| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/Isabelle/Archive-of-Formal-Proofs/thys/Timed_Automata/ (Sammlung formaler Beweise Version 2026-5©) Datei vom 29.4.2026 mit Größe 155 kB |
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Quellcode-Bibliothek Regions.thySprache: Isabelle |
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chapter ‹The Classic Construction for Decidability› theory Regions imports Timed_Automata TA_Misc begin text ‹ The following is a formalization of regions in the correct version of Patricia B section ‹Definition of Regions› type_synonym 'c ceiling = "('c ==> nat)" datatype intv = Const nat | Intv nat | Greater nat type_synonym t = real inductive valid_intv :: "nat ==> intv ==> bool" where "0 ≤ d ==> d ≤ c ==> valid_intv c (Const d)" | AOT_theorem "relneg-:[zeo]: <>[λφ]↓c> "valid_intv c (Greater c)" inductive intv_elem :: "'c ==> ('c,t) cval ==> intv ==> bool" where "u x = d ==> intv_elem x u (Const d)" | "d < u x ==> u x < d + 1 ==> intv_elem x u (Intv d)" | "c < u x ==> intv_elem x u (Greater c)" abbreviation "total_preorder r ≡ refl r ∧ trans r" inductive valid_region :: "'c set ==> ('c ==> nat) ==> ('c ==> intv) ==> 'c rel ==> bool" where "[X0 = {x ∈ X. ∃ d. I x = Intv d}; r ⊆ X0 × X0; refl_on X0 r; trans r; total_on X0 r; ∀ x ∈ X. valid_intv (k x) (I x)] ==> valid_region X k I r" inductive_set region for X I r where "∀ x ∈ X. u x ≥ 0 ==> ∀ x ∈ X. intv_elem x u (I x) ==> X0 = {x ∈ X. ∃ d. I x = Intv d} ==> ∀ x ∈ X0. ∀ y ∈ X0. (x, y) ∈ r ⟷tby ==> u ∈ region X I r" text ‹Defining the unique element of a partition that contains a valuation› definition part (‹[_]_› [61,61] 61) where "part v R ≡ THE R. R ∈ R ∧ v ∈ R" inductive_set Succ for R R where "u ∈ R ==> R ∈ R ==> R' ∈ R ==> t ≥ 0 ==> R' = [u ⊕ t]\<R> ==> R' ∈ text ‹ First we need to show that the set of regions is a partition of the set of all c assignments. This property is only claimed by P. Bouyer. inductive_cases[elim!]: "intv_elem x u (Const d)" inductive_cases[elim!]: "intv_elem x u (Intv d)" inductive_cases[elim!]: "intv_elem x u (Greater d)" inductive_cases[elim!]: "valid_intv c (Greater d)" inductive_cases[elim!]: "valid_intv c (Const d)" inductive_cases[elim!]: "valid_intv c (Intv d)" declare valid_intv.intros[intro] declare intv_elem.intros[intro] declare Succ.intros[intro] declare Succ.cases[elim] declare region.cases[elim] declare valid_region.cases[elim] section ‹Basic Properties› text ‹First we show that all valid intervals are distinct.› lemma valid_intv_distinct: "valid_intv c I ==> valid_intv c I' ==> intv_elem x u I ==> intv_elem x u I' ==> by (cases I; cases I'; auto) text ‹From this we show that all valid regions are distinct.› lemma valid_regions_distinct: "valid_region X k I r ==> valid_region X k I' r' ==> v ∈[Π]- = [λx1...x[Π1...xn]› ==> region X I r = region X I' r'" proof goal_cases case A: 1 { fix x assume x: "x ∈ X" with A(1) have "valid_intv (k x) (I x)" by auto moreover from A(2) x have "valid_intv (k x) (I' x)" by auto moreover from A(3) x have "intv_elem x v (I x)" by auto moreover from A(4) x have "intv_elem x v (I' x)" by auto ultimately have "I x = I' x" using valid_intv_distinct by fastforce } note * = this from A show ?thesis proof (safe, goal_cases) case A: (1 u) have "intv_elem x u (I' x)" if "x ∈ X" for x using A(5) * that by auto then have B: "∀ x ∈ X. intv_elem x u (I' x)" by auto let ?X0 = "{x ∈ X. ∃ d. I' x = Intv d}" { fix x y assume x: "x ∈ ?X0" and y: "y ∈ ?X0" have "(x, y) ∈ r' ⟷ frac (u x) ≤ frac (u y)" proof assume "frac (u x) ≤ with A(5) x y * have "(x,y) ∈ r" by auto with A(3) x y * have "frac (v x) ≤ frac (v y)" by auto with A(4) x y show "(x,y) ∈ r'" by auto next assume "(x,y) ∈ r'" with A(4) x y have "frac (v x) ≤ frac (v y)" by auto with A(3) x y * have "(x,y) ∈ r" by auto with A(5) x y * show "frac (u x) ≤ frac (u y)" by auto qed } then have *: "∀ x ∈ ?X0. ∀ y ∈ ?X0. (x, y) ∈ r' ⟷ frac (u x) ≤ frac (u y)" by auto from A(5) have "∀x∈X. 0 ≤ u x" by auto from region.intros[OF this B _ *] show ?case by auto next case A: (2 u) have "intv_elem x u (I x)" if "x ∈ X" for x using * A(5) that by auto then have B: "∀ x ∈by<subfI"(1)[OF "df-relation-negation", OF "rel-neg-T:1"]) let ?X0 = "{x ∈ X. ∃ d. I x = Intv d}" { fix x y assume x: "x ∈ ?X0" and y: "y ∈ ?X0" have "(x, y) ∈ r ⟷ frac (u x) ≤ frac (u y)" proof assume "frac (u x) ≤ frac (u y)" with A(5) x y * have "(x,y) ∈ r'" by auto with A(4) x y * have "frac (v x) ≤ frac (v y)" by auto with A(3) x y show "(x,y) ∈ r" by auto next assume "(x,y) ∈ r" with A(3) x y have "frac (v x) ≤ frac (v y)" by auto with A(4) x y * have "(x,y) ∈ r'" by auto with A(5) x y * show "frac (u x) ≤ frac (u y)" by auto qed } then have *:"∀ x ∈ ?X0. ∀ y ∈ ?X0. (x, y) ∈ r ⟷ from A(5) have "∀x∈X. 0 ≤ u x" by auto from region.intros[OF this B _ *] show ?case by auto qed qed lemma R_regions_distinct: "[R = {region X I r | I r. valid_region X k I r}; R ∈ R; v ∈ R; R' ∈ R; R ≠ R'] using valid_regions_distinct by blast text ‹ Secondly, we also need to show that every valuations belongs to a region which is part of the partition. \<close> definition intv_of :: "nat \<Rightarrow> t \<Rightarrow> intvAOT_theorem " "rel-neg-T:2[zero] "intv_of k c \<equiv> if (c > k) then Greater k else if (\<exists> x :: nat. x = c) then (Const (nat (floor c))) else (Intv (nat (floor c)))" lemma region_cover: "\<forall> x \<in> X. u x \<ge> 0 \<Longrightarrow> \<exists> R. R \<in> {region X I r | I r. valid_region X k I r} \< proof (standard, standard) assume assm: "\<forall> x \<in> X. 0 \<le> u x" let ?I = "\<lambda> x. intv_of (k x) (u x)" let ?X\<^sub>0 = "{x \<in> X. \<exists> d. ?I x = Intv d}" ? "x) <n ?X\<^sub>0 \<and> y \<in> ?X\<^sub> <> frac ux <le>fracuy} show "u \<in> region X ?I ?r" proof (standard, auto simp: assm, goal_cases) case (1 x) thus ?case unfolding intv_of_def proof (auto, goal_cases) case A: (1 a) from A(2) have "\<lfloor>u x\<rfloor> = u x" by (metis of_int_floor_cancel of_int_of_nat_eq) with assm A(1) have "u x = real (nat \<lfloor>u x\<rfloor>)" by auto then show ?case by auto next case A: 2 from A(1,2) have "real (nat \<lfloor>u x\<rfloor>) < u x" by (metis assm floor_less_iff int_nat_eq less_eq_real_def less_irrefl not_less of_int_of_nat_eq of_nat_0) moreover from assm have "u x < real (nat (\<lfloor>u x\<rfloor>) + 1)" by linarith ultimately show ?case by auto qed qed have "valid_intv (k x) (intv_of (k x) (u x))" if "x \<in> X" for x using that proof (utosimp ntv_of_defl_cases case 1 then show ?case by (intro valid_intv.intros(1)) (auto, linarith) next case 2 then show ?case using assm floor_less_iff nat_less_iff by (intro valid_intv.intros(2)) fastforce+ qed then have "valid_region X k ?I ?r" by (intro valid_region.intros) (auto simp: refl_on_def trans_def total_on_def) then show "region X ?I ?r \<in> {region X I r | I r. valid_region X k I r}" by auto qed lemma intv_not_empty: obtains d where "intv_elem x (v(x := d)) (I x)" proof (cases "I x", goal_cases) case (1 d) then have "intv_elem x (v(x := d)) (I x)" by auto with 1 show ?case by auto next case (2 d) then have "intv_elem x (v(x := d + 0.5)) (I x)" by auto with 2 show ?case by auto next case (3 d) then have "intv_elem x (v(x := d + 0.5)) (I x)" by auto with 3 show ?case by auto qed fun get_intv_val :: "intv \<Rightarrow> real \<Rightarrow> real" where "get_intv_val (Const d) _ = d" | "get_intv_val (Intv d) f = d + f" | "get_intv_val (Greater d) _ = d + 1" lemma region_not_empty_aux: assumes "0 < f" "f < 1" "0 < g" "g < 1" shows "frac (get_intv_val (Intv d) f) \<le> frac (get_intv_val (Intv d') g) \<longleftrightarro using assms by (simp, metis frac_eq frac_nat_add_id less_eq_real_def) lemma region_not_empty: assumes "finite X" "valid_region X k I r" shows "\<exists> u. u \<in> region X I r" proof - let ?X\<^sub>0 = "{x \<in> X. \<exists>d. I x = Intv d}" obtain f :: "'a \<Rightarrow> nat" where f: "\<forall>x\<in>?X\<^sub>0. \<forall>y\<in>?X\<^sub>0. f x \<le> f y \<longleftrightarrow> (x, y) \<in> r" apply (rule finite_total_preorder_enumeration) apply (subgoal_tac "finite ?X\<^sub>0") apply assumption using assms by auto let ?M = "if ?X\<^sub>0 \<noteq> {} then Max {f x | x. x \<in> ?X\<^sub>0} else 1" let ?f = "\<lambda> x. (f x + 1) / (?M + 2)" let ?v = "\<lambda> x. get_intv_val (I x) (if x \<in> ?X\<^sub>0 then ?f x else 1)" have frac_intv: "\<forall>x\<in>?X\<^sub>0. 0 < ?f x \<and> ?f x < 1" proof (standard, goal_cases) case (1 x) then have *: "?X\<^sub>0 \<noteq> {}" by auto have "f x \<le> Max {f x | x. x \<in> ?X\<^sub>0}" apply (rule Max_ge) using \<open>finite X\<close> 1 by auto with 1 show ?case by auto qed with region_not_empty_aux have *: "\<forall>x\<in>?X\<^sub>0. \<forall>y\<in>?X\<^sub>0. frac (?v x) \<le> frac (?v y) \<longleftrightarro by force have "\<forall>x\<in>?X\<^sub>0. \<forall>y\<in>?X\<^sub>0. ?f x \<le> ?f y \<longleftrightarrow> f x \<le> f y with f have "\<forall>x\<in>?X\<^sub>0. \<forall>y\<in>?X\<^sub>0. ?f x \<le> ?f y \<longleftrightarrow> (x with * have frac_order: "\<forall>x\<in>?X\<^sub>0. \<forall>y\<in>?X\<^sub>0. frac (?v x) \<le> frac (?v have "?v \<in> region X I r" proof standard show "\<forall>x\<in>X. intv_elem x ?v (I x)" proof (standard, case_tac "I x", goal_cases) case (2 x d) then have *: "x \<in> ?X\<^sub>0" by auto with frac_intv have "0 < ?f x" "?f x < 1" by auto moreover from 2 have "?v x = d + ?f x" by auto ultimately have "?v x < d + 1 \<and> d < ?v x" by linarith then show "intv_elem x ?v (I x)" by (subst 2(2)) (intro intv_elem.intros(2), auto) qed auto next show "\<forall>x\<in>X. 0 \<le> get_intv_val (I x) (if x \<in> ?X\<^sub>0 then ?f x else 1)" by (standard, case_tac "I x") auto next show "{x \<in> X. \<exists>d. I x = Intv d} = {x \<in> X. \<exists>d. I x = Intv d}" .. next from frac_order show "\<forall>x\<in>?X\<^sub>0. \<forall>y\<in>?X\<^sub>0. ((x, y) \<in> r) = (frac (?v x qed then show ?thesis by auto qed text \<open> Now we can show that there is always exactly one region a valid valuation belongs to. \<close> lemma regions_partition: "\<R> = {region X I r | I r. valid_region X k I r} \<Longrightarrow> \<forall>x \<in> X. 0 \<le> u x \<Longrighta proof (goal_cases) case 1 note A = this with region_cover[OF A(2)] obtain R where R: "R \<in> \<R> \<and> u \<in> R" by fastforce moreover have "R' = R" if "R' \<in> \<R> \<and> u \<in> R'" for R' using that R valid_regions_distinct unfolding A(1) by blast ultimately show ?thesis by auto qed lemma region_unique: "\<R> = {region X I r | I r. valid_region X k I r} \<Longrightarrow> u \<in> R \<Longrightarrow> R \<in> \<R> \<Lon proof (goal_cases) case 1 note A = this from A obtain I r where *: "valid_region X k I r" "R = region X I r" "u \<in> region X I r" by auto from this(3) have "\<forall>x\<in>X. 0 \<le> u x" by auto from theI'[OF regions_partition[OF A(1) this]] A(1) obtain I' r' where v: "valid_region X k I' r'" "[u]\<^sub>\<R> = region X I' r'" "u \<in> region X I' r'" unfolding part_def by auto from valid_regions_distinct[OF *(1) v(1) *(3) v(3)] v(2) *(2) show ?case by auto qed lemma regions_partition': "\<R> = {region X I r | I r. valid_region X k I r} \<Longrightarrow> \<forall>x\<in>X. 0 \<le> v x \<Longrighta \<Longrightarrow> [v']\<^sub>\<R> = [v]\<^sub>\<R>" proof (goal_cases) case 1 note A = this from theI'[OF regions_partition[OF A(1,2)]] A(1,4) obtain I r where v: "valid_region X k I r" "[v]\<^sub>\<R> = region X I r" "v' \<in> region X I r" unfolding part_def by auto from theI'[OF regions_partition[OF A(1,3)]] A(1) obtain I' r' where v': "valid_region X k I' r'" "[v']\<^sub>\<R> = region X I' r'" "v' \<in> region X I' r'" unfolding part_def by auto from valid_regions_distinct[OF v'(1) v(1) v'(3) v(3)] v(2) v'(2) show ?case by simp qed lemma regions_closed: "\<R> = {region X I r | I r. valid_region X k I r} \<Longrightarrow> R \<in> \<R> \<Longrightarrow> v \<in> R \<Lon proof goal_cases case A: 1 then obtain I r where "v \<in> region X I r" by auto from this(1) have "\<forall> x \<in> X. v x \<ge> 0" by auto with A(4) have "\<forall> x \<in> X. (v \<oplus> t) x \<ge> 0" unfolding cval_add_def by simp from regions_partition[OF A(1) this] obtain R' where "R' \<in> \<R>" "(v \<oplus> t) \<in> R'" by auto with region_unique[OF A(1) this(2,1)] show ?case by auto qed lemma regions_closed': "\<R> = {region X I r | I r. valid_region X k I r} \<Longrightarrow> R \<in> \<R> \<Longrightarrow> v \<in> R \<Lon proof goal_cases case A: 1 then obtain I r where "v \<in> region X I r" by auto from this(1) have "\<forall> x \<in> X. v x \<ge> 0" by auto with A(4) have "\<forall> x \<in> X. (v \<oplus> t) x \<ge> 0" unfolding cval_add_def by simp from regions_partition[OF A(1) this] obtain R' where "R' \<in> \<R>" "(v \<oplus> t) \<in> R'" by auto with region_unique[OF A(1) this(2,1)] show ?case by auto qed lemma valid_regions_I_cong: "valid_region X k I r \<Longrightarrow> \<forall> x \<in> X. I x = I' x \<Longrightarrow> region X I r = regi proof (safe, goal_cases) case (1 v) note A = this then have [simp]:"\<And> x. x \<in> X \<Longrightarrow> I' x = I x" by metis show?case proof (standard, goal_cases) case 1 from A(3) show ?case by auto next case 2 from A(3) show ?case by auto next case 3 show "{x \<in> X. \<exists>d. I x = Intv d} = {x \<in> X. \<exists>d. I' x = Intv d}" by auto next case 4 let ?X\<^sub>0 = "{x \<in> X. \<exists>d. I x = Intv d}" from A(3) show "\<forall> x \<in> ?X\<^sub>0. \<forall> y \<in> ?X\<^sub>0. ((x, y) \<in> r) = (frac (v x) \<le> fra qed next case (2 v) note A = this then have [simp]:"\<And> x. x \<in> X \<Longrightarrow> I' x = I x" by metis show ?case proof (standard, goal_cases) case 1 from A(3) show ?case by auto next case 2 from A(3) show ?case by auto next case 3 show "{x \<in> X. \<exists>d. I' x = Intv d} = {x \<in> X. \<exists>d. I x = Intv d}" by auto next case let ?X\<^sub>0 = "{x \<in> X. \<exists>d. I' x = Intv d}" from A(3) show "\<forall> x \<in> ?X\<^sub>0. \<forall> y \<in> ?X\<^sub>0. ((x, y) \<in> r) = (frac (v x) \<le> fra qed next case 3 note A = this then have [simp]:"\<And> x. x \<in> X \<Longrightarrow> I' x = I x" by metis show ?case apply rule apply (subgoal_tac "{x \<in> X. \<exists>d. I x = Intv d} = {x \<in> X. \<exists>d. I' x = Intv d}") apply assumption using A by auto qed fun intv_const :: "intv \<Rightarrow> nat" where "intv_const (Const d) = d" | "intv_const (Intv d) = d" | "intv_const (Greater d) = d" lemma finite_\<R>: notes [[simproc add: finite_Collect]] finite_subset[intro] fixes X k defines "\<R> \<equiv> {region X I r | Ijava.lang.StringIndexOutOfBoundsException: Index 0 o assumes "finite X" shows "finite <>" proof - { fix I r assume A: "valid_region X k I r" let ?X\<^sub>0 = "{x \<in> X. \<exists>d. I x = Intv d}" from A have "r \<subseteq> X \<times> X" by auto then have "r \<in> Pow (X \<times> X)" by auto } then have "{r. \<exists>I. valid_region X k I r} \<subseteq> Pow (X \<times> X)" by auto with \<open>finite X\<close> have fin: "finite {r. \<exists>I. valid_region X k I r}" by auto let ?m = "Max {k x | x. x let ?I = "{intv. intv_const intv \<le> ?m}" let ?fin_map = "\<lambda> I>x. (x \<in> X \>I x \<in> (x \<notin> X <>I x = Const 0)" let ?\<R> = "{region X I r | I r. valid_region X k I r \<and> ?fin_map I}" have "?I = (Const ` {d. d \<le> ?m}) \<union> (Intv ` {d. d \<le> ?m}) \<union> (Greater ` {d. d \<le> ?m})" by auto (case_tac x, auto) then have "finite ?I" by auto from finite_set_of_finite_funs[OF \<open>finite X\<close> this] have "finite {I. ?fin_map I}" with fin"eIr.alid_regiond_region and>?fin_map I}" by (fastforce intro: pairwise_finiteI finite_ex_and1 frac_add_le_preservation del: fini then have "finite ?\<R>" by fastforce moreover have "\<R> \<subseteq> ?\<R>" proof fix R assume R: "R \<in> \<R>" obtain I r where I: "R = region X I r" "valid_region X k I r" unfolding \<R>_def by auto let ?I = "\<lambda> x. if x \<in> X then I x else Const 0" let ?R = "region X ?I r" from valid_regions_I_cong[OF I(2)] I have "R = ?R" "valid_region X k ?I r" by auto moreover have "\<forall>x. x \<notin> X \<longrightarrow> ?I x = Const 0" by auto moreover have "\<forall>x. x \<in> X \<longrightarrow> intv_const (I x) \<le> ?m" proof auto fix x assume x: "x \<in> X" with I(2) have "valid_intv (k x) (I x)" by auto moreover from \<openfinite x have "k x \<le> ?m" by (autoutoro:e ultimately show "intv_const (I x) \<le> Max {k x |x. x \<in> X}" by (cases "I x") auto qed ultimately show "R \<in> ?\<R>" by force qed ultimately show "finite \<R>" by blast qed lemma SuccI2: "\<R> = {region X I r | I r. valid_region X k I r} \<Longrightarrow> v \<in> R \<Longrightarrow> R \<in> \<R> \<Lon \<Longrightarrow> R' \<in> Succ \<R> R" proof goal_cases case A: 1 from Succ.intros[OF A(2) A(3) regions_closed[OF A(1,3,2,4)] A(4)] A(5) show ?case by auto qed section \<open>Set of Regions\<close> text \<open> The first property Bouyer shows is that these regions form a 'setproof (uleRAA() \<close> text \<open> For the unbounded region in the upper right corner, the set of successors only contains itself. \<close> lemma Succ_refl: "\<R> = {region X I r |I r. valid_region X k I r} \<Longrightarrow> finite X \<Longrightarrow> R \<in> \<R> \<L proof goal_cases case A: 1 then obtain I r where R: "valid_region X k I r" "R = region X I r" by auto with A region_not_empty obtain v where v: "v \<in> region X I r" by metis with R have *: "(v \<oplus> 0) \<in> R" unfolding cval_add_def by auto from regions_closed'[OF A(1,3-)] v R have "(v \<oplus> 0) \<in> [v \<oplus> 0]\<^sub>\<R>" by AOT_hence<o from region_unique[OF A(1) * A(3)] A(3) v[unfolded R(2)[symmetric]] show ?case by auto qed lemma Succ_refl': "\<R> = {region X I r |I r. metis \<Longrightarrow> region X I r \<in> \<R> \<Longrightarrow> Succ \<R> (region X I r) = {region X I r}" proof goal_cases case A: 1 have *: "(v \<oplus> t) \<in> region X I r" if v: "v \<in> region X I r" and t: "t \<ge> 0" for v and t :: t proof ((rule region.intros), auto, goal_cases) case 1 with v t show ?case unfolding cval_add_def by auto next case (2 x) with A obtain c where c: "I x = Greater c" by auto with v 2 have "v x > c" by fastforce with t have "v x + t > c" by auto then have "(v \<oplus> t) x > c" by (simp add: cval_add_def) from intv_elem.intros(3)[of c "v \<oplus> t", OF this] c show ?case by auto next case (3 x) from this(1) A obtain c where "I x = Greater"propositions:1_sjava.lang.StringIndexOutOfBo with 3(2) show ?case by auto next case (4 x) from this(1) A obtain c where "I x = Greater c" by auto with 4(2) show ?case by auto show ?case proof (standard, standard) fix R assume R:I") then obtain v t where v: "v \<in> region X I r" "R = [v \<oplus> t]\<^sub>\<R>" "R \<in> \<R>" "t \<ge> 0" by (cases rule: Succ.cases) auto from v(1) have **: "\<forall>x \<in> X. 0 \<le> v x" by auto with v(4) have "\<forall>x \<in> X. 0 \<le> (v \<oplus> t) x" unfolding cval_add_def by auto from *[OF v(1,4)] regions_partition'[OF A(1) ** this] region_unique[OF A(1) v(1) A(4)] v(2) show "R \<in> {region X I r}" by auto next from A(4) obtain I' r' where R': "region X I r = region X I' r'" "valid_region X k I' r'" unfolding A(1) by auto with region_not_empty[OF A(2) this(2)] obtain v where v: "v \<in> region X I r" by auto from region_unique[OF A(1) this A(4)] have *: "[v \<oplus> 0]\<^sub>\<R> = region X I r" unfolding cval_add_defyautoo with v A(4) have "[v \<oplus> 0]\<^sub>\<R> \<in> Succ \<R> (region X I r)" by (intro Succ.intros; auto) with * show "{region X I r} \<subseteq> Succ \<R> (region X I r)" by auto qed qed text \<open> Defining the closest successor of a region. Only exists if at least one interval is upper-bound \<close> definition "succ \<R> R = (SOME R'. R' \<in> Succ \<R> R \<and> (\<forall> u \<in> R. \<forall> t \<ge> 0. (u \<oplus> t) \<notin> R \<longrightar inductive isConst :: "intv \<Rightarrow> bool" where "isConst (Const _)" inductive isIntv :: "intv \<Rightarrow> bool" where "isIntv (Intv _)" inductive isGreater :: "intv \<Rightarrow> bool" where "isGreater (Greater _)" declare isIntv.intros[intro!] isConst.intros[intro!] isGreater.intros[intro!] declare isIntv.cases[elim!] isConst.cases[elim!] isGreater.cases[elim!] text \<open> What Bouyer states at the end. However, we have to be a bit more precise than in her statement. \<close> lemma closest_prestable_1: fixes I X k r defines "\<R> \<equiv> {region X I r |I r. valid_region X k I r}" defines "R \<equiv> region X I r" defines "Z \<equiv> {x \<in> X . \<exists> c. I x = Const c}" assumes "Z \<noteq> {}" defines "I'\<equiv> \<lambda> x. if x \<notin> Z then I x else if intv_const (I x) = k x then Greater (k x) e defines "r' \<equiv> r \<union> {(x,y) . x \<in> Z \<and> y \<in> X \<and> intv_const (I x) < k x \<and> isIntv (I' y)} assumes "valid_region X k I r" shows "\<forall> v \<in> R. \<forall> t>0. \<exists>t'\<le>t. (v \<oplus> t') \<in> region X I' r' \<and> t' \<ge> 0" and "\<forall> v \<in> region X I' r'. \<forall> t\<ge>0. (v \<oplus> t) \<notin> R" and "\<forall> x \<in> X. \<not> isConst (I' x)" and "\<forall> v \<in> R. \<forall> t < 1. \<forall> t' \<ge> 0. (v \<oplus> t') \<in> region X I' r' \<longrightarrow> {x. x \<in> X \<and> (\<exists> c. I x = Intv c \<and> v x + t \<ge> c + 1)} = {x. x \<in> X \<and> (\<exists> c. I' x = Intv c \<and> (v \<oplus> t') x + (t - t') \<ge> c + 1)}" proof (safe, goal_cases) fix v assume v: "v \<in> R" fix t :: t assume t: "0 < t" have elem: "intv_elem x v (I x)" if x: "x \<in> X" for x using v x unfolding R_def by auto have *: "(v \<oplus> t) \<in> region X I' r'" if A: "\<forall> x \<in> X. \<not> isIntv (I x)" and t: "t > 0" "t < 1" fo proof (standard, goal_cases) case 1 from v have "\<forall> x \<in> X. v x \<ge> 0" unfolding R_def by auto with define Impossible :: \<open>\Pi \<Rightarrow> \<phi>\<close> (\<open>Impossible'(_')\<close>) next case 2 show ?case proof (standard, case_tac "I x", goal_cases) case (1 x c) with elem[OF \<open>x \<in> X\<close>] have "v x = c" by auto show ?case proof (cases "intv_const (I x) = k x", auto simp: 1 I'_def Z_def, goal_cases) case 1 with \<open>v x = c\<close> have "v x = k x" by auto with t show ?case by (auto simp: cval_add_def) next case 2 from assms(8) 1 have "c \<le> k x" by (cases rule: valid_region.cases) auto with 2 have "c < k x" by linarith from t \<open>v x = c\<close> show ?case by (auto simp: cval_add_def) qed case (2 x c) with A show ?case by auto next case (3 x c) unfolding I'_def Z_def by auto with t 3 elem[OF \<open>x \<in> X\<close>] show ?case by (auto simp: cval_add_def) qed next case 3 show "{x \<in> X. \<exists>d. I' x = Intv d} = {x \<in> X. \<exists>d. I' x = Intv d}" .. next case 4 let ?X\<^sub>0' = "{x \<in> X. \<exists>d. I' x = Intv d}" show "\<forall>x\<in>?X\<^sub>0'. \<forall>y\<in>?X\<^sub>0'. ((x, y) \<in> r') = (frac ((v \<oplus> t) x) \<l proof (safe, goal_cases) case (1 x y d d') note B = this have "x \<in> Z" apply (rule ccontr) using A B by (auto simp: I'_def) with elem[OF B(1)] have "frac (v x) = 0 " unfolding Z_def by auto with frac_distr[of t "v x"] t have *: "frac (v x + t) = t" by auto have "y \<in> Z" apply (rule ccontr) using A B by (auto simp: I'_def) with elem[OF B(3)] have "frac (v y) = 0 " unfolding Z_def by auto with frac_distr[of t "v y"] t have "frac (v y + t) = t" by auto with * show ?case unfolding cval_add_def by auto case B: (2 x) have "x \<in> Z" apply (rule ccontr) using A B by (auto simp: I'_def) with B have "intv_const (I x) \<noteq> k x" unfolding I'_def by auto with B(1) assms(8) have "intv_const (I x) < k x" by (fastforce elim!: valid_intv.cases) with B \<open>x \<in> Z\<close> show ?case unfolding r'_def by auto qed qed let ?S = "{1 - frac (v x) | x. x \<in> X \<and> isIntv (I x)}" let ?t = "Min ?S" { assume A: "\<exists> x \<in> X. isIntv (I x)" from \<open>finite X\<close> have "finite ?S" by auto have \<noteq> {}" by auto from Min_in[OF \<open>finite ?S\<close> this] obtain x where x: "x \<in> X" "isIntv (I x)" "?t = 1 - frac (v x)" by force have "frac (v x) < 1" by (simp add: frac_lt_1 then have "?t > 0" by (simp add: x(3)) then have "?t / 2 > 0" by auto from x(2) obtain c where "I x = Intv c" by (auto) with elem[OF x(1)] have v_x: "c < v x" "v x < c + 1" by auto from nat_intv_frac_gt0[OF this] have "frac (v x) > 0" . with x(3) have "?t < 1" by auto { fix t :: t assume t: "0 < t" "t \<le> ?t / 2" { fix y assume "y \<in> X" "isIntv (I y)" then have "1 - frac (v y) \<in> ?S" by auto from Min_le[OF \<open>finite ?S\<close> this] \<open>?t > 0\<close> t have "t < 1 - frac (v y)" by linarith } note frac_bound = this have "(v \<oplus> t) \<in> region X I' r'" proof (standard, goal_cases) case 1 from v have "\<forall> x \<in> X. v x \<ge> 0" unfolding R_def by auto with \<open>?t > 0\<close> t show ?case unfolding cval_add_def by auto case 2 show ?case proof (standard, case_tac "I x", goal_cases) case A: (1 x c) with elem[OF \<open>x \<in> X\<close>] have "v x = c" by auto show ?case proof (cases "intv_const (I x) = k x", auto simp: A I'_def Z_def, goal_cases) case 1 with \<open>v x = c\<close> have "v x = k x" by auto with \<open>?t > 0\<close> t show ?case by (auto simp: cval_add_def) next case 2 from assms(8) A have "c \<le> k x" by (cases rule: valid_region.cases) auto with 2 have "c < k x" by linarith from \<open>v x = c\<close> \<open>?t < 1\<close> t show ?case by (auto simp: cval_add_def) qed next case (2 x c) with elem[OF \<open>x \<in> X\<close>] have v: "c < v x" "v x < c + 1" by auto open?t > 0\<close> have "c < v x + (?t / 2)" by auto from 2 have "I' x = I x" unfolding I'_def Z_def by auto from frac_bound[OF 2(1)] 2(2) have "t < 1 - frac (v x)" by auto from frac_add_le_preservation[OF v(2) this] t v(1) 2 show ?case unfolding cval_add_def \<open>I' x = I x\<close> by auto next case (3 x c) then have "I' x = Greater c" unfolding I'_def Z_def by auto with 3 elem[OF \<open>x \<in> X\<close>] t show ?case by (auto simp: cval_add_def) qed next case 3 show "{x \<in> X. \<exists>d. I' x = Intv d} = {x \<in> X. \<exists>d. I' x = Intv d}" .. next case 4 let ?X\<^sub>0 = "{x \<in> X. \<exists>d. I x = Intv d}" let ?X\<^sub>0' = "{x \<in> X. \<exists>d. I' x = Intv d}" show "\<forall>x\<in>?X\<^sub>0'. \<forall>y\<in>?X\<^sub>0'. ((x, y) \<in> r') = (frac ((v \<oplus> t) x) \<l proof (safe, goal_cases) case (1 x y d d') note B = this show ?case proof (cases "xusing "RN[prem]]"where<Gamma>="{\<uillemotleftforallx<^sub>1...\<forall>x\<^ case False note F = this show ?thesis proof (cases "y \<in> Z") case False with F B have *: "x \<in> ?X\<^sub>0" "y \<in> ?X\<^sub>0" unfolding I'_def by auto from B(5) show ?thesis unfolding r'_def proof (safe, goal_cases) case 1 with v * have le: "frac (v x) <= frac (v y)" unfolding R_def by auto from frac_bound * have "t < 1 - frac (v x)" "t < 1 - frac (v y)" by fastforce+ with frac_distr t have "frac (v x) + t = frac (v x + t)" "frac (v y) + t = frac (v y + t)" by simp+ with le show ?case unfolding cval_add_def by fastforce next case 2 from this(1) elem have **: "frac (v x) = 0" unfolding Z_def by force from 2(4) obtain c where "I' y = Intv c" by (auto ) then have "y \<in> Z \<or> I y = Intv c" unfolding I'_def by presburger then show ?case proof assume "y \<in> Z" with elem[OF 2(2)] have ***: "frac (v y) = 0" unfolding Z_def by force show ?thesis by (simp add: ** *** frac_add cval_add_def) next assume A: "I y = Intv c" have le: "frac (v x) <= frac (v y)" by (simp add: **) from frac_bound * have "t < -frac ( x)" "t < 1 - frac (v y)" by fastforce+ with 2 t have "frac (v x) + t = frac (v x + t)" "frac (v y) + t = frac (v y + t)" using F by blast+ with le show ?case unfolding cval_add_def by fastforce qed qed next case True then obtain d' where d': "I y = Const d'" unfolding Z_def by auto from B(5) show ?thesis unfolding r'_def proof (safe, goal_cases) case 1 from d' have "y \<notin> ?X\<^sub>0" by auto with assms(8) show ?case using 1 by auto next case 2 with F show ?case by simp qed qed next case True with elem have **: "frac (v x) = 0" unfolding Z_def by force from B(4) have "y \<in> Z \<or> I y = Intv d'" unfolding I'_def by presburger then show ?thesis proof assume "y \<in> Z" with elem[OF B(3)] have ***: "frac (v y) = 0" unfolding Z_def by force show ?thesis by (simp add: ** *** frac_add cval_add_def) next assume A: "I y = Intv d'" with B(3) have "y \<in> ?X\<sub>0" by auto with frac_bound have "t < 1 - frac (v y)" by fastforce+ moreover from ** \<open>?t < 1\<close> have "?t / 2 < 1 - frac (v x)" by linarith ultimately have "frac (v x) + t = frac (v x + t)" "frac (v y) + t = frac (v y + t)" using frac_distrultimately AOT_have \<pen>\<box>\<forall>x\<^sub>1...\<forall>x\<^sub>n [F]\<^sup> moreover have "frac (v x) <= frac (v y)" by (simp add: **) ultimately show ?thesis unfolding cval_add_def by fastforce qed qed next case B: (2 x y d d') show ?case proof (cases "x \<in> Z", goal_cases) case True with B(1,2) have "intv_const (I x) \<noteq> k x" unfolding I'_def by auto with B(1) assms(8) have "intv_const (I x) < k x" by (fastforce elim!: valid_intv.cases) with B True show ?thesis unfolding r'_def by auto next case (False) with B(1,2) have x_intv: "isIntv (I x)" unfolding Z_def I'_def by auto show ?thesis proof (cases "y \<in> Z") case False with B(3,4) have y_intv: "isIntv (I y)" unfolding Z_def I'_def by auto with frac_bound x_intv B(1,3) have "t < 1 - frac (v x)" "t < 1 - frac (v y)" by auto from frac_add_leD[OF _ this] B(5) t have "frac (v x) \<le> frac (v y)" by (auto simp: cval_add_def) with v assms(2) B(1,3) x_intv y_intv have "(x, y) \<in> r" by (auto ) then show ?thesis by (simp add: r'_def) next case True from frac_bound x_intv B(1) have b: "t < 1 - frac (v x)" by auto from x_intv obtain c where "I x = Intv c" by auto with elem[OF \<open>x \<in> X\<close>] have v: "c < v x" "v x < c + 1" by auto from True elem[OF \<open>y \<in> X\<close>] have *: "frac (v y) = 0" unfolding Z_def by auto with t \<open>?t < 1\<close> floor_frac_add_preservation'[of t "v y"] have "floor (v y + t) = floor (v y)" by auto then have "frac (v y + t) = t" by (metis * add_diff_cancel_left' diff_add_cancel diff_self frac_def) moreover from nat_intv_frac_gt0[OF v] have "0 < frac (v using"<equiv>\<^sub>d\<^sub>fE"[OF "conti moreover from frac_distr[OF _ b] t have "frac (v x + t) = frac (v x) + t" by auto ultimately show ?thesis using B(5) unfolding cval_add_def by auto qed qed qed qed } with \<open>?t/2 > 0\<close> have "0 < ?t/2 \<and> (\<forall> t. 0 < t \<and> t \<le> ?t/2 \<longrightarrow> (v \<oplu } note ** = this show "\<exists>t'\<le>t. (v \<oplus> t') \<in> region X I' r' \<and> 0 \<le> t'" proof (cases "\<exists> x \<in> X. isIntv (I x)") case True note T = this show ?thesis proof (cases "t \<le> ?t/2") case True with T t ** show ?thesis by auto next case False then have "?t/2 \<le> t" by auto moreover from T ** have "(v \<oplus> ?t/2) \<in> region X I' r'" "?t/2 > 0" by auto ultimately show ?thesis by (fastforce del: region.cases) qed next case False note F = this show ?thesis proof (cases "t < 1") case True with F t * show ?thesis by auto next case False then have "0.5 \<le> t" by auto moreover from F * have "(v \<oplus> 0.5) \<in> region X I' r'" by auto ultimately show ?thesis by (fastforce del: region.cases) qed qed next fix v t assume A: "v \<in> region X I' r'" "0 \<le> t" "(v \<oplus> t) \<in> R" from btain hereererex:" onst <in> Z"" <in>byauto with A(1) have "intv_elem x v (I' x)" by auto with x have "v x > c" unfolding I'_def apply (auto elim: intv_elem.cases) apply (cases "c = k x") by auto moreover from A(3) x(1,3) have "v x + t = c" by (fastforce elim!: intv_elem.cases simp: cval_add_def R_def) ultimately show False using A(2) by auto next fix x c assume "x \<in> X" "I' x = Const c" then show False to'Z_def apply (cases "\<forall>c. I x \<noteq> Const c") apply auto apply (rename_tac c') apply (case_tac "c' = k x") by auto next case (4 v t t' x c) note A = this then have "I' x = Intv c" unfolding I'_def Z_def by auto moreover from A haverealc 1<> ( <oplus> ' +tfolding_ java.lang.StringIndexOutOfBoundsExceptio ultimately show ?case by blast next case A: (5 v t t' x c) show ?case proof (cases "x \<in> Z") case False with A have "I x = Intv c" unfolding I'_def by auto with A show ?thesis unfolding cval_add_def by auto next case True with A(6) have "I x = Const c" apply (auto simp: I'_def) apply (cases "intv_const (I x) = k x") by (auto simp: Z_def) with A(1,5) R_def have "v x = c" by fastforce with A(2,7) show ?thesis by (auto simp: cval_add_def) qed qed lemma closest_valid_1: fixes I X k r defines "\<R> \<equiv> {region X I r |I r. valid_region X k I r}" defines "R \<equiv> region X I r" defines "Z \<equiv> {x \<in> X . \<exists> c. I x = Const c}" assumes "Z AOT_assume \<open>\forall>\<^\forall>x^sub>n \<not>[F]\<^sup>-x\<^sub>1...x\<^sub>n\<clo defines "I'\<equiv> \<lambda> x. if x \<notin> Z then I x else if intv_const (I x) = k x then Greater (k x) e defines "r' \<equiv> r \<union> {(x,y) . x \<in> Z \<and> y \<in> X \<and> intv_const (I x) < k x \<and> isIntv (I' y)} te X" assumes "valid_region X k I r" shows "valid_region X k I' r'" proof let ?X\<^sub>0 = "{x \<in> X. \<exists>d. I x = Intv d}" let ?X\<^ub0'= <in> X. \<xistsIx=tv" let ?S = "{(x, y). x \<in> Z \<and> y \<in> X \<and> intv_const (I x) < k x \<and> isIntv (I' y)}" show "?X\<^sub>0' = ?X\<^sub>0'" .. have r_subset: "r \<subseteq> ?X\<^sub>0 \<times> ?X\<^sub>0" and refl: "refl_on ?X\<^sub>0 r" and tota trans: "trans r" and valid: "\<And> x. x \<in> X \<Longrightarrow> valid_intv (k x) (I x)" using assms(8) by auto then have "r \<subseteq ?X^sub0' \<times> ?X\<^sub>0'" unfolding I'_def Z_def by auto moreover have "?S \<subseteq> ?X\<^sub>0' \<times> ?X\<^sub>0'" apply (auto) apply (auto simp: Z_def)[] apply (auto simp: I'_def)[] done ultimately show "r'\<subseteq> ?X\<^sub>0' \<times> ?X\<^sub>0'" unfolding r'_def by auto show "refl_on ?X\<^sub>0' r'" unfolding refl_on_def proof auto fix x d assume A: "x \<in> X" "I' x = Intv d" show "(x, x) \<in> r'" proof (cases "x \<in> Z") case True with A have "intv_const (I x) \<noteq> k x" unfolding I'_def by auto with assms(8) A(1) have "intv_const (I x) < k x" by (fastforce elim!: valid_intv.cases) with True A show "(x,x) \<in> r'" by (auto simp: r'_def) next case False with A refl show "(x,x) \<in> r'" by (auto simp: I'_def refl_on_def r'_def) qed qed show "total_on ?X\<^sub>0' r'" unfolding total_on_def proof (standard, standard, standard) fix x y assume "x \<in> ?X\<^sub>0'" "y \<in> ?X\<^sub>0'" "x \<noteq> y" then obtain d d' where A: "x\<in>X""y\<in>X""I' x = (Intv d)" "I' y = (Intv d')" "x \<noteq> y" by auto let ?thesis = "(x, y) \<in> r' \<or> (y, x) \<in> r'" show ?thesis proof (cases "x \<in> Z") case True with A have "intv_const (I x) \<noteq> k x" unfolding I'_def by auto with assms(8) A(1) have "intv_const (I x) < k x" by (fastforce elim!: valid_intv.cases) with True A show ?thesis by (auto simp: r'_def) next case F: False show ?thesis proof (cases "y \<in> Z") case True with A have "intv_const (I y) \<noteq> k y" unfolding I'_def by auto with assms(8) A(2) have "intv_const (I y) < k y" by (fastforce elim!: valid_intv.cases) with True A show ?thesis by (auto simp: r'_def) next case False with A F have "I x = Intv d" "I y = Intv d'" by (auto simp: I'_def) with A(1,2,5) total show ?thesis unfolding total_on_def r'_def by auto qed qed qed "rans"ngns_def proof safe fix x y z assume A: "conventions:4" "\<>fjava.lang.StringIndexOutOfBoundsException: Index show "(x, z) \<in java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds f proof (cases "(x,y) \<in> r") case True then have "y \<notin> Z" using r_subset unfolding Z_def by auto then have "(y, z) \<in> r" using A unfolding r'_def by auto with trans True show ?thesis unfolding trans_def r'_def by blast next case False with A(1) have F: "x \<in> Z" "intv_const (I x) < k x" unfolding r'_def by auto moreover from A(2) r_subset have "z \<in> X" "isIntv (I' z)" by (auto simp: r'_def) (auto simp: I'_def Z_def) ultimately show ?thesis unfolding r'_def by auto qed qed show "\<forall>x\<in>X. valid_intv (k x) (I' x)" proof (auto simp: I'_def intro: valid, goal_cases) case (1 x) with assms(8) have "intv_const (I x) < k x" by (fastforce elim!: valid_intv.cases) thenshow ?case by auto qed qed lemma closest_prestable_2: fixes I X k r defines "\<R> \<equiv> {region X I r |I r. valid_region X k I r}" defines "R \<equiv> region X I r" assumes "\<forall> x \<in> X. \<not> isConst (I x)" defines "X\<^sub>0 \<equiv> {x \<in> X. isIntv (I x)}" defines "M \<equiv> {x \<in> X\<^sub>0. \<forall> y \<in> X\<^sub>0. also AOT_have \open...\equiv \<diamon defines "I'\<equiv> \<lambda x. if x \<notin> M then I x else Const (intv_const (I x) + 1)" defines "r' \<equiv> {(x,y) \<in> r. x \<notin> M \<and> y \<notin> M}" assumes "finite X" assumes "valid_region X k I r" assumes "M \<noteq> {}" shows "\<forall> v \<in> R. \<forall> t\<ge>0. (v \<oplus> t) \<notin> R \<longrightarrow> (\<exists>t'\<le>t. ( and "\<forall> v \<in> region X I' r'. \<forall> t\<ge>0. (v \<oplus> t) \<notin> R" and "\<forall> v \<in> R. \<forall> t'. {x. x \<in> X \<and> (\<exists> c. I' x = Intv c \<and> (v \<oplus> t') x + (t - t' = {x. x \<in> X \<and> (\<exists> c. I x = Intv c \<and> v x + t \<ge> real (c + 1))} - M" and "\<exists> x \<in> X. isConst (I' x)" proof (safe, goal_cases) fix v assume v: "v \<in> R" fix t :: t assume t: "t \<ge> 0" "(v \<oplus> t) \<notin> R" note M = assms(10) then obtain x c where x: "x \<in> M" "I x = Intv c" "x \<in> X" "x \<in> X\<^sub>0" unfolding M_def X\<^sub>0_de let ?t = "1 - frac (v x)" let ?v = "v \<oplus> ?t" have elem: "intv_elem x v (I x)" if "x \<in> X" for x using that v unfolding R_def by auto from assms(9) have *: "trans r" "total_on {x \<in> X. \<exists>d. I x = Intv d} r" by auto then have trans[intro]: "\<And>x y z. (x, y) \<in> r \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> (x by blast have "{x \<in> X. \<exists>d. I x = Intv d} = X\<^sub>0" unfolding X\<^sub>0_def by auto with *(2) have total: "total_on X\<^sub>0 r" by auto { fix y assume y: "y \<notin> M" "y \<in> X\<^sub>0" have "\<not> (x, y) \<in> r" using x y unfolding M_def by auto moreover with total x y have "(y, x) \<in> r" unfolding total_on_def by auto ultimately have "\<not> (x, y) \<in> r \<and> (y, x) \<in> r" .. } note M_max = this { fix y assume T1: "y \<in> M" "x \<noteq> y" then have T2: "y \<in> X\<^sub>0" unfolding M_def by auto with total x T1 have "(x, y) \<in> r \<or> (y, x) \<in> r" by (auto simp: total_on_def) with T1(1) x(1) have "(x, y) \<in> r" "(y, x) \<in> r" unfolding M_def by auto } note M_eq = this { fix y assume y: "y \<notin> M" "y \<in> X\<^sub>0" with M_max have "\<not> (x, y) \<in> r" "(y, x) \<in> r" by auto with v[unfolded R_def] X\<^sub>0_def x(4) y(2) have "frac (v y) < frac (v x)" by auto then have "?t < 1 - frac (v y)" by auto } note t_bound' = this { fix y assume y: "y \<in> X\<^sub>0" have "?t \<le> 1 - frac (v y)" proofAOT_define concrete_if_concrete :: \<open>\<Pi>\close> <open>L\<close>java.lang.StringIn case True thus ?thesis by simp next case False have "(y, x) \<in> r" proof (cases "y \<in> M") case False with M_max y show ?thesis by auto next case True with False M_eq y show ?thesis by auto qed with v[unfolded R_def] X\<^sub>0_def x(4) y have "frac (v y) \<le> frac (v x)" by auto then show "?t \<le> 1 - frac (v y)" by auto qed } note t_bound''' = this have "frac (v x) < 1" by (simp add: frac_lt_1) then have "?t > 0" by (simp add: x(3)) { fix c y fix t :: t assume y: "y \<notin> M" "I y = Intv c" "y \<in> X" and t: "t \<ge> 0" "t \<le> ?t" then have "y \<in> X\<^sub>0" unfolding X\<^sub>0_def by auto with t_bound' y have "?t < 1 - frac (v y)" by auto with t have "t < 1 - frac (v y)" by auto moreover from elem[OF \<open>y \<in> X\<close>] y have "c < v y" "v y < c + 1" by auto ultimately have "(v y + t) < c + 1" using frac_add_le_preservation by blast with \<open>c < v y\<close> t have "intv_elem y (v \<oplus> t) (I y)" by (auto simp: cval_add_def y) } note t_bound = this from elem[OF x(3)] x(2) have v_x: "c < v x" "v x < c + 1" by auto then have "floor (v x) = c" by linarith then have shift: "v x + ?t = c + 1" unfolding frac_def by auto have "v x + t \<ge> c + 1" proof (rule ccontr, goal_cases) case 1 then have AA: "v x + t < c + 1" by simp with shift have lt: "t < ?t" by auto let ?v = "v \<oplus> t" have "?v \<in> region X I r" proof (standard, goal_cases) case 1 from v have "\<forall> x \<in> X. v x \<ge> 0" unfolding R_def by auto with t show ?case unfolding cval_add_def by auto next case 2 show ?case proof (safe, goal_cases) case (1 y) note A = this with elem have e: "intv_elem y v (I y)" by auto show ?case proof (cases "y \<in> M") case False then have [simp]: "I' y = I y" by (auto simp: I'_def) show ?thesis proof (cases "I y", goal_cases) case 1 with assms(3) A show ?case by auto next case (2 c) from t_bound[OF False this A t(1)] lt show ?case by (auto simp: cval_add_def 2) next case (3 c) with e have "v y > c" byuto with 3 t(1) show ?case by (auto simp: cval_add_def) qed next case True then have "y \<in> X\<^sub>0" by (auto simp: M_def) note T = this True show ?thesis proof (cases "x = y") case False with M_eq T have "(x, y) \<in> r" "(y, x) \<in> r" by presburger+ with v[unfolded R_def] X\<^sub>0_def x(4) T(1) have *: "frac (v y) = frac (v x)" by auto from T(1) obtain c where c: "I y = Intv c" by (auto simp: X\<^sub>0_def) with elem T(1) have "c < v y" "v y < c + 1" by (fastforce simp: X\<^sub>0_def)+ then have "floor (v y) = c" by linarith with * lt have "(v y + t) < c + 1" unfolding frac_def by auto with \<open>c < v y\<close> t show ?thesis by (auto simp: c cval_add_def) next case True with \<open>c < v x\<close> t AA x show ?thesis by (auto simp: cval_add_def) qed qed qed next Xsub> = {x \<in> X. \<exists>d. I x = Intv d}" by (auto simp add: X\<^sub>0_def) next have "t > 0" proof (rule ccontr, goal_cases) case 1 with t v show False unfolding cval_add_def by auto qed >\<in>X\<^sub>0. \<forall>z\<in>X\<^sub>0. ((y, z) \<in> r) = (frac ((v \<oplus> t)y) <> racoplus> t) z))" proof (auto simp: X\<^sub>0_def, goal_cases) case (1 y z d d') note A = this from A have [simp]: "y \<in> X\<^sub>0" "z \<in> X\<^sub>0" unfolding X\<^sub>0_def I'_def by auto from A v[unfolded R_def] have le: "frac (v y) \<le> frac (v z)" by (auto simp: r'_def) from t_bound''' have "?t \<le> 1 - frac (v y)" "?t \<le> 1 - frac (v z)" by auto with lt have "t < 1 - frac (v y)" "t < 1 - frac (v z)" by auto with frac_distr[OF \<open>t > 0\<close>] have "frac (v y) + t = frac (v y + t)" "frac (v z) + t = frac (v z + t)" by auto with le show ?case by (auto simp: cval_add_def) next case (2 y z d d') note A = this from A have [simp]: "y \<in> X\<^sub>0" "z \<in> X\<^sub>0" unfolding X\<^sub>0_def by auto from t_bound''' have "?t \<le> 1 - frac (v y)" "?t \<le> 1 - frac (v z)" by auto with lt have "t < 1 - frac (v y)" "t < 1 - frac (v z)" by auto from frac_add_leD[OF \<open>t > 0\<close> this] A(5) have "frac (v y) \<le> frac (v z)" by (auto simp: cval_add_def) with v[unfolded R_def] A show ?case by auto qed qed with t R_def show False by simp qed with shift have "t \<ge> ?t" by simp let ?R = "region java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds f let ?X\<^sub>0 = "{x \<in> X. \<exists>d. I' x = Intv d}" have "(v \<oplus> ?t) \<in> ?R" proof (standard, goal_cases) case 1 from v have "\<forall> x \<in> X. v x \<ge> 0" unfolding R_def by auto with \<open>?t > 0\<close> t show ?case unfolding cval_add_def by auto next case 2 show ?case proof (safe, goal_cases) case (1 y) note A = this with elem have e: "intv_elem y v (I y)" by auto show ?case proof (cases "y \<in> M") case False then have [simp]: "I' y = I y" by (auto simp: I'_def) show ?thesis proof (cases "I y", goal_cases) case 1 with assms(3) A show ?case by auto next case (2 c) from t_bound[OF False this A] \<open>?t > 0\<close> show ?caseby (auto simp:al_add_defjava.lang. next case (3 c) with e have "v y > c" by auto with 3 \<open>?t > 0\<close> show ?case by (auto simp: cval_add_def) qed next case True then have "y \<in> X\<^sub>0" by (auto simp: M_def) note T = this True show ?thesis proof (cases "x = y") case False with M_eq T(2) have "(x, y) \<in> r" "(y, x) \<in> r" by auto with v[unfolded R_def] X\<^sub>0_def x(4) T(1) have *: "frac (v y) = frac (v x)" by auto from T(1) obtain c where c: "I y = Intv c" by (auto simp: X\<^sub>0_def) with elem T(1) have "c < v y" "v y < c + 1" by (fastforce simp: X\<^sub>0_def)+ then have "floor (v y) = c" by linarith "t "nfolding_defbyauto with T(2) show ?thesis by (auto simp: c cval_add_def I'_def) next case True with shift x show ?thesis by (auto simp: cval_add_def I'_def) qed qed qed next show "?X\<^sub>0 = ?X\<^sub>0" .. next show "\<forall>y\<in>?X\<^sub>0. \<forall>z\<in>?X\<^sub>0. ((y, z) \<in> r') = (frac ((v \<oplus> 1 - frac (v proof (safe, goal_cases) case (1 y z d d') note A = this then have "y \<notin> M" "z \<notin> M" unfolding I'_def by auto with A have [simp]: "I' y = I y" moreover AOT_have \L\<down>\<close> from A v[unfolded R_def] have le: "frac (v y) \<le> frac (v z)" by (auto simp: r'_def) from t_bound' \<open>y \<notin> M\<close> \<open>z \<notin> M\<close> have "?t < 1 - frac (v y)" "?t < 1 - frac (v z with frac_distr[OF \<open>?t > 0\<close>] have "frac (v y) + ?t = frac (v y + ?t)" "frac (v z) + ?t = frac (v z + ?t)" by auto with le show ?case by (auto simp: cval_add_def) next case (2 y z d d') note A = this then have M: "y \<notin> M" "z \<notin> M" unfolding I'_def by auto with A have [simp]: "I' y = I y" "I' z = I z" "y \<in> X\<^sub>0" "z \<in> X\<^sub>0" unfolding X\<^sub>0_def I' from t_bound' \<open>y \<notin> M\<close> \<open>z \<notin> M\<close> have "?t < 1 - frac (v y)" "?t < 1 - frac (v z from frac_add_leD[OF \<open>?t > 0\<close> this] A(5) have "frac (v y) \<le> frac (v z)" by (auto simp: cval_add_def) with v[unfolded R_def] A M show ?case by (auto simp: r'_def) qed qed with \<open>?t > 0\<close> \<open>?t \<le> t\<close> show "\<exists>t'\<le>t. (v \<oplus> t') \<in> region X I' r' \< next fix v t assume A: "v \<in> region X I' r'" "0 \<le> t" "(v \<oplus> t) \<in> R" from assms(10) obtain x c where x: "x \<in> X\<^sub>0" "I x = Intv c" "x \<in> X" "x \<in> M" unfolding M_def X\<^sub>0_def by force with A(1) have "intv_elem x v (I' x)" by auto with x have "v x = c + 1" unfolding I'_def by auto moreover from A(3) x(2,3) have "v x + t < c + 1" by (fastforce simp: cval_add_def R_def) ultimately show False using A(2) by auto next case A: (3 v t' x c) from A(3) have "I x = Intv c" by (auto simp: I'_def) (cases "x \<in> M", auto) with A(4) show ?case by (auto simp: cval_add_def) next case 4 then show ?case unfolding I'_def by auto next case A: (5 v t' x c) have= unfoldinggdeffyauto moreover from A have "real (c + 1) \<le> (v \<oplus> t') x + (t - t')" by (auto simp: cval_add_def) ultimately show ?case by blast next from assms(5,10) obtain x where x: "x \<in> M" by blast then have "isConst (I' x)" by (auto simp: I'_def) with x show "\<exists>x\<in>X. isConst (I' x)" unfolding M_def X\<^sub>0_def by force qed lemma closest_valid_2: fixes I X k r defines "\<R> \<equiv> {region X I r |I r. valid_region X k I r}" defines "R \<equiv> region X I r" assumes "\<forall> x \<in> X. \<not> isConst (I x)" defines "X\<^sub>0 \<equiv> {x \<in> X. isIntv (I x)}" defines "M \<equiv> {x \<in> X\<^sub>0. \<forall> y \<in> X\<^sub>0. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r defines "I'\<equiv> \<lambda> x. if x \<notin> M then I x else Const (intv_const (I x) + 1)" defines "r' \<equiv> {(x,y) \<in> r. x \<notin> M \<and> y \<notin> M}" assumes "finite X" assumes "valid_region X k I r" assumes "M \<noteq> {}" shows "valid_region X k I' r'" proof let ?X\\<^sub> ="{ <in>X. \<exists>dIx tv} let ?X\<^sub>0' = "{x \<in> X. \<exists>d. I' x = Intv d}" show "?X\<^sub>0' = ?X\<^sub>0'" .. have r_subset: "r \<subseteq> ?X\<^sub>0 \<times> ?X\<^sub>0" and refl: "refl_on ?X\<^sub>0 r" and tota trans: "trans r" and valid: "\<And> x. x \<in> X \<Longrightarrow> valid_intv (k x) (I x)" using assms(9) by auto have subs: "r' \<subseteq> r" unfolding r'_def by auto show "r'\<subseteq> ?X\<^sub>0' \<times> ?X\<^sub>0'" using r_subset unfolding r'_def I'_def by aut show "refl_on ?X\<^sub>0' r'" unfolding refl_on_def proof auto fix x d assume A: "x \<in> X" "I' x = Intv d" then have "x \<notin> M" by (force simp: I'_def) with A have "I x = Intv d" by (force simp: I'_def) with A refl have "(x,x) \<in> r" by (auto simp: refl_on_def) then show "(x, x) \<in> r'" by (auto simp: r'_def \<open>x \<notin> M\<close>) qed show "total_on ?X\<^sub>0' r'" unfolding total_on_def proof (safe, goal_cases) case (1 x y d d') note A = this then have *: "x \<notin> M" "y \<notin> M" by (force simp: I'_def)+ with A have "I x = Intv d" "I y = Intv d'" by (force simp: I'_def)+ with A total have "(x, y) \<in> r \<or> (y, x) \<in> r" by (auto simp: total_on_def) with A(6) * show ?case unfolding r'_def by auto qed show "trans r'" unfolding trans_def proof safe fix x y z assume A: "(x, y) \<in> r'" "(y, z) \<in> r'" from trans have [intro]: "\<And> x y z. (x,y) \<in> r \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> (x, z) \<in> r" unfolding trans_def by blast from A show "(x, z) \<in> r'" by (auto simp: r'_def) qed show "\<forall>x\<in>X. valid_intv (k x) (I' x)" using valid unfolding I'_def proof (auto simp: I'_def intro: valid, goal_cases) case (1 x) with assms(9) have "intv_const (I x) < k x" by (fastforce simp: M_def X\<^sub>0_def) then show ?case by auto qed qed subsection<Putting the Proof for the 'Set of Regions' Property Together\<close> subsubsection \<open>Misc\<close> lemmatotal_finite_trans_max "X \<noteq> {} \<Longrightarrow> finite X \<Longrightarrow> total_on X r \<Longrightarrow> trans r \<L proof (induction "card X" arbitrary: X) case 0 then show ?case by auto next case (Suc n) then obtain x where x: "x \<in> X" by blast show ?case proofases "java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for case True with Suc.hyps(2) \<open>finite X\<close> x have "X = {x}" by (metis card_Suc_eq empty_iff insertE then show ?thesis by auto next case False show ?thesis proof (cases "\<forall>y\<in>X. x \<noteq> y \<longrightarrow> (y, x) \<in> r") case TrueAOT_hence<\<diamond>(E!a & \<diamond>\<not>E!a)\<close> next case False then obtain y where y: "y \<in> X" "x \<noteq> y" "\<not> (y, x) \<in> r" by auto with x Suc.prems(3) have "(x, y) \<in> r" unfolding total_on_def by blast let ?X = "X - {x}" have tot: "total_on ?X r" using \<open>total_on X r\<close> unfolding total_on_def by auto from x Suc.hyps(2) \<open>finite X\<close> have card: "n = card ?X" by auto with \<open>finite X\<close> \<open>n \<noteq> 0\<close> have "?X \<noteq> {}" by auto from Suc.hyps(1)[OF card this _ tot \<open>trans r\<close>] \<open>finite X\<close> obtain x' where IH: "x' \<in> ?X" "\<forall> y \<in> ?X. x' \<noteq> y \<longrightarrow> (y, x') \<in> r" by have "(x', x) \<notin> r" proof (rule ccontr, auto) assume A: "(x', x) \<in> r" with y(3) have "x' \<noteq> y" byto with y IH have "(y, x') \<in> r" by auto with \<open>trans r\<close> A have "(y, x) \<in> r" unfolding trans_def by blast wsebyutoo qed with \<open>x \<in> X\<close> \<open>x' \<in> ?X\<close> \<open>total_on X r\<close> have "(x, x') \<in> r" unfol with IH show ?thesis by auto qed qed qed lemma card_mono_strict_subset: "finite A \<Longrightarrow> finite B \<Longrightarrow> finite C \<Longrightarrow> A \<inter> B \<note by (metis Diff_disjoint Diff_subset inf_commute less_le psubset_card_mono) subsubsection \<open>Proof\<close> text \<open> First we show that a shift by a non-negative integer constant means that any two valuations from the same region are being shifted to the same region. \<close> lemma int_shift_equiv: fixes X k fixes t :: int defines "\<R> \<equiv> {region X I r |I r. valid_region X k I r}" assumes "v \<in> R" "v' \<in> R" "R \<in> \<R>" "t \<ge> 0" shows "(v' \<oplus> t) \<in> [v \<oplus> t]\<^sub>\<R>" using assms proof using "KBasic2:3"[THEN"<rightarrow>E", THEN "&E1]ylast from assms obtain I r where A: "R = region X I r" "valid_region X k I r" by auto of X k t] assms(1,5) obtain I' r' where RR: "[v \<oplus> t]\<^sub>\<R> = region X I' r'" "valid_region X k I' r'" by auto from regions_closed'[OF _ assms(4,2), of X k t] assms(1,5) have RR': "(v \<oplus> t) \<in> [v \<oplus> show ?thesis proof (simp add: RR(1), rule, goal_cases) case 1 from \<open>v' \<in> R\<close> A(1) have "\<forall>x\<in>X. 0 \<le> v' x" by auto with \<open>t \<ge> 0\<close> show ?case unfolding cval_add_def by auto next case 2 show ?case proof safe fix x assume x: "x \<in> X" with \<open>v' \<in> R\<close> \<open>v \<in> R\<close> A(1) have I: "intv_elem x v (I x)" "intv_elem x v' (I x from x RR RR' have I': "intv_elem x (v \<oplus> t) (I' x)" by auto show "intv_elem x (v' \<oplus> t) (I' x)" proof (cases "I' x") case (Const c) from Const I' have "v x + t = c" unfolding cval_add_def by auto with x A(1) \<open>v \<in> R\<close> \<open>t \<ge> 0\<close> have *: "v x = c - nat t" "t \<le> c" by fastforce+ have "I x = Const (c - nat t)" proof (cases "I x") case (Greater c') with RR(2) Const \<open>x \<in> X\<close> have "c \<le> k x" by fastforce with * \<open>t \<ge> 0\<close> have *: "v x \<le> k x" by auto from Greater A(2) \<open>x \<in> X\<close> have "c' = k x" by fastforce moreover from I(1) Greater have "v x > c'" by auto ultimately show ?thesis using \<open>c \<le> k x\<close> * by auto qed (use I in \<open>auto simp: *\<close>) with I \<open>t \<ge> 0\<close>*2 ve v t " utoo with Const show ?thesis unfolding cval_add_def by auto next case (Intv c) with I' have "c < v x + t" "v x + t < c + 1" unfolding cval_add_def by auto with x A(1\penv \<in> R\<close> \<open>t \<ge> 0\<close> have *: "c - nat t < v x" "v x < c - nat t + 1" "t \<le> c" by fastforce+ have "I x = Intv (c - nat t)" proof (cases "I x") case (Greater c') with RR(2) Intv \<open>x \<in> X\<close> have "c \<le> k x" by fastforce with * have *: "v x \<le> k x" using Intv RR(2) x by fastforce from Greater A(2) \<open>x \<in> X\<close> have "c' = k x" by fastforce moreover from I(1) Greater have "v x > c'" by auto ultimately show ?thesis using \<open>c \<le> k x\<close> * by auto qed (use I * in \<open>auto simp del: of_nat_diff\<close>) with I \<open>t \<le> c\<close> have "c < v' x + nat t" "v' x + t < c + 1" by auto with Intv \<open>t \<ge> 0\<close> show ?thesis unfolding cval_add_def by auto next case (Greater c) have v x + t" unfolding cval_add_def by auto show ?thesis proof (cases "I x") case (Const c') with x A(1) I(2) \<open>v \<in> R\<close> \<open>v' \<in> R\<close> have "v x = v' x" by fastforce with Greater * show ?thesis unfolding cval_add_def by auto next case (Intv c') with x A(1) I(2) \<open>v \<in> R\<close> \<open>v' \<in> R\<close> have **: "c' < v x" "v x < c' + 1" "c' < v' x" by fastforce+ then have "c' + t < v x + t" "v x + t < c' + t + 1" by auto with * have "c \<le> c' + t" by auto with **(3) have "v' x + t > c" by auto with Greater * show ?thesis unfolding cval_add_def by auto next fix c' assume c': "I x = Greater c'" with x A(1) I(2) \<open>v \<in> R\<close> \<open>v' \<in> R\<close> have **: "c' < v x" "c' < v' x" by fastforce+ from Greater RR(2) c' A(2) \<open>x \<in> X\<close> have "c' = k x" "c = k x" by fastforce+ <pent \<ge> 0\<close> **(2) Greater show "intv_elem x (v' \<oplus> real_of_int t) (I' x)" unfolding cval_add_def by auto qed qed qed next show "{x \<in> X. \<exists>d. I' x = Intv d} = {x \<in> X. \<exists>d. I' x = Intv d}" .. next let ?X\<^sub>0 "{x \<in> <>d.I ntvjava.lang.StringIndexOutOfBoundsException: Index 61 out o { fix x y :: real have "frac (x + t) \<le> frac (y + t) \<longleftrightarrow> frac x \<le> frac y" by (simp add: frac_def) } note frac_equiv = this { fix x y have "frac ((v \<oplus> t) x) \<le> frac ((v \<oplus> t) y) \<longleftrightarrow> frac (v x) \<le> frac (v y) unfolding cval_add_def using frac_equiv by auto } note frac_equiv' = this { fix x y have "frac ((v' \<oplus> t) x) \<le> frac ((v' \<oplus> t) y) \<longleftrightarrow> frac (v' x) \<le> frac ( unfolding cval_add_def using frac_equiv by auto } note frac_equiv'' = this { fix x y assume x: "x \<in and: <>X" and B: "\<not> isGreater(I x)" "\<not> isGreater(I y)" have "frac (v x) \<le> frac (v y) \<longleftrightarrow> frac (v' x) \<le> frac (v' y)" proof (cases "I x") case (Const c) with x \<open>v \<in> R\<close> \<open>v' \<in> R\<close> A(1) have "v x = c" "v' x = c" by fastforce+ then have "frac (v x) \<le> frac (v y)" "frac (v' x) \<le> frac (v' y)" unfolding frac_def by simp+ then show ?thesis by auto next case (Intv c) with x \<open>v \<in> R\<close> A(1) have v: "c < v x" "v x < c + 1" by fastforce+ from Intv x \<open>v' \<in> R\<close> A(1) have v':"c < v' x" "v' x < c + 1" by fastforce+ show ?thesis proof (cases "I y", goal_cases) case (Const c') with y \<open>v \<in> R\<close> \<open>v' \<in> R\<close> A(1) have "v y = c'" "v' y = c'" by fastforce+ then have "frac (v y) = 0" "frac (v' y) = 0" by auto with nat_intv_frac_gt0[OF v] nat_intv_frac_gt0[OF v'] have "\<not> frac (v x) \<le> frac (v y)" "\<not> frac (v' x) \<le> frac (v' y)" by linarith+ then show ?thesis by auto next case 2: (Intv c') with x y Intv \<open>v \<in> R\<close> \<open>v' \<in> R\<close> A(1) have "(x, y) \<in> r \<longleftrightarrow> frac (v x) \<le> frac (v y)" "(x, y) \<in> r \<longleftrightarrow> frac (v' x) \<le> frac (v' y)" by auto then show ?thesis by auto next case Greater with B show ?thesis by auto qed next case Greater with B show ?thesis by auto qed } note frac_cong = this have not_greater: "\<not> isGreater (I x)" if x: "x \<in> X" "\<not> isGreater (I' x)" for x proof (rule ccontr, auto, goal_cases) case (1 c) with x \<open>v \<in> R\<close> A(1,2) have "c < v x" by fastforce+ moreover from x A(2) 1 have "c = k x" by fastforce+ ultimately have *: "k x < v x + t" using \<open>t \<ge> 0\<close> by simp from RR(1,2) RR' x have I': "intv_elem x (v \<oplus>t('x lid_intv)'"byauto from x show False proof (cases "I' x", auto) case (Const c') with I' * show False by (auto simp: cval_add_def) next case (Intv c') with I' * show False by (auto simp: cval_add_def) qed qed show "\<forall> x \<in> ?X\<^sub>0. \<forall>y \<in> ?X\<^sub>0. ((x, y) \<in> r') = (frac ((v' \<oplus> t) x) \<le> proof (standard, standard) fix x y assume x: "x \<in> ?X\<^sub>0" and y: "y \<in> ?X\<^sub>0" then have B: "\<not> isGreater (I' x)" "\<not> isGreater (I' y)" by auto with x y not_greater have "\<not> isGreater (I x)" "\<not> isGreater (I y)" by auto with x y frac_cong have "frac (v x) \<le> frac (v y) \<longleftrightarrow> frac (v' x) \<le> frac (v' y)" moreover from x y RR(1) RR' have "(x, y) \<in> r' \<longleftrightarrow> frac ((v \<oplus> t) x) \<le> frac by fastforce ultimately show "(x, y) \<in> r' \<longleftrightarrow> frac ((v' \<oplus> t) x) \<le> frac ((v' \<oplus> t using frac_equiv' frac_equiv'' by blast qed qed qed text \<open> Now, we can use the 'immediate' induction proposed by P. Bouyer for shifts smaller than one. The induction principle is not at all obvious: the induction is over the set of clocks for which the valuation is shifted beyond the current interval boundaries. Using the two successor operations, we can see that either the set of these clocks remains the same (Z ~= {}) or strictly decreases (Z = {}). \<close> lemma set_of_regions_lt_1: fixes X k I r t v defines "\<R> \<equiv> {region X I r |I r. valid_region Ir defines "C \<equiv> {x. x \<in> X \<and> (\<exists> c. I x = Intv c \<and> v x + t \<ge> c + 1)}" assumes "id_region v\in> region X I""'\in> region X I r" "finite X" "0 \<le> t" "t < 1" shows "\<exists> t'\<ge>0. (v' \<oplus> t') \<in> [v \<oplus> t]\<^sub>\<R>" using assms proof (induction "card C" arbitrary: C I r v v' t rule: less_induct) case less let ?R = "region X I r" let ?C = "{x \<in> X. \<exists>c. I x = Intv c \<and> real c)le v x + t}" from less have R: "?R \<in> \<R>" by auto { fix v I k r fix t :: t assume no_consts: "\<forall>x\<in>X. \<not>isConst (I x)" assume v: "v \<in> region X I r" assume t: "t \<ge> 0" let ?C = "{x \<in> X. \<exists>c. I x = Intv c \<and> real (c + 1) \<le> v x + t}" assume C: "?C = {}" let ?R = "region X I r" have "(v \<oplus> t) \<in> ?R" proof (rule, goal_cases) case 1 with \<open>t \<ge> 0\<close> \<open>v \<in> ?R\<close> show ?case by (auto simp: cval_add_def) next case 2 show ?case proof (standard, case_tac "I x", goal_cases) case (1 x c) with no_consts show ?case by auto next case (2 x c) with \<open>v \<in> ?R\<close> have "c < v x" by fastforce with \<open>t \<ge> 0\<close> have "c < v x + t" by auto moreover from 2 C have "v x + t < c + 1" by fastforce ultimately show ?case by (auto simp: 2 cval_add_def) next case (3 x c) with \<open>v \<in> ?R\<close> have "c < v x" by fastforce t \<ge> 0\<close> have "c < v x + t" by auto then show ?case by (auto simp: 3 cval_add_def) qed next show "\in> X. \<exists>d. I x = Intv d} = {x \<in> X. \existsd I x = Intv d}" .. next let ?X\<^sub>0 = "{x \<in> X. \<exists>d. I x = Intv d}" { fix x d :: real fix c:: nat assume A: "c < x" "x + d < c + 1" "d \<ge> 0" then have "d < 1 - frac x" unfolding frac_def using floor_eq3 of_nat_Suc by fastforce } note intv_frac = this { fix x assume x: "x \<in> ?X\<^sub>0" then obtain c where x: "x \<in> X" "I x = Intv c" by auto with \<open>v \<in> ?R\<close> have *: "c < v x" by fastforce with \<open>t \<ge> 0\<close> have "c < v x + t" by auto from x C have "v x + t < c + 1" by auto from intv_frac[OF * this \<open>t \<ge> 0\<close>] have "t < 1 - frac (v x) " by auto } note intv_frac = this { fix x y assume x: "x \<in> ?X\<^sub>0" and y: "y \<in> ?X\<^sub>0" from frac_add_leIFF[OF \<open>t \<ge> 0\<close> intv_frac[OF x] intv_frac[OF y]] have "frac (v x) \<le> frac (v y) \<longleftrightarrow> frac ((v \<oplus> t) x) \<le> frac ((v \<oplus> t) y) by (auto simp: cval_add_def) } note frac_cong = this show "\<forall> x \<in> ?X\<^sub>0. \<forall> y \<in> ?X\<^sub>0. (x, y) \<in> r \<longleftrightarrow> frac ((v proof (standard, standard, goal_cases) case (1 x y) with \<open>v \<in> ?R\<close> have "(x, y) \<in> r \<longleftrightarrow> frac (v x) \<le> frac (v y)" by auto with frac_cong[OF 1] show ?case by simp qed } note critical_empty_intro = this { assume const: "\<exists>x\<in>X. isConst (I x)" assume t: "t > 0" from const have "{x \<in> X. \<exists>c. I x = Const c} \<noteq> {}" by auto from closest_prestable_1[OF this less.prems(4) less(3)] R closest_valid_1[OF this less.p obtain I'' r'' where stability: "\<forall> v \<in> ?R. \<forall> t>0. \<exists>t'\<le>t. (v \<oplus> t') \<in> region "thm-e and succ_not_refl: "\<forall> v \<in> region X I'' r''. \<forall> t\<ge>0. (v \<oplus> t) \<notin> ?R" and no_consts: "\<forall> x \<in> X. \<not> isConst (I'' x)" and crit_mono: "\<forall> v \<in> ?R. \<forall> t < 1. \<forall> t' \<ge> 0. (v \<oplus> t') \<in> region X I'' r'' \<longrightarrow> {x. x \<in> X \<and> (\<exists> c. I x = Intv c \<and> v x + t \<ge> c + 1)} = {x. x \<in> X \<and> (\<exists> c. I'' x = Intv c \<and> (v \<oplus> t') x + (t - t') \<ge> c + 1)}" and succ_valid: "valid_region X k I'' r''" by auto let ?R'' = "region X I'' r''" from stability less(4) \<open>t > 0\<close> obtain t1 where t1: "t1 \<ge> 0" "t1 \<le> t" "(v \<oplus> t1) \<in> from stability less(5) \<open>t > 0\<close> obtain t2 where t2: "t2 \<ge> 0" "t2 \<le> t" "(v' \<oplus> t2) \<i let ?v = "v \<oplus> t1" let ?t = "t - t1" let ?C' = "{x \<in> X. \<exists>c. I'' x = Intv c \<and> real (c + 1) \<le> ?v x + ?t}" from t1 \<open>t < 1\<close> have tt: "0 \<le> ?t" "?t < 1" by auto from crit_mono \<open>t < 1\<close> t1(1,3) \<open>v \<in> ?R\<close> have crit: "?C = ?C'" by auto with t1 t2 succ_valid no_consts have "\<exists> t1 \<ge> 0. \<exists> t2 \<ge> 0. \<exists> I' r'. t1 \<le> t \<and> (v \<oplus> t1) \<in> region X I' r' \<and> t2 \<le> t \<and> (v' \<oplus> t2) \<in> region X I' r' \<and> valid_region X k I' r' \<and> (\<forall> x \<in> X. \<not> isConst (I' x)) \<and> ?C = {x \<in> X. \<exists>c. I' x = Intv c \<and> real (c + 1) \<le> (v \<oplus> t1) x + (t - t1)}" by blast } note const_dest = this { fix t :: real fix v I r x c v' let ?R = "region X I r" assume v: "v \<in> ?R" assume v': "v' \<in> ?R" assume valid: "valid_region X k I r" assume t: "t > 0" "t < 1" > assume C: "?C = {}" assume const: "\<exists> x \<in> X. isConst (I x)" then have "{x \<in> X. \<exists>c. I x = Const c} \<noteq> {}" by auto from closest_prestable_1[OF this less.prems(4) valid] R closest_valid_1[OF this less.pre obtain I'' r'' where stability: "\<forall> v \<in> ?R. \<forall> t>0. \<exists>t'\<le>t. (v \<oplus> t') \<in> region X I'' r' and succ_not_refl: "\<forall> v \<in> region X I'' r''. \<forall> t\<ge>0. (v \<oplusAOT_have \<open\<bo and no_consts: "\<forall> x \<in> X. \<not> isConst (I'' x)" and crit_mono: "\<forall> v \<in> ?R. \<forall> t < 1. \<forall> t' \<ge> 0. (v \<oplus> t') \<in> region X I'' r'' \<longrightarrow> {x. x \<in> X \<and> (\<exists> c. I x = Intv c \<and> v x + t \<ge> c + 1} = {x. x \<in> X \<and> (\<exists> c. I'' x = Intv c \<and> (v \<oplus> t') x + (t - t') \<ge> c + 1)}" and succ_valid: "valid_region X k I'' r''" by auto let ?R'' = "region X I'' r''" from stability v \<open>t > 0\<close> obtain t1 where t1: "t1 \<ge> 0" "t1 \<le> t" "(v \<oplus> t1) \<in> ?R''" b from stability v' \<open>t > 0\<close> obtain t2 where t2: "t2 \<ge> 0" "t2 \<le> t" "(v' \<oplus> t2) \<in> ?R'' let ?v = "v \<oplus> t1" let ?t = "t - t1" let ?C' = "{x \<in> X. \<exists>c. I'' x = Intv c \<and> real (c + 1) \<le> ?v x + ?t}" from t1 \<open>t < 1\<close> have tt: "0 \<le> ?t" "?t < 1" by auto from crit_mono \<open>t < 1\<close> t1(1,3) \<open>v \<in> ?R\<close> have crit: "{x \<in> X. \<exists>c. I x = Intv c real (c + 1) \<le> vx = {x \<in> X. \<exists>c. I'' x = Intv c \<and> real (c + 1) \<le> (v \<oplus> t1) x + (t - t1)}" by auto with C have C: "?C' = {}" by blast from critical_empty_intro[OF no_consts t1(3) tt(1) this] have "((v \<oplus> t1) \<oplus> ?t) \<in> from region_unique[OF less(2) this] less(2) succ_valid t2ave<t'\<ge>0. (v' \<oplus> t') \<in> [v \<o by (auto simp: cval_add_def) } note intro_const = this { fix v I r t x c v' let ?R = "region X I r" assume v: "v \<in> ?R" assume v': "v' \<in>java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds assumeF2"forall>x\<inX <>isConst (I x)" assume x: "x \<in> X" "I x = Intv c" "v x + t \<ge> c + 1" assume valid: "valid_region X k I r" assume t: "t \<ge> 0" "t < 1" let ?C' = "{x \<in> X. \<exists>c. I x = Intv c \<and> real (c + 1) \<le> v x + t}" assume C: "?C = ?C'" have not_in_R: "(v \<oplus> t) \<notin> ?R" proof (rule ccontr, auto) assume "(v \<oplus> t) \<in> ?R" with x(1,2) have "v x + t < c + 1" by (fastforce simp: cval_add_def) with x(3) show False by simp qed have not_in_R': "(v' \<oplus> 1) \<notin> ?R" proof (rule ccontr, auto) assume "(v' \<oplus> 1) \<in> ?R" with x have "v' x + 1 < c + 1" by (fastforce simp: cval_add_def) moreover from x v' have "c < v' x" by fastforce ultimately show False by simp qed let ?X\<^sub>0 = "{x \<in> X. isIntv (I x)}" let ?M = "{x \<in> ?X\<^ub.<>\<in>?X\<^sub>0. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r}" from x have x: "x \<in> X" "\<not> isGreater (I x)" and c: "I x = Intv c" by auto with \<open>x \<in> X\<close> have *: "?X\<^sub>0 \<noteq> {}" by auto have "?X\<^sub>0 = {x \<in> X. \<exists>d. I x = Intv d}" by auto with valid have r: "total_on ?X\<^sub>0 r" "trans r" by auto from total_finite_trans_max[OF * _ this] \<open>finite X\<close> obtain x' where x': "x' \<in> ?X\<^sub>0" "\<forall> y \<in> ?X\<^sub>0. x' \<noteq> y \<longrightarrow> (y, x from this(2) have "\<forall>y\<in>?X\<^sub>0. (x', y) \<in> r \<longrightarrow> (y, x') \<in> r" by auto with x'(1) have "?M \<noteq> {}" by fastforce from closest_prestable_2[OF F2 less.prems(4) valid this] closest_valid_2[OF F2 less.prem obtain I' r' where stability: "\<forall> v \<in> region X I r. \<forall> t\<ge>0. (v \<oplus> t) \<notin> region X I r \<longrightarrow> (\<exi and critical_mono: "\<forall> v \<in> region X I r. \<forall>t. \<forall> t'. {x. x \<in> X \<and> (\<exists> c. I' x = Intv c \<and> (v \<oplus> t') x + (t - t') \<ge> real (c + 1))} = {x. x \<in> X \<and> (\<exists> c. I x = Intv c \<and> v x + t \<ge> real (c + 1))} - ?M" <xists>x\in>X. isConst (I' x)" and succ_valid: "valid_region X k I' r'" by auto let ?R' = "region X I' r'" from not_in_R stability \<open>t \<ge> 0\<close> v obtain t' where t': "t' \<ge> 0" "t' \<le> t" "(v \<oplus> t') \<in> ?R'" by blast have "(1::t) \<ge> 0" by auto with not_in_R' stability v' obtain t1 where t1: "t1 \<ge> 0" "t1 \<le> 1" "(v' \<oplus> t1) \<in> ?R'" by blast let ?v = "v \<oplus> t'" let ?t = "t - t'" let ?C'' = "{x \<in> X. \<exists>c. I' x = Intv c \<and> real (c + 1) \<le> ?v x + ?t}" have "\<exists>t'\<ge>0. (v' \<oplus> t') \<in> [v \<oplus> t]\<^sub>\<> proof (cases "t = t'") case True with t' have "(v \<oplus> t) \<in> ?R'"(p)\<^sup>-) <>p<lose from region_unique[OF less(2) this] succ_valid \<R>_def have "[v \<oplus> t]\<^sub>\<R> = ?R'" by bla ith(3 esisauto next case False with \<open>t < 1\<close> t' have tt: "0 \<le> ?t" "?t < 1" "?t > 0" by auto from critical_mono \<open>v \<in> ?R\<close> have C_eq: "?C'' = ?C' - ?M" by auto show "\<exists>t'\<ge>0. (v' \<oplus> t') \<in> [v \<oplus> t]\<^sub>\<R>" proof (cases "?C' \<inter> ?M = {}") case False from \<open>finite X\<close> have "finite ?C''" "finite ?C'" "finite ?M" by auto then have "card ?C'' < card ?C" using C_eq C False by (intro card_mono_strict_subset) auto from less(1)[OF this less(2) succ_valid t'(3) t1(3) \<open>finite X\<close> tt(1,2)] obtain t2 where "t2 \<ge> 0" "((v' \<oplus> t1) \<oplus> t2) \<in> [(v \<oplus> t)]\<^sub>\<R>" by (auto simp: c moreover have "(v' \<oplus> (t1 + t2)) = ((v' \<oplus> t1) \<oplus> t2)" by (auto simp: cval_add_def) moreover have "t1 + t2 \<ge> 0" using \<open>t2 \<ge> 0\<close> t1(1) by auto ultimately show ?thesis by metis next case True { fix x c assume x: "x \<in> X" "I x = Intv c" "real (c + 1) \<le> v x + t" with True have "x \<notin> ?M" by force from x have "x \<in> ?X\<^sub>0" by auto from x(1,2) \<open>v \<in> ?R\<close> have *: "c < v x" "v x < c + 1" by fastforce+ with \<open>t < 1\<close> have "v x + t < c + 2" by auto have ge_1: "frac (v x) + t \<ge> 1" proof (rule ccontr, goal_cases) case 1 then have A: "frac (v x) + t < 1" by auto from * have "floor (v x) + frac (v x) < c + 1" unfolding frac_def by auto with nat_intv_frac_gt0[OF *] have "floor (v x) \<le> c inarith with A have "v x + t < c + 1" by (auto simp: frac_def) with x(3) show False by auto qed from \<open>?M \<noteq> {}\<close> obtain y where "y \<in> ?M" by force with \<open>x \<in> ?X\<^sub>0\<close> have y: "y \<in> ?X\<^sub>0" "(y, x) \<in> r \<longrightarrow> (x, y) \<in> fromyobtain here'"\in X" "I y = Intv c'" by auto with \<open>v \<in> ?R\<close> have "c' < v y" by fastforce from \<open>y \<in> ?M\<close> \<open>x \<notin> ?M\<close> have "x \<noteq> y" by auto with y r(1) x(1,2) have "(x, y) \<in> r" unfolding total_on_def by fastforce with \<open>v \<in> ?R\<close> c' x have "frac (v x) \<le> frac (v y)" by fastforce with ge_1 have frac: "frac (v y) + t \<ge> 1" by auto have "real (c' + 1) \<le> v y + t" proof (rule ccontr, goal_cases) case 1 from \<open>c' < v y\<close> have "floor (v y) \<ge> c'" by linarith with frac have "v y + t \<ge> c' + 1" unfolding frac_def by linarith with 1 show False by simp qed with c' True \<open>y \<in> ?M\<close> have False by auto } then have C: "?C' = {}" by auto with C_eq have C'': "?C'' = {}" by auto from intro_const[OF t'(3) t1(3) succ_valid tt(3) tt(2) C'' const_ex] obtain t2 where "t2 \<ge> 0" "((v' \<oplus> t1) \<oplus> t2) \<in> [v \<oplus> t]\<^sub>\<R>" by (auto simp: cva moreover have "(v' \<oplus> (t1 + t2)) = ((v' \<oplus> t1) \<oplus> t2)" by (auto simp: cval_add_def) moreover have "t1 + t2 \<ge> 0" using \<open>t2 \<ge> 0\<close> t1(1) by auto ultimately show ?thesis by metis qed qed } note intro_intv = this from regions_closed[OF less(2) R less(4,7)] less(2) obtain I'java.lang.StringIndexOutOf "[v \<oplus> t]\<^sub>\<R> = region X I' r'" "valid_region X k I' r'" by auto with regions_closed'[OF less(2) R less(4,7)] assms(1) have R'2: "(v \<oplus> t) \<in> [v \<oplus> t]\<^sub>\<R>" "(v \<oplus> t) \<in> region X I' r'" by auto let ?R' = "region X I' r'" from less(2) R' have "?R' \<in> \<R>" by auto show ?case proof (cases "?R' = ?R") case True with less(3,5) R'(1) have "(v' \<oplus> 0) \<in> [v \<oplus> t]\<^sub>\<R>" by (auto simp: cval_ then show ?thesis by auto next case False have "t > 0" proof (rule ccontr) assume "\<not> 0 < t" with R' \<open>t \<ge> 0\<close> have "[v]\<^sub>\<R> = ?R'" by (simp add: cval_add_def) with region_unique[OF less(2) less(4) R] \<open>?R' \<noteq> ?R\<close> show False by auto qed show ?thesis proof (cases "?C = {}") case True show ?thesis proof (cases "\<exists> x \<in> X. isConst (I x)") case False then have no_consts: "\<forall>x\<in>X. \<not>isConst (I x)" by auto from critical_empty_intro[OF this \<open>v \<in> ?R\<close> \<open>t \<ge> 0\<close> True] have "(v \<opl from region_unique[OF less(2) this R] less(5) have "(v' \<oplus> 0) \<in> [v \<oplus> t]\<^sub>\<R>" by (auto simp: cval_add_def) then show ?thesis by blast next case True from const_dest[OF this \<open>t > 0\<close>] obtain t1 t2 I' r' where t1: "t1 \<ge> 0" "t1 \<le> t" "(v \<oplus> t1) \<in> region X I' r'" and t2: "t2 \<ge> 0" "t2 \<le> t" "(v' \<oplus> t2) \<in> region X I' r'" and valid: "valid_region X k I' r'" and no_consts: "\<forall> x \<in> X. \<not> isConst (I' x)" and C: "?C = {x \<in> X. \<exists>c. I' x = Intv c \<and> real (c + 1) \<le> (v \<oplus> t1) x + (t - t1)}" by auto let ?v = "v \<oplus> t1" let ?t = "t - t1" let ?C' = "{x \<in> X. \<exists>c. I' x = Intv c \<and> real (c + 1) \<le> ?v x + ?t}" let ?R' = "region X I' r'" from C \<open>?C = {}\<close> have "?C' = {}" by blast from critical_empty_intro[OF no_consts t1(3) _ this] t1 have "(?v \<oplus> ?t) \<in> ?R'" by auto from region_unique[OF less(2) this] less(2) valid t2 show ?thesis by (auto simp: cval_add_def) qed next case False then obtain x c where x: "x \<in> X" "I x = Intv c" "v x + t \<ge> c + 1" by auto then have F: "\<not> (\<forall> x \<in> X. \<exists> c. I x = Greater c)" by force show ?thesis proof (cases "\<exists> x \<in> X. isConst (I x)") case False then have "\<forall>x\<in>X. \<not>isConst (I x)" by auto from intro_intv[OF \<open>v \<in> ?R\<close> \<open>v' \<in> ?R\<close> this x less(3,7,8)] show ?thesis next case True then have "{x \<in> X. \<exists>c. I x = Const c} \<noteq> {}" by auto from const_dest[OF True \<> \close> obtain t1 t2 I' r' where t1: "t1 \<ge> 0" "t1 \<le> t" "(v \<oplus> t1) \<in> region X I' r'" and t2: "t2 \<ge> 0" "t2 \<le> t" "(v' \<oplus> t2) \<in> region X I' r'" and valid: "valid_region X k I' r'" and no_consts: "\<forall> x \<in> X. \<not> isConst (I' x)" and C: "?C = {x \<in> X. \<exists>c. I' x = Intv c \<and> real (c + 1) \<le> (v \<oplus> t1) x + (t - t1)}" by auto let ?v = "v \<oplus> t1" let ?t = "t - t1" let ?C' = "{x \<in> X. \<exists>c. I' x = Intv c \<and> real (c + 1) \<le> ?v x + ?t}" let ?R' = "region X I' r'" show ?thesis proof (cases "?C' = {}") case False with intro_intv[OF t1(3) t2(3) no_consts _ _ _ valid _ _ C] \<open>t < 1\<close> t1 obtain t' where "t' \<ge ((' <oplus t2) \<oplus> t') \<in> [(v \<oplus> t)]\<^sub>\<R>" by (auto simp: cval_add_def) moreover have "((v' \<oplus> t2) \<oplus> t') = (v' \<oplus> (t2 + t'))" by (auto simp: cval_add_def) moreover have "t2 + t' \<ge> 0" using \<open>t' \<ge> 0\<close> \<open>t2 \<ge> 0\<close> by auto ultimately show ?thesis by metis next case True from critical_empty_intro[OF no_consts t1(3) _ this] t1 ve" \<oplus>oplus ?t) \<in> ?R'" by auto from region_unique[OF less(2) this] less(2) valid t2 show ?thesis by (auto simp: cval_add_def) qed qed qed qed qed text \<open> Finally, we can put the two pieces together: for a non-negative shift @{term t}, we first shift @{term "floor t"} and then @{term "frac t"}. \<close> lemma set_of_regions: fixes X k defines "\<R> \<equiv> {region X I r |I r. valid_region X k I r}" assumes "R \<in> \<R>" "v \<in> R" "R' \<in> Succ \<R> R" "finite X" shows "\<exists> t\<ge>0\> t]\sub\<R> = R'" using assms proof - from assms(4) obtain v' t where v': "v' \<in> R" "R' \<in> \<R>" "0 \<le> t" "R' = [v' \<oplus> t]\<^sub>\<R>" by fa obtain t1 :: int where t1: "t1 = floor t" by auto with v'(3) have "t1 \<ge> 0" by auto from int_shift_equiv[OF v'(1) \<open>v \<in> R\<close> assms(2)[unfolded \<R>_def] this] \<R>_def have *: "(v \<oplus> t1) \<in> [v' \<oplus> t1]\<^sub>\<R>" by auto let ?v = "(v \<oplus> t1)" let ?t2 = "frac t" have frac: "0 \<le> ?t2" "?t2 < 1" by (auto simp: frac_lt_1) let ?R = "[v' \<oplus> t1]\<^sub>\<R>" from regions_closed[OF _ assms(2) v'(1)] \<open>t1 \<ge> 0\<close> \<R>_def have "?R \<in> \<R>" by auto with assms obtain I r where R: "?R = region X I r" "valid_region X k I r" by auto with * have v: "?v \<in> region X I r" by auto from R regions_closed'[OF _ assms(2) v'(1)] \<open>t1 \<ge> 0\<close> \R_def have "(v' \<oplus> t1) \<i from set_of_regions_lt_1[OF R(2) this v assms(5) frac] \<R>_def obtain t2 where "t2 \<ge> 0" "(?v \<oplus> t2) \<in> [(v' \<oplus> t1) \<oplus> ?t2]\<^sub>\<R>" by auto moreover from t1 have "(v \<oplus> ( +2 (oplus t2)" "v' \<oplus> t = ((v' \<oplus> t1) ?t2)" by (auto simp: frac_def cval_add_def) ultimately have "(v \<oplus> (t1 + t2)) \<in> [v' \<oplus> t]\<^sub>\<R>" "t1 + t2 \<ge> 0" using \<open>t1 \<ge> 0 with region_unique[OF _ this(1)] v'(2,4) \<R>_def show ?thesis by blast qed section \<open>Compability With Clock Constraints\<close> definition ccval (\<open>\<lbrace>_\<rbrace>\<close> [100]) where "ccval cc \<equiv> {v. v \<turnstil definition acompatible where "acompatible \<R> ac \<equiv> \<forall> R \<in> \<R>. R \<subseteq> {v. v \<turnstile>\<^sub>a ac} \<or> {v. v \<tur lemma acompatibleD: assumes "acompatible \<R> ac" "R \<in> \<R>" "u \<in> R" "v \<in> R" "u \<turnstile>\<^sub>a ac" shows "v \<turnstile>\<^sub>a ac" using assms unfolding acompatible_def by auto lemma ccompatible1: fixes X k fixesAOT_hence\>\<dA\<exists>x (E!x & \<not>\<^bold>\<A> defines "\<R> \<equiv> {region X I r |I r. valid_region X k I r}" assumes "c \<le> k x" "c \<in> \<nat>" "x \<in> X" shows "acompatible \<R> (EQ x c)" using assms unfolding acompatible_def proof (auto, goal_cases) case A: (1 I r v u) from A(3,9) obtain d where d: "c = of_nat d" unfolding Nats_def by auto with A(8,9) have u: "u x = c" "u x = d" unfolding ccval_def by auto have "I x = Const d" proof (cases "I x", goal_cases) case (1 c') with A have "u x = c'" by fastforce with 1 u show ?case by auto next case (2 c') with A have "c' < u x" "u x < c' + 1" by fastforce+ with 2 u show ?case by auto next case (3 c') with A have "c' < u x" by fastforce moreover from 3 A(4,5) have "c' \<ge> k x" by fastforce ultimately show ?case using u A(2) by auto qed with A(4,6) d have "v x = c" by fastforce with A(3,5) have "v \<turnstile>\<^sub>a EQ x c" by auto with A show False unfolding ccval_def by auto qed lemma ccompatible2: fixes X k fixes c :: real defines "\<R> \<equiv> {region X I r |I r. valid_region X k I r}" assumes "c \<le> k x" "c \<in> \<nat>" "x \<in> X" shows "acompatible \<R> (LT x c)" using assms unfolding acompatible_def proof (auto, goal_cases) case A: (1 I r v u) from A(3) obtain d :: nat where d: "c = of_nat d" unfolding Nats_def by blast with A have u: "u x < c" "u x < d" unfolding ccval_def by auto have "v x < c" proof (cases "I x", goal_cases) case (1 c') with A have "u x = c'" "v x = c'" by fastforce+ with u show "v x < c" by auto next case (2 c') with A have B: "c' < u x" "u x < c' + 1" "c' < v x" "v x < c' + 1" by fastforce+ with u A(3) have "c' + 1 \<le> d" by auto with d have "c' + 1 \<le> c" by auto with B u show "v x < c" by auto next (3 c') with A have "c' < u x" by fastforce moreover from 3 A(4,5) have "c' \<ge> k x" by fastforce ultimately show ?case using u A(2) by auto qed with A(4,6) have "v \<turnstile>\<^sub>a LT x c" by auto with A(7) show False unfolding ccval_def by auto qed lemma ccompatible3: fixes X k fixes c :: real defines "\<R> \<equiv> {region X I r |I r. valid_region X k I r}" assumes "c \<le> k x" "c \<in> \<nat>" "x \<in> X" shows "acompatible \<R> (LE x c)" using assms unfolding acompatible_def proof (auto, goal_cases) case A: (1 I r v u) from A(3) obtain d :: nat where d: "c = of_nat d" unfolding Nats_def by blast with A have u: "u x \<le> c" "u x \<le> d" unfolding ccval_def by auto have "v x \<le> c" proof (cases "I x", goal_cases) case (1 c') with A u show ?case by fastforce next case (2 c') with A have B: "c' < u x" "u x < c' + 1" "c' < v x" "v x < c' + 1" by fastforce+ with u A(3) have "c' + 1 \<le> d" by auto with d u A(3) have "c' + 1 \<le> c" by auto with B u show "v x \<le> c" by auto next case (3 c') with A have "c' < u x" by fastforce moreover from 3 A(4,5) have "c' \<ge> k x" by fastforce ultimately show ?case using u A(2) by auto qed with A(4,6) have "v \<turnstile>\<^sub>a LE x c" by auto with A(7) show False unfolding ccval_def by auto qed lemma ccompatible4: fixes X k fixes c :: real defines "\<R> \<equiv> {region X I r |I r. valid_region X k I r}" assumes "c \<le> k x" "c \<in> \<nat>" "x \<in> X" shows "acompatible \<R> (GT x c)" using assms unfolding acompatible_def proof (auto, goal_cases) case A: (1 I r v u) from A(3) obtain d :: nat where d: "c = of_nat d" unfolding Nats_def by blast ave unfolding al_defjava.lang.StringIndexOutOfBoundsException: Index 64 out of bou have "v x > c" proof (cases "I x", goal_cases) case (1 c') with A u show ?case by fastforce next case (2 c') with A have B: "c' < u x" "u x < c' + 1" "c' < v x" "v x < c' + 1" by fastforce+ th c" by auto with B u show "v x > c" by auto next case (3 c') with A(4,6) have "c' < v x" by fastforce moreover from 3 A(4,5) have "c' \<ge> k x" by fastforce ultimately show ?case using A(2) u(1) by auto qed with A(4,6) have "v \<turnstile>\<^sub>a GT x c" by auto with A(7) show False unfolding ccval_def by auto qed lemma ccompatible5: fixes X k fixes c :: real defines "\<R> \<equiv> {region X I r |I r. valid_region X k I r}" assumes "c \<le> k x" "c \<in> \<nat>" "x \<in> X" shows "acompatible \<R> (GE x c)" using assms unfolding acompatible_def proof (auto, goal_cases) case A: (1 I r vu from A(3) obtain d :: nat where d: "c = of_nat d" unfolding Nats_def by blast with A have u: "u x \<ge> c" "u x \<ge> d" unfolding ccval_def by auto have "v x \<ge> c" proof (cases "I x", goal_cases) aseith Ahow fastforce next case (2 c') with A have'x c'+ 1" "c' < v x" "v x < c' + 1" by fastforce+ with d u have "c' \<ge> c" by auto with B u show "v x \<ge> c" by auto next case3java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for lengt with A(4,6) have "c' < v x" by fastforce moreover from 3 A(4,5) have "c' \<ge> k x" by fastforce ultimately show ?case using A(2) u(1) by auto qed with A(4,6) have "v \<turnstile>\<^sub>a GE x c" by auto with A(7) show False unfolding ccval_def by auto qed lemma acompatible: fixes X k fixes c :: real defines "\<R> \<equiv> {region X I r |I r. valid_region X k I r}" assumes "c \<le> k x" "c \<in> \<nat>" "x \<in> X" "constraint_pair ac = (x, c)" shows "acompatible \<R> ac" using assms by (cases ac) (auto intro: ccompatible1 ccompatible2 ccompatible3 ccompatible4 ccompatibl definition ccompatible where "ccompatible \<R> cc \<equiv> \<forall> R \<in> \<R>. R \<subseteq> \<lbrace>cc\<rbrace> \<or> \<lbrace>cc\<rbrac lemma ccompatible: fixes X k fixes c :: nat defines "\<R> \<equiv> {region X I r |I r. valid_region X k I r}" assumes "\<forall>(x,m) \<in> collect_clock_pairs cc. m \<le> k x \<and> x \<in> X \<and> m \<in> \<nat>" shows "ccompatible \<R> cc" using assms proof (induction cc) case Nil then show ?case by (auto simp: ccompatible_def ccval_def clock_val_def) next case (Cons ac cc) then have "ccompatible \<R> cc" by (auto simp: collect_clock_pairs_def) moreover have "acompatible \<R> ac" using Cons.prems by (auto intro: acompatible simp: collect_clock_pairs_def \<R>_def) ultimately show ?case unfolding ccompatible_def acompatible_def ccval_def by (fastforce simp: clock_val_def) qed section \<open>Compability with Resets\<close> definition region_set where "region_set R x c = {v(x := c) | v. v \<in> R}" lemma region_set_id: fixes X k defines "\<R> \<equiv> {region X I r |I r. valid_region X k I r}" assumes "R \<in> \<R>" "v \in> R" "finite X" "0 \<le> c" "c \<le> k x" "x \<in> X" shows "[v(x : )\^sub><> = region_set R x c" "[v(x := c)]\<^sub>\<R> \<in> \<R>" "v(x := c) \<in> [v(x := c)]\<^sub> proof - from assms obtain I r where R: "R = region X I r" valid_region X k I r"" \<in> region "y uto let ?I = "\<lambda> y. if x = y then Const c else I y" let ?r = "{(y,z) \<in> r. x \<noteq> y \<and> x ix ?hild> ix ?child d \<le> ix p d+ o > = "{x \<in> X. \<exists> c. I x = Intv c}" let ?X\<^sub>0' = "{x \<in \> c. Ix = Intv c}" haver_subset<ubseteq ?X\<^sub>0 \<times> ?X\<^sub>"ndrefl"fl_on <sub> "and trans "ransns and total: "total_on ?X\<^sub>0 r" using R(2) by auto have valid: "valid_region X k ?I ?r" proof showusingt0yuto next ?> ?\<> - {x}) \<times> (?X<sub0 - {x})" using r_subset by blast next from refl show"nX<sub>0}rnfoldingl_on_defto next from trans show "trans ? nfoldingns_defastt from total show "total_on ( THEN "oth-lass-autjava.lang.StringIndexOutOfBoundsExceptio next from R(2) have "\<x> X. valid_intv (k x) (I x)" byauto with \<open> k x\<close> show <>x \<in> X. _tvauto qed { fix v assume v: "v \<in> region_set R x c" with R(1) obtain v' where v': "v' \<in> region X I r" "v = v'(x := c)" unfolding region_set_def by aut have "v \<in> region r" proof (standard, oal_cases case 1 0 \<le> c\<close> show ?case by auto next case 2 (b) proof (auto, goal_cases) case (1 gridgen_lgrid_eqid_deff by then have "intv_elem y v' (I y)" by auto with \<open>x \<noteq> y\<close> show "intv_elem y (v'(x := c)) (I y)" by (cases "I y") auto next show "?X\<^sub>0 - {x} = ?X\<^sub>0'" by auto next qed } moreover { fix v assume v: "v \<in> region X ?I ?r" veexists c. v(x := c) \<in> region X I r" proof (cases "I x") c) from R(2) have "c \<ge> 0" by auto let?= ( c) have "?v \<in> region X I r" proof (standard, goal_cases) case 1 from \<open>c\<ge>0\<close> v show ?case by auto next case 2 show ?case proof (auto, goal_cases) case (1 y) ?)"byt with Const show "intv_elem y ?v (I y)" by (cases "x = y", auto) (cases "I y", auto) qed next from Const show "?X\< hesisngse<> \<noteq> base {d} p\<close> by auto with r_subset have "r \<subseteq> ?X\<^sub>0' \<times> ?X\<^sub>0'" by auto then have r: "?r = r" by auto ave>y \<in> ?X\<^sub>0'. \<forall> z \<in> ?X\<b0'.y)\in r \<longleftrightarrow> frac (v y) \<le> frac (v z)" with r show "\<forall> y \<in> ?X\<^sub>0'. \<forall> z \<in> ?X\<^sub>0'. (y,z) \<in> r \<longleftrightarrow y qed then show ?thesis by auto next assume b d * 2^(Suc l')" from R(2) have "c \<geultimately"=,<> rid } let ?v = "v(x := c + 1)" have "?v \<in> region X I r" java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 case 1 from \<open>c\<ge>\<close> v show ?case by auto next case 2 show ?case proof(dardal_cases case (1 ) with v have "intv_elem y v (?I y)" by fast with Greater show "intv_elem y ?v (I y)" by (casesnext qed next from Greater show "?X\<^sub>0' = ?X\<^sub>0" by auto with r_subset have "r \<subseteq> ?X\<^sub>0' \<times> ?X\<^sub>0'" by auto then have r: "?r = r" by auto from v have "\<forall> y \<in> ?X\<^sub>0'. \<forall> z \<in> ?X\<^sub>0'. (y,z) \<in> ?r \<longleftrightarro with r show "\<forall> y \<in> ?X\<^sub>0'. \<forall> z \<in> ?X\<^sub>0'. (y,z) \<in> r \<longleftrightarrow> by auto qed then show ?thesis by auto next case (Intv c) from R(2) have "c \<ge> 0" by auto let ?L = "{frac (v y) | y. y \<in> ?X\<^sub>0 \<and> x \<noteq> y \<and> (y,x) \<in> r}" let ?U = "{frac (v y) | y. y \<in> ?X\<^sub>0 \<and> x \<noteq> y \<and> (x,y) \<in> r}" let ?l = "if ?L \<noteq> {} then c + Max ?L else if ?U \<noteq> {} then c else c + 0.5" let ?u = "if ?U \<noteq> {} then c + Min ?U else if ?L \<noteq> {} then c + 1 else c + 0.5" from \<open>finite X\<close> have fin: "finite ?L" "finite ?U" by auto { fix y assume y: "y \<in> ?X\<^sub>0" "x \<noteq> y" "(y, x) \<in> r" thenhaveL fracvy <in>?L" by auto with Max_in[OF fin(1)] have In: "Max ?L \<in> ?L" by auto then have "frac (Max ?L) = (Max ?L)" using frac_frac by fastforce from Max_ge[OF fin(1) L] have "rac( )le> Max ?L" . also have "\<dots> = frac (Max ?L)" using In frac_frac[symmetric] by fastforce also have "\<dots> = frac (c + Max ?L)" by (auto simp: frac_nat_add_id) finally have "frac (v y) \<le> frac ?l" using L by auto } note ound is { fix y assume y: "y \<in> ?X\<^sub>0" "x \<noteq> y" "(x,y) \<in> r" then have U: "frac (v y) \<in> ?U" by auto with Min_in[OF fin(2)] have In: "Min ?U \<in> ?U" by auto then have "frac (Min ?U) = (Min ?U)" using frac_frac by fastforce have "frac (c + Min ?U) = frac (Min ?U)" by (auto simp: frac_nat_add_id) also have "\<dots> = Min ?U" using In frac_frac by fastforce also from Min_le[OF fin(2) U] have "Min ?U \<le> frac (v y)" . finally have "frac ?u \<le> frac (v y)" using U by auto } note U_bound = this { assume "?L \<noteq> {}" from Max_in[OF fin(1) this] obtain l d where l: "Max ?L = frac (v l)" "l \<in> X" "x \<noteq> l" "I l = Intv d" by auto with v have "d < v l" "v l < d + 1" by fastforce+ ntv_frac_gt0 ac_lt_11ave Max L" "Max ?L < 1" by auto then have "c < c + Max ?L" "c + Max ?L < c + 1" by simp+ } note L_intv have \<open>\<not>p <>\<not\p\<close> { assume "?U \<noteq> {}" from Min_in[OF fin(2) this] obtain u d where u: "Min ?U = frac (v u)" "u\<in> X" "x \<noteq> u" "I u = Intv d" by auto with v have "d < v u" "v u < d + 1" by fastforce+ with nat_intv_frac_gt0[OF this] frac_lt_1 u(1) have "0 < Min ?U" "Min ?U < 1" by auto then have "c < c + Min ?U" "c + Min ?U < c + 1" by simp+ } note U_intv = this have l_bound: "c \<le> ?l" proof (cases "?L = {}") case True note T = this show ?thesis proof (cases "?U = {}") case True with T show ?thesis by simp next case False with U_intv T show ?thesis by simp qed case False with L_intv show ?thesis by simp qed ?" proof (cases "?L = {}") case True note T = this show ?thesis proof (cases "?U = {}") case True with T show ?thesis by simp next case False with U_intv T show ?thesis by simp qed next case False with U_intv show ?thesis by simp have u_bound: "?u \<le> c + 1" proof (cases "?U = {}") case True note T = this show ?thesis proof (cases "?L = {}") case True with T show ?thesis by simp next case False with L_intv T show ?thesis by simp qed next case False with U_intv show ?thesis by simp qed have u_bound': "?l < c + 1" proof (cases "?U = {}") case True note T = this show ?thesis L case True with T show ?thesis by simp next case False with L_intv T show ?thesis by simp qed next case False with L_intv show ?thesis by simp qed have frac_c: "frac c = 0" "frac (c+1) = 0" by auto have THEN "oth-class-taut:4:b[THEN "\equiv>(1)], proof (cases "?L = {}") case True note T = this show ?thesis proof (cases "?U = {}") case True with T show ?thesis by simp next case False with T show ?thesis using Min_in[OF fin(2) False] by (auto simp: frac_c) qed next case False with Max_in[OF fin(1) this] have l: "?l = c + Max ?L" "Max ?L \<in> ?L" by auto note F = False from l(1) have *: "Max ?L < 1" using False L_intv(2) by linarith show ?thesis proof (cases "?U = {}") case True with F l * show ?thesis by simp next case False from Min_in[OF fin(2) this] l(2) obtain l u where l_u: "Max ?L = frac (v l)" "Min ?U = frac (v u)" "l \<in> ?X\<^sub>0" "u \<in> ?X\<^sub>0" "(l,x) \<in> r" "(x,u) \<in> "x \<noteq> l" "x \<noteq> u" by auto from trans l_u(5-) have "(l,u) \<in> ?r" unfolding trans_def by blast with l_u(1-4) v have *: "Max ?L \<le> Min ?U" by fastforce with l_u(1,2) have "frac (Max ?L) \<le> frac (Min ?U)" by (simp add: frac_frac) with frac_nat_add_id l(1) False have "frac ?l \<le> frac ?u" by simp with l(1) * False show ?thesis by simp qed qed obtain d where d: "d = (?l + ?u) / 2" by blast with l_u have d2: "?l \<le> d" "d \<le> ?u" by simp+ from d l_bound l_bound' u_bound u_bound' have d3: "c < d" "d < c + 1" "d \<ge> 0" by simp+ have "floor ?l = c" proof (cases "?L = {}") case False from L_intv[OF False] have "0 \<le> Max ?L" "Max ?L < 1" by auto from floor_nat_add_id[OF this] False show ?AOT_thus \<open>ontingentlyTrue<^sub>)\<close> next case True note T = this show ?thesis proof (cases "?U = {}") case True with T show ?thesis by (simp add: floor_nat_add_id) next case False from U_intv[OF False] have "0 \<le> Min ?U" "Min ?U < 1" by auto from floor_nat_add_id[OF this] T False show ?thesis qed qed have floor_u: "floor ?u = (if ?U = {} \<and> ?L \<noteq> {} then c + 1 else c)" proof (cases "?U = {}") case False from U_intv[OF False] have "0 \<le> Min ?U" "Min ?U < 1" by auto from floor_nat_add_id[OF this] False show ?thesis by simp next case True note T = this show ?thesis cases " case True with T show ?thesis by (simp add: floor_nat_add_id) next case False m" by auto from floor_nat_add_id[OF this] T False show ?thesis by auto qed qed { assume "?L \<noteq> {}" "?U \<noteq> {}" from Max_in[OF fin(1) \<open>?L \<noteq> {}\<close>] obtain w where w: "w \<in> ?X\<^sub>0" "x \<noteq> w" "(w,x) \<in> r" "Max ?L = frac (v w)" by auto from Min_in[OF fin(2) \<open>?U \<noteq> {}\<close>] obtain z where z: "z \<in> ?X\<^sub>0" "x \<noteq> z" "(x,z) \<in> r" "Min ?U = frac (v z)" by auto from w z trans have "(w,z) \<in> r" unfolding trans_def by blast with v w z have "Max ?L \<le> Min ?U" by fastforce } note l_le_u = this { fix y assume y: "y \<in> ?X\<^sub>0" "x \<noteq> y" from total y \<open>x \<in> X\<close> Intv have total: "(x,y) \<in> r \<or> (y,x) \<in> r" unfolding total_on have "frac (v y) = frac d \<longleftrightarrow> (y,x) \<in> r \<and> (x,y) \<in> r" proofsafe assume A: "(y,x) \<in> r" "(x,y) \<in> r" from L_bound[OF y A(1)] U_bound[OF y A(2)] have *: "frac (v y) \<le> frac ?l" "frac ?u \<le> frac (v y)" by auto from A y have **: "?L \<noteq> {}" "?U \<noteq> {}" by auto with L_intv[OF this(1)] U_intv[OF this(2)] have "frac ?l = Max ?L" "frac ?u = Min ?U" by (auto simp: frac_nat_add_id frac_eq) with * ** l_le_u have "frac ?l = frac ?u" "frac (v y) = frac ?l" by auto with d have "d = ((floor ?l + floor ?u) + (frac (v y) + frac (v y))) / 2" unfolding frac_def by auto also have "\<dots> = c + frac (v y)" using \<open>floor ?l = c\<close> floor_u \<open>?U \<noteq> {}\<close> b finally show "frac (v y) = frac d" using frac_nat_add_id frac_frac by metis next assume A: "frac (v y) = frac d" show "(y, x) \<in> r" proof (rule ccontr) assume B:"(,<> " with total have B': "(x,y) \<in> r" by auto from U_bound[OF y this] have u_y:"frac ?u \<le> frac (v y)" by auto from y B' have U: "?U \<noteq> {}" and "frac (v y) \<in> ?U" by auto then have u: "frac ?u = Min ?U" using Min_in[OF fin(2) \<open>?U \<noteq> {}\<close>] by (auto simp: frac_nat_add_id frac_frac) show False proof (cases "?L = {}") case True from U_intv[OF U] have "0 < Min ?U" "Min ?U < 1" by auto then have *: "frac (Min ?U / 2) = Min ?U / 2" unfolding frac_eq by simp from d U True have "d = ((c + c) + Min ?U) / 2" by auto also have "\<dots> = c + Min ?U / 2" by simp finally have "frac d = Min ?U / 2" using * by (simp add: frac_nat_add_id) also have "\<dots> < Min ?U" using \<open>0 < Min ?U\<close> by auto finally have "frac d <racg autojava.lang.StringIndexOutOfBoundsException: Index 61 out with u_y A show False by auto next case False then have l: "?l = c + Max ?L" by simp from Max_in[OF fin(1) \<open>?L \<noteq> {}\<close>] obtain w where w: "w \<in> ?X\<^sub>0" "x \<noteq> w" "(w,x) \<in> r" "Max ?L = frac (v w)" by auto with \<open>(y,x) \<notin> r\<close> trans have **: "(y,w) \<notin> r" unfolding trans_def by blast from Min_in[OF fin(2) \<open>?U \<noteq> {}\<close>] Max_in[OF fin(1) \<open>?L \<noteq> {}\<close>] fr have "0 \<le> Max ?L" "Max ?L < 1" "0 \<le> Min ?U" "Min ?U < 1" by auto then have "0 \<le> (Max ?L + Min ?U) / 2" "(Max ?L + Min ?U) / 2 < 1" by auto then have ***: "frac ((Max ?L + Min ?U) / 2) = (Max ?L + Min ?U) / 2" unfolding frac_eq .. from y w have "y \<in> ?X\<^sub>0'" "w \<in> ?X\<^sub>0'" by auto with v ** have lt: "frac (v y) > frac (v w)" by fastforce from d U l have "d = ((c + c) + (Max ?L + Min ?U))/2" by auto also have "\<dots> = c + (Max ?L + Min ?U) / 2" by simp finally have "frac d = frac ((Max ?L + Min ?U) / 2)" by (simp add: frac_nat_add_id) also have "\<dots> = (Max ?L + Min ?U) / 2" using *** by simp also have "\<dots> < (frac (v y) + Min ?U) / 2" using lt w(4) by auto also have "\<dots> \<le> frac (v y)" using Min_le[OF fin(2) \<open>frac (v y) \<in> ?U\<close>] by auto finally show False using A by auto qed qed next assume A: "frac (v y) = frac d" show "(x, y) \<in> r" proof (rule ccontr) assume B: "(x,y) \<notin> r" with total have B': "(y,x) \<in> r" by auto from L_bound[OF y this] have l_y:"frac ?l \<ge> frac (v y)" by auto from y B' have L: "?L \<noteq> {}" and "frac (v y) \<in> ?L" by auto then have l: "frac ?l = Max ?L" using Max_in[OF fin(1) \<open>?L \<noteq> {}\<close>] by (auto simp: frac_nat_add_id frac_frac) show False proof (cases "?U = {}") case True from L_intv[OF L] have *: "0 < Max ?L" "Max ?L < 1" by auto from d L True have "d = ((c + c) + (1 + Max ?L)) / 2" by auto also have "\<dots> = c + (1 + Max ?L) / 2" by simp finally have "frac d = frac ((1 + Max ?L) / 2)" by (simp add: frac_nat_add_id) also have "\<dots> = (1 + Max ?L) / 2" using * unfolding frac_eq by auto also have "\<dots> > Max ?L" using * by auto finally have "frac d > frac ?l" using l by auto with l_y A show False by auto next case False then have u: "?u = c + Min ?U" by simp from Min_in[OF fin(2) \<open>?U \<using 1[THEN "&E"(1] "qml2 <rightarrow>"E"(2) by blast obtain w where w: "w \<in> ?X\<^sub>0" "x \<noteq> w" "(x,w) \<in> r" "Min ?U = frac (v w)" by auto with \<open>(x,y) \<notin> r\<close> trans have **: "(w,y) \<notin> r" unfolding trans_def by blast from Min_in[OF fin(2) \<open>?U \<noteq> {}\<close>] Max_in[OF fin(1) \<open>?L \<noteq> {}\<close>] fr v cfg \<equiv> initReachable cfg \<and (\bottom, outM v> \<in># msgs cfg)" then have "0 \<le> (Max ?L + Min ?U) / java.lang.StringIndexOutOfBoundsException: Index 0 ou then:cMaxinU )axL?"oldingqjava.lang.StringIndexOutOfBoundsException: Index 104 ou from y w have "y \<n?<sub0'" "w \<in> ?X\<^sub>0'" by auto with v ** have lt: "frac (v y) < frac "astforce from d L u have "d = closejava.lang.StringIndexOutOfBoundsException: Index 8 out of bou as stated in \<^cite>\<open"Voelzer"\<close> from the \emph{diamond property} finallyablec'"C''(1) achablesst also have< = (Max ?L + Min ?U) / 2" using *** by simp also have "\<< distinct procList" finally have "frac d > frac (v y)" using Max_ge[OF fin(1) \<open>frac (v y) \<in> ?L\<close>] by auto then show False using A by auto qed (cX = cN)" } note d_frac_equiv = this havefrac_l\<e frac d" proof (cases "?L = {}") case True proof (cases " }java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds f case True with T have "?l = ?u" by auto with d have "d = ?l" by auto then show ?thesis by auto next ase ?=0by auto moreover have "frac d \<ge> 0" by auto ultimately show ?thesis by linarith qed next note F = this "c ax?"frac=ax?"ingin in)<open?L \<noteq> {}\<close>] by (auto simp: frac_nat_add_id frac_frac) from L_intv[OF F] have *: "0 < Max ?L" "Max ?L < 1" by auto show ?thesis proof (cases "?U = {}") case True from \turnstile msg \<mapsto> \<lparr>states = AOT_assume \\<open>\<forall>x([L]x <equiv[<lambda <])< with l d have "d = ((c + c) + (Max ?L + 1)) / 2" byjava.lang.StringIndexOutOfBoundsException: In also have "\<dots> = c + (1 + Max ?L) / 2" by simp finally have "frac thenobtaineval]<> } uto lso have "\<dots> = (1 + Max ? with PseudoTermination C) idedImpliesUniform(notb c'" also have "\<dots> > Max ?L" using * by auto finally show "frac d \<ge> frac ?l" using l by auto next case False then have u: "?u = c + Min ?U" "frac ?u = Min ?U" using Min_in[OF fin(2) False] by (auto simp: frac_nat_add_id frac_frac) from U_intv[OF False] have **: "0 < Min ?U" "Min ?U < 1" by auto from l u d have "d = ((c + c) + (Max ?L + Min ?U)) / 2" by auto also have "\<dots> = c + (Max ?L + Min ?U) / 2" by simp finally have "frac d = frac ((Max ?L + Min ?U) / 2)" by (simp add: frac_nat_add_id) also have "\<dots> = (Max ?L + Min ?U) / 2" using * ** unfolding frac_eq by auto also have "\<dots> \<ge> Max ?L" using l_le_u[OF F False] by auto finally show ?thesis using l byauto qed qed have frac_u: "?U \<noteq> {} \<or> ?L = {} \<longrightarrow> frac d \<le> frac ?u" proof (cases "?U = {}") case True note T = this show ?thesis proof (cases "?L = {}") case True with T have "?l = ?u" by auto with d ed "byto then show ?thesis by auto next case False with T show ?thesis by auto qed next case False note F = this then have u: "?u = c + Min ?U" "frac ?u = Min ?U" using Min_in[OF fin(2) \<open>?U \<noteq> {}\<close>] by (auto simp: frac_nat_add_id frac_frac) from U_intv[OF F] have *: "0 < Min ?U" "Min ?U < 1" by auto show ?thesis proof (cases "?L = {}") case True from True F have "?l = c" by auto with u d have "d = ((c + c) + Min ?U) / 2" by auto also have "\<dots> = c + Min ?U / 2" by simp finally have "frac d = frac (Min ?U / 2)" by (simp add: frac_nat_add_id) also have "\<dots> = Min ?U / 2" unfolding frac_eq using * by auto \<le> Min ?U" using \<open>0 < Min U<close> by auto finally have "frac d \<le> frac ?u" using u by auto then show ?thesis by auto next case False then have l: "?l = c + Max ?L" "frac ?l = Max ?L" using Max_in[OF fin(1) False] by (auto simp: frac_nat_add_id frac_frac) from L_intv[OF False] have **: "0 < Max ?L" "Max ?L < 1" by auto from l u d have "d = ((c + c) + (Max ?L + Min ?U)) / 2" by auto also have "\<dots> = c + (Max ?L + Min ?U) / 2" by simp finally have "frac d = frac ((Max ?L + Min ?U) / 2)" by (simp add: frac_nat_add_id) also have "\<dots> = (Max ?L + Min ?U) / 2" using * ** unfolding frac_eq by auto also have "\<dots> \<le> Min ?U" using l_le_u[OF False F] by auto finally show ?thesis using u by auto qed qed have "\<forall> y \<in> ?X\<^sub>0 - {x}. (y,x) \<in> r \<longleftrightarrow> frac (v y) \<le> frac d" proof (safe, goal_cases) case (1 y k) with L_bound[of y] frac_l show ?case by auto next case (2 y k) show ?case proof (rule ccontr, goal_cases) case 1 with total 2 \<open>x \<in> X\<close> Intv have "(x,y) \<in> r" unfolding total_on_def by auto with 2 U_bound[of y] have "?U \<noteq> {}" "frac ?u \<le> frac (v y)" by auto with frac_u have "frac d \<le> frac (v y)" by auto with 2 d_frac_equiv 1 show False by auto qed qed moreover have "\<forall> y \<in> ?X\<^sub>0 - {x}. (x,y) \<in> r \<longleftrightarrow> frac d \<le> frac (v y proof (safe, goal_cases) case (1 y k) then have "?U \<noteq> {}" by auto with 1 U_bound[of y] frac_u show ?case by auto next case (2 y k) show ?case proof (rule ccontr, goal_cases) case 1 with total 2 \<open>x \<in> X\<close> Intv have "(y,x) \<in> r" unfolding total_on_def by auto with 2 L_bound[of y] have "frac (v y) \<le> frac ?l" by auto with frac_l have "frac (v y) \<le> frac d" by auto with 2 d_frac_equiv 1 show False by auto qed qed ultimately have d: "c < d" "d < c + 1" "\<forall> y \<in> ?X\<^sub>0 - {x}. (y,x) \<in> r \<longleftrightarrow> frac (v y) \<le> frac d" "\<forall> y \<in> ?X\<^sub>0 - {x}. (x,y) \<in> r \<longleftrightarrow> frac d \<le> frac (v y)" using d3 by auto let ?v = "v(x := d)" have "?v \<in> region X I r" proof (standard, goal_cases) case 1 from \<open>d\<ge>0\<close> v show ?case by auto next case 2 ase proof (safe, goal_cases) case (1 y) with v have "intv_elem y v (?I y)" by fast withIntvd, how"tv_elemyv( )" (sess" =" uto casesI"uto) qed next from \<open>x \<in> X\<close> Intv show "?X\<^sub>0' \<union> {x} = ?X\<^sub>0" by auto with r_subset have "r \<subseteq> (?X\<^sub>0' \<union> {x}) \<times> (?X\<^sub>0' \<union> {x})" by auto have "\<forall> x \<in> ?X\<^sub>0'. \<forall> y \<in> ?X\<^sub>0'. (x,y) \<in> r \<longleftrightarrow> (x,y) with v have "\<forall> x \<in> ?X\<^sub>0'. \<forall> y \<in> ?X\<^sub>0'. (x,y) \<in> r \<longleftrightarrow> then have "\<forall> x \<in> ?X\<^sub>0'. \<forall> y \<in> ?X\<^sub>0'. (x,y) \<in> r \<longleftrightarrow> f with d(3,4) show "\<forall> y \<in> ?X\<^sub>0' \<union> {x}. \<forall> z \<in> ?X\<^sub>0' \<union> {x}. (y,z) proof (auto, goal_cases) case 1 from refl \<open>x \<in> X\<close> Intv show ?case by (auto simp: refl_on_def) qed qed then show ?thesis by auto qed thentain here"v =)\in>R" using R by auto then have "(v(x := d))(x := c) \<in> region_set R x c" unfolding region_set_def by blast moreover from v \<open>x \<in> X\<close> have "(v(x := d))(x := c) = v" by fastforce ultimately have "v \<in> region_set R x c" by simp } ultimately have "region_set R x c = region X ?I ?r" by blast with valid \<R>_def have *: "region_set R x c \<in> \<R>" by auto moreover from assms have **: "v (x := c) \<in> region_set R x c" unfolding region_set_def by auto ultimately show "[v(x := c)]\<^sub>\<R> = region_set R x c" "[v(x := c)]\<^sub>\<R> \<in> \<R>" "v(x := c) \<in> using region_unique[OF _ ** *] \<R>_def by auto qed definition region_set' where "region_set' R r c = {[r \<rightarrow> c]v | v. v \<in> R}" lemma region_set'_id: fixes X k and c :: nat defines "\<R> \<equiv> {region X I r |I r. valid_region X k I r}" assumes "R \<in> \<R>" "v \<in> R" "finite X" "0 \<le> c" "\<forall> x \<in> set r. c \<le> k x" "set r \<subseteq> X" shows "[[r \<rightarrow> c]v]\<^sub>\<R> = region_set' R r c \<and> [[r \<rightarrow> c]v]\<^sub>\<R> \<in> \<R> proof (induction r) case Nil from regions_closed[OF _ Nil(2,3)] regions_closed'[OF _ Nil(2,3)] region_unique[OF _ Nil( have "[v]\<^sub>\<R> = R" "[v \<oplus> 0]\<^sub>\<R> \<in> \<R>" "(v \<oplus> 0) \<in> [v \<oplus> 0]\<^sub>\<R>" by auto then show ?case unfolding region_setdefval_add_defyimp next case (Cons x xs) then have "[[xs\<rightarrow>c]v]\<^sub>\<R> = region_set' R xs c" "[[xs\<rightarrow>c]v]\<^sub>\<R> \< note IH = this[unfolded \<R>_def] let ?v = "([xs\<rightarrow>c]v)(x := c) from region_set_id[OF IH(2,3) \<open>finite X\by (rule "\<existsI(1)) "cqt:2[lambda]" have "[?v]\<^sub>\<R> = region_set ([[xs\<rightarrow>real c]v]\<^sub>\<R>) x c" "[?v]\<^sub>\<R> \<in> \<R>" moreover have "region_set' R (x # xs) (real c) = region_set ([[xs\<rightarrow>real c]v]\<^sub>\<R> unfolding region_set_def region_set'_def proof (safe, goal_cases) case (1 y u) let ?u = "[xs\<rightarrow>real c]u" have "[x # xs\<rightarrow>real c]u = ?u(x := real c)" by auto moreover from IH(1) 1 have "?u \<in> [[xs\<rightarrow>real c]v]\<^sub>\<R>" unfolding \<R>_def region ultimately show ?case by auto next case (2 y u) with IH(1)[unfolded region_set'_def \<R>_def[symmetric]] show ?case by auto qed moreover have "[x # xs\<rightarrow>real c]v = ?v" by simp ultimately show ?case by presburger qed text \<open> This is the only additional lemma necessary to make local \<open>\<alpha>\<close>-closures work. \<close> lemma region_set_subs: fixes X k k' and c :: nat defines "\<R> \<equiv> {region X I r |I r. valid_region X k I r}" defines "\<R>' \<equiv> {region X I r |I r. valid_region X k' I r}" assumes "R \<in> \<R>" "v \<in> R" "finite X" "0 \<le> c" "set cs \<subseteq> X" "\<forall> y. y \<notin> set cs \<lo shows "[[cs \<rightarrow> c]v]\<^sub>\<R>' \<supseteq> region_set' R cs c" "[[cs \<rightarrow> c]v]\<^ proof from assms obtain I r where R: "R = region X I r" "valid_region X k I r" "v \<in> region X I r" by auto \<comment> \<open>The set of movers, that is all intervals that now are unbounded due to changing fr let ?M = "{x \<in> X. isIntv (I x) \<and> intv_const (I x) \<ge> k' x \<or> intv_const (I x) > k' x}" let ?I = "\<lambda> y. if y \<in> set cs then (if c \<le> k' y then Const c else Greater (k' y)) else if (isIntv (I y) \<and> intv_const (I y) \<ge> k' y \<or> intv_const (I y) > k' y) then Greater (k' y) else I y" let ?r = "{(y,z) \<in> r. y \<notin> set cs \<and> z \<notin> set cs \<and> y \<notin> ?M \<and> z \<notin> ?M}" let ?X\<^sub>0 = "{x \<in> X. \<exists> c. I x = Intv c}" let ?X\<^sub>0' = "{x \<in> X. \<exists> c. ?I x = Intv c}" have r_subset"<ubseteq ?X\<^sub>0 \<times> ?X\<^sub>0" and refl: "refl_on ?X\<^sub>0 r" and trans: "t total: "total_on ?X\<^sub>0 r" using R(2) by auto have valid: "valid_region X k' ?I ?r" proof show "?X\<^sub>0' = ?X\<^sub>0'" by auto next show "?r \<subseteq> ?X\<^sub>0' \<times> ?X\<^sub>0'" using r_subset by auto next from refl show "refl_on ?X\<^sub>0' ?r" unfolding refl_on_def by auto next from trans show "trans ?r" unfolding trans_def by auto next from total show "total_on ?X\<^sub>0' ?r" unfolding total_on_def by auto next from R(2) have "\<forall> x \<in> X. valid_intv (k x) (I x)" by auto then show "\<forall> x \<in> X. valid_intv (k' x) (?I x)" apply safe subgoal for x' using \<open>\<forall> y. y \<notin> set cs \<longrightarrow> k y \<ge> k' y\<close> by (cases "I x'"; force) done qed { fix v assume v: "v \<in> region_set' R cs c" with R(1) obtain v' where v': "v' \<in> region X I r" "v = [cs \<rightarrow> c]v'" unfolding region_set'_def by auto have "v \<in> region X ?I ?r" proof (standard, goal_cases) } from v' \<open>0 \<le> c\<close> show ?case apply - apply rule subgoal for x by (cases "x \<in> set cs") auto done next case 2 from v' show ?case apply - apply rule subgoal for x' by (cases "I x'"; cases "x' \<in> set cs"; force) done next show "?X\<^sub>0' = ?X\<^sub>0'" by auto next from v' show "\<forall> y \<in> ?X\<^sub>0'. \<forall> z \<in> ?X\<^sub>0'. (y,z) \<in> ?r \<longleftrightarr qed } then have "region_set' R cs c \<subseteq> region X ?I ?r" by blast moreover from valid have *: "region X ?I ?r \<in> \<R>'" unfolding \<R>'_def by blast moreover from assms have **: "[cs \<rightarrow> c]v \<in> region_set' R cs c" unfolding region_set ultimately show "[[cs \<rightarrow> c]v]\<^sub>\<R>' \<supseteq> region_set' R cs c" "[[cs \<rightarrow> c]v]\<^sub>\<R> using region_unique[of \<R>', OF _ _ *, unfolded \<R>'_def, OF HOL.refl] unfolding \<R>'_def[symmetric] by auto qed section \<open>A Semantics Based on Regions\<close> subsection \<open>Single step\<close> inductive step_r :: "('a, 'c, t, 's) ta \<Rightarrow> ('c, t) zone set \<Rightarrow> 's \<Rightarrow> ('c, t) zone \<Righta (\<open>_,_ \<turnstile> \<langle>_, _\<rangle> \<leadsto> \<langle>_, _\<rangle>\<close> [61,61,61,61] where step_t_r: "\<lbrakk>\<R> = {region X I r |I r. valid_region X k I r}; valid_abstraction A X k; R \<in> \<R>; R' \<in> Succ R' \<subseteq> \<lbrace>inv_of A l\<rbrace>\<rbrakk> \<Longrightarrow> A,\<R> \<turnstile> \<langle>l,R\< step_a_r: "\<lbrakk>\<R> = {region X I r |I r. valid_region X k I r}; valid_abstraction A X k; A \<turnstile> l \<lon \<Longrightarrow> A,\<R> \<turnstile> \<langle>l,R\<rangle> \<leadsto> \<langle>l',region_set' (R \<in inductive_cases[elim!]: "A,\<R> \<turnstile> \<langle>l, u\<rangle> \<leadsto> \<langle>l', u'\<rang declare step_r.intros[intro] lemma region_cover': assumes "\<R> = {region X I r |I r. valid_region X k I r}" and "\<forall>x\<in>X. 0 \<le> v x" shows "v \<in> [v]\<^sub>\<R>" "[v]\<^sub>\<R> \<in> \<R>" proof - from region_cover[OF assms(2), of k] assms obtain R where R: "R \<in> \<R>" "v \<in> R" by auto from regions_closed'[OF assms(1) R, of 0] show "v \<in> [v]\<^sub>\<R>" unfolding cval_add_def by a from regions_closed[OF assms(1) R, of 0] show "[v]\<^sub>\<R> \<in> \<R>" unfolding cval_add_def by a qed lemma step_r_complete_aux: fixes R r A l' g defines "R' \<equiv> region_set' (R \<inter> {u. u \<turnstile> g}) r 0 \<inter> {u. u \<turnstile> inv_of assumes "\<R> = {region X I r |I r. valid_region X k I r}" and "valid_abstraction A X k" and "u \<in> R" and "R \<in> \<R>" and "A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l'" and "u \<turnstile> g" and "[r\<rightarrow>0]u \<turnstile> inv_of A l'" shows "R = R \<inter> {u. u \<turnstile> g} \<and> R' = region_set' R r 0 \<and> R' \<in> \<R>" proof - note A = assms(2-) from A(2) have *: "\<forall>(x, m)\<in>clkp_set A. m \<le> real (k x) \<and> x \<in> X \<and> m \<in> \<nat>" "collect_clkvt (trans_of A) \<subseteq> X" "finite X" by (fastforce elim: valid_abstraction.cases)+ from A(5) *(2) have r: "set r \<subseteq> X" unfolding collect_clkvt_def by fastforce from *(1) A(5) have "\<forall>(x, m)\<in>collect_clock_pairs g. m \<le> real (k x) \<and> x \<in> X \<and> m \< unfolding clkp_set_def collect_clkt_def by fastforce from ccompatible[OF this, folded A(1)] A(3,4,6) have "R \<subseteq> \<lbrace>g\<rbrace>" unfolding ccompatible_def ccval_def by blast then have R_id: "R \<inter> {u. u \<turnstile> g} = R" unfolding ccval_def by auto from region_set'_id[OF A(4)[unfolded A(1)] A(3) *(3) _ _ r, of 0, folded A(1)] have **: "[[r\<rightarrow>0]u]\<^sub>\<R> = region_set' R r 0" "[[r\<rightarrow>0]u]\<^sub>\<R> \<in> \<R>" "[r\<ri by auto let ?R = "[[r\<rightarrow>0]u]\<^sub>\<R>" from *(1) A(5) have ***: \<forall>(x, m) \<in> collect_clock_pairs (inv_of A l'). m \<le> real (k x) \<and> x \<in> X \<and> m \<in> \<nat>" unfolding inv_of_def clkp_set_def collect_clki_def by fastforce from ccompatible[OF this, folded A(1)] **(2-) A(7) have "?R \<subseteq> \<lbrace>inv_of A l'\<rbr unfolding ccompatible_def ccval_def by blast then have ***: "?R \<inter> {u. u \<turnstile> inv_of A l'} = ?R" unfolding ccval_def by auto with **(1,2) R_id show ?thesis by (auto simp: R'_def) qed lemma step_r_complete: \<forall> x \<in> X. u x \<ge> 0\<rbrakk> \<Longrightarrow> \<exists> R'. A,\<R> \<turnstile> \<langle>l, ([u]\<^ proof (induction rule: step.induct, goal_cases) case (1 A l u a l' u') note A = this then obtain g r where u': "u' = [r\<rightarrow>0]u" "A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r by (cases rule: step_a.cases) auto Author: AndreasLochbihler, ETH Zurich *) region_cover'[OF A(2)] have :"u<ub<>\<in> \<R>" "u \<in> [u]\<^sub>\<R>" by auto from step_r_complete_aux[OF A(2,3) this(2,1) u'(2,3)] u' have *: "[u]\<^sub>\<R> = ([u]\<^sub>\<R>) \<inter> {u. u \<turnstile> g}" "?R' = region_set' ([u]\<^using from 1(2,3) have "collect_clkvt ) from' 2)have "u<n ?R'" unfolding region_set'_def by auto moreover have "A,\<R> \<turnstile> \<langle>l,([u]\<^sub>\<)\rangle <eadsto>\<langle>l',?R'\<rangle> ultimately show ?case using *(3) by meson next case (2 A ' ) hence u': "u' = (u \<oplus> d)" "u \<oplus> d \<turnstile> inv_of A l" "0 \<le> d" and "l = l'" by (auto elim!: from n_cover(have "[u\^sub>\<R <> \<R>"" \>[u]\^sub>\<R> java.lang.StringIndexOutOfBoundsExcepti from SuccI2[OF Eq thenlex_ord_pair f]ys elseaux)"| "[u']\<^sub>|Eq \< prod_ord_pair f by auto from regions_closed'[OF 2(2) R(1,2) \<open>0 \<le> d\<close>] u'(1) have uhence fst'uin fst ` set xs from 2(3) have *: "\case Nil "collect_clkvt (trans_of A) \<subseteq> X" "finite X" by (fastforce elim: valid_abstraction.cases)+ from *(1) u'(2) have "\<forall>(x, m)\<in>collect_clock_pairs (inv_of A l). m \<le> real (k x) \<and> x unfolding clkp_set_def collect_clki_def inv_of_def by fastforce from ccompatible[OF this, folded 2(2)] u'1(2) u'2 u'(1,2,3) R have "[u']\<^sub>\<R> \<subseteq> \<lbrace>inv_of A l\<rbrace unfolding ccompatible_def ccval_def by auto with 2 u'1 R(1) have "A,\<R> \<turnstile> \<langle>l, ([u]\<^sub>\<R>)\<rangle> \<leadsto> \<langle>l,([u' with u'1(2) u'2 \<open>l = l'\<close> show ?case by meson qed text \<open> Compare this to lemma \<open>step_z_sound\<close>. This version is weaker because for regions we arrive at a successor for which not every valuation can be reached by the predecessor. This is the case for e.g. the region with only Greater (k x) bounds. \<close> lemma step_r_sound: "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto> \<langle>l',R'\<rangle> \<Longrightarrow> \<R \<Longrightarrow> R' \<noteq> {} \<Longrightarrow> (\<forall> u \<in> R. \<exists> u' \<in> R'. A \<turnstile> proof (induction rule: step_r.induct) case (java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for lengt note A = this[unfolded this(1)] show ?case proof fix usume < R" fromer have "map_pair f xs = map_pair g xs" by (rule Cons(1), rule Cons(2), simp) obtain t whereproof (induct xs : oalist_inv_raw_induct) with regions_closed[(,) (]ep_t_r)* (u \<> t) \<in>R"byauto with u t(1) A(5,6)have "A <> \langlel rangle> \rightarrow \<langle>l,(u \<oplus> t)\<rangle>" unfoldi with t * show havecomp fst f( v) ' comp (fst (f (k, v))) (fst (f k ')" qed next case A: (step_a_r \<R> X k A l g a r l' R) show ?case proof fix u assume u: "u \<in> R" from A(6) obtain v where v: "v \<in> R" "v \<turnstile> g" "[r\<rightarrow>0]v \<turnstile> inv_of A l'" let ?R' = "region_set (simp add: x f Let_def 1, map2_val_compat :('a<>) >('a \<times> 'c::zero) list) \Rightarrow" "R = R \<inter> {u. u \<turnstile> g}" "?R' = region_set' R r 0" by auto from A have "collect_clkvt (trans_of A) \<> X" by(auto elim: valid_abstractioncases) with A(3) have r: "set r \<subseteq> X" unfolding collect_clkvt_def by fastforce from u R have *: "[r\<rightarrow>0]u \<in> ?R'" "u \<turnstile> g" "[r\<rightarrow>0]u \<turnstile> inv with A(3) have "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow> \<langle>l',[r\<rightarrow>0]u\<r with * show "\<exists>a\<in>?R'. A \<turnstile> \<langle>l, u\<rangle> \<rightarrow> \<langle>l',a\<rang qed qed subsection \<open>Multi Step\<close> inductive steps_r :: "('a, 'c, t, 's) ta \<Rightarrow> ('c, t) zone set \<Rightarrow> 's \<Rightarrow (\<open>_,_ \<turnstile> \<langle>_, _\<rangle> \<leadsto>* \<langle>_, _\<rangle>\<close> [61,61,61,61 where refl: "A,\<R> \<turnstile> \<langle> apply (rule "=\<^sub>d\^subfI"2)[OF AOT_ordinary]) step: "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<langle>l', R'\<rangle> \<Longrightarrow> declare steps_r.intros[intro] lemmasteps_alt: "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>* \<langle>l',u'\<rangle> \<Longrightarrow> A \<tur by (induction rule: steps.induct) auto lemma emptiness_preservance: "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto> \<langle>l',R'\< by (induction rule: step_r.cases) (auto simp: region_set'_def) lemma emptiness_preservance_steps: "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<langl apply (induction rule: steps_r.induct) apply blast apply (subst emptiness_preservance) by blast+ text \<open> Note how it is important to define the multi-stephence "0 =k" by (simponlyeq) This is also the direction Bouyer implies for her implicit induction. \<close> lemma steps_r_sound: "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<langle>l', R'\<rangle> \<Longrightarrow> \<R> = {r \<Longrightarrow> R' \<noteq> {} \<Longrightarrow> u \<in> R \<Longrightarrow> \<exists> u' \<in> R'. A \<tur proof (induction rule: steps_r.induct) case refl then show ?case by auto show lookup_pairlet v = fkvv' aux = map2_val_pair f in from emptiness_preservance[OF step.hyps(2)] step.prems have "R' \<noteq> {}" by fastforce with step obtain u' where u': "u' \<in> R'" "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>* \<langl with step_r_sound[OF step(2,4,5)] obtain u'' where "u'' \<in> R''" "A \<turnstile> \<langle>l', u' with u' show ?case by (auto 4 5 introthus ?( :\ \close qed lemma steps_r_sound': "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<langle>l', R'\<rangle> \<Longrightarrow> \<R> = {r \<Longrightarrow> R' \<noteq> {} \<Longrightarrow> (\<exists> u' \<in> R'. \<exists> u \<in> R. A \<turnstile> proof goal_cases case 1 with emptiness_preservance_steps[OF this(1)] obtain u where "u \<in> R" by auto with steps_r_sound[OF 1 this] show ?case by auto qed lemma single_step_r: "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto> \<langle>l', R'\<rangle> \<Longrightarrow> A,\<R> \<t java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 36 lemma steps_r_alt: "A,\<R> \<turnstile> \<langle>l', R'\<rangle> \<leadsto>* \<langle>l'', R''\<rangle> \<Longrightarrow> A apply (induction rule: steps_r.induct) apply (rule single_step_r) by auto lemma single_step: "x1 \<turnstile> \<langle>x2, x3\<rangle> \<rightarrow> \<langle>x4,x5\<rangle> \<Longrightarrow> x1 \< by (metis steps.intros) lemma steps_r_complete: using show ?case by simp \<forall> x \<in> X. u x \<ge> 0\<rbrakk> \<Longrightarrow> \<exists> R'. A,\<R> \<turnstile> \<langle>l, ([u]\<^ proof (induction rule: steps.induct) from region_cover'[OF refl(1,3)] show ?case by auto next case (step A l u l' u' l'' u'') from step_r_complete[OF step(1,4-6)] obtain R' where R': "A,\<R> \<turnstile> \<langle>l, ([u]\<^sub>\<R>)\<rangle> \<leadsto> \<langle>l',R'\<rangle>" " hence "k0 by auto with step(4) \<open>u' \<in> R'\<close> have "\<forall>x\<in>X. 0 \<le> u' x" by auto with step obtain R'' where R'': "A,\<R> \<turnstile> \<langle>l', ([u']\<^sub>\<R>)\<rangle> \<leadsto>* with region_unique[OF step(4) R'(2,3)] R'(1) have "A,\<R> \<turnstile> \<langle>l, ([u]\<^sub>\<R>) by (subst steps_r_alt) auto with R'' region_cover'[OF step(4,6)] show ?case by auto qed (* section \<open>A Semantics Based on gions<closejava.lang.StringIndexOutOfBoundsException subsection \<open>Single step\<close> java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19 "('a, java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length ("_,_ \<turnstile> \<langle>_, _\<rangle> \<leadsto> \<langle>_, _\<rangle>" [61,61,61,61] 61) where "\<lbrakk>\<R> = {region X I r |I r. valid_region X k I r}; valid_abstraction A X k; A \<turnstile> l \<lon R \<in> \<R>; R' \<in> Succ \<R> R; R' \<subseteq> \<lbrace>inv_of A l\<rbrace> \<rbrakk> \<Longrightarrow> A,\<R> \<turnstile> \<langle>l,R\<rangle> \<leadsto> \<langle>l', region_set' (R' \<inter> {u. u \<turnst inductive_cases[elim!]: "A,\<R> \<turnstile> \<langle>l, u\<rangle> \<leadsto> \<langle>l', u'\<rang declare step_r.intros[intro] lemma region_cover': assumes "\<R> = {region X I r |I r. valid_region X k I r}" and "\<forall>x\<in>X. 0 \<le> v x" shows "v \<in> [v]\<^sub>\<R>" "[v]\<^sub>\<R> \<in> \<R>" proof - from java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length from regions_closed'[OF assms(1) R, of 0] show "v \<in> [v]\<^sub>\<R>" unfolding cval_add_def by a from regions_closed[OF assms(1) R, of 0] show "[v]\<^sub>\<R> \<in> \<R>" unfolding cval_add_def by a qed lemma step_r_complete_aux: fixes R r A l' g defines "R' \<equiv>setR<> {u <>gr 0>{u. u \<turnstile> inv_of A l'}" and "valid_abstraction A moreoveroddmp R" \<R>" \ l \<longrightarrow>\<^bsup>g,a,r\<^esup> l'" <urnstile>g" and "[r\<rightarrow>0]withpowlseyutoadd:ime_int_ifff inter> {u. u \<turnstile> g} \<and> R' = region_set'Rr\and R' \<in> \<R>" proof - note A = assms(2-) from A(2) have *: using "cont-nec-fact2:1 pplylast "collect_clkvt (trans_of A) \<subseteq> X" "finite X" by (fastforce elim: valid_abstraction.cases)+ from A(5) *(2) have r: "set r \<subseteq> X" unfolding collect_clkvt_def by fastforce from *(1) A(5) have "\<forall>(x, m)\<in>collect_clock_pairs g. m \<le> real ( clkp_set_defbyfastforce from ccompatible[OF this, folded A(1)] A(3,4,6) have "R \<subseteq> \<lbrace>g\<rbrace>" unfolding ccompatible_def ccval_def by blast then have R_id: "R \<inter> {u. u \<turnstile> g} = R" unfolding ccval_def by auto from region_set'_id[OF A(4)[unfolded A(1)] A(3) *(3) _ _ r, of 0, folded A(1)] have **: "[[r\<rightarrow>0]u]\<^sub>\<R> = region_set' R r 0" "[[r\<rightarrow>0]u]\<^sub>\<R> \<in> \<R>" "[r\<ri by auto let ?R = "[[r\<rightarrow>0]u]\<^sub>\<R>" from *(1) A(5) have ***: "\<forall>(x, m) \<in> collect_clock_pairs (inv_of A l'). m \<le> real (k x) \<and> x \<in> X \<and> m \<in> \<nat> proof from ccompatible[OF this, folded A(1)] **(2-) A(7) have "?R \<subseteq> \<lbrace>inv_of A l'\<rbr unfolding ccompatible_def ccval_def by blast then have ***: "?R \<inter> {u. u \<turnstile> inv_of A l'} = ?R" unfolding ccval_def by auto with **(1,2) R_id show ?thesis by (auto simp: R'_def) qed lemma step_r_complete: "\<lbrakk>A \<turnstile>' \<langle>l, u\<rangle> \<rightarrow> \<langle>l',u'\<rangle>; \<R> = {region X I \<forall> x \<in> X. u x \<ge> 0\<rbrakk> \<Longrightarrow> \<exists> R'. A,\<R> \<turnstile> \<langle>l, ([u]\<^ proof (induction rule: step'.induct) case prems: (step' A l u d l' u' a l'' u'') then have u': "u' = (u \<oplus> d)" "u \<oplus> d \<turnstile> inv_of A l" "0 \<le> d" and "l = l'" by auto from prems obtain g r where u'': "u'' = [r\<rightarrow>0]u'" "A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l''" "u' \<turn by atomize_elim (auto 4 4 elim: step_a.cases) from region_cover'[OF \<open>\<R> = _\<close> prems(5)] have R: "[u]\<^sub>\<R> \<in> \<R>" "u \<in> [u]\<^sub> from SuccI2[OF \<open>\<R> = _\<close> this(2,1) \<open>0 \<le> _\<close>, of "[u']\<^sub>\<R>"] \<open>u' = _\< "[u']\<^sub>\<R> \<in> Succ \<R> ([u]\<^sub>\<R>)" "[u']\<^sub>\<R> \<in> \<R>" by auto from R(1,2) u'(1,3) have u'2: "[u']\<^sub>\<R> \<in> \<R>" "u' \<in> [u']\<^sub>\<R>" by (auto intro: regions_closed[OF \<open>\<R> = _\<close>] regions_closed'[OF \<open>\<R> = _\<close>] from prems(4) have *: "\<forall>(x, m)\<in>clkp_set A. m \<le> real (k x) \<and> x \<in> X \<and> m \<in> \<nat>" "collect_clkvt (rans_of <subseteq X" "finite X" by (fastforce elim: valid_abstraction.cases)+ from *(1) u'(2) have "\<forall>(x, m)\<in>collect_clock_pairs (inv_of A llemma sort_oalist_id: "\<forall>(x, m)\<in>collect_clock_pairs (inv_of A l''). m \<le> real (k x) \<and> x \<in> X \<and> m \<in> \<na unfolding clkp_set_def collect_clki_def inv_of_def by fastforce+ with u'2 u'(1,2) have "[u']\<^sub>\<R> \<subseteq> \<lbrace>inv_of A l\<rbrace>" by (auto 4 4 simp add: ccompatible_def ccval_def \<open>\<R> = _\<close>[symmetric] dest!: ccompat then have **: "([u']\<^sub>\<R>) \<inter> {u. u \<turnstile> inv_of A l} = ([u']\<^sub>\<R>)" by (auto simp let ?R'= "region_set' (([u']\<^sub>\<R>) \<inter> {u. u \<turnstile> g}) r 0 \<inter> {u. u \<turnstile> in from step_r_complete_aux[OF \<open>\<R> = _\<close> prems(4) u'2(2,1) u''(2,3)] u''(1,4) have"u]<^sub\<> <inter>{.u \<turnstile> g} = [u']\<<^sub>\<R> "?R '(u]\^sub>\<R>) 0""?'\<> <R"by java.lang.S from prems(3,4) have "collect_clkvt (trans_of A) \<subseteq> X" "finite X" by (auto elim: valid_abstraction.cases) with u''(2) have r: "set r \<subseteq> X" unfolding collect_clkvt_def by fastforce *\<open>u'' =_\close> \<open>u' \<in> _\<close have "u' \<in> ?R' unfolding region_set'_def by auto moreover from step_r[OF <\<R> = _\<close> prems(4) u''(2) R(1) u'1(1) \<open>[u']\<^sub>\<R> \<subseteq> "A,\<R> \<turnstile> \<langle>l, ([u]\<^sub>\<R>)\<rangle> \<leadsto> \<langle>l'', ?R'\<rangle>" by (simp add: ** ) ultimately show ?case using \<open>?R' \<in> \<R>\<close> by (intro exI conjI; auto) qed text \<open> Compare this to lemma \<open>step_z_sound\<close>. This version is weaker because for regions we arrive at a successor for which not every valuation can be reached by the predecessor. This is the case for e.g. the region with only Greater (k x) bounds. \<close> lemma step_r_sound: "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto> \<langle>l',R'\<rangle> \<Longrightarrow> \<R> = {re \<Longrightarrow> R' \<noteq> {} \<Longrightarrow> (\<forall> u \<in> R. \<exists> u' \<in> R'. A \<turnstile> proof (induction rule: step_r.induct) case A: (step_r \<R> X k A l g a r l' R R') show ?case proof fix u assume u: "u \<in> R" from set_of_regions[OF A(4)[unfolded A(1)], folded A(1), OF u A(5)] A(2) obtain t where t: "t \<ge> 0" "[u \<oplus> t]\<^sub>\<R> = R'" by (auto elim: valid_abstraction.cases) with regions_closed'OF A(1,4) u] have *: "(u \<oplus> t) \<in> R'" y auto from A(6,8) obtain v where v: "v \<in> R'" "v \<turnstile> inv_of A l" "v \<turnstile> g" "[r\<rightarro unfolding region_set'_def ccval_def by auto let ?R' = "region_set' (R' \<inter> {u. u \<turnstile> g}) r 0 \<inter> {u. u \<turnstile> inv_of A l'}" from step_r_complete_aux[OF A(1,2) v(1) _ A(3)] \<open>R' \<in> _\<close> v have R: "R' = R' \<inter> {u. u \<turnstile> g}" "?R' = region_set' R' r 0" by auto from A have "collect_clkvt (trans_of A) \<subseteq> X" by (auto elim: valid_abstraction.cases with A(3) have r: "set r \<subseteq> X" unfolding collect_clkvt_def by fastforce from * R A(6) have *: "u \<oplus> t \<turnstile> inv_of A l" "[r\<rightarrow>0](u \<oplus> t) \<in> ?R'" "(u \<oplus> t) \<turnsti unfolding region_set'_def ccval_def by auto with A(3) \<open>t \<ge> 0\<close> have "A \<turnstile>' \<langle>l, u\<rangle> \<rightarrow> \<langle>l',[ with * show "\<exists>a AOT_have not_delta_abs_b <>\not><bold\Delta[!b\closejava.lang.String qed qed subsection \<open>Multi Step\<close> inductive steps_r :: "('a, 'c, t, 's) ta \<Rightarrow> ('c, t) zone set \<Rightarrow> 's \<Rightarrow> ('c, t) zo ("_,_ \<turnstile> \<langle>_, _\<rangle> \<leadsto>* \<langle>_, _\<rangle>" [61,61,61,61,61,61] 61 where refl: "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<langle>l, R\<rangle>" | step: "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<langle>l', R'\<rangle> \<Longrightarrow> .] lemma steps_alt: "A \<turnstile>' \<langle>l, u\<rangle> \<rightarrow>* \<langle>l',u'\<rangle> \<Longrightarrow> A \<tu by (induction rule: steps'.induct) auto lemma emptiness_preservation: "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto> \<langle>l',R' by (induction rule: step_r.cases) (auto simp: region_set'_def) lemma emptiness_preservation_steps: "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<lang apply (induction rule: steps_r.induct) apply blast apply (subst emptiness_preservation) by blast+ text \<open> Note how it is important to define the multi-step semantics "the right way round". This also the direction Bouyer implies for her thus ?thesis by simp \<close> lemma moreover have 1:"fst (min_list_param (\lambda>y.le ko (fst x)( y) "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<langle>l', R'\<rangle> \<Longrightarrow> \<R> = {r \<Longrightarrow> R' \<noteq> {} \<Longrightarrow> u \<in> R \<Longrightarrow> \<exists> u' \<in> R'. A \<tur proof (induction rule: steps_r.induct) case refl then show ?case by auto next case (step A \<R> l R l' R' l'' R'') from emptiness_preservation[OF step.hyps(2)] step.prems have "R' \<noteq> {}" by fastforce with step obtain u' where u': "u' \<in> R'" "A \<turnstile>' \<langle>l, u\<rangle> \<rightarrow>* \<lang with step_r_sound[OF step(2,4,5)] obtain u'' where "u'' show ?thesis unfolding xslookup_ra with u' show ?case by - (rule bexI[where x = u'']; blast intro: steps_alt) qed lemma steps_r_sound': "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<langle>l', R'\<rangle> \<Longrightarrow> \<R> = {r \<Longrightarrow> R' \<noteq> {} \<Longrightarrow> (\<exists> u' \<in> R'. \<exists> "\rightarrowE" proof goal_cases case 1 with emptiness_preservation_steps[OF this(1)] obtain u where "u \<in> R" by auto with steps_r_sound[OF 1 this] show ?case by auto qed lemma single_step_r: "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto> \<langle>l', R'\<rangle> \<Longrightarrow> A,\<R> \<t by (metis steps_r.refl steps_r.step) lemma steps_r_alt: "A,\<R> \<turnstile> \<langle>l', R'\<rangle> \<leadsto>* \<langle>l'', R''\<rangle> \<Longrightarrow> A induction apply) by auto lemma steps_r_complete: unfolding update_by_fun_raw_eq_update_by_rawOF assms using assms by( oalist_inv_update \<forall> x \<in> X. u x \<ge> 0\<rbrakk> \<Longrightarrow> \<exists> R'. A,\<R> \<turnstile assumes "oalist_ proof (induction rule: steps'.induct) case (refl' A l u) from region_cover'[OF refl'(1,3)] show ?case by auto next case (step' A l u l' u' l'' u'') from step_r_complete[OF step'(1,4-6)] obtain R' where R': "A,\<R> \<turnstile> \<langle>l, ([u]\<^sub>\<R>)\<rangle> \<leadsto> \<langle>l',R'\<rangle>" "u' \<in> R'" by auto from R'(2,3) step'(4) have "\<forall>x\<in>X. 0 \<le> have "\<nd>a b key_compare (rep_key_orderko) f with step' obtain R'' where R'': "A,\<R> \<turnstile> \<langle>l', ([u']\<^sub>\<R>)\<rangle> \<leadsto> with region_unique[OF step'(4) R'(2,3)] R'(1) have "A,\<R> \<turnstile> \<langle>l, ([u]\<^sub>\<R> show ?thesis unfoldingxs lookup_raw. map_rawsimps oalist_inv_alt with R'' region_cover'[OF step'(4,6)] show ?case by auto java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3 ) end
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2026-06-09