x" in exI, auto)

 apply (case_tac "xa=UU",auto)

 apply (drule stream_exhaust_eq [THEN iffD1],auto)

 apply (drule streams_prefix_lemma1,simp+)

by (simp add: sconc_def)

lemma ft_sconc: "x ~= UU ==> ft$(x ooo y) = ft$x"

by (cases x, auto)

lemma sconc_assoc: "(x ooo y) ooo z = x ooo y ooo z"

apply (case_tac "#x")

 apply (rule stream_finite_ind [of x],auto simp del: scons_sconc)

  apply (simp add: stream.finite_def del: scons_sconc)

  apply (drule slen_take_lemma1,auto simp del: scons_sconc)

 apply (case_tac "a = UU", auto)

by (simp add: sconc_def)

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

lemma cont_sconc_lemma1: "stream_finite x \<Longrightarrow> cont (\<lambda>y. x ooo y)"

by (erule stream_finite_ind, simp_all)

lemma cont_sconc_lemma2: "\<not> stream_finite x \<Longrightarrow> cont (\<lambda>y. x ooo y)"

by (simp add: sconc_def slen_def)

lemma cont_sconc: "cont (\<lambda>y. x ooo y)"

apply (cases "stream_finite x")

apply (erule cont_sconc_lemma1)

apply (erule cont_sconc_lemma2)

done

lemma sconc_mono: "y << y' ==> x ooo y << x ooo y'"

by (rule cont_sconc [THEN cont2mono, THEN monofunE])

lemma sconc_mono1 [simp]: "x << x ooo y"

by (rule sconc_mono [of UU, simplified])

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

lemma empty_sconc [simp]: "(x ooo y = UU) = (x = UU & y = UU)"

apply (case_tac "#x",auto)

   apply (insert sconc_mono1 [of x y])

   by auto

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

lemma rt_sconc [rule_format, simp]: "s~=UU --> rt$(s ooo x) = rt$s ooo x"

by (cases s, auto)

lemma i_th_sconc_lemma [rule_format]:

  "ALL x y. Fin n < #x --> i_th n (x ooo y) = i_th n x"

apply (induct_tac n, auto)

apply (simp add: Fin_0 i_th_def)

apply (simp add: slen_empty_eq ft_sconc)

apply (simp add: i_th_def)

apply (case_tac "x=UU",auto)

apply (drule stream_exhaust_eq [THEN iffD1], auto)

apply (erule_tac x="ya" in allE)

apply (case_tac "#ya") by simp_all

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

lemma sconc_lemma [rule_format, simp]: "ALL s. stream_take n$s ooo i_rt n s = s"

apply (induct_tac n,auto)

apply (case_tac "s=UU",auto)

by (drule stream_exhaust_eq [THEN iffD1],auto)

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

   subsection "pointwise equality"

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

lemma ex_last_stream_take_scons: "stream_take (Suc n)$s =

                     stream_take n$s ooo i_rt n (stream_take (Suc n)$s)"

by (insert sconc_lemma [of n "stream_take (Suc n)$s"],simp)

lemma i_th_stream_take_eq:

"!!n. ALL n. i_th n s1 = i_th n s2 ==> stream_take n$s1 = stream_take n$s2"

apply (induct_tac n,auto)

apply (subgoal_tac "stream_take (Suc na)$s1 =

                    stream_take na$s1 ooo i_rt na (stream_take (Suc na)$s1)")

 apply (subgoal_tac "i_rt na (stream_take (Suc na)$s1) =

                    i_rt na (stream_take (Suc na)$s2)")

  apply (subgoal_tac "stream_take (Suc na)$s2 =

                    stream_take na$s2 ooo i_rt na (stream_take (Suc na)$s2)")

   apply (insert ex_last_stream_take_scons,simp)

  apply blast

 apply (erule_tac x="na" in allE)

 apply (insert i_th_last_eq [of _ s1 s2])

by blast+

lemma pointwise_eq_lemma[rule_format]: "ALL n. i_th n s1 = i_th n s2 ==> s1 = s2"

by (insert i_th_stream_take_eq [THEN stream.take_lemma],blast)

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

   subsection "finiteness"

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

lemma slen_sconc_finite1:

  "[| #(x ooo y) = Infty; Fin n = #x |] ==> #y = Infty"

apply (case_tac "#y ~= Infty",auto)

apply (drule_tac y=y in rt_sconc1)

apply (insert stream_finite_i_rt [of n "x ooo y"])

by (simp add: slen_infinite)

lemma slen_sconc_infinite1: "#x=Infty ==> #(x ooo y) = Infty"

by (simp add: sconc_def)

lemma slen_sconc_infinite2: "#y=Infty ==> #(x ooo y) = Infty"

apply (case_tac "#x")

 apply (simp add: sconc_def)

 apply (rule someI2_ex)

  apply (drule ex_sconc,auto)

 apply (erule contrapos_pp)

 apply (insert stream_finite_i_rt)

 apply (fastsimp simp add: slen_infinite,auto)

by (simp add: sconc_def)

lemma sconc_finite: "(#x~=Infty & #y~=Infty) = (#(x ooo y)~=Infty)"

apply auto

  apply (metis not_Infty_eq slen_sconc_finite1)

 apply (metis not_Infty_eq slen_sconc_infinite1)

apply (metis not_Infty_eq slen_sconc_infinite2)

done

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

lemma slen_sconc_mono3: "[| Fin n = #x; Fin k = #(x ooo y) |] ==> n <= k"

apply (insert slen_mono [of "x" "x ooo y"])

apply (cases "#x") apply simp_all

apply (cases "#(x ooo y)") apply simp_all

done

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

   subsection "finite slen"

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

lemma slen_sconc: "[| Fin n = #x; Fin m = #y |] ==> #(x ooo y) = Fin (n + m)"

apply (case_tac "#(x ooo y)")

 apply (frule_tac y=y in rt_sconc1)

 apply (insert take_i_rt_len [of "THE j. Fin j = #(x ooo y)" "x ooo y" n n m],simp)

 apply (insert slen_sconc_mono3 [of n x _ y],simp)

by (insert sconc_finite [of x y],auto)

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

   subsection "flat prefix"

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

lemma sconc_prefix: "(s1::'a::flat stream) << s2 ==> EX t. s1 ooo t = s2"

apply (case_tac "#s1")

 apply (subgoal_tac "stream_take nat$s1 = stream_take nat$s2")

  apply (rule_tac x="i_rt nat s2" in exI)

  apply (simp add: sconc_def)

  apply (rule someI2_ex)

   apply (drule ex_sconc)

   apply (simp,clarsimp,drule streams_prefix_lemma1)

   apply (simp+,rule slen_take_lemma3 [of _ s1 s2])

  apply (simp+,rule_tac x="UU" in exI)

apply (insert slen_take_lemma3 [of _ s1 s2])

by (rule stream.take_lemma,simp)

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

   subsection "continuity"

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

lemma chain_sconc: "chain S ==> chain (%i. (x ooo S i))"

by (simp add: chain_def,auto simp add: sconc_mono)

lemma chain_scons: "chain S ==> chain (%i. a && S i)"

apply (simp add: chain_def,auto)

by (rule monofun_cfun_arg,simp)

lemma contlub_scons_lemma: "chain S ==> (LUB i. a && S i) = a && (LUB i. S i)"

by (rule cont2contlubE [OF cont_Rep_cfun2, symmetric])

lemma finite_lub_sconc: "chain Y ==> (stream_finite x) ==>

                        (LUB i. x ooo Y i) = (x ooo (LUB i. Y i))"

apply (rule stream_finite_ind [of x])

 apply (auto)

apply (subgoal_tac "(LUB i. a && (s ooo Y i)) = a && (LUB i. s ooo Y i)")

 by (force,blast dest: contlub_scons_lemma chain_sconc)

lemma contlub_sconc_lemma:

  "chain Y ==> (LUB i. x ooo Y i) = (x ooo (LUB i. Y i))"

apply (case_tac "#x=Infty")

 apply (simp add: sconc_def)

apply (drule finite_lub_sconc,auto simp add: slen_infinite)

done

lemma monofun_sconc: "monofun (%y. x ooo y)"

by (simp add: monofun_def sconc_mono)

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

   section "constr_sconc"

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

lemma constr_sconc_UUs [simp]: "constr_sconc UU s = s"

by (simp add: constr_sconc_def zero_inat_def)

lemma "x ooo y = constr_sconc x y"

apply (case_tac "#x")

 apply (rule stream_finite_ind [of x],auto simp del: scons_sconc)

  defer 1

  apply (simp add: constr_sconc_def del: scons_sconc)

  apply (case_tac "#s")

   apply (simp add: iSuc_Fin)

   apply (case_tac "a=UU",auto simp del: scons_sconc)

   apply (simp)

  apply (simp add: sconc_def)

 apply (simp add: constr_sconc_def)

apply (simp add: stream.finite_def)

by (drule slen_take_lemma1,auto)

end