--- a/src/HOL/Library/Library.thy Wed Oct 08 09:09:12 2014 +0200
+++ b/src/HOL/Library/Library.thy Wed Oct 08 10:22:00 2014 +0200
@@ -36,6 +36,7 @@
Lattice_Algebras
Lattice_Syntax
Lattice_Constructions
+ Linear_Temporal_Logic_on_Streams
ListVector
Lubs_Glbs
Mapping
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Linear_Temporal_Logic_on_Streams.thy Wed Oct 08 10:22:00 2014 +0200
@@ -0,0 +1,391 @@
+(* Title: HOL/Library/Linear_Temporal_Logic_on_Streams.thy
+ Author: Andrei Popescu, TU Muenchen
+ Author: Dmitriy Traytel, TU Muenchen
+*)
+
+header {* Linear Temporal Logic on Streams *}
+
+theory Linear_Temporal_Logic_on_Streams
+ imports Stream Sublist
+begin
+
+section{* Preliminaries *}
+
+lemma shift_prefix:
+assumes "xl @- xs = yl @- ys" and "length xl \<le> length yl"
+shows "prefixeq xl yl"
+using assms proof(induct xl arbitrary: yl xs ys)
+ case (Cons x xl yl xs ys)
+ thus ?case by (cases yl) auto
+qed auto
+
+lemma shift_prefix_cases:
+assumes "xl @- xs = yl @- ys"
+shows "prefixeq xl yl \<or> prefixeq yl xl"
+using shift_prefix[OF assms] apply(cases "length xl \<le> length yl")
+by (metis, metis assms nat_le_linear shift_prefix)
+
+
+section{* Linear temporal logic *}
+
+(* Propositional connectives: *)
+abbreviation (input) IMPL (infix "impl" 60)
+where "\<phi> impl \<psi> \<equiv> \<lambda> xs. \<phi> xs \<longrightarrow> \<psi> xs"
+
+abbreviation (input) OR (infix "or" 60)
+where "\<phi> or \<psi> \<equiv> \<lambda> xs. \<phi> xs \<or> \<psi> xs"
+
+abbreviation (input) AND (infix "aand" 60)
+where "\<phi> aand \<psi> \<equiv> \<lambda> xs. \<phi> xs \<and> \<psi> xs"
+
+abbreviation (input) "not \<phi> \<equiv> \<lambda> xs. \<not> \<phi> xs"
+
+abbreviation (input) "true \<equiv> \<lambda> xs. True"
+
+abbreviation (input) "false \<equiv> \<lambda> xs. False"
+
+lemma impl_not_or: "\<phi> impl \<psi> = (not \<phi>) or \<psi>"
+by blast
+
+lemma not_or: "not (\<phi> or \<psi>) = (not \<phi>) aand (not \<psi>)"
+by blast
+
+lemma not_aand: "not (\<phi> aand \<psi>) = (not \<phi>) or (not \<psi>)"
+by blast
+
+lemma non_not[simp]: "not (not \<phi>) = \<phi>" by simp
+
+(* Temporal (LTL) connectives: *)
+fun holds where "holds P xs \<longleftrightarrow> P (shd xs)"
+fun nxt where "nxt \<phi> xs = \<phi> (stl xs)"
+
+inductive ev for \<phi> where
+base: "\<phi> xs \<Longrightarrow> ev \<phi> xs"
+|
+step: "ev \<phi> (stl xs) \<Longrightarrow> ev \<phi> xs"
+
+coinductive alw for \<phi> where
+alw: "\<lbrakk>\<phi> xs; alw \<phi> (stl xs)\<rbrakk> \<Longrightarrow> alw \<phi> xs"
+
+(* weak until: *)
+coinductive UNTIL (infix "until" 60) for \<phi> \<psi> where
+base: "\<psi> xs \<Longrightarrow> (\<phi> until \<psi>) xs"
+|
+step: "\<lbrakk>\<phi> xs; (\<phi> until \<psi>) (stl xs)\<rbrakk> \<Longrightarrow> (\<phi> until \<psi>) xs"
+
+lemma holds_mono:
+assumes holds: "holds P xs" and 0: "\<And> x. P x \<Longrightarrow> Q x"
+shows "holds Q xs"
+using assms by auto
+
+lemma holds_aand:
+"(holds P aand holds Q) steps \<longleftrightarrow> holds (\<lambda> step. P step \<and> Q step) steps" by auto
+
+lemma nxt_mono:
+assumes nxt: "nxt \<phi> xs" and 0: "\<And> xs. \<phi> xs \<Longrightarrow> \<psi> xs"
+shows "nxt \<psi> xs"
+using assms by auto
+
+lemma ev_mono:
+assumes ev: "ev \<phi> xs" and 0: "\<And> xs. \<phi> xs \<Longrightarrow> \<psi> xs"
+shows "ev \<psi> xs"
+using ev by induct (auto intro: ev.intros simp: 0)
+
+lemma alw_mono:
+assumes alw: "alw \<phi> xs" and 0: "\<And> xs. \<phi> xs \<Longrightarrow> \<psi> xs"
+shows "alw \<psi> xs"
+using alw by coinduct (auto elim: alw.cases simp: 0)
+
+lemma until_monoL:
+assumes until: "(\<phi>1 until \<psi>) xs" and 0: "\<And> xs. \<phi>1 xs \<Longrightarrow> \<phi>2 xs"
+shows "(\<phi>2 until \<psi>) xs"
+using until by coinduct (auto elim: UNTIL.cases simp: 0)
+
+lemma until_monoR:
+assumes until: "(\<phi> until \<psi>1) xs" and 0: "\<And> xs. \<psi>1 xs \<Longrightarrow> \<psi>2 xs"
+shows "(\<phi> until \<psi>2) xs"
+using until by coinduct (auto elim: UNTIL.cases simp: 0)
+
+lemma until_mono:
+assumes until: "(\<phi>1 until \<psi>1) xs" and
+0: "\<And> xs. \<phi>1 xs \<Longrightarrow> \<phi>2 xs" "\<And> xs. \<psi>1 xs \<Longrightarrow> \<psi>2 xs"
+shows "(\<phi>2 until \<psi>2) xs"
+using until by coinduct (auto elim: UNTIL.cases simp: 0)
+
+lemma until_false: "\<phi> until false = alw \<phi>"
+proof-
+ {fix xs assume "(\<phi> until false) xs" hence "alw \<phi> xs"
+ apply coinduct by (auto elim: UNTIL.cases)
+ }
+ moreover
+ {fix xs assume "alw \<phi> xs" hence "(\<phi> until false) xs"
+ apply coinduct by (auto elim: alw.cases)
+ }
+ ultimately show ?thesis by blast
+qed
+
+lemma ev_nxt: "ev \<phi> = (\<phi> or nxt (ev \<phi>))"
+apply(rule ext) by (metis ev.simps nxt.simps)
+
+lemma alw_nxt: "alw \<phi> = (\<phi> aand nxt (alw \<phi>))"
+apply(rule ext) by (metis alw.simps nxt.simps)
+
+lemma ev_ev[simp]: "ev (ev \<phi>) = ev \<phi>"
+proof-
+ {fix xs
+ assume "ev (ev \<phi>) xs" hence "ev \<phi> xs"
+ by induct (auto intro: ev.intros)
+ }
+ thus ?thesis by (auto intro: ev.intros)
+qed
+
+lemma alw_alw[simp]: "alw (alw \<phi>) = alw \<phi>"
+proof-
+ {fix xs
+ assume "alw \<phi> xs" hence "alw (alw \<phi>) xs"
+ by coinduct (auto elim: alw.cases)
+ }
+ thus ?thesis by (auto elim: alw.cases)
+qed
+
+lemma ev_shift:
+assumes "ev \<phi> xs"
+shows "ev \<phi> (xl @- xs)"
+using assms by (induct xl) (auto intro: ev.intros)
+
+lemma ev_imp_shift:
+assumes "ev \<phi> xs" shows "\<exists> xl xs2. xs = xl @- xs2 \<and> \<phi> xs2"
+using assms by induct (metis shift.simps(1), metis shift.simps(2) stream.collapse)+
+
+lemma alw_ev_shift: "alw \<phi> xs1 \<Longrightarrow> ev (alw \<phi>) (xl @- xs1)"
+by (auto intro: ev_shift ev.intros)
+
+lemma alw_shift:
+assumes "alw \<phi> (xl @- xs)"
+shows "alw \<phi> xs"
+using assms by (induct xl) (auto elim: alw.cases)
+
+lemma ev_ex_nxt:
+assumes "ev \<phi> xs"
+shows "\<exists> n. (nxt ^^ n) \<phi> xs"
+using assms proof induct
+ case (base xs) thus ?case by (intro exI[of _ 0]) auto
+next
+ case (step xs)
+ then obtain n where "(nxt ^^ n) \<phi> (stl xs)" by blast
+ thus ?case by (intro exI[of _ "Suc n"]) (metis funpow.simps(2) nxt.simps o_def)
+qed
+
+lemma alw_sdrop:
+assumes "alw \<phi> xs" shows "alw \<phi> (sdrop n xs)"
+by (metis alw_shift assms stake_sdrop)
+
+lemma nxt_sdrop: "(nxt ^^ n) \<phi> xs \<longleftrightarrow> \<phi> (sdrop n xs)"
+by (induct n arbitrary: xs) auto
+
+definition "wait \<phi> xs \<equiv> LEAST n. (nxt ^^ n) \<phi> xs"
+
+lemma nxt_wait:
+assumes "ev \<phi> xs" shows "(nxt ^^ (wait \<phi> xs)) \<phi> xs"
+unfolding wait_def using ev_ex_nxt[OF assms] by(rule LeastI_ex)
+
+lemma nxt_wait_least:
+assumes ev: "ev \<phi> xs" and nxt: "(nxt ^^ n) \<phi> xs" shows "wait \<phi> xs \<le> n"
+unfolding wait_def using ev_ex_nxt[OF ev] by (metis Least_le nxt)
+
+lemma sdrop_wait:
+assumes "ev \<phi> xs" shows "\<phi> (sdrop (wait \<phi> xs) xs)"
+using nxt_wait[OF assms] unfolding nxt_sdrop .
+
+lemma sdrop_wait_least:
+assumes ev: "ev \<phi> xs" and nxt: "\<phi> (sdrop n xs)" shows "wait \<phi> xs \<le> n"
+using assms nxt_wait_least unfolding nxt_sdrop by auto
+
+lemma nxt_ev: "(nxt ^^ n) \<phi> xs \<Longrightarrow> ev \<phi> xs"
+by (induct n arbitrary: xs) (auto intro: ev.intros)
+
+lemma not_ev: "not (ev \<phi>) = alw (not \<phi>)"
+proof(rule ext, safe)
+ fix xs assume "not (ev \<phi>) xs" thus "alw (not \<phi>) xs"
+ by (coinduct) (auto intro: ev.intros)
+next
+ fix xs assume "ev \<phi> xs" and "alw (not \<phi>) xs" thus False
+ by (induct) (auto elim: alw.cases)
+qed
+
+lemma not_alw: "not (alw \<phi>) = ev (not \<phi>)"
+proof-
+ have "not (alw \<phi>) = not (alw (not (not \<phi>)))" by simp
+ also have "... = ev (not \<phi>)" unfolding not_ev[symmetric] by simp
+ finally show ?thesis .
+qed
+
+lemma not_ev_not[simp]: "not (ev (not \<phi>)) = alw \<phi>"
+unfolding not_ev by simp
+
+lemma not_alw_not[simp]: "not (alw (not \<phi>)) = ev \<phi>"
+unfolding not_alw by simp
+
+lemma alw_ev_sdrop:
+assumes "alw (ev \<phi>) (sdrop m xs)"
+shows "alw (ev \<phi>) xs"
+using assms
+by coinduct (metis alw_nxt ev_shift funpow_swap1 nxt.simps nxt_sdrop stake_sdrop)
+
+lemma ev_alw_imp_alw_ev:
+assumes "ev (alw \<phi>) xs" shows "alw (ev \<phi>) xs"
+using assms apply induct
+ apply (metis (full_types) alw_mono ev.base)
+ by (metis alw alw_nxt ev.step)
+
+lemma alw_aand: "alw (\<phi> aand \<psi>) = alw \<phi> aand alw \<psi>"
+proof-
+ {fix xs assume "alw (\<phi> aand \<psi>) xs" hence "(alw \<phi> aand alw \<psi>) xs"
+ by (auto elim: alw_mono)
+ }
+ moreover
+ {fix xs assume "(alw \<phi> aand alw \<psi>) xs" hence "alw (\<phi> aand \<psi>) xs"
+ by coinduct (auto elim: alw.cases)
+ }
+ ultimately show ?thesis by blast
+qed
+
+lemma ev_or: "ev (\<phi> or \<psi>) = ev \<phi> or ev \<psi>"
+proof-
+ {fix xs assume "(ev \<phi> or ev \<psi>) xs" hence "ev (\<phi> or \<psi>) xs"
+ by (auto elim: ev_mono)
+ }
+ moreover
+ {fix xs assume "ev (\<phi> or \<psi>) xs" hence "(ev \<phi> or ev \<psi>) xs"
+ by induct (auto intro: ev.intros)
+ }
+ ultimately show ?thesis by blast
+qed
+
+lemma ev_alw_aand:
+assumes \<phi>: "ev (alw \<phi>) xs" and \<psi>: "ev (alw \<psi>) xs"
+shows "ev (alw (\<phi> aand \<psi>)) xs"
+proof-
+ obtain xl xs1 where xs1: "xs = xl @- xs1" and \<phi>\<phi>: "alw \<phi> xs1"
+ using \<phi> by (metis ev_imp_shift)
+ moreover obtain yl ys1 where xs2: "xs = yl @- ys1" and \<psi>\<psi>: "alw \<psi> ys1"
+ using \<psi> by (metis ev_imp_shift)
+ ultimately have 0: "xl @- xs1 = yl @- ys1" by auto
+ hence "prefixeq xl yl \<or> prefixeq yl xl" using shift_prefix_cases by auto
+ thus ?thesis proof
+ assume "prefixeq xl yl"
+ then obtain yl1 where yl: "yl = xl @ yl1" by (elim prefixeqE)
+ have xs1': "xs1 = yl1 @- ys1" using 0 unfolding yl by simp
+ have "alw \<phi> ys1" using \<phi>\<phi> unfolding xs1' by (metis alw_shift)
+ hence "alw (\<phi> aand \<psi>) ys1" using \<psi>\<psi> unfolding alw_aand by auto
+ thus ?thesis unfolding xs2 by (auto intro: alw_ev_shift)
+ next
+ assume "prefixeq yl xl"
+ then obtain xl1 where xl: "xl = yl @ xl1" by (elim prefixeqE)
+ have ys1': "ys1 = xl1 @- xs1" using 0 unfolding xl by simp
+ have "alw \<psi> xs1" using \<psi>\<psi> unfolding ys1' by (metis alw_shift)
+ hence "alw (\<phi> aand \<psi>) xs1" using \<phi>\<phi> unfolding alw_aand by auto
+ thus ?thesis unfolding xs1 by (auto intro: alw_ev_shift)
+ qed
+qed
+
+lemma ev_alw_alw_impl:
+assumes "ev (alw \<phi>) xs" and "alw (alw \<phi> impl ev \<psi>) xs"
+shows "ev \<psi> xs"
+using assms by induct (auto intro: ev.intros elim: alw.cases)
+
+lemma ev_alw_stl[simp]: "ev (alw \<phi>) (stl x) \<longleftrightarrow> ev (alw \<phi>) x"
+by (metis (full_types) alw_nxt ev_nxt nxt.simps)
+
+lemma alw_alw_impl_ev:
+"alw (alw \<phi> impl ev \<psi>) = (ev (alw \<phi>) impl alw (ev \<psi>))" (is "?A = ?B")
+proof-
+ {fix xs assume "?A xs \<and> ev (alw \<phi>) xs" hence "alw (ev \<psi>) xs"
+ apply coinduct using ev_nxt by (auto elim: ev_alw_alw_impl alw.cases intro: ev.intros)
+ }
+ moreover
+ {fix xs assume "?B xs" hence "?A xs"
+ apply coinduct by (auto elim: alw.cases intro: ev.intros)
+ }
+ ultimately show ?thesis by blast
+qed
+
+lemma ev_alw_impl:
+assumes "ev \<phi> xs" and "alw (\<phi> impl \<psi>) xs" shows "ev \<psi> xs"
+using assms by induct (auto intro: ev.intros elim: alw.cases)
+
+lemma ev_alw_impl_ev:
+assumes "ev \<phi> xs" and "alw (\<phi> impl ev \<psi>) xs" shows "ev \<psi> xs"
+using ev_alw_impl[OF assms] by simp
+
+lemma alw_mp:
+assumes "alw \<phi> xs" and "alw (\<phi> impl \<psi>) xs"
+shows "alw \<psi> xs"
+proof-
+ {assume "alw \<phi> xs \<and> alw (\<phi> impl \<psi>) xs" hence ?thesis
+ apply coinduct by (auto elim: alw.cases)
+ }
+ thus ?thesis using assms by auto
+qed
+
+lemma all_imp_alw:
+assumes "\<And> xs. \<phi> xs" shows "alw \<phi> xs"
+proof-
+ {assume "\<forall> xs. \<phi> xs"
+ hence ?thesis by coinduct auto
+ }
+ thus ?thesis using assms by auto
+qed
+
+lemma alw_impl_ev_alw:
+assumes "alw (\<phi> impl ev \<psi>) xs"
+shows "alw (ev \<phi> impl ev \<psi>) xs"
+using assms by coinduct (auto elim: alw.cases dest: ev_alw_impl intro: ev.intros)
+
+lemma ev_holds_sset:
+"ev (holds P) xs \<longleftrightarrow> (\<exists> x \<in> sset xs. P x)" (is "?L \<longleftrightarrow> ?R")
+proof safe
+ assume ?L thus ?R by induct (metis holds.simps stream.set_sel(1), metis stl_sset)
+next
+ fix x assume "x \<in> sset xs" "P x"
+ thus ?L by (induct rule: sset_induct) (simp_all add: ev.base ev.step)
+qed
+
+(* LTL as a program logic: *)
+lemma alw_invar:
+assumes "\<phi> xs" and "alw (\<phi> impl nxt \<phi>) xs"
+shows "alw \<phi> xs"
+proof-
+ {assume "\<phi> xs \<and> alw (\<phi> impl nxt \<phi>) xs" hence ?thesis
+ apply coinduct by(auto elim: alw.cases)
+ }
+ thus ?thesis using assms by auto
+qed
+
+lemma variance:
+assumes 1: "\<phi> xs" and 2: "alw (\<phi> impl (\<psi> or nxt \<phi>)) xs"
+shows "(alw \<phi> or ev \<psi>) xs"
+proof-
+ {assume "\<not> ev \<psi> xs" hence "alw (not \<psi>) xs" unfolding not_ev[symmetric] .
+ moreover have "alw (not \<psi> impl (\<phi> impl nxt \<phi>)) xs"
+ using 2 by coinduct (auto elim: alw.cases)
+ ultimately have "alw (\<phi> impl nxt \<phi>) xs" by(auto dest: alw_mp)
+ with 1 have "alw \<phi> xs" by(rule alw_invar)
+ }
+ thus ?thesis by blast
+qed
+
+lemma ev_alw_imp_nxt:
+assumes e: "ev \<phi> xs" and a: "alw (\<phi> impl (nxt \<phi>)) xs"
+shows "ev (alw \<phi>) xs"
+proof-
+ obtain xl xs1 where xs: "xs = xl @- xs1" and \<phi>: "\<phi> xs1"
+ using e by (metis ev_imp_shift)
+ have "\<phi> xs1 \<and> alw (\<phi> impl (nxt \<phi>)) xs1" using a \<phi> unfolding xs by (metis alw_shift)
+ hence "alw \<phi> xs1" by(coinduct xs1 rule: alw.coinduct) (auto elim: alw.cases)
+ thus ?thesis unfolding xs by (auto intro: alw_ev_shift)
+qed
+
+
+
+end
\ No newline at end of file