# Theory Stream

theory Stream
imports HOLCF Extended_Nat
```(*  Title:      HOL/HOLCF/Library/Stream.thy
Author:     Franz Regensburger, David von Oheimb, Borislav Gajanovic
*)

section ‹General Stream domain›

theory Stream
imports HOLCF "HOL-Library.Extended_Nat"
begin

default_sort pcpo

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

definition
smap :: "('a → 'b) → 'a stream → 'b stream" where
"smap = fix⋅(Λ h f s. case s of x && xs ⇒ f⋅x && h⋅f⋅xs)"

definition
sfilter :: "('a → tr) → 'a stream → 'a stream" where
"sfilter = fix⋅(Λ h p s. case s of x && xs ⇒
If p⋅x then x && h⋅p⋅xs else h⋅p⋅xs)"

definition
slen :: "'a stream ⇒ enat"  ("#_" [1000] 1000) where
"#s = (if stream_finite s then enat (LEAST n. stream_take n⋅s = s) else ∞)"

(* concatenation *)

definition
i_rt :: "nat ⇒ 'a stream ⇒ 'a stream" where (* chops the first i elements *)
"i_rt = (λi s. iterate i⋅rt⋅s)"

definition
i_th :: "nat ⇒ 'a stream ⇒ 'a" where (* the i-th element *)
"i_th = (λi s. ft⋅(i_rt i s))"

definition
sconc :: "'a stream ⇒ 'a stream ⇒ 'a stream"  (infixr "ooo" 65) where
"s1 ooo s2 = (case #s1 of
enat n ⇒ (SOME s. (stream_take n⋅s = s1) ∧ (i_rt n s = s2))
| ∞     ⇒ 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
enat n ⇒ constr_sconc' n s1 s2
| ∞    ⇒ 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 ⟷ (∃a y. a ≠ UU ∧ x = a && y)"
by (cases x, auto)

lemma stream_neq_UU: "x ≠ UU ⟹ ∃a a_s. x = a && a_s ∧ a ≠ UU"
by (simp add: stream_exhaust_eq,auto)

lemma stream_prefix:
"⟦a && s ⊑ t; a ≠ UU⟧ ⟹ ∃b tt. t = b && tt ∧ b ≠ UU ∧ s ⊑ tt"
by (cases t, auto)

lemma stream_prefix':
"b ≠ UU ⟹ x ⊑ b && z =
(x = UU ∨ (∃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)
apply (drule ax_flat,simp)
done

(* ----------------------------------------------------------------------- *)
(* 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 and 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 (λi. 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)
apply (simp only: chain_stream_take)
done

lemma stream_take_more [rule_format]:
"∀x. stream_take n⋅x = x ⟶ stream_take (Suc n)⋅x = x"
apply (induct_tac n,auto)
apply (case_tac "x=UU",auto)
apply (drule stream_exhaust_eq [THEN iffD1],auto)
done

lemma stream_take_lemma3 [rule_format]:
"∀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)
apply (erule_tac x="xs" in allE,simp)
done

lemma stream_take_lemma4:
"∀x xs. stream_take n⋅xs = xs ⟶ stream_take (Suc n)⋅(x && xs) = x && xs"
by auto

lemma stream_take_idempotent [rule_format, simp]:
"∀s. stream_take n⋅(stream_take n⋅s) = stream_take n⋅s"
apply (induct_tac n, auto)
apply (case_tac "s=UU", auto)
apply (drule stream_exhaust_eq [THEN iffD1], auto)
done

lemma stream_take_take_Suc [rule_format, simp]:
"∀s. stream_take n⋅(stream_take (Suc n)⋅s) = stream_take n⋅s"
apply (induct_tac n, auto)
apply (case_tac "s=UU", auto)
apply (drule stream_exhaust_eq [THEN iffD1], auto)
done

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]:
"∀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)
apply (drule mono_stream_take_pred,simp)
done

lemma stream_take_le_mono : "k ≤ n ⟹ stream_take k⋅s1 ⊑ stream_take n⋅s1"
apply (insert chain_stream_take [of s1])
apply (drule chain_mono,auto)
done

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

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

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)
apply auto
done

(* ----------------------------------------------------------------------- *)
(* 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)
apply (simp add: stream_take_lemma4)
done

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)
apply (erule stream_take_lemma3,simp)
done

lemma stream_finite_rt_eq: "stream_finite (rt⋅s) = stream_finite s"
apply (cases s, auto)
apply (rule stream_finite_lemma1, simp)
apply (rule stream_finite_lemma2,simp)
apply assumption
done

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)
apply (rule stream_finite_lemma1, simp)
done

lemma stream_take_finite [simp]: "stream_finite (stream_take n⋅s)"
apply (simp add: stream.finite_def)
apply (rule_tac x="n" in exI,simp)
done

lemma adm_not_stream_finite: "adm (λx. ¬ stream_finite x)"
apply (erule contrapos_nn)
apply (erule (1) stream_finite_less [rule_format])
done

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

section "slen"

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

lemma slen_scons [simp]: "x ≠ ⊥ ⟹ #(x && xs) = eSuc (#xs)"
apply (case_tac "stream_finite (x && xs)")
apply (simp add: slen_def, auto)
apply (simp add: stream.finite_def, auto simp add: eSuc_enat)
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)
apply (drule stream_finite_lemma1,auto)
done

lemma slen_less_1_eq: "(#x < enat (Suc 0)) = (x = ⊥)"
by (cases x) (auto simp add: enat_0 eSuc_enat[THEN sym])

lemma slen_empty_eq: "(#x = 0) = (x = ⊥)"
by (cases x) auto

lemma slen_scons_eq: "(enat (Suc n) < #x) = (? a y. x = a && y ∧ a ≠ ⊥ ∧ enat 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_eSuc: "#x = eSuc n ⟶ (∃a y. x = a && y ∧ a ≠ ⊥ ∧ #y = n)"
by (cases x) auto

lemma slen_stream_take_finite [simp]: "#(stream_take n⋅s) ≠ ∞"
by (simp add: slen_def)

lemma slen_scons_eq_rev: "#x < enat (Suc (Suc n)) ⟷ (∀a y. x ≠ a && y ∨ a = ⊥ ∨ #y < enat (Suc n))"
apply (cases x, auto)
apply (simp add: zero_enat_def)
apply (case_tac "#stream") apply (simp_all add: eSuc_enat)
apply (case_tac "#stream") apply (simp_all add: eSuc_enat)
done

lemma slen_take_lemma4 [rule_format]:
"∀s. stream_take n⋅s ≠ s ⟶ #(stream_take n⋅s) = enat n"
apply (induct n, auto simp add: enat_0)
apply (case_tac "s=UU", simp)
apply (drule stream_exhaust_eq [THEN iffD1], auto simp add: eSuc_enat)
done

(*
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: "∀x. enat n < #x ⟷ stream_take n⋅x ≠ x"
apply (induct_tac n, auto)
apply (simp add: enat_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 eSuc_enat)
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")
apply simp_all
done

lemma slen_take_eq_rev: "#x ≤ enat n ⟷ stream_take n⋅x = x"
by (simp add: linorder_not_less [symmetric] slen_take_eq)

lemma slen_take_lemma1: "#x = enat 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 (cases s1)
apply (cases s2, simp+)+
done

lemma slen_take_lemma5: "#(stream_take n⋅s) ≤ enat n"
apply (case_tac "stream_take n⋅s = s")
apply (simp add: slen_take_eq_rev)
apply (simp add: slen_take_lemma4)
done

lemma slen_take_lemma2: "∀x. ¬ stream_finite x ⟶ #(stream_take i⋅x) = enat i"
apply (simp add: stream.finite_def, auto)
apply (simp add: slen_take_lemma4)
done

lemma slen_infinite: "stream_finite x ⟷ #x ≠ ∞"
by (simp add: slen_def)

lemma slen_mono_lemma: "stream_finite s ⟹ ∀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)
apply (simp add: slen_def)
done

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. enat (i + j) ≤ #x ⟶ enat j ≤ #(iterate i⋅rt⋅x)"
apply (induct i, auto)
apply (case_tac "x = UU", auto simp add: zero_enat_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: eSuc_enat)
apply (simp add: iterate_lemma)
done

lemma slen_take_lemma3 [rule_format]:
"∀(x::'a::flat stream) y. enat 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: zero_enat_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)
apply (drule ax_flat, simp)
done

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)
apply (simp add: slen_take_lemma4 eSuc_enat)
done

(* ----------------------------------------------------------------------- *)
(* 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 [where 'a='a and 'b='b, THEN eq_reflection, 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)"
by (insert sfilter_def [where 'a='a, THEN eq_reflection, THEN fix_eq2], auto)

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

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"
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: "enat (i + j) ≤ #x ⟹ enat 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)
apply (drule slen_rt_mono,simp)
done

lemma i_rt_take_lemma1 [rule_format]: "∀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)
apply (drule stream_exhaust_eq [THEN iffD1],auto)
done

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_enat_def)
apply (simp add: slen_take_eq)
apply (cases "#s")
using i_rt_take_lemma1 [of n s]
apply (simp_all add: zero_enat_def)
done

lemma i_rt_lemma_slen: "#s=enat 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)
apply (simp add: i_rt_Suc_back stream_finite_rt_eq)+
done

lemma take_i_rt_len_lemma: "∀sl x j t. enat sl = #x ∧ n ≤ sl ∧
#(stream_take n⋅x) = enat t ∧ #(i_rt n x) = enat j
⟶ enat (j + t) = #x"
apply (induct n, auto)
apply (simp add: zero_enat_def)
apply (case_tac "x=UU",auto)
apply (simp add: zero_enat_def)
apply (drule stream_exhaust_eq [THEN iffD1],clarsimp)
apply (subgoal_tac "∃k. enat 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: eSuc_def split: enat.splits)
apply (simp add: eSuc_def split: enat.splits)
apply (simp only: the_equality)
apply (simp add: eSuc_def split: enat.splits)
apply force
apply (simp add: eSuc_def split: enat.splits)
done

lemma take_i_rt_len:
"⟦enat sl = #x; n ≤ sl; #(stream_take n⋅x) = enat t; #(i_rt n x) = enat j⟧ ⟹
enat (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]:
"∀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)
apply (simp add: i_th_def i_rt_Suc_forw)
done

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])
apply auto
done

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)
apply (blast intro: stream_take_lemma10)
done

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)
apply (drule mono_stream_take_pred,simp)
done

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)
apply simp
done

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)
apply (erule take_i_rt_prefix_lemma,simp)
done

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)
apply blast+
done

(* ----------------------------------------------------------------------- *)
section "sconc"
(* ----------------------------------------------------------------------- *)

lemma UU_sconc [simp]: " UU ooo s = s "
by (simp add: sconc_def zero_enat_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_enat_def eSuc_def split: enat.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]:
"∀k y. #x = enat k ⟶ (∃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_enat_def eSuc_def split: enat.splits)
apply (erule_tac x="y" in allE,auto)
apply (rule_tac x="a && w" in exI,auto)
done

lemma rt_sconc1: "enat n = #x ⟹ i_rt n (x ooo y) = y"
apply (simp add: sconc_def split: enat.splits, arith?,auto)
apply (rule someI2_ex,auto)
apply (drule ex_sconc,simp)
done

lemma sconc_inj2: "⟦enat n = #x; x ooo y = x ooo z⟧ ⟹ y = z"
apply (frule_tac y=y in rt_sconc1)
apply (auto elim: rt_sconc1)
done

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)
apply (simp add: sconc_def)
done

lemma stream_take_sconc [simp]: "enat n = #x ⟹ stream_take n⋅(x ooo y) = x"
apply (simp add: sconc_def)
apply (cases "#x")
apply auto
apply (rule someI2_ex, auto)
apply (drule ex_sconc,simp)
done

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 eSuc_enat)
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 && xa" in exI, auto)
apply (case_tac "xaa=UU",auto)
apply (drule stream_exhaust_eq [THEN iffD1],auto)
apply (drule streams_prefix_lemma1,simp+)
apply (simp add: sconc_def)
done

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 ⟹ cont (λy. x ooo y)"
by (erule stream_finite_ind, simp_all)

lemma cont_sconc_lemma2: "¬ stream_finite x ⟹ cont (λy. x ooo y)"
by (simp add: sconc_def slen_def)

lemma cont_sconc: "cont (λy. 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])
apply auto
done

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

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]:
"∀x y. enat n < #x ⟶ i_th n (x ooo y) = i_th n x"
apply (induct_tac n, auto)
apply (simp add: enat_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")
apply simp_all
done

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

lemma sconc_lemma [rule_format, simp]: "∀s. stream_take n⋅s ooo i_rt n s = s"
apply (induct_tac n,auto)
apply (case_tac "s=UU",auto)
apply (drule stream_exhaust_eq [THEN iffD1],auto)
done

(* ----------------------------------------------------------------------- *)
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. ∀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]: "∀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) = ∞; enat n = #x⟧ ⟹ #y = ∞"
apply (case_tac "#y ≠ ∞",auto)
apply (drule_tac y=y in rt_sconc1)
apply (insert stream_finite_i_rt [of n "x ooo y"])
apply (simp add: slen_infinite)
done

lemma slen_sconc_infinite1: "#x=∞ ⟹ #(x ooo y) = ∞"
by (simp add: sconc_def)

lemma slen_sconc_infinite2: "#y=∞ ⟹ #(x ooo y) = ∞"
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 (fastforce simp add: slen_infinite,auto)
by (simp add: sconc_def)

lemma sconc_finite: "#x ≠ ∞ ∧ #y ≠ ∞ ⟷ #(x ooo y) ≠ ∞"
apply auto
apply (metis not_infinity_eq slen_sconc_finite1)
apply (metis not_infinity_eq slen_sconc_infinite1)
apply (metis not_infinity_eq slen_sconc_infinite2)
done

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

lemma slen_sconc_mono3: "⟦enat n = #x; enat 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: "⟦enat n = #x; enat m = #y⟧ ⟹ #(x ooo y) = enat (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. enat j = #(x ooo y)" "x ooo y" n n m],simp)
apply (insert slen_sconc_mono3 [of n x _ y],simp)
apply (insert sconc_finite [of x y],auto)
done

(* ----------------------------------------------------------------------- *)
subsection "flat prefix"
(* ----------------------------------------------------------------------- *)

lemma sconc_prefix: "(s1::'a::flat stream) << s2 ⟹ ∃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])
apply (rule stream.take_lemma,simp)
done

(* ----------------------------------------------------------------------- *)
subsection "continuity"
(* ----------------------------------------------------------------------- *)

lemma chain_sconc: "chain S ⟹ chain (λi. (x ooo S i))"
by (simp add: chain_def,auto simp add: sconc_mono)

lemma chain_scons: "chain S ⟹ chain (λi. a && S i)"
apply (simp add: chain_def,auto)
apply (rule monofun_cfun_arg,simp)
done

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)")
apply (force,blast dest: contlub_scons_lemma chain_sconc)
done

lemma contlub_sconc_lemma:
"chain Y ⟹ (LUB i. x ooo Y i) = (x ooo (LUB i. Y i))"
apply (case_tac "#x=∞")
apply (simp add: sconc_def)
apply (drule finite_lub_sconc,auto simp add: slen_infinite)
done

lemma monofun_sconc: "monofun (λy. 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_enat_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: eSuc_enat)
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)
apply (drule slen_take_lemma1,auto)
done

end
```