|
444
|
1 |
theory ClosedForms imports
|
|
|
2 |
"BasicIdentities"
|
|
443
|
3 |
begin
|
|
|
4 |
|
|
453
|
5 |
lemma map_concat_cons:
|
|
|
6 |
shows "map f rsa @ f a # rs = map f (rsa @ [a]) @ rs"
|
|
|
7 |
by simp
|
|
|
8 |
|
|
|
9 |
lemma neg_removal_element_of:
|
|
|
10 |
shows " \<not> a \<notin> aset \<Longrightarrow> a \<in> aset"
|
|
|
11 |
by simp
|
|
|
12 |
|
|
465
|
13 |
|
|
|
14 |
|
|
|
15 |
|
|
|
16 |
|
|
478
|
17 |
|
|
465
|
18 |
|
|
|
19 |
|
|
478
|
20 |
|
|
465
|
21 |
|
|
|
22 |
lemma flts_middle0:
|
|
|
23 |
shows "rflts (rsa @ RZERO # rsb) = rflts (rsa @ rsb)"
|
|
|
24 |
apply(induct rsa)
|
|
|
25 |
apply simp
|
|
|
26 |
by (metis append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
|
|
|
27 |
|
|
|
28 |
lemma flts_middle01:
|
|
|
29 |
shows "rflts (rsa @ [RZERO] @ rsb) = rflts (rsa @ rsb)"
|
|
|
30 |
by (simp add: flts_middle0)
|
|
|
31 |
|
|
|
32 |
lemma flts_append1:
|
|
|
33 |
shows "\<lbrakk>a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1; a \<in> set rs\<rbrakk> \<Longrightarrow>
|
|
|
34 |
rflts (rsa @ [a] @ rsb) = rflts rsa @ [a] @ (rflts rsb)"
|
|
|
35 |
apply(induct rsa arbitrary: rsb)
|
|
|
36 |
apply simp
|
|
|
37 |
using rflts_def_idiot apply presburger
|
|
|
38 |
apply(case_tac aa)
|
|
|
39 |
apply simp+
|
|
|
40 |
done
|
|
|
41 |
|
|
478
|
42 |
|
|
465
|
43 |
|
|
|
44 |
lemma simp_removes_duplicate1:
|
|
|
45 |
shows " a \<in> set rsa \<Longrightarrow> rsimp (RALTS (rsa @ [a])) = rsimp (RALTS (rsa))"
|
|
|
46 |
and " rsimp (RALTS (a1 # rsa @ [a1])) = rsimp (RALTS (a1 # rsa))"
|
|
|
47 |
apply(induct rsa arbitrary: a1)
|
|
|
48 |
apply simp
|
|
|
49 |
apply simp
|
|
|
50 |
prefer 2
|
|
|
51 |
apply(case_tac "a = aa")
|
|
|
52 |
apply simp
|
|
|
53 |
apply simp
|
|
|
54 |
apply (metis Cons_eq_appendI Cons_eq_map_conv distinct_removes_duplicate_flts list.set_intros(2))
|
|
|
55 |
apply (metis append_Cons append_Nil distinct_removes_duplicate_flts list.set_intros(1) list.simps(8) list.simps(9))
|
|
|
56 |
by (metis (mono_tags, lifting) append_Cons distinct_removes_duplicate_flts list.set_intros(1) list.simps(8) list.simps(9) map_append rsimp.simps(2))
|
|
|
57 |
|
|
|
58 |
lemma simp_removes_duplicate2:
|
|
|
59 |
shows "a \<in> set rsa \<Longrightarrow> rsimp (RALTS (rsa @ [a] @ rsb)) = rsimp (RALTS (rsa @ rsb))"
|
|
|
60 |
apply(induct rsb arbitrary: rsa)
|
|
|
61 |
apply simp
|
|
|
62 |
using distinct_removes_duplicate_flts apply auto[1]
|
|
|
63 |
by (metis append.assoc head_one_more_simp rsimp.simps(2) simp_flatten simp_removes_duplicate1(1))
|
|
|
64 |
|
|
|
65 |
lemma simp_removes_duplicate3:
|
|
|
66 |
shows "a \<in> set rsa \<Longrightarrow> rsimp (RALTS (rsa @ a # rsb)) = rsimp (RALTS (rsa @ rsb))"
|
|
|
67 |
using simp_removes_duplicate2 by auto
|
|
|
68 |
|
|
|
69 |
lemma distinct_removes_middle4:
|
|
|
70 |
shows "a \<in> set rsa \<Longrightarrow> rdistinct (rsa @ [a] @ rsb) rset = rdistinct (rsa @ rsb) rset"
|
|
|
71 |
using distinct_removes_middle(1) by fastforce
|
|
|
72 |
|
|
|
73 |
lemma distinct_removes_middle_list:
|
|
|
74 |
shows "\<forall>a \<in> set x. a \<in> set rsa \<Longrightarrow> rdistinct (rsa @ x @ rsb) rset = rdistinct (rsa @ rsb) rset"
|
|
|
75 |
apply(induct x)
|
|
|
76 |
apply simp
|
|
|
77 |
by (simp add: distinct_removes_middle3)
|
|
|
78 |
|
|
|
79 |
|
|
|
80 |
lemma distinct_removes_duplicate_flts2:
|
|
|
81 |
shows " a \<in> set rsa
|
|
|
82 |
\<Longrightarrow> rdistinct (rflts (rsa @ [a] @ rsb)) {} =
|
|
|
83 |
rdistinct (rflts (rsa @ rsb)) {}"
|
|
|
84 |
apply(induct a arbitrary: rsb)
|
|
|
85 |
using flts_middle01 apply presburger
|
|
|
86 |
apply(subgoal_tac "rflts (rsa @ [RONE] @ rsb) = rflts rsa @ [RONE] @ rflts rsb")
|
|
|
87 |
prefer 2
|
|
|
88 |
using flts_append1 apply blast
|
|
|
89 |
apply simp
|
|
|
90 |
apply(subgoal_tac "RONE \<in> set (rflts rsa)")
|
|
|
91 |
prefer 2
|
|
|
92 |
using rflts_def_idiot2 apply blast
|
|
|
93 |
apply(subst distinct_removes_middle3)
|
|
|
94 |
apply simp
|
|
|
95 |
using flts_append apply presburger
|
|
|
96 |
apply simp
|
|
|
97 |
apply (metis distinct_removes_middle3 flts_append in_set_conv_decomp rflts.simps(5))
|
|
|
98 |
apply (metis distinct_removes_middle(1) flts_append flts_append1 rflts_def_idiot2 rrexp.distinct(25) rrexp.distinct(5))
|
|
|
99 |
apply(subgoal_tac "rflts (rsa @ [RALTS x] @ rsb) = rflts rsa @ x @ rflts rsb")
|
|
|
100 |
prefer 2
|
|
|
101 |
apply (simp add: flts_append)
|
|
|
102 |
apply (simp only:)
|
|
|
103 |
|
|
|
104 |
apply(subgoal_tac "\<forall>r1 \<in> set x. r1 \<in> set (rflts rsa)")
|
|
|
105 |
prefer 2
|
|
|
106 |
using spilled_alts_contained apply blast
|
|
|
107 |
apply(subst flts_append)
|
|
|
108 |
using distinct_removes_middle_list apply blast
|
|
|
109 |
using distinct_removes_middle2 flts_append rflts_def_idiot2 by fastforce
|
|
|
110 |
|
|
|
111 |
|
|
|
112 |
lemma simp_removes_duplicate:
|
|
|
113 |
shows "a \<in> set rsa \<Longrightarrow> rsimp (rsimp_ALTs (rsa @ a # rs)) = rsimp (rsimp_ALTs (rsa @ rs))"
|
|
|
114 |
apply(subgoal_tac "rsimp (rsimp_ALTs (rsa @ a # rs)) = rsimp (RALTS (rsa @ a # rs))")
|
|
|
115 |
prefer 2
|
|
|
116 |
apply (smt (verit, best) Cons_eq_append_conv append_is_Nil_conv empty_set equals0D list.distinct(1) rsimp_ALTs.elims)
|
|
|
117 |
apply(simp only:)
|
|
|
118 |
apply simp
|
|
|
119 |
apply(subgoal_tac "(rdistinct (rflts (map rsimp rsa @ rsimp a # map rsimp rs)) {}) = (rdistinct (rflts (map rsimp rsa @ map rsimp rs)) {})")
|
|
|
120 |
apply(simp only:)
|
|
|
121 |
prefer 2
|
|
|
122 |
apply(subgoal_tac "rsimp a \<in> set (map rsimp rsa)")
|
|
|
123 |
prefer 2
|
|
|
124 |
apply simp
|
|
|
125 |
using distinct_removes_duplicate_flts2 apply force
|
|
|
126 |
apply(case_tac rsa)
|
|
|
127 |
apply simp
|
|
|
128 |
apply(case_tac rs)
|
|
|
129 |
apply simp
|
|
|
130 |
apply(case_tac list)
|
|
|
131 |
apply simp
|
|
|
132 |
using idem_after_simp1 apply presburger
|
|
|
133 |
apply simp+
|
|
|
134 |
apply(subgoal_tac "rsimp_ALTs (aa # list @ aaa # lista) = RALTS (aa # list @ aaa # lista)")
|
|
|
135 |
apply simp
|
|
|
136 |
using rsimpalts_conscons by presburger
|
|
467
|
137 |
|
|
|
138 |
|
|
|
139 |
|
|
|
140 |
|
|
468
|
141 |
|
|
467
|
142 |
|
|
|
143 |
|
|
471
|
144 |
|
|
|
145 |
inductive frewrite:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ \<leadsto>f _" [10, 10] 10)
|
|
|
146 |
where
|
|
|
147 |
"(RZERO # rs) \<leadsto>f rs"
|
|
|
148 |
| "((RALTS rs) # rsa) \<leadsto>f (rs @ rsa)"
|
|
|
149 |
| "rs1 \<leadsto>f rs2 \<Longrightarrow> (r # rs1) \<leadsto>f (r # rs2)"
|
|
|
150 |
|
|
|
151 |
|
|
|
152 |
inductive
|
|
|
153 |
frewrites:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ \<leadsto>f* _" [10, 10] 10)
|
|
|
154 |
where
|
|
473
|
155 |
[intro, simp]:"rs \<leadsto>f* rs"
|
|
|
156 |
| [intro]: "\<lbrakk>rs1 \<leadsto>f* rs2; rs2 \<leadsto>f rs3\<rbrakk> \<Longrightarrow> rs1 \<leadsto>f* rs3"
|
|
|
157 |
|
|
|
158 |
inductive grewrite:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ \<leadsto>g _" [10, 10] 10)
|
|
|
159 |
where
|
|
|
160 |
"(RZERO # rs) \<leadsto>g rs"
|
|
|
161 |
| "((RALTS rs) # rsa) \<leadsto>g (rs @ rsa)"
|
|
|
162 |
| "rs1 \<leadsto>g rs2 \<Longrightarrow> (r # rs1) \<leadsto>g (r # rs2)"
|
|
|
163 |
| "rsa @ [a] @ rsb @ [a] @ rsc \<leadsto>g rsa @ [a] @ rsb @ rsc"
|
|
|
164 |
|
|
478
|
165 |
lemma grewrite_variant1:
|
|
|
166 |
shows "a \<in> set rs1 \<Longrightarrow> rs1 @ a # rs \<leadsto>g rs1 @ rs"
|
|
|
167 |
apply (metis append.assoc append_Cons append_Nil grewrite.intros(4) split_list_first)
|
|
|
168 |
done
|
|
|
169 |
|
|
473
|
170 |
|
|
|
171 |
inductive
|
|
|
172 |
grewrites:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ \<leadsto>g* _" [10, 10] 10)
|
|
|
173 |
where
|
|
|
174 |
[intro, simp]:"rs \<leadsto>g* rs"
|
|
|
175 |
| [intro]: "\<lbrakk>rs1 \<leadsto>g* rs2; rs2 \<leadsto>g rs3\<rbrakk> \<Longrightarrow> rs1 \<leadsto>g* rs3"
|
|
478
|
176 |
|
|
|
177 |
|
|
|
178 |
|
|
473
|
179 |
(*
|
|
|
180 |
inductive
|
|
|
181 |
frewrites2:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ <\<leadsto>f* _" [10, 10] 10)
|
|
|
182 |
where
|
|
|
183 |
[intro]: "\<lbrakk>rs1 \<leadsto>f* rs2; rs2 \<leadsto>f* rs1\<rbrakk> \<Longrightarrow> rs1 <\<leadsto>f* rs2"
|
|
|
184 |
*)
|
|
471
|
185 |
|
|
|
186 |
lemma fr_in_rstar : "r1 \<leadsto>f r2 \<Longrightarrow> r1 \<leadsto>f* r2"
|
|
|
187 |
using frewrites.intros(1) frewrites.intros(2) by blast
|
|
|
188 |
|
|
|
189 |
lemma freal_trans[trans]:
|
|
|
190 |
assumes a1: "r1 \<leadsto>f* r2" and a2: "r2 \<leadsto>f* r3"
|
|
|
191 |
shows "r1 \<leadsto>f* r3"
|
|
|
192 |
using a2 a1
|
|
|
193 |
apply(induct r2 r3 arbitrary: r1 rule: frewrites.induct)
|
|
|
194 |
apply(auto)
|
|
|
195 |
done
|
|
|
196 |
|
|
|
197 |
|
|
|
198 |
lemma many_steps_later: "\<lbrakk>r1 \<leadsto>f r2; r2 \<leadsto>f* r3 \<rbrakk> \<Longrightarrow> r1 \<leadsto>f* r3"
|
|
|
199 |
by (meson fr_in_rstar freal_trans)
|
|
|
200 |
|
|
|
201 |
|
|
475
|
202 |
lemma gr_in_rstar : "r1 \<leadsto>g r2 \<Longrightarrow> r1 \<leadsto>g* r2"
|
|
|
203 |
using grewrites.intros(1) grewrites.intros(2) by blast
|
|
|
204 |
|
|
|
205 |
lemma greal_trans[trans]:
|
|
|
206 |
assumes a1: "r1 \<leadsto>g* r2" and a2: "r2 \<leadsto>g* r3"
|
|
|
207 |
shows "r1 \<leadsto>g* r3"
|
|
|
208 |
using a2 a1
|
|
|
209 |
apply(induct r2 r3 arbitrary: r1 rule: grewrites.induct)
|
|
|
210 |
apply(auto)
|
|
|
211 |
done
|
|
|
212 |
|
|
|
213 |
|
|
|
214 |
lemma gmany_steps_later: "\<lbrakk>r1 \<leadsto>g r2; r2 \<leadsto>g* r3 \<rbrakk> \<Longrightarrow> r1 \<leadsto>g* r3"
|
|
|
215 |
by (meson gr_in_rstar greal_trans)
|
|
|
216 |
|
|
478
|
217 |
lemma gstar_rdistinct_general:
|
|
|
218 |
shows "rs1 @ rs \<leadsto>g* rs1 @ (rdistinct rs (set rs1))"
|
|
|
219 |
apply(induct rs arbitrary: rs1)
|
|
|
220 |
apply simp
|
|
|
221 |
apply(case_tac " a \<in> set rs1")
|
|
|
222 |
apply simp
|
|
|
223 |
apply(subgoal_tac "rs1 @ a # rs \<leadsto>g rs1 @ rs")
|
|
|
224 |
using gmany_steps_later apply auto[1]
|
|
|
225 |
apply (metis append.assoc append_Cons append_Nil grewrite.intros(4) split_list_first)
|
|
|
226 |
apply simp
|
|
|
227 |
apply(drule_tac x = "rs1 @ [a]" in meta_spec)
|
|
|
228 |
by simp
|
|
|
229 |
|
|
|
230 |
|
|
|
231 |
lemma gstar_rdistinct:
|
|
|
232 |
shows "rs \<leadsto>g* rdistinct rs {}"
|
|
|
233 |
apply(induct rs)
|
|
|
234 |
apply simp
|
|
|
235 |
by (metis append.left_neutral empty_set gstar_rdistinct_general)
|
|
|
236 |
|
|
475
|
237 |
|
|
|
238 |
|
|
471
|
239 |
lemma frewrite_append:
|
|
|
240 |
shows "\<lbrakk> rsa \<leadsto>f rsb \<rbrakk> \<Longrightarrow> rs @ rsa \<leadsto>f rs @ rsb"
|
|
|
241 |
apply(induct rs)
|
|
472
|
242 |
apply simp+
|
|
|
243 |
using frewrite.intros(3) by blast
|
|
475
|
244 |
|
|
|
245 |
lemma grewrite_append:
|
|
|
246 |
shows "\<lbrakk> rsa \<leadsto>g rsb \<rbrakk> \<Longrightarrow> rs @ rsa \<leadsto>g rs @ rsb"
|
|
|
247 |
apply(induct rs)
|
|
|
248 |
apply simp+
|
|
|
249 |
using grewrite.intros(3) by blast
|
|
472
|
250 |
|
|
471
|
251 |
|
|
|
252 |
|
|
|
253 |
lemma frewrites_cons:
|
|
|
254 |
shows "\<lbrakk> rsa \<leadsto>f* rsb \<rbrakk> \<Longrightarrow> r # rsa \<leadsto>f* r # rsb"
|
|
472
|
255 |
apply(induct rsa rsb rule: frewrites.induct)
|
|
|
256 |
apply simp
|
|
|
257 |
using frewrite.intros(3) by blast
|
|
471
|
258 |
|
|
|
259 |
|
|
475
|
260 |
lemma grewrites_cons:
|
|
|
261 |
shows "\<lbrakk> rsa \<leadsto>g* rsb \<rbrakk> \<Longrightarrow> r # rsa \<leadsto>g* r # rsb"
|
|
|
262 |
apply(induct rsa rsb rule: grewrites.induct)
|
|
|
263 |
apply simp
|
|
|
264 |
using grewrite.intros(3) by blast
|
|
|
265 |
|
|
|
266 |
|
|
471
|
267 |
lemma frewrites_append:
|
|
|
268 |
shows " \<lbrakk>rsa \<leadsto>f* rsb\<rbrakk> \<Longrightarrow> (rs @ rsa) \<leadsto>f* (rs @ rsb)"
|
|
472
|
269 |
apply(induct rs)
|
|
471
|
270 |
apply simp
|
|
472
|
271 |
by (simp add: frewrites_cons)
|
|
471
|
272 |
|
|
475
|
273 |
lemma grewrites_append:
|
|
|
274 |
shows " \<lbrakk>rsa \<leadsto>g* rsb\<rbrakk> \<Longrightarrow> (rs @ rsa) \<leadsto>g* (rs @ rsb)"
|
|
|
275 |
apply(induct rs)
|
|
|
276 |
apply simp
|
|
|
277 |
by (simp add: grewrites_cons)
|
|
|
278 |
|
|
471
|
279 |
|
|
|
280 |
|
|
|
281 |
lemma frewrites_concat:
|
|
|
282 |
shows "\<lbrakk>rs1 \<leadsto>f rs2; rsa \<leadsto>f* rsb \<rbrakk> \<Longrightarrow> (rs1 @ rsa) \<leadsto>f* (rs2 @ rsb)"
|
|
|
283 |
apply(induct rs1 rs2 rule: frewrite.induct)
|
|
|
284 |
apply(simp)
|
|
|
285 |
apply(subgoal_tac "(RZERO # rs @ rsa) \<leadsto>f (rs @ rsa)")
|
|
|
286 |
prefer 2
|
|
|
287 |
using frewrite.intros(1) apply blast
|
|
|
288 |
apply(subgoal_tac "(rs @ rsa) \<leadsto>f* (rs @ rsb)")
|
|
|
289 |
using many_steps_later apply blast
|
|
472
|
290 |
apply (simp add: frewrites_append)
|
|
|
291 |
apply (metis append.assoc append_Cons frewrite.intros(2) frewrites_append many_steps_later)
|
|
|
292 |
using frewrites_cons by auto
|
|
471
|
293 |
|
|
475
|
294 |
lemma grewrites_concat:
|
|
|
295 |
shows "\<lbrakk>rs1 \<leadsto>g rs2; rsa \<leadsto>g* rsb \<rbrakk> \<Longrightarrow> (rs1 @ rsa) \<leadsto>g* (rs2 @ rsb)"
|
|
|
296 |
apply(induct rs1 rs2 rule: grewrite.induct)
|
|
|
297 |
apply(simp)
|
|
|
298 |
apply(subgoal_tac "(RZERO # rs @ rsa) \<leadsto>g (rs @ rsa)")
|
|
|
299 |
prefer 2
|
|
|
300 |
using grewrite.intros(1) apply blast
|
|
|
301 |
apply(subgoal_tac "(rs @ rsa) \<leadsto>g* (rs @ rsb)")
|
|
|
302 |
using gmany_steps_later apply blast
|
|
|
303 |
apply (simp add: grewrites_append)
|
|
|
304 |
apply (metis append.assoc append_Cons grewrite.intros(2) grewrites_append gmany_steps_later)
|
|
|
305 |
using grewrites_cons apply auto
|
|
|
306 |
apply(subgoal_tac "rsaa @ a # rsba @ a # rsc @ rsa \<leadsto>g* rsaa @ a # rsba @ a # rsc @ rsb")
|
|
|
307 |
using grewrite.intros(4) grewrites.intros(2) apply force
|
|
|
308 |
using grewrites_append by auto
|
|
|
309 |
|
|
|
310 |
|
|
|
311 |
lemma grewritess_concat:
|
|
|
312 |
shows "\<lbrakk>rsa \<leadsto>g* rsb; rsc \<leadsto>g* rsd \<rbrakk> \<Longrightarrow> (rsa @ rsc) \<leadsto>g* (rsb @ rsd)"
|
|
|
313 |
apply(induct rsa rsb rule: grewrites.induct)
|
|
|
314 |
apply(case_tac rs)
|
|
|
315 |
apply simp
|
|
|
316 |
using grewrites_append apply blast
|
|
|
317 |
by (meson greal_trans grewrites.simps grewrites_concat)
|
|
|
318 |
|
|
476
|
319 |
fun alt_set:: "rrexp \<Rightarrow> rrexp set"
|
|
|
320 |
where
|
|
|
321 |
"alt_set (RALTS rs) = set rs \<union> \<Union> (alt_set ` (set rs))"
|
|
|
322 |
| "alt_set r = {r}"
|
|
|
323 |
|
|
|
324 |
lemma alt_set_has_all:
|
|
|
325 |
shows "RALTS rs \<in> alt_set rx \<Longrightarrow> set rs \<subseteq> alt_set rx"
|
|
|
326 |
apply(induct rx arbitrary: rs)
|
|
|
327 |
apply simp_all
|
|
|
328 |
apply(rename_tac rSS rss)
|
|
|
329 |
using in_mono by fastforce
|
|
|
330 |
|
|
|
331 |
|
|
|
332 |
|
|
|
333 |
|
|
|
334 |
lemma grewrite_equal_rsimp:
|
|
|
335 |
shows "\<lbrakk>rs1 \<leadsto>g rs2; rsimp_ALTs (rdistinct (rflts (map rsimp rs1)) (rset \<union> \<Union>(alt_set ` rset))) =
|
|
|
336 |
rsimp_ALTs (rdistinct (rflts (map rsimp rs2)) (rset \<union> \<Union>(alt_set ` rset)))\<rbrakk>
|
|
|
337 |
\<Longrightarrow> rsimp_ALTs (rdistinct (rflts (rsimp r # map rsimp rs1)) (rset \<union> \<Union>(alt_set ` rset))) =
|
|
|
338 |
rsimp_ALTs (rdistinct (rflts (rsimp r # map rsimp rs2)) (rset \<union> \<Union>(alt_set ` rset)))"
|
|
|
339 |
apply(induct rs1 rs2 arbitrary:rset rule: grewrite.induct)
|
|
|
340 |
apply simp
|
|
|
341 |
apply (metis append_Cons append_Nil flts_middle0)
|
|
|
342 |
apply(case_tac "rsimp r \<in> rset")
|
|
|
343 |
apply simp
|
|
|
344 |
oops
|
|
|
345 |
|
|
478
|
346 |
lemma grewrite_cases_middle:
|
|
|
347 |
shows "rs1 \<leadsto>g rs2 \<Longrightarrow>
|
|
|
348 |
(\<exists>rsa rsb rsc. rs1 = (rsa @ [RALTS rsb] @ rsc) \<and> rs2 = (rsa @ rsb @ rsc)) \<or>
|
|
|
349 |
(\<exists>rsa rsc. rs1 = rsa @ [RZERO] @ rsc \<and> rs2 = rsa @ rsc) \<or>
|
|
|
350 |
(\<exists>rsa rsb rsc a. rs1 = rsa @ [a] @ rsb @ [a] @ rsc \<and> rs2 = rsa @ [a] @ rsb @ rsc)"
|
|
|
351 |
apply( induct rs1 rs2 rule: grewrite.induct)
|
|
|
352 |
apply simp
|
|
|
353 |
apply blast
|
|
|
354 |
apply (metis append_Cons append_Nil)
|
|
|
355 |
apply (metis append_Cons)
|
|
|
356 |
by blast
|
|
476
|
357 |
|
|
|
358 |
|
|
|
359 |
lemma grewrite_equal_rsimp:
|
|
|
360 |
shows "rs1 \<leadsto>g rs2 \<Longrightarrow> rsimp (RALTS rs1) = rsimp (RALTS rs2)"
|
|
478
|
361 |
apply(frule grewrite_cases_middle)
|
|
|
362 |
apply(case_tac "(\<exists>rsa rsb rsc. rs1 = rsa @ [RALTS rsb] @ rsc \<and> rs2 = rsa @ rsb @ rsc)")
|
|
|
363 |
using simp_flatten3 apply auto[1]
|
|
|
364 |
apply(case_tac "(\<exists>rsa rsc. rs1 = rsa @ [RZERO] @ rsc \<and> rs2 = rsa @ rsc)")
|
|
|
365 |
apply (metis (mono_tags, opaque_lifting) append_Cons append_Nil list.set_intros(1) list.simps(9) rflts.simps(2) rsimp.simps(2) rsimp.simps(3) simp_removes_duplicate3)
|
|
|
366 |
by (smt (verit) append.assoc append_Cons append_Nil in_set_conv_decomp simp_removes_duplicate3)
|
|
476
|
367 |
|
|
|
368 |
|
|
475
|
369 |
lemma grewrites_equal_rsimp:
|
|
|
370 |
shows "rs1 \<leadsto>g* rs2 \<Longrightarrow> rsimp (RALTS rs1) = rsimp (RALTS rs2)"
|
|
476
|
371 |
apply (induct rs1 rs2 rule: grewrites.induct)
|
|
475
|
372 |
apply simp
|
|
476
|
373 |
using grewrite_equal_rsimp by presburger
|
|
|
374 |
|
|
|
375 |
|
|
|
376 |
|
|
|
377 |
|
|
|
378 |
lemma grewrites_equal_simp_2:
|
|
|
379 |
shows "rsimp (RALTS rs1) = rsimp (RALTS rs2) \<Longrightarrow> rs1 \<leadsto>g* rs2"
|
|
478
|
380 |
oops
|
|
|
381 |
|
|
|
382 |
|
|
|
383 |
|
|
475
|
384 |
|
|
476
|
385 |
lemma grewrites_last:
|
|
|
386 |
shows "r # [RALTS rs] \<leadsto>g* r # rs"
|
|
|
387 |
by (metis gr_in_rstar grewrite.intros(2) grewrite.intros(3) self_append_conv)
|
|
|
388 |
|
|
|
389 |
lemma simp_flatten2:
|
|
|
390 |
shows "rsimp (RALTS (r # [RALTS rs])) = rsimp (RALTS (r # rs))"
|
|
|
391 |
using grewrites_equal_rsimp grewrites_last by blast
|
|
475
|
392 |
|
|
473
|
393 |
lemma frewrites_middle:
|
|
|
394 |
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow> r # (RALTS rs # rs1) \<leadsto>f* r # (rs @ rs1)"
|
|
|
395 |
by (simp add: fr_in_rstar frewrite.intros(2) frewrite.intros(3))
|
|
471
|
396 |
|
|
473
|
397 |
lemma frewrites_alt:
|
|
|
398 |
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow> (RALT r1 r2) # rs1 \<leadsto>f* r1 # r2 # rs2"
|
|
|
399 |
by (metis Cons_eq_appendI append_self_conv2 frewrite.intros(2) frewrites_cons many_steps_later)
|
|
471
|
400 |
|
|
|
401 |
lemma early_late_der_frewrites:
|
|
|
402 |
shows "map (rder x) (rflts rs) \<leadsto>f* rflts (map (rder x) rs)"
|
|
|
403 |
apply(induct rs)
|
|
|
404 |
apply simp
|
|
|
405 |
apply(case_tac a)
|
|
|
406 |
apply simp+
|
|
|
407 |
using frewrite.intros(1) many_steps_later apply blast
|
|
|
408 |
apply(case_tac "x = x3")
|
|
472
|
409 |
apply simp
|
|
|
410 |
using frewrites_cons apply presburger
|
|
|
411 |
using frewrite.intros(1) many_steps_later apply fastforce
|
|
|
412 |
apply(case_tac "rnullable x41")
|
|
473
|
413 |
apply simp+
|
|
|
414 |
apply (simp add: frewrites_alt)
|
|
|
415 |
apply (simp add: frewrites_cons)
|
|
|
416 |
apply (simp add: frewrites_append)
|
|
|
417 |
by (simp add: frewrites_cons)
|
|
471
|
418 |
|
|
|
419 |
|
|
|
420 |
|
|
|
421 |
|
|
473
|
422 |
lemma with_wo0_distinct:
|
|
|
423 |
shows "rdistinct rs rset \<leadsto>f* rdistinct rs (insert RZERO rset)"
|
|
|
424 |
apply(induct rs arbitrary: rset)
|
|
|
425 |
apply simp
|
|
|
426 |
apply(case_tac a)
|
|
|
427 |
apply(case_tac "RZERO \<in> rset")
|
|
|
428 |
apply simp+
|
|
|
429 |
using fr_in_rstar frewrite.intros(1) apply presburger
|
|
|
430 |
apply (case_tac "RONE \<in> rset")
|
|
|
431 |
apply simp+
|
|
|
432 |
using frewrites_cons apply presburger
|
|
|
433 |
apply(case_tac "a \<in> rset")
|
|
|
434 |
apply simp
|
|
|
435 |
apply (simp add: frewrites_cons)
|
|
|
436 |
apply(case_tac "a \<in> rset")
|
|
|
437 |
apply simp
|
|
|
438 |
apply (simp add: frewrites_cons)
|
|
|
439 |
apply(case_tac "a \<in> rset")
|
|
|
440 |
apply simp
|
|
|
441 |
apply (simp add: frewrites_cons)
|
|
|
442 |
apply(case_tac "a \<in> rset")
|
|
|
443 |
apply simp
|
|
|
444 |
apply (simp add: frewrites_cons)
|
|
|
445 |
done
|
|
|
446 |
|
|
476
|
447 |
(*Interesting lemma: not obvious but easily proven by sledgehammer*)
|
|
473
|
448 |
|
|
476
|
449 |
|
|
|
450 |
|
|
|
451 |
|
|
|
452 |
(*lemma induction last rule not go through
|
|
|
453 |
example:
|
|
|
454 |
r #
|
|
|
455 |
rdistinct rs1
|
|
|
456 |
(insert RZERO
|
|
|
457 |
(insert r
|
|
|
458 |
(rset \<union>
|
|
|
459 |
\<Union> (alt_set `
|
|
|
460 |
rset)))) \<leadsto>g* r #
|
|
|
461 |
rdistinct rs2
|
|
|
462 |
(insert RZERO (insert r (rset \<union> \<Union> (alt_set ` rset))))
|
|
|
463 |
rs2 = [+rs] rs3 = rs,
|
|
|
464 |
r = +rs
|
|
|
465 |
[] \<leadsto>g* rs which is wrong
|
|
|
466 |
*)
|
|
|
467 |
|
|
469
|
468 |
|
|
|
469 |
|
|
476
|
470 |
|
|
|
471 |
|
|
|
472 |
|
|
|
473 |
|
|
|
474 |
lemma frewrite_simpeq:
|
|
|
475 |
shows "rs1 \<leadsto>f rs2 \<Longrightarrow> rsimp (RALTS rs1) = rsimp (RALTS rs2)"
|
|
|
476 |
apply(induct rs1 rs2 rule: frewrite.induct)
|
|
|
477 |
apply simp
|
|
|
478 |
using simp_flatten apply presburger
|
|
478
|
479 |
by (metis (no_types, opaque_lifting) grewrites_equal_rsimp grewrites_last list.simps(9) rsimp.simps(2))
|
|
|
480 |
|
|
|
481 |
lemma gstar0:
|
|
|
482 |
shows "rsa @ (rdistinct rs (set rsa)) \<leadsto>g* rsa @ (rdistinct rs (insert RZERO (set rsa)))"
|
|
|
483 |
apply(induct rs arbitrary: rsa)
|
|
|
484 |
apply simp
|
|
|
485 |
apply(case_tac "a = RZERO")
|
|
|
486 |
apply simp
|
|
|
487 |
|
|
|
488 |
using gr_in_rstar grewrite.intros(1) grewrites_append apply presburger
|
|
|
489 |
apply(case_tac "a \<in> set rsa")
|
|
|
490 |
apply simp+
|
|
|
491 |
apply(drule_tac x = "rsa @ [a]" in meta_spec)
|
|
|
492 |
by simp
|
|
|
493 |
|
|
|
494 |
lemma gstar01:
|
|
|
495 |
shows "rdistinct rs {} \<leadsto>g* rdistinct rs {RZERO}"
|
|
|
496 |
by (metis empty_set gstar0 self_append_conv2)
|
|
|
497 |
|
|
|
498 |
|
|
|
499 |
lemma grewrite_rdistinct_aux:
|
|
|
500 |
shows "rs @ rdistinct rsa rset \<leadsto>g* rs @ rdistinct rsa (rset \<union> set rs)"
|
|
|
501 |
sorry
|
|
|
502 |
|
|
|
503 |
lemma grewrite_rdistinct_worth1:
|
|
|
504 |
shows "(rsb @ [a]) @ rdistinct rs set1 \<leadsto>g* (rsb @ [a]) @ rdistinct rs (insert a set1)"
|
|
|
505 |
by (metis append.assoc empty_set grewrite_rdistinct_aux grewrites_append inf_sup_aci(5) insert_is_Un list.simps(15))
|
|
|
506 |
|
|
|
507 |
lemma grewrite_rdisitinct:
|
|
|
508 |
shows "rs @ rdistinct rsa {RALTS rs} \<leadsto>g* rs @ rdistinct rsa (insert (RALTS rs) (set rs))"
|
|
|
509 |
apply(induct rsa arbitrary: rs)
|
|
|
510 |
apply simp
|
|
|
511 |
apply(case_tac "a = RALTS rs")
|
|
|
512 |
apply simp
|
|
|
513 |
apply(case_tac "a \<in> set rs")
|
|
|
514 |
apply simp
|
|
|
515 |
apply(subgoal_tac "rs @
|
|
|
516 |
a # rdistinct rsa {RALTS rs, a} \<leadsto>g rs @ rdistinct rsa {RALTS rs, a}")
|
|
|
517 |
apply(subgoal_tac
|
|
|
518 |
"rs @ rdistinct rsa {RALTS rs, a} \<leadsto>g* rs @ rdistinct rsa (insert (RALTS rs) (set rs))")
|
|
|
519 |
using gmany_steps_later apply blast
|
|
|
520 |
apply(subgoal_tac
|
|
|
521 |
" rs @ rdistinct rsa {RALTS rs, a} \<leadsto>g* rs @ rdistinct rsa ({RALTS rs, a} \<union> set rs)")
|
|
|
522 |
apply (simp add: insert_absorb)
|
|
|
523 |
using grewrite_rdistinct_aux apply blast
|
|
|
524 |
using grewrite_variant1 apply blast
|
|
|
525 |
by (metis grewrite_rdistinct_aux insert_is_Un)
|
|
|
526 |
|
|
|
527 |
|
|
|
528 |
lemma frewrite_rd_grewrites_general:
|
|
|
529 |
shows "\<lbrakk>rs1 \<leadsto>f rs2; \<And>rs. \<exists>rs3.
|
|
|
530 |
(rs @ (rdistinct rs1 (set rs)) \<leadsto>g* rs3) \<and> (rs @ (rdistinct rs2 (set rs)) \<leadsto>g* rs3)\<rbrakk>
|
|
|
531 |
\<Longrightarrow>
|
|
|
532 |
\<exists>rs3. (rs @ (r # rdistinct rs1 (set rs \<union> {r})) \<leadsto>g* rs3) \<and> (rs @ (r # rdistinct rs2 (set rs \<union> {r})) \<leadsto>g* rs3)"
|
|
|
533 |
apply(drule_tac x = "rs @ [r]" in meta_spec )
|
|
|
534 |
by simp
|
|
|
535 |
|
|
|
536 |
|
|
|
537 |
lemma grewrites_middle_distinct:
|
|
|
538 |
shows "RALTS rs \<in> set rsb \<Longrightarrow>
|
|
|
539 |
rsb @
|
|
|
540 |
rdistinct ( rs @ rsa)
|
|
|
541 |
(set rsb) \<leadsto>g* rsb @ rdistinct rsa (set rsb)"
|
|
|
542 |
sorry
|
|
|
543 |
|
|
|
544 |
|
|
|
545 |
|
|
|
546 |
lemma frewrite_rd_grewrites_aux:
|
|
|
547 |
shows " rsb @
|
|
|
548 |
rdistinct (RALTS rs # rsa)
|
|
|
549 |
(set rsb) \<leadsto>g* rsb @
|
|
|
550 |
rdistinct rs (set rsb) @ rdistinct rsa (insert (RALTS rs) (set rs) \<union> set rsb)"
|
|
|
551 |
|
|
|
552 |
|
|
|
553 |
sorry
|
|
|
554 |
|
|
|
555 |
lemma flts_gstar:
|
|
|
556 |
shows "rs \<leadsto>g* rflts rs"
|
|
|
557 |
sorry
|
|
|
558 |
|
|
|
559 |
lemma list_dlist_union:
|
|
|
560 |
shows "set rs \<subseteq> set rsb \<union> set (rdistinct rs (set rsb))"
|
|
|
561 |
by (metis rdistinct_concat_general rdistinct_set_equality set_append sup_ge2)
|
|
|
562 |
|
|
|
563 |
lemma subset_distinct_rewrite1:
|
|
|
564 |
shows "set1 \<subseteq> set rsb \<Longrightarrow> rsb @ rs \<leadsto>g* rsb @ (rdistinct rs set1)"
|
|
|
565 |
apply(induct rs arbitrary: rsb)
|
|
|
566 |
apply simp
|
|
|
567 |
apply(case_tac "a \<in> set1")
|
|
|
568 |
apply simp
|
|
|
569 |
|
|
|
570 |
using gmany_steps_later grewrite_variant1 apply blast
|
|
|
571 |
apply simp
|
|
|
572 |
apply(drule_tac x = "rsb @ [a]" in meta_spec)
|
|
|
573 |
apply(subgoal_tac "set1 \<subseteq> set (rsb @ [a])")
|
|
|
574 |
apply (simp only:)
|
|
|
575 |
apply(subgoal_tac "(rsb @ [a]) @ rdistinct rs set1 \<leadsto>g* (rsb @ [a]) @ rdistinct rs (insert a set1)")
|
|
|
576 |
apply (metis (no_types, opaque_lifting) append.assoc append_Cons append_Nil greal_trans)
|
|
|
577 |
apply (metis append.assoc empty_set grewrite_rdistinct_aux grewrites_append inf_sup_aci(5) insert_is_Un list.simps(15))
|
|
|
578 |
by auto
|
|
|
579 |
|
|
|
580 |
|
|
|
581 |
lemma subset_distinct_rewrite:
|
|
|
582 |
shows "set rsb' \<subseteq> set rsb \<Longrightarrow> rsb @ rs \<leadsto>g* rsb @ (rdistinct rs (set rsb'))"
|
|
|
583 |
by (simp add: subset_distinct_rewrite1)
|
|
|
584 |
|
|
|
585 |
|
|
|
586 |
|
|
|
587 |
lemma distinct_3list:
|
|
|
588 |
shows "rsb @ (rdistinct rs (set rsb)) @ rsa \<leadsto>g*
|
|
|
589 |
rsb @ (rdistinct rs (set rsb)) @ (rdistinct rsa (set rs))"
|
|
|
590 |
by (metis append.assoc list_dlist_union set_append subset_distinct_rewrite)
|
|
|
591 |
|
|
|
592 |
|
|
|
593 |
|
|
|
594 |
|
|
|
595 |
lemma grewrites_shape1:
|
|
|
596 |
shows " RALTS rs \<notin> set rsb \<Longrightarrow>
|
|
|
597 |
rsb @
|
|
|
598 |
RALTS rs #
|
|
|
599 |
rdistinct rsa
|
|
|
600 |
(
|
|
|
601 |
(set rsb)) \<leadsto>g* rsb @
|
|
|
602 |
rdistinct rs (set rsb) @
|
|
|
603 |
rdistinct (rflts (rdistinct rsa ( (set rsb \<union> set rs)))) (set rs)"
|
|
|
604 |
|
|
|
605 |
|
|
|
606 |
apply (subgoal_tac " rsb @
|
|
|
607 |
RALTS rs #
|
|
|
608 |
rdistinct rsa
|
|
|
609 |
(
|
|
|
610 |
(set rsb)) \<leadsto>g* rsb @
|
|
|
611 |
rs @
|
|
|
612 |
rdistinct rsa
|
|
|
613 |
(
|
|
|
614 |
(set rsb)) ")
|
|
|
615 |
prefer 2
|
|
|
616 |
using gr_in_rstar grewrite.intros(2) grewrites_append apply presburger
|
|
|
617 |
apply(subgoal_tac "rsb @ rs @ rdistinct rsa ( (set rsb)) \<leadsto>g* rsb @
|
|
|
618 |
(rdistinct rs (set rsb) @ rdistinct rsa ( (set rsb)))")
|
|
|
619 |
prefer 2
|
|
|
620 |
apply (metis append_assoc grewrites.intros(1) grewritess_concat gstar_rdistinct_general)
|
|
|
621 |
apply(subgoal_tac " rsb @ rdistinct rs (set rsb) @ rdistinct rsa ( (set rsb))
|
|
|
622 |
\<leadsto>g* rsb @ rdistinct rs (set rsb) @ rdistinct rsa ( (set rsb) \<union> (set rs))")
|
|
|
623 |
prefer 2
|
|
|
624 |
apply (smt (verit, best) append.assoc append_assoc boolean_algebra_cancel.sup2 grewrite_rdistinct_aux inf_sup_aci(5) insert_is_Un rdistinct_concat_general rdistinct_set_equality set_append sup.commute sup.right_idem sup_commute)
|
|
|
625 |
apply(subgoal_tac "rdistinct rsa ( (set rsb) \<union> set rs) \<leadsto>g*
|
|
|
626 |
rflts (rdistinct rsa ( (set rsb) \<union> set rs))")
|
|
|
627 |
apply(subgoal_tac "rsb @ (rdistinct rs (set rsb)) @ rflts (rdistinct rsa ( (set rsb) \<union> set rs)) \<leadsto>g*
|
|
|
628 |
rsb @ (rdistinct rs (set rsb)) @ (rdistinct (rflts (rdistinct rsa ( (set rsb) \<union> set rs))) (set rs))")
|
|
|
629 |
apply (smt (verit, ccfv_SIG) Un_insert_left greal_trans grewrites_append)
|
|
|
630 |
using distinct_3list apply presburger
|
|
|
631 |
using flts_gstar apply blast
|
|
|
632 |
done
|
|
|
633 |
|
|
|
634 |
lemma r_finite1:
|
|
|
635 |
shows "r = RALTS (r # rs) = False"
|
|
|
636 |
apply(induct r)
|
|
|
637 |
apply simp+
|
|
|
638 |
apply (metis list.set_intros(1))
|
|
|
639 |
by blast
|
|
|
640 |
|
|
|
641 |
|
|
|
642 |
|
|
|
643 |
lemma grewrite_singleton:
|
|
|
644 |
shows "[r] \<leadsto>g r # rs \<Longrightarrow> rs = []"
|
|
|
645 |
apply (induct "[r]" "r # rs" rule: grewrite.induct)
|
|
|
646 |
apply simp
|
|
|
647 |
apply (metis r_finite1)
|
|
|
648 |
using grewrite.simps apply blast
|
|
|
649 |
by simp
|
|
|
650 |
|
|
|
651 |
lemma impossible_grewrite1:
|
|
|
652 |
shows "\<not>( [RONE] \<leadsto>g [])"
|
|
|
653 |
using grewrite.cases by fastforce
|
|
|
654 |
|
|
|
655 |
|
|
|
656 |
lemma impossible_grewrite2:
|
|
|
657 |
shows "\<not> ([RALTS rs] \<leadsto>g (RALTS rs) # a # rs)"
|
|
|
658 |
|
|
|
659 |
using grewrite_singleton by blast
|
|
|
660 |
thm grewrite.cases
|
|
|
661 |
lemma impossible_grewrite3:
|
|
|
662 |
shows "\<not> (RALTS rs # rs1 \<leadsto>g (RALTS rs) # a # rs1)"
|
|
|
663 |
oops
|
|
|
664 |
|
|
|
665 |
|
|
|
666 |
lemma grewrites_singleton:
|
|
|
667 |
shows "[r] \<leadsto>g* r # rs \<Longrightarrow> rs = []"
|
|
|
668 |
apply(induct "[r]" "r # rs" rule: grewrites.induct)
|
|
|
669 |
apply simp
|
|
|
670 |
|
|
|
671 |
oops
|
|
|
672 |
|
|
|
673 |
lemma grewrite_nonequal_elem:
|
|
|
674 |
shows "r # rs2 \<leadsto>g r # rs3 \<Longrightarrow> rs2 \<leadsto>g rs3"
|
|
|
675 |
oops
|
|
|
676 |
|
|
|
677 |
lemma grewrites_nonequal_elem:
|
|
|
678 |
shows "r # rs2 \<leadsto>g* r # rs3 \<Longrightarrow> rs2 \<leadsto>g* rs3"
|
|
|
679 |
apply(induct r)
|
|
|
680 |
|
|
|
681 |
oops
|
|
|
682 |
|
|
|
683 |
|
|
|
684 |
|
|
|
685 |
|
|
|
686 |
lemma :
|
|
|
687 |
shows "rs1 @ rs2 \<leadsto>g* rs1 @ rs3 \<Longrightarrow> rs2 \<leadsto>g* rs3"
|
|
|
688 |
apply(induct rs1 arbitrary: rs2 rs3 rule: rev_induct)
|
|
|
689 |
apply simp
|
|
|
690 |
apply(drule_tac x = "[x] @ rs2" in meta_spec)
|
|
|
691 |
apply(drule_tac x = "[x] @ rs3" in meta_spec)
|
|
|
692 |
apply(simp)
|
|
|
693 |
|
|
|
694 |
oops
|
|
|
695 |
|
|
|
696 |
|
|
|
697 |
|
|
|
698 |
lemma grewrites_shape3_aux:
|
|
|
699 |
shows "rs @ (rdistinct rsa (insert (RALTS rs) rsc)) \<leadsto>g* rs @ rdistinct (rflts (rdistinct rsa rsc)) (set rs)"
|
|
|
700 |
apply(induct rsa arbitrary: rsc rs)
|
|
|
701 |
apply simp
|
|
|
702 |
apply(case_tac "a \<in> rsc")
|
|
|
703 |
apply simp
|
|
|
704 |
apply(case_tac "a = RALTS rs")
|
|
|
705 |
apply simp
|
|
|
706 |
apply(subgoal_tac " rdistinct (rs @ rflts (rdistinct rsa (insert (RALTS rs) rsc))) (set rs) \<leadsto>g*
|
|
|
707 |
rdistinct (rflts (rdistinct rsa (insert (RALTS rs) rsc))) (set rs)")
|
|
|
708 |
apply (metis insertI1 insert_absorb rdistinct_concat2)
|
|
|
709 |
apply (simp add: rdistinct_concat)
|
|
|
710 |
|
|
|
711 |
apply simp
|
|
|
712 |
apply(case_tac "a = RZERO")
|
|
|
713 |
apply (metis gmany_steps_later grewrite.intros(1) grewrite_append rflts.simps(2))
|
|
|
714 |
apply(case_tac "\<exists>rs1. a = RALTS rs1")
|
|
|
715 |
prefer 2
|
|
|
716 |
apply simp
|
|
|
717 |
apply(subgoal_tac "rflts (a # rdistinct rsa (insert a rsc)) = a # rflts (rdistinct rsa (insert a rsc))")
|
|
|
718 |
apply (simp only:)
|
|
|
719 |
apply(case_tac "a \<notin> set rs")
|
|
|
720 |
apply simp
|
|
|
721 |
apply(drule_tac x = "insert a rsc" in meta_spec)
|
|
|
722 |
apply(drule_tac x = "rs " in meta_spec)
|
|
|
723 |
|
|
|
724 |
apply(erule exE)
|
|
|
725 |
apply simp
|
|
|
726 |
apply(subgoal_tac "RALTS rs1 #
|
|
|
727 |
rdistinct rsa
|
|
|
728 |
(insert (RALTS rs)
|
|
|
729 |
(insert (RALTS rs1)
|
|
|
730 |
rsc)) \<leadsto>g* rs1 @
|
|
|
731 |
rdistinct rsa
|
|
|
732 |
(insert (RALTS rs)
|
|
|
733 |
(insert (RALTS rs1)
|
|
|
734 |
rsc)) ")
|
|
|
735 |
apply(subgoal_tac " rs1 @
|
|
|
736 |
rdistinct rsa
|
|
|
737 |
(insert (RALTS rs)
|
|
|
738 |
(insert (RALTS rs1)
|
|
|
739 |
rsc)) \<leadsto>g*
|
|
|
740 |
rs1 @
|
|
|
741 |
rdistinct rsa
|
|
|
742 |
(insert (RALTS rs)
|
|
|
743 |
(insert (RALTS rs1)
|
|
|
744 |
rsc))")
|
|
|
745 |
|
|
|
746 |
apply(case_tac "a \<in> set rs")
|
|
|
747 |
|
|
|
748 |
|
|
|
749 |
|
|
|
750 |
sorry
|
|
|
751 |
|
|
|
752 |
|
|
|
753 |
lemma grewrites_shape3:
|
|
|
754 |
shows " RALTS rs \<notin> set rsb \<Longrightarrow>
|
|
|
755 |
rsb @
|
|
|
756 |
RALTS rs #
|
|
|
757 |
rdistinct rsa
|
|
|
758 |
(insert (RALTS rs)
|
|
|
759 |
(set rsb)) \<leadsto>g* rsb @
|
|
|
760 |
rdistinct rs (set rsb) @
|
|
|
761 |
rdistinct (rflts (rdistinct rsa (set rsb \<union> set rs))) (set rs)"
|
|
|
762 |
apply(subgoal_tac "rsb @ RALTS rs # rdistinct rsa (insert (RALTS rs) (set rsb)) \<leadsto>g*
|
|
|
763 |
rsb @ rs @ rdistinct rsa (insert (RALTS rs) (set rsb))")
|
|
|
764 |
prefer 2
|
|
|
765 |
using gr_in_rstar grewrite.intros(2) grewrites_append apply presburger
|
|
|
766 |
apply(subgoal_tac "rsb @ rs @ rdistinct rsa (insert (RALTS rs) (set rsb )) \<leadsto>g*
|
|
|
767 |
rsb @ rs @ rdistinct rsa (insert (RALTS rs) (set rsb \<union> set rs))")
|
|
|
768 |
prefer 2
|
|
|
769 |
apply (metis Un_insert_left grewrite_rdistinct_aux grewrites_append)
|
|
|
770 |
|
|
|
771 |
apply(subgoal_tac "rsb @ rs @ rdistinct rsa (insert (RALTS rs) (set rsb \<union> set rs)) \<leadsto>g*
|
|
|
772 |
rsb @ rs @ rdistinct (rflts (rdistinct rsa (set rsb \<union> set rs))) (set rs)")
|
|
|
773 |
prefer 2
|
|
|
774 |
using grewrites_append grewrites_shape3_aux apply presburger
|
|
|
775 |
apply(subgoal_tac "rsb @ rs \<leadsto>g* rsb @ rdistinct rs (set rsb)")
|
|
|
776 |
apply (smt (verit, ccfv_SIG) append_eq_appendI greal_trans grewrites.simps grewritess_concat)
|
|
|
777 |
using gstar_rdistinct_general by blast
|
|
|
778 |
|
|
|
779 |
|
|
|
780 |
lemma grewrites_shape2:
|
|
|
781 |
shows " RALTS rs \<notin> set rsb \<Longrightarrow>
|
|
|
782 |
rsb @
|
|
|
783 |
rdistinct (rs @ rsa)
|
|
|
784 |
(set rsb) \<leadsto>g* rsb @
|
|
|
785 |
rdistinct rs (set rsb) @
|
|
|
786 |
rdistinct (rflts (rdistinct rsa (set rsb \<union> set rs))) (set rs)"
|
|
|
787 |
|
|
|
788 |
(* by (smt (z3) append.assoc distinct_3list flts_gstar greal_trans grewrites_append rdistinct_concat_general same_append_eq set_append)
|
|
|
789 |
*)
|
|
|
790 |
sorry
|
|
|
791 |
|
|
|
792 |
|
|
|
793 |
|
|
|
794 |
|
|
476
|
795 |
|
|
|
796 |
lemma frewrite_rd_grewrites:
|
|
|
797 |
shows "rs1 \<leadsto>f rs2 \<Longrightarrow>
|
|
478
|
798 |
\<exists>rs3. (rs @ (rdistinct rs1 (set rs)) \<leadsto>g* rs3) \<and> (rs @ (rdistinct rs2 (set rs)) \<leadsto>g* rs3) "
|
|
|
799 |
apply(induct rs1 rs2 arbitrary: rs rule: frewrite.induct)
|
|
|
800 |
apply(rule_tac x = "rsa @ (rdistinct rs ({RZERO} \<union> set rsa))" in exI)
|
|
476
|
801 |
apply(rule conjI)
|
|
478
|
802 |
apply(case_tac "RZERO \<in> set rsa")
|
|
|
803 |
apply simp+
|
|
|
804 |
using gstar0 apply fastforce
|
|
|
805 |
apply (simp add: gr_in_rstar grewrite.intros(1) grewrites_append)
|
|
|
806 |
apply (simp add: gstar0)
|
|
|
807 |
prefer 2
|
|
|
808 |
apply(case_tac "r \<in> set rs")
|
|
|
809 |
apply simp
|
|
|
810 |
apply(drule_tac x = "rs @ [r]" in meta_spec)
|
|
|
811 |
apply(erule exE)
|
|
|
812 |
apply(rule_tac x = "rs3" in exI)
|
|
|
813 |
apply simp
|
|
|
814 |
apply(case_tac "RALTS rs \<in> set rsb")
|
|
|
815 |
apply simp
|
|
|
816 |
apply(rule_tac x = "rflts rsb @ rdistinct rsa (set rsb)" in exI)
|
|
|
817 |
apply(rule conjI)
|
|
|
818 |
apply (simp add: flts_gstar grewritess_concat)
|
|
|
819 |
apply (meson flts_gstar greal_trans grewrites.intros(1) grewrites_middle_distinct grewritess_concat)
|
|
|
820 |
apply(simp)
|
|
|
821 |
apply(rule_tac x =
|
|
|
822 |
"rsb @ (rdistinct rs (set rsb)) @
|
|
|
823 |
(rdistinct (rflts (rdistinct rsa ( (set rsb \<union> set rs)) ) ) (set rs))" in exI)
|
|
|
824 |
apply(rule conjI)
|
|
|
825 |
prefer 2
|
|
|
826 |
using grewrites_shape2 apply force
|
|
|
827 |
using grewrites_shape3 by auto
|
|
|
828 |
|
|
|
829 |
|
|
|
830 |
|
|
|
831 |
lemma frewrite_simprd:
|
|
|
832 |
shows "rs1 \<leadsto>f rs2 \<Longrightarrow> rsimp (RALTS rs1) = rsimp (RALTS rs2)"
|
|
|
833 |
by (meson frewrite_simpeq)
|
|
473
|
834 |
|
|
476
|
835 |
|
|
|
836 |
lemma frewrites_rd_grewrites:
|
|
|
837 |
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow>
|
|
478
|
838 |
rsimp (RALTS rs1) = rsimp (RALTS rs2)"
|
|
476
|
839 |
apply(induct rs1 rs2 rule: frewrites.induct)
|
|
|
840 |
apply simp
|
|
478
|
841 |
using frewrite_simprd by presburger
|
|
476
|
842 |
|
|
|
843 |
|
|
|
844 |
|
|
|
845 |
|
|
|
846 |
lemma frewrite_simpeq2:
|
|
|
847 |
shows "rs1 \<leadsto>f rs2 \<Longrightarrow> rsimp (RALTS (rdistinct rs1 {})) = rsimp (RALTS (rdistinct rs2 {}))"
|
|
478
|
848 |
apply(subgoal_tac "\<exists> rs3. (rdistinct rs1 {} \<leadsto>g* rs3) \<and> (rdistinct rs2 {} \<leadsto>g* rs3)")
|
|
|
849 |
using grewrites_equal_rsimp apply fastforce
|
|
|
850 |
using frewrite_rd_grewrites by presburger
|
|
|
851 |
|
|
473
|
852 |
(*a more refined notion of \<leadsto>* is needed,
|
|
|
853 |
this lemma fails when rs1 contains some RALTS rs where elements
|
|
|
854 |
of rs appear in later parts of rs1, which will be picked up by rs2
|
|
|
855 |
and deduplicated*)
|
|
476
|
856 |
lemma frewrites_simpeq:
|
|
473
|
857 |
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow>
|
|
476
|
858 |
rsimp (RALTS (rdistinct rs1 {})) = rsimp (RALTS ( rdistinct rs2 {})) "
|
|
|
859 |
apply(induct rs1 rs2 rule: frewrites.induct)
|
|
|
860 |
apply simp
|
|
478
|
861 |
using frewrite_simpeq2 by presburger
|
|
476
|
862 |
|
|
473
|
863 |
|
|
|
864 |
lemma frewrite_single_step:
|
|
|
865 |
shows " rs2 \<leadsto>f rs3 \<Longrightarrow> rsimp (RALTS rs2) = rsimp (RALTS rs3)"
|
|
|
866 |
apply(induct rs2 rs3 rule: frewrite.induct)
|
|
|
867 |
apply simp
|
|
|
868 |
using simp_flatten apply blast
|
|
|
869 |
by (metis (no_types, opaque_lifting) list.simps(9) rsimp.simps(2) simp_flatten2)
|
|
|
870 |
|
|
|
871 |
lemma frewrites_equivalent_simp:
|
|
|
872 |
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow> rsimp (RALTS rs1) = rsimp (RALTS rs2)"
|
|
|
873 |
apply(induct rs1 rs2 rule: frewrites.induct)
|
|
|
874 |
apply simp
|
|
|
875 |
using frewrite_single_step by presburger
|
|
|
876 |
|
|
476
|
877 |
lemma grewrite_simpalts:
|
|
|
878 |
shows " rs2 \<leadsto>g rs3 \<Longrightarrow> rsimp (rsimp_ALTs rs2) = rsimp (rsimp_ALTs rs3)"
|
|
|
879 |
apply(induct rs2 rs3 rule : grewrite.induct)
|
|
|
880 |
using identity_wwo0 apply presburger
|
|
|
881 |
apply (metis frewrite.intros(1) frewrite_single_step identity_wwo0 rsimp_ALTs.simps(3) simp_flatten)
|
|
|
882 |
apply (smt (verit, ccfv_SIG) gmany_steps_later grewrites.intros(1) grewrites_cons grewrites_equal_rsimp identity_wwo0 rsimp_ALTs.simps(3))
|
|
|
883 |
apply simp
|
|
|
884 |
apply(subst rsimp_alts_equal)
|
|
|
885 |
apply(case_tac "rsa = [] \<and> rsb = [] \<and> rsc = []")
|
|
|
886 |
apply(subgoal_tac "rsa @ a # rsb @ rsc = [a]")
|
|
|
887 |
apply (simp only:)
|
|
|
888 |
apply (metis append_Nil frewrite.intros(1) frewrite_single_step identity_wwo0 rsimp_ALTs.simps(3) simp_removes_duplicate1(2))
|
|
|
889 |
apply simp
|
|
|
890 |
by (smt (verit, best) append.assoc append_Cons frewrite.intros(1) frewrite_single_step identity_wwo0 in_set_conv_decomp rsimp_ALTs.simps(3) simp_removes_duplicate3)
|
|
|
891 |
|
|
|
892 |
|
|
|
893 |
lemma grewrites_simpalts:
|
|
|
894 |
shows " rs2 \<leadsto>g* rs3 \<Longrightarrow> rsimp (rsimp_ALTs rs2) = rsimp (rsimp_ALTs rs3)"
|
|
|
895 |
apply(induct rs2 rs3 rule: grewrites.induct)
|
|
|
896 |
apply simp
|
|
|
897 |
using grewrite_simpalts by presburger
|
|
|
898 |
(*
|
|
473
|
899 |
lemma frewrites_dB_wwo0_simp:
|
|
|
900 |
shows "rdistinct rs1 {RZERO} \<leadsto>f* rdistinct rs2 {RZERO}
|
|
|
901 |
\<Longrightarrow> rsimp (RALTS (rdistinct rs1 {})) = rsimp (RALTS (rdistinct rs2 {}))"
|
|
|
902 |
|
|
|
903 |
sorry
|
|
476
|
904 |
*)
|
|
|
905 |
lemma "rsimp (rsimp_ALTs (RZERO # rdistinct (map (rder x) (rflts rs)) {RZERO})) =
|
|
|
906 |
rsimp (rsimp_ALTs (rdistinct (map (rder x) (rflts rs)) {})) "
|
|
|
907 |
|
|
|
908 |
sorry
|
|
473
|
909 |
|
|
|
910 |
|
|
|
911 |
|
|
467
|
912 |
lemma simp_der_flts:
|
|
471
|
913 |
shows "rsimp (RALTS (rdistinct (map (rder x) (rflts rs)) {})) =
|
|
|
914 |
rsimp (RALTS (rdistinct (rflts (map (rder x) rs)) {}))"
|
|
473
|
915 |
apply(subgoal_tac "map (rder x) (rflts rs) \<leadsto>f* rflts (map (rder x) rs)")
|
|
478
|
916 |
using frewrites_simpeq apply presburger
|
|
|
917 |
using early_late_der_frewrites by auto
|
|
|
918 |
|
|
|
919 |
|
|
476
|
920 |
|
|
|
921 |
|
|
|
922 |
|
|
|
923 |
|
|
|
924 |
|
|
|
925 |
|
|
|
926 |
|
|
|
927 |
|
|
|
928 |
lemma simp_der_pierce_flts_prelim:
|
|
|
929 |
shows "rsimp (rsimp_ALTs (rdistinct (map (rder x) (rflts rs)) {}))
|
|
|
930 |
= rsimp (rsimp_ALTs (rdistinct (rflts (map (rder x) rs)) {}))"
|
|
478
|
931 |
by (metis append.right_neutral grewrite.intros(2) grewrite_simpalts rsimp_ALTs.simps(2) simp_der_flts)
|
|
467
|
932 |
|
|
|
933 |
|
|
465
|
934 |
lemma simp_der_pierce_flts:
|
|
471
|
935 |
shows " rsimp (
|
|
|
936 |
rsimp_ALTs (rdistinct (map (rder x) (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {})
|
|
|
937 |
) =
|
|
|
938 |
rsimp (
|
|
|
939 |
rsimp_ALTs (rdistinct (rflts (map (rder x) (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {})
|
|
|
940 |
)"
|
|
476
|
941 |
using simp_der_pierce_flts_prelim by presburger
|
|
465
|
942 |
|
|
453
|
943 |
|
|
|
944 |
|
|
|
945 |
lemma simp_more_distinct:
|
|
465
|
946 |
shows "rsimp (rsimp_ALTs (rsa @ rs)) = rsimp (rsimp_ALTs (rsa @ (rdistinct rs (set rsa)))) "
|
|
467
|
947 |
|
|
|
948 |
|
|
465
|
949 |
|
|
453
|
950 |
|
|
|
951 |
sorry
|
|
|
952 |
|
|
|
953 |
lemma non_empty_list:
|
|
|
954 |
shows "a \<in> set as \<Longrightarrow> as \<noteq> []"
|
|
|
955 |
by (metis empty_iff empty_set)
|
|
|
956 |
|
|
456
|
957 |
lemma distinct_comp:
|
|
|
958 |
shows "rdistinct (rs1@rs2) {} = (rdistinct rs1 {}) @ (rdistinct rs2 (set rs1))"
|
|
|
959 |
apply(induct rs2 arbitrary: rs1)
|
|
|
960 |
apply simp
|
|
|
961 |
apply(subgoal_tac "rs1 @ a # rs2 = (rs1 @ [a]) @ rs2")
|
|
|
962 |
apply(simp only:)
|
|
|
963 |
apply(case_tac "a \<in> set rs1")
|
|
|
964 |
apply simp
|
|
|
965 |
oops
|
|
453
|
966 |
|
|
456
|
967 |
lemma instantiate1:
|
|
|
968 |
shows "\<lbrakk>\<And>ab rset1. rdistinct (ab # as) rset1 = rdistinct (ab # as @ [ab]) rset1\<rbrakk> \<Longrightarrow>
|
|
|
969 |
rdistinct (aa # as) rset = rdistinct (aa # as @ [aa]) rset"
|
|
|
970 |
apply(drule_tac x = "aa" in meta_spec)
|
|
|
971 |
apply(drule_tac x = "rset" in meta_spec)
|
|
453
|
972 |
apply simp
|
|
456
|
973 |
done
|
|
|
974 |
|
|
|
975 |
|
|
|
976 |
lemma not_head_elem:
|
|
|
977 |
shows " \<lbrakk>aa \<in> set (a # as); aa \<notin> (set as)\<rbrakk> \<Longrightarrow> a = aa"
|
|
|
978 |
|
|
|
979 |
by fastforce
|
|
|
980 |
|
|
|
981 |
(*
|
|
|
982 |
apply simp
|
|
|
983 |
apply (metis append_Cons)
|
|
|
984 |
apply(case_tac "ab \<in> rset1")
|
|
|
985 |
apply (metis (no_types, opaque_lifting) Un_insert_left append_Cons insert_iff rdistinct.simps(2) sup_bot_left)
|
|
|
986 |
apply(subgoal_tac "rdistinct (ab # (aa # as) @ [ab]) rset1 =
|
|
|
987 |
ab # (rdistinct ((aa # as) @ [ab]) (insert ab rset1))")
|
|
|
988 |
apply(simp only:)
|
|
|
989 |
apply(subgoal_tac "rdistinct (ab # aa # as) rset1 = ab # (rdistinct (aa # as) (insert ab rset1))")
|
|
|
990 |
apply(simp only:)
|
|
|
991 |
apply(subgoal_tac "rdistinct ((aa # as) @ [ab]) (insert ab rset1) = rdistinct (aa # as) (insert ab rset1)")
|
|
|
992 |
apply blast
|
|
|
993 |
*)
|
|
|
994 |
|
|
453
|
995 |
|
|
|
996 |
lemma flts_identity1:
|
|
|
997 |
shows "rflts (rs @ [RONE]) = rflts rs @ [RONE] "
|
|
|
998 |
apply(induct rs)
|
|
|
999 |
apply simp+
|
|
|
1000 |
apply(case_tac a)
|
|
|
1001 |
apply simp
|
|
|
1002 |
apply simp+
|
|
|
1003 |
done
|
|
|
1004 |
|
|
|
1005 |
lemma flts_identity10:
|
|
|
1006 |
shows " rflts (rs @ [RCHAR c]) = rflts rs @ [RCHAR c]"
|
|
|
1007 |
apply(induct rs)
|
|
|
1008 |
apply simp+
|
|
|
1009 |
apply(case_tac a)
|
|
|
1010 |
apply simp+
|
|
|
1011 |
done
|
|
|
1012 |
|
|
|
1013 |
lemma flts_identity11:
|
|
|
1014 |
shows " rflts (rs @ [RSEQ r1 r2]) = rflts rs @ [RSEQ r1 r2]"
|
|
|
1015 |
apply(induct rs)
|
|
|
1016 |
apply simp+
|
|
|
1017 |
apply(case_tac a)
|
|
|
1018 |
apply simp+
|
|
|
1019 |
done
|
|
|
1020 |
|
|
|
1021 |
lemma flts_identity12:
|
|
|
1022 |
shows " rflts (rs @ [RSTAR r0]) = rflts rs @ [RSTAR r0]"
|
|
|
1023 |
apply(induct rs)
|
|
|
1024 |
apply simp+
|
|
|
1025 |
apply(case_tac a)
|
|
|
1026 |
apply simp+
|
|
|
1027 |
done
|
|
|
1028 |
|
|
|
1029 |
lemma flts_identity2:
|
|
|
1030 |
shows "a \<noteq> RZERO \<and> (\<forall>rs. a \<noteq> RALTS rs) \<Longrightarrow> rflts (rs @ [a]) = rflts rs @ [a]"
|
|
|
1031 |
apply(case_tac a)
|
|
|
1032 |
apply simp
|
|
|
1033 |
using flts_identity1 apply auto[1]
|
|
|
1034 |
using flts_identity10 apply blast
|
|
|
1035 |
using flts_identity11 apply auto[1]
|
|
|
1036 |
apply blast
|
|
|
1037 |
using flts_identity12 by presburger
|
|
456
|
1038 |
|
|
|
1039 |
lemma flts_identity3:
|
|
|
1040 |
shows "a = RZERO \<Longrightarrow> rflts (rs @ [a]) = rflts rs"
|
|
|
1041 |
apply simp
|
|
|
1042 |
apply(induct rs)
|
|
|
1043 |
apply simp+
|
|
|
1044 |
apply(case_tac aa)
|
|
|
1045 |
apply simp+
|
|
|
1046 |
done
|
|
|
1047 |
|
|
|
1048 |
lemma distinct_removes_last3:
|
|
465
|
1049 |
shows "\<lbrakk>a \<in> set as\<rbrakk>
|
|
456
|
1050 |
\<Longrightarrow> rdistinct as {} = rdistinct (as @ [a]) {}"
|
|
465
|
1051 |
by (simp add: distinct_removes_last2)
|
|
456
|
1052 |
|
|
|
1053 |
lemma set_inclusion_with_flts1:
|
|
|
1054 |
shows " \<lbrakk>RONE \<in> set rs\<rbrakk> \<Longrightarrow> RONE \<in> set (rflts rs)"
|
|
|
1055 |
apply(induct rs)
|
|
|
1056 |
apply simp
|
|
|
1057 |
apply(case_tac " RONE \<in> set rs")
|
|
|
1058 |
apply simp
|
|
|
1059 |
apply (metis Un_upper2 insert_absorb insert_subset list.set_intros(2) rflts.simps(2) rflts.simps(3) rflts_def_idiot set_append)
|
|
|
1060 |
apply(case_tac "RONE = a")
|
|
|
1061 |
apply simp
|
|
|
1062 |
apply simp
|
|
|
1063 |
done
|
|
|
1064 |
|
|
|
1065 |
lemma set_inclusion_with_flts10:
|
|
|
1066 |
shows " \<lbrakk>RCHAR x \<in> set rs\<rbrakk> \<Longrightarrow> RCHAR x \<in> set (rflts rs)"
|
|
|
1067 |
apply(induct rs)
|
|
|
1068 |
apply simp
|
|
|
1069 |
apply(case_tac " RCHAR x \<in> set rs")
|
|
|
1070 |
apply simp
|
|
|
1071 |
apply (metis Un_upper2 insert_absorb insert_subset rflts.simps(2) rflts.simps(3) rflts_def_idiot set_append set_subset_Cons)
|
|
|
1072 |
apply(case_tac "RCHAR x = a")
|
|
|
1073 |
apply simp
|
|
|
1074 |
apply fastforce
|
|
|
1075 |
apply simp
|
|
|
1076 |
done
|
|
|
1077 |
|
|
|
1078 |
lemma set_inclusion_with_flts11:
|
|
|
1079 |
shows " \<lbrakk>RSEQ r1 r2 \<in> set rs\<rbrakk> \<Longrightarrow> RSEQ r1 r2 \<in> set (rflts rs)"
|
|
|
1080 |
apply(induct rs)
|
|
|
1081 |
apply simp
|
|
|
1082 |
apply(case_tac " RSEQ r1 r2 \<in> set rs")
|
|
|
1083 |
apply simp
|
|
|
1084 |
apply (metis Un_upper2 insert_absorb insert_subset rflts.simps(2) rflts.simps(3) rflts_def_idiot set_append set_subset_Cons)
|
|
|
1085 |
apply(case_tac "RSEQ r1 r2 = a")
|
|
|
1086 |
apply simp
|
|
|
1087 |
apply fastforce
|
|
|
1088 |
apply simp
|
|
|
1089 |
done
|
|
|
1090 |
|
|
|
1091 |
|
|
|
1092 |
lemma set_inclusion_with_flts:
|
|
|
1093 |
shows " \<lbrakk>a \<in> set as; rsimp a \<in> set (map rsimp as); rsimp a = RONE\<rbrakk> \<Longrightarrow> rsimp a \<in> set (rflts (map rsimp as))"
|
|
|
1094 |
by (simp add: set_inclusion_with_flts1)
|
|
453
|
1095 |
|
|
456
|
1096 |
lemma "\<And>x5. \<lbrakk>a \<in> set as; rsimp a \<in> set (map rsimp as); rsimp a = RALTS x5\<rbrakk>
|
|
|
1097 |
\<Longrightarrow> rsimp_ALTs (rdistinct (rflts (map rsimp as @ [rsimp a])) {}) =
|
|
|
1098 |
rsimp_ALTs (rdistinct (rflts (map rsimp as @ x5)) {})"
|
|
|
1099 |
|
|
465
|
1100 |
sorry
|
|
|
1101 |
|
|
453
|
1102 |
|
|
|
1103 |
lemma last_elem_dup1:
|
|
|
1104 |
shows " a \<in> set as \<Longrightarrow> rsimp (RALTS (as @ [a] )) = rsimp (RALTS (as ))"
|
|
|
1105 |
apply simp
|
|
|
1106 |
apply(subgoal_tac "rsimp a \<in> set (map rsimp as)")
|
|
|
1107 |
prefer 2
|
|
|
1108 |
apply simp
|
|
456
|
1109 |
apply(case_tac "rsimp a")
|
|
|
1110 |
apply simp
|
|
|
1111 |
|
|
|
1112 |
using flts_identity3 apply presburger
|
|
|
1113 |
apply(subst flts_identity2)
|
|
|
1114 |
using rrexp.distinct(1) rrexp.distinct(15) apply presburger
|
|
|
1115 |
apply(subst distinct_removes_last3[symmetric])
|
|
|
1116 |
using set_inclusion_with_flts apply blast
|
|
|
1117 |
apply simp
|
|
|
1118 |
apply (metis distinct_removes_last3 flts_identity10 set_inclusion_with_flts10)
|
|
|
1119 |
apply (metis distinct_removes_last3 flts_identity11 set_inclusion_with_flts11)
|
|
453
|
1120 |
sorry
|
|
|
1121 |
|
|
|
1122 |
lemma last_elem_dup:
|
|
|
1123 |
shows " a \<in> set as \<Longrightarrow> rsimp (rsimp_ALTs (as @ [a] )) = rsimp (rsimp_ALTs (as ))"
|
|
|
1124 |
apply(induct as rule: rev_induct)
|
|
|
1125 |
apply simp
|
|
|
1126 |
apply simp
|
|
|
1127 |
apply(subgoal_tac "xs \<noteq> []")
|
|
|
1128 |
prefer 2
|
|
|
1129 |
|
|
|
1130 |
|
|
|
1131 |
|
|
|
1132 |
|
|
|
1133 |
sorry
|
|
|
1134 |
|
|
|
1135 |
lemma appeared_before_remove_later:
|
|
|
1136 |
shows "a \<in> set as \<Longrightarrow> rsimp (rsimp_ALTs ( as @ a # rs)) = rsimp (rsimp_ALTs (as @ rs))"
|
|
|
1137 |
and "a \<in> set as \<Longrightarrow> rsimp (rsimp_ALTs as ) = rsimp (rsimp_ALTs (as @ [a]))"
|
|
|
1138 |
apply(induct rs arbitrary: as)
|
|
|
1139 |
apply simp
|
|
|
1140 |
|
|
|
1141 |
|
|
|
1142 |
sorry
|
|
|
1143 |
|
|
|
1144 |
lemma distinct_remove_later:
|
|
|
1145 |
shows "\<lbrakk>rder x a \<in> rder x ` set rsa\<rbrakk>
|
|
|
1146 |
\<Longrightarrow> rsimp (rsimp_ALTs (map (rder x) rsa @ rder x a # map (rder x) (rdistinct rs (insert a (set rsa))))) =
|
|
|
1147 |
rsimp (rsimp_ALTs (map (rder x) rsa @ map (rder x) (rdistinct rs (set rsa))))"
|
|
451
|
1148 |
|
|
|
1149 |
sorry
|
|
|
1150 |
|
|
|
1151 |
|
|
453
|
1152 |
lemma distinct_der_general:
|
|
|
1153 |
shows "rsimp (rsimp_ALTs (map (rder x) (rsa @ (rdistinct rs (set rsa))))) =
|
|
|
1154 |
rsimp ( rsimp_ALTs ((map (rder x) rsa)@(rdistinct (map (rder x) rs) (set (map (rder x) rsa)))) )"
|
|
|
1155 |
apply(induct rs arbitrary: rsa)
|
|
|
1156 |
apply simp
|
|
|
1157 |
apply(case_tac "a \<in> set rsa")
|
|
|
1158 |
apply(subgoal_tac "rder x a \<in> set (map (rder x) rsa)")
|
|
|
1159 |
apply simp
|
|
|
1160 |
apply simp
|
|
|
1161 |
apply(case_tac "rder x a \<notin> set (map (rder x) rsa)")
|
|
|
1162 |
apply(simp)
|
|
|
1163 |
apply(subst map_concat_cons)+
|
|
|
1164 |
apply(drule_tac x = "rsa @ [a]" in meta_spec)
|
|
|
1165 |
apply simp
|
|
|
1166 |
apply(drule neg_removal_element_of)
|
|
|
1167 |
apply simp
|
|
|
1168 |
apply(subst distinct_remove_later)
|
|
|
1169 |
apply simp
|
|
|
1170 |
apply(drule_tac x = "rsa" in meta_spec)
|
|
|
1171 |
by blast
|
|
|
1172 |
|
|
|
1173 |
|
|
|
1174 |
|
|
|
1175 |
|
|
451
|
1176 |
lemma distinct_der:
|
|
|
1177 |
shows "rsimp (rsimp_ALTs (map (rder x) (rdistinct rs {}))) = rsimp ( rsimp_ALTs (rdistinct (map (rder x) rs) {}))"
|
|
453
|
1178 |
by (metis distinct_der_general list.simps(8) self_append_conv2 set_empty)
|
|
451
|
1179 |
|
|
453
|
1180 |
|
|
|
1181 |
|
|
|
1182 |
|
|
|
1183 |
lemma rders_simp_lambda:
|
|
|
1184 |
shows " rsimp \<circ> rder x \<circ> (\<lambda>r. rders_simp r xs) = (\<lambda>r. rders_simp r (xs @ [x]))"
|
|
|
1185 |
using rders_simp_append by auto
|
|
451
|
1186 |
|
|
453
|
1187 |
lemma rders_simp_nonempty_simped:
|
|
|
1188 |
shows "xs \<noteq> [] \<Longrightarrow> rsimp \<circ> (\<lambda>r. rders_simp r xs) = (\<lambda>r. rders_simp r xs)"
|
|
|
1189 |
using rders_simp_same_simpders rsimp_idem by auto
|
|
|
1190 |
|
|
|
1191 |
lemma repeated_altssimp:
|
|
|
1192 |
shows "\<forall>r \<in> set rs. rsimp r = r \<Longrightarrow> rsimp (rsimp_ALTs (rdistinct (rflts rs) {})) =
|
|
|
1193 |
rsimp_ALTs (rdistinct (rflts rs) {})"
|
|
|
1194 |
by (metis map_idI rsimp.simps(2) rsimp_idem)
|
|
451
|
1195 |
|
|
465
|
1196 |
|
|
|
1197 |
lemma add0_isomorphic:
|
|
|
1198 |
shows "rsimp_ALTs (rdistinct (rflts [rsimp r, RZERO]) {}) = rsimp r"
|
|
|
1199 |
sorry
|
|
|
1200 |
|
|
|
1201 |
|
|
|
1202 |
lemma distinct_append_simp:
|
|
|
1203 |
shows " rsimp (rsimp_ALTs rs1) = rsimp (rsimp_ALTs rs2) \<Longrightarrow>
|
|
|
1204 |
rsimp (rsimp_ALTs (f a # rs1)) =
|
|
|
1205 |
rsimp (rsimp_ALTs (f a # rs2))"
|
|
|
1206 |
apply(case_tac rs1)
|
|
|
1207 |
apply simp
|
|
|
1208 |
apply(case_tac rs2)
|
|
|
1209 |
apply simp
|
|
|
1210 |
apply simp
|
|
|
1211 |
prefer 2
|
|
|
1212 |
apply(case_tac list)
|
|
|
1213 |
apply(case_tac rs2)
|
|
|
1214 |
apply simp
|
|
|
1215 |
using add0_isomorphic apply blast
|
|
|
1216 |
apply simp
|
|
467
|
1217 |
oops
|
|
465
|
1218 |
|
|
444
|
1219 |
lemma alts_closed_form: shows
|
|
|
1220 |
"rsimp (rders_simp (RALTS rs) s) =
|
|
|
1221 |
rsimp (RALTS (map (\<lambda>r. rders_simp r s) rs))"
|
|
|
1222 |
apply(induct s rule: rev_induct)
|
|
|
1223 |
apply simp
|
|
|
1224 |
apply simp
|
|
|
1225 |
apply(subst rders_simp_append)
|
|
|
1226 |
apply(subgoal_tac " rsimp (rders_simp (rders_simp (RALTS rs) xs) [x]) =
|
|
|
1227 |
rsimp(rders_simp (rsimp_ALTs (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})) [x])")
|
|
|
1228 |
prefer 2
|
|
|
1229 |
apply (metis inside_simp_removal rders_simp_one_char)
|
|
|
1230 |
apply(simp only: )
|
|
451
|
1231 |
apply(subst rders_simp_one_char)
|
|
|
1232 |
apply(subst rsimp_idem)
|
|
|
1233 |
apply(subgoal_tac "rsimp (rder x (rsimp_ALTs (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {}))) =
|
|
|
1234 |
rsimp ((rsimp_ALTs (map (rder x) (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})))) ")
|
|
|
1235 |
prefer 2
|
|
|
1236 |
using rder_rsimp_ALTs_commute apply presburger
|
|
|
1237 |
apply(simp only:)
|
|
|
1238 |
apply(subgoal_tac "rsimp (rsimp_ALTs (map (rder x) (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})))
|
|
|
1239 |
= rsimp (rsimp_ALTs (rdistinct (map (rder x) (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {}))")
|
|
|
1240 |
prefer 2
|
|
|
1241 |
|
|
|
1242 |
using distinct_der apply presburger
|
|
|
1243 |
apply(simp only:)
|
|
453
|
1244 |
apply(subgoal_tac " rsimp (rsimp_ALTs (rdistinct (map (rder x) (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {})) =
|
|
|
1245 |
rsimp (rsimp_ALTs (rdistinct ( (rflts (map (rder x) (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)))) {}))")
|
|
|
1246 |
apply(simp only:)
|
|
|
1247 |
apply(subgoal_tac " rsimp (rsimp_ALTs (rdistinct (rflts (map (rder x) (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs))) {})) =
|
|
|
1248 |
rsimp (rsimp_ALTs (rdistinct (rflts ( (map (rsimp \<circ> (rder x) \<circ> (\<lambda>r. rders_simp r xs)) rs))) {}))")
|
|
|
1249 |
apply(simp only:)
|
|
|
1250 |
apply(subst rders_simp_lambda)
|
|
|
1251 |
apply(subst rders_simp_nonempty_simped)
|
|
|
1252 |
apply simp
|
|
|
1253 |
apply(subgoal_tac "\<forall>r \<in> set (map (\<lambda>r. rders_simp r (xs @ [x])) rs). rsimp r = r")
|
|
|
1254 |
prefer 2
|
|
|
1255 |
apply (simp add: rders_simp_same_simpders rsimp_idem)
|
|
|
1256 |
apply(subst repeated_altssimp)
|
|
|
1257 |
apply simp
|
|
|
1258 |
apply fastforce
|
|
465
|
1259 |
apply (metis inside_simp_removal list.map_comp rder.simps(4) rsimp.simps(2) rsimp_idem)
|
|
476
|
1260 |
using simp_der_pierce_flts by blast
|
|
443
|
1261 |
|
|
444
|
1262 |
lemma alts_closed_form_variant: shows
|
|
|
1263 |
"s \<noteq> [] \<Longrightarrow> rders_simp (RALTS rs) s =
|
|
|
1264 |
rsimp (RALTS (map (\<lambda>r. rders_simp r s) rs))"
|
|
478
|
1265 |
by (metis alts_closed_form comp_apply rders_simp_nonempty_simped)
|
|
|
1266 |
|
|
443
|
1267 |
|
|
|
1268 |
|
|
|
1269 |
|
|
444
|
1270 |
lemma star_closed_form:
|
|
|
1271 |
shows "rders_simp (RSTAR r0) (c#s) =
|
|
|
1272 |
rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates s r0 [[c]]) ) ))"
|
|
|
1273 |
apply(induct s)
|
|
|
1274 |
apply simp
|
|
|
1275 |
sorry
|
|
443
|
1276 |
|
|
|
1277 |
|
|
|
1278 |
|
|
|
1279 |
lemma seq_closed_form: shows
|
|
|
1280 |
"rsimp (rders_simp (RSEQ r1 r2) s) =
|
|
|
1281 |
rsimp ( RALTS ( (RSEQ (rders_simp r1 s) r2) #
|
|
445
|
1282 |
(map (rders_simp r2) (vsuf s r1))
|
|
443
|
1283 |
)
|
|
|
1284 |
)"
|
|
|
1285 |
apply(induct s)
|
|
|
1286 |
apply simp
|
|
|
1287 |
sorry
|
|
|
1288 |
|
|
|
1289 |
|
|
444
|
1290 |
lemma seq_closed_form_variant: shows
|
|
|
1291 |
"s \<noteq> [] \<Longrightarrow> (rders_simp (RSEQ r1 r2) s) =
|
|
|
1292 |
rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) # (map (rders_simp r2) (vsuf s r1))))"
|
|
445
|
1293 |
apply(induct s rule: rev_induct)
|
|
|
1294 |
apply simp
|
|
|
1295 |
apply(subst rders_simp_append)
|
|
|
1296 |
apply(subst rders_simp_one_char)
|
|
|
1297 |
apply(subst rsimp_idem[symmetric])
|
|
|
1298 |
apply(subst rders_simp_one_char[symmetric])
|
|
|
1299 |
apply(subst rders_simp_append[symmetric])
|
|
|
1300 |
apply(insert seq_closed_form)
|
|
|
1301 |
apply(subgoal_tac "rsimp (rders_simp (RSEQ r1 r2) (xs @ [x]))
|
|
|
1302 |
= rsimp (RALTS (RSEQ (rders_simp r1 (xs @ [x])) r2 # map (rders_simp r2) (vsuf (xs @ [x]) r1)))")
|
|
|
1303 |
apply force
|
|
|
1304 |
by presburger
|
|
443
|
1305 |
|
|
476
|
1306 |
lemma simp_helps_der_pierce:
|
|
|
1307 |
shows " rsimp
|
|
|
1308 |
(rder x
|
|
|
1309 |
(rsimp_ALTs rs)) =
|
|
|
1310 |
rsimp
|
|
|
1311 |
(rsimp_ALTs
|
|
|
1312 |
(map (rder x )
|
|
|
1313 |
rs
|
|
|
1314 |
)
|
|
|
1315 |
)"
|
|
|
1316 |
sorry
|
|
|
1317 |
|
|
444
|
1318 |
end |