--- a/src/HOLCF/ex/Stream.thy Mon May 24 11:29:49 2010 -0700
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,965 +0,0 @@
-(* Title: HOLCF/ex/Stream.thy
- Author: Franz Regensburger, David von Oheimb, Borislav Gajanovic
-*)
-
-header {* General Stream domain *}
-
-theory Stream
-imports HOLCF Nat_Infinity
-begin
-
-domain 'a stream = scons (ft::'a) (lazy rt::"'a stream") (infixr "&&" 65)
-
-definition
- smap :: "('a \<rightarrow> 'b) \<rightarrow> 'a stream \<rightarrow> 'b stream" where
- "smap = fix\<cdot>(\<Lambda> h f s. case s of x && xs \<Rightarrow> f\<cdot>x && h\<cdot>f\<cdot>xs)"
-
-definition
- sfilter :: "('a \<rightarrow> tr) \<rightarrow> 'a stream \<rightarrow> 'a stream" where
- "sfilter = fix\<cdot>(\<Lambda> h p s. case s of x && xs \<Rightarrow>
- If p\<cdot>x then x && h\<cdot>p\<cdot>xs else h\<cdot>p\<cdot>xs fi)"
-
-definition
- slen :: "'a stream \<Rightarrow> inat" ("#_" [1000] 1000) where
- "#s = (if stream_finite s then Fin (LEAST n. stream_take n\<cdot>s = s) else \<infinity>)"
-
-
-(* concatenation *)
-
-definition
- i_rt :: "nat => 'a stream => 'a stream" where (* chops the first i elements *)
- "i_rt = (%i s. iterate i$rt$s)"
-
-definition
- i_th :: "nat => 'a stream => 'a" where (* the i-th element *)
- "i_th = (%i s. ft$(i_rt i s))"
-
-definition
- sconc :: "'a stream => 'a stream => 'a stream" (infixr "ooo" 65) where
- "s1 ooo s2 = (case #s1 of
- Fin n \<Rightarrow> (SOME s. (stream_take n$s=s1) & (i_rt n s = s2))
- | \<infinity> \<Rightarrow> s1)"
-
-primrec constr_sconc' :: "nat => 'a stream => 'a stream => 'a stream"
-where
- constr_sconc'_0: "constr_sconc' 0 s1 s2 = s2"
-| constr_sconc'_Suc: "constr_sconc' (Suc n) s1 s2 = ft$s1 &&
- constr_sconc' n (rt$s1) s2"
-
-definition
- constr_sconc :: "'a stream => 'a stream => 'a stream" where (* constructive *)
- "constr_sconc s1 s2 = (case #s1 of
- Fin n \<Rightarrow> constr_sconc' n s1 s2
- | \<infinity> \<Rightarrow> s1)"
-
-
-(* ----------------------------------------------------------------------- *)
-(* theorems about scons *)
-(* ----------------------------------------------------------------------- *)
-
-
-section "scons"
-
-lemma scons_eq_UU: "(a && s = UU) = (a = UU)"
-by simp
-
-lemma scons_not_empty: "[| a && x = UU; a ~= UU |] ==> R"
-by simp
-
-lemma stream_exhaust_eq: "(x ~= UU) = (EX a y. a ~= UU & x = a && y)"
-by (cases x, auto)
-
-lemma stream_neq_UU: "x~=UU ==> EX a a_s. x=a&&a_s & a~=UU"
-by (simp add: stream_exhaust_eq,auto)
-
-lemma stream_prefix:
- "[| a && s << t; a ~= UU |] ==> EX b tt. t = b && tt & b ~= UU & s << tt"
-by (cases t, auto)
-
-lemma stream_prefix':
- "b ~= UU ==> x << b && z =
- (x = UU | (EX a y. x = a && y & a ~= UU & a << b & y << z))"
-by (cases x, auto)
-
-
-(*
-lemma stream_prefix1: "[| x<<y; xs<<ys |] ==> x&&xs << y&&ys"
-by (insert stream_prefix' [of y "x&&xs" ys],force)
-*)
-
-lemma stream_flat_prefix:
- "[| x && xs << y && ys; (x::'a::flat) ~= UU|] ==> x = y & xs << ys"
-apply (case_tac "y=UU",auto)
-by (drule ax_flat,simp)
-
-
-
-
-(* ----------------------------------------------------------------------- *)
-(* theorems about stream_when *)
-(* ----------------------------------------------------------------------- *)
-
-section "stream_when"
-
-
-lemma stream_when_strictf: "stream_when$UU$s=UU"
-by (cases s, auto)
-
-
-
-(* ----------------------------------------------------------------------- *)
-(* theorems about ft and rt *)
-(* ----------------------------------------------------------------------- *)
-
-
-section "ft & rt"
-
-
-lemma ft_defin: "s~=UU ==> ft$s~=UU"
-by simp
-
-lemma rt_strict_rev: "rt$s~=UU ==> s~=UU"
-by auto
-
-lemma surjectiv_scons: "(ft$s)&&(rt$s)=s"
-by (cases s, auto)
-
-lemma monofun_rt_mult: "x << s ==> iterate i$rt$x << iterate i$rt$s"
-by (rule monofun_cfun_arg)
-
-
-
-(* ----------------------------------------------------------------------- *)
-(* theorems about stream_take *)
-(* ----------------------------------------------------------------------- *)
-
-
-section "stream_take"
-
-
-lemma stream_reach2: "(LUB i. stream_take i$s) = s"
-by (rule stream.reach)
-
-lemma chain_stream_take: "chain (%i. stream_take i$s)"
-by simp
-
-lemma stream_take_prefix [simp]: "stream_take n$s << s"
-apply (insert stream_reach2 [of s])
-apply (erule subst) back
-apply (rule is_ub_thelub)
-by (simp only: chain_stream_take)
-
-lemma stream_take_more [rule_format]:
- "ALL x. stream_take n$x = x --> stream_take (Suc n)$x = x"
-apply (induct_tac n,auto)
-apply (case_tac "x=UU",auto)
-by (drule stream_exhaust_eq [THEN iffD1],auto)
-
-lemma stream_take_lemma3 [rule_format]:
- "ALL x xs. x~=UU --> stream_take n$(x && xs) = x && xs --> stream_take n$xs=xs"
-apply (induct_tac n,clarsimp)
-(*apply (drule sym, erule scons_not_empty, simp)*)
-apply (clarify, rule stream_take_more)
-apply (erule_tac x="x" in allE)
-by (erule_tac x="xs" in allE,simp)
-
-lemma stream_take_lemma4:
- "ALL x xs. stream_take n$xs=xs --> stream_take (Suc n)$(x && xs) = x && xs"
-by auto
-
-lemma stream_take_idempotent [rule_format, simp]:
- "ALL s. stream_take n$(stream_take n$s) = stream_take n$s"
-apply (induct_tac n, auto)
-apply (case_tac "s=UU", auto)
-by (drule stream_exhaust_eq [THEN iffD1], auto)
-
-lemma stream_take_take_Suc [rule_format, simp]:
- "ALL s. stream_take n$(stream_take (Suc n)$s) =
- stream_take n$s"
-apply (induct_tac n, auto)
-apply (case_tac "s=UU", auto)
-by (drule stream_exhaust_eq [THEN iffD1], auto)
-
-lemma mono_stream_take_pred:
- "stream_take (Suc n)$s1 << stream_take (Suc n)$s2 ==>
- stream_take n$s1 << stream_take n$s2"
-by (insert monofun_cfun_arg [of "stream_take (Suc n)$s1"
- "stream_take (Suc n)$s2" "stream_take n"], auto)
-(*
-lemma mono_stream_take_pred:
- "stream_take (Suc n)$s1 << stream_take (Suc n)$s2 ==>
- stream_take n$s1 << stream_take n$s2"
-by (drule mono_stream_take [of _ _ n],simp)
-*)
-
-lemma stream_take_lemma10 [rule_format]:
- "ALL k<=n. stream_take n$s1 << stream_take n$s2
- --> stream_take k$s1 << stream_take k$s2"
-apply (induct_tac n,simp,clarsimp)
-apply (case_tac "k=Suc n",blast)
-apply (erule_tac x="k" in allE)
-by (drule mono_stream_take_pred,simp)
-
-lemma stream_take_le_mono : "k<=n ==> stream_take k$s1 << stream_take n$s1"
-apply (insert chain_stream_take [of s1])
-by (drule chain_mono,auto)
-
-lemma mono_stream_take: "s1 << s2 ==> stream_take n$s1 << stream_take n$s2"
-by (simp add: monofun_cfun_arg)
-
-(*
-lemma stream_take_prefix [simp]: "stream_take n$s << s"
-apply (subgoal_tac "s=(LUB n. stream_take n$s)")
- apply (erule ssubst, rule is_ub_thelub)
- apply (simp only: chain_stream_take)
-by (simp only: stream_reach2)
-*)
-
-lemma stream_take_take_less:"stream_take k$(stream_take n$s) << stream_take k$s"
-by (rule monofun_cfun_arg,auto)
-
-
-(* ------------------------------------------------------------------------- *)
-(* special induction rules *)
-(* ------------------------------------------------------------------------- *)
-
-
-section "induction"
-
-lemma stream_finite_ind:
- "[| stream_finite x; P UU; !!a s. [| a ~= UU; P s |] ==> P (a && s) |] ==> P x"
-apply (simp add: stream.finite_def,auto)
-apply (erule subst)
-by (drule stream.finite_induct [of P _ x], auto)
-
-lemma stream_finite_ind2:
-"[| P UU; !! x. x ~= UU ==> P (x && UU); !! y z s. [| y ~= UU; z ~= UU; P s |] ==> P (y && z && s )|] ==>
- !s. P (stream_take n$s)"
-apply (rule nat_less_induct [of _ n],auto)
-apply (case_tac n, auto)
-apply (case_tac nat, auto)
-apply (case_tac "s=UU",clarsimp)
-apply (drule stream_exhaust_eq [THEN iffD1],clarsimp)
-apply (case_tac "s=UU",clarsimp)
-apply (drule stream_exhaust_eq [THEN iffD1],clarsimp)
-apply (case_tac "y=UU",clarsimp)
-by (drule stream_exhaust_eq [THEN iffD1],clarsimp)
-
-lemma stream_ind2:
-"[| adm P; P UU; !!a. a ~= UU ==> P (a && UU); !!a b s. [| a ~= UU; b ~= UU; P s |] ==> P (a && b && s) |] ==> P x"
-apply (insert stream.reach [of x],erule subst)
-apply (erule admD, rule chain_stream_take)
-apply (insert stream_finite_ind2 [of P])
-by simp
-
-
-
-(* ----------------------------------------------------------------------- *)
-(* simplify use of coinduction *)
-(* ----------------------------------------------------------------------- *)
-
-
-section "coinduction"
-
-lemma stream_coind_lemma2: "!s1 s2. R s1 s2 --> ft$s1 = ft$s2 & R (rt$s1) (rt$s2) ==> stream_bisim R"
- apply (simp add: stream.bisim_def,clarsimp)
- apply (drule spec, drule spec, drule (1) mp)
- apply (case_tac "x", simp)
- apply (case_tac "x'", simp)
-by auto
-
-
-
-(* ----------------------------------------------------------------------- *)
-(* theorems about stream_finite *)
-(* ----------------------------------------------------------------------- *)
-
-
-section "stream_finite"
-
-lemma stream_finite_UU [simp]: "stream_finite UU"
-by (simp add: stream.finite_def)
-
-lemma stream_finite_UU_rev: "~ stream_finite s ==> s ~= UU"
-by (auto simp add: stream.finite_def)
-
-lemma stream_finite_lemma1: "stream_finite xs ==> stream_finite (x && xs)"
-apply (simp add: stream.finite_def,auto)
-apply (rule_tac x="Suc n" in exI)
-by (simp add: stream_take_lemma4)
-
-lemma stream_finite_lemma2: "[| x ~= UU; stream_finite (x && xs) |] ==> stream_finite xs"
-apply (simp add: stream.finite_def, auto)
-apply (rule_tac x="n" in exI)
-by (erule stream_take_lemma3,simp)
-
-lemma stream_finite_rt_eq: "stream_finite (rt$s) = stream_finite s"
-apply (cases s, auto)
-apply (rule stream_finite_lemma1, simp)
-by (rule stream_finite_lemma2,simp)
-
-lemma stream_finite_less: "stream_finite s ==> !t. t<<s --> stream_finite t"
-apply (erule stream_finite_ind [of s], auto)
-apply (case_tac "t=UU", auto)
-apply (drule stream_exhaust_eq [THEN iffD1],auto)
-apply (erule_tac x="y" in allE, simp)
-by (rule stream_finite_lemma1, simp)
-
-lemma stream_take_finite [simp]: "stream_finite (stream_take n$s)"
-apply (simp add: stream.finite_def)
-by (rule_tac x="n" in exI,simp)
-
-lemma adm_not_stream_finite: "adm (%x. ~ stream_finite x)"
-apply (rule adm_upward)
-apply (erule contrapos_nn)
-apply (erule (1) stream_finite_less [rule_format])
-done
-
-
-
-(* ----------------------------------------------------------------------- *)
-(* theorems about stream length *)
-(* ----------------------------------------------------------------------- *)
-
-
-section "slen"
-
-lemma slen_empty [simp]: "#\<bottom> = 0"
-by (simp add: slen_def stream.finite_def zero_inat_def Least_equality)
-
-lemma slen_scons [simp]: "x ~= \<bottom> ==> #(x&&xs) = iSuc (#xs)"
-apply (case_tac "stream_finite (x && xs)")
-apply (simp add: slen_def, auto)
-apply (simp add: stream.finite_def, auto simp add: iSuc_Fin)
-apply (rule Least_Suc2, auto)
-(*apply (drule sym)*)
-(*apply (drule sym scons_eq_UU [THEN iffD1],simp)*)
-apply (erule stream_finite_lemma2, simp)
-apply (simp add: slen_def, auto)
-by (drule stream_finite_lemma1,auto)
-
-lemma slen_less_1_eq: "(#x < Fin (Suc 0)) = (x = \<bottom>)"
-by (cases x, auto simp add: Fin_0 iSuc_Fin[THEN sym])
-
-lemma slen_empty_eq: "(#x = 0) = (x = \<bottom>)"
-by (cases x, auto)
-
-lemma slen_scons_eq: "(Fin (Suc n) < #x) = (? a y. x = a && y & a ~= \<bottom> & Fin n < #y)"
-apply (auto, case_tac "x=UU",auto)
-apply (drule stream_exhaust_eq [THEN iffD1], auto)
-apply (case_tac "#y") apply simp_all
-apply (case_tac "#y") apply simp_all
-done
-
-lemma slen_iSuc: "#x = iSuc n --> (? a y. x = a&&y & a ~= \<bottom> & #y = n)"
-by (cases x, auto)
-
-lemma slen_stream_take_finite [simp]: "#(stream_take n$s) ~= \<infinity>"
-by (simp add: slen_def)
-
-lemma slen_scons_eq_rev: "(#x < Fin (Suc (Suc n))) = (!a y. x ~= a && y | a = \<bottom> | #y < Fin (Suc n))"
- apply (cases x, auto)
- apply (simp add: zero_inat_def)
- apply (case_tac "#stream") apply (simp_all add: iSuc_Fin)
- apply (case_tac "#stream") apply (simp_all add: iSuc_Fin)
-done
-
-lemma slen_take_lemma4 [rule_format]:
- "!s. stream_take n$s ~= s --> #(stream_take n$s) = Fin n"
-apply (induct n, auto simp add: Fin_0)
-apply (case_tac "s=UU", simp)
-by (drule stream_exhaust_eq [THEN iffD1], auto simp add: iSuc_Fin)
-
-(*
-lemma stream_take_idempotent [simp]:
- "stream_take n$(stream_take n$s) = stream_take n$s"
-apply (case_tac "stream_take n$s = s")
-apply (auto,insert slen_take_lemma4 [of n s]);
-by (auto,insert slen_take_lemma1 [of "stream_take n$s" n],simp)
-
-lemma stream_take_take_Suc [simp]: "stream_take n$(stream_take (Suc n)$s) =
- stream_take n$s"
-apply (simp add: po_eq_conv,auto)
- apply (simp add: stream_take_take_less)
-apply (subgoal_tac "stream_take n$s = stream_take n$(stream_take n$s)")
- apply (erule ssubst)
- apply (rule_tac monofun_cfun_arg)
- apply (insert chain_stream_take [of s])
-by (simp add: chain_def,simp)
-*)
-
-lemma slen_take_eq: "ALL x. (Fin n < #x) = (stream_take n\<cdot>x ~= x)"
-apply (induct_tac n, auto)
-apply (simp add: Fin_0, clarsimp)
-apply (drule not_sym)
-apply (drule slen_empty_eq [THEN iffD1], simp)
-apply (case_tac "x=UU", simp)
-apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)
-apply (erule_tac x="y" in allE, auto)
-apply (simp_all add: not_less iSuc_Fin)
-apply (case_tac "#y") apply simp_all
-apply (case_tac "x=UU", simp)
-apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)
-apply (erule_tac x="y" in allE, simp)
-apply (case_tac "#y") by simp_all
-
-lemma slen_take_eq_rev: "(#x <= Fin n) = (stream_take n\<cdot>x = x)"
-by (simp add: linorder_not_less [symmetric] slen_take_eq)
-
-lemma slen_take_lemma1: "#x = Fin n ==> stream_take n\<cdot>x = x"
-by (rule slen_take_eq_rev [THEN iffD1], auto)
-
-lemma slen_rt_mono: "#s2 <= #s1 ==> #(rt$s2) <= #(rt$s1)"
-apply (cases s1)
- by (cases s2, simp+)+
-
-lemma slen_take_lemma5: "#(stream_take n$s) <= Fin n"
-apply (case_tac "stream_take n$s = s")
- apply (simp add: slen_take_eq_rev)
-by (simp add: slen_take_lemma4)
-
-lemma slen_take_lemma2: "!x. ~stream_finite x --> #(stream_take i\<cdot>x) = Fin i"
-apply (simp add: stream.finite_def, auto)
-by (simp add: slen_take_lemma4)
-
-lemma slen_infinite: "stream_finite x = (#x ~= Infty)"
-by (simp add: slen_def)
-
-lemma slen_mono_lemma: "stream_finite s ==> ALL t. s << t --> #s <= #t"
-apply (erule stream_finite_ind [of s], auto)
-apply (case_tac "t=UU", auto)
-apply (drule stream_exhaust_eq [THEN iffD1], auto)
-done
-
-lemma slen_mono: "s << t ==> #s <= #t"
-apply (case_tac "stream_finite t")
-apply (frule stream_finite_less)
-apply (erule_tac x="s" in allE, simp)
-apply (drule slen_mono_lemma, auto)
-by (simp add: slen_def)
-
-lemma iterate_lemma: "F$(iterate n$F$x) = iterate n$F$(F$x)"
-by (insert iterate_Suc2 [of n F x], auto)
-
-lemma slen_rt_mult [rule_format]: "!x. Fin (i + j) <= #x --> Fin j <= #(iterate i$rt$x)"
-apply (induct i, auto)
-apply (case_tac "x=UU", auto simp add: zero_inat_def)
-apply (drule stream_exhaust_eq [THEN iffD1], auto)
-apply (erule_tac x="y" in allE, auto)
-apply (simp add: not_le) apply (case_tac "#y") apply (simp_all add: iSuc_Fin)
-by (simp add: iterate_lemma)
-
-lemma slen_take_lemma3 [rule_format]:
- "!(x::'a::flat stream) y. Fin n <= #x --> x << y --> stream_take n\<cdot>x = stream_take n\<cdot>y"
-apply (induct_tac n, auto)
-apply (case_tac "x=UU", auto)
-apply (simp add: zero_inat_def)
-apply (simp add: Suc_ile_eq)
-apply (case_tac "y=UU", clarsimp)
-apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)+
-apply (erule_tac x="ya" in allE, simp)
-by (drule ax_flat, simp)
-
-lemma slen_strict_mono_lemma:
- "stream_finite t ==> !s. #(s::'a::flat stream) = #t & s << t --> s = t"
-apply (erule stream_finite_ind, auto)
-apply (case_tac "sa=UU", auto)
-apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)
-by (drule ax_flat, simp)
-
-lemma slen_strict_mono: "[|stream_finite t; s ~= t; s << (t::'a::flat stream) |] ==> #s < #t"
-by (auto simp add: slen_mono less_le dest: slen_strict_mono_lemma)
-
-lemma stream_take_Suc_neq: "stream_take (Suc n)$s ~=s ==>
- stream_take n$s ~= stream_take (Suc n)$s"
-apply auto
-apply (subgoal_tac "stream_take n$s ~=s")
- apply (insert slen_take_lemma4 [of n s],auto)
-apply (cases s, simp)
-by (simp add: slen_take_lemma4 iSuc_Fin)
-
-(* ----------------------------------------------------------------------- *)
-(* theorems about smap *)
-(* ----------------------------------------------------------------------- *)
-
-
-section "smap"
-
-lemma smap_unfold: "smap = (\<Lambda> f t. case t of x&&xs \<Rightarrow> f$x && smap$f$xs)"
-by (insert smap_def [where 'a='a and 'b='b, THEN eq_reflection, THEN fix_eq2], auto)
-
-lemma smap_empty [simp]: "smap\<cdot>f\<cdot>\<bottom> = \<bottom>"
-by (subst smap_unfold, simp)
-
-lemma smap_scons [simp]: "x~=\<bottom> ==> smap\<cdot>f\<cdot>(x&&xs) = (f\<cdot>x)&&(smap\<cdot>f\<cdot>xs)"
-by (subst smap_unfold, force)
-
-
-
-(* ----------------------------------------------------------------------- *)
-(* theorems about sfilter *)
-(* ----------------------------------------------------------------------- *)
-
-section "sfilter"
-
-lemma sfilter_unfold:
- "sfilter = (\<Lambda> p s. case s of x && xs \<Rightarrow>
- If p\<cdot>x then x && sfilter\<cdot>p\<cdot>xs else sfilter\<cdot>p\<cdot>xs fi)"
-by (insert sfilter_def [where 'a='a, THEN eq_reflection, 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"
- by (induct n) (simp_all add: i_rt_def)
-
-lemma i_rt_0 [simp]: "i_rt 0 s = s"
-by (simp add: i_rt_def)
-
-lemma i_rt_Suc [simp]: "a ~= UU ==> i_rt (Suc n) (a&&s) = i_rt n s"
-by (simp add: i_rt_def iterate_Suc2 del: iterate_Suc)
-
-lemma i_rt_Suc_forw: "i_rt (Suc n) s = i_rt n (rt$s)"
-by (simp only: i_rt_def iterate_Suc2)
-
-lemma i_rt_Suc_back:"i_rt (Suc n) s = rt$(i_rt n s)"
-by (simp only: i_rt_def,auto)
-
-lemma i_rt_mono: "x << s ==> i_rt n x << i_rt n s"
-by (simp add: i_rt_def monofun_rt_mult)
-
-lemma i_rt_ij_lemma: "Fin (i + j) <= #x ==> Fin j <= #(i_rt i x)"
-by (simp add: i_rt_def slen_rt_mult)
-
-lemma slen_i_rt_mono: "#s2 <= #s1 ==> #(i_rt n s2) <= #(i_rt n s1)"
-apply (induct_tac n,auto)
-apply (simp add: i_rt_Suc_back)
-by (drule slen_rt_mono,simp)
-
-lemma i_rt_take_lemma1 [rule_format]: "ALL s. i_rt n (stream_take n$s) = UU"
-apply (induct_tac n)
- apply (simp add: i_rt_Suc_back,auto)
-apply (case_tac "s=UU",auto)
-by (drule stream_exhaust_eq [THEN iffD1],auto)
-
-lemma i_rt_slen: "(i_rt n s = UU) = (stream_take n$s = s)"
-apply auto
- apply (insert i_rt_ij_lemma [of n "Suc 0" s])
- apply (subgoal_tac "#(i_rt n s)=0")
- apply (case_tac "stream_take n$s = s",simp+)
- apply (insert slen_take_eq [rule_format,of n s],simp)
- apply (cases "#s") apply (simp_all add: zero_inat_def)
- apply (simp add: slen_take_eq)
- apply (cases "#s")
- using i_rt_take_lemma1 [of n s]
- apply (simp_all add: zero_inat_def)
- done
-
-lemma i_rt_lemma_slen: "#s=Fin n ==> i_rt n s = UU"
-by (simp add: i_rt_slen slen_take_lemma1)
-
-lemma stream_finite_i_rt [simp]: "stream_finite (i_rt n s) = stream_finite s"
-apply (induct_tac n, auto)
- apply (cases s, auto simp del: i_rt_Suc)
-by (simp add: i_rt_Suc_back stream_finite_rt_eq)+
-
-lemma take_i_rt_len_lemma: "ALL sl x j t. Fin sl = #x & n <= sl &
- #(stream_take n$x) = Fin t & #(i_rt n x)= Fin j
- --> Fin (j + t) = #x"
-apply (induct n, auto)
- apply (simp add: zero_inat_def)
-apply (case_tac "x=UU",auto)
- apply (simp add: zero_inat_def)
-apply (drule stream_exhaust_eq [THEN iffD1],clarsimp)
-apply (subgoal_tac "EX k. Fin k = #y",clarify)
- apply (erule_tac x="k" in allE)
- apply (erule_tac x="y" in allE,auto)
- apply (erule_tac x="THE p. Suc p = t" in allE,auto)
- apply (simp add: iSuc_def split: inat.splits)
- apply (simp add: iSuc_def split: inat.splits)
- apply (simp only: the_equality)
- apply (simp add: iSuc_def split: inat.splits)
- apply force
-apply (simp add: iSuc_def split: inat.splits)
-done
-
-lemma take_i_rt_len:
-"[| Fin sl = #x; n <= sl; #(stream_take n$x) = Fin t; #(i_rt n x) = Fin j |] ==>
- Fin (j + t) = #x"
-by (blast intro: take_i_rt_len_lemma [rule_format])
-
-
-(* ----------------------------------------------------------------------- *)
- section "i_th"
-(* ----------------------------------------------------------------------- *)
-
-lemma i_th_i_rt_step:
-"[| i_th n s1 << i_th n s2; i_rt (Suc n) s1 << i_rt (Suc n) s2 |] ==>
- i_rt n s1 << i_rt n s2"
-apply (simp add: i_th_def i_rt_Suc_back)
-apply (cases "i_rt n s1", simp)
-apply (cases "i_rt n s2", auto)
-done
-
-lemma i_th_stream_take_Suc [rule_format]:
- "ALL s. i_th n (stream_take (Suc n)$s) = i_th n s"
-apply (induct_tac n,auto)
- apply (simp add: i_th_def)
- apply (case_tac "s=UU",auto)
- apply (drule stream_exhaust_eq [THEN iffD1],auto)
-apply (case_tac "s=UU",simp add: i_th_def)
-apply (drule stream_exhaust_eq [THEN iffD1],auto)
-by (simp add: i_th_def i_rt_Suc_forw)
-
-lemma i_th_last: "i_th n s && UU = i_rt n (stream_take (Suc n)$s)"
-apply (insert surjectiv_scons [of "i_rt n (stream_take (Suc n)$s)"])
-apply (rule i_th_stream_take_Suc [THEN subst])
-apply (simp add: i_th_def i_rt_Suc_back [symmetric])
-by (simp add: i_rt_take_lemma1)
-
-lemma i_th_last_eq:
-"i_th n s1 = i_th n s2 ==> i_rt n (stream_take (Suc n)$s1) = i_rt n (stream_take (Suc n)$s2)"
-apply (insert i_th_last [of n s1])
-apply (insert i_th_last [of n s2])
-by auto
-
-lemma i_th_prefix_lemma:
-"[| k <= n; stream_take (Suc n)$s1 << stream_take (Suc n)$s2 |] ==>
- i_th k s1 << i_th k s2"
-apply (insert i_th_stream_take_Suc [of k s1, THEN sym])
-apply (insert i_th_stream_take_Suc [of k s2, THEN sym],auto)
-apply (simp add: i_th_def)
-apply (rule monofun_cfun, auto)
-apply (rule i_rt_mono)
-by (blast intro: stream_take_lemma10)
-
-lemma take_i_rt_prefix_lemma1:
- "stream_take (Suc n)$s1 << stream_take (Suc n)$s2 ==>
- i_rt (Suc n) s1 << i_rt (Suc n) s2 ==>
- i_rt n s1 << i_rt n s2 & stream_take n$s1 << stream_take n$s2"
-apply auto
- apply (insert i_th_prefix_lemma [of n n s1 s2])
- apply (rule i_th_i_rt_step,auto)
-by (drule mono_stream_take_pred,simp)
-
-lemma take_i_rt_prefix_lemma:
-"[| stream_take n$s1 << stream_take n$s2; i_rt n s1 << i_rt n s2 |] ==> s1 << s2"
-apply (case_tac "n=0",simp)
-apply (auto)
-apply (subgoal_tac "stream_take 0$s1 << stream_take 0$s2 &
- i_rt 0 s1 << i_rt 0 s2")
- defer 1
- apply (rule zero_induct,blast)
- apply (blast dest: take_i_rt_prefix_lemma1)
-by simp
-
-lemma streams_prefix_lemma: "(s1 << s2) =
- (stream_take n$s1 << stream_take n$s2 & i_rt n s1 << i_rt n s2)"
-apply auto
- apply (simp add: monofun_cfun_arg)
- apply (simp add: i_rt_mono)
-by (erule take_i_rt_prefix_lemma,simp)
-
-lemma streams_prefix_lemma1:
- "[| stream_take n$s1 = stream_take n$s2; i_rt n s1 = i_rt n s2 |] ==> s1 = s2"
-apply (simp add: po_eq_conv,auto)
- apply (insert streams_prefix_lemma)
- by blast+
-
-
-(* ----------------------------------------------------------------------- *)
- section "sconc"
-(* ----------------------------------------------------------------------- *)
-
-lemma UU_sconc [simp]: " UU ooo s = s "
-by (simp add: sconc_def zero_inat_def)
-
-lemma scons_neq_UU: "a~=UU ==> a && s ~=UU"
-by auto
-
-lemma singleton_sconc [rule_format, simp]: "x~=UU --> (x && UU) ooo y = x && y"
-apply (simp add: sconc_def zero_inat_def iSuc_def split: inat.splits, auto)
-apply (rule someI2_ex,auto)
- apply (rule_tac x="x && y" in exI,auto)
-apply (simp add: i_rt_Suc_forw)
-apply (case_tac "xa=UU",simp)
-by (drule stream_exhaust_eq [THEN iffD1],auto)
-
-lemma ex_sconc [rule_format]:
- "ALL k y. #x = Fin k --> (EX w. stream_take k$w = x & i_rt k w = y)"
-apply (case_tac "#x")
- apply (rule stream_finite_ind [of x],auto)
- apply (simp add: stream.finite_def)
- apply (drule slen_take_lemma1,blast)
- apply (simp_all add: zero_inat_def iSuc_def split: inat.splits)
-apply (erule_tac x="y" in allE,auto)
-by (rule_tac x="a && w" in exI,auto)
-
-lemma rt_sconc1: "Fin n = #x ==> i_rt n (x ooo y) = y"
-apply (simp add: sconc_def split: inat.splits, arith?,auto)
-apply (rule someI2_ex,auto)
-by (drule ex_sconc,simp)
-
-lemma sconc_inj2: "\<lbrakk>Fin n = #x; x ooo y = x ooo z\<rbrakk> \<Longrightarrow> y = z"
-apply (frule_tac y=y in rt_sconc1)
-by (auto elim: rt_sconc1)
-
-lemma sconc_UU [simp]:"s ooo UU = s"
-apply (case_tac "#s")
- apply (simp add: sconc_def)
- apply (rule someI2_ex)
- apply (rule_tac x="s" in exI)
- apply auto
- apply (drule slen_take_lemma1,auto)
- apply (simp add: i_rt_lemma_slen)
- apply (drule slen_take_lemma1,auto)
- apply (simp add: i_rt_slen)
-by (simp add: sconc_def)
-
-lemma stream_take_sconc [simp]: "Fin n = #x ==> stream_take n$(x ooo y) = x"
-apply (simp add: sconc_def)
-apply (cases "#x")
-apply auto
-apply (rule someI2_ex, auto)
-by (drule ex_sconc,simp)
-
-lemma scons_sconc [rule_format,simp]: "a~=UU --> (a && x) ooo y = a && x ooo y"
-apply (cases "#x",auto)
- apply (simp add: sconc_def iSuc_Fin)
- apply (rule someI2_ex)
- apply (drule ex_sconc, simp)
- apply (rule someI2_ex, auto)
- apply (simp add: i_rt_Suc_forw)
- apply (rule_tac x="a && x" in exI, auto)
- apply (case_tac "xa=UU",auto)
- apply (drule stream_exhaust_eq [THEN iffD1],auto)
- apply (drule streams_prefix_lemma1,simp+)
-by (simp add: sconc_def)
-
-lemma ft_sconc: "x ~= UU ==> ft$(x ooo y) = ft$x"
-by (cases x, auto)
-
-lemma sconc_assoc: "(x ooo y) ooo z = x ooo y ooo z"
-apply (case_tac "#x")
- apply (rule stream_finite_ind [of x],auto simp del: scons_sconc)
- apply (simp add: stream.finite_def del: scons_sconc)
- apply (drule slen_take_lemma1,auto simp del: scons_sconc)
- apply (case_tac "a = UU", auto)
-by (simp add: sconc_def)
-
-
-(* ----------------------------------------------------------------------- *)
-
-lemma cont_sconc_lemma1: "stream_finite x \<Longrightarrow> cont (\<lambda>y. x ooo y)"
-by (erule stream_finite_ind, simp_all)
-
-lemma cont_sconc_lemma2: "\<not> stream_finite x \<Longrightarrow> cont (\<lambda>y. x ooo y)"
-by (simp add: sconc_def slen_def)
-
-lemma cont_sconc: "cont (\<lambda>y. x ooo y)"
-apply (cases "stream_finite x")
-apply (erule cont_sconc_lemma1)
-apply (erule cont_sconc_lemma2)
-done
-
-lemma sconc_mono: "y << y' ==> x ooo y << x ooo y'"
-by (rule cont_sconc [THEN cont2mono, THEN monofunE])
-
-lemma sconc_mono1 [simp]: "x << x ooo y"
-by (rule sconc_mono [of UU, simplified])
-
-(* ----------------------------------------------------------------------- *)
-
-lemma empty_sconc [simp]: "(x ooo y = UU) = (x = UU & y = UU)"
-apply (case_tac "#x",auto)
- apply (insert sconc_mono1 [of x y])
- by auto
-
-(* ----------------------------------------------------------------------- *)
-
-lemma rt_sconc [rule_format, simp]: "s~=UU --> rt$(s ooo x) = rt$s ooo x"
-by (cases s, auto)
-
-lemma i_th_sconc_lemma [rule_format]:
- "ALL x y. Fin n < #x --> i_th n (x ooo y) = i_th n x"
-apply (induct_tac n, auto)
-apply (simp add: Fin_0 i_th_def)
-apply (simp add: slen_empty_eq ft_sconc)
-apply (simp add: i_th_def)
-apply (case_tac "x=UU",auto)
-apply (drule stream_exhaust_eq [THEN iffD1], auto)
-apply (erule_tac x="ya" in allE)
-apply (case_tac "#ya") by simp_all
-
-
-
-(* ----------------------------------------------------------------------- *)
-
-lemma sconc_lemma [rule_format, simp]: "ALL s. stream_take n$s ooo i_rt n s = s"
-apply (induct_tac n,auto)
-apply (case_tac "s=UU",auto)
-by (drule stream_exhaust_eq [THEN iffD1],auto)
-
-(* ----------------------------------------------------------------------- *)
- subsection "pointwise equality"
-(* ----------------------------------------------------------------------- *)
-
-lemma ex_last_stream_take_scons: "stream_take (Suc n)$s =
- stream_take n$s ooo i_rt n (stream_take (Suc n)$s)"
-by (insert sconc_lemma [of n "stream_take (Suc n)$s"],simp)
-
-lemma i_th_stream_take_eq:
-"!!n. ALL n. i_th n s1 = i_th n s2 ==> stream_take n$s1 = stream_take n$s2"
-apply (induct_tac n,auto)
-apply (subgoal_tac "stream_take (Suc na)$s1 =
- stream_take na$s1 ooo i_rt na (stream_take (Suc na)$s1)")
- apply (subgoal_tac "i_rt na (stream_take (Suc na)$s1) =
- i_rt na (stream_take (Suc na)$s2)")
- apply (subgoal_tac "stream_take (Suc na)$s2 =
- stream_take na$s2 ooo i_rt na (stream_take (Suc na)$s2)")
- apply (insert ex_last_stream_take_scons,simp)
- apply blast
- apply (erule_tac x="na" in allE)
- apply (insert i_th_last_eq [of _ s1 s2])
-by blast+
-
-lemma pointwise_eq_lemma[rule_format]: "ALL n. i_th n s1 = i_th n s2 ==> s1 = s2"
-by (insert i_th_stream_take_eq [THEN stream.take_lemma],blast)
-
-(* ----------------------------------------------------------------------- *)
- subsection "finiteness"
-(* ----------------------------------------------------------------------- *)
-
-lemma slen_sconc_finite1:
- "[| #(x ooo y) = Infty; Fin n = #x |] ==> #y = Infty"
-apply (case_tac "#y ~= Infty",auto)
-apply (drule_tac y=y in rt_sconc1)
-apply (insert stream_finite_i_rt [of n "x ooo y"])
-by (simp add: slen_infinite)
-
-lemma slen_sconc_infinite1: "#x=Infty ==> #(x ooo y) = Infty"
-by (simp add: sconc_def)
-
-lemma slen_sconc_infinite2: "#y=Infty ==> #(x ooo y) = Infty"
-apply (case_tac "#x")
- apply (simp add: sconc_def)
- apply (rule someI2_ex)
- apply (drule ex_sconc,auto)
- apply (erule contrapos_pp)
- apply (insert stream_finite_i_rt)
- apply (fastsimp simp add: slen_infinite,auto)
-by (simp add: sconc_def)
-
-lemma sconc_finite: "(#x~=Infty & #y~=Infty) = (#(x ooo y)~=Infty)"
-apply auto
- apply (metis not_Infty_eq slen_sconc_finite1)
- apply (metis not_Infty_eq slen_sconc_infinite1)
-apply (metis not_Infty_eq slen_sconc_infinite2)
-done
-
-(* ----------------------------------------------------------------------- *)
-
-lemma slen_sconc_mono3: "[| Fin n = #x; Fin k = #(x ooo y) |] ==> n <= k"
-apply (insert slen_mono [of "x" "x ooo y"])
-apply (cases "#x") apply simp_all
-apply (cases "#(x ooo y)") apply simp_all
-done
-
-(* ----------------------------------------------------------------------- *)
- subsection "finite slen"
-(* ----------------------------------------------------------------------- *)
-
-lemma slen_sconc: "[| Fin n = #x; Fin m = #y |] ==> #(x ooo y) = Fin (n + m)"
-apply (case_tac "#(x ooo y)")
- apply (frule_tac y=y in rt_sconc1)
- apply (insert take_i_rt_len [of "THE j. Fin j = #(x ooo y)" "x ooo y" n n m],simp)
- apply (insert slen_sconc_mono3 [of n x _ y],simp)
-by (insert sconc_finite [of x y],auto)
-
-(* ----------------------------------------------------------------------- *)
- subsection "flat prefix"
-(* ----------------------------------------------------------------------- *)
-
-lemma sconc_prefix: "(s1::'a::flat stream) << s2 ==> EX t. s1 ooo t = s2"
-apply (case_tac "#s1")
- apply (subgoal_tac "stream_take nat$s1 = stream_take nat$s2")
- apply (rule_tac x="i_rt nat s2" in exI)
- apply (simp add: sconc_def)
- apply (rule someI2_ex)
- apply (drule ex_sconc)
- apply (simp,clarsimp,drule streams_prefix_lemma1)
- apply (simp+,rule slen_take_lemma3 [of _ s1 s2])
- apply (simp+,rule_tac x="UU" in exI)
-apply (insert slen_take_lemma3 [of _ s1 s2])
-by (rule stream.take_lemma,simp)
-
-(* ----------------------------------------------------------------------- *)
- subsection "continuity"
-(* ----------------------------------------------------------------------- *)
-
-lemma chain_sconc: "chain S ==> chain (%i. (x ooo S i))"
-by (simp add: chain_def,auto simp add: sconc_mono)
-
-lemma chain_scons: "chain S ==> chain (%i. a && S i)"
-apply (simp add: chain_def,auto)
-by (rule monofun_cfun_arg,simp)
-
-lemma contlub_scons_lemma: "chain S ==> (LUB i. a && S i) = a && (LUB i. S i)"
-by (rule cont2contlubE [OF cont_Rep_CFun2, symmetric])
-
-lemma finite_lub_sconc: "chain Y ==> (stream_finite x) ==>
- (LUB i. x ooo Y i) = (x ooo (LUB i. Y i))"
-apply (rule stream_finite_ind [of x])
- apply (auto)
-apply (subgoal_tac "(LUB i. a && (s ooo Y i)) = a && (LUB i. s ooo Y i)")
- by (force,blast dest: contlub_scons_lemma chain_sconc)
-
-lemma contlub_sconc_lemma:
- "chain Y ==> (LUB i. x ooo Y i) = (x ooo (LUB i. Y i))"
-apply (case_tac "#x=Infty")
- apply (simp add: sconc_def)
-apply (drule finite_lub_sconc,auto simp add: slen_infinite)
-done
-
-lemma monofun_sconc: "monofun (%y. x ooo y)"
-by (simp add: monofun_def sconc_mono)
-
-
-(* ----------------------------------------------------------------------- *)
- section "constr_sconc"
-(* ----------------------------------------------------------------------- *)
-
-lemma constr_sconc_UUs [simp]: "constr_sconc UU s = s"
-by (simp add: constr_sconc_def zero_inat_def)
-
-lemma "x ooo y = constr_sconc x y"
-apply (case_tac "#x")
- apply (rule stream_finite_ind [of x],auto simp del: scons_sconc)
- defer 1
- apply (simp add: constr_sconc_def del: scons_sconc)
- apply (case_tac "#s")
- apply (simp add: iSuc_Fin)
- apply (case_tac "a=UU",auto simp del: scons_sconc)
- apply (simp)
- apply (simp add: sconc_def)
- apply (simp add: constr_sconc_def)
-apply (simp add: stream.finite_def)
-by (drule slen_take_lemma1,auto)
-
-end