15 | "code (Right v) = S # (code v)" |
15 | "code (Right v) = S # (code v)" |
16 | "code (Seq v1 v2) = (code v1) @ (code v2)" |
16 | "code (Seq v1 v2) = (code v1) @ (code v2)" |
17 | "code (Stars []) = [S]" |
17 | "code (Stars []) = [S]" |
18 | "code (Stars (v # vs)) = (Z # code v) @ code (Stars vs)" |
18 | "code (Stars (v # vs)) = (Z # code v) @ code (Stars vs)" |
19 |
19 |
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20 fun sz where |
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21 "sz ZERO = 0" |
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22 | "sz ONE = 0" |
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23 | "sz (CH _) = 0" |
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24 | "sz (SEQ r1 r2) = 1 + sz r1 + sz r2" |
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25 | "sz (ALT r1 r2) = 1 + sz r1 + sz r2" |
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26 | "sz (STAR r) = 1 + sz r" |
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27 | "sz (NTIMES r n) = 1 + (n + 1) + sz r" |
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28 |
20 |
29 |
21 fun |
30 fun |
22 Stars_add :: "val \<Rightarrow> val \<Rightarrow> val" |
31 Stars_add :: "val \<Rightarrow> val \<Rightarrow> val" |
23 where |
32 where |
24 "Stars_add v (Stars vs) = Stars (v # vs)" |
33 "Stars_add v (Stars vs) = Stars (v # vs)" |
25 |
34 |
26 function |
35 function (sequential) |
27 decode' :: "bit list \<Rightarrow> rexp \<Rightarrow> (val * bit list)" |
36 decode' :: "bit list \<Rightarrow> rexp \<Rightarrow> (val * bit list)" |
28 where |
37 where |
29 "decode' bs ZERO = (undefined, bs)" |
38 "decode' bs ZERO = (undefined, bs)" |
30 | "decode' bs ONE = (Void, bs)" |
39 | "decode' bs ONE = (Void, bs)" |
31 | "decode' bs (CH d) = (Char d, bs)" |
40 | "decode' bs (CH d) = (Char d, bs)" |
37 | "decode' [] (STAR r) = (Void, [])" |
46 | "decode' [] (STAR r) = (Void, [])" |
38 | "decode' (S # bs) (STAR r) = (Stars [], bs)" |
47 | "decode' (S # bs) (STAR r) = (Stars [], bs)" |
39 | "decode' (Z # bs) (STAR r) = (let (v, bs') = decode' bs r in |
48 | "decode' (Z # bs) (STAR r) = (let (v, bs') = decode' bs r in |
40 let (vs, bs'') = decode' bs' (STAR r) |
49 let (vs, bs'') = decode' bs' (STAR r) |
41 in (Stars_add v vs, bs''))" |
50 in (Stars_add v vs, bs''))" |
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51 | "decode' bs (NTIMES r n) = decode' bs (STAR r)" |
42 by pat_completeness auto |
52 by pat_completeness auto |
43 |
53 |
44 lemma decode'_smaller: |
54 lemma decode'_smaller: |
45 assumes "decode'_dom (bs, r)" |
55 assumes "decode'_dom (bs, r)" |
46 shows "length (snd (decode' bs r)) \<le> length bs" |
56 shows "length (snd (decode' bs r)) \<le> length bs" |
47 using assms |
57 using assms |
48 apply(induct bs r) |
58 apply(induct bs r) |
49 apply(auto simp add: decode'.psimps split: prod.split) |
59 apply(auto simp add: decode'.psimps split: prod.split) |
50 using dual_order.trans apply blast |
60 using dual_order.trans apply blast |
51 by (meson dual_order.trans le_SucI) |
61 apply (meson dual_order.trans le_SucI) |
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62 done |
52 |
63 |
53 termination "decode'" |
64 termination "decode'" |
54 apply(relation "inv_image (measure(%cs. size cs) <*lex*> measure(%s. size s)) (%(ds,r). (r,ds))") |
65 apply(relation "inv_image (measure(%cs. sz cs) <*lex*> measure(%s. size s)) (%(ds,r). (r,ds))") |
55 apply(auto dest!: decode'_smaller) |
66 apply(auto dest!: decode'_smaller) |
56 by (metis less_Suc_eq_le snd_conv) |
67 apply (metis less_Suc_eq_le snd_conv) |
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68 done |
57 |
69 |
58 definition |
70 definition |
59 decode :: "bit list \<Rightarrow> rexp \<Rightarrow> val option" |
71 decode :: "bit list \<Rightarrow> rexp \<Rightarrow> val option" |
60 where |
72 where |
61 "decode ds r \<equiv> (let (v, ds') = decode' ds r |
73 "decode ds r \<equiv> (let (v, ds') = decode' ds r |
66 shows "decode' (code (Stars vs) @ ds) (STAR r) = (Stars vs, ds)" |
78 shows "decode' (code (Stars vs) @ ds) (STAR r) = (Stars vs, ds)" |
67 using assms |
79 using assms |
68 apply(induct vs) |
80 apply(induct vs) |
69 apply(auto) |
81 apply(auto) |
70 done |
82 done |
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83 |
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84 lemma decode'_code_NTIMES: |
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85 assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> (\<forall>x. decode' (code v @ x) r = (v, x))" |
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86 shows "decode' (code (Stars vs) @ ds) (NTIMES r n) = (Stars vs, ds)" |
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87 using assms |
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88 apply(induct vs arbitrary: n r ds) |
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89 apply(auto) |
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90 done |
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91 |
71 |
92 |
72 lemma decode'_code: |
93 lemma decode'_code: |
73 assumes "\<Turnstile> v : r" |
94 assumes "\<Turnstile> v : r" |
74 shows "decode' ((code v) @ ds) r = (v, ds)" |
95 shows "decode' ((code v) @ ds) r = (v, ds)" |
75 using assms |
96 using assms |
76 apply(induct v r arbitrary: ds) |
97 apply(induct v r arbitrary: ds rule: Prf.induct) |
77 apply(auto) |
98 apply(auto)[6] |
78 using decode'_code_Stars by blast |
99 using decode'_code_Stars apply blast |
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100 apply(rule decode'_code_NTIMES) |
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101 apply(simp) |
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102 apply(auto) |
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103 done |
79 |
104 |
80 lemma decode_code: |
105 lemma decode_code: |
81 assumes "\<Turnstile> v : r" |
106 assumes "\<Turnstile> v : r" |
82 shows "decode (code v) r = Some v" |
107 shows "decode (code v) r = Some v" |
83 using assms unfolding decode_def |
108 using assms unfolding decode_def |
91 | AONE "bit list" |
116 | AONE "bit list" |
92 | ACHAR "bit list" char |
117 | ACHAR "bit list" char |
93 | ASEQ "bit list" arexp arexp |
118 | ASEQ "bit list" arexp arexp |
94 | AALTs "bit list" "arexp list" |
119 | AALTs "bit list" "arexp list" |
95 | ASTAR "bit list" arexp |
120 | ASTAR "bit list" arexp |
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121 | ANTIMES "bit list" arexp nat |
96 |
122 |
97 abbreviation |
123 abbreviation |
98 "AALT bs r1 r2 \<equiv> AALTs bs [r1, r2]" |
124 "AALT bs r1 r2 \<equiv> AALTs bs [r1, r2]" |
99 |
125 |
100 fun asize :: "arexp \<Rightarrow> nat" where |
126 fun asize :: "arexp \<Rightarrow> nat" where |
102 | "asize (AONE cs) = 1" |
128 | "asize (AONE cs) = 1" |
103 | "asize (ACHAR cs c) = 1" |
129 | "asize (ACHAR cs c) = 1" |
104 | "asize (AALTs cs rs) = Suc (sum_list (map asize rs))" |
130 | "asize (AALTs cs rs) = Suc (sum_list (map asize rs))" |
105 | "asize (ASEQ cs r1 r2) = Suc (asize r1 + asize r2)" |
131 | "asize (ASEQ cs r1 r2) = Suc (asize r1 + asize r2)" |
106 | "asize (ASTAR cs r) = Suc (asize r)" |
132 | "asize (ASTAR cs r) = Suc (asize r)" |
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133 | "asize (ANTIMES cs r n) = Suc (asize r) + n" |
107 |
134 |
108 fun |
135 fun |
109 erase :: "arexp \<Rightarrow> rexp" |
136 erase :: "arexp \<Rightarrow> rexp" |
110 where |
137 where |
111 "erase AZERO = ZERO" |
138 "erase AZERO = ZERO" |
114 | "erase (AALTs _ []) = ZERO" |
141 | "erase (AALTs _ []) = ZERO" |
115 | "erase (AALTs _ [r]) = (erase r)" |
142 | "erase (AALTs _ [r]) = (erase r)" |
116 | "erase (AALTs bs (r#rs)) = ALT (erase r) (erase (AALTs bs rs))" |
143 | "erase (AALTs bs (r#rs)) = ALT (erase r) (erase (AALTs bs rs))" |
117 | "erase (ASEQ _ r1 r2) = SEQ (erase r1) (erase r2)" |
144 | "erase (ASEQ _ r1 r2) = SEQ (erase r1) (erase r2)" |
118 | "erase (ASTAR _ r) = STAR (erase r)" |
145 | "erase (ASTAR _ r) = STAR (erase r)" |
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146 | "erase (ANTIMES _ r n) = NTIMES (erase r) n" |
119 |
147 |
120 |
148 |
121 fun fuse :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp" where |
149 fun fuse :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp" where |
122 "fuse bs AZERO = AZERO" |
150 "fuse bs AZERO = AZERO" |
123 | "fuse bs (AONE cs) = AONE (bs @ cs)" |
151 | "fuse bs (AONE cs) = AONE (bs @ cs)" |
124 | "fuse bs (ACHAR cs c) = ACHAR (bs @ cs) c" |
152 | "fuse bs (ACHAR cs c) = ACHAR (bs @ cs) c" |
125 | "fuse bs (AALTs cs rs) = AALTs (bs @ cs) rs" |
153 | "fuse bs (AALTs cs rs) = AALTs (bs @ cs) rs" |
126 | "fuse bs (ASEQ cs r1 r2) = ASEQ (bs @ cs) r1 r2" |
154 | "fuse bs (ASEQ cs r1 r2) = ASEQ (bs @ cs) r1 r2" |
127 | "fuse bs (ASTAR cs r) = ASTAR (bs @ cs) r" |
155 | "fuse bs (ASTAR cs r) = ASTAR (bs @ cs) r" |
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156 | "fuse bs (ANTIMES cs r n) = ANTIMES (bs @ cs) r n" |
128 |
157 |
129 lemma fuse_append: |
158 lemma fuse_append: |
130 shows "fuse (bs1 @ bs2) r = fuse bs1 (fuse bs2 r)" |
159 shows "fuse (bs1 @ bs2) r = fuse bs1 (fuse bs2 r)" |
131 apply(induct r) |
160 apply(induct r) |
132 apply(auto) |
161 apply(auto) |
139 | "intern (CH c) = ACHAR [] c" |
168 | "intern (CH c) = ACHAR [] c" |
140 | "intern (ALT r1 r2) = AALT [] (fuse [Z] (intern r1)) |
169 | "intern (ALT r1 r2) = AALT [] (fuse [Z] (intern r1)) |
141 (fuse [S] (intern r2))" |
170 (fuse [S] (intern r2))" |
142 | "intern (SEQ r1 r2) = ASEQ [] (intern r1) (intern r2)" |
171 | "intern (SEQ r1 r2) = ASEQ [] (intern r1) (intern r2)" |
143 | "intern (STAR r) = ASTAR [] (intern r)" |
172 | "intern (STAR r) = ASTAR [] (intern r)" |
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173 | "intern (NTIMES r n) = ANTIMES [] (intern r) n" |
144 |
174 |
145 |
175 |
146 fun retrieve :: "arexp \<Rightarrow> val \<Rightarrow> bit list" where |
176 fun retrieve :: "arexp \<Rightarrow> val \<Rightarrow> bit list" where |
147 "retrieve (AONE bs) Void = bs" |
177 "retrieve (AONE bs) Void = bs" |
148 | "retrieve (ACHAR bs c) (Char d) = bs" |
178 | "retrieve (ACHAR bs c) (Char d) = bs" |
151 | "retrieve (AALTs bs (r#rs)) (Right v) = bs @ retrieve (AALTs [] rs) v" |
181 | "retrieve (AALTs bs (r#rs)) (Right v) = bs @ retrieve (AALTs [] rs) v" |
152 | "retrieve (ASEQ bs r1 r2) (Seq v1 v2) = bs @ retrieve r1 v1 @ retrieve r2 v2" |
182 | "retrieve (ASEQ bs r1 r2) (Seq v1 v2) = bs @ retrieve r1 v1 @ retrieve r2 v2" |
153 | "retrieve (ASTAR bs r) (Stars []) = bs @ [S]" |
183 | "retrieve (ASTAR bs r) (Stars []) = bs @ [S]" |
154 | "retrieve (ASTAR bs r) (Stars (v#vs)) = |
184 | "retrieve (ASTAR bs r) (Stars (v#vs)) = |
155 bs @ [Z] @ retrieve r v @ retrieve (ASTAR [] r) (Stars vs)" |
185 bs @ [Z] @ retrieve r v @ retrieve (ASTAR [] r) (Stars vs)" |
156 |
186 | "retrieve (ANTIMES bs r 0) (Stars []) = bs @ [S]" |
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187 | "retrieve (ANTIMES bs r (Suc n)) (Stars (v#vs)) = |
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188 bs @ [Z] @ retrieve r v @ retrieve (ANTIMES [] r n) (Stars vs)" |
157 |
189 |
158 |
190 |
159 fun |
191 fun |
160 bnullable :: "arexp \<Rightarrow> bool" |
192 bnullable :: "arexp \<Rightarrow> bool" |
161 where |
193 where |
163 | "bnullable (AONE bs) = True" |
195 | "bnullable (AONE bs) = True" |
164 | "bnullable (ACHAR bs c) = False" |
196 | "bnullable (ACHAR bs c) = False" |
165 | "bnullable (AALTs bs rs) = (\<exists>r \<in> set rs. bnullable r)" |
197 | "bnullable (AALTs bs rs) = (\<exists>r \<in> set rs. bnullable r)" |
166 | "bnullable (ASEQ bs r1 r2) = (bnullable r1 \<and> bnullable r2)" |
198 | "bnullable (ASEQ bs r1 r2) = (bnullable r1 \<and> bnullable r2)" |
167 | "bnullable (ASTAR bs r) = True" |
199 | "bnullable (ASTAR bs r) = True" |
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200 | "bnullable (ANTIMES bs r n) = (if n = 0 then True else bnullable r)" |
168 |
201 |
169 abbreviation |
202 abbreviation |
170 bnullables :: "arexp list \<Rightarrow> bool" |
203 bnullables :: "arexp list \<Rightarrow> bool" |
171 where |
204 where |
172 "bnullables rs \<equiv> (\<exists>r \<in> set rs. bnullable r)" |
205 "bnullables rs \<equiv> (\<exists>r \<in> set rs. bnullable r)" |
173 |
206 |
174 fun |
207 function (sequential) |
175 bmkeps :: "arexp \<Rightarrow> bit list" and |
208 bmkeps :: "arexp \<Rightarrow> bit list" |
176 bmkepss :: "arexp list \<Rightarrow> bit list" |
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177 where |
209 where |
178 "bmkeps(AONE bs) = bs" |
210 "bmkeps(AONE bs) = bs" |
179 | "bmkeps(ASEQ bs r1 r2) = bs @ (bmkeps r1) @ (bmkeps r2)" |
211 | "bmkeps(ASEQ bs r1 r2) = bs @ (bmkeps r1) @ (bmkeps r2)" |
180 | "bmkeps(AALTs bs rs) = bs @ (bmkepss rs)" |
212 | "bmkeps(AALTs bs (r#rs)) = |
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213 (if bnullable(r) then (bs @ bmkeps r) else (bmkeps (AALTs bs rs)))" |
181 | "bmkeps(ASTAR bs r) = bs @ [S]" |
214 | "bmkeps(ASTAR bs r) = bs @ [S]" |
182 | "bmkepss (r # rs) = (if bnullable(r) then (bmkeps r) else (bmkepss rs))" |
215 | "bmkeps(ANTIMES bs r n) = |
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216 (if n = 0 then bs @ [S] else bs @ [Z] @ (bmkeps r) @ bmkeps(ANTIMES [] r (n - 1)))" |
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217 apply(pat_completeness) |
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218 apply(auto) |
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219 done |
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220 |
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221 termination "bmkeps" |
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222 apply(relation "measure asize") |
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223 apply(auto) |
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224 using asize.elims by force |
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225 |
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226 fun |
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227 bmkepss :: "arexp list \<Rightarrow> bit list" |
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228 where |
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229 "bmkepss (r # rs) = (if bnullable(r) then (bmkeps r) else (bmkepss rs))" |
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230 |
183 |
231 |
184 lemma bmkepss1: |
232 lemma bmkepss1: |
185 assumes "\<not> bnullables rs1" |
233 assumes "\<not> bnullables rs1" |
186 shows "bmkepss (rs1 @ rs2) = bmkepss rs2" |
234 shows "bmkepss (rs1 @ rs2) = bmkepss rs2" |
187 using assms |
235 using assms |
188 by (induct rs1) (auto) |
236 by(induct rs1) (auto) |
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237 |
189 |
238 |
190 lemma bmkepss2: |
239 lemma bmkepss2: |
191 assumes "bnullables rs1" |
240 assumes "bnullables rs1" |
192 shows "bmkepss (rs1 @ rs2) = bmkepss rs1" |
241 shows "bmkepss (rs1 @ rs2) = bmkepss rs1" |
193 using assms |
242 using assms |
204 | "bder c (ASEQ bs r1 r2) = |
253 | "bder c (ASEQ bs r1 r2) = |
205 (if bnullable r1 |
254 (if bnullable r1 |
206 then AALT bs (ASEQ [] (bder c r1) r2) (fuse (bmkeps r1) (bder c r2)) |
255 then AALT bs (ASEQ [] (bder c r1) r2) (fuse (bmkeps r1) (bder c r2)) |
207 else ASEQ bs (bder c r1) r2)" |
256 else ASEQ bs (bder c r1) r2)" |
208 | "bder c (ASTAR bs r) = ASEQ (bs @ [Z]) (bder c r) (ASTAR [] r)" |
257 | "bder c (ASTAR bs r) = ASEQ (bs @ [Z]) (bder c r) (ASTAR [] r)" |
209 |
258 | "bder c (ANTIMES bs r n) = (if n = 0 then AZERO else ASEQ (bs @ [Z]) (bder c r) (ANTIMES [] r (n - 1)))" |
210 |
259 |
211 fun |
260 fun |
212 bders :: "arexp \<Rightarrow> string \<Rightarrow> arexp" |
261 bders :: "arexp \<Rightarrow> string \<Rightarrow> arexp" |
213 where |
262 where |
214 "bders r [] = r" |
263 "bders r [] = r" |
262 using assms |
311 using assms |
263 apply(induct vs) |
312 apply(induct vs) |
264 apply(simp_all) |
313 apply(simp_all) |
265 done |
314 done |
266 |
315 |
267 lemma retrieve_codestars2: |
316 lemma retrieve_encode_NTIMES: |
268 assumes "\<forall>v \<in> set vs. \<Turnstile> v : r \<and> code v = retrieve (intern r) v" |
317 assumes "\<forall>v\<in>set vs. \<Turnstile> v : r \<and> code v = retrieve (intern r) v" "length vs = n" |
269 shows "retrieve (ASTAR bs (intern r)) (Stars []) = bs @ [S]" |
318 shows "code (Stars vs) = retrieve (ANTIMES [] (intern r) n) (Stars vs)" |
270 apply simp |
319 using assms |
271 done |
320 apply(induct vs arbitrary: n) |
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321 apply(simp_all) |
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322 by force |
272 |
323 |
273 |
324 |
274 lemma retrieve_fuse2: |
325 lemma retrieve_fuse2: |
275 assumes "\<Turnstile> v : (erase r)" |
326 assumes "\<Turnstile> v : (erase r)" |
276 shows "retrieve (fuse bs r) v = bs @ retrieve r v" |
327 shows "retrieve (fuse bs r) v = bs @ retrieve r v" |
305 lemma retrieve_code: |
362 lemma retrieve_code: |
306 assumes "\<Turnstile> v : r" |
363 assumes "\<Turnstile> v : r" |
307 shows "code v = retrieve (intern r) v" |
364 shows "code v = retrieve (intern r) v" |
308 using assms |
365 using assms |
309 apply(induct v r ) |
366 apply(induct v r ) |
310 apply(simp_all add: retrieve_fuse retrieve_encode_STARS) |
367 apply(simp_all add: retrieve_fuse retrieve_encode_STARS) |
311 done |
368 apply(subst retrieve_encode_NTIMES) |
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369 apply(auto) |
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370 done |
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371 |
312 |
372 |
313 |
373 |
314 lemma retrieve_AALTs_bnullable1: |
374 lemma retrieve_AALTs_bnullable1: |
315 assumes "bnullable r" |
375 assumes "bnullable r" |
316 shows "retrieve (AALTs bs (r # rs)) (mkeps (erase (AALTs bs (r # rs)))) |
376 shows "retrieve (AALTs bs (r # rs)) (mkeps (erase (AALTs bs (r # rs)))) |
345 using retrieve_AALTs_bnullable1 apply presburger |
405 using retrieve_AALTs_bnullable1 apply presburger |
346 apply (metis retrieve_AALTs_bnullable2) |
406 apply (metis retrieve_AALTs_bnullable2) |
347 apply (simp add: retrieve_AALTs_bnullable1) |
407 apply (simp add: retrieve_AALTs_bnullable1) |
348 by (metis retrieve_AALTs_bnullable2) |
408 by (metis retrieve_AALTs_bnullable2) |
349 |
409 |
350 |
410 lemma bmkeps_retrieve_ANTIMES: |
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411 assumes "if n = 0 then True else bmkeps r = retrieve r (mkeps (erase r))" |
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412 and "bnullable (ANTIMES bs r n)" |
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413 shows "bmkeps (ANTIMES bs r n) = retrieve (ANTIMES bs r n) (Stars (replicate n (mkeps (erase r))))" |
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414 using assms |
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415 apply(induct n arbitrary: r bs) |
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416 apply(auto)[1] |
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417 apply(simp) |
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418 done |
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419 |
351 lemma bmkeps_retrieve: |
420 lemma bmkeps_retrieve: |
352 assumes "bnullable r" |
421 assumes "bnullable r" |
353 shows "bmkeps r = retrieve r (mkeps (erase r))" |
422 shows "bmkeps r = retrieve r (mkeps (erase r))" |
354 using assms |
423 using assms |
355 apply(induct r) |
424 apply(induct r rule: bmkeps.induct) |
356 apply(auto) |
425 apply(auto) |
357 using bmkeps_retrieve_AALTs by auto |
426 apply (simp add: retrieve_AALTs_bnullable1) |
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427 using retrieve_AALTs_bnullable1 apply force |
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428 apply(metis retrieve_AALTs_bnullable2) |
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429 by (metis Cons_eq_appendI One_nat_def Suc_diff_1 eq_Nil_appendI replicate_Suc retrieve.simps(10)) |
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430 |
358 |
431 |
359 lemma bder_retrieve: |
432 lemma bder_retrieve: |
360 assumes "\<Turnstile> v : der c (erase r)" |
433 assumes "\<Turnstile> v : der c (erase r)" |
361 shows "retrieve (bder c r) v = retrieve r (injval (erase r) c v)" |
434 shows "retrieve (bder c r) v = retrieve r (injval (erase r) c v)" |
362 using assms |
435 using assms |
393 apply(simp) |
466 apply(simp) |
394 apply(erule Prf_elims) |
467 apply(erule Prf_elims) |
395 apply(clarify) |
468 apply(clarify) |
396 apply(erule Prf_elims) |
469 apply(erule Prf_elims) |
397 apply(clarify) |
470 apply(clarify) |
398 by (simp add: retrieve_fuse2) |
471 apply (simp add: retrieve_fuse2) |
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472 (* ANTIMES case *) |
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473 apply(auto) |
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474 apply(erule Prf_elims) |
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475 apply(erule Prf_elims) |
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476 apply(clarify) |
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477 apply(erule Prf_elims) |
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478 apply(clarify) |
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479 by (metis (full_types) Suc_pred append_assoc injval.simps(8) retrieve.simps(10) retrieve.simps(6)) |
399 |
480 |
400 |
481 |
401 lemma MAIN_decode: |
482 lemma MAIN_decode: |
402 assumes "\<Turnstile> v : ders s r" |
483 assumes "\<Turnstile> v : ders s r" |
403 shows "Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r" |
484 shows "Some (flex r id s v) = decode (retrieve (bders (intern r) s) v) r" |