src/HOL/Library/Linear_Temporal_Logic_on_Streams.thy
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```     1 (*  Title:      HOL/Library/Linear_Temporal_Logic_on_Streams.thy
```
```     2     Author:     Andrei Popescu, TU Muenchen
```
```     3     Author:     Dmitriy Traytel, TU Muenchen
```
```     4 *)
```
```     5
```
```     6 section \<open>Linear Temporal Logic on Streams\<close>
```
```     7
```
```     8 theory Linear_Temporal_Logic_on_Streams
```
```     9   imports Stream Sublist Extended_Nat Infinite_Set
```
```    10 begin
```
```    11
```
```    12 section\<open>Preliminaries\<close>
```
```    13
```
```    14 lemma shift_prefix:
```
```    15 assumes "xl @- xs = yl @- ys" and "length xl \<le> length yl"
```
```    16 shows "prefixeq xl yl"
```
```    17 using assms proof(induct xl arbitrary: yl xs ys)
```
```    18   case (Cons x xl yl xs ys)
```
```    19   thus ?case by (cases yl) auto
```
```    20 qed auto
```
```    21
```
```    22 lemma shift_prefix_cases:
```
```    23 assumes "xl @- xs = yl @- ys"
```
```    24 shows "prefixeq xl yl \<or> prefixeq yl xl"
```
```    25 using shift_prefix[OF assms] apply(cases "length xl \<le> length yl")
```
```    26 by (metis, metis assms nat_le_linear shift_prefix)
```
```    27
```
```    28
```
```    29 section\<open>Linear temporal logic\<close>
```
```    30
```
```    31 (* Propositional connectives: *)
```
```    32 abbreviation (input) IMPL (infix "impl" 60)
```
```    33 where "\<phi> impl \<psi> \<equiv> \<lambda> xs. \<phi> xs \<longrightarrow> \<psi> xs"
```
```    34
```
```    35 abbreviation (input) OR (infix "or" 60)
```
```    36 where "\<phi> or \<psi> \<equiv> \<lambda> xs. \<phi> xs \<or> \<psi> xs"
```
```    37
```
```    38 abbreviation (input) AND (infix "aand" 60)
```
```    39 where "\<phi> aand \<psi> \<equiv> \<lambda> xs. \<phi> xs \<and> \<psi> xs"
```
```    40
```
```    41 abbreviation (input) "not \<phi> \<equiv> \<lambda> xs. \<not> \<phi> xs"
```
```    42
```
```    43 abbreviation (input) "true \<equiv> \<lambda> xs. True"
```
```    44
```
```    45 abbreviation (input) "false \<equiv> \<lambda> xs. False"
```
```    46
```
```    47 lemma impl_not_or: "\<phi> impl \<psi> = (not \<phi>) or \<psi>"
```
```    48 by blast
```
```    49
```
```    50 lemma not_or: "not (\<phi> or \<psi>) = (not \<phi>) aand (not \<psi>)"
```
```    51 by blast
```
```    52
```
```    53 lemma not_aand: "not (\<phi> aand \<psi>) = (not \<phi>) or (not \<psi>)"
```
```    54 by blast
```
```    55
```
```    56 lemma non_not[simp]: "not (not \<phi>) = \<phi>" by simp
```
```    57
```
```    58 (* Temporal (LTL) connectives: *)
```
```    59 fun holds where "holds P xs \<longleftrightarrow> P (shd xs)"
```
```    60 fun nxt where "nxt \<phi> xs = \<phi> (stl xs)"
```
```    61
```
```    62 definition "HLD s = holds (\<lambda>x. x \<in> s)"
```
```    63
```
```    64 abbreviation HLD_nxt (infixr "\<cdot>" 65) where
```
```    65   "s \<cdot> P \<equiv> HLD s aand nxt P"
```
```    66
```
```    67 inductive ev for \<phi> where
```
```    68 base: "\<phi> xs \<Longrightarrow> ev \<phi> xs"
```
```    69 |
```
```    70 step: "ev \<phi> (stl xs) \<Longrightarrow> ev \<phi> xs"
```
```    71
```
```    72 coinductive alw for \<phi> where
```
```    73 alw: "\<lbrakk>\<phi> xs; alw \<phi> (stl xs)\<rbrakk> \<Longrightarrow> alw \<phi> xs"
```
```    74
```
```    75 (* weak until: *)
```
```    76 coinductive UNTIL (infix "until" 60) for \<phi> \<psi> where
```
```    77 base: "\<psi> xs \<Longrightarrow> (\<phi> until \<psi>) xs"
```
```    78 |
```
```    79 step: "\<lbrakk>\<phi> xs; (\<phi> until \<psi>) (stl xs)\<rbrakk> \<Longrightarrow> (\<phi> until \<psi>) xs"
```
```    80
```
```    81 lemma holds_mono:
```
```    82 assumes holds: "holds P xs" and 0: "\<And> x. P x \<Longrightarrow> Q x"
```
```    83 shows "holds Q xs"
```
```    84 using assms by auto
```
```    85
```
```    86 lemma holds_aand:
```
```    87 "(holds P aand holds Q) steps \<longleftrightarrow> holds (\<lambda> step. P step \<and> Q step) steps" by auto
```
```    88
```
```    89 lemma HLD_iff: "HLD s \<omega> \<longleftrightarrow> shd \<omega> \<in> s"
```
```    90   by (simp add: HLD_def)
```
```    91
```
```    92 lemma HLD_Stream[simp]: "HLD X (x ## \<omega>) \<longleftrightarrow> x \<in> X"
```
```    93   by (simp add: HLD_iff)
```
```    94
```
```    95 lemma nxt_mono:
```
```    96 assumes nxt: "nxt \<phi> xs" and 0: "\<And> xs. \<phi> xs \<Longrightarrow> \<psi> xs"
```
```    97 shows "nxt \<psi> xs"
```
```    98 using assms by auto
```
```    99
```
```   100 declare ev.intros[intro]
```
```   101 declare alw.cases[elim]
```
```   102
```
```   103 lemma ev_induct_strong[consumes 1, case_names base step]:
```
```   104   "ev \<phi> x \<Longrightarrow> (\<And>xs. \<phi> xs \<Longrightarrow> P xs) \<Longrightarrow> (\<And>xs. ev \<phi> (stl xs) \<Longrightarrow> \<not> \<phi> xs \<Longrightarrow> P (stl xs) \<Longrightarrow> P xs) \<Longrightarrow> P x"
```
```   105   by (induct rule: ev.induct) auto
```
```   106
```
```   107 lemma alw_coinduct[consumes 1, case_names alw stl]:
```
```   108   "X x \<Longrightarrow> (\<And>x. X x \<Longrightarrow> \<phi> x) \<Longrightarrow> (\<And>x. X x \<Longrightarrow> \<not> alw \<phi> (stl x) \<Longrightarrow> X (stl x)) \<Longrightarrow> alw \<phi> x"
```
```   109   using alw.coinduct[of X x \<phi>] by auto
```
```   110
```
```   111 lemma ev_mono:
```
```   112 assumes ev: "ev \<phi> xs" and 0: "\<And> xs. \<phi> xs \<Longrightarrow> \<psi> xs"
```
```   113 shows "ev \<psi> xs"
```
```   114 using ev by induct (auto intro: ev.intros simp: 0)
```
```   115
```
```   116 lemma alw_mono:
```
```   117 assumes alw: "alw \<phi> xs" and 0: "\<And> xs. \<phi> xs \<Longrightarrow> \<psi> xs"
```
```   118 shows "alw \<psi> xs"
```
```   119 using alw by coinduct (auto elim: alw.cases simp: 0)
```
```   120
```
```   121 lemma until_monoL:
```
```   122 assumes until: "(\<phi>1 until \<psi>) xs" and 0: "\<And> xs. \<phi>1 xs \<Longrightarrow> \<phi>2 xs"
```
```   123 shows "(\<phi>2 until \<psi>) xs"
```
```   124 using until by coinduct (auto elim: UNTIL.cases simp: 0)
```
```   125
```
```   126 lemma until_monoR:
```
```   127 assumes until: "(\<phi> until \<psi>1) xs" and 0: "\<And> xs. \<psi>1 xs \<Longrightarrow> \<psi>2 xs"
```
```   128 shows "(\<phi> until \<psi>2) xs"
```
```   129 using until by coinduct (auto elim: UNTIL.cases simp: 0)
```
```   130
```
```   131 lemma until_mono:
```
```   132 assumes until: "(\<phi>1 until \<psi>1) xs" and
```
```   133 0: "\<And> xs. \<phi>1 xs \<Longrightarrow> \<phi>2 xs" "\<And> xs. \<psi>1 xs \<Longrightarrow> \<psi>2 xs"
```
```   134 shows "(\<phi>2 until \<psi>2) xs"
```
```   135 using until by coinduct (auto elim: UNTIL.cases simp: 0)
```
```   136
```
```   137 lemma until_false: "\<phi> until false = alw \<phi>"
```
```   138 proof-
```
```   139   {fix xs assume "(\<phi> until false) xs" hence "alw \<phi> xs"
```
```   140    apply coinduct by (auto elim: UNTIL.cases)
```
```   141   }
```
```   142   moreover
```
```   143   {fix xs assume "alw \<phi> xs" hence "(\<phi> until false) xs"
```
```   144    apply coinduct by (auto elim: alw.cases)
```
```   145   }
```
```   146   ultimately show ?thesis by blast
```
```   147 qed
```
```   148
```
```   149 lemma ev_nxt: "ev \<phi> = (\<phi> or nxt (ev \<phi>))"
```
```   150 apply(rule ext) by (metis ev.simps nxt.simps)
```
```   151
```
```   152 lemma alw_nxt: "alw \<phi> = (\<phi> aand nxt (alw \<phi>))"
```
```   153 apply(rule ext) by (metis alw.simps nxt.simps)
```
```   154
```
```   155 lemma ev_ev[simp]: "ev (ev \<phi>) = ev \<phi>"
```
```   156 proof-
```
```   157   {fix xs
```
```   158    assume "ev (ev \<phi>) xs" hence "ev \<phi> xs"
```
```   159    by induct (auto intro: ev.intros)
```
```   160   }
```
```   161   thus ?thesis by (auto intro: ev.intros)
```
```   162 qed
```
```   163
```
```   164 lemma alw_alw[simp]: "alw (alw \<phi>) = alw \<phi>"
```
```   165 proof-
```
```   166   {fix xs
```
```   167    assume "alw \<phi> xs" hence "alw (alw \<phi>) xs"
```
```   168    by coinduct (auto elim: alw.cases)
```
```   169   }
```
```   170   thus ?thesis by (auto elim: alw.cases)
```
```   171 qed
```
```   172
```
```   173 lemma ev_shift:
```
```   174 assumes "ev \<phi> xs"
```
```   175 shows "ev \<phi> (xl @- xs)"
```
```   176 using assms by (induct xl) (auto intro: ev.intros)
```
```   177
```
```   178 lemma ev_imp_shift:
```
```   179 assumes "ev \<phi> xs"  shows "\<exists> xl xs2. xs = xl @- xs2 \<and> \<phi> xs2"
```
```   180 using assms by induct (metis shift.simps(1), metis shift.simps(2) stream.collapse)+
```
```   181
```
```   182 lemma alw_ev_shift: "alw \<phi> xs1 \<Longrightarrow> ev (alw \<phi>) (xl @- xs1)"
```
```   183 by (auto intro: ev_shift ev.intros)
```
```   184
```
```   185 lemma alw_shift:
```
```   186 assumes "alw \<phi> (xl @- xs)"
```
```   187 shows "alw \<phi> xs"
```
```   188 using assms by (induct xl) (auto elim: alw.cases)
```
```   189
```
```   190 lemma ev_ex_nxt:
```
```   191 assumes "ev \<phi> xs"
```
```   192 shows "\<exists> n. (nxt ^^ n) \<phi> xs"
```
```   193 using assms proof induct
```
```   194   case (base xs) thus ?case by (intro exI[of _ 0]) auto
```
```   195 next
```
```   196   case (step xs)
```
```   197   then obtain n where "(nxt ^^ n) \<phi> (stl xs)" by blast
```
```   198   thus ?case by (intro exI[of _ "Suc n"]) (metis funpow.simps(2) nxt.simps o_def)
```
```   199 qed
```
```   200
```
```   201 lemma alw_sdrop:
```
```   202 assumes "alw \<phi> xs"  shows "alw \<phi> (sdrop n xs)"
```
```   203 by (metis alw_shift assms stake_sdrop)
```
```   204
```
```   205 lemma nxt_sdrop: "(nxt ^^ n) \<phi> xs \<longleftrightarrow> \<phi> (sdrop n xs)"
```
```   206 by (induct n arbitrary: xs) auto
```
```   207
```
```   208 definition "wait \<phi> xs \<equiv> LEAST n. (nxt ^^ n) \<phi> xs"
```
```   209
```
```   210 lemma nxt_wait:
```
```   211 assumes "ev \<phi> xs"  shows "(nxt ^^ (wait \<phi> xs)) \<phi> xs"
```
```   212 unfolding wait_def using ev_ex_nxt[OF assms] by(rule LeastI_ex)
```
```   213
```
```   214 lemma nxt_wait_least:
```
```   215 assumes ev: "ev \<phi> xs" and nxt: "(nxt ^^ n) \<phi> xs"  shows "wait \<phi> xs \<le> n"
```
```   216 unfolding wait_def using ev_ex_nxt[OF ev] by (metis Least_le nxt)
```
```   217
```
```   218 lemma sdrop_wait:
```
```   219 assumes "ev \<phi> xs"  shows "\<phi> (sdrop (wait \<phi> xs) xs)"
```
```   220 using nxt_wait[OF assms] unfolding nxt_sdrop .
```
```   221
```
```   222 lemma sdrop_wait_least:
```
```   223 assumes ev: "ev \<phi> xs" and nxt: "\<phi> (sdrop n xs)"  shows "wait \<phi> xs \<le> n"
```
```   224 using assms nxt_wait_least unfolding nxt_sdrop by auto
```
```   225
```
```   226 lemma nxt_ev: "(nxt ^^ n) \<phi> xs \<Longrightarrow> ev \<phi> xs"
```
```   227 by (induct n arbitrary: xs) (auto intro: ev.intros)
```
```   228
```
```   229 lemma not_ev: "not (ev \<phi>) = alw (not \<phi>)"
```
```   230 proof(rule ext, safe)
```
```   231   fix xs assume "not (ev \<phi>) xs" thus "alw (not \<phi>) xs"
```
```   232   by (coinduct) (auto intro: ev.intros)
```
```   233 next
```
```   234   fix xs assume "ev \<phi> xs" and "alw (not \<phi>) xs" thus False
```
```   235   by (induct) (auto elim: alw.cases)
```
```   236 qed
```
```   237
```
```   238 lemma not_alw: "not (alw \<phi>) = ev (not \<phi>)"
```
```   239 proof-
```
```   240   have "not (alw \<phi>) = not (alw (not (not \<phi>)))" by simp
```
```   241   also have "... = ev (not \<phi>)" unfolding not_ev[symmetric] by simp
```
```   242   finally show ?thesis .
```
```   243 qed
```
```   244
```
```   245 lemma not_ev_not[simp]: "not (ev (not \<phi>)) = alw \<phi>"
```
```   246 unfolding not_ev by simp
```
```   247
```
```   248 lemma not_alw_not[simp]: "not (alw (not \<phi>)) = ev \<phi>"
```
```   249 unfolding not_alw by simp
```
```   250
```
```   251 lemma alw_ev_sdrop:
```
```   252 assumes "alw (ev \<phi>) (sdrop m xs)"
```
```   253 shows "alw (ev \<phi>) xs"
```
```   254 using assms
```
```   255 by coinduct (metis alw_nxt ev_shift funpow_swap1 nxt.simps nxt_sdrop stake_sdrop)
```
```   256
```
```   257 lemma ev_alw_imp_alw_ev:
```
```   258 assumes "ev (alw \<phi>) xs"  shows "alw (ev \<phi>) xs"
```
```   259 using assms apply induct
```
```   260   apply (metis (full_types) alw_mono ev.base)
```
```   261   by (metis alw alw_nxt ev.step)
```
```   262
```
```   263 lemma alw_aand: "alw (\<phi> aand \<psi>) = alw \<phi> aand alw \<psi>"
```
```   264 proof-
```
```   265   {fix xs assume "alw (\<phi> aand \<psi>) xs" hence "(alw \<phi> aand alw \<psi>) xs"
```
```   266    by (auto elim: alw_mono)
```
```   267   }
```
```   268   moreover
```
```   269   {fix xs assume "(alw \<phi> aand alw \<psi>) xs" hence "alw (\<phi> aand \<psi>) xs"
```
```   270    by coinduct (auto elim: alw.cases)
```
```   271   }
```
```   272   ultimately show ?thesis by blast
```
```   273 qed
```
```   274
```
```   275 lemma ev_or: "ev (\<phi> or \<psi>) = ev \<phi> or ev \<psi>"
```
```   276 proof-
```
```   277   {fix xs assume "(ev \<phi> or ev \<psi>) xs" hence "ev (\<phi> or \<psi>) xs"
```
```   278    by (auto elim: ev_mono)
```
```   279   }
```
```   280   moreover
```
```   281   {fix xs assume "ev (\<phi> or \<psi>) xs" hence "(ev \<phi> or ev \<psi>) xs"
```
```   282    by induct (auto intro: ev.intros)
```
```   283   }
```
```   284   ultimately show ?thesis by blast
```
```   285 qed
```
```   286
```
```   287 lemma ev_alw_aand:
```
```   288 assumes \<phi>: "ev (alw \<phi>) xs" and \<psi>: "ev (alw \<psi>) xs"
```
```   289 shows "ev (alw (\<phi> aand \<psi>)) xs"
```
```   290 proof-
```
```   291   obtain xl xs1 where xs1: "xs = xl @- xs1" and \<phi>\<phi>: "alw \<phi> xs1"
```
```   292   using \<phi> by (metis ev_imp_shift)
```
```   293   moreover obtain yl ys1 where xs2: "xs = yl @- ys1" and \<psi>\<psi>: "alw \<psi> ys1"
```
```   294   using \<psi> by (metis ev_imp_shift)
```
```   295   ultimately have 0: "xl @- xs1 = yl @- ys1" by auto
```
```   296   hence "prefixeq xl yl \<or> prefixeq yl xl" using shift_prefix_cases by auto
```
```   297   thus ?thesis proof
```
```   298     assume "prefixeq xl yl"
```
```   299     then obtain yl1 where yl: "yl = xl @ yl1" by (elim prefixeqE)
```
```   300     have xs1': "xs1 = yl1 @- ys1" using 0 unfolding yl by simp
```
```   301     have "alw \<phi> ys1" using \<phi>\<phi> unfolding xs1' by (metis alw_shift)
```
```   302     hence "alw (\<phi> aand \<psi>) ys1" using \<psi>\<psi> unfolding alw_aand by auto
```
```   303     thus ?thesis unfolding xs2 by (auto intro: alw_ev_shift)
```
```   304   next
```
```   305     assume "prefixeq yl xl"
```
```   306     then obtain xl1 where xl: "xl = yl @ xl1" by (elim prefixeqE)
```
```   307     have ys1': "ys1 = xl1 @- xs1" using 0 unfolding xl by simp
```
```   308     have "alw \<psi> xs1" using \<psi>\<psi> unfolding ys1' by (metis alw_shift)
```
```   309     hence "alw (\<phi> aand \<psi>) xs1" using \<phi>\<phi> unfolding alw_aand by auto
```
```   310     thus ?thesis unfolding xs1 by (auto intro: alw_ev_shift)
```
```   311   qed
```
```   312 qed
```
```   313
```
```   314 lemma ev_alw_alw_impl:
```
```   315 assumes "ev (alw \<phi>) xs" and "alw (alw \<phi> impl ev \<psi>) xs"
```
```   316 shows "ev \<psi> xs"
```
```   317 using assms by induct (auto intro: ev.intros elim: alw.cases)
```
```   318
```
```   319 lemma ev_alw_stl[simp]: "ev (alw \<phi>) (stl x) \<longleftrightarrow> ev (alw \<phi>) x"
```
```   320 by (metis (full_types) alw_nxt ev_nxt nxt.simps)
```
```   321
```
```   322 lemma alw_alw_impl_ev:
```
```   323 "alw (alw \<phi> impl ev \<psi>) = (ev (alw \<phi>) impl alw (ev \<psi>))" (is "?A = ?B")
```
```   324 proof-
```
```   325   {fix xs assume "?A xs \<and> ev (alw \<phi>) xs" hence "alw (ev \<psi>) xs"
```
```   326    apply coinduct using ev_nxt by (auto elim: ev_alw_alw_impl alw.cases intro: ev.intros)
```
```   327   }
```
```   328   moreover
```
```   329   {fix xs assume "?B xs" hence "?A xs"
```
```   330    apply coinduct by (auto elim: alw.cases intro: ev.intros)
```
```   331   }
```
```   332   ultimately show ?thesis by blast
```
```   333 qed
```
```   334
```
```   335 lemma ev_alw_impl:
```
```   336 assumes "ev \<phi> xs" and "alw (\<phi> impl \<psi>) xs"  shows "ev \<psi> xs"
```
```   337 using assms by induct (auto intro: ev.intros elim: alw.cases)
```
```   338
```
```   339 lemma ev_alw_impl_ev:
```
```   340 assumes "ev \<phi> xs" and "alw (\<phi> impl ev \<psi>) xs"  shows "ev \<psi> xs"
```
```   341 using ev_alw_impl[OF assms] by simp
```
```   342
```
```   343 lemma alw_mp:
```
```   344 assumes "alw \<phi> xs" and "alw (\<phi> impl \<psi>) xs"
```
```   345 shows "alw \<psi> xs"
```
```   346 proof-
```
```   347   {assume "alw \<phi> xs \<and> alw (\<phi> impl \<psi>) xs" hence ?thesis
```
```   348    apply coinduct by (auto elim: alw.cases)
```
```   349   }
```
```   350   thus ?thesis using assms by auto
```
```   351 qed
```
```   352
```
```   353 lemma all_imp_alw:
```
```   354 assumes "\<And> xs. \<phi> xs"  shows "alw \<phi> xs"
```
```   355 proof-
```
```   356   {assume "\<forall> xs. \<phi> xs"
```
```   357    hence ?thesis by coinduct auto
```
```   358   }
```
```   359   thus ?thesis using assms by auto
```
```   360 qed
```
```   361
```
```   362 lemma alw_impl_ev_alw:
```
```   363 assumes "alw (\<phi> impl ev \<psi>) xs"
```
```   364 shows "alw (ev \<phi> impl ev \<psi>) xs"
```
```   365 using assms by coinduct (auto elim: alw.cases dest: ev_alw_impl intro: ev.intros)
```
```   366
```
```   367 lemma ev_holds_sset:
```
```   368 "ev (holds P) xs \<longleftrightarrow> (\<exists> x \<in> sset xs. P x)" (is "?L \<longleftrightarrow> ?R")
```
```   369 proof safe
```
```   370   assume ?L thus ?R by induct (metis holds.simps stream.set_sel(1), metis stl_sset)
```
```   371 next
```
```   372   fix x assume "x \<in> sset xs" "P x"
```
```   373   thus ?L by (induct rule: sset_induct) (simp_all add: ev.base ev.step)
```
```   374 qed
```
```   375
```
```   376 (* LTL as a program logic: *)
```
```   377 lemma alw_invar:
```
```   378 assumes "\<phi> xs" and "alw (\<phi> impl nxt \<phi>) xs"
```
```   379 shows "alw \<phi> xs"
```
```   380 proof-
```
```   381   {assume "\<phi> xs \<and> alw (\<phi> impl nxt \<phi>) xs" hence ?thesis
```
```   382    apply coinduct by(auto elim: alw.cases)
```
```   383   }
```
```   384   thus ?thesis using assms by auto
```
```   385 qed
```
```   386
```
```   387 lemma variance:
```
```   388 assumes 1: "\<phi> xs" and 2: "alw (\<phi> impl (\<psi> or nxt \<phi>)) xs"
```
```   389 shows "(alw \<phi> or ev \<psi>) xs"
```
```   390 proof-
```
```   391   {assume "\<not> ev \<psi> xs" hence "alw (not \<psi>) xs" unfolding not_ev[symmetric] .
```
```   392    moreover have "alw (not \<psi> impl (\<phi> impl nxt \<phi>)) xs"
```
```   393    using 2 by coinduct (auto elim: alw.cases)
```
```   394    ultimately have "alw (\<phi> impl nxt \<phi>) xs" by(auto dest: alw_mp)
```
```   395    with 1 have "alw \<phi> xs" by(rule alw_invar)
```
```   396   }
```
```   397   thus ?thesis by blast
```
```   398 qed
```
```   399
```
```   400 lemma ev_alw_imp_nxt:
```
```   401 assumes e: "ev \<phi> xs" and a: "alw (\<phi> impl (nxt \<phi>)) xs"
```
```   402 shows "ev (alw \<phi>) xs"
```
```   403 proof-
```
```   404   obtain xl xs1 where xs: "xs = xl @- xs1" and \<phi>: "\<phi> xs1"
```
```   405   using e by (metis ev_imp_shift)
```
```   406   have "\<phi> xs1 \<and> alw (\<phi> impl (nxt \<phi>)) xs1" using a \<phi> unfolding xs by (metis alw_shift)
```
```   407   hence "alw \<phi> xs1" by(coinduct xs1 rule: alw.coinduct) (auto elim: alw.cases)
```
```   408   thus ?thesis unfolding xs by (auto intro: alw_ev_shift)
```
```   409 qed
```
```   410
```
```   411
```
```   412 inductive ev_at :: "('a stream \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> 'a stream \<Rightarrow> bool" for P :: "'a stream \<Rightarrow> bool" where
```
```   413   base: "P \<omega> \<Longrightarrow> ev_at P 0 \<omega>"
```
```   414 | step:"\<not> P \<omega> \<Longrightarrow> ev_at P n (stl \<omega>) \<Longrightarrow> ev_at P (Suc n) \<omega>"
```
```   415
```
```   416 inductive_simps ev_at_0[simp]: "ev_at P 0 \<omega>"
```
```   417 inductive_simps ev_at_Suc[simp]: "ev_at P (Suc n) \<omega>"
```
```   418
```
```   419 lemma ev_at_imp_snth: "ev_at P n \<omega> \<Longrightarrow> P (sdrop n \<omega>)"
```
```   420   by (induction n arbitrary: \<omega>) auto
```
```   421
```
```   422 lemma ev_at_HLD_imp_snth: "ev_at (HLD X) n \<omega> \<Longrightarrow> \<omega> !! n \<in> X"
```
```   423   by (auto dest!: ev_at_imp_snth simp: HLD_iff)
```
```   424
```
```   425 lemma ev_at_HLD_single_imp_snth: "ev_at (HLD {x}) n \<omega> \<Longrightarrow> \<omega> !! n = x"
```
```   426   by (drule ev_at_HLD_imp_snth) simp
```
```   427
```
```   428 lemma ev_at_unique: "ev_at P n \<omega> \<Longrightarrow> ev_at P m \<omega> \<Longrightarrow> n = m"
```
```   429 proof (induction arbitrary: m rule: ev_at.induct)
```
```   430   case (base \<omega>) then show ?case
```
```   431     by (simp add: ev_at.simps[of _ _ \<omega>])
```
```   432 next
```
```   433   case (step \<omega> n) from step.prems step.hyps step.IH[of "m - 1"] show ?case
```
```   434     by (auto simp add: ev_at.simps[of _ _ \<omega>])
```
```   435 qed
```
```   436
```
```   437 lemma ev_iff_ev_at: "ev P \<omega> \<longleftrightarrow> (\<exists>n. ev_at P n \<omega>)"
```
```   438 proof
```
```   439   assume "ev P \<omega>" then show "\<exists>n. ev_at P n \<omega>"
```
```   440     by (induction rule: ev_induct_strong) (auto intro: ev_at.intros)
```
```   441 next
```
```   442   assume "\<exists>n. ev_at P n \<omega>"
```
```   443   then obtain n where "ev_at P n \<omega>"
```
```   444     by auto
```
```   445   then show "ev P \<omega>"
```
```   446     by induction auto
```
```   447 qed
```
```   448
```
```   449 lemma ev_at_shift: "ev_at (HLD X) i (stake (Suc i) \<omega> @- \<omega>' :: 's stream) \<longleftrightarrow> ev_at (HLD X) i \<omega>"
```
```   450   by (induction i arbitrary: \<omega>) (auto simp: HLD_iff)
```
```   451
```
```   452 lemma ev_iff_ev_at_unqiue: "ev P \<omega> \<longleftrightarrow> (\<exists>!n. ev_at P n \<omega>)"
```
```   453   by (auto intro: ev_at_unique simp: ev_iff_ev_at)
```
```   454
```
```   455 lemma alw_HLD_iff_streams: "alw (HLD X) \<omega> \<longleftrightarrow> \<omega> \<in> streams X"
```
```   456 proof
```
```   457   assume "alw (HLD X) \<omega>" then show "\<omega> \<in> streams X"
```
```   458   proof (coinduction arbitrary: \<omega>)
```
```   459     case (streams \<omega>) then show ?case by (cases \<omega>) auto
```
```   460   qed
```
```   461 next
```
```   462   assume "\<omega> \<in> streams X" then show "alw (HLD X) \<omega>"
```
```   463   proof (coinduction arbitrary: \<omega>)
```
```   464     case (alw \<omega>) then show ?case by (cases \<omega>) auto
```
```   465   qed
```
```   466 qed
```
```   467
```
```   468 lemma not_HLD: "not (HLD X) = HLD (- X)"
```
```   469   by (auto simp: HLD_iff)
```
```   470
```
```   471 lemma not_alw_iff: "\<not> (alw P \<omega>) \<longleftrightarrow> ev (not P) \<omega>"
```
```   472   using not_alw[of P] by (simp add: fun_eq_iff)
```
```   473
```
```   474 lemma not_ev_iff: "\<not> (ev P \<omega>) \<longleftrightarrow> alw (not P) \<omega>"
```
```   475   using not_alw_iff[of "not P" \<omega>, symmetric]  by simp
```
```   476
```
```   477 lemma ev_Stream: "ev P (x ## s) \<longleftrightarrow> P (x ## s) \<or> ev P s"
```
```   478   by (auto elim: ev.cases)
```
```   479
```
```   480 lemma alw_ev_imp_ev_alw:
```
```   481   assumes "alw (ev P) \<omega>" shows "ev (P aand alw (ev P)) \<omega>"
```
```   482 proof -
```
```   483   have "ev P \<omega>" using assms by auto
```
```   484   from this assms show ?thesis
```
```   485     by induct auto
```
```   486 qed
```
```   487
```
```   488 lemma ev_False: "ev (\<lambda>x. False) \<omega> \<longleftrightarrow> False"
```
```   489 proof
```
```   490   assume "ev (\<lambda>x. False) \<omega>" then show False
```
```   491     by induct auto
```
```   492 qed auto
```
```   493
```
```   494 lemma alw_False: "alw (\<lambda>x. False) \<omega> \<longleftrightarrow> False"
```
```   495   by auto
```
```   496
```
```   497 lemma ev_iff_sdrop: "ev P \<omega> \<longleftrightarrow> (\<exists>m. P (sdrop m \<omega>))"
```
```   498 proof safe
```
```   499   assume "ev P \<omega>" then show "\<exists>m. P (sdrop m \<omega>)"
```
```   500     by (induct rule: ev_induct_strong) (auto intro: exI[of _ 0] exI[of _ "Suc n" for n])
```
```   501 next
```
```   502   fix m assume "P (sdrop m \<omega>)" then show "ev P \<omega>"
```
```   503     by (induct m arbitrary: \<omega>) auto
```
```   504 qed
```
```   505
```
```   506 lemma alw_iff_sdrop: "alw P \<omega> \<longleftrightarrow> (\<forall>m. P (sdrop m \<omega>))"
```
```   507 proof safe
```
```   508   fix m assume "alw P \<omega>" then show "P (sdrop m \<omega>)"
```
```   509     by (induct m arbitrary: \<omega>) auto
```
```   510 next
```
```   511   assume "\<forall>m. P (sdrop m \<omega>)" then show "alw P \<omega>"
```
```   512     by (coinduction arbitrary: \<omega>) (auto elim: allE[of _ 0] allE[of _ "Suc n" for n])
```
```   513 qed
```
```   514
```
```   515 lemma infinite_iff_alw_ev: "infinite {m. P (sdrop m \<omega>)} \<longleftrightarrow> alw (ev P) \<omega>"
```
```   516   unfolding infinite_nat_iff_unbounded_le alw_iff_sdrop ev_iff_sdrop
```
```   517   by simp (metis le_Suc_ex le_add1)
```
```   518
```
```   519 lemma alw_inv:
```
```   520   assumes stl: "\<And>s. f (stl s) = stl (f s)"
```
```   521   shows "alw P (f s) \<longleftrightarrow> alw (\<lambda>x. P (f x)) s"
```
```   522 proof
```
```   523   assume "alw P (f s)" then show "alw (\<lambda>x. P (f x)) s"
```
```   524     by (coinduction arbitrary: s rule: alw_coinduct)
```
```   525        (auto simp: stl)
```
```   526 next
```
```   527   assume "alw (\<lambda>x. P (f x)) s" then show "alw P (f s)"
```
```   528     by (coinduction arbitrary: s rule: alw_coinduct) (auto simp: stl[symmetric])
```
```   529 qed
```
```   530
```
```   531 lemma ev_inv:
```
```   532   assumes stl: "\<And>s. f (stl s) = stl (f s)"
```
```   533   shows "ev P (f s) \<longleftrightarrow> ev (\<lambda>x. P (f x)) s"
```
```   534 proof
```
```   535   assume "ev P (f s)" then show "ev (\<lambda>x. P (f x)) s"
```
```   536     by (induction "f s" arbitrary: s) (auto simp: stl)
```
```   537 next
```
```   538   assume "ev (\<lambda>x. P (f x)) s" then show "ev P (f s)"
```
```   539     by induction (auto simp: stl[symmetric])
```
```   540 qed
```
```   541
```
```   542 lemma alw_smap: "alw P (smap f s) \<longleftrightarrow> alw (\<lambda>x. P (smap f x)) s"
```
```   543   by (rule alw_inv) simp
```
```   544
```
```   545 lemma ev_smap: "ev P (smap f s) \<longleftrightarrow> ev (\<lambda>x. P (smap f x)) s"
```
```   546   by (rule ev_inv) simp
```
```   547
```
```   548 lemma alw_cong:
```
```   549   assumes P: "alw P \<omega>" and eq: "\<And>\<omega>. P \<omega> \<Longrightarrow> Q1 \<omega> \<longleftrightarrow> Q2 \<omega>"
```
```   550   shows "alw Q1 \<omega> \<longleftrightarrow> alw Q2 \<omega>"
```
```   551 proof -
```
```   552   from eq have "(alw P aand Q1) = (alw P aand Q2)" by auto
```
```   553   then have "alw (alw P aand Q1) \<omega> = alw (alw P aand Q2) \<omega>" by auto
```
```   554   with P show "alw Q1 \<omega> \<longleftrightarrow> alw Q2 \<omega>"
```
```   555     by (simp add: alw_aand)
```
```   556 qed
```
```   557
```
```   558 lemma ev_cong:
```
```   559   assumes P: "alw P \<omega>" and eq: "\<And>\<omega>. P \<omega> \<Longrightarrow> Q1 \<omega> \<longleftrightarrow> Q2 \<omega>"
```
```   560   shows "ev Q1 \<omega> \<longleftrightarrow> ev Q2 \<omega>"
```
```   561 proof -
```
```   562   from P have "alw (\<lambda>xs. Q1 xs \<longrightarrow> Q2 xs) \<omega>" by (rule alw_mono) (simp add: eq)
```
```   563   moreover from P have "alw (\<lambda>xs. Q2 xs \<longrightarrow> Q1 xs) \<omega>" by (rule alw_mono) (simp add: eq)
```
```   564   moreover note ev_alw_impl[of Q1 \<omega> Q2] ev_alw_impl[of Q2 \<omega> Q1]
```
```   565   ultimately show "ev Q1 \<omega> \<longleftrightarrow> ev Q2 \<omega>"
```
```   566     by auto
```
```   567 qed
```
```   568
```
```   569 lemma alwD: "alw P x \<Longrightarrow> P x"
```
```   570   by auto
```
```   571
```
```   572 lemma alw_alwD: "alw P \<omega> \<Longrightarrow> alw (alw P) \<omega>"
```
```   573   by simp
```
```   574
```
```   575 lemma alw_ev_stl: "alw (ev P) (stl \<omega>) \<longleftrightarrow> alw (ev P) \<omega>"
```
```   576   by (auto intro: alw.intros)
```
```   577
```
```   578 lemma holds_Stream: "holds P (x ## s) \<longleftrightarrow> P x"
```
```   579   by simp
```
```   580
```
```   581 lemma holds_eq1[simp]: "holds (op = x) = HLD {x}"
```
```   582   by rule (auto simp: HLD_iff)
```
```   583
```
```   584 lemma holds_eq2[simp]: "holds (\<lambda>y. y = x) = HLD {x}"
```
```   585   by rule (auto simp: HLD_iff)
```
```   586
```
```   587 lemma not_holds_eq[simp]: "holds (- op = x) = not (HLD {x})"
```
```   588   by rule (auto simp: HLD_iff)
```
```   589
```
```   590 text \<open>Strong until\<close>
```
```   591
```
```   592 inductive suntil (infix "suntil" 60) for \<phi> \<psi> where
```
```   593   base: "\<psi> \<omega> \<Longrightarrow> (\<phi> suntil \<psi>) \<omega>"
```
```   594 | step: "\<phi> \<omega> \<Longrightarrow> (\<phi> suntil \<psi>) (stl \<omega>) \<Longrightarrow> (\<phi> suntil \<psi>) \<omega>"
```
```   595
```
```   596 inductive_simps suntil_Stream: "(\<phi> suntil \<psi>) (x ## s)"
```
```   597
```
```   598 lemma suntil_induct_strong[consumes 1, case_names base step]:
```
```   599   "(\<phi> suntil \<psi>) x \<Longrightarrow>
```
```   600     (\<And>\<omega>. \<psi> \<omega> \<Longrightarrow> P \<omega>) \<Longrightarrow>
```
```   601     (\<And>\<omega>. \<phi> \<omega> \<Longrightarrow> \<not> \<psi> \<omega> \<Longrightarrow> (\<phi> suntil \<psi>) (stl \<omega>) \<Longrightarrow> P (stl \<omega>) \<Longrightarrow> P \<omega>) \<Longrightarrow> P x"
```
```   602   using suntil.induct[of \<phi> \<psi> x P] by blast
```
```   603
```
```   604 lemma ev_suntil: "(\<phi> suntil \<psi>) \<omega> \<Longrightarrow> ev \<psi> \<omega>"
```
```   605   by (induct rule: suntil.induct) (auto intro: ev.intros)
```
```   606
```
```   607 lemma suntil_inv:
```
```   608   assumes stl: "\<And>s. f (stl s) = stl (f s)"
```
```   609   shows "(P suntil Q) (f s) \<longleftrightarrow> ((\<lambda>x. P (f x)) suntil (\<lambda>x. Q (f x))) s"
```
```   610 proof
```
```   611   assume "(P suntil Q) (f s)" then show "((\<lambda>x. P (f x)) suntil (\<lambda>x. Q (f x))) s"
```
```   612     by (induction "f s" arbitrary: s) (auto simp: stl intro: suntil.intros)
```
```   613 next
```
```   614   assume "((\<lambda>x. P (f x)) suntil (\<lambda>x. Q (f x))) s" then show "(P suntil Q) (f s)"
```
```   615     by induction (auto simp: stl[symmetric] intro: suntil.intros)
```
```   616 qed
```
```   617
```
```   618 lemma suntil_smap: "(P suntil Q) (smap f s) \<longleftrightarrow> ((\<lambda>x. P (smap f x)) suntil (\<lambda>x. Q (smap f x))) s"
```
```   619   by (rule suntil_inv) simp
```
```   620
```
```   621 lemma hld_smap: "HLD x (smap f s) = holds (\<lambda>y. f y \<in> x) s"
```
```   622   by (simp add: HLD_def)
```
```   623
```
```   624 lemma suntil_mono:
```
```   625   assumes eq: "\<And>\<omega>. P \<omega> \<Longrightarrow> Q1 \<omega> \<Longrightarrow> Q2 \<omega>" "\<And>\<omega>. P \<omega> \<Longrightarrow> R1 \<omega> \<Longrightarrow> R2 \<omega>"
```
```   626   assumes *: "(Q1 suntil R1) \<omega>" "alw P \<omega>" shows "(Q2 suntil R2) \<omega>"
```
```   627   using * by induct (auto intro: eq suntil.intros)
```
```   628
```
```   629 lemma suntil_cong:
```
```   630   "alw P \<omega> \<Longrightarrow> (\<And>\<omega>. P \<omega> \<Longrightarrow> Q1 \<omega> \<longleftrightarrow> Q2 \<omega>) \<Longrightarrow> (\<And>\<omega>. P \<omega> \<Longrightarrow> R1 \<omega> \<longleftrightarrow> R2 \<omega>) \<Longrightarrow>
```
```   631     (Q1 suntil R1) \<omega> \<longleftrightarrow> (Q2 suntil R2) \<omega>"
```
```   632   using suntil_mono[of P Q1 Q2 R1 R2 \<omega>] suntil_mono[of P Q2 Q1 R2 R1 \<omega>] by auto
```
```   633
```
```   634 lemma ev_suntil_iff: "ev (P suntil Q) \<omega> \<longleftrightarrow> ev Q \<omega>"
```
```   635 proof
```
```   636   assume "ev (P suntil Q) \<omega>" then show "ev Q \<omega>"
```
```   637    by induct (auto dest: ev_suntil)
```
```   638 next
```
```   639   assume "ev Q \<omega>" then show "ev (P suntil Q) \<omega>"
```
```   640     by induct (auto intro: suntil.intros)
```
```   641 qed
```
```   642
```
```   643 lemma true_suntil: "((\<lambda>_. True) suntil P) = ev P"
```
```   644   by (simp add: suntil_def ev_def)
```
```   645
```
```   646 lemma suntil_lfp: "(\<phi> suntil \<psi>) = lfp (\<lambda>P s. \<psi> s \<or> (\<phi> s \<and> P (stl s)))"
```
```   647   by (simp add: suntil_def)
```
```   648
```
```   649 lemma sfilter_P[simp]: "P (shd s) \<Longrightarrow> sfilter P s = shd s ## sfilter P (stl s)"
```
```   650   using sfilter_Stream[of P "shd s" "stl s"] by simp
```
```   651
```
```   652 lemma sfilter_not_P[simp]: "\<not> P (shd s) \<Longrightarrow> sfilter P s = sfilter P (stl s)"
```
```   653   using sfilter_Stream[of P "shd s" "stl s"] by simp
```
```   654
```
```   655 lemma sfilter_eq:
```
```   656   assumes "ev (holds P) s"
```
```   657   shows "sfilter P s = x ## s' \<longleftrightarrow>
```
```   658     P x \<and> (not (holds P) suntil (HLD {x} aand nxt (\<lambda>s. sfilter P s = s'))) s"
```
```   659   using assms
```
```   660   by (induct rule: ev_induct_strong)
```
```   661      (auto simp add: HLD_iff intro: suntil.intros elim: suntil.cases)
```
```   662
```
```   663 lemma sfilter_streams:
```
```   664   "alw (ev (holds P)) \<omega> \<Longrightarrow> \<omega> \<in> streams A \<Longrightarrow> sfilter P \<omega> \<in> streams {x\<in>A. P x}"
```
```   665 proof (coinduction arbitrary: \<omega>)
```
```   666   case (streams \<omega>)
```
```   667   then have "ev (holds P) \<omega>" by blast
```
```   668   from this streams show ?case
```
```   669     by (induct rule: ev_induct_strong) (auto elim: streamsE)
```
```   670 qed
```
```   671
```
```   672 lemma alw_sfilter:
```
```   673   assumes *: "alw (ev (holds P)) s"
```
```   674   shows "alw Q (sfilter P s) \<longleftrightarrow> alw (\<lambda>x. Q (sfilter P x)) s"
```
```   675 proof
```
```   676   assume "alw Q (sfilter P s)" with * show "alw (\<lambda>x. Q (sfilter P x)) s"
```
```   677   proof (coinduction arbitrary: s rule: alw_coinduct)
```
```   678     case (stl s)
```
```   679     then have "ev (holds P) s"
```
```   680       by blast
```
```   681     from this stl show ?case
```
```   682       by (induct rule: ev_induct_strong) auto
```
```   683   qed auto
```
```   684 next
```
```   685   assume "alw (\<lambda>x. Q (sfilter P x)) s" with * show "alw Q (sfilter P s)"
```
```   686   proof (coinduction arbitrary: s rule: alw_coinduct)
```
```   687     case (stl s)
```
```   688     then have "ev (holds P) s"
```
```   689       by blast
```
```   690     from this stl show ?case
```
```   691       by (induct rule: ev_induct_strong) auto
```
```   692   qed auto
```
```   693 qed
```
```   694
```
```   695 lemma ev_sfilter:
```
```   696   assumes *: "alw (ev (holds P)) s"
```
```   697   shows "ev Q (sfilter P s) \<longleftrightarrow> ev (\<lambda>x. Q (sfilter P x)) s"
```
```   698 proof
```
```   699   assume "ev Q (sfilter P s)" from this * show "ev (\<lambda>x. Q (sfilter P x)) s"
```
```   700   proof (induction "sfilter P s" arbitrary: s rule: ev_induct_strong)
```
```   701     case (step s)
```
```   702     then have "ev (holds P) s"
```
```   703       by blast
```
```   704     from this step show ?case
```
```   705       by (induct rule: ev_induct_strong) auto
```
```   706   qed auto
```
```   707 next
```
```   708   assume "ev (\<lambda>x. Q (sfilter P x)) s" then show "ev Q (sfilter P s)"
```
```   709   proof (induction rule: ev_induct_strong)
```
```   710     case (step s) then show ?case
```
```   711       by (cases "P (shd s)") auto
```
```   712   qed auto
```
```   713 qed
```
```   714
```
```   715 lemma holds_sfilter:
```
```   716   assumes "ev (holds Q) s" shows "holds P (sfilter Q s) \<longleftrightarrow> (not (holds Q) suntil (holds (Q aand P))) s"
```
```   717 proof
```
```   718   assume "holds P (sfilter Q s)" with assms show "(not (holds Q) suntil (holds (Q aand P))) s"
```
```   719     by (induct rule: ev_induct_strong) (auto intro: suntil.intros)
```
```   720 next
```
```   721   assume "(not (holds Q) suntil (holds (Q aand P))) s" then show "holds P (sfilter Q s)"
```
```   722     by induct auto
```
```   723 qed
```
```   724
```
```   725 lemma suntil_aand_nxt:
```
```   726   "(\<phi> suntil (\<phi> aand nxt \<psi>)) \<omega> \<longleftrightarrow> (\<phi> aand nxt (\<phi> suntil \<psi>)) \<omega>"
```
```   727 proof
```
```   728   assume "(\<phi> suntil (\<phi> aand nxt \<psi>)) \<omega>" then show "(\<phi> aand nxt (\<phi> suntil \<psi>)) \<omega>"
```
```   729     by induction (auto intro: suntil.intros)
```
```   730 next
```
```   731   assume "(\<phi> aand nxt (\<phi> suntil \<psi>)) \<omega>"
```
```   732   then have "(\<phi> suntil \<psi>) (stl \<omega>)" "\<phi> \<omega>"
```
```   733     by auto
```
```   734   then show "(\<phi> suntil (\<phi> aand nxt \<psi>)) \<omega>"
```
```   735     by (induction "stl \<omega>" arbitrary: \<omega>)
```
```   736        (auto elim: suntil.cases intro: suntil.intros)
```
```   737 qed
```
```   738
```
```   739 lemma alw_sconst: "alw P (sconst x) \<longleftrightarrow> P (sconst x)"
```
```   740 proof
```
```   741   assume "P (sconst x)" then show "alw P (sconst x)"
```
```   742     by coinduction auto
```
```   743 qed auto
```
```   744
```
```   745 lemma ev_sconst: "ev P (sconst x) \<longleftrightarrow> P (sconst x)"
```
```   746 proof
```
```   747   assume "ev P (sconst x)" then show "P (sconst x)"
```
```   748     by (induction "sconst x") auto
```
```   749 qed auto
```
```   750
```
```   751 lemma suntil_sconst: "(\<phi> suntil \<psi>) (sconst x) \<longleftrightarrow> \<psi> (sconst x)"
```
```   752 proof
```
```   753   assume "(\<phi> suntil \<psi>) (sconst x)" then show "\<psi> (sconst x)"
```
```   754     by (induction "sconst x") auto
```
```   755 qed (auto intro: suntil.intros)
```
```   756
```
```   757 lemma hld_smap': "HLD x (smap f s) = HLD (f -` x) s"
```
```   758   by (simp add: HLD_def)
```
```   759
```
`   760 end`