# Theory Stream

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theory Stream
imports HOLCF Nat_Infinity
begin

    (*  Title:      HOLCF/ex/Stream.thy
        ID:         $Id: Stream.thy,v 1.13 2005/09/06 17:28:59 wenzelm Exp $
        Author:     Franz Regensburger, David von Oheimb, Borislav Gajanovic
    *)

    header {* General Stream domain *}

    theory Stream
    imports HOLCF Nat_Infinity
    begin

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

    consts

      smap          :: "('a -> 'b) -> 'a stream -> 'b stream"
      sfilter       :: "('a -> tr) -> 'a stream -> 'a stream"
      slen          :: "'a stream => inat"                   ("#_" [1000] 1000)

    defs

      smap_def:     "smap    ≡ fix·(Λ h f s. case s of x && xs => f·x && h·f·xs)"
      sfilter_def:  "sfilter ≡ fix·(Λ h p s. case s of x && xs =>
                                         If p·x then x && h·p·xs else h·p·xs fi)"
      slen_def:     "#s ≡ if stream_finite s
                          then Fin (LEAST n. stream_take n·s = s) else ∞"

    (* concatenation *)

    consts

      i_rt :: "nat => 'a stream => 'a stream" (* chops the first i elements *)
      i_th :: "nat => 'a stream => 'a"        (* the i-th element *)

      sconc         :: "'a stream => 'a stream => 'a stream" (infixr "ooo" 65)
      constr_sconc  :: "'a stream => 'a stream => 'a stream" (* constructive *)
      constr_sconc' :: "nat => 'a stream => 'a stream => 'a stream"

    defs
      i_rt_def: "i_rt == %i s. iterate i rt s"
      i_th_def: "i_th == %i s. ft$(i_rt i s)"

      sconc_def: "s1 ooo s2 == case #s1 of
                           Fin n => (SOME s. (stream_take n$s=s1) & (i_rt n s = s2))
                         | ∞     => s1"

      constr_sconc_def: "constr_sconc s1 s2 == case #s1 of
                                                 Fin n => constr_sconc' n s1 s2
                                               | ∞    => s1"
    primrec
      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"

    declare stream.rews [simp add]

    (* ----------------------------------------------------------------------- *)
    (* theorems about scons                                                    *)
    (* ----------------------------------------------------------------------- *)

    section "scons"

    lemma scons_eq_UU: "(a && s = UU) = (a = UU)"
    by (auto, erule contrapos_pp, simp)

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

    lemma stream_exhaust_eq: "(x ~= UU) = (EX a y. a ~= UU &  x = a && y)"
    by (auto,insert stream.exhaust [of x],auto)

    lemma stream_neq_UU: "x~=UU ==> EX a as. x=a&&as & a~=UU"
    by (simp add: stream_exhaust_eq,auto)

    lemma stream_inject_eq [simp]:
      "[| a ~= UU; b ~= UU |] ==> (a && s = b && t) = (a = b &  s = t)"
    by (insert stream.injects [of a s b t], auto)

    lemma stream_prefix:
      "[| a && s << t; a ~= UU  |] ==> EX b tt. t = b && tt &  b ~= UU &  s << tt"
    apply (insert stream.exhaust [of t], auto)
    apply (drule eq_UU_iff [THEN iffD2], simp)
    by (drule stream.inverts, auto)

    lemma stream_prefix':
      "b ~= UU ==> x << b && z =
       (x = UU |  (EX a y. x = a && y &  a ~= UU &  a << b &  y << z))"
    apply (case_tac "x=UU",auto)
    apply (drule stream_exhaust_eq [THEN iffD1],auto)
    apply (drule stream.inverts,auto)
    by (intro monofun_cfun,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)
    apply (drule eq_UU_iff [THEN iffD2],auto)
    apply (drule stream.inverts,auto)
    apply (drule ax_flat [rule_format],simp)
    by (drule stream.inverts,auto)

    (* ----------------------------------------------------------------------- *)
    (* theorems about stream_when                                              *)
    (* ----------------------------------------------------------------------- *)

    section "stream_when"

    lemma stream_when_strictf: "stream_when$UU$s=UU"
    by (rule stream.casedist [of s], auto)

    (* ----------------------------------------------------------------------- *)
    (* theorems about ft and rt                                                *)
    (* ----------------------------------------------------------------------- *)

    section "ft & rt"

    lemma ft_defin: "s~=UU ==> ft$s~=UU"
    by (drule stream_exhaust_eq [THEN iffD1],auto)

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

    lemma surjectiv_scons: "(ft$s)&&(rt$s)=s"
    by (rule stream.casedist [of s], auto)

    lemma monofun_rt_mult: "x << s ==> iterate i rt x << iterate i rt s"
    by (insert monofun_iterate2 [of i "rt"], simp add: monofun_def, auto)

    (* ----------------------------------------------------------------------- *)
    (* theorems about stream_take                                              *)
    (* ----------------------------------------------------------------------- *)

    section "stream_take"

    lemma stream_reach2: "(LUB i. stream_take i$s) = s"
    apply (insert stream.reach [of s], erule subst) back
    apply (simp add: fix_def2 stream.take_def)
    apply (insert contlub_cfun_fun [of "%i. iterate i stream_copy UU" s,THEN sym])
    by (simp add: chain_iterate)

    lemma chain_stream_take: "chain (%i. stream_take i$s)"
    apply (rule chainI)
    apply (rule monofun_cfun_fun)
    apply (simp add: stream.take_def del: iterate_Suc)
    by (rule chainE, simp add: chain_iterate)

    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_mono3,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_ind [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_induct2 [of _ n],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 (frule adm_impl_admw, rule wfix_ind, auto)
    apply (rule adm_subst [THEN adm_impl_admw],auto)
    apply (insert stream_finite_ind2 [of P])
    by (simp add: stream.take_def)

    (* ----------------------------------------------------------------------- *)
    (* 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 (case_tac "x=UU",clarsimp)
    apply (erule_tac x="UU" in allE,simp)
    apply (case_tac "x'=UU",simp)
    apply (drule stream_exhaust_eq [THEN iffD1],auto)+
    apply (case_tac "x'=UU",auto)
    apply (erule_tac x="a && y" in allE)
    apply (erule_tac x="UU" in allE)+
    apply (auto,drule stream_exhaust_eq [THEN iffD1],clarsimp)
    apply (erule_tac x="a && y" in allE)
    apply (erule_tac x="aa && ya" in allE)
    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 (rule stream.casedist [of 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])
    apply (clarsimp, drule eq_UU_iff [THEN iffD2], auto)
    apply (case_tac "t=UU", auto)
    apply (drule stream_exhaust_eq [THEN iffD1],auto)
    apply (drule stream.inverts, 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 admI2, auto)
    apply (drule stream_finite_less,drule is_ub_thelub)
    by auto

    (* ----------------------------------------------------------------------- *)
    (* theorems about stream length                                            *)
    (* ----------------------------------------------------------------------- *)

    section "slen"

    lemma slen_empty [simp]: "#⊥ = 0"
    apply (simp add: slen_def stream.finite_def)
    by (simp add: inat_defs Least_equality)

    lemma slen_scons [simp]: "x ~= ⊥ ==> #(x&&xs) = iSuc (#xs)"
    apply (case_tac "stream_finite (x && xs)")
    apply (simp add: slen_def, auto)
    apply (simp add: stream.finite_def, auto)
    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 = ⊥)"
    by (rule stream.casedist [of x], auto simp del: iSuc_Fin
        simp add: Fin_0 iSuc_Fin[THEN sym] i0_iless_iSuc iSuc_mono)

    lemma slen_empty_eq: "(#x = 0) = (x = ⊥)"
    by (rule stream.casedist [of x], auto)

    lemma slen_scons_eq: "(Fin (Suc n) < #x) = (? a y. x = a && y &  a ~= ⊥ &  Fin n < #y)"
    apply (auto, case_tac "x=UU",auto)
    apply (drule stream_exhaust_eq [THEN iffD1], auto)
    apply (rule_tac x="a" in exI)
    apply (rule_tac x="y" in exI, simp)
    by (simp add: inat_defs split:inat_splits)+

    lemma slen_iSuc: "#x = iSuc n --> (? a y. x = a&&y &  a ~= ⊥ &  #y = n)"
    by (rule stream.casedist [of x], auto)

    lemma slen_stream_take_finite [simp]: "#(stream_take n$s) ~= ∞"
    by (simp add: slen_def)

    lemma slen_scons_eq_rev: "(#x < Fin (Suc (Suc n))) = (!a y. x ~= a && y |  a = ⊥ |  #y < Fin (Suc n))"
    apply (rule stream.casedist [of x], auto)
    apply ((*drule sym,*) drule scons_eq_UU [THEN iffD1],auto)
    apply (simp add: inat_defs split:inat_splits)
    apply (subgoal_tac "s=y & aa=a",simp)
    apply (simp add: inat_defs split:inat_splits)
    apply (case_tac "aa=UU",auto)
    apply (erule_tac x="a" in allE, simp)
    by (simp add: inat_defs split:inat_splits)

    lemma slen_take_lemma4 [rule_format]:
      "!s. stream_take n$s ~= s --> #(stream_take n$s) = Fin n"
    apply (induct_tac n,auto simp add: Fin_0)
    apply (case_tac "s=UU",simp)
    by (drule stream_exhaust_eq [THEN iffD1], auto)

    (*
    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·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 add: inat_defs split:inat_splits)
    apply (case_tac "x=UU", simp)
    apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)
    apply (erule_tac x="y" in allE, simp)
    by (simp add: inat_defs split:inat_splits)

    lemma slen_take_eq_rev: "(#x <= Fin n) = (stream_take n·x = x)"
    by (simp add: ile_def slen_take_eq)

    lemma slen_take_lemma1: "#x = Fin n ==> stream_take n·x = x"
    by (rule slen_take_eq_rev [THEN iffD1], auto)

    lemma slen_rt_mono: "#s2 <= #s1 ==> #(rt$s2) <= #(rt$s1)"
    apply (rule stream.casedist [of s1])
     by (rule stream.casedist [of 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·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 eq_UU_iff [THEN iffD2])
    apply (drule scons_eq_UU [THEN iffD2], simp)
    apply (drule stream_exhaust_eq [THEN iffD1], auto)
    apply (erule_tac x="y" in allE, auto)
    by (drule stream.inverts, auto)

    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_tac i, auto)
    apply (case_tac "x=UU", auto)
    apply (simp add: inat_defs)
    apply (drule stream_exhaust_eq [THEN iffD1], auto)
    apply (erule_tac x="y" in allE, auto)
    apply (simp add: inat_defs split:inat_splits)
    by (simp add: iterate_lemma)

    lemma slen_take_lemma3 [rule_format]:
      "!(x::'a::flat stream) y. Fin n <= #x --> x << y --> stream_take n·x = stream_take n·y"
    apply (induct_tac n, auto)
    apply (case_tac "x=UU", auto)
    apply (simp add: inat_defs)
    apply (simp add: Suc_ile_eq)
    apply (case_tac "y=UU", clarsimp)
    apply (drule eq_UU_iff [THEN iffD2],simp)
    apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)+
    apply (erule_tac x="ya" in allE, simp)
    apply (drule stream.inverts,auto)
    by (drule ax_flat [rule_format], 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 (drule eq_UU_iff [THEN iffD2], simp)
    apply (case_tac "sa=UU", auto)
    apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)
    apply (drule stream.inverts, simp, simp, clarsimp)
    by (drule ax_flat [rule_format], simp)

    lemma slen_strict_mono: "[|stream_finite t; s ~= t; s << (t::'a::flat stream) |] ==> #s < #t"
    apply (intro ilessI1, auto)
    apply (simp add: slen_mono)
    by (drule slen_strict_mono_lemma, auto)

    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 (rule stream.casedist [of s],simp)
    apply (simp add: inat_defs split:inat_splits)
    by (simp add: slen_take_lemma4)

    (* ----------------------------------------------------------------------- *)
    (* theorems about smap                                                     *)
    (* ----------------------------------------------------------------------- *)

    section "smap"

    lemma smap_unfold: "smap = (Λ f t. case t of x&&xs => f$x && smap$f$xs)"
    by (insert smap_def [THEN fix_eq2], auto)

    lemma smap_empty [simp]: "smap·f·⊥ = ⊥"
    by (subst smap_unfold, simp)

    lemma smap_scons [simp]: "x~=⊥ ==> smap·f·(x&&xs) = (f·x)&&(smap·f·xs)"
    by (subst smap_unfold, force)

    (* ----------------------------------------------------------------------- *)
    (* theorems about sfilter                                                  *)
    (* ----------------------------------------------------------------------- *)

    section "sfilter"

    lemma sfilter_unfold:
     "sfilter = (Λ p s. case s of x && xs =>
      If p·x then x && sfilter·p·xs else sfilter·p·xs fi)"
    by (insert sfilter_def [THEN fix_eq2], auto)

    lemma strict_sfilter: "sfilter·⊥ = ⊥"
    apply (rule ext_cfun)
    apply (subst sfilter_unfold, auto)
    apply (case_tac "x=UU", auto)
    by (drule stream_exhaust_eq [THEN iffD1], auto)

    lemma sfilter_empty [simp]: "sfilter·f·⊥ = ⊥"
    by (subst sfilter_unfold, force)

    lemma sfilter_scons [simp]:
      "x ~= ⊥ ==> sfilter·f·(x && xs) =
                               If f·x then x && sfilter·f·xs else sfilter·f·xs fi"
    by (subst sfilter_unfold, force)

    (* ----------------------------------------------------------------------- *)
       section "i_rt"
    (* ----------------------------------------------------------------------- *)

    lemma i_rt_UU [simp]: "i_rt n UU = UU"
    apply (simp add: i_rt_def)
    by (rule iterate.induct,auto)

    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 (simp add: inat_defs split:inat_splits)
     apply (simp add: slen_take_eq )
    by (simp, insert i_rt_take_lemma1 [of n s],simp)

    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 (rule stream.casedist [of "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_tac n,auto)
     apply (simp add: inat_defs)
    apply (case_tac "x=UU",auto)
     apply (simp add: inat_defs)
    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: inat_defs split:inat_splits)
      apply (simp add: inat_defs split:inat_splits)
      apply (simp only: the_equality)
     apply (simp add: inat_defs split:inat_splits)
     apply force
    by (simp add: inat_defs split:inat_splits)

    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 (rule stream.casedist [of "i_rt n s1"],simp)
    apply (rule stream.casedist [of "i_rt n s2"],auto)
    apply (drule eq_UU_iff [THEN iffD2], simp add: scons_eq_UU)
    by (intro monofun_cfun, auto)

    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 (insert neq0_conv [of n])
    apply (insert not0_implies_Suc [of n],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 inat_defs)

    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 inat_defs 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 add: inat_defs 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 inat_defs split:inat_splits, arith?,auto)
    apply (rule someI2_ex,auto)
    by (drule ex_sconc,simp)

    lemma sconc_inj2: "[|Fin n = #x; x ooo y = x ooo z|] ==> 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 inat_defs)
     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 inat_defs)

    lemma stream_take_sconc [simp]: "Fin n = #x ==> stream_take n$(x ooo y) = x"
    apply (simp add: sconc_def)
    apply (simp add: inat_defs split:inat_splits,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 (case_tac "#x",auto)
     apply (simp add: sconc_def)
     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_tac s="stream_take nat$x" in scons_neq_UU)
      apply (simp add: i_rt_Suc_forw)*)
     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 (rule stream.casedist [of 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 sconc_mono: "y << y' ==> x ooo y << x ooo y'"
    apply (case_tac "#x")
     apply (rule stream_finite_ind [of "x"])
       apply (auto simp add: stream.finite_def)
      apply (drule slen_take_lemma1,blast)
     by (simp add: stream_prefix',auto simp add: sconc_def)

    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 (insert eq_UU_iff [THEN iffD2, of x],auto)

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

    lemma rt_sconc [rule_format, simp]: "s~=UU --> rt$(s ooo x) = rt$s ooo x"
    by (rule stream.casedist,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)
    by (simp add: inat_defs split:inat_splits)

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

    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_lemmas],blast)

    (* ----------------------------------------------------------------------- *)
       subsection "finiteness"
    (* ----------------------------------------------------------------------- *)

    lemma slen_sconc_finite1:
      "[| #(x ooo y) = Infty; Fin n = #x |] ==> #y = Infty"
    apply (case_tac "#y ~= Infty",auto)
    apply (simp only: slen_infinite [symmetric])
    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 (simp add: slen_infinite,auto)
    by (simp add: sconc_def)

    lemma sconc_finite: "(#x~=Infty & #y~=Infty) = (#(x ooo y)~=Infty)"
    apply auto
      apply (case_tac "#x",auto)
      apply (erule contrapos_pp,simp)
      apply (erule slen_sconc_finite1,simp)
     apply (drule slen_sconc_infinite1 [of _ y],simp)
    by (drule slen_sconc_infinite2 [of _ x],simp)

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

    lemma slen_sconc_mono3: "[| Fin n = #x; Fin k = #(x ooo y) |] ==> n <= k"
    apply (insert slen_mono [of "x" "x ooo y"])
    by (simp add: inat_defs split: inat_splits)

    (* ----------------------------------------------------------------------- *)
       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_lemmas,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: "contlub (%x. a && x)"
    by (simp add: contlub_Rep_CFun2)

    lemma contlub_scons_lemma: "chain S ==> (LUB i. a && S i) = a && (LUB i. S i)"
    apply (insert contlub_scons [of a])
    by (simp only: contlub_def)

    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)
     prefer 2
     apply (drule finite_lub_sconc,auto simp add: slen_infinite)
    apply (simp add: slen_def)
    apply (insert lub_const [of x] unique_lub [of _ x _])
    by (auto simp add: lub)

    lemma contlub_sconc: "contlub (%y. x ooo y)"
    by (rule contlubI, insert contlub_sconc_lemma [of _ x], simp)

    lemma monofun_sconc: "monofun (%y. x ooo y)"
    by (simp add: monofun_def sconc_mono)

    lemma cont_sconc: "cont (%y. x ooo y)"
    apply (rule  monocontlub2cont)
     apply (rule monofunI, simp add: sconc_mono)
    by (rule contlub_sconc)

    (* ----------------------------------------------------------------------- *)
       section "constr_sconc"
    (* ----------------------------------------------------------------------- *)

    lemma constr_sconc_UUs [simp]: "constr_sconc UU s = s"
    by (simp add: constr_sconc_def inat_defs)

    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: inat_defs)
       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)

    declare eq_UU_iff [THEN sym, simp add]

    end

## scons

lemma scons_eq_UU:

      (a && s = UU) = (a = UU)

lemma scons_not_empty:

      [| a && x = UU; a ≠ UU |] ==> R

lemma stream_exhaust_eq:

      (x ≠ UU) = (∃a y. a ≠ UU ∧ x = a && y)

lemma stream_neq_UU:

      x ≠ UU ==> ∃a as. x = a && as ∧ a ≠ UU

lemma stream_inject_eq:

      [| a ≠ UU; b ≠ UU |] ==> (a && s = b && t) = (a = b ∧ s = t)

lemma stream_prefix:

      [| a && s << t; a ≠ UU |] ==> ∃b tt. t = b && tt ∧ b ≠ UU ∧ s << tt

lemma stream_prefix':

      b ≠ UU
      ==> x << b && z = (x = UU ∨ (∃a y. x = a && y ∧ a ≠ UU ∧ a << b ∧ y << z))

lemma stream_flat_prefix:

      [| x && xs << y && ys; x ≠ UU |] ==> x = y ∧ xs << ys

## stream_when

lemma stream_when_strictf:

      stream_when·UU·s = UU

## ft & rt

lemma ft_defin:

      s ≠ UU ==> ft·s ≠ UU

lemma rt_strict_rev:

      rt·s ≠ UU ==> s ≠ UU

lemma surjectiv_scons:

      ft·s && rt·s = s

lemma monofun_rt_mult:

      x << s ==> iterate i rt x << iterate i rt s

## stream_take

lemma stream_reach2:

      (LUB i. stream_take i·s) = s

lemma chain_stream_take:

      chain (%i. stream_take i·s)

lemma stream_take_prefix:

      stream_take n·s << s

lemma stream_take_more:

      stream_take n·x = x ==> stream_take (Suc n)·x = x

lemma stream_take_lemma3:

      [| x ≠ UU; stream_take n·(x && xs) = x && xs |] ==> stream_take n·xs = xs

lemma stream_take_lemma4:

      ∀x xs. stream_take n·xs = xs --> stream_take (Suc n)·(x && xs) = x && xs

lemma stream_take_idempotent:

      stream_take n·(stream_take n·s) = stream_take n·s

lemma stream_take_take_Suc:

      stream_take n·(stream_take (Suc n)·s) = stream_take n·s

lemma mono_stream_take_pred:

      stream_take (Suc n)·s1.0 << stream_take (Suc n)·s2.0
      ==> stream_take n·s1.0 << stream_take n·s2.0

lemma stream_take_lemma10:

      [| k ≤ n; stream_take n·s1.0 << stream_take n·s2.0 |]
      ==> stream_take k·s1.0 << stream_take k·s2.0

lemma stream_take_le_mono:

      k ≤ n ==> stream_take k·s1.0 << stream_take n·s1.0

lemma mono_stream_take:

      s1.0 << s2.0 ==> stream_take n·s1.0 << stream_take n·s2.0

lemma stream_take_take_less:

      stream_take k·(stream_take n·s) << stream_take k·s

## induction

lemma stream_finite_ind:

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

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)

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

## coinduction

lemma stream_coind_lemma2:

      ∀s1 s2. R s1 s2 --> ft·s1 = ft·s2 ∧ R (rt·s1) (rt·s2) ==> stream_bisim R

## stream_finite

lemma stream_finite_UU:

      stream_finite UU

lemma stream_finite_UU_rev:

      ¬ stream_finite s ==> s ≠ UU

lemma stream_finite_lemma1:

      stream_finite xs ==> stream_finite (x && xs)

lemma stream_finite_lemma2:

      [| x ≠ UU; stream_finite (x && xs) |] ==> stream_finite xs

lemma stream_finite_rt_eq:

      stream_finite (rt·s) = stream_finite s

lemma stream_finite_less:

      stream_finite s ==> ∀t. t << s --> stream_finite t

lemma stream_take_finite:

      stream_finite (stream_take n·s)

lemma adm_not_stream_finite:

      adm (%x. ¬ stream_finite x)

## slen

lemma slen_empty:

      #UU = 0

lemma slen_scons:

      x ≠ UU ==> #(x && xs) = iSuc #xs

lemma slen_less_1_eq:

      (#x < Fin (Suc 0)) = (x = UU)

lemma slen_empty_eq:

      (#x = 0) = (x = UU)

lemma slen_scons_eq:

      (Fin (Suc n) < #x) = (∃a y. x = a && y ∧ a ≠ UU ∧ Fin n < #y)

lemma slen_iSuc:

      #x = iSuc n --> (∃a y. x = a && y ∧ a ≠ UU ∧ #y = n)

lemma slen_stream_take_finite:

      #(stream_take n·s) ≠ ∞

lemma slen_scons_eq_rev:

      (#x < Fin (Suc (Suc n))) = (∀a y. x ≠ a && y ∨ a = UU ∨ #y < Fin (Suc n))

lemma slen_take_lemma4:

      stream_take n·s ≠ s ==> #(stream_take n·s) = Fin n

lemma slen_take_eq:

      ∀x. (Fin n < #x) = (stream_take n·x ≠ x)

lemma slen_take_eq_rev:

      (#x ≤ Fin n) = (stream_take n·x = x)

lemma slen_take_lemma1:

      #x = Fin n ==> stream_take n·x = x

lemma slen_rt_mono:

      #s2.0 ≤ #s1.0 ==> #(rt·s2.0) ≤ #(rt·s1.0)

lemma slen_take_lemma5:

      #(stream_take n·s) ≤ Fin n

lemma slen_take_lemma2:

      ∀x. ¬ stream_finite x --> #(stream_take i·x) = Fin i

lemma slen_infinite:

      stream_finite x = (#x ≠ ∞)

lemma slen_mono_lemma:

      stream_finite s ==> ∀t. s << t --> #s ≤ #t

lemma slen_mono:

      s << t ==> #s ≤ #t

lemma iterate_lemma:

      F·(iterate n F x) = iterate n F (F·x)

lemma slen_rt_mult:

      Fin (i + j) ≤ #x ==> Fin j ≤ #(iterate i rt x)

lemma slen_take_lemma3:

      [| Fin n ≤ #x; x << y |] ==> stream_take n·x = stream_take n·y

lemma slen_strict_mono_lemma:

      stream_finite t ==> ∀s. #s = #t ∧ s << t --> s = t

lemma slen_strict_mono:

      [| stream_finite t; s ≠ t; s << t |] ==> #s < #t

lemma stream_take_Suc_neq:

      stream_take (Suc n)·s ≠ s ==> stream_take n·s ≠ stream_take (Suc n)·s

## smap

lemma smap_unfold:

      smap = (LAM f t. case t of x && xs => f·x && smap·f·xs)

lemma smap_empty:

      smap·f·UU = UU

lemma smap_scons:

      x ≠ UU ==> smap·f·(x && xs) = f·x && smap·f·xs

## sfilter

lemma sfilter_unfold:

      sfilter =
      (LAM p s.
          case s of x && xs => If p·x then x && sfilter·p·xs else sfilter·p·xs fi)

lemma strict_sfilter:

      sfilter·UU = UU

lemma sfilter_empty:

      sfilter·f·UU = UU

lemma sfilter_scons:

      x ≠ UU
      ==> sfilter·f·(x && xs) = If f·x then x && sfilter·f·xs else sfilter·f·xs fi

## i_rt

lemma i_rt_UU:

      i_rt n UU = UU

lemma i_rt_0:

      i_rt 0 s = s

lemma i_rt_Suc:

      a ≠ UU ==> i_rt (Suc n) (a && s) = i_rt n s

lemma i_rt_Suc_forw:

      i_rt (Suc n) s = i_rt n (rt·s)

lemma i_rt_Suc_back:

      i_rt (Suc n) s = rt·(i_rt n s)

lemma i_rt_mono:

      x << s ==> i_rt n x << i_rt n s

lemma i_rt_ij_lemma:

      Fin (i + j) ≤ #x ==> Fin j ≤ #(i_rt i x)

lemma slen_i_rt_mono:

      #s2.0 ≤ #s1.0 ==> #(i_rt n s2.0) ≤ #(i_rt n s1.0)

lemma i_rt_take_lemma1:

      i_rt n (stream_take n·s) = UU

lemma i_rt_slen:

      (i_rt n s = UU) = (stream_take n·s = s)

lemma i_rt_lemma_slen:

      #s = Fin n ==> i_rt n s = UU

lemma stream_finite_i_rt:

      stream_finite (i_rt n s) = stream_finite s

lemma take_i_rt_len_lemma:

      ∀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

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

## i_th

lemma i_th_i_rt_step:

      [| i_th n s1.0 << i_th n s2.0; i_rt (Suc n) s1.0 << i_rt (Suc n) s2.0 |]
      ==> i_rt n s1.0 << i_rt n s2.0

lemma i_th_stream_take_Suc:

      i_th n (stream_take (Suc n)·s) = i_th n s

lemma i_th_last:

      i_th n s && UU = i_rt n (stream_take (Suc n)·s)

lemma i_th_last_eq:

      i_th n s1.0 = i_th n s2.0
      ==> i_rt n (stream_take (Suc n)·s1.0) = i_rt n (stream_take (Suc n)·s2.0)

lemma i_th_prefix_lemma:

      [| k ≤ n; stream_take (Suc n)·s1.0 << stream_take (Suc n)·s2.0 |]
      ==> i_th k s1.0 << i_th k s2.0

lemma take_i_rt_prefix_lemma1:

      [| stream_take (Suc n)·s1.0 << stream_take (Suc n)·s2.0;
         i_rt (Suc n) s1.0 << i_rt (Suc n) s2.0 |]
      ==> i_rt n s1.0 << i_rt n s2.0 ∧ stream_take n·s1.0 << stream_take n·s2.0

lemma take_i_rt_prefix_lemma:

      [| stream_take n·s1.0 << stream_take n·s2.0; i_rt n s1.0 << i_rt n s2.0 |]
      ==> s1.0 << s2.0

lemma streams_prefix_lemma:

      s1.0 << s2.0 =
      (stream_take n·s1.0 << stream_take n·s2.0 ∧ i_rt n s1.0 << i_rt n s2.0)

lemma streams_prefix_lemma1:

      [| stream_take n·s1.0 = stream_take n·s2.0; i_rt n s1.0 = i_rt n s2.0 |]
      ==> s1.0 = s2.0

## sconc

lemma UU_sconc:

      UU ooo s = s

lemma scons_neq_UU:

      a ≠ UU ==> a && s ≠ UU

lemma singleton_sconc:

      x ≠ UU ==> (x && UU) ooo y = x && y

lemma ex_sconc:

      #x = Fin k ==> ∃w. stream_take k·w = x ∧ i_rt k w = y

lemma rt_sconc1:

      Fin n = #x ==> i_rt n (x ooo y) = y

lemma sconc_inj2:

      [| Fin n = #x; x ooo y = x ooo z |] ==> y = z

lemma sconc_UU:

      s ooo UU = s

lemma stream_take_sconc:

      Fin n = #x ==> stream_take n·(x ooo y) = x

lemma scons_sconc:

      a ≠ UU ==> (a && x) ooo y = a && x ooo y

lemma ft_sconc:

      x ≠ UU ==> ft·(x ooo y) = ft·x

lemma sconc_assoc:

      (x ooo y) ooo z = x ooo y ooo z

lemma sconc_mono:

      y << y' ==> x ooo y << x ooo y'

lemma sconc_mono1:

      x << x ooo y

lemma empty_sconc:

      (x ooo y = UU) = (x = UU ∧ y = UU)

lemma rt_sconc:

      s ≠ UU ==> rt·(s ooo x) = rt·s ooo x

lemma i_th_sconc_lemma:

      Fin n < #x ==> i_th n (x ooo y) = i_th n x

lemma sconc_lemma:

      stream_take n·s ooo i_rt n s = s

### 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)

lemma i_th_stream_take_eq:

      ∀n. i_th n s1.0 = i_th n s2.0 ==> stream_take n·s1.0 = stream_take n·s2.0

lemma pointwise_eq_lemma:

      (!!n. i_th n s1.0 = i_th n s2.0) ==> s1.0 = s2.0

### finiteness

lemma slen_sconc_finite1:

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

lemma slen_sconc_infinite1:

      #x = ∞ ==> #(x ooo y) = ∞

lemma slen_sconc_infinite2:

      #y = ∞ ==> #(x ooo y) = ∞

lemma sconc_finite:

      (#x ≠ ∞ ∧ #y ≠ ∞) = (#(x ooo y) ≠ ∞)

lemma slen_sconc_mono3:

      [| Fin n = #x; Fin k = #(x ooo y) |] ==> n ≤ k

### finite slen

lemma slen_sconc:

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

### flat prefix

lemma sconc_prefix:

      s1.0 << s2.0 ==> ∃t. s1.0 ooo t = s2.0

### continuity

lemma chain_sconc:

      chain S ==> chain (%i. x ooo S i)

lemma chain_scons:

      chain S ==> chain (%i. a && S i)

lemma contlub_scons:

      contlub (Rep_CFun (op &&·a))

lemma contlub_scons_lemma:

      chain S ==> (LUB i. a && S i) = a && lub (range S)

lemma finite_lub_sconc:

      [| chain Y; stream_finite x |] ==> (LUB i. x ooo Y i) = x ooo lub (range Y)

lemma contlub_sconc_lemma:

      chain Y ==> (LUB i. x ooo Y i) = x ooo lub (range Y)

lemma contlub_sconc:

      contlub (op ooo x)

lemma monofun_sconc:

      monofun (op ooo x)

lemma cont_sconc:

      cont (op ooo x)

## constr_sconc

lemma constr_sconc_UUs:

      constr_sconc UU s = s

lemma

      x ooo y = constr_sconc x y