--- a/src/HOLCF/ex/Stream.thy Tue Sep 07 15:59:16 2004 +0200
+++ b/src/HOLCF/ex/Stream.thy Tue Sep 07 16:02:42 2004 +0200
@@ -1,29 +1,1012 @@
(* Title: HOLCF/ex/Stream.thy
ID: $Id$
- Author: Franz Regensburger, David von Oheimb
+ Author: Franz Regensburger, David von Oheimb, Borislav Gajanovic
General Stream domain.
-TODO: should be integrated with HOLCF/Streams
*)
-Stream = HOLCF + Nat_Infinity +
+theory Stream = HOLCF + Nat_Infinity:
-domain 'a stream = "&&" (ft::'a) (lazy rt::'a stream) (infixr 65)
+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)
+ smap :: "('a \<rightarrow> 'b) \<rightarrow> 'a stream \<rightarrow> 'b stream"
+ sfilter :: "('a \<rightarrow> tr) \<rightarrow> 'a stream \<rightarrow> 'a stream"
+ slen :: "'a stream \<Rightarrow> inat" ("#_" [1000] 1000)
defs
- smap_def "smap == fix\\<cdot>(\\<Lambda> h f s. case s of x && xs => f\\<cdot>x && h\\<cdot>f\\<cdot>xs)"
- sfilter_def "sfilter == fix\\<cdot>(\\<Lambda> h p s. case s of x && xs =>
- If p\\<cdot>x then x && h\\<cdot>p\\<cdot>xs else h\\<cdot>p\\<cdot>xs fi)"
- slen_def "#s == if stream_finite s
- then Fin (LEAST n. stream_take n\\<cdot>s = s) else \\<infinity>"
+ smap_def: "smap \<equiv> fix\<cdot>(\<Lambda> h f s. case s of x && xs \<Rightarrow> f\<cdot>x && h\<cdot>f\<cdot>xs)"
+ sfilter_def: "sfilter \<equiv> 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 fi)"
+ slen_def: "#s \<equiv> if stream_finite s
+ then Fin (LEAST n. stream_take n\<cdot>s = s) else \<infinity>"
+
+(* concatenation *)
+
+consts
+
+ i_rt :: "nat => 'a stream => 'a stream" (* chops the first i elements *)
+ i_th :: "nat => 'a stream => 'a" (* the i-th element ä*)
+
+ 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$(i_rt i s)"
+
+ sconc_def: "s1 ooo s2 == case #s1 of
+ Fin n \<Rightarrow> (SOME s. (stream_take n$s=s1) & (i_rt n s = s2))
+ | \<infinity> \<Rightarrow> s1"
+
+ constr_sconc_def: "constr_sconc s1 s2 == case #s1 of
+ Fin n \<Rightarrow> constr_sconc' n s1 s2
+ | \<infinity> \<Rightarrow> 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, 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$s)"
+apply (simp add: stream.finite_def)
+by (rule_tac x="n" in exI,simp)
+
+lemma adm_not_stream_finite: "adm (%x. ~ stream_finite x)"
+apply (rule admI2, auto)
+apply (drule stream_finite_less,drule is_ub_thelub)
+by auto
+
+
+
+(* ----------------------------------------------------------------------- *)
+(* theorems about stream length *)
+(* ----------------------------------------------------------------------- *)
+
+
+section "slen"
+
+lemma slen_empty [simp]: "#\<bottom> = 0"
+apply (simp add: slen_def stream.finite_def)
+by (simp add: inat_defs Least_equality)
+
+lemma slen_scons [simp]: "x ~= \<bottom> ==> #(x&&xs) = iSuc (#xs)"
+apply (case_tac "stream_finite (x && xs)")
+apply (simp add: slen_def, auto)
+apply (simp add: stream.finite_def, auto)
+apply (rule Least_Suc2,auto)
+apply (drule sym)
+apply (drule sym scons_eq_UU [THEN iffD1],simp)
+apply (erule stream_finite_lemma2, simp)
+apply (simp add: slen_def, auto)
+by (drule stream_finite_lemma1,auto)
+
+lemma slen_less_1_eq: "(#x < Fin (Suc 0)) = (x = \<bottom>)"
+by (rule stream.casedist [of x], auto simp del: iSuc_Fin
+ simp add: Fin_0 iSuc_Fin[THEN sym] i0_iless_iSuc iSuc_mono)
+
+lemma slen_empty_eq: "(#x = 0) = (x = \<bottom>)"
+by (rule stream.casedist [of x], auto)
+
+lemma slen_scons_eq: "(Fin (Suc n) < #x) = (? a y. x = a && y & a ~= \<bottom> & Fin n < #y)"
+apply (auto, case_tac "x=UU",auto)
+apply (drule stream_exhaust_eq [THEN iffD1], auto)
+apply (rule_tac x="a" in exI)
+apply (rule_tac x="y" in exI, simp)
+by (simp add: inat_defs split:inat_splits)+
+
+lemma slen_iSuc: "#x = iSuc n --> (? a y. x = a&&y & a ~= \<bottom> & #y = n)"
+by (rule stream.casedist [of x], auto)
+
+lemma slen_stream_take_finite [simp]: "#(stream_take n$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 (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$s)")
+ apply (erule ssubst)
+ apply (rule_tac monofun_cfun_arg)
+ apply (insert chain_stream_take [of s])
+by (simp add: chain_def,simp)
+*)
+
+lemma slen_take_eq: "ALL x. (Fin n < #x) = (stream_take n\<cdot>x ~= x)"
+apply (induct_tac n, auto)
+apply (simp add: Fin_0, clarsimp)
+apply (drule not_sym)
+apply (drule slen_empty_eq [THEN iffD1], simp)
+apply (case_tac "x=UU", simp)
+apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)
+apply (erule_tac x="y" in allE, auto)
+apply (simp add: inat_defs split:inat_splits)
+apply (case_tac "x=UU", simp)
+apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)
+apply (erule_tac x="y" in allE, simp)
+by (simp add: inat_defs split:inat_splits)
+
+lemma slen_take_eq_rev: "(#x <= Fin n) = (stream_take n\<cdot>x = x)"
+by (simp add: ile_def slen_take_eq)
+
+lemma slen_take_lemma1: "#x = Fin n ==> stream_take n\<cdot>x = x"
+by (rule slen_take_eq_rev [THEN iffD1], auto)
+
+lemma slen_rt_mono: "#s2 <= #s1 ==> #(rt$s2) <= #(rt$s1)"
+apply (rule stream.casedist [of s1])
+ by (rule stream.casedist [of s2],simp+)+
+
+lemma slen_take_lemma5: "#(stream_take n$s) <= Fin n";
+apply (case_tac "stream_take n$s = s")
+ apply (simp add: slen_take_eq_rev)
+by (simp add: slen_take_lemma4)
+
+lemma slen_take_lemma2: "!x. ~stream_finite x --> #(stream_take i\<cdot>x) = Fin i"
+apply (simp add: stream.finite_def, auto)
+by (simp add: slen_take_lemma4)
+
+lemma slen_infinite: "stream_finite x = (#x ~= Infty)"
+by (simp add: slen_def)
+
+lemma slen_mono_lemma: "stream_finite s ==> ALL t. s << t --> #s <= #t"
+apply (erule stream_finite_ind [of s], auto)
+apply (case_tac "t=UU", auto)
+apply (drule eq_UU_iff [THEN iffD2])
+apply (drule scons_eq_UU [THEN iffD2], simp)
+apply (drule stream_exhaust_eq [THEN iffD1], auto)
+apply (erule_tac x="y" in allE, auto)
+by (drule stream.inverts, auto)
+
+lemma slen_mono: "s << t ==> #s <= #t"
+apply (case_tac "stream_finite t")
+apply (frule stream_finite_less)
+apply (erule_tac x="s" in allE, simp)
+apply (drule slen_mono_lemma, auto)
+by (simp add: slen_def)
+
+lemma iterate_lemma: "F$(iterate n F x) = iterate n F (F$x)"
+by (insert iterate_Suc2 [of n F x], auto)
+
+lemma slen_rt_mult [rule_format]: "!x. Fin (i + j) <= #x --> Fin j <= #(iterate i rt x)"
+apply (induct_tac i, auto)
+apply (case_tac "x=UU", auto)
+apply (simp add: inat_defs)
+apply (drule stream_exhaust_eq [THEN iffD1], auto)
+apply (erule_tac x="y" in allE, auto)
+apply (simp add: inat_defs split:inat_splits)
+by (simp add: iterate_lemma)
+
+lemma slen_take_lemma3 [rule_format]:
+ "!(x::'a::flat stream) y. Fin n <= #x --> x << y --> stream_take n\<cdot>x = stream_take n\<cdot>y"
+apply (induct_tac n, auto)
+apply (case_tac "x=UU", auto)
+apply (simp add: inat_defs)
+apply (simp add: Suc_ile_eq)
+apply (case_tac "y=UU", clarsimp)
+apply (drule eq_UU_iff [THEN iffD2],simp)
+apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)+
+apply (erule_tac x="ya" in allE, simp)
+apply (drule stream.inverts,auto)
+by (drule ax_flat [rule_format], simp)
+
+lemma slen_strict_mono_lemma:
+ "stream_finite t ==> !s. #(s::'a::flat stream) = #t & s << t --> s = t"
+apply (erule stream_finite_ind, auto)
+apply (drule eq_UU_iff [THEN iffD2], simp)
+apply (case_tac "sa=UU", auto)
+apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)
+apply (drule stream.inverts, simp, simp, clarsimp)
+by (drule ax_flat [rule_format], simp)
+
+lemma slen_strict_mono: "[|stream_finite t; s ~= t; s << (t::'a::flat stream) |] ==> #s < #t"
+apply (intro ilessI1, auto)
+apply (simp add: slen_mono)
+by (drule slen_strict_mono_lemma, auto)
+
+lemma stream_take_Suc_neq: "stream_take (Suc n)$s ~=s ==>
+ stream_take n$s ~= stream_take (Suc n)$s"
+apply auto
+apply (subgoal_tac "stream_take n$s ~=s")
+ apply (insert slen_take_lemma4 [of n s],auto)
+apply (rule stream.casedist [of s],simp)
+apply (simp add: inat_defs split:inat_splits)
+by (simp add: slen_take_lemma4)
+
+(* ----------------------------------------------------------------------- *)
+(* theorems about smap *)
+(* ----------------------------------------------------------------------- *)
+
+
+section "smap"
+
+lemma smap_unfold: "smap = (\<Lambda> f t. case t of x&&xs \<Rightarrow> f$x && smap$f$xs)"
+by (insert smap_def [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 fi)"
+by (insert sfilter_def [THEN fix_eq2], auto)
+
+lemma strict_sfilter: "sfilter\<cdot>\<bottom> = \<bottom>"
+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\<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 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$w = x & i_rt k w = y)"
+apply (case_tac "#x")
+ apply (rule stream_finite_ind [of x],auto)
+ apply (simp add: stream.finite_def)
+ apply (drule slen_take_lemma1,blast)
+ apply (simp add: inat_defs split:inat_splits)+
+apply (erule_tac x="y" in allE,auto)
+by (rule_tac x="a && w" in exI,auto)
+
+lemma rt_sconc1: "Fin n = #x ==> i_rt n (x ooo y) = y";
+apply (simp add: sconc_def inat_defs split:inat_splits, arith?,auto)
+apply (rule someI2_ex,auto)
+by (drule ex_sconc,simp)
+
+lemma sconc_inj2: "\<lbrakk>Fin n = #x; x ooo y = x ooo z\<rbrakk> \<Longrightarrow> y = z"
+apply (frule_tac y=y in rt_sconc1)
+by (auto elim: rt_sconc1)
+
+lemma sconc_UU [simp]:"s ooo UU = s"
+apply (case_tac "#s")
+ apply (simp add: sconc_def inat_defs)
+ apply (rule someI2_ex)
+ apply (rule_tac x="s" in exI)
+ apply auto
+ apply (drule slen_take_lemma1,auto)
+ apply (simp add: i_rt_lemma_slen)
+ apply (drule slen_take_lemma1,auto)
+ apply (simp add: i_rt_slen)
+by (simp add: sconc_def inat_defs)
+
+lemma stream_take_sconc [simp]: "Fin n = #x ==> stream_take n$(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)
+
+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 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
-
-