author | immler |
Sun, 27 Oct 2019 21:51:14 -0400 | |
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parent 67613 | ce654b0e6d69 |
child 80914 | d97fdabd9e2b |
permissions | -rw-r--r-- |
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(* Title: HOL/HOLCF/Library/Stream.thy |
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Author: Franz Regensburger, David von Oheimb, Borislav Gajanovic |
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*) |
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||
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section \<open>General Stream domain\<close> |
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|
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theory Stream |
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imports HOLCF "HOL-Library.Extended_Nat" |
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begin |
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default_sort pcpo |
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domain (unsafe) 'a stream = scons (ft::'a) (lazy rt::"'a stream") (infixr "&&" 65) |
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definition |
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smap :: "('a \<rightarrow> 'b) \<rightarrow> 'a stream \<rightarrow> 'b stream" where |
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"smap = fix\<cdot>(\<Lambda> h f s. case s of x && xs \<Rightarrow> f\<cdot>x && h\<cdot>f\<cdot>xs)" |
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definition |
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sfilter :: "('a \<rightarrow> tr) \<rightarrow> 'a stream \<rightarrow> 'a stream" where |
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"sfilter = fix\<cdot>(\<Lambda> h p s. case s of x && xs \<Rightarrow> |
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If p\<cdot>x then x && h\<cdot>p\<cdot>xs else h\<cdot>p\<cdot>xs)" |
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definition |
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slen :: "'a stream \<Rightarrow> enat" ("#_" [1000] 1000) where |
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"#s = (if stream_finite s then enat (LEAST n. stream_take n\<cdot>s = s) else \<infinity>)" |
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|
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(* concatenation *) |
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|
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definition |
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i_rt :: "nat \<Rightarrow> 'a stream \<Rightarrow> 'a stream" where (* chops the first i elements *) |
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"i_rt = (\<lambda>i s. iterate i\<cdot>rt\<cdot>s)" |
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definition |
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i_th :: "nat \<Rightarrow> 'a stream \<Rightarrow> 'a" where (* the i-th element *) |
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"i_th = (\<lambda>i s. ft\<cdot>(i_rt i s))" |
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definition |
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sconc :: "'a stream \<Rightarrow> 'a stream \<Rightarrow> 'a stream" (infixr "ooo" 65) where |
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"s1 ooo s2 = (case #s1 of |
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enat n \<Rightarrow> (SOME s. (stream_take n\<cdot>s = s1) \<and> (i_rt n s = s2)) |
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| \<infinity> \<Rightarrow> s1)" |
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primrec constr_sconc' :: "nat \<Rightarrow> 'a stream \<Rightarrow> 'a stream \<Rightarrow> 'a stream" |
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where |
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constr_sconc'_0: "constr_sconc' 0 s1 s2 = s2" |
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| constr_sconc'_Suc: "constr_sconc' (Suc n) s1 s2 = ft\<cdot>s1 && constr_sconc' n (rt\<cdot>s1) s2" |
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|
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definition |
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constr_sconc :: "'a stream \<Rightarrow> 'a stream \<Rightarrow> 'a stream" where (* constructive *) |
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"constr_sconc s1 s2 = (case #s1 of |
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enat n \<Rightarrow> constr_sconc' n s1 s2 |
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| \<infinity> \<Rightarrow> s1)" |
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||
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(* ----------------------------------------------------------------------- *) |
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(* theorems about scons *) |
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(* ----------------------------------------------------------------------- *) |
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section "scons" |
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63 |
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lemma scons_eq_UU: "(a && s = UU) = (a = UU)" |
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by simp |
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lemma scons_not_empty: "\<lbrakk>a && x = UU; a \<noteq> UU\<rbrakk> \<Longrightarrow> R" |
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by simp |
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|
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lemma stream_exhaust_eq: "x \<noteq> UU \<longleftrightarrow> (\<exists>a y. a \<noteq> UU \<and> x = a && y)" |
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by (cases x, auto) |
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lemma stream_neq_UU: "x \<noteq> UU \<Longrightarrow> \<exists>a a_s. x = a && a_s \<and> a \<noteq> UU" |
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by (simp add: stream_exhaust_eq,auto) |
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|
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lemma stream_prefix: |
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"\<lbrakk>a && s \<sqsubseteq> t; a \<noteq> UU\<rbrakk> \<Longrightarrow> \<exists>b tt. t = b && tt \<and> b \<noteq> UU \<and> s \<sqsubseteq> tt" |
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by (cases t, auto) |
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lemma stream_prefix': |
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"b \<noteq> UU \<Longrightarrow> x \<sqsubseteq> b && z = |
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(x = UU \<or> (\<exists>a y. x = a && y \<and> a \<noteq> UU \<and> a \<sqsubseteq> b \<and> y \<sqsubseteq> z))" |
|
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by (cases x, auto) |
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|
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|
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(* |
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lemma stream_prefix1: "\<lbrakk>x \<sqsubseteq> y; xs \<sqsubseteq> ys\<rbrakk> \<Longrightarrow> x && xs \<sqsubseteq> y && ys" |
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by (insert stream_prefix' [of y "x && xs" ys],force) |
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*) |
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lemma stream_flat_prefix: |
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"\<lbrakk>x && xs \<sqsubseteq> y && ys; (x::'a::flat) \<noteq> UU\<rbrakk> \<Longrightarrow> x = y \<and> xs \<sqsubseteq> ys" |
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apply (case_tac "y = UU",auto) |
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apply (drule ax_flat,simp) |
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done |
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|
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|
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(* ----------------------------------------------------------------------- *) |
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(* theorems about stream_case *) |
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(* ----------------------------------------------------------------------- *) |
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|
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section "stream_case" |
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|
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lemma stream_case_strictf: "stream_case\<cdot>UU\<cdot>s = UU" |
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by (cases s, auto) |
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|
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|
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|
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(* ----------------------------------------------------------------------- *) |
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(* theorems about ft and rt *) |
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(* ----------------------------------------------------------------------- *) |
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|
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|
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section "ft and rt" |
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|
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|
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lemma ft_defin: "s \<noteq> UU \<Longrightarrow> ft\<cdot>s \<noteq> UU" |
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by simp |
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lemma rt_strict_rev: "rt\<cdot>s \<noteq> UU \<Longrightarrow> s \<noteq> UU" |
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by auto |
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lemma surjectiv_scons: "(ft\<cdot>s) && (rt\<cdot>s) = s" |
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by (cases s, auto) |
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|
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lemma monofun_rt_mult: "x \<sqsubseteq> s \<Longrightarrow> iterate i\<cdot>rt\<cdot>x \<sqsubseteq> iterate i\<cdot>rt\<cdot>s" |
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by (rule monofun_cfun_arg) |
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|
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|
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|
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(* ----------------------------------------------------------------------- *) |
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(* theorems about stream_take *) |
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(* ----------------------------------------------------------------------- *) |
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|
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|
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section "stream_take" |
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|
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lemma stream_reach2: "(LUB i. stream_take i\<cdot>s) = s" |
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by (rule stream.reach) |
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|
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lemma chain_stream_take: "chain (\<lambda>i. stream_take i\<cdot>s)" |
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by simp |
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|
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lemma stream_take_prefix [simp]: "stream_take n\<cdot>s \<sqsubseteq> s" |
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apply (insert stream_reach2 [of s]) |
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apply (erule subst) back |
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apply (rule is_ub_thelub) |
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apply (simp only: chain_stream_take) |
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done |
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|
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lemma stream_take_more [rule_format]: |
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"\<forall>x. stream_take n\<cdot>x = x \<longrightarrow> stream_take (Suc n)\<cdot>x = x" |
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apply (induct_tac n,auto) |
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integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
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apply (case_tac "x=UU",auto) |
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apply (drule stream_exhaust_eq [THEN iffD1],auto) |
160 |
done |
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161 |
|
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lemma stream_take_lemma3 [rule_format]: |
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"\<forall>x xs. x \<noteq> UU \<longrightarrow> stream_take n\<cdot>(x && xs) = x && xs \<longrightarrow> stream_take n\<cdot>xs = xs" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
164 |
apply (induct_tac n,clarsimp) |
16745 | 165 |
(*apply (drule sym, erule scons_not_empty, simp)*) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
166 |
apply (clarify, rule stream_take_more) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
167 |
apply (erule_tac x="x" in allE) |
49521 | 168 |
apply (erule_tac x="xs" in allE,simp) |
169 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
170 |
|
17291 | 171 |
lemma stream_take_lemma4: |
63549 | 172 |
"\<forall>x xs. stream_take n\<cdot>xs = xs \<longrightarrow> stream_take (Suc n)\<cdot>(x && xs) = x && xs" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
173 |
by auto |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
174 |
|
17291 | 175 |
lemma stream_take_idempotent [rule_format, simp]: |
63549 | 176 |
"\<forall>s. stream_take n\<cdot>(stream_take n\<cdot>s) = stream_take n\<cdot>s" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
177 |
apply (induct_tac n, auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
178 |
apply (case_tac "s=UU", auto) |
49521 | 179 |
apply (drule stream_exhaust_eq [THEN iffD1], auto) |
180 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
181 |
|
17291 | 182 |
lemma stream_take_take_Suc [rule_format, simp]: |
63549 | 183 |
"\<forall>s. stream_take n\<cdot>(stream_take (Suc n)\<cdot>s) = stream_take n\<cdot>s" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
184 |
apply (induct_tac n, auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
185 |
apply (case_tac "s=UU", auto) |
49521 | 186 |
apply (drule stream_exhaust_eq [THEN iffD1], auto) |
187 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
188 |
|
17291 | 189 |
lemma mono_stream_take_pred: |
63549 | 190 |
"stream_take (Suc n)\<cdot>s1 \<sqsubseteq> stream_take (Suc n)\<cdot>s2 \<Longrightarrow> |
191 |
stream_take n\<cdot>s1 \<sqsubseteq> stream_take n\<cdot>s2" |
|
192 |
by (insert monofun_cfun_arg [of "stream_take (Suc n)\<cdot>s1" |
|
193 |
"stream_take (Suc n)\<cdot>s2" "stream_take n"], auto) |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
194 |
(* |
17291 | 195 |
lemma mono_stream_take_pred: |
63549 | 196 |
"stream_take (Suc n)\<cdot>s1 \<sqsubseteq> stream_take (Suc n)\<cdot>s2 \<Longrightarrow> |
197 |
stream_take n\<cdot>s1 \<sqsubseteq> stream_take n\<cdot>s2" |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
198 |
by (drule mono_stream_take [of _ _ n],simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
199 |
*) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
200 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
201 |
lemma stream_take_lemma10 [rule_format]: |
63549 | 202 |
"\<forall>k\<le>n. stream_take n\<cdot>s1 \<sqsubseteq> stream_take n\<cdot>s2 \<longrightarrow> stream_take k\<cdot>s1 \<sqsubseteq> stream_take k\<cdot>s2" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
203 |
apply (induct_tac n,simp,clarsimp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
204 |
apply (case_tac "k=Suc n",blast) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
205 |
apply (erule_tac x="k" in allE) |
49521 | 206 |
apply (drule mono_stream_take_pred,simp) |
207 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
208 |
|
63549 | 209 |
lemma stream_take_le_mono : "k \<le> n \<Longrightarrow> stream_take k\<cdot>s1 \<sqsubseteq> stream_take n\<cdot>s1" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
210 |
apply (insert chain_stream_take [of s1]) |
49521 | 211 |
apply (drule chain_mono,auto) |
212 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
213 |
|
63549 | 214 |
lemma mono_stream_take: "s1 \<sqsubseteq> s2 \<Longrightarrow> stream_take n\<cdot>s1 \<sqsubseteq> stream_take n\<cdot>s2" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
215 |
by (simp add: monofun_cfun_arg) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
216 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
217 |
(* |
63549 | 218 |
lemma stream_take_prefix [simp]: "stream_take n\<cdot>s \<sqsubseteq> s" |
219 |
apply (subgoal_tac "s=(LUB n. stream_take n\<cdot>s)") |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
220 |
apply (erule ssubst, rule is_ub_thelub) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
221 |
apply (simp only: chain_stream_take) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
222 |
by (simp only: stream_reach2) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
223 |
*) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
224 |
|
63549 | 225 |
lemma stream_take_take_less:"stream_take k\<cdot>(stream_take n\<cdot>s) \<sqsubseteq> stream_take k\<cdot>s" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
226 |
by (rule monofun_cfun_arg,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
227 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
228 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
229 |
(* ------------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
230 |
(* special induction rules *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
231 |
(* ------------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
232 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
233 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
234 |
section "induction" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
235 |
|
17291 | 236 |
lemma stream_finite_ind: |
63549 | 237 |
"\<lbrakk>stream_finite x; P UU; \<And>a s. \<lbrakk>a \<noteq> UU; P s\<rbrakk> \<Longrightarrow> P (a && s)\<rbrakk> \<Longrightarrow> P x" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
238 |
apply (simp add: stream.finite_def,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
239 |
apply (erule subst) |
49521 | 240 |
apply (drule stream.finite_induct [of P _ x], auto) |
241 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
242 |
|
17291 | 243 |
lemma stream_finite_ind2: |
63549 | 244 |
"\<lbrakk>P UU; \<And>x. x \<noteq> UU \<Longrightarrow> P (x && UU); \<And>y z s. \<lbrakk>y \<noteq> UU; z \<noteq> UU; P s\<rbrakk> \<Longrightarrow> P (y && z && s)\<rbrakk> \<Longrightarrow> |
245 |
\<forall>s. P (stream_take n\<cdot>s)" |
|
29855
e0ab6cf95539
Repaired a proof that did, after all, refer to the theorem nat_induct2.
paulson
parents:
29530
diff
changeset
|
246 |
apply (rule nat_less_induct [of _ n],auto) |
e0ab6cf95539
Repaired a proof that did, after all, refer to the theorem nat_induct2.
paulson
parents:
29530
diff
changeset
|
247 |
apply (case_tac n, auto) |
e0ab6cf95539
Repaired a proof that did, after all, refer to the theorem nat_induct2.
paulson
parents:
29530
diff
changeset
|
248 |
apply (case_tac nat, auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
249 |
apply (case_tac "s=UU",clarsimp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
250 |
apply (drule stream_exhaust_eq [THEN iffD1],clarsimp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
251 |
apply (case_tac "s=UU",clarsimp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
252 |
apply (drule stream_exhaust_eq [THEN iffD1],clarsimp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
253 |
apply (case_tac "y=UU",clarsimp) |
49521 | 254 |
apply (drule stream_exhaust_eq [THEN iffD1],clarsimp) |
255 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
256 |
|
17291 | 257 |
lemma stream_ind2: |
63549 | 258 |
"\<lbrakk> adm P; P UU; \<And>a. a \<noteq> UU \<Longrightarrow> P (a && UU); \<And>a b s. \<lbrakk>a \<noteq> UU; b \<noteq> UU; P s\<rbrakk> \<Longrightarrow> P (a && b && s)\<rbrakk> \<Longrightarrow> P x" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
259 |
apply (insert stream.reach [of x],erule subst) |
35494
45c9a8278faf
domain package no longer generates copy functions; all proofs use take functions instead
huffman
parents:
35444
diff
changeset
|
260 |
apply (erule admD, rule chain_stream_take) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
261 |
apply (insert stream_finite_ind2 [of P]) |
35494
45c9a8278faf
domain package no longer generates copy functions; all proofs use take functions instead
huffman
parents:
35444
diff
changeset
|
262 |
by simp |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
263 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
264 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
265 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
266 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
267 |
(* simplify use of coinduction *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
268 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
269 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
270 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
271 |
section "coinduction" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
272 |
|
63549 | 273 |
lemma stream_coind_lemma2: "\<forall>s1 s2. R s1 s2 \<longrightarrow> ft\<cdot>s1 = ft\<cdot>s2 \<and> R (rt\<cdot>s1) (rt\<cdot>s2) \<Longrightarrow> stream_bisim R" |
30807 | 274 |
apply (simp add: stream.bisim_def,clarsimp) |
35497 | 275 |
apply (drule spec, drule spec, drule (1) mp) |
276 |
apply (case_tac "x", simp) |
|
40025 | 277 |
apply (case_tac "y", simp) |
49521 | 278 |
apply auto |
279 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
280 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
281 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
282 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
283 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
284 |
(* theorems about stream_finite *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
285 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
286 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
287 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
288 |
section "stream_finite" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
289 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
290 |
lemma stream_finite_UU [simp]: "stream_finite UU" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
291 |
by (simp add: stream.finite_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
292 |
|
63549 | 293 |
lemma stream_finite_UU_rev: "\<not> stream_finite s \<Longrightarrow> s \<noteq> UU" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
294 |
by (auto simp add: stream.finite_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
295 |
|
63549 | 296 |
lemma stream_finite_lemma1: "stream_finite xs \<Longrightarrow> stream_finite (x && xs)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
297 |
apply (simp add: stream.finite_def,auto) |
35557
5da670d57118
uniformly use variable names m and n in take-related lemmas; use export_without_context where appropriate
huffman
parents:
35524
diff
changeset
|
298 |
apply (rule_tac x="Suc n" in exI) |
49521 | 299 |
apply (simp add: stream_take_lemma4) |
300 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
301 |
|
63549 | 302 |
lemma stream_finite_lemma2: "\<lbrakk>x \<noteq> UU; stream_finite (x && xs)\<rbrakk> \<Longrightarrow> stream_finite xs" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
303 |
apply (simp add: stream.finite_def, auto) |
35557
5da670d57118
uniformly use variable names m and n in take-related lemmas; use export_without_context where appropriate
huffman
parents:
35524
diff
changeset
|
304 |
apply (rule_tac x="n" in exI) |
49521 | 305 |
apply (erule stream_take_lemma3,simp) |
306 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
307 |
|
63549 | 308 |
lemma stream_finite_rt_eq: "stream_finite (rt\<cdot>s) = stream_finite s" |
35781
b7738ab762b1
renamed some lemmas generated by the domain package
huffman
parents:
35642
diff
changeset
|
309 |
apply (cases s, auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
310 |
apply (rule stream_finite_lemma1, simp) |
49521 | 311 |
apply (rule stream_finite_lemma2,simp) |
312 |
apply assumption |
|
313 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
314 |
|
63549 | 315 |
lemma stream_finite_less: "stream_finite s \<Longrightarrow> \<forall>t. t \<sqsubseteq> s \<longrightarrow> stream_finite t" |
19440 | 316 |
apply (erule stream_finite_ind [of s], auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
317 |
apply (case_tac "t=UU", auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
318 |
apply (drule stream_exhaust_eq [THEN iffD1],auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
319 |
apply (erule_tac x="y" in allE, simp) |
49521 | 320 |
apply (rule stream_finite_lemma1, simp) |
321 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
322 |
|
63549 | 323 |
lemma stream_take_finite [simp]: "stream_finite (stream_take n\<cdot>s)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
324 |
apply (simp add: stream.finite_def) |
49521 | 325 |
apply (rule_tac x="n" in exI,simp) |
326 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
327 |
|
63549 | 328 |
lemma adm_not_stream_finite: "adm (\<lambda>x. \<not> stream_finite x)" |
25833 | 329 |
apply (rule adm_upward) |
330 |
apply (erule contrapos_nn) |
|
331 |
apply (erule (1) stream_finite_less [rule_format]) |
|
332 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
333 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
334 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
335 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
336 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
337 |
(* theorems about stream length *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
338 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
339 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
340 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
341 |
section "slen" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
342 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
343 |
lemma slen_empty [simp]: "#\<bottom> = 0" |
43919 | 344 |
by (simp add: slen_def stream.finite_def zero_enat_def Least_equality) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
345 |
|
63549 | 346 |
lemma slen_scons [simp]: "x \<noteq> \<bottom> \<Longrightarrow> #(x && xs) = eSuc (#xs)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
347 |
apply (case_tac "stream_finite (x && xs)") |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
348 |
apply (simp add: slen_def, auto) |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
349 |
apply (simp add: stream.finite_def, auto simp add: eSuc_enat) |
27111 | 350 |
apply (rule Least_Suc2, auto) |
16745 | 351 |
(*apply (drule sym)*) |
352 |
(*apply (drule sym scons_eq_UU [THEN iffD1],simp)*) |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
353 |
apply (erule stream_finite_lemma2, simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
354 |
apply (simp add: slen_def, auto) |
49521 | 355 |
apply (drule stream_finite_lemma1,auto) |
356 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
357 |
|
43924 | 358 |
lemma slen_less_1_eq: "(#x < enat (Suc 0)) = (x = \<bottom>)" |
49521 | 359 |
by (cases x) (auto simp add: enat_0 eSuc_enat[THEN sym]) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
360 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
361 |
lemma slen_empty_eq: "(#x = 0) = (x = \<bottom>)" |
49521 | 362 |
by (cases x) auto |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
363 |
|
67613 | 364 |
lemma slen_scons_eq: "(enat (Suc n) < #x) = (\<exists>a y. x = a && y \<and> a \<noteq> \<bottom> \<and> enat n < #y)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
365 |
apply (auto, case_tac "x=UU",auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
366 |
apply (drule stream_exhaust_eq [THEN iffD1], auto) |
27111 | 367 |
apply (case_tac "#y") apply simp_all |
368 |
apply (case_tac "#y") apply simp_all |
|
369 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
370 |
|
63549 | 371 |
lemma slen_eSuc: "#x = eSuc n \<longrightarrow> (\<exists>a y. x = a && y \<and> a \<noteq> \<bottom> \<and> #y = n)" |
49521 | 372 |
by (cases x) auto |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
373 |
|
63549 | 374 |
lemma slen_stream_take_finite [simp]: "#(stream_take n\<cdot>s) \<noteq> \<infinity>" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
375 |
by (simp add: slen_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
376 |
|
63549 | 377 |
lemma slen_scons_eq_rev: "#x < enat (Suc (Suc n)) \<longleftrightarrow> (\<forall>a y. x \<noteq> a && y \<or> a = \<bottom> \<or> #y < enat (Suc n))" |
35781
b7738ab762b1
renamed some lemmas generated by the domain package
huffman
parents:
35642
diff
changeset
|
378 |
apply (cases x, auto) |
43919 | 379 |
apply (simp add: zero_enat_def) |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
380 |
apply (case_tac "#stream") apply (simp_all add: eSuc_enat) |
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
381 |
apply (case_tac "#stream") apply (simp_all add: eSuc_enat) |
27111 | 382 |
done |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
383 |
|
17291 | 384 |
lemma slen_take_lemma4 [rule_format]: |
63549 | 385 |
"\<forall>s. stream_take n\<cdot>s \<noteq> s \<longrightarrow> #(stream_take n\<cdot>s) = enat n" |
43924 | 386 |
apply (induct n, auto simp add: enat_0) |
27111 | 387 |
apply (case_tac "s=UU", simp) |
49521 | 388 |
apply (drule stream_exhaust_eq [THEN iffD1], auto simp add: eSuc_enat) |
389 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
390 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
391 |
(* |
17291 | 392 |
lemma stream_take_idempotent [simp]: |
63549 | 393 |
"stream_take n\<cdot>(stream_take n\<cdot>s) = stream_take n\<cdot>s" |
394 |
apply (case_tac "stream_take n\<cdot>s = s") |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
395 |
apply (auto,insert slen_take_lemma4 [of n s]); |
63549 | 396 |
by (auto,insert slen_take_lemma1 [of "stream_take n\<cdot>s" n],simp) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
397 |
|
63549 | 398 |
lemma stream_take_take_Suc [simp]: "stream_take n\<cdot>(stream_take (Suc n)\<cdot>s) = |
399 |
stream_take n\<cdot>s" |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
400 |
apply (simp add: po_eq_conv,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
401 |
apply (simp add: stream_take_take_less) |
63549 | 402 |
apply (subgoal_tac "stream_take n\<cdot>s = stream_take n\<cdot>(stream_take n\<cdot>s)") |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
403 |
apply (erule ssubst) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
404 |
apply (rule_tac monofun_cfun_arg) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
405 |
apply (insert chain_stream_take [of s]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
406 |
by (simp add: chain_def,simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
407 |
*) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
408 |
|
63549 | 409 |
lemma slen_take_eq: "\<forall>x. enat n < #x \<longleftrightarrow> stream_take n\<cdot>x \<noteq> x" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
410 |
apply (induct_tac n, auto) |
43924 | 411 |
apply (simp add: enat_0, clarsimp) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
412 |
apply (drule not_sym) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
413 |
apply (drule slen_empty_eq [THEN iffD1], simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
414 |
apply (case_tac "x=UU", simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
415 |
apply (drule stream_exhaust_eq [THEN iffD1], clarsimp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
416 |
apply (erule_tac x="y" in allE, auto) |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
417 |
apply (simp_all add: not_less eSuc_enat) |
27111 | 418 |
apply (case_tac "#y") apply simp_all |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
419 |
apply (case_tac "x=UU", simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
420 |
apply (drule stream_exhaust_eq [THEN iffD1], clarsimp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
421 |
apply (erule_tac x="y" in allE, simp) |
49521 | 422 |
apply (case_tac "#y") |
423 |
apply simp_all |
|
424 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
425 |
|
63549 | 426 |
lemma slen_take_eq_rev: "#x \<le> enat n \<longleftrightarrow> stream_take n\<cdot>x = x" |
26102 | 427 |
by (simp add: linorder_not_less [symmetric] slen_take_eq) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
428 |
|
63549 | 429 |
lemma slen_take_lemma1: "#x = enat n \<Longrightarrow> stream_take n\<cdot>x = x" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
430 |
by (rule slen_take_eq_rev [THEN iffD1], auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
431 |
|
63549 | 432 |
lemma slen_rt_mono: "#s2 \<le> #s1 \<Longrightarrow> #(rt\<cdot>s2) \<le> #(rt\<cdot>s1)" |
35781
b7738ab762b1
renamed some lemmas generated by the domain package
huffman
parents:
35642
diff
changeset
|
433 |
apply (cases s1) |
49521 | 434 |
apply (cases s2, simp+)+ |
435 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
436 |
|
63549 | 437 |
lemma slen_take_lemma5: "#(stream_take n\<cdot>s) \<le> enat n" |
438 |
apply (case_tac "stream_take n\<cdot>s = s") |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
439 |
apply (simp add: slen_take_eq_rev) |
49521 | 440 |
apply (simp add: slen_take_lemma4) |
441 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
442 |
|
63549 | 443 |
lemma slen_take_lemma2: "\<forall>x. \<not> stream_finite x \<longrightarrow> #(stream_take i\<cdot>x) = enat i" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
444 |
apply (simp add: stream.finite_def, auto) |
49521 | 445 |
apply (simp add: slen_take_lemma4) |
446 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
447 |
|
63549 | 448 |
lemma slen_infinite: "stream_finite x \<longleftrightarrow> #x \<noteq> \<infinity>" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
449 |
by (simp add: slen_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
450 |
|
63549 | 451 |
lemma slen_mono_lemma: "stream_finite s \<Longrightarrow> \<forall>t. s \<sqsubseteq> t \<longrightarrow> #s \<le> #t" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
452 |
apply (erule stream_finite_ind [of s], auto) |
63549 | 453 |
apply (case_tac "t = UU", auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
454 |
apply (drule stream_exhaust_eq [THEN iffD1], auto) |
30807 | 455 |
done |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
456 |
|
63549 | 457 |
lemma slen_mono: "s \<sqsubseteq> t \<Longrightarrow> #s \<le> #t" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
458 |
apply (case_tac "stream_finite t") |
17291 | 459 |
apply (frule stream_finite_less) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
460 |
apply (erule_tac x="s" in allE, simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
461 |
apply (drule slen_mono_lemma, auto) |
49521 | 462 |
apply (simp add: slen_def) |
463 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
464 |
|
63549 | 465 |
lemma iterate_lemma: "F\<cdot>(iterate n\<cdot>F\<cdot>x) = iterate n\<cdot>F\<cdot>(F\<cdot>x)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
466 |
by (insert iterate_Suc2 [of n F x], auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
467 |
|
63549 | 468 |
lemma slen_rt_mult [rule_format]: "\<forall>x. enat (i + j) \<le> #x \<longrightarrow> enat j \<le> #(iterate i\<cdot>rt\<cdot>x)" |
27111 | 469 |
apply (induct i, auto) |
63549 | 470 |
apply (case_tac "x = UU", auto simp add: zero_enat_def) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
471 |
apply (drule stream_exhaust_eq [THEN iffD1], auto) |
63549 | 472 |
apply (erule_tac x = "y" in allE, auto) |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
473 |
apply (simp add: not_le) apply (case_tac "#y") apply (simp_all add: eSuc_enat) |
49521 | 474 |
apply (simp add: iterate_lemma) |
475 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
476 |
|
17291 | 477 |
lemma slen_take_lemma3 [rule_format]: |
63549 | 478 |
"\<forall>(x::'a::flat stream) y. enat n \<le> #x \<longrightarrow> x \<sqsubseteq> y \<longrightarrow> stream_take n\<cdot>x = stream_take n\<cdot>y" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
479 |
apply (induct_tac n, auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
480 |
apply (case_tac "x=UU", auto) |
43919 | 481 |
apply (simp add: zero_enat_def) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
482 |
apply (simp add: Suc_ile_eq) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
483 |
apply (case_tac "y=UU", clarsimp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
484 |
apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)+ |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
485 |
apply (erule_tac x="ya" in allE, simp) |
25920 | 486 |
by (drule ax_flat, simp) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
487 |
|
17291 | 488 |
lemma slen_strict_mono_lemma: |
63549 | 489 |
"stream_finite t \<Longrightarrow> \<forall>s. #(s::'a::flat stream) = #t \<and> s \<sqsubseteq> t \<longrightarrow> s = t" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
490 |
apply (erule stream_finite_ind, auto) |
63549 | 491 |
apply (case_tac "sa = UU", auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
492 |
apply (drule stream_exhaust_eq [THEN iffD1], clarsimp) |
49521 | 493 |
apply (drule ax_flat, simp) |
494 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
495 |
|
63549 | 496 |
lemma slen_strict_mono: "\<lbrakk>stream_finite t; s \<noteq> t; s \<sqsubseteq> (t::'a::flat stream)\<rbrakk> \<Longrightarrow> #s < #t" |
27111 | 497 |
by (auto simp add: slen_mono less_le dest: slen_strict_mono_lemma) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
498 |
|
63549 | 499 |
lemma stream_take_Suc_neq: "stream_take (Suc n)\<cdot>s \<noteq> s \<Longrightarrow> |
500 |
stream_take n\<cdot>s \<noteq> stream_take (Suc n)\<cdot>s" |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
501 |
apply auto |
63549 | 502 |
apply (subgoal_tac "stream_take n\<cdot>s \<noteq> s") |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
503 |
apply (insert slen_take_lemma4 [of n s],auto) |
35781
b7738ab762b1
renamed some lemmas generated by the domain package
huffman
parents:
35642
diff
changeset
|
504 |
apply (cases s, simp) |
49521 | 505 |
apply (simp add: slen_take_lemma4 eSuc_enat) |
506 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
507 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
508 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
509 |
(* theorems about smap *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
510 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
511 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
512 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
513 |
section "smap" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
514 |
|
63549 | 515 |
lemma smap_unfold: "smap = (\<Lambda> f t. case t of x && xs \<Rightarrow> f\<cdot>x && smap\<cdot>f\<cdot>xs)" |
29530
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
27361
diff
changeset
|
516 |
by (insert smap_def [where 'a='a and 'b='b, THEN eq_reflection, THEN fix_eq2], auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
517 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
518 |
lemma smap_empty [simp]: "smap\<cdot>f\<cdot>\<bottom> = \<bottom>" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
519 |
by (subst smap_unfold, simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
520 |
|
63549 | 521 |
lemma smap_scons [simp]: "x \<noteq> \<bottom> \<Longrightarrow> smap\<cdot>f\<cdot>(x && xs) = (f\<cdot>x) && (smap\<cdot>f\<cdot>xs)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
522 |
by (subst smap_unfold, force) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
523 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
524 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
525 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
526 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
527 |
(* theorems about sfilter *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
528 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
529 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
530 |
section "sfilter" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
531 |
|
17291 | 532 |
lemma sfilter_unfold: |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
533 |
"sfilter = (\<Lambda> p s. case s of x && xs \<Rightarrow> |
40322
707eb30e8a53
make syntax of continuous if-then-else consistent with HOL if-then-else
huffman
parents:
40213
diff
changeset
|
534 |
If p\<cdot>x then x && sfilter\<cdot>p\<cdot>xs else sfilter\<cdot>p\<cdot>xs)" |
29530
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
27361
diff
changeset
|
535 |
by (insert sfilter_def [where 'a='a, THEN eq_reflection, THEN fix_eq2], auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
536 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
537 |
lemma strict_sfilter: "sfilter\<cdot>\<bottom> = \<bottom>" |
40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
37110
diff
changeset
|
538 |
apply (rule cfun_eqI) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
539 |
apply (subst sfilter_unfold, auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
540 |
apply (case_tac "x=UU", auto) |
49521 | 541 |
apply (drule stream_exhaust_eq [THEN iffD1], auto) |
542 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
543 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
544 |
lemma sfilter_empty [simp]: "sfilter\<cdot>f\<cdot>\<bottom> = \<bottom>" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
545 |
by (subst sfilter_unfold, force) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
546 |
|
17291 | 547 |
lemma sfilter_scons [simp]: |
63549 | 548 |
"x \<noteq> \<bottom> \<Longrightarrow> sfilter\<cdot>f\<cdot>(x && xs) = |
40322
707eb30e8a53
make syntax of continuous if-then-else consistent with HOL if-then-else
huffman
parents:
40213
diff
changeset
|
549 |
If f\<cdot>x then x && sfilter\<cdot>f\<cdot>xs else sfilter\<cdot>f\<cdot>xs" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
550 |
by (subst sfilter_unfold, force) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
551 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
552 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
553 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
554 |
section "i_rt" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
555 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
556 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
557 |
lemma i_rt_UU [simp]: "i_rt n UU = UU" |
34941 | 558 |
by (induct n) (simp_all add: i_rt_def) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
559 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
560 |
lemma i_rt_0 [simp]: "i_rt 0 s = s" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
561 |
by (simp add: i_rt_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
562 |
|
63549 | 563 |
lemma i_rt_Suc [simp]: "a \<noteq> UU \<Longrightarrow> i_rt (Suc n) (a&&s) = i_rt n s" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
564 |
by (simp add: i_rt_def iterate_Suc2 del: iterate_Suc) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
565 |
|
63549 | 566 |
lemma i_rt_Suc_forw: "i_rt (Suc n) s = i_rt n (rt\<cdot>s)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
567 |
by (simp only: i_rt_def iterate_Suc2) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
568 |
|
63549 | 569 |
lemma i_rt_Suc_back: "i_rt (Suc n) s = rt\<cdot>(i_rt n s)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
570 |
by (simp only: i_rt_def,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
571 |
|
63549 | 572 |
lemma i_rt_mono: "x << s \<Longrightarrow> i_rt n x << i_rt n s" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
573 |
by (simp add: i_rt_def monofun_rt_mult) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
574 |
|
63549 | 575 |
lemma i_rt_ij_lemma: "enat (i + j) \<le> #x \<Longrightarrow> enat j \<le> #(i_rt i x)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
576 |
by (simp add: i_rt_def slen_rt_mult) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
577 |
|
63549 | 578 |
lemma slen_i_rt_mono: "#s2 \<le> #s1 \<Longrightarrow> #(i_rt n s2) \<le> #(i_rt n s1)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
579 |
apply (induct_tac n,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
580 |
apply (simp add: i_rt_Suc_back) |
49521 | 581 |
apply (drule slen_rt_mono,simp) |
582 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
583 |
|
63549 | 584 |
lemma i_rt_take_lemma1 [rule_format]: "\<forall>s. i_rt n (stream_take n\<cdot>s) = UU" |
17291 | 585 |
apply (induct_tac n) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
586 |
apply (simp add: i_rt_Suc_back,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
587 |
apply (case_tac "s=UU",auto) |
49521 | 588 |
apply (drule stream_exhaust_eq [THEN iffD1],auto) |
589 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
590 |
|
63549 | 591 |
lemma i_rt_slen: "i_rt n s = UU \<longleftrightarrow> stream_take n\<cdot>s = s" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
592 |
apply auto |
17291 | 593 |
apply (insert i_rt_ij_lemma [of n "Suc 0" s]) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
594 |
apply (subgoal_tac "#(i_rt n s)=0") |
63549 | 595 |
apply (case_tac "stream_take n\<cdot>s = s",simp+) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
596 |
apply (insert slen_take_eq [rule_format,of n s],simp) |
43919 | 597 |
apply (cases "#s") apply (simp_all add: zero_enat_def) |
27111 | 598 |
apply (simp add: slen_take_eq) |
599 |
apply (cases "#s") |
|
600 |
using i_rt_take_lemma1 [of n s] |
|
43919 | 601 |
apply (simp_all add: zero_enat_def) |
27111 | 602 |
done |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
603 |
|
63549 | 604 |
lemma i_rt_lemma_slen: "#s=enat n \<Longrightarrow> i_rt n s = UU" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
605 |
by (simp add: i_rt_slen slen_take_lemma1) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
606 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
607 |
lemma stream_finite_i_rt [simp]: "stream_finite (i_rt n s) = stream_finite s" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
608 |
apply (induct_tac n, auto) |
35781
b7738ab762b1
renamed some lemmas generated by the domain package
huffman
parents:
35642
diff
changeset
|
609 |
apply (cases s, auto simp del: i_rt_Suc) |
49521 | 610 |
apply (simp add: i_rt_Suc_back stream_finite_rt_eq)+ |
611 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
612 |
|
63549 | 613 |
lemma take_i_rt_len_lemma: "\<forall>sl x j t. enat sl = #x \<and> n \<le> sl \<and> |
614 |
#(stream_take n\<cdot>x) = enat t \<and> #(i_rt n x) = enat j |
|
615 |
\<longrightarrow> enat (j + t) = #x" |
|
27111 | 616 |
apply (induct n, auto) |
43919 | 617 |
apply (simp add: zero_enat_def) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
618 |
apply (case_tac "x=UU",auto) |
43919 | 619 |
apply (simp add: zero_enat_def) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
620 |
apply (drule stream_exhaust_eq [THEN iffD1],clarsimp) |
63549 | 621 |
apply (subgoal_tac "\<exists>k. enat k = #y",clarify) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
622 |
apply (erule_tac x="k" in allE) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
623 |
apply (erule_tac x="y" in allE,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
624 |
apply (erule_tac x="THE p. Suc p = t" in allE,auto) |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
625 |
apply (simp add: eSuc_def split: enat.splits) |
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
626 |
apply (simp add: eSuc_def split: enat.splits) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
627 |
apply (simp only: the_equality) |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
628 |
apply (simp add: eSuc_def split: enat.splits) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
629 |
apply force |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
630 |
apply (simp add: eSuc_def split: enat.splits) |
27111 | 631 |
done |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
632 |
|
17291 | 633 |
lemma take_i_rt_len: |
63549 | 634 |
"\<lbrakk>enat sl = #x; n \<le> sl; #(stream_take n\<cdot>x) = enat t; #(i_rt n x) = enat j\<rbrakk> \<Longrightarrow> |
43924 | 635 |
enat (j + t) = #x" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
636 |
by (blast intro: take_i_rt_len_lemma [rule_format]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
637 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
638 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
639 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
640 |
section "i_th" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
641 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
642 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
643 |
lemma i_th_i_rt_step: |
17291 | 644 |
"[| i_th n s1 << i_th n s2; i_rt (Suc n) s1 << i_rt (Suc n) s2 |] ==> |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
645 |
i_rt n s1 << i_rt n s2" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
646 |
apply (simp add: i_th_def i_rt_Suc_back) |
35781
b7738ab762b1
renamed some lemmas generated by the domain package
huffman
parents:
35642
diff
changeset
|
647 |
apply (cases "i_rt n s1", simp) |
b7738ab762b1
renamed some lemmas generated by the domain package
huffman
parents:
35642
diff
changeset
|
648 |
apply (cases "i_rt n s2", auto) |
30807 | 649 |
done |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
650 |
|
17291 | 651 |
lemma i_th_stream_take_Suc [rule_format]: |
63549 | 652 |
"\<forall>s. i_th n (stream_take (Suc n)\<cdot>s) = i_th n s" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
653 |
apply (induct_tac n,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
654 |
apply (simp add: i_th_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
655 |
apply (case_tac "s=UU",auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
656 |
apply (drule stream_exhaust_eq [THEN iffD1],auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
657 |
apply (case_tac "s=UU",simp add: i_th_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
658 |
apply (drule stream_exhaust_eq [THEN iffD1],auto) |
49521 | 659 |
apply (simp add: i_th_def i_rt_Suc_forw) |
660 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
661 |
|
63549 | 662 |
lemma i_th_last: "i_th n s && UU = i_rt n (stream_take (Suc n)\<cdot>s)" |
663 |
apply (insert surjectiv_scons [of "i_rt n (stream_take (Suc n)\<cdot>s)"]) |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
664 |
apply (rule i_th_stream_take_Suc [THEN subst]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
665 |
apply (simp add: i_th_def i_rt_Suc_back [symmetric]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
666 |
by (simp add: i_rt_take_lemma1) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
667 |
|
17291 | 668 |
lemma i_th_last_eq: |
63549 | 669 |
"i_th n s1 = i_th n s2 \<Longrightarrow> i_rt n (stream_take (Suc n)\<cdot>s1) = i_rt n (stream_take (Suc n)\<cdot>s2)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
670 |
apply (insert i_th_last [of n s1]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
671 |
apply (insert i_th_last [of n s2]) |
49521 | 672 |
apply auto |
673 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
674 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
675 |
lemma i_th_prefix_lemma: |
63549 | 676 |
"\<lbrakk>k \<le> n; stream_take (Suc n)\<cdot>s1 << stream_take (Suc n)\<cdot>s2\<rbrakk> \<Longrightarrow> |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
677 |
i_th k s1 << i_th k s2" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
678 |
apply (insert i_th_stream_take_Suc [of k s1, THEN sym]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
679 |
apply (insert i_th_stream_take_Suc [of k s2, THEN sym],auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
680 |
apply (simp add: i_th_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
681 |
apply (rule monofun_cfun, auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
682 |
apply (rule i_rt_mono) |
49521 | 683 |
apply (blast intro: stream_take_lemma10) |
684 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
685 |
|
17291 | 686 |
lemma take_i_rt_prefix_lemma1: |
63549 | 687 |
"stream_take (Suc n)\<cdot>s1 << stream_take (Suc n)\<cdot>s2 \<Longrightarrow> |
688 |
i_rt (Suc n) s1 << i_rt (Suc n) s2 \<Longrightarrow> |
|
689 |
i_rt n s1 << i_rt n s2 \<and> stream_take n\<cdot>s1 << stream_take n\<cdot>s2" |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
690 |
apply auto |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
691 |
apply (insert i_th_prefix_lemma [of n n s1 s2]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
692 |
apply (rule i_th_i_rt_step,auto) |
49521 | 693 |
apply (drule mono_stream_take_pred,simp) |
694 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
695 |
|
17291 | 696 |
lemma take_i_rt_prefix_lemma: |
63549 | 697 |
"\<lbrakk>stream_take n\<cdot>s1 << stream_take n\<cdot>s2; i_rt n s1 << i_rt n s2\<rbrakk> \<Longrightarrow> s1 << s2" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
698 |
apply (case_tac "n=0",simp) |
25161 | 699 |
apply (auto) |
63549 | 700 |
apply (subgoal_tac "stream_take 0\<cdot>s1 << stream_take 0\<cdot>s2 \<and> i_rt 0 s1 << i_rt 0 s2") |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
701 |
defer 1 |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
702 |
apply (rule zero_induct,blast) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
703 |
apply (blast dest: take_i_rt_prefix_lemma1) |
49521 | 704 |
apply simp |
705 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
706 |
|
63549 | 707 |
lemma streams_prefix_lemma: "s1 << s2 \<longleftrightarrow> |
708 |
(stream_take n\<cdot>s1 << stream_take n\<cdot>s2 \<and> i_rt n s1 << i_rt n s2)" |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
709 |
apply auto |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
710 |
apply (simp add: monofun_cfun_arg) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
711 |
apply (simp add: i_rt_mono) |
49521 | 712 |
apply (erule take_i_rt_prefix_lemma,simp) |
713 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
714 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
715 |
lemma streams_prefix_lemma1: |
63549 | 716 |
"\<lbrakk>stream_take n\<cdot>s1 = stream_take n\<cdot>s2; i_rt n s1 = i_rt n s2\<rbrakk> \<Longrightarrow> s1 = s2" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
717 |
apply (simp add: po_eq_conv,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
718 |
apply (insert streams_prefix_lemma) |
49521 | 719 |
apply blast+ |
720 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
721 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
722 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
723 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
724 |
section "sconc" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
725 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
726 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
727 |
lemma UU_sconc [simp]: " UU ooo s = s " |
43919 | 728 |
by (simp add: sconc_def zero_enat_def) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
729 |
|
63549 | 730 |
lemma scons_neq_UU: "a \<noteq> UU \<Longrightarrow> a && s \<noteq> UU" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
731 |
by auto |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
732 |
|
63549 | 733 |
lemma singleton_sconc [rule_format, simp]: "x \<noteq> UU \<longrightarrow> (x && UU) ooo y = x && y" |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
734 |
apply (simp add: sconc_def zero_enat_def eSuc_def split: enat.splits, auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
735 |
apply (rule someI2_ex,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
736 |
apply (rule_tac x="x && y" in exI,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
737 |
apply (simp add: i_rt_Suc_forw) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
738 |
apply (case_tac "xa=UU",simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
739 |
by (drule stream_exhaust_eq [THEN iffD1],auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
740 |
|
17291 | 741 |
lemma ex_sconc [rule_format]: |
63549 | 742 |
"\<forall>k y. #x = enat k \<longrightarrow> (\<exists>w. stream_take k\<cdot>w = x \<and> i_rt k w = y)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
743 |
apply (case_tac "#x") |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
744 |
apply (rule stream_finite_ind [of x],auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
745 |
apply (simp add: stream.finite_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
746 |
apply (drule slen_take_lemma1,blast) |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
747 |
apply (simp_all add: zero_enat_def eSuc_def split: enat.splits) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
748 |
apply (erule_tac x="y" in allE,auto) |
49521 | 749 |
apply (rule_tac x="a && w" in exI,auto) |
750 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
751 |
|
63549 | 752 |
lemma rt_sconc1: "enat n = #x \<Longrightarrow> i_rt n (x ooo y) = y" |
43919 | 753 |
apply (simp add: sconc_def split: enat.splits, arith?,auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
754 |
apply (rule someI2_ex,auto) |
49521 | 755 |
apply (drule ex_sconc,simp) |
756 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
757 |
|
43924 | 758 |
lemma sconc_inj2: "\<lbrakk>enat n = #x; x ooo y = x ooo z\<rbrakk> \<Longrightarrow> y = z" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
759 |
apply (frule_tac y=y in rt_sconc1) |
49521 | 760 |
apply (auto elim: rt_sconc1) |
761 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
762 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
763 |
lemma sconc_UU [simp]:"s ooo UU = s" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
764 |
apply (case_tac "#s") |
27111 | 765 |
apply (simp add: sconc_def) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
766 |
apply (rule someI2_ex) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
767 |
apply (rule_tac x="s" in exI) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
768 |
apply auto |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
769 |
apply (drule slen_take_lemma1,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
770 |
apply (simp add: i_rt_lemma_slen) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
771 |
apply (drule slen_take_lemma1,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
772 |
apply (simp add: i_rt_slen) |
49521 | 773 |
apply (simp add: sconc_def) |
774 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
775 |
|
63549 | 776 |
lemma stream_take_sconc [simp]: "enat n = #x \<Longrightarrow> stream_take n\<cdot>(x ooo y) = x" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
777 |
apply (simp add: sconc_def) |
27111 | 778 |
apply (cases "#x") |
779 |
apply auto |
|
780 |
apply (rule someI2_ex, auto) |
|
49521 | 781 |
apply (drule ex_sconc,simp) |
782 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
783 |
|
63549 | 784 |
lemma scons_sconc [rule_format,simp]: "a \<noteq> UU \<longrightarrow> (a && x) ooo y = a && x ooo y" |
27111 | 785 |
apply (cases "#x",auto) |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
786 |
apply (simp add: sconc_def eSuc_enat) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
787 |
apply (rule someI2_ex) |
27111 | 788 |
apply (drule ex_sconc, simp) |
789 |
apply (rule someI2_ex, auto) |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
790 |
apply (simp add: i_rt_Suc_forw) |
57492
74bf65a1910a
Hypsubst preserves equality hypotheses
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
49521
diff
changeset
|
791 |
apply (rule_tac x="a && xa" in exI, auto) |
74bf65a1910a
Hypsubst preserves equality hypotheses
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
49521
diff
changeset
|
792 |
apply (case_tac "xaa=UU",auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
793 |
apply (drule stream_exhaust_eq [THEN iffD1],auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
794 |
apply (drule streams_prefix_lemma1,simp+) |
49521 | 795 |
apply (simp add: sconc_def) |
796 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
797 |
|
63549 | 798 |
lemma ft_sconc: "x \<noteq> UU \<Longrightarrow> ft\<cdot>(x ooo y) = ft\<cdot>x" |
49521 | 799 |
by (cases x) auto |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
800 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
801 |
lemma sconc_assoc: "(x ooo y) ooo z = x ooo y ooo z" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
802 |
apply (case_tac "#x") |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
803 |
apply (rule stream_finite_ind [of x],auto simp del: scons_sconc) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
804 |
apply (simp add: stream.finite_def del: scons_sconc) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
805 |
apply (drule slen_take_lemma1,auto simp del: scons_sconc) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
806 |
apply (case_tac "a = UU", auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
807 |
by (simp add: sconc_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
808 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
809 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
810 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
811 |
|
25833 | 812 |
lemma cont_sconc_lemma1: "stream_finite x \<Longrightarrow> cont (\<lambda>y. x ooo y)" |
813 |
by (erule stream_finite_ind, simp_all) |
|
814 |
||
815 |
lemma cont_sconc_lemma2: "\<not> stream_finite x \<Longrightarrow> cont (\<lambda>y. x ooo y)" |
|
816 |
by (simp add: sconc_def slen_def) |
|
817 |
||
818 |
lemma cont_sconc: "cont (\<lambda>y. x ooo y)" |
|
819 |
apply (cases "stream_finite x") |
|
820 |
apply (erule cont_sconc_lemma1) |
|
821 |
apply (erule cont_sconc_lemma2) |
|
822 |
done |
|
823 |
||
63549 | 824 |
lemma sconc_mono: "y << y' \<Longrightarrow> x ooo y << x ooo y'" |
25833 | 825 |
by (rule cont_sconc [THEN cont2mono, THEN monofunE]) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
826 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
827 |
lemma sconc_mono1 [simp]: "x << x ooo y" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
828 |
by (rule sconc_mono [of UU, simplified]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
829 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
830 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
831 |
|
63549 | 832 |
lemma empty_sconc [simp]: "x ooo y = UU \<longleftrightarrow> x = UU \<and> y = UU" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
833 |
apply (case_tac "#x",auto) |
17291 | 834 |
apply (insert sconc_mono1 [of x y]) |
49521 | 835 |
apply auto |
836 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
837 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
838 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
839 |
|
63549 | 840 |
lemma rt_sconc [rule_format, simp]: "s \<noteq> UU \<longrightarrow> rt\<cdot>(s ooo x) = rt\<cdot>s ooo x" |
35781
b7738ab762b1
renamed some lemmas generated by the domain package
huffman
parents:
35642
diff
changeset
|
841 |
by (cases s, auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
842 |
|
17291 | 843 |
lemma i_th_sconc_lemma [rule_format]: |
63549 | 844 |
"\<forall>x y. enat n < #x \<longrightarrow> i_th n (x ooo y) = i_th n x" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
845 |
apply (induct_tac n, auto) |
43924 | 846 |
apply (simp add: enat_0 i_th_def) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
847 |
apply (simp add: slen_empty_eq ft_sconc) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
848 |
apply (simp add: i_th_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
849 |
apply (case_tac "x=UU",auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
850 |
apply (drule stream_exhaust_eq [THEN iffD1], auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
851 |
apply (erule_tac x="ya" in allE) |
49521 | 852 |
apply (case_tac "#ya") |
853 |
apply simp_all |
|
854 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
855 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
856 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
857 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
858 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
859 |
|
63549 | 860 |
lemma sconc_lemma [rule_format, simp]: "\<forall>s. stream_take n\<cdot>s ooo i_rt n s = s" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
861 |
apply (induct_tac n,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
862 |
apply (case_tac "s=UU",auto) |
49521 | 863 |
apply (drule stream_exhaust_eq [THEN iffD1],auto) |
864 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
865 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
866 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
867 |
subsection "pointwise equality" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
868 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
869 |
|
63549 | 870 |
lemma ex_last_stream_take_scons: "stream_take (Suc n)\<cdot>s = |
871 |
stream_take n\<cdot>s ooo i_rt n (stream_take (Suc n)\<cdot>s)" |
|
872 |
by (insert sconc_lemma [of n "stream_take (Suc n)\<cdot>s"],simp) |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
873 |
|
17291 | 874 |
lemma i_th_stream_take_eq: |
63549 | 875 |
"\<And>n. \<forall>n. i_th n s1 = i_th n s2 \<Longrightarrow> stream_take n\<cdot>s1 = stream_take n\<cdot>s2" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
876 |
apply (induct_tac n,auto) |
63549 | 877 |
apply (subgoal_tac "stream_take (Suc na)\<cdot>s1 = |
878 |
stream_take na\<cdot>s1 ooo i_rt na (stream_take (Suc na)\<cdot>s1)") |
|
879 |
apply (subgoal_tac "i_rt na (stream_take (Suc na)\<cdot>s1) = |
|
880 |
i_rt na (stream_take (Suc na)\<cdot>s2)") |
|
881 |
apply (subgoal_tac "stream_take (Suc na)\<cdot>s2 = |
|
882 |
stream_take na\<cdot>s2 ooo i_rt na (stream_take (Suc na)\<cdot>s2)") |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
883 |
apply (insert ex_last_stream_take_scons,simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
884 |
apply blast |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
885 |
apply (erule_tac x="na" in allE) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
886 |
apply (insert i_th_last_eq [of _ s1 s2]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
887 |
by blast+ |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
888 |
|
63549 | 889 |
lemma pointwise_eq_lemma[rule_format]: "\<forall>n. i_th n s1 = i_th n s2 \<Longrightarrow> s1 = s2" |
35642
f478d5a9d238
generate separate qualified theorem name for each type's reach and take_lemma
huffman
parents:
35557
diff
changeset
|
890 |
by (insert i_th_stream_take_eq [THEN stream.take_lemma],blast) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
891 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
892 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
893 |
subsection "finiteness" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
894 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
895 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
896 |
lemma slen_sconc_finite1: |
63549 | 897 |
"\<lbrakk>#(x ooo y) = \<infinity>; enat n = #x\<rbrakk> \<Longrightarrow> #y = \<infinity>" |
898 |
apply (case_tac "#y \<noteq> \<infinity>",auto) |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
899 |
apply (drule_tac y=y in rt_sconc1) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
900 |
apply (insert stream_finite_i_rt [of n "x ooo y"]) |
49521 | 901 |
apply (simp add: slen_infinite) |
902 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
903 |
|
63549 | 904 |
lemma slen_sconc_infinite1: "#x=\<infinity> \<Longrightarrow> #(x ooo y) = \<infinity>" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
905 |
by (simp add: sconc_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
906 |
|
63549 | 907 |
lemma slen_sconc_infinite2: "#y=\<infinity> \<Longrightarrow> #(x ooo y) = \<infinity>" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
908 |
apply (case_tac "#x") |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
909 |
apply (simp add: sconc_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
910 |
apply (rule someI2_ex) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
911 |
apply (drule ex_sconc,auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
912 |
apply (erule contrapos_pp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
913 |
apply (insert stream_finite_i_rt) |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44019
diff
changeset
|
914 |
apply (fastforce simp add: slen_infinite,auto) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
915 |
by (simp add: sconc_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
916 |
|
63549 | 917 |
lemma sconc_finite: "#x \<noteq> \<infinity> \<and> #y \<noteq> \<infinity> \<longleftrightarrow> #(x ooo y) \<noteq> \<infinity>" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
918 |
apply auto |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
919 |
apply (metis not_infinity_eq slen_sconc_finite1) |
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
920 |
apply (metis not_infinity_eq slen_sconc_infinite1) |
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
921 |
apply (metis not_infinity_eq slen_sconc_infinite2) |
31084 | 922 |
done |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
923 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
924 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
925 |
|
63549 | 926 |
lemma slen_sconc_mono3: "\<lbrakk>enat n = #x; enat k = #(x ooo y)\<rbrakk> \<Longrightarrow> n \<le> k" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
927 |
apply (insert slen_mono [of "x" "x ooo y"]) |
27111 | 928 |
apply (cases "#x") apply simp_all |
929 |
apply (cases "#(x ooo y)") apply simp_all |
|
930 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
931 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
932 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
933 |
subsection "finite slen" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
934 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
935 |
|
63549 | 936 |
lemma slen_sconc: "\<lbrakk>enat n = #x; enat m = #y\<rbrakk> \<Longrightarrow> #(x ooo y) = enat (n + m)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
937 |
apply (case_tac "#(x ooo y)") |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
938 |
apply (frule_tac y=y in rt_sconc1) |
43924 | 939 |
apply (insert take_i_rt_len [of "THE j. enat j = #(x ooo y)" "x ooo y" n n m],simp) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
940 |
apply (insert slen_sconc_mono3 [of n x _ y],simp) |
49521 | 941 |
apply (insert sconc_finite [of x y],auto) |
942 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
943 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
944 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
945 |
subsection "flat prefix" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
946 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
947 |
|
63549 | 948 |
lemma sconc_prefix: "(s1::'a::flat stream) << s2 \<Longrightarrow> \<exists>t. s1 ooo t = s2" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
949 |
apply (case_tac "#s1") |
63549 | 950 |
apply (subgoal_tac "stream_take nat\<cdot>s1 = stream_take nat\<cdot>s2") |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
951 |
apply (rule_tac x="i_rt nat s2" in exI) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
952 |
apply (simp add: sconc_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
953 |
apply (rule someI2_ex) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
954 |
apply (drule ex_sconc) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
955 |
apply (simp,clarsimp,drule streams_prefix_lemma1) |
17291 | 956 |
apply (simp+,rule slen_take_lemma3 [of _ s1 s2]) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
957 |
apply (simp+,rule_tac x="UU" in exI) |
17291 | 958 |
apply (insert slen_take_lemma3 [of _ s1 s2]) |
49521 | 959 |
apply (rule stream.take_lemma,simp) |
960 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
961 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
962 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
963 |
subsection "continuity" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
964 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
965 |
|
63549 | 966 |
lemma chain_sconc: "chain S \<Longrightarrow> chain (\<lambda>i. (x ooo S i))" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
967 |
by (simp add: chain_def,auto simp add: sconc_mono) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
968 |
|
63549 | 969 |
lemma chain_scons: "chain S \<Longrightarrow> chain (\<lambda>i. a && S i)" |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
970 |
apply (simp add: chain_def,auto) |
49521 | 971 |
apply (rule monofun_cfun_arg,simp) |
972 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
973 |
|
63549 | 974 |
lemma contlub_scons_lemma: "chain S \<Longrightarrow> (LUB i. a && S i) = a && (LUB i. S i)" |
40327 | 975 |
by (rule cont2contlubE [OF cont_Rep_cfun2, symmetric]) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
976 |
|
63549 | 977 |
lemma finite_lub_sconc: "chain Y \<Longrightarrow> stream_finite x \<Longrightarrow> |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
978 |
(LUB i. x ooo Y i) = (x ooo (LUB i. Y i))" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
979 |
apply (rule stream_finite_ind [of x]) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
980 |
apply (auto) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
981 |
apply (subgoal_tac "(LUB i. a && (s ooo Y i)) = a && (LUB i. s ooo Y i)") |
49521 | 982 |
apply (force,blast dest: contlub_scons_lemma chain_sconc) |
983 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
984 |
|
17291 | 985 |
lemma contlub_sconc_lemma: |
63549 | 986 |
"chain Y \<Longrightarrow> (LUB i. x ooo Y i) = (x ooo (LUB i. Y i))" |
43921 | 987 |
apply (case_tac "#x=\<infinity>") |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
988 |
apply (simp add: sconc_def) |
18075 | 989 |
apply (drule finite_lub_sconc,auto simp add: slen_infinite) |
990 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
991 |
|
63549 | 992 |
lemma monofun_sconc: "monofun (\<lambda>y. x ooo y)" |
16218 | 993 |
by (simp add: monofun_def sconc_mono) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
994 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
995 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
996 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
997 |
section "constr_sconc" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
998 |
(* ----------------------------------------------------------------------- *) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
999 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1000 |
lemma constr_sconc_UUs [simp]: "constr_sconc UU s = s" |
43919 | 1001 |
by (simp add: constr_sconc_def zero_enat_def) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1002 |
|
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1003 |
lemma "x ooo y = constr_sconc x y" |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1004 |
apply (case_tac "#x") |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1005 |
apply (rule stream_finite_ind [of x],auto simp del: scons_sconc) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1006 |
defer 1 |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1007 |
apply (simp add: constr_sconc_def del: scons_sconc) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1008 |
apply (case_tac "#s") |
44019
ee784502aed5
Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents:
43924
diff
changeset
|
1009 |
apply (simp add: eSuc_enat) |
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1010 |
apply (case_tac "a=UU",auto simp del: scons_sconc) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1011 |
apply (simp) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1012 |
apply (simp add: sconc_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1013 |
apply (simp add: constr_sconc_def) |
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1014 |
apply (simp add: stream.finite_def) |
49521 | 1015 |
apply (drule slen_take_lemma1,auto) |
1016 |
done |
|
15188
9d57263faf9e
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
oheimb
parents:
14981
diff
changeset
|
1017 |
|
2570 | 1018 |
end |