--- a/src/HOLCF/ex/Stream.thy Tue Sep 06 19:22:31 2005 +0200
+++ b/src/HOLCF/ex/Stream.thy Tue Sep 06 19:28:58 2005 +0200
@@ -1,53 +1,55 @@
-(* Title: HOLCF/ex/Stream.thy
+(* Title: HOLCF/ex/Stream.thy
ID: $Id$
- Author: Franz Regensburger, David von Oheimb, Borislav Gajanovic
-
-General Stream domain.
+ Author: Franz Regensburger, David von Oheimb, Borislav Gajanovic
*)
-theory Stream imports HOLCF Nat_Infinity begin
+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 \<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)
+ 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 \<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>"
+ 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 *)
- sconc :: "'a stream => 'a stream => 'a stream" (infixr "ooo" 65)
+ 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"
+ 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$(i_rt i s)"
+ i_rt_def: "i_rt == %i s. iterate i rt s"
+ i_th_def: "i_th == %i s. ft$(i_rt i s)"
- sconc_def: "s1 ooo s2 == case #s1 of
+ sconc_def: "s1 ooo s2 == case #s1 of
Fin n \<Rightarrow> (SOME s. (stream_take n$s=s1) & (i_rt n s = s2))
- | \<infinity> \<Rightarrow> s1"
+ | \<infinity> \<Rightarrow> s1"
- constr_sconc_def: "constr_sconc s1 s2 == case #s1 of
- Fin n \<Rightarrow> constr_sconc' n s1 s2
+ constr_sconc_def: "constr_sconc s1 s2 == case #s1 of
+ Fin n \<Rightarrow> constr_sconc' n s1 s2
| \<infinity> \<Rightarrow> s1"
-primrec
+primrec
constr_sconc'_0: "constr_sconc' 0 s1 s2 = s2"
- constr_sconc'_Suc: "constr_sconc' (Suc n) s1 s2 = ft$s1 &&
+ constr_sconc'_Suc: "constr_sconc' (Suc n) s1 s2 = ft$s1 &&
constr_sconc' n (rt$s1) s2"
@@ -76,14 +78,14 @@
"[| 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:
+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 =
+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)
@@ -95,7 +97,7 @@
by (insert stream_prefix' [of y "x&&xs" ys],force)
*)
-lemma stream_flat_prefix:
+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)
@@ -144,7 +146,7 @@
(* ----------------------------------------------------------------------- *)
-section "stream_take";
+section "stream_take"
lemma stream_reach2: "(LUB i. stream_take i$s) = s"
@@ -154,7 +156,7 @@
by (simp add: chain_iterate)
lemma chain_stream_take: "chain (%i. stream_take i$s)"
-apply (rule chainI)
+apply (rule chainI)
apply (rule monofun_cfun_fun)
apply (simp add: stream.take_def del: iterate_Suc)
by (rule chainE, simp add: chain_iterate)
@@ -165,13 +167,13 @@
apply (rule is_ub_thelub)
by (simp only: chain_stream_take)
-lemma stream_take_more [rule_format]:
+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]:
+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)*)
@@ -179,37 +181,37 @@
apply (erule_tac x="x" in allE)
by (erule_tac x="xs" in allE,simp)
-lemma stream_take_lemma4:
+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]:
+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)$s) =
+lemma stream_take_take_Suc [rule_format, simp]:
+ "ALL s. stream_take n$(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:
+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)$s1"
+by (insert monofun_cfun_arg [of "stream_take (Suc n)$s1"
"stream_take (Suc n)$s2" "stream_take n"], auto)
(*
-lemma mono_stream_take_pred:
+lemma mono_stream_take_pred:
"stream_take (Suc n)$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$s2
+ "ALL k<=n. stream_take n$s1 << stream_take n$s2
--> stream_take k$s1 << stream_take k$s2"
apply (induct_tac n,simp,clarsimp)
apply (case_tac "k=Suc n",blast)
@@ -242,14 +244,14 @@
section "induction"
-lemma stream_finite_ind:
+lemma stream_finite_ind:
"[| stream_finite x; P UU; !!a s. [| a ~= UU; P s |] ==> P (a && s) |] ==> P x"
apply (simp add: stream.finite_def,auto)
apply (erule subst)
by (drule stream.finite_ind [of P _ x], auto)
-lemma stream_finite_ind2:
-"[| P UU; !! x. x ~= UU ==> P (x && UU); !! y z s. [| y ~= UU; z ~= UU; P s |] ==> P (y && z && s )|] ==>
+lemma stream_finite_ind2:
+"[| P UU; !! x. x ~= UU ==> P (x && UU); !! y z s. [| y ~= UU; z ~= UU; P s |] ==> P (y && z && s )|] ==>
!s. P (stream_take n$s)"
apply (rule nat_induct2 [of _ n],auto)
apply (case_tac "s=UU",clarsimp)
@@ -259,7 +261,7 @@
apply (case_tac "y=UU",clarsimp)
by (drule stream_exhaust_eq [THEN iffD1],clarsimp)
-lemma stream_ind2:
+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)
@@ -363,7 +365,7 @@
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
+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>)"
@@ -386,26 +388,26 @@
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 (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]:
+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]:
+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)$s) =
+lemma stream_take_take_Suc [simp]: "stream_take n$(stream_take (Suc n)$s) =
stream_take n$s"
apply (simp add: po_eq_conv,auto)
apply (simp add: stream_take_take_less)
@@ -440,7 +442,7 @@
apply (rule stream.casedist [of s1])
by (rule stream.casedist [of s2],simp+)+
-lemma slen_take_lemma5: "#(stream_take n$s) <= Fin n";
+lemma slen_take_lemma5: "#(stream_take n$s) <= Fin n"
apply (case_tac "stream_take n$s = s")
apply (simp add: slen_take_eq_rev)
by (simp add: slen_take_lemma4)
@@ -463,12 +465,12 @@
lemma slen_mono: "s << t ==> #s <= #t"
apply (case_tac "stream_finite t")
-apply (frule stream_finite_less)
+apply (frule stream_finite_less)
apply (erule_tac x="s" in allE, simp)
apply (drule slen_mono_lemma, auto)
by (simp add: slen_def)
-lemma iterate_lemma: "F$(iterate n F x) = iterate n F (F$x)"
+lemma iterate_lemma: "F$(iterate n F x) = iterate n F (F$x)"
by (insert iterate_Suc2 [of n F x], auto)
lemma slen_rt_mult [rule_format]: "!x. Fin (i + j) <= #x --> Fin j <= #(iterate i rt x)"
@@ -480,7 +482,7 @@
apply (simp add: inat_defs split:inat_splits)
by (simp add: iterate_lemma)
-lemma slen_take_lemma3 [rule_format]:
+lemma slen_take_lemma3 [rule_format]:
"!(x::'a::flat stream) y. Fin n <= #x --> x << y --> stream_take n\<cdot>x = stream_take n\<cdot>y"
apply (induct_tac n, auto)
apply (case_tac "x=UU", auto)
@@ -493,7 +495,7 @@
apply (drule stream.inverts,auto)
by (drule ax_flat [rule_format], simp)
-lemma slen_strict_mono_lemma:
+lemma slen_strict_mono_lemma:
"stream_finite t ==> !s. #(s::'a::flat stream) = #t & s << t --> s = t"
apply (erule stream_finite_ind, auto)
apply (drule eq_UU_iff [THEN iffD2], simp)
@@ -507,7 +509,7 @@
apply (simp add: slen_mono)
by (drule slen_strict_mono_lemma, auto)
-lemma stream_take_Suc_neq: "stream_take (Suc n)$s ~=s ==>
+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")
@@ -540,7 +542,7 @@
section "sfilter"
-lemma sfilter_unfold:
+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)
@@ -554,9 +556,9 @@
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"
+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)
@@ -592,14 +594,14 @@
by (drule slen_rt_mono,simp)
lemma i_rt_take_lemma1 [rule_format]: "ALL s. i_rt n (stream_take n$s) = UU"
-apply (induct_tac n);
+apply (induct_tac n)
apply (simp add: i_rt_Suc_back,auto)
apply (case_tac "s=UU",auto)
by (drule stream_exhaust_eq [THEN iffD1],auto)
lemma i_rt_slen: "(i_rt n s = UU) = (stream_take n$s = s)"
apply auto
- apply (insert i_rt_ij_lemma [of n "Suc 0" s]);
+ apply (insert i_rt_ij_lemma [of n "Suc 0" s])
apply (subgoal_tac "#(i_rt n s)=0")
apply (case_tac "stream_take n$s = s",simp+)
apply (insert slen_take_eq [rule_format,of n s],simp)
@@ -616,7 +618,7 @@
by (simp add: i_rt_Suc_back stream_finite_rt_eq)+
lemma take_i_rt_len_lemma: "ALL sl x j t. Fin sl = #x & n <= sl &
- #(stream_take n$x) = Fin t & #(i_rt n x)= Fin j
+ #(stream_take n$x) = Fin t & #(i_rt n x)= Fin j
--> Fin (j + t) = #x"
apply (induct_tac n,auto)
apply (simp add: inat_defs)
@@ -634,7 +636,7 @@
apply force
by (simp add: inat_defs split:inat_splits)
-lemma take_i_rt_len:
+lemma take_i_rt_len:
"[| Fin sl = #x; n <= sl; #(stream_take n$x) = Fin t; #(i_rt n x) = Fin j |] ==>
Fin (j + t) = #x"
by (blast intro: take_i_rt_len_lemma [rule_format])
@@ -645,7 +647,7 @@
(* ----------------------------------------------------------------------- *)
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_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)
@@ -653,7 +655,7 @@
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]:
+lemma i_th_stream_take_Suc [rule_format]:
"ALL s. i_th n (stream_take (Suc n)$s) = i_th n s"
apply (induct_tac n,auto)
apply (simp add: i_th_def)
@@ -669,14 +671,14 @@
apply (simp add: i_th_def i_rt_Suc_back [symmetric])
by (simp add: i_rt_take_lemma1)
-lemma i_th_last_eq:
+lemma i_th_last_eq:
"i_th n s1 = i_th n s2 ==> i_rt n (stream_take (Suc n)$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)$s2 |] ==>
+"[| k <= n; stream_take (Suc n)$s1 << stream_take (Suc n)$s2 |] ==>
i_th k s1 << i_th k s2"
apply (insert i_th_stream_take_Suc [of k s1, THEN sym])
apply (insert i_th_stream_take_Suc [of k s2, THEN sym],auto)
@@ -685,29 +687,29 @@
apply (rule i_rt_mono)
by (blast intro: stream_take_lemma10)
-lemma take_i_rt_prefix_lemma1:
+lemma take_i_rt_prefix_lemma1:
"stream_take (Suc n)$s1 << stream_take (Suc n)$s2 ==>
- i_rt (Suc n) s1 << i_rt (Suc n) s2 ==>
+ i_rt (Suc n) s1 << i_rt (Suc n) s2 ==>
i_rt n s1 << i_rt n s2 & stream_take n$s1 << stream_take n$s2"
apply auto
apply (insert i_th_prefix_lemma [of n n s1 s2])
apply (rule i_th_i_rt_step,auto)
by (drule mono_stream_take_pred,simp)
-lemma take_i_rt_prefix_lemma:
+lemma take_i_rt_prefix_lemma:
"[| stream_take n$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$s2 &
+apply (subgoal_tac "stream_take 0$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$s2 & i_rt n s1 << i_rt n s2)";
+lemma streams_prefix_lemma: "(s1 << s2) =
+ (stream_take n$s1 << stream_take n$s2 & i_rt n s1 << i_rt n s2)"
apply auto
apply (simp add: monofun_cfun_arg)
apply (simp add: i_rt_mono)
@@ -738,7 +740,7 @@
apply (case_tac "xa=UU",simp)
by (drule stream_exhaust_eq [THEN iffD1],auto)
-lemma ex_sconc [rule_format]:
+lemma ex_sconc [rule_format]:
"ALL k y. #x = Fin k --> (EX w. stream_take k$w = x & i_rt k w = y)"
apply (case_tac "#x")
apply (rule stream_finite_ind [of x],auto)
@@ -748,7 +750,7 @@
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";
+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)
@@ -777,7 +779,7 @@
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 (simp add: sconc_def)
apply (rule someI2_ex)
apply (drule ex_sconc,simp)
apply (rule someI2_ex,auto)
@@ -818,7 +820,7 @@
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 (insert sconc_mono1 [of x y])
by (insert eq_UU_iff [THEN iffD2, of x],auto)
(* ----------------------------------------------------------------------- *)
@@ -826,7 +828,7 @@
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]:
+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)
@@ -850,18 +852,18 @@
subsection "pointwise equality"
(* ----------------------------------------------------------------------- *)
-lemma ex_last_stream_take_scons: "stream_take (Suc n)$s =
+lemma ex_last_stream_take_scons: "stream_take (Suc n)$s =
stream_take n$s ooo i_rt n (stream_take (Suc n)$s)"
by (insert sconc_lemma [of n "stream_take (Suc n)$s"],simp)
-lemma i_th_stream_take_eq:
+lemma i_th_stream_take_eq:
"!!n. ALL n. i_th n s1 = i_th n s2 ==> stream_take n$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)$s1) =
+ apply (subgoal_tac "i_rt na (stream_take (Suc na)$s1) =
i_rt na (stream_take (Suc na)$s2)")
- apply (subgoal_tac "stream_take (Suc na)$s2 =
+ apply (subgoal_tac "stream_take (Suc na)$s2 =
stream_take na$s2 ooo i_rt na (stream_take (Suc na)$s2)")
apply (insert ex_last_stream_take_scons,simp)
apply blast
@@ -928,15 +930,15 @@
lemma sconc_prefix: "(s1::'a::flat stream) << s2 ==> EX t. s1 ooo t = s2"
apply (case_tac "#s1")
- apply (subgoal_tac "stream_take nat$s1 = stream_take nat$s2");
+ apply (subgoal_tac "stream_take nat$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 slen_take_lemma3 [of _ s1 s2])
apply (simp+,rule_tac x="UU" in exI)
-apply (insert slen_take_lemma3 [of _ s1 s2]);
+apply (insert slen_take_lemma3 [of _ s1 s2])
by (rule stream.take_lemmas,simp)
(* ----------------------------------------------------------------------- *)
@@ -957,14 +959,14 @@
apply (insert contlub_scons [of a])
by (simp only: contlub_def)
-lemma finite_lub_sconc: "chain Y ==> (stream_finite x) ==>
+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:
+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)
@@ -974,8 +976,8 @@
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 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)
@@ -983,7 +985,7 @@
lemma cont_sconc: "cont (%y. x ooo y)"
apply (rule monocontlub2cont)
apply (rule monofunI, simp add: sconc_mono)
-by (rule contlub_sconc);
+by (rule contlub_sconc)
(* ----------------------------------------------------------------------- *)