1 (* Title: HOLCF/FOCUS/Fstreams.thy
2 Author: Borislav Gajanovic
4 FOCUS flat streams (with lifted elements).
6 TODO: integrate this with Fstream.
15 types 'a fstream = "('a lift) stream"
18 fsingleton :: "'a => 'a fstream" ("<_>" [1000] 999) where
19 fsingleton_def2: "fsingleton = (%a. Def a && UU)"
22 fsfilter :: "'a set \<Rightarrow> 'a fstream \<rightarrow> 'a fstream" where
23 "fsfilter A = sfilter\<cdot>(flift2 (\<lambda>x. x\<in>A))"
26 fsmap :: "('a => 'b) => 'a fstream -> 'b fstream" where
27 "fsmap f = smap$(flift2 f)"
30 jth :: "nat => 'a fstream => 'a" where
31 "jth = (%n s. if Fin n < #s then THE a. i_th n s = Def a else undefined)"
34 first :: "'a fstream => 'a" where
35 "first = (%s. jth 0 s)"
38 last :: "'a fstream => 'a" where
39 "last = (%s. case #s of Fin n => (if n~=0 then jth (THE k. Suc k = n) s else undefined))"
43 emptystream :: "'a fstream" ("<>") where
47 fsfilter' :: "'a set \<Rightarrow> 'a fstream \<Rightarrow> 'a fstream" ("(_'(C')_)" [64,63] 63) where
48 "A(C)s == fsfilter A\<cdot>s"
51 fsfilter' ("(_\<copyright>_)" [64,63] 63)
54 lemma ft_fsingleton[simp]: "ft$(<a>) = Def a"
55 by (simp add: fsingleton_def2)
57 lemma slen_fsingleton[simp]: "#(<a>) = Fin 1"
58 by (simp add: fsingleton_def2 inat_defs)
60 lemma slen_fstreams[simp]: "#(<a> ooo s) = iSuc (#s)"
61 by (simp add: fsingleton_def2)
63 lemma slen_fstreams2[simp]: "#(s ooo <a>) = iSuc (#s)"
65 apply (auto simp add: iSuc_Fin)
66 apply (insert slen_sconc [of _ s "Suc 0" "<a>"], auto)
67 by (simp add: sconc_def)
69 lemma j_th_0_fsingleton[simp]:"jth 0 (<a>) = a"
70 apply (simp add: fsingleton_def2 jth_def)
71 by (simp add: i_th_def Fin_0)
73 lemma jth_0[simp]: "jth 0 (<a> ooo s) = a"
74 apply (simp add: fsingleton_def2 jth_def)
75 by (simp add: i_th_def Fin_0)
77 lemma first_sconc[simp]: "first (<a> ooo s) = a"
78 by (simp add: first_def)
80 lemma first_fsingleton[simp]: "first (<a>) = a"
81 by (simp add: first_def)
83 lemma jth_n[simp]: "Fin n = #s ==> jth n (s ooo <a>) = a"
84 apply (simp add: jth_def, auto)
85 apply (simp add: i_th_def rt_sconc1)
86 by (simp add: inat_defs split: inat_splits)
88 lemma last_sconc[simp]: "Fin n = #s ==> last (s ooo <a>) = a"
89 apply (simp add: last_def)
90 apply (simp add: inat_defs split:inat_splits)
93 lemma last_fsingleton[simp]: "last (<a>) = a"
94 by (simp add: last_def)
96 lemma first_UU[simp]: "first UU = undefined"
97 by (simp add: first_def jth_def)
99 lemma last_UU[simp]:"last UU = undefined"
100 by (simp add: last_def jth_def inat_defs)
102 lemma last_infinite[simp]:"#s = Infty ==> last s = undefined"
103 by (simp add: last_def)
105 lemma jth_slen_lemma1:"n <= k & Fin n = #s ==> jth k s = undefined"
106 by (simp add: jth_def inat_defs split:inat_splits, auto)
108 lemma jth_UU[simp]:"jth n UU = undefined"
109 by (simp add: jth_def)
111 lemma ext_last:"[|s ~= UU; Fin (Suc n) = #s|] ==> (stream_take n$s) ooo <(last s)> = s"
112 apply (simp add: last_def)
113 apply (case_tac "#s", auto)
114 apply (simp add: fsingleton_def2)
115 apply (subgoal_tac "Def (jth n s) = i_th n s")
116 apply (auto simp add: i_th_last)
117 apply (drule slen_take_lemma1, auto)
118 apply (simp add: jth_def)
119 apply (case_tac "i_th n s = UU")
121 apply (simp add: i_th_def)
122 apply (case_tac "i_rt n s = UU", auto)
123 apply (drule i_rt_slen [THEN iffD1])
124 apply (drule slen_take_eq_rev [rule_format, THEN iffD2],auto)
125 by (drule not_Undef_is_Def [THEN iffD1], auto)
128 lemma fsingleton_lemma1[simp]: "(<a> = <b>) = (a=b)"
129 by (simp add: fsingleton_def2)
131 lemma fsingleton_lemma2[simp]: "<a> ~= <>"
132 by (simp add: fsingleton_def2)
134 lemma fsingleton_sconc:"<a> ooo s = Def a && s"
135 by (simp add: fsingleton_def2)
138 "[| adm P; P <>; !!a s. P s ==> P (<a> ooo s) |] ==> P x"
139 apply (simp add: fsingleton_def2)
140 apply (rule stream.induct, auto)
141 by (drule not_Undef_is_Def [THEN iffD1], auto)
144 "[| adm P; P <>; !!a. P (<a>); !!a b s. P s ==> P (<a> ooo <b> ooo s) |] ==> P x"
145 apply (simp add: fsingleton_def2)
146 apply (rule stream_ind2, auto)
147 by (drule not_Undef_is_Def [THEN iffD1], auto)+
149 lemma fstreams_take_Suc[simp]: "stream_take (Suc n)$(<a> ooo s) = <a> ooo stream_take n$s"
150 by (simp add: fsingleton_def2)
152 lemma fstreams_not_empty[simp]: "<a> ooo s ~= <>"
153 by (simp add: fsingleton_def2)
155 lemma fstreams_not_empty2[simp]: "s ooo <a> ~= <>"
156 by (case_tac "s=UU", auto)
158 lemma fstreams_exhaust: "x = UU | (EX a s. x = <a> ooo s)"
159 apply (simp add: fsingleton_def2, auto)
160 apply (erule contrapos_pp, auto)
161 apply (drule stream_exhaust_eq [THEN iffD1], auto)
162 by (drule not_Undef_is_Def [THEN iffD1], auto)
164 lemma fstreams_cases: "[| x = UU ==> P; !!a y. x = <a> ooo y ==> P |] ==> P"
165 by (insert fstreams_exhaust [of x], auto)
167 lemma fstreams_exhaust_eq: "(x ~= UU) = (? a y. x = <a> ooo y)"
168 apply (simp add: fsingleton_def2, auto)
169 apply (drule stream_exhaust_eq [THEN iffD1], auto)
170 by (drule not_Undef_is_Def [THEN iffD1], auto)
172 lemma fstreams_inject: "(<a> ooo s = <b> ooo t) = (a=b & s=t)"
173 by (simp add: fsingleton_def2)
175 lemma fstreams_prefix: "<a> ooo s << t ==> EX tt. t = <a> ooo tt & s << tt"
176 apply (simp add: fsingleton_def2)
177 apply (insert stream_prefix [of "Def a" s t], auto)
180 lemma fstreams_prefix': "x << <a> ooo z = (x = <> | (EX y. x = <a> ooo y & y << z))"
181 apply (auto, case_tac "x=UU", auto)
182 apply (drule stream_exhaust_eq [THEN iffD1], auto)
183 apply (simp add: fsingleton_def2, auto)
184 apply (drule ax_flat, simp)
185 by (erule sconc_mono)
187 lemma ft_fstreams[simp]: "ft$(<a> ooo s) = Def a"
188 by (simp add: fsingleton_def2)
190 lemma rt_fstreams[simp]: "rt$(<a> ooo s) = s"
191 by (simp add: fsingleton_def2)
193 lemma ft_eq[simp]: "(ft$s = Def a) = (EX t. s = <a> ooo t)"
194 apply (cases s, auto)
195 by ((*drule sym,*) auto simp add: fsingleton_def2)
197 lemma surjective_fstreams: "(<d> ooo y = x) = (ft$x = Def d & rt$x = y)"
200 lemma fstreams_mono: "<a> ooo b << <a> ooo c ==> b << c"
201 by (simp add: fsingleton_def2)
203 lemma fsmap_UU[simp]: "fsmap f$UU = UU"
204 by (simp add: fsmap_def)
206 lemma fsmap_fsingleton_sconc: "fsmap f$(<x> ooo xs) = <(f x)> ooo (fsmap f$xs)"
207 by (simp add: fsmap_def fsingleton_def2 flift2_def)
209 lemma fsmap_fsingleton[simp]: "fsmap f$(<x>) = <(f x)>"
210 by (simp add: fsmap_def fsingleton_def2 flift2_def)
213 lemma fstreams_chain_lemma[rule_format]:
214 "ALL s x y. stream_take n$(s::'a fstream) << x & x << y & y << s & x ~= y --> stream_take (Suc n)$s << y"
215 apply (induct_tac n, auto)
216 apply (case_tac "s=UU", auto)
217 apply (drule stream_exhaust_eq [THEN iffD1], auto)
218 apply (case_tac "y=UU", auto)
219 apply (drule stream_exhaust_eq [THEN iffD1], auto)
220 apply (simp add: flat_below_iff)
221 apply (case_tac "s=UU", auto)
222 apply (drule stream_exhaust_eq [THEN iffD1], auto)
223 apply (erule_tac x="ya" in allE)
224 apply (drule stream_prefix, auto)
225 apply (case_tac "y=UU",auto)
226 apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)
228 apply (simp add: flat_below_iff)
229 apply (erule_tac x="tt" in allE)
230 apply (erule_tac x="yb" in allE, auto)
231 apply (simp add: flat_below_iff)
232 by (simp add: flat_below_iff)
234 lemma fstreams_lub_lemma1: "[| chain Y; (LUB i. Y i) = <a> ooo s |] ==> EX j t. Y j = <a> ooo t"
235 apply (subgoal_tac "(LUB i. Y i) ~= UU")
236 apply (drule chain_UU_I_inverse2, auto)
237 apply (drule_tac x="i" in is_ub_thelub, auto)
238 by (drule fstreams_prefix' [THEN iffD1], auto)
241 "[| chain Y; (LUB i. Y i) = <a> ooo s |]
242 ==> (EX j t. Y j = <a> ooo t) & (EX X. chain X & (ALL i. EX j. <a> ooo X i << Y j) & (LUB i. X i) = s)"
243 apply (auto simp add: fstreams_lub_lemma1)
244 apply (rule_tac x="%n. stream_take n$s" in exI, auto)
245 apply (induct_tac i, auto)
246 apply (drule fstreams_lub_lemma1, auto)
247 apply (rule_tac x="j" in exI, auto)
248 apply (case_tac "max_in_chain j Y")
249 apply (frule lub_finch1 [THEN lub_eqI], auto)
250 apply (rule_tac x="j" in exI)
251 apply (erule subst) back back
252 apply (simp add: below_prod_def sconc_mono)
253 apply (simp add: max_in_chain_def, auto)
254 apply (rule_tac x="ja" in exI)
255 apply (subgoal_tac "Y j << Y ja")
256 apply (drule fstreams_prefix, auto)+
257 apply (rule sconc_mono)
258 apply (rule fstreams_chain_lemma, auto)
259 apply (subgoal_tac "Y ja << (LUB i. (Y i))", clarsimp)
260 apply (drule fstreams_mono, simp)
261 apply (rule is_ub_thelub, simp)
262 apply (blast intro: chain_mono)
263 by (rule stream_reach2)
266 lemma lub_Pair_not_UU_lemma:
267 "[| chain Y; (LUB i. Y i) = ((a::'a::flat), b); a ~= UU; b ~= UU |]
268 ==> EX j c d. Y j = (c, d) & c ~= UU & d ~= UU"
269 apply (frule lub_prod, clarsimp)
270 apply (drule chain_UU_I_inverse2, clarsimp)
271 apply (case_tac "Y i", clarsimp)
272 apply (case_tac "max_in_chain i Y")
273 apply (drule maxinch_is_thelub, auto)
274 apply (rule_tac x="i" in exI, auto)
275 apply (simp add: max_in_chain_def, auto)
276 apply (subgoal_tac "Y i << Y j",auto)
277 apply (simp add: below_prod_def, clarsimp)
278 apply (drule ax_flat, auto)
279 apply (case_tac "snd (Y j) = UU",auto)
280 apply (case_tac "Y j", auto)
281 apply (rule_tac x="j" in exI)
282 apply (case_tac "Y j",auto)
283 by (drule chain_mono, auto)
285 lemma fstreams_lub_lemma2:
286 "[| chain Y; (LUB i. Y i) = (a, <m> ooo ms); (a::'a::flat) ~= UU |] ==> EX j t. Y j = (a, <m> ooo t)"
287 apply (frule lub_Pair_not_UU_lemma, auto)
288 apply (drule_tac x="j" in is_ub_thelub, auto)
289 apply (drule ax_flat, clarsimp)
290 by (drule fstreams_prefix' [THEN iffD1], auto)
293 "[| chain Y; (LUB i. Y i) = (a, <m> ooo ms); (a::'a::flat) ~= UU |]
294 ==> (EX j t. Y j = (a, <m> ooo t)) & (EX X. chain X & (ALL i. EX j. (a, <m> ooo X i) << Y j) & (LUB i. X i) = ms)"
295 apply (auto simp add: fstreams_lub_lemma2)
296 apply (rule_tac x="%n. stream_take n$ms" in exI, auto)
297 apply (induct_tac i, auto)
298 apply (drule fstreams_lub_lemma2, auto)
299 apply (rule_tac x="j" in exI, auto)
300 apply (case_tac "max_in_chain j Y")
301 apply (frule lub_finch1 [THEN lub_eqI], auto)
302 apply (rule_tac x="j" in exI)
303 apply (erule subst) back back
304 apply (simp add: sconc_mono)
305 apply (simp add: max_in_chain_def, auto)
306 apply (rule_tac x="ja" in exI)
307 apply (subgoal_tac "Y j << Y ja")
308 apply (simp add: below_prod_def, auto)
309 apply (drule below_trans)
310 apply (simp add: ax_flat, auto)
311 apply (drule fstreams_prefix, auto)+
312 apply (rule sconc_mono)
313 apply (subgoal_tac "tt ~= tta" "tta << ms")
314 apply (blast intro: fstreams_chain_lemma)
315 apply (frule lub_prod, auto)
316 apply (subgoal_tac "snd (Y ja) << (LUB i. snd (Y i))", clarsimp)
317 apply (drule fstreams_mono, simp)
318 apply (rule is_ub_thelub chainI)
319 apply (simp add: chain_def below_prod_def)
320 apply (subgoal_tac "fst (Y j) ~= fst (Y ja) | snd (Y j) ~= snd (Y ja)", simp)
321 apply (drule ax_flat, simp)+
322 apply (drule prod_eqI, auto)
323 apply (simp add: chain_mono)
324 by (rule stream_reach2)
327 lemma cpo_cont_lemma:
328 "[| monofun (f::'a::cpo => 'b::cpo); (!Y. chain Y --> f (lub(range Y)) << (LUB i. f (Y i))) |] ==> cont f"
329 by (erule contI2, simp)