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1 theory Sets = Main: |
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2 |
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3 section{*Sets, Functions and Relations*} |
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4 |
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5 subsection{*Set Notation*} |
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6 |
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7 term "A \<union> B" |
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8 term "A \<inter> B" |
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9 term "A - B" |
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10 term "a \<in> A" |
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11 term "b \<notin> A" |
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12 term "{a,b}" |
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13 term "{x. P x}" |
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14 term "{x+y+eps |x y. x < y}" |
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15 term "\<Union> M" |
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16 term "\<Union>a \<in> A. F a" |
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17 |
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18 subsection{*Functions*} |
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19 |
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20 thm id_def |
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21 thm o_assoc |
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22 thm image_Int |
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23 thm vimage_Compl |
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24 |
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25 |
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26 subsection{*Relations*} |
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27 |
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28 thm Id_def |
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29 thm converse_comp |
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30 thm Image_def |
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31 thm relpow.simps |
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32 thm rtrancl_idemp |
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33 thm trancl_converse |
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34 |
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35 subsection{*Wellfoundedness*} |
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36 |
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37 thm wf_def |
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38 thm wf_iff_no_infinite_down_chain |
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39 |
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40 |
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41 subsection{*Fixed Point Operators*} |
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42 |
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43 thm lfp_def gfp_def |
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44 thm lfp_unfold |
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45 thm lfp_induct |
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46 |
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47 |
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48 subsection{*Case Study: Verified Model Checking*} |
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49 |
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50 |
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51 typedecl state |
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52 |
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53 consts M :: "(state \<times> state)set"; |
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54 |
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55 typedecl atom |
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56 |
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57 consts L :: "state \<Rightarrow> atom set" |
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58 |
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59 datatype formula = Atom atom |
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60 | Neg formula |
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61 | And formula formula |
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62 | AX formula |
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63 | EF formula |
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64 |
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65 consts valid :: "state \<Rightarrow> formula \<Rightarrow> bool" ("(_ \<Turnstile> _)" [80,80] 80) |
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66 |
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67 primrec |
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68 "s \<Turnstile> Atom a = (a \<in> L s)" |
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69 "s \<Turnstile> Neg f = (\<not>(s \<Turnstile> f))" |
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70 "s \<Turnstile> And f g = (s \<Turnstile> f \<and> s \<Turnstile> g)" |
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71 "s \<Turnstile> AX f = (\<forall>t. (s,t) \<in> M \<longrightarrow> t \<Turnstile> f)" |
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72 "s \<Turnstile> EF f = (\<exists>t. (s,t) \<in> M\<^sup>* \<and> t \<Turnstile> f)"; |
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73 |
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74 consts mc :: "formula \<Rightarrow> state set"; |
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75 primrec |
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76 "mc(Atom a) = {s. a \<in> L s}" |
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77 "mc(Neg f) = -mc f" |
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78 "mc(And f g) = mc f \<inter> mc g" |
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79 "mc(AX f) = {s. \<forall>t. (s,t) \<in> M \<longrightarrow> t \<in> mc f}" |
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80 "mc(EF f) = lfp(\<lambda>T. mc f \<union> (M\<inverse> `` T))" |
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81 |
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82 lemma mono_ef: "mono(\<lambda>T. A \<union> (M\<inverse> `` T))" |
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83 apply(rule monoI) |
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84 apply blast |
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85 done |
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86 |
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87 lemma EF_lemma: |
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88 "lfp(\<lambda>T. A \<union> (M\<inverse> `` T)) = {s. \<exists>t. (s,t) \<in> M\<^sup>* \<and> t \<in> A}" |
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89 apply(rule equalityI) |
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90 thm lfp_lowerbound |
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91 apply(rule lfp_lowerbound) |
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92 apply(blast intro: rtrancl_trans); |
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93 apply(rule subsetI) |
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94 apply(simp, clarify) |
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95 apply(erule converse_rtrancl_induct) |
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96 thm lfp_unfold[OF mono_ef] |
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97 apply(subst lfp_unfold[OF mono_ef]) |
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98 apply(blast) |
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99 apply(subst lfp_unfold[OF mono_ef]) |
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100 apply(blast) |
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101 done |
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102 |
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103 theorem "mc f = {s. s \<Turnstile> f}"; |
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104 apply(induct_tac f); |
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105 apply(auto simp add:EF_lemma); |
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106 done; |
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107 |
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108 text{* |
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109 \begin{exercise} |
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110 @{term AX} has a dual operator @{term EN}\footnote{We cannot use the customary @{text EX} |
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111 as that is the \textsc{ascii}-equivalent of @{text"\<exists>"}} |
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112 (``there exists a next state such that'') with the intended semantics |
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113 @{prop[display]"(s \<Turnstile> EN f) = (EX t. (s,t) : M & t \<Turnstile> f)"} |
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114 Fortunately, @{term"EN f"} can already be expressed as a PDL formula. How? |
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115 |
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116 Show that the semantics for @{term EF} satisfies the following recursion equation: |
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117 @{prop[display]"(s \<Turnstile> EF f) = (s \<Turnstile> f | s \<Turnstile> EN(EF f))"} |
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118 \end{exercise} |
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119 *} |
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120 |
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121 end |
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122 |
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123 |