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Dynamic_model.html
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<html>
<head>
<title>Dynamic_model.thy</title>
</head>
<body bgcolor="#ffffff">
<pre><font color="#000000"><span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 1 </font></span><font color="#006699"><strong>theory</strong></font> Dynamic_model
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 2 </font></span><font color="#009966"><strong>imports</strong></font> Main
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 3 </font></span><font color="#009966"><strong>begin</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 4 </font></span><font color="#006699"><strong>subsection</strong></font> <font color="#6600cc">{* Security State Machine *}</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 5 </font></span><font color="#006699"><strong>locale</strong></font> SM <font color="#000000"><strong>=</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 6 </font></span> <font color="#009966"><strong>fixes</strong></font> s0 <font color="#000000"><strong>::</strong></font> 's
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 7 </font></span> <font color="#009966"><strong>fixes</strong></font> step <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"'s ⇒ 'e ⇒ 's"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 8 </font></span> <font color="#009966"><strong>fixes</strong></font> domain <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"'e ⇒ ('d option)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 9 </font></span> <font color="#009966"><strong>fixes</strong></font> vpeq <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"'s ⇒ 'd ⇒ 's ⇒ bool"</font> <font color="#000000"><strong>(</strong></font><font color="#ff00cc">"(_ ∼ _ ∼ _)"</font><font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 10 </font></span> <font color="#009966"><strong>fixes</strong></font> interferes <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"'d ⇒ 's ⇒ 'd ⇒ bool"</font> <font color="#000000"><strong>(</strong></font><font color="#ff00cc">"(_ @ _ ↝_)"</font><font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 11 </font></span> <font color="#009966"><strong>assumes</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 12 </font></span> vpeq_transitive_lemma <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀ s t r d. (s ∼ d ∼ t) ∧ (t ∼ d ∼ r) ⟶ (s ∼ d ∼ r)"</font> <font color="#009966"><strong>and</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 13 </font></span> vpeq_symmetric_lemma <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀ s t d. (s ∼ d ∼ t) ⟶ (t ∼ d ∼ s)"</font> <font color="#009966"><strong>and</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 14 </font></span> vpeq_reflexive_lemma <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀ s d. (s ∼ d ∼ s)"</font> <font color="#009966"><strong>and</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 15 </font></span> interf_reflexive<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀d s. (d @ s ↝ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 16 </font></span><font color="#009966"><strong>begin</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 17 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 18 </font></span> <font color="#006699"><strong>definition</strong></font> non_interferes <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"'d ⇒ 's ⇒ 'd ⇒ bool"</font> <font color="#000000"><strong>(</strong></font><font color="#ff00cc">"(_ @ _ ∖↝ _)"</font><font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 19 </font></span> <font color="#009966"><strong>where</strong></font> <font color="#ff00cc">"(u @ s ∖↝ v) ≡ (u @ s ↝ v)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 20 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 21 </font></span> <font color="#006699"><strong>definition</strong></font> ivpeq <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"'s ⇒ 'd set ⇒ 's ⇒ bool"</font> <font color="#000000"><strong>(</strong></font><font color="#ff00cc">"(_ ≈ _ ≈ _)"</font><font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 22 </font></span> <font color="#009966"><strong>where</strong></font> <font color="#ff00cc">"ivpeq s D t ≡ ∀ d ∈ D. (s ∼ d ∼ t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 23 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 24 </font></span> <font color="#006699"><strong>primrec</strong></font> run <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"'s ⇒ 'e list ⇒ 's"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 25 </font></span> <font color="#009966"><strong>where</strong></font> run_Nil<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"run s [] = s"</font> <font color="#000000"><strong>|</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 26 </font></span> run_Cons<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"run s (a#as) = run (step s a) as "</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 27 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 28 </font></span> <font color="#006699"><strong>definition</strong></font> reachable <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"'s ⇒ 's ⇒ bool"</font> <font color="#000000"><strong>(</strong></font><font color="#ff00cc">"(_ ↪ _)"</font> <font color="#000000"><strong>[</strong></font>70<font color="#000000"><strong>,</strong></font>71<font color="#000000"><strong>]</strong></font> 60<font color="#000000"><strong>)</strong></font> <font color="#009966"><strong>where</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 29 </font></span> <font color="#ff00cc">"reachable s1 s2 ≡ (∃as. run s1 as = s2 )"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 30 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 31 </font></span> <font color="#006699"><strong>definition</strong></font> reachable0 <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"'s ⇒ bool"</font> <font color="#009966"><strong>where</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 32 </font></span> <font color="#ff00cc">"reachable0 s ≡ reachable s0 s"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 33 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 34 </font></span> <font color="#006699"><strong>declare</strong></font> non_interferes_def<font color="#000000"><strong>[</strong></font>cong<font color="#000000"><strong>]</strong></font> <font color="#009966"><strong>and</strong></font> ivpeq_def<font color="#000000"><strong>[</strong></font>cong<font color="#000000"><strong>]</strong></font> <font color="#009966"><strong>and</strong></font> reachable_def<font color="#000000"><strong>[</strong></font>cong<font color="#000000"><strong>]</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 35 </font></span> <font color="#009966"><strong>and</strong></font> reachable0_def<font color="#000000"><strong>[</strong></font>cong<font color="#000000"><strong>]</strong></font> <font color="#009966"><strong>and</strong></font> run.simps<font color="#000000"><strong>(</strong></font>1<font color="#000000"><strong>)</strong></font><font color="#000000"><strong>[</strong></font>cong<font color="#000000"><strong>]</strong></font> <font color="#009966"><strong>and</strong></font> run.simps<font color="#000000"><strong>(</strong></font>2<font color="#000000"><strong>)</strong></font><font color="#000000"><strong>[</strong></font>cong<font color="#000000"><strong>]</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 36 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 37 </font></span> <font color="#006699"><strong>lemma</strong></font> reachable_s0 <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 s0"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 38 </font></span> <font color="#006699"><strong>by</strong></font> <font color="#000000"><strong>(</strong></font>metis SM.reachable_def SM_axioms reachable0_def run.simps<font color="#000000"><strong>(</strong></font>1<font color="#000000"><strong>)</strong></font><font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 39 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 40 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 41 </font></span> <font color="#006699"><strong>lemma</strong></font> reachable_self <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable s s"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 42 </font></span> <font color="#006699"><strong>using</strong></font> reachable_def run.simps<font color="#000000"><strong>(</strong></font>1<font color="#000000"><strong>)</strong></font> <font color="#006699"><strong>by</strong></font> fastforce
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 43 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 44 </font></span> <font color="#006699"><strong>lemma</strong></font> reachable_step <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"s' = step s a ⟹ reachable s s'"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 45 </font></span> <font color="#006699"><strong>proof</strong></font>-
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 46 </font></span> <font color="#0099ff"><strong>assume</strong></font> a0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"s' = step s a"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 47 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#006699"><strong>have</strong></font> <font color="#ff00cc">"s' = run s [a]"</font> <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 48 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>using</strong></font> reachable_def <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 49 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 50 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 51 </font></span> <font color="#006699"><strong>lemma</strong></font> run_trans <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀C T V as bs. T = run C as ∧ V = run T bs ⟶ V = run C (as@bs)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 52 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 53 </font></span> <font color="#006699"><strong>{</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 54 </font></span> <font color="#0099ff"><strong>fix</strong></font> T V as bs
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 55 </font></span> <font color="#006699"><strong>have</strong></font> <font color="#ff00cc">"∀C. T = run C as ∧ V = run T bs ⟶ V = run C (as@bs)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 56 </font></span> <font color="#006699"><strong>proof</strong></font><font color="#000000"><strong>(</strong></font>induct as<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 57 </font></span> <font color="#0099ff"><strong>case</strong></font> Nil <font color="#0099ff"><strong>show</strong></font> ?case <font color="#006699"><strong>by</strong></font> simp
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 58 </font></span> <font color="#006699"><strong>next</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 59 </font></span> <font color="#0099ff"><strong>case</strong></font> <font color="#000000"><strong>(</strong></font>Cons c cs<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 60 </font></span> <font color="#0099ff"><strong>assume</strong></font> a0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀C. T = run C cs ∧ V = run T bs ⟶ V = run C (cs @ bs)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 61 </font></span> <font color="#0099ff"><strong>show</strong></font> ?case
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 62 </font></span> <font color="#006699"><strong>proof</strong></font>-
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 63 </font></span> <font color="#006699"><strong>{</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 64 </font></span> <font color="#0099ff"><strong>fix</strong></font> C
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 65 </font></span> <font color="#006699"><strong>have</strong></font> <font color="#ff00cc">"T = run C (c # cs) ∧ V = run T bs ⟶ V = run C ((c # cs) @ bs) "</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 66 </font></span> <font color="#006699"><strong>proof</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 67 </font></span> <font color="#0099ff"><strong>assume</strong></font> b0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"T = run C (c # cs) ∧ V = run T bs"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 68 </font></span> <font color="#006699"><strong>from</strong></font> b0 <font color="#0099ff"><strong>obtain</strong></font> C' <font color="#009966"><strong>where</strong></font> b2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"C' = step C c ∧ T = run C' cs"</font> <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 69 </font></span> <font color="#006699"><strong>with</strong></font> a0 b0 <font color="#006699"><strong>have</strong></font> <font color="#ff00cc">"V = run C' (cs@bs)"</font> <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 70 </font></span> <font color="#006699"><strong>with</strong></font> b2 <font color="#0099ff"><strong>show</strong></font> <font color="#ff00cc">"V = run C ((c # cs) @ bs)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 71 </font></span> <font color="#006699"><strong>using</strong></font> append_Cons run_Cons <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 72 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 73 </font></span> <font color="#006699"><strong>}</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 74 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 75 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 76 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 77 </font></span> <font color="#006699"><strong>}</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 78 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 79 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 80 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 81 </font></span> <font color="#006699"><strong>lemma</strong></font> reachable_trans <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"⟦reachable C T; reachable T V⟧ ⟹ reachable C V"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 82 </font></span> <font color="#006699"><strong>proof</strong></font>-
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 83 </font></span> <font color="#0099ff"><strong>assume</strong></font> a0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable C T"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 84 </font></span> <font color="#0099ff"><strong>assume</strong></font> a1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable T V"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 85 </font></span> <font color="#006699"><strong>from</strong></font> a0 <font color="#006699"><strong>have</strong></font> <font color="#ff00cc">"C = T ∨ (∃as. T = run C as)"</font> <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 86 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 87 </font></span> <font color="#006699"><strong>proof</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 88 </font></span> <font color="#0099ff"><strong>assume</strong></font> b0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"C = T"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 89 </font></span> <font color="#0099ff"><strong>show</strong></font> ?thesis
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 90 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 91 </font></span> <font color="#006699"><strong>from</strong></font> a1 <font color="#006699"><strong>have</strong></font> <font color="#ff00cc">"T = V ∨ (∃as. V = run T as)"</font> <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 92 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 93 </font></span> <font color="#006699"><strong>proof</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 94 </font></span> <font color="#0099ff"><strong>assume</strong></font> c0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"T=V"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 95 </font></span> <font color="#006699"><strong>with</strong></font> a0 <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 96 </font></span> <font color="#006699"><strong>next</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 97 </font></span> <font color="#0099ff"><strong>assume</strong></font> c0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(∃as. V = run T as)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 98 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>using</strong></font> a1 b0 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 99 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 100 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 101 </font></span> <font color="#006699"><strong>next</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 102 </font></span> <font color="#0099ff"><strong>assume</strong></font> b0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∃as. T = run C as"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 103 </font></span> <font color="#0099ff"><strong>show</strong></font> ?thesis
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 104 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 105 </font></span> <font color="#006699"><strong>from</strong></font> a1 <font color="#006699"><strong>have</strong></font> <font color="#ff00cc">"T = V ∨ (∃as. V = run T as)"</font> <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 106 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 107 </font></span> <font color="#006699"><strong>proof</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 108 </font></span> <font color="#0099ff"><strong>assume</strong></font> c0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"T=V"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 109 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>using</strong></font> a0 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 110 </font></span> <font color="#006699"><strong>next</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 111 </font></span> <font color="#0099ff"><strong>assume</strong></font> c0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(∃as. V = run T as)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 112 </font></span> <font color="#006699"><strong>from</strong></font> b0 <font color="#0099ff"><strong>obtain</strong></font> as <font color="#009966"><strong>where</strong></font> d0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"T = run C as"</font> <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 113 </font></span> <font color="#006699"><strong>from</strong></font> c0 <font color="#0099ff"><strong>obtain</strong></font> bs <font color="#009966"><strong>where</strong></font> d1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"V = run T bs"</font> <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 114 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>using</strong></font> d0 run_trans <font color="#006699"><strong>by</strong></font> fastforce
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 115 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 116 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 117 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 118 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 119 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 120 </font></span> <font color="#006699"><strong>lemma</strong></font> reachableStep <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"⟦reachable0 C; C' = step C a⟧ ⟹ reachable0 C'"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 121 </font></span> <font color="#006699"><strong>apply</strong></font> <font color="#000000"><strong>(</strong></font>simp add<font color="#000000"><strong>:</strong></font> reachable0_def<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 122 </font></span> <font color="#006699"><strong>using</strong></font> reachable_step reachable_trans <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 123 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 124 </font></span> <font color="#006699"><strong>lemma</strong></font> reachable0_reach <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"⟦reachable0 C; reachable C C'⟧ ⟹ reachable0 C'"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 125 </font></span> <font color="#006699"><strong>using</strong></font> reachable_trans <font color="#006699"><strong>by</strong></font> fastforce
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 126 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 127 </font></span> <font color="#006699"><strong>declare</strong></font> reachable_def<font color="#000000"><strong>[</strong></font>cong del<font color="#000000"><strong>]</strong></font> <font color="#009966"><strong>and</strong></font> reachable0_def<font color="#000000"><strong>[</strong></font>cong del<font color="#000000"><strong>]</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 128 </font></span><font color="#009966"><strong>end</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 129 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 130 </font></span><font color="#006699"><strong>subsection</strong></font><font color="#6600cc">{* Information flow security properties *}</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 131 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 132 </font></span><font color="#006699"><strong>locale</strong></font> SM_enabled <font color="#000000"><strong>=</strong></font> SM s0 step domain vpeq interferes
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 133 </font></span> <font color="#009966"><strong>for</strong></font> s0 <font color="#000000"><strong>::</strong></font> 's <font color="#009966"><strong>and</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 134 </font></span> step <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"'s ⇒ 'e ⇒ 's"</font> <font color="#009966"><strong>and</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 135 </font></span> domain <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"'e ⇒ ('d option)"</font> <font color="#009966"><strong>and</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 136 </font></span> vpeq <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"'s ⇒ 'd ⇒ 's ⇒ bool"</font> <font color="#000000"><strong>(</strong></font><font color="#ff00cc">"(_ ∼ _ ∼ _)"</font><font color="#000000"><strong>)</strong></font> <font color="#009966"><strong>and</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 137 </font></span> interferes <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"'d ⇒ 's ⇒ 'd ⇒ bool"</font> <font color="#000000"><strong>(</strong></font><font color="#ff00cc">"(_ @ _ ↝_)"</font><font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 138 </font></span> <font color="#000000"><strong>+</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 139 </font></span> <font color="#009966"><strong>assumes</strong></font> enabled0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀s a. reachable0 s ⟶ (∃ s'. s' = step s a)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 140 </font></span> <font color="#009966"><strong>and</strong></font> policy_respect<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀v u s t. (s ∼ u ∼ t)</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 141 </font></span><font color="#ff00cc"> ⟶ (interferes v s u = interferes v t u)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 142 </font></span><font color="#009966"><strong>begin</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 143 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 144 </font></span> <font color="#006699"><strong>lemma</strong></font> enabled <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 s ⟹ (∃ s'. s' = step s a)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 145 </font></span> <font color="#006699"><strong>using</strong></font> enabled0 <font color="#006699"><strong>by</strong></font> simp
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 146 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 147 </font></span> <font color="#006699"><strong>primrec</strong></font> sources <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"'e list ⇒ 'd ⇒ 's ⇒ 'd set"</font> <font color="#009966"><strong>where</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 148 </font></span> sources_Nil<font color="#000000"><strong>:</strong></font><font color="#ff00cc">"sources [] d s = {d}"</font> <font color="#000000"><strong>|</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 149 </font></span> sources_Cons<font color="#000000"><strong>:</strong></font><font color="#ff00cc">"sources (a # as) d s = (⋃{sources as d (step s a)}) ∪ </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 150 </font></span><font color="#ff00cc"> {w . w = the (domain a) ∧ (∃v . interferes w s v ∧ v∈sources as d (step s a))}"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 151 </font></span> <font color="#006699"><strong>declare</strong></font> sources_Nil <font color="#000000"><strong>[</strong></font>simp del<font color="#000000"><strong>]</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 152 </font></span> <font color="#006699"><strong>declare</strong></font> sources_Cons <font color="#000000"><strong>[</strong></font>simp del<font color="#000000"><strong>]</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 153 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 154 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 155 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 156 </font></span> <font color="#006699"><strong>primrec</strong></font> ipurge <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"'e list ⇒ 'd ⇒ 's ⇒ 'e list"</font> <font color="#009966"><strong>where</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 157 </font></span> ipurge_Nil<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"ipurge [] u s = []"</font> <font color="#000000"><strong>|</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 158 </font></span> ipurge_Cons<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"ipurge (a#as) u s = (if (the (domain a) ∈ (sources (a#as) u s))</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 159 </font></span><font color="#ff00cc"> then</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 160 </font></span><font color="#ff00cc"> a # ipurge as u (step s a)</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 161 </font></span><font color="#ff00cc"> else</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 162 </font></span><font color="#ff00cc"> ipurge as u (step s a)</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 163 </font></span><font color="#ff00cc"> )"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 164 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 165 </font></span> <font color="#006699"><strong>definition</strong></font> observ_equivalence <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"'s ⇒ 'e list ⇒ 's ⇒ </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 166 </font></span><font color="#ff00cc"> 'e list ⇒ 'd ⇒ bool"</font> <font color="#000000"><strong>(</strong></font><font color="#ff00cc">"(_ ⊲ _ ≅ _ ⊲ _ @ _)"</font><font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 167 </font></span> <font color="#009966"><strong>where</strong></font> <font color="#ff00cc">"observ_equivalence s as t bs d ≡ </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 168 </font></span><font color="#ff00cc"> ((run s as) ∼ d ∼ (run t bs))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 169 </font></span> <font color="#006699"><strong>declare</strong></font> observ_equivalence_def<font color="#000000"><strong>[</strong></font>cong<font color="#000000"><strong>]</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 170 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 171 </font></span> <font color="#006699"><strong>lemma</strong></font> observ_equiv_sym<font color="#000000"><strong>:</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 172 </font></span> <font color="#ff00cc">"(s ⊲ as ≅ t ⊲ bs @ d) ⟹ (t ⊲ bs ≅ s ⊲ as @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 173 </font></span> <font color="#006699"><strong>using</strong></font> observ_equivalence_def vpeq_symmetric_lemma <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 174 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 175 </font></span> <font color="#006699"><strong>lemma</strong></font> observ_equiv_trans<font color="#000000"><strong>:</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 176 </font></span> <font color="#ff00cc">"⟦reachable0 t; (s ⊲ as ≅ t ⊲ bs @ d); (t ⊲ bs ≅ x ⊲ cs @ d)⟧ ⟹ (s ⊲ as ≅ x ⊲ cs @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 177 </font></span> <font color="#006699"><strong>using</strong></font> observ_equivalence_def vpeq_transitive_lemma <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 178 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 179 </font></span> <font color="#006699"><strong>definition</strong></font> noninterference_r <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"bool"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 180 </font></span> <font color="#009966"><strong>where</strong></font> <font color="#ff00cc">"noninterference_r ≡ ∀d as s. reachable0 s ⟶ (s ⊲ as ≅ s ⊲ (ipurge as d s) @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 181 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 182 </font></span> <font color="#006699"><strong>definition</strong></font> noninterference <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"bool"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 183 </font></span> <font color="#009966"><strong>where</strong></font> <font color="#ff00cc">"noninterference ≡ ∀d as. (s0 ⊲ as ≅ s0 ⊲ (ipurge as d s0) @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 184 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 185 </font></span> <font color="#006699"><strong>definition</strong></font> weak_noninterference <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"bool"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 186 </font></span> <font color="#009966"><strong>where</strong></font> <font color="#ff00cc">"weak_noninterference ≡ ∀d as bs. ipurge as d s0 = ipurge bs d s0</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 187 </font></span><font color="#ff00cc"> ⟶ (s0 ⊲ as ≅ s0 ⊲ bs @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 188 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 189 </font></span> <font color="#006699"><strong>definition</strong></font> weak_noninterference_r <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"bool"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 190 </font></span> <font color="#009966"><strong>where</strong></font> <font color="#ff00cc">"weak_noninterference_r ≡ ∀d as bs s. reachable0 s ∧ ipurge as d s = ipurge bs d s</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 191 </font></span><font color="#ff00cc"> ⟶ (s ⊲ as ≅ s ⊲ bs @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 192 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 193 </font></span> <font color="#006699"><strong>definition</strong></font> noninfluence<font color="#000000"><strong>::</strong></font><font color="#ff00cc">"bool"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 194 </font></span> <font color="#009966"><strong>where</strong></font> <font color="#ff00cc">"noninfluence ≡ ∀ d as s t. reachable0 s ∧ reachable0 t</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 195 </font></span><font color="#ff00cc"> ∧ (s ≈ (sources as d s) ≈ t)</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 196 </font></span><font color="#ff00cc"> ⟶ (s ⊲ as ≅ t ⊲ (ipurge as d t) @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 197 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 198 </font></span> <font color="#006699"><strong>definition</strong></font> weak_noninfluence <font color="#000000"><strong>::</strong></font><font color="#ff00cc">"bool"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 199 </font></span> <font color="#009966"><strong>where</strong></font> <font color="#ff00cc">"weak_noninfluence ≡ ∀ d as bs s t . reachable0 s ∧ reachable0 t ∧ (s ≈ (sources as d s) ≈ t) </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 200 </font></span><font color="#ff00cc"> ∧ ipurge as d t = ipurge bs d t</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 201 </font></span><font color="#ff00cc"> ⟶ (s ⊲ as ≅ t ⊲ bs @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 202 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 203 </font></span> <font color="#006699"><strong>definition</strong></font> weak_noninfluence2 <font color="#000000"><strong>::</strong></font><font color="#ff00cc">"bool"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 204 </font></span> <font color="#009966"><strong>where</strong></font> <font color="#ff00cc">"weak_noninfluence2 ≡ ∀ d as bs s t . reachable0 s ∧ reachable0 t ∧ (s ≈ (sources as d s) ≈ t) </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 205 </font></span><font color="#ff00cc"> ∧ ipurge as d s = ipurge bs d t</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 206 </font></span><font color="#ff00cc"> ⟶ (s ⊲ as ≅ t ⊲ bs @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 207 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 208 </font></span> <font color="#006699"><strong>definition</strong></font> nonleakage <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"bool"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 209 </font></span> <font color="#009966"><strong>where</strong></font> <font color="#ff00cc">"nonleakage ≡ ∀d as s t. reachable0 s ∧ reachable0 t</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 210 </font></span><font color="#ff00cc"> ∧ (s ≈ (sources as d s) ≈ t) ⟶ (s ⊲ as ≅ t ⊲ as @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 211 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 212 </font></span> <font color="#006699"><strong>declare</strong></font> noninterference_r_def<font color="#000000"><strong>[</strong></font>cong<font color="#000000"><strong>]</strong></font> <font color="#009966"><strong>and</strong></font> noninterference_def<font color="#000000"><strong>[</strong></font>cong<font color="#000000"><strong>]</strong></font> <font color="#009966"><strong>and</strong></font> weak_noninterference_def<font color="#000000"><strong>[</strong></font>cong<font color="#000000"><strong>]</strong></font> <font color="#009966"><strong>and</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 213 </font></span> weak_noninterference_r_def<font color="#000000"><strong>[</strong></font>cong<font color="#000000"><strong>]</strong></font> <font color="#009966"><strong>and</strong></font> noninfluence_def<font color="#000000"><strong>[</strong></font>cong<font color="#000000"><strong>]</strong></font> <font color="#009966"><strong>and</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 214 </font></span> weak_noninfluence_def<font color="#000000"><strong>[</strong></font>cong<font color="#000000"><strong>]</strong></font> <font color="#009966"><strong>and</strong></font> weak_noninfluence2_def<font color="#000000"><strong>[</strong></font>cong<font color="#000000"><strong>]</strong></font> <font color="#009966"><strong>and</strong></font> nonleakage_def<font color="#000000"><strong>[</strong></font>cong<font color="#000000"><strong>]</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 215 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 216 </font></span><font color="#006699"><strong>subsection</strong></font><font color="#6600cc">{* Unwinding conditions*}</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 217 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 218 </font></span> <font color="#006699"><strong>definition</strong></font> dynamic_step_consistent <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"bool"</font> <font color="#009966"><strong>where</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 219 </font></span> <font color="#ff00cc">"dynamic_step_consistent ≡ ∀a d s t. reachable0 s ∧ reachable0 t ∧ (s ∼ d ∼ t) ∧ </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 220 </font></span><font color="#ff00cc"> (((the (domain a)) @ s ↝ d) ⟶ (s ∼ (the (domain a)) ∼ t))</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 221 </font></span><font color="#ff00cc"> ⟶ ((step s a) ∼ d ∼ (step t a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 222 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 223 </font></span> <font color="#006699"><strong>definition</strong></font> dynamic_weakly_step_consistent <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"bool"</font> <font color="#009966"><strong>where</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 224 </font></span> <font color="#ff00cc">"dynamic_weakly_step_consistent ≡ ∀a d s t. reachable0 s ∧ reachable0 t ∧ (s ∼ d ∼ t) ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 225 </font></span><font color="#ff00cc"> ((the (domain a)) @ s ↝ d) ∧ (s ∼ (the (domain a)) ∼ t) </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 226 </font></span><font color="#ff00cc"> ⟶ ((step s a) ∼ d ∼ (step t a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 227 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 228 </font></span> <font color="#006699"><strong>definition</strong></font> dynamic_weakly_step_consistent_e <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"'e ⇒ bool"</font> <font color="#009966"><strong>where</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 229 </font></span> <font color="#ff00cc">"dynamic_weakly_step_consistent_e a ≡ ∀d s t. reachable0 s ∧ reachable0 t ∧ (s ∼ d ∼ t) ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 230 </font></span><font color="#ff00cc"> ((the (domain a)) @ s ↝ d) ∧ (s ∼ (the (domain a)) ∼ t) </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 231 </font></span><font color="#ff00cc"> ⟶ ((step s a) ∼ d ∼ (step t a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 232 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 233 </font></span> <font color="#006699"><strong>lemma</strong></font> dynamic_weakly_step_consistent_all_evt <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"dynamic_weakly_step_consistent = (∀a. dynamic_weakly_step_consistent_e a)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 234 </font></span> <font color="#006699"><strong>by</strong></font> <font color="#000000"><strong>(</strong></font>simp add<font color="#000000"><strong>:</strong></font> dynamic_weakly_step_consistent_def dynamic_weakly_step_consistent_e_def<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 235 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 236 </font></span> <font color="#006699"><strong>definition</strong></font> dynamic_local_respect <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"bool"</font> <font color="#009966"><strong>where</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 237 </font></span> <font color="#ff00cc">"dynamic_local_respect ≡ ∀a d s. reachable0 s ∧ ¬((the (domain a)) @ s ↝ d) ⟶ (s ∼ d ∼ (step s a)) "</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 238 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 239 </font></span> <font color="#006699"><strong>definition</strong></font> dynamic_local_respect_e <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"'e ⇒ bool"</font> <font color="#009966"><strong>where</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 240 </font></span> <font color="#ff00cc">"dynamic_local_respect_e a ≡ ∀d s. reachable0 s ∧ ¬((the (domain a)) @ s ↝ d) ⟶ (s ∼ d ∼ (step s a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 241 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 242 </font></span> <font color="#006699"><strong>lemma</strong></font> dynamic_local_respect_all_evt <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"dynamic_local_respect = (∀a. dynamic_local_respect_e a)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 243 </font></span> <font color="#006699"><strong>by</strong></font> <font color="#000000"><strong>(</strong></font>simp add<font color="#000000"><strong>:</strong></font> dynamic_local_respect_def dynamic_local_respect_e_def<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 244 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 245 </font></span> <font color="#006699"><strong>declare</strong></font> dynamic_step_consistent_def <font color="#000000"><strong>[</strong></font>cong<font color="#000000"><strong>]</strong></font> <font color="#009966"><strong>and</strong></font> dynamic_weakly_step_consistent_def <font color="#000000"><strong>[</strong></font>cong<font color="#000000"><strong>]</strong></font> <font color="#009966"><strong>and</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 246 </font></span> dynamic_local_respect_def <font color="#000000"><strong>[</strong></font>cong<font color="#000000"><strong>]</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 247 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 248 </font></span> <font color="#006699"><strong>lemma</strong></font> step_cons_impl_weak <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"dynamic_step_consistent ⟹ dynamic_weakly_step_consistent"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 249 </font></span> <font color="#006699"><strong>using</strong></font> dynamic_step_consistent_def dynamic_weakly_step_consistent_def <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 250 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 251 </font></span> <font color="#006699"><strong>definition</strong></font> lemma_local <font color="#000000"><strong>::</strong></font> <font color="#ff00cc">"bool"</font> <font color="#009966"><strong>where</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 252 </font></span> <font color="#ff00cc">"lemma_local ≡ ∀s a as u. the (domain a) ∉ sources (a # as) u s ⟶ (s ≈ (sources (a # as) u s) ≈ (step s a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 253 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 254 </font></span> <font color="#006699"><strong>lemma</strong></font> weak_with_step_cons<font color="#000000"><strong>:</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 255 </font></span> <font color="#009966"><strong>assumes</strong></font> p1<font color="#000000"><strong>:</strong></font> dynamic_weakly_step_consistent
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 256 </font></span> <font color="#009966"><strong>and</strong></font> p2<font color="#000000"><strong>:</strong></font> dynamic_local_respect
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 257 </font></span> <font color="#009966"><strong>shows</strong></font> dynamic_step_consistent
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 258 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 259 </font></span> <font color="#006699"><strong>{</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 260 </font></span> <font color="#0099ff"><strong>fix</strong></font> a d s t
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 261 </font></span> <font color="#006699"><strong>have</strong></font> <font color="#ff00cc">"reachable0 s ∧ reachable0 t ∧ (s ∼ d ∼ t) ∧ </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 262 </font></span><font color="#ff00cc"> (((the (domain a)) @ s ↝ d) ⟶ (s ∼ (the (domain a)) ∼ t))</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 263 </font></span><font color="#ff00cc"> ⟶ ((step s a) ∼ d ∼ (step t a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 264 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 265 </font></span> <font color="#006699"><strong>{</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 266 </font></span> <font color="#0099ff"><strong>assume</strong></font> a0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 s"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 267 </font></span> <font color="#0099ff"><strong>assume</strong></font> a1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 t"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 268 </font></span> <font color="#0099ff"><strong>assume</strong></font> a2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ∼ d ∼ t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 269 </font></span> <font color="#0099ff"><strong>assume</strong></font> a3<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(((the (domain a)) @ s ↝ d) ⟶ (s ∼ (the (domain a)) ∼ t))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 270 </font></span> <font color="#006699"><strong>have</strong></font> <font color="#ff00cc">"((step s a) ∼ d ∼ (step t a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 271 </font></span> <font color="#006699"><strong>proof</strong></font> <font color="#000000"><strong>(</strong></font>cases <font color="#ff00cc">"((the (domain a)) @ s ↝ d)"</font><font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 272 </font></span> <font color="#0099ff"><strong>assume</strong></font> b0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"((the (domain a)) @ s ↝ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 273 </font></span> <font color="#006699"><strong>have</strong></font> b1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ∼ (the (domain a)) ∼ t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 274 </font></span> <font color="#006699"><strong>using</strong></font> b0 a3 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 275 </font></span> <font color="#006699"><strong>have</strong></font> b2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"((step s a) ∼ d ∼ (step t a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 276 </font></span> <font color="#006699"><strong>using</strong></font> a0 a1 a2 b0 b1 p1 dynamic_weakly_step_consistent_def <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 277 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 278 </font></span> <font color="#006699"><strong>next</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 279 </font></span> <font color="#0099ff"><strong>assume</strong></font> b0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"¬((the (domain a)) @ s ↝ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 280 </font></span> <font color="#006699"><strong>have</strong></font> b1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"¬((the (domain a)) @ t ↝ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 281 </font></span> <font color="#006699"><strong>using</strong></font> a0 a1 a2 b0 policy_respect <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 282 </font></span> <font color="#006699"><strong>have</strong></font> b2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"s ∼ d ∼ (step s a)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 283 </font></span> <font color="#006699"><strong>using</strong></font> b0 p2 a0 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 284 </font></span> <font color="#006699"><strong>have</strong></font> b3<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"t ∼ d ∼ (step t a)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 285 </font></span> <font color="#006699"><strong>using</strong></font> b1 p2 a1 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 286 </font></span> <font color="#006699"><strong>have</strong></font> b4<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"((step s a) ∼ d ∼ (step t a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 287 </font></span> <font color="#006699"><strong>using</strong></font> b2 b3 a2 vpeq_symmetric_lemma vpeq_transitive_lemma <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 288 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 289 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 290 </font></span> <font color="#006699"><strong>}</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 291 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 292 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 293 </font></span> <font color="#006699"><strong>}</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 294 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 295 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 296 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 297 </font></span><font color="#006699"><strong>subsection</strong></font><font color="#6600cc">{* Lemmas for the inference framework *}</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 298 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 299 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 300 </font></span> <font color="#006699"><strong>lemma</strong></font> sources_refl<font color="#000000"><strong>:</strong></font><font color="#ff00cc">"reachable0 s ⟹ u ∈ sources as u s"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 301 </font></span> <font color="#006699"><strong>apply</strong></font><font color="#000000"><strong>(</strong></font>induct as arbitrary<font color="#000000"><strong>:</strong></font> s<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 302 </font></span> <font color="#006699"><strong>apply</strong></font><font color="#000000"><strong>(</strong></font>simp add<font color="#000000"><strong>:</strong></font> sources_Nil<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 303 </font></span> <font color="#006699"><strong>apply</strong></font><font color="#000000"><strong>(</strong></font>simp add<font color="#000000"><strong>:</strong></font> sources_Cons<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 304 </font></span> <font color="#006699"><strong>using</strong></font> enabled reachableStep
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 305 </font></span> <font color="#006699"><strong>by</strong></font> metis
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 306 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 307 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 308 </font></span> <font color="#006699"><strong>lemma</strong></font> lemma_1_sub_1 <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"⟦reachable0 s ;</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 309 </font></span><font color="#ff00cc"> dynamic_local_respect;</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 310 </font></span><font color="#ff00cc"> the (domain a) ∉ sources (a # as) u s;</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 311 </font></span><font color="#ff00cc"> (s ≈ (sources (a # as) u s) ≈ t)⟧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 312 </font></span><font color="#ff00cc"> ⟹ (s ≈ (sources as u (step s a)) ≈ (step s a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 313 </font></span> <font color="#006699"><strong>apply</strong></font> <font color="#000000"><strong>(</strong></font>simp add<font color="#000000"><strong>:</strong></font>dynamic_local_respect_def sources_Cons<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 314 </font></span> <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 315 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 316 </font></span> <font color="#006699"><strong>lemma</strong></font> lemma_1_sub_2 <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"⟦reachable0 s ;</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 317 </font></span><font color="#ff00cc"> reachable0 t ;</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 318 </font></span><font color="#ff00cc"> dynamic_local_respect;</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 319 </font></span><font color="#ff00cc"> the (domain a) ∉ sources (a # as) u s; </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 320 </font></span><font color="#ff00cc"> (s ≈ (sources (a # as) u s) ≈ t)⟧ </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 321 </font></span><font color="#ff00cc"> ⟹ (t ≈ (sources as u (step s a)) ≈ (step t a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 322 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 323 </font></span> <font color="#0099ff"><strong>assume</strong></font> a1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 s"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 324 </font></span> <font color="#0099ff"><strong>assume</strong></font> a2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 t"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 325 </font></span> <font color="#0099ff"><strong>assume</strong></font> a3<font color="#000000"><strong>:</strong></font> dynamic_local_respect
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 326 </font></span> <font color="#0099ff"><strong>assume</strong></font> a6<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"the (domain a) ∉ sources (a # as) u s"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 327 </font></span> <font color="#0099ff"><strong>assume</strong></font> a7<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ≈ (sources (a # as) u s) ≈ t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 328 </font></span> <font color="#006699"><strong>have</strong></font> b1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀v. v∈sources as u (step s a) ⟶ ¬interferes (the (domain a)) s v"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 329 </font></span> <font color="#006699"><strong>using</strong></font> a6 sources_Cons <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 330 </font></span> <font color="#006699"><strong>have</strong></font> b2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"sources (a # as) u s = sources as u (step s a)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 331 </font></span> <font color="#006699"><strong>using</strong></font> a6 sources_Cons <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 332 </font></span> <font color="#006699"><strong>have</strong></font> b3<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀v. v∈sources as u (step s a) ⟶ (s ∼ v ∼ t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 333 </font></span> <font color="#006699"><strong>using</strong></font> a7 b2 ivpeq_def <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 334 </font></span> <font color="#006699"><strong>have</strong></font> b4<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀v. v∈sources as u (step s a) ⟶ ¬interferes (the (domain a)) t v"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 335 </font></span> <font color="#006699"><strong>using</strong></font> a1 a2 policy_respect b1 b3 <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 336 </font></span> <font color="#006699"><strong>have</strong></font> b5<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀v. v∈sources as u (step s a) ⟶ (t ∼ v ∼ (step t a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 337 </font></span> <font color="#006699"><strong>using</strong></font> a2 a3 b4 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 338 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 339 </font></span> <font color="#006699"><strong>using</strong></font> ivpeq_def <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 340 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 341 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 342 </font></span> <font color="#006699"><strong>lemma</strong></font> lemma_1_sub_3 <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"⟦</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 343 </font></span><font color="#ff00cc"> the (domain a) ∉ sources (a # as) u s;</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 344 </font></span><font color="#ff00cc"> (s ≈ (sources (a # as) u s) ≈ t)⟧ </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 345 </font></span><font color="#ff00cc"> ⟹ (s ≈ (sources as u (step s a)) ≈ t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 346 </font></span> <font color="#006699"><strong>apply</strong></font> <font color="#000000"><strong>(</strong></font>simp add<font color="#000000"><strong>:</strong></font>sources_Cons<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 347 </font></span> <font color="#006699"><strong>apply</strong></font> <font color="#000000"><strong>(</strong></font>simp add<font color="#000000"><strong>:</strong></font>sources_Cons<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 348 </font></span> <font color="#006699"><strong>done</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 349 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 350 </font></span> <font color="#006699"><strong>lemma</strong></font> lemma_1_sub_4 <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"⟦(s ≈ (sources as u (step s a)) ≈ t);</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 351 </font></span><font color="#ff00cc"> (s ≈ (sources as u (step s a)) ≈ (step s a));</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 352 </font></span><font color="#ff00cc"> (t ≈ (sources as u (step s a)) ≈ (step t a)) ⟧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 353 </font></span><font color="#ff00cc"> ⟹ ((step s a) ≈(sources as u (step s a)) ≈ (step t a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 354 </font></span> <font color="#006699"><strong>by</strong></font> <font color="#000000"><strong>(</strong></font>meson ivpeq_def vpeq_symmetric_lemma vpeq_transitive_lemma<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 355 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 356 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 357 </font></span> <font color="#006699"><strong>lemma</strong></font> lemma_1 <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"⟦reachable0 s;</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 358 </font></span><font color="#ff00cc"> reachable0 t;</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 359 </font></span><font color="#ff00cc"> dynamic_step_consistent;</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 360 </font></span><font color="#ff00cc"> dynamic_local_respect;</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 361 </font></span><font color="#ff00cc"> (s ≈ (sources (a # as) u s) ≈ t)⟧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 362 </font></span><font color="#ff00cc"> ⟹ ((step s a) ≈ (sources as u (step s a)) ≈ (step t a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 363 </font></span> <font color="#006699"><strong>apply</strong></font> <font color="#000000"><strong>(</strong></font>case_tac <font color="#ff00cc">"the (domain a)∈sources (a # as) u s"</font><font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 364 </font></span> <font color="#006699"><strong>apply</strong></font> <font color="#000000"><strong>(</strong></font>simp add<font color="#000000"><strong>:</strong></font> dynamic_step_consistent_def<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 365 </font></span> <font color="#006699"><strong>apply</strong></font> <font color="#000000"><strong>(</strong></font>simp add<font color="#000000"><strong>:</strong></font> sources_Cons<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 366 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 367 </font></span> <font color="#0099ff"><strong>assume</strong></font> a1<font color="#000000"><strong>:</strong></font> dynamic_local_respect
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 368 </font></span> <font color="#0099ff"><strong>assume</strong></font> a4<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"the (domain a) ∉ sources (a # as) u s"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 369 </font></span> <font color="#0099ff"><strong>assume</strong></font> a5<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ≈ (sources (a # as) u s) ≈ t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 370 </font></span> <font color="#0099ff"><strong>assume</strong></font> b0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 s"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 371 </font></span> <font color="#0099ff"><strong>assume</strong></font> b1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 t"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 372 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 373 </font></span> <font color="#006699"><strong>have</strong></font> a6<font color="#000000"><strong>:</strong></font><font color="#ff00cc">"(s ≈ (sources as u (step s a)) ≈ t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 374 </font></span> <font color="#006699"><strong>using</strong></font> a1 policy_respect a4 a5 lemma_1_sub_3 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 375 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#006699"><strong>have</strong></font> a7<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ≈ (sources as u (step s a)) ≈ (step s a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 376 </font></span> <font color="#006699"><strong>using</strong></font> b0 a1 policy_respect a4 a5 lemma_1_sub_1 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 377 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#006699"><strong>have</strong></font> a8<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(t ≈ (sources as u (step s a)) ≈ (step t a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 378 </font></span> <font color="#006699"><strong>using</strong></font> b1 b0 a1 policy_respect a4 a5 lemma_1_sub_2 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 379 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> <font color="#ff00cc">" ((step s a) ≈(sources as u (step s a)) ≈ (step t a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 380 </font></span> <font color="#006699"><strong>using</strong></font> a6 a7 lemma_1_sub_4 <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 381 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 382 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 383 </font></span> <font color="#006699"><strong>lemma</strong></font> lemma_2 <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"⟦reachable0 s;</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 384 </font></span><font color="#ff00cc"> dynamic_local_respect;</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 385 </font></span><font color="#ff00cc"> the (domain a) ∉ sources (a # as) u s⟧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 386 </font></span><font color="#ff00cc"> ⟹ (s ≈ (sources as u (step s a)) ≈ (step s a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 387 </font></span> <font color="#006699"><strong>apply</strong></font> <font color="#000000"><strong>(</strong></font>simp add<font color="#000000"><strong>:</strong></font>dynamic_local_respect_def<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 388 </font></span> <font color="#006699"><strong>apply</strong></font> <font color="#000000"><strong>(</strong></font>simp add<font color="#000000"><strong>:</strong></font>sources_Cons<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 389 </font></span> <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 390 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 391 </font></span> <font color="#006699"><strong>lemma</strong></font> sources_eq1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀s t as u. reachable0 s ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 392 </font></span><font color="#ff00cc"> reachable0 t ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 393 </font></span><font color="#ff00cc"> dynamic_step_consistent ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 394 </font></span><font color="#ff00cc"> dynamic_local_respect ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 395 </font></span><font color="#ff00cc"> (s ≈ (sources as u s) ≈ t)</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 396 </font></span><font color="#ff00cc"> ⟶ (sources as u s) = (sources as u t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 397 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 398 </font></span> <font color="#006699"><strong>{</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 399 </font></span> <font color="#0099ff"><strong>fix</strong></font> as
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 400 </font></span> <font color="#006699"><strong>have</strong></font> <font color="#ff00cc">"∀s t u. reachable0 s ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 401 </font></span><font color="#ff00cc"> reachable0 t ∧ </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 402 </font></span><font color="#ff00cc"> dynamic_step_consistent ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 403 </font></span><font color="#ff00cc"> dynamic_local_respect ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 404 </font></span><font color="#ff00cc"> (s ≈ (sources as u s) ≈ t)</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 405 </font></span><font color="#ff00cc"> ⟶ (sources as u s) = (sources as u t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 406 </font></span> <font color="#006699"><strong>proof</strong></font><font color="#000000"><strong>(</strong></font>induct as<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 407 </font></span> <font color="#0099ff"><strong>case</strong></font> Nil <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?case <font color="#006699"><strong>by</strong></font> <font color="#000000"><strong>(</strong></font>simp add<font color="#000000"><strong>:</strong></font> sources_Nil<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 408 </font></span> <font color="#006699"><strong>next</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 409 </font></span> <font color="#0099ff"><strong>case</strong></font> <font color="#000000"><strong>(</strong></font>Cons b bs<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 410 </font></span> <font color="#0099ff"><strong>assume</strong></font> p0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀s t u.((reachable0 s) </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 411 </font></span><font color="#ff00cc"> ∧ (reachable0 t) </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 412 </font></span><font color="#ff00cc"> ∧ dynamic_step_consistent </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 413 </font></span><font color="#ff00cc"> ∧ dynamic_local_respect </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 414 </font></span><font color="#ff00cc"> ∧ (s ≈ (sources bs u s) ≈ t)) ⟶</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 415 </font></span><font color="#ff00cc"> (sources bs u s) = (sources bs u t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 416 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?case
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 417 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 418 </font></span> <font color="#006699"><strong>{</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 419 </font></span> <font color="#0099ff"><strong>fix</strong></font> s t u
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 420 </font></span> <font color="#0099ff"><strong>assume</strong></font> p1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 s"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 421 </font></span> <font color="#0099ff"><strong>assume</strong></font> p2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 t"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 422 </font></span> <font color="#0099ff"><strong>assume</strong></font> p3<font color="#000000"><strong>:</strong></font> dynamic_step_consistent
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 423 </font></span> <font color="#0099ff"><strong>assume</strong></font> p5<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"dynamic_local_respect"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 424 </font></span> <font color="#0099ff"><strong>assume</strong></font> p9<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ≈ (sources (b # bs) u s) ≈ t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 425 </font></span> <font color="#006699"><strong>have</strong></font> a2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"((step s b) ≈ (sources bs u (step s b)) ≈ (step t b))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 426 </font></span> <font color="#006699"><strong>using</strong></font> lemma_1 p1 p2 p3 policy_respect p5 p9 <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 427 </font></span> <font color="#006699"><strong>have</strong></font> a3<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"sources (b # bs) u s = sources (b # bs) u t"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 428 </font></span> <font color="#006699"><strong>proof</strong></font> <font color="#000000"><strong>(</strong></font>cases <font color="#ff00cc">"the (domain b) ∈ (sources (b # bs) u s)"</font><font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 429 </font></span> <font color="#0099ff"><strong>assume</strong></font> b0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"the (domain b) ∈ (sources (b # bs) u s)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 430 </font></span> <font color="#006699"><strong>have</strong></font> b1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"s ∼ (the(domain b)) ∼ t"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 431 </font></span> <font color="#006699"><strong>using</strong></font> b0 p9 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 432 </font></span> <font color="#006699"><strong>have</strong></font> b3<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"interferes (the (domain b)) s u = interferes (the (domain b)) t u "</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 433 </font></span> <font color="#006699"><strong>using</strong></font> p1 p2 policy_respect p9 sources_refl <font color="#006699"><strong>by</strong></font> fastforce
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 434 </font></span> <font color="#006699"><strong>have</strong></font> b4<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(sources bs u (step s b)) = (sources bs u (step t b))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 435 </font></span> <font color="#006699"><strong>using</strong></font> a2 p0 p1 p2 p3 p5 reachableStep <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 436 </font></span> <font color="#006699"><strong>have</strong></font> b5<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀v. v∈sources bs u (step s b) </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 437 </font></span><font color="#ff00cc"> ⟶ interferes (the (domain b)) s v = interferes (the (domain b)) t v "</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 438 </font></span> <font color="#006699"><strong>using</strong></font> p1 p2 ivpeq_def policy_respect p9 sources_Cons <font color="#006699"><strong>by</strong></font> fastforce
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 439 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> <font color="#ff00cc">"sources (b # bs) u s = sources (b # bs) u t"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 440 </font></span> <font color="#006699"><strong>using</strong></font> b4 b5 sources_Cons <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 441 </font></span> <font color="#006699"><strong>next</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 442 </font></span> <font color="#0099ff"><strong>assume</strong></font> b0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"the (domain b) ∉ (sources (b # bs) u s)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 443 </font></span> <font color="#006699"><strong>have</strong></font> b1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"sources (b # bs) u s = sources bs u (step s b)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 444 </font></span> <font color="#006699"><strong>using</strong></font> b0 sources_Cons <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 445 </font></span> <font color="#006699"><strong>have</strong></font> b2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(sources bs u (step s b)) = (sources bs u (step t b))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 446 </font></span> <font color="#006699"><strong>using</strong></font> a2 p0 p1 p2 p3 p5 reachableStep <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 447 </font></span> <font color="#006699"><strong>have</strong></font> b3<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀v. v∈sources bs u (step s b)⟶¬ interferes (the (domain b)) s v "</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 448 </font></span> <font color="#006699"><strong>using</strong></font> b0 sources_Cons <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 449 </font></span> <font color="#006699"><strong>have</strong></font> b4<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀v. v∈sources bs u (step s b)⟶¬ interferes (the (domain b)) t v "</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 450 </font></span> <font color="#006699"><strong>using</strong></font> b1 b3 p1 p2 p9 policy_respect <font color="#006699"><strong>by</strong></font> fastforce
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 451 </font></span> <font color="#006699"><strong>have</strong></font> b5<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀v. v∈sources bs u (step t b)⟶¬ interferes (the (domain b)) t v "</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 452 </font></span> <font color="#006699"><strong>by</strong></font> <font color="#000000"><strong>(</strong></font>simp add<font color="#000000"><strong>:</strong></font> b2 b4<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 453 </font></span> <font color="#006699"><strong>have</strong></font> b6<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"the (domain b) ∉ (sources (b # bs) u t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 454 </font></span> <font color="#006699"><strong>using</strong></font> b0 b2 b5 sources.simps<font color="#000000"><strong>(</strong></font>2<font color="#000000"><strong>)</strong></font> <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 455 </font></span> <font color="#006699"><strong>have</strong></font> b7<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"sources (b # bs) u t = sources bs u (step t b)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 456 </font></span> <font color="#006699"><strong>using</strong></font> b6 sources_Cons <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 457 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 458 </font></span> <font color="#006699"><strong>by</strong></font> <font color="#000000"><strong>(</strong></font>simp add<font color="#000000"><strong>:</strong></font> b1 b2<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 459 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 460 </font></span> <font color="#006699"><strong>}</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 461 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 462 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 463 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 464 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 465 </font></span> <font color="#006699"><strong>}</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 466 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 467 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 468 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 469 </font></span> <font color="#006699"><strong>lemma</strong></font> ipurge_eq<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀s t as u. reachable0 s ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 470 </font></span><font color="#ff00cc"> reachable0 t ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 471 </font></span><font color="#ff00cc"> dynamic_step_consistent ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 472 </font></span><font color="#ff00cc"> dynamic_local_respect ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 473 </font></span><font color="#ff00cc"> (s ≈ (sources as u s) ≈ t)</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 474 </font></span><font color="#ff00cc"> ⟶ (ipurge as u s) = (ipurge as u t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 475 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 476 </font></span> <font color="#006699"><strong>{</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 477 </font></span> <font color="#0099ff"><strong>fix</strong></font> as
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 478 </font></span> <font color="#006699"><strong>have</strong></font> <font color="#ff00cc">"∀s t u. reachable0 s ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 479 </font></span><font color="#ff00cc"> reachable0 t ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 480 </font></span><font color="#ff00cc"> dynamic_step_consistent ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 481 </font></span><font color="#ff00cc"> dynamic_local_respect ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 482 </font></span><font color="#ff00cc"> (s ≈ (sources as u s) ≈ t)</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 483 </font></span><font color="#ff00cc"> ⟶ (ipurge as u s) = (ipurge as u t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 484 </font></span> <font color="#006699"><strong>proof</strong></font><font color="#000000"><strong>(</strong></font>induct as<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 485 </font></span> <font color="#0099ff"><strong>case</strong></font> Nil <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?case <font color="#006699"><strong>by</strong></font> <font color="#000000"><strong>(</strong></font>simp add<font color="#000000"><strong>:</strong></font> sources_Nil<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 486 </font></span> <font color="#006699"><strong>next</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 487 </font></span> <font color="#0099ff"><strong>case</strong></font> <font color="#000000"><strong>(</strong></font>Cons b bs<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 488 </font></span> <font color="#0099ff"><strong>assume</strong></font> p0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀s t u.((reachable0 s) </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 489 </font></span><font color="#ff00cc"> ∧ (reachable0 t) </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 490 </font></span><font color="#ff00cc"> ∧ dynamic_step_consistent </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 491 </font></span><font color="#ff00cc"> ∧ dynamic_local_respect </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 492 </font></span><font color="#ff00cc"> ∧ (s ≈ (sources bs u s) ≈ t))</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 493 </font></span><font color="#ff00cc"> ⟶ (ipurge bs u s) = (ipurge bs u t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 494 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?case
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 495 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 496 </font></span> <font color="#006699"><strong>{</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 497 </font></span> <font color="#0099ff"><strong>fix</strong></font> s t u
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 498 </font></span> <font color="#0099ff"><strong>assume</strong></font> p1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 s"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 499 </font></span> <font color="#0099ff"><strong>assume</strong></font> p2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 t"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 500 </font></span> <font color="#0099ff"><strong>assume</strong></font> p3<font color="#000000"><strong>:</strong></font> dynamic_step_consistent
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 501 </font></span> <font color="#0099ff"><strong>assume</strong></font> p5<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"dynamic_local_respect"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 502 </font></span> <font color="#0099ff"><strong>assume</strong></font> p9<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ≈ (sources (b # bs) u s) ≈ t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 503 </font></span> <font color="#006699"><strong>have</strong></font> a1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"((step s b) ≈ (sources bs u (step s b)) ≈ (step t b))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 504 </font></span> <font color="#006699"><strong>using</strong></font> lemma_1 p1 p2 p3 p5 p9 <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 505 </font></span> <font color="#006699"><strong>have</strong></font> a2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(ipurge bs u (step s b)) = (ipurge bs u (step t b))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 506 </font></span> <font color="#006699"><strong>using</strong></font> a1 p0 p1 p2 p3 p5 p9 reachableStep <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 507 </font></span> <font color="#006699"><strong>have</strong></font> a3<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"sources (b # bs) u s = sources (b # bs) u t"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 508 </font></span> <font color="#006699"><strong>using</strong></font> p1 p2 p3 p5 p9 sources_eq1 <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 509 </font></span> <font color="#006699"><strong>have</strong></font> a4<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"ipurge (b # bs) u s = ipurge (b # bs) u t"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 510 </font></span> <font color="#006699"><strong>proof</strong></font> <font color="#000000"><strong>(</strong></font>cases <font color="#ff00cc">"the (domain b) ∈ (sources (b # bs) u s)"</font><font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 511 </font></span> <font color="#0099ff"><strong>assume</strong></font> b0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"the (domain b) ∈ (sources (b # bs) u s)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 512 </font></span> <font color="#006699"><strong>have</strong></font> b1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"s ∼ (the(domain b)) ∼ t"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 513 </font></span> <font color="#006699"><strong>using</strong></font> b0 p9 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 514 </font></span> <font color="#006699"><strong>have</strong></font> b3<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"the (domain b) ∈ (sources (b # bs) u t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 515 </font></span> <font color="#006699"><strong>using</strong></font> a3 b0 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 516 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 517 </font></span> <font color="#006699"><strong>using</strong></font> a2 b0 ipurge_Cons <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 518 </font></span> <font color="#006699"><strong>next</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 519 </font></span> <font color="#0099ff"><strong>assume</strong></font> b0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"the (domain b) ∉ (sources (b # bs) u s)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 520 </font></span> <font color="#006699"><strong>have</strong></font> b1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"sources (b # bs) u s = sources bs u (step s b)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 521 </font></span> <font color="#006699"><strong>using</strong></font> b0 sources_Cons <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 522 </font></span> <font color="#006699"><strong>have</strong></font> b3<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀v. v∈sources bs u (step s b)⟶¬ interferes (the (domain b)) s v "</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 523 </font></span> <font color="#006699"><strong>using</strong></font> b0 sources_Cons <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 524 </font></span> <font color="#006699"><strong>have</strong></font> b4<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀v. v∈sources bs u (step s b)⟶¬ interferes (the (domain b)) t v "</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 525 </font></span> <font color="#006699"><strong>using</strong></font> b1 b3 p1 p2 p9 policy_respect <font color="#006699"><strong>by</strong></font> fastforce
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 526 </font></span> <font color="#006699"><strong>have</strong></font> b5<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"the (domain b) ∉ (sources (b # bs) u t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 527 </font></span> <font color="#006699"><strong>using</strong></font> a3 b1 b4 interf_reflexive <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 528 </font></span> <font color="#006699"><strong>have</strong></font> b6<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"ipurge (b # bs) u s = ipurge bs u (step s b)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 529 </font></span> <font color="#006699"><strong>using</strong></font> b0 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 530 </font></span> <font color="#006699"><strong>have</strong></font> b7<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"ipurge (b # bs) u t = ipurge bs u (step t b)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 531 </font></span> <font color="#006699"><strong>using</strong></font> b5 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 532 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 533 </font></span> <font color="#006699"><strong>using</strong></font> b6 b7 a2 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 534 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 535 </font></span> <font color="#006699"><strong>}</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 536 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 537 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 538 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 539 </font></span> <font color="#006699"><strong>}</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 540 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 541 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 542 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 543 </font></span> <font color="#006699"><strong>lemma</strong></font> non_influgence_lemma<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀s t as u. reachable0 s ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 544 </font></span><font color="#ff00cc"> reachable0 t ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 545 </font></span><font color="#ff00cc"> dynamic_step_consistent ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 546 </font></span><font color="#ff00cc"> dynamic_local_respect ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 547 </font></span><font color="#ff00cc"> (s ≈ (sources as u s) ≈ t)</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 548 </font></span><font color="#ff00cc"> ⟶ ((s ⊲ as ≅ t ⊲ (ipurge as u t) @ u))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 549 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 550 </font></span> <font color="#006699"><strong>{</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 551 </font></span> <font color="#0099ff"><strong>fix</strong></font> as
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 552 </font></span> <font color="#006699"><strong>have</strong></font> <font color="#ff00cc">"∀s t u. reachable0 s ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 553 </font></span><font color="#ff00cc"> reachable0 t ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 554 </font></span><font color="#ff00cc"> dynamic_step_consistent ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 555 </font></span><font color="#ff00cc"> dynamic_local_respect ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 556 </font></span><font color="#ff00cc"> (s ≈ (sources as u s) ≈ t)</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 557 </font></span><font color="#ff00cc"> ⟶ ((s ⊲ as ≅ t ⊲ (ipurge as u t) @ u))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 558 </font></span> <font color="#006699"><strong>proof</strong></font> <font color="#000000"><strong>(</strong></font>induct as<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 559 </font></span> <font color="#0099ff"><strong>case</strong></font> Nil <font color="#0099ff"><strong>show</strong></font> ?case <font color="#006699"><strong>using</strong></font> sources_Nil <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 560 </font></span> <font color="#006699"><strong>next</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 561 </font></span> <font color="#0099ff"><strong>case</strong></font> <font color="#000000"><strong>(</strong></font>Cons b bs<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 562 </font></span> <font color="#0099ff"><strong>assume</strong></font> p0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀s t u.((reachable0 s) </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 563 </font></span><font color="#ff00cc"> ∧ (reachable0 t) </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 564 </font></span><font color="#ff00cc"> ∧ dynamic_step_consistent </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 565 </font></span><font color="#ff00cc"> ∧ dynamic_local_respect </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 566 </font></span><font color="#ff00cc"> ∧ (s ≈ (sources bs u s) ≈ t)) ⟶</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 567 </font></span><font color="#ff00cc"> ((s ⊲ bs ≅ t ⊲ (ipurge bs u t) @ u))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 568 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?case
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 569 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 570 </font></span> <font color="#006699"><strong>{</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 571 </font></span> <font color="#0099ff"><strong>fix</strong></font> s t u
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 572 </font></span> <font color="#0099ff"><strong>assume</strong></font> p1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 s"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 573 </font></span> <font color="#0099ff"><strong>assume</strong></font> p2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 t"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 574 </font></span> <font color="#0099ff"><strong>assume</strong></font> p3<font color="#000000"><strong>:</strong></font> dynamic_step_consistent
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 575 </font></span> <font color="#0099ff"><strong>assume</strong></font> p4<font color="#000000"><strong>:</strong></font> dynamic_local_respect
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 576 </font></span> <font color="#0099ff"><strong>assume</strong></font> p8<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ≈ (sources (b # bs) u s) ≈ t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 577 </font></span> <font color="#006699"><strong>have</strong></font> a1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"((step s b) ≈ (sources bs u (step s b)) ≈ (step t b))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 578 </font></span> <font color="#006699"><strong>using</strong></font> lemma_1 p1 p2 p3 p4 p8 <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 579 </font></span> <font color="#006699"><strong>have</strong></font> <font color="#ff00cc">"s ⊲ b # bs ≅ t ⊲ ipurge (b # bs) u t @ u"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 580 </font></span> <font color="#006699"><strong>proof</strong></font> <font color="#000000"><strong>(</strong></font>cases <font color="#ff00cc">"the (domain b) ∈ sources (b # bs) u s"</font><font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 581 </font></span> <font color="#0099ff"><strong>assume</strong></font> b0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"the (domain b) ∈ sources (b # bs) u s"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 582 </font></span> <font color="#006699"><strong>have</strong></font> b1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"interferes (the (domain b)) s u = interferes (the (domain b)) t u "</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 583 </font></span> <font color="#006699"><strong>using</strong></font> p1 p2 policy_respect p8 sources_refl <font color="#006699"><strong>by</strong></font> fastforce
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 584 </font></span> <font color="#006699"><strong>have</strong></font> b2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀v. v∈sources bs u (step s b) </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 585 </font></span><font color="#ff00cc"> ⟶ interferes (the (domain b)) s v = interferes (the (domain b)) t v "</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 586 </font></span> <font color="#006699"><strong>using</strong></font> p1 p2 ivpeq_def policy_respect p8 sources_Cons <font color="#006699"><strong>by</strong></font> fastforce
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 587 </font></span> <font color="#006699"><strong>have</strong></font> b3<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"ipurge (b # bs) u t = b # (ipurge bs u (step t b))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 588 </font></span> <font color="#006699"><strong>by</strong></font> <font color="#000000"><strong>(</strong></font>metis b0 ipurge_Cons p1 p2 p3 p4 p8 sources_eq1<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 589 </font></span> <font color="#006699"><strong>have</strong></font> b4<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(((step s b) ⊲ bs ≅ (step t b) ⊲ (ipurge bs u (step t b)) @ u))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 590 </font></span> <font color="#006699"><strong>using</strong></font> a1 p0 p1 p2 p3 p4 reachableStep <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 591 </font></span> <font color="#0099ff"><strong>show</strong></font> ?thesis
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 592 </font></span> <font color="#006699"><strong>using</strong></font> b3 b4 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 593 </font></span> <font color="#006699"><strong>next</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 594 </font></span> <font color="#0099ff"><strong>assume</strong></font> b0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"the (domain b) ∉ sources (b # bs) u s"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 595 </font></span> <font color="#006699"><strong>have</strong></font> b1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"ipurge (b # bs) u t = (ipurge bs u (step t b))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 596 </font></span> <font color="#006699"><strong>by</strong></font> <font color="#000000"><strong>(</strong></font>metis a1 b0 ipurge_Cons ipurge_eq p1 p2 p3 p4 p8 reachableStep<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 597 </font></span> <font color="#006699"><strong>have</strong></font> b2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ≈ (sources bs u (step s b)) ≈ (step s b))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 598 </font></span> <font color="#006699"><strong>using</strong></font> b0 lemma_2 p1 p4 <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 599 </font></span> <font color="#006699"><strong>have</strong></font> b3<font color="#000000"><strong>:</strong></font><font color="#ff00cc">"(s ≈ (sources bs u (step s b)) ≈ t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 600 </font></span> <font color="#006699"><strong>using</strong></font> b0 lemma_1_sub_3 p8 <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 601 </font></span> <font color="#006699"><strong>have</strong></font> b4<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"((step s b) ≈ (sources bs u (step s b)) ≈ t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 602 </font></span> <font color="#006699"><strong>by</strong></font> <font color="#000000"><strong>(</strong></font>meson b3 b2 ivpeq_def vpeq_symmetric_lemma vpeq_transitive_lemma<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 603 </font></span> <font color="#006699"><strong>have</strong></font> b5<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(((step s b) ⊲ bs ≅ t ⊲ (ipurge bs u t) @ u))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 604 </font></span> <font color="#006699"><strong>using</strong></font> b4 p0 p1 p2 p3 p4 reachableStep <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 605 </font></span> <font color="#006699"><strong>have</strong></font> b6<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(t ≈ (sources bs u (step s b)) ≈ (step t b))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 606 </font></span> <font color="#006699"><strong>using</strong></font> p1 p2 b0 lemma_1_sub_2 p4 p8 <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 607 </font></span> <font color="#006699"><strong>have</strong></font> b7<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"ipurge bs u t = ipurge bs u (step t b)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 608 </font></span> <font color="#006699"><strong>by</strong></font> <font color="#000000"><strong>(</strong></font>metis a1 b4 ipurge_eq p1 p2 p3 p4 reachableStep<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 609 </font></span> <font color="#006699"><strong>have</strong></font> b8<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(((step s b) ⊲ bs ≅ t ⊲ (ipurge bs u (step t b)) @ u))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 610 </font></span> <font color="#006699"><strong>using</strong></font> b5 b7 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 611 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 612 </font></span> <font color="#006699"><strong>using</strong></font> b1 observ_equivalence_def run_Cons <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 613 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 614 </font></span> <font color="#006699"><strong>}</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 615 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 616 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 617 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 618 </font></span> <font color="#006699"><strong>}</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 619 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 620 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 621 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 622 </font></span><font color="#006699"><strong>subsection</strong></font><font color="#6600cc">{* Interference framework of information flow security properties *}</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 623 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 624 </font></span> <font color="#006699"><strong>theorem</strong></font> nonintf_impl_weak<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"noninterference ⟹ weak_noninterference"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 625 </font></span> <font color="#006699"><strong>by</strong></font> <font color="#000000"><strong>(</strong></font>metis noninterference_def observ_equiv_sym observ_equiv_trans reachable_s0 weak_noninterference_def<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 626 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 627 </font></span> <font color="#006699"><strong>theorem</strong></font> wk_nonintf_r_impl_wk_nonintf<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"weak_noninterference_r ⟹ weak_noninterference"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 628 </font></span> <font color="#006699"><strong>using</strong></font> reachable_s0 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 629 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 630 </font></span> <font color="#006699"><strong>theorem</strong></font> nonintf_r_impl_noninterf<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"noninterference_r ⟹ noninterference"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 631 </font></span> <font color="#006699"><strong>using</strong></font> noninterference_def noninterference_r_def reachable_s0 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 632 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 633 </font></span> <font color="#006699"><strong>theorem</strong></font> nonintf_r_impl_wk_nonintf_r<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"noninterference_r ⟹ weak_noninterference_r"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 634 </font></span> <font color="#006699"><strong>by</strong></font> <font color="#000000"><strong>(</strong></font>metis noninterference_r_def observ_equiv_sym observ_equiv_trans weak_noninterference_r_def<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 635 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 636 </font></span> <font color="#006699"><strong>lemma</strong></font> noninf_impl_nonintf_r<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"noninfluence ⟹ noninterference_r"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 637 </font></span> <font color="#006699"><strong>using</strong></font> ivpeq_def noninfluence_def noninterference_r_def vpeq_reflexive_lemma <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 638 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 639 </font></span> <font color="#006699"><strong>lemma</strong></font> noninf_impl_nonlk<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"noninfluence ⟹ nonleakage"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 640 </font></span> <font color="#006699"><strong>using</strong></font> noninterference_r_def nonleakage_def observ_equiv_sym
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 641 </font></span> observ_equiv_trans noninfluence_def noninf_impl_nonintf_r <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 642 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 643 </font></span> <font color="#006699"><strong>lemma</strong></font> wk_noninfl_impl_nonlk<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"weak_noninfluence ⟹ nonleakage"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 644 </font></span> <font color="#006699"><strong>using</strong></font> weak_noninfluence_def nonleakage_def <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 645 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 646 </font></span> <font color="#006699"><strong>lemma</strong></font> wk_noninfl_impl_wk_nonintf_r<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"weak_noninfluence ⟹ weak_noninterference_r"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 647 </font></span> <font color="#006699"><strong>using</strong></font> ivpeq_def weak_noninfluence_def vpeq_reflexive_lemma weak_noninterference_r_def <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 648 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 649 </font></span> <font color="#006699"><strong>lemma</strong></font> sources_step2<font color="#000000"><strong>:</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 650 </font></span> <font color="#ff00cc">"⟦reachable0 s; (the (domain a))@s ↝ d⟧ ⟹ sources [a] d s = {the (domain a),d}"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 651 </font></span> <font color="#006699"><strong>apply</strong></font><font color="#000000"><strong>(</strong></font>auto simp<font color="#000000"><strong>:</strong></font> sources_Cons sources_Nil enabled dest<font color="#000000"><strong>:</strong></font> enabled<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 652 </font></span> <font color="#006699"><strong>done</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 653 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 654 </font></span> <font color="#006699"><strong>lemma</strong></font> exec_equiv_both<font color="#000000"><strong>:</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 655 </font></span> <font color="#ff00cc">"⟦reachable0 C1; reachable0 C2;(step C1 a) ⊲ as ≅ (step C2 b) ⊲ bs @ u⟧ </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 656 </font></span><font color="#ff00cc"> ⟹ (C1 ⊲ (a # as) ≅ C2 ⊲ (b # bs) @ u)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 657 </font></span> <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 658 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 659 </font></span> <font color="#006699"><strong>lemma</strong></font> sources_unwinding_step<font color="#000000"><strong>:</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 660 </font></span> <font color="#ff00cc">"⟦reachable0 s; reachable0 t;s ≈(sources (a#as) d s)≈ t; dynamic_step_consistent⟧ </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 661 </font></span><font color="#ff00cc"> ⟹ ((step s a) ≈(sources as d (step s a))≈ (step t a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 662 </font></span> <font color="#006699"><strong>apply</strong></font><font color="#000000"><strong>(</strong></font>clarsimp simp<font color="#000000"><strong>:</strong></font> ivpeq_def sources_Cons<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 663 </font></span> <font color="#006699"><strong>using</strong></font> UnionI dynamic_step_consistent_def <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 664 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 665 </font></span> <font color="#006699"><strong>lemma</strong></font> nonlk_imp_sc<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"nonleakage ⟹ dynamic_step_consistent"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 666 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 667 </font></span> <font color="#0099ff"><strong>assume</strong></font> p0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"nonleakage"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 668 </font></span> <font color="#006699"><strong>have</strong></font> p1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀as d s t. reachable0 s ∧ reachable0 t</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 669 </font></span><font color="#ff00cc"> ∧ (s ≈ (sources as d s) ≈ t) ⟶ (s ⊲ as ≅ t ⊲ as @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 670 </font></span> <font color="#006699"><strong>using</strong></font> p0 nonleakage_def <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 671 </font></span> <font color="#006699"><strong>have</strong></font> p2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀a d s t. reachable0 s ∧ reachable0 t ∧ (s ∼ d ∼ t) ∧ </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 672 </font></span><font color="#ff00cc"> (((the (domain a)) @ s ↝ d) ⟶ (s ∼ (the (domain a)) ∼ t))</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 673 </font></span><font color="#ff00cc"> ⟶ ((step s a) ∼ d ∼ (step t a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 674 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 675 </font></span> <font color="#006699"><strong>{</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 676 </font></span> <font color="#0099ff"><strong>fix</strong></font> a d s t
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 677 </font></span> <font color="#0099ff"><strong>assume</strong></font> a0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 s ∧ reachable0 t ∧ (s ∼ d ∼ t) ∧ </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 678 </font></span><font color="#ff00cc"> (((the (domain a)) @ s ↝ d) ⟶ (s ∼ (the (domain a)) ∼ t))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 679 </font></span> <font color="#006699"><strong>have</strong></font> a4<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"s ≈ (sources [] d s) ≈ t"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 680 </font></span> <font color="#006699"><strong>using</strong></font> a0 sources_Nil <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 681 </font></span> <font color="#006699"><strong>have</strong></font> a5<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ⊲ [] ≅ t ⊲ [] @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 682 </font></span> <font color="#006699"><strong>using</strong></font> a4 a0 p1 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 683 </font></span> <font color="#006699"><strong>have</strong></font> a6<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"((step s a) ∼ d ∼ (step t a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 684 </font></span> <font color="#006699"><strong>proof</strong></font> <font color="#000000"><strong>(</strong></font>cases <font color="#ff00cc">"(the (domain a))@s ↝ d"</font><font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 685 </font></span> <font color="#0099ff"><strong>assume</strong></font> b0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(the (domain a))@s ↝ d"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 686 </font></span> <font color="#006699"><strong>have</strong></font> b1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"sources [a] d s = {d, (the(domain a))}"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 687 </font></span> <font color="#006699"><strong>using</strong></font> b0 sources_Cons sources_Nil <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 688 </font></span> <font color="#006699"><strong>have</strong></font> c0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ∼ (the (domain a)) ∼ t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 689 </font></span> <font color="#006699"><strong>using</strong></font> b0 a0 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 690 </font></span> <font color="#006699"><strong>have</strong></font> b2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"s ≈ (sources [a] d s) ≈ t"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 691 </font></span> <font color="#006699"><strong>using</strong></font> b1 a0 c0 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 692 </font></span> <font color="#006699"><strong>have</strong></font> b3<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ⊲ [a] ≅ t ⊲ [a] @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 693 </font></span> <font color="#006699"><strong>using</strong></font> b2 a0 p1 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 694 </font></span> <font color="#006699"><strong>have</strong></font> b4<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"((step s a) ∼ d ∼ (step t a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 695 </font></span> <font color="#006699"><strong>using</strong></font> b3 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 696 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 697 </font></span> <font color="#006699"><strong>next</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 698 </font></span> <font color="#0099ff"><strong>assume</strong></font> b0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"¬((the (domain a))@s ↝ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 699 </font></span> <font color="#006699"><strong>have</strong></font> b1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"sources [a] d s = {d}"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 700 </font></span> <font color="#006699"><strong>using</strong></font> b0 sources_Cons sources_Nil <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 701 </font></span> <font color="#006699"><strong>have</strong></font> b2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ≈ (sources [a] d s) ≈ t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 702 </font></span> <font color="#006699"><strong>using</strong></font> b1 a0 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 703 </font></span> <font color="#006699"><strong>have</strong></font> b3<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ⊲ [a] ≅ t ⊲ [a] @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 704 </font></span> <font color="#006699"><strong>using</strong></font> b2 a0 p1 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 705 </font></span> <font color="#006699"><strong>have</strong></font> b4<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"((step s a) ∼ d ∼ (step t a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 706 </font></span> <font color="#006699"><strong>using</strong></font> b3 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 707 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 708 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 709 </font></span> <font color="#006699"><strong>}</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 710 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 711 </font></span> <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 712 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 713 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 714 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 715 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 716 </font></span> <font color="#006699"><strong>lemma</strong></font> sc_imp_nonlk<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"dynamic_step_consistent ⟹ nonleakage"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 717 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 718 </font></span> <font color="#0099ff"><strong>assume</strong></font> p0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"dynamic_step_consistent"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 719 </font></span> <font color="#006699"><strong>have</strong></font> p1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀a d s t. reachable0 s ∧ reachable0 t ∧ (s ∼ d ∼ t) ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 720 </font></span><font color="#ff00cc"> (s ∼ (the (domain a)) ∼ t) ⟶ ((step s a) ∼ d ∼ (step t a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 721 </font></span> <font color="#006699"><strong>using</strong></font> p0 dynamic_step_consistent_def <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 722 </font></span> <font color="#006699"><strong>have</strong></font> p2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀as d s t. reachable0 s ∧ reachable0 t</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 723 </font></span><font color="#ff00cc"> ∧ (s ≈ (sources as d s) ≈ t) ⟶ (s ⊲ as ≅ t ⊲ as @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 724 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 725 </font></span> <font color="#006699"><strong>{</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 726 </font></span> <font color="#0099ff"><strong>fix</strong></font> as
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 727 </font></span> <font color="#006699"><strong>have</strong></font> <font color="#ff00cc">"∀d s t. reachable0 s ∧ reachable0 t</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 728 </font></span><font color="#ff00cc"> ∧ (s ≈ (sources as d s) ≈ t) ⟶ (s ⊲ as ≅ t ⊲ as @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 729 </font></span> <font color="#006699"><strong>proof</strong></font> <font color="#000000"><strong>(</strong></font>induct as<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 730 </font></span> <font color="#0099ff"><strong>case</strong></font> Nil <font color="#0099ff"><strong>show</strong></font> ?case <font color="#006699"><strong>using</strong></font> sources_refl <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 731 </font></span> <font color="#006699"><strong>next</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 732 </font></span> <font color="#0099ff"><strong>case</strong></font> <font color="#000000"><strong>(</strong></font>Cons b bs<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 733 </font></span> <font color="#0099ff"><strong>assume</strong></font> a0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀d s t. reachable0 s ∧ reachable0 t</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 734 </font></span><font color="#ff00cc"> ∧ (s ≈ (sources bs d s) ≈ t) ⟶ (s ⊲ bs ≅ t ⊲ bs @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 735 </font></span> <font color="#0099ff"><strong>show</strong></font> ?case
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 736 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 737 </font></span> <font color="#006699"><strong>{</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 738 </font></span> <font color="#0099ff"><strong>fix</strong></font> d s t
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 739 </font></span> <font color="#0099ff"><strong>assume</strong></font> b0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 s ∧ reachable0 t"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 740 </font></span> <font color="#0099ff"><strong>assume</strong></font> b1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ≈ (sources (b#bs) d s) ≈ t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 741 </font></span> <font color="#006699"><strong>have</strong></font> b2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"((step s b) ≈(sources bs d (step s b))≈ (step t b))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 742 </font></span> <font color="#006699"><strong>using</strong></font> b0 b1 p0 sources_unwinding_step <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 743 </font></span> <font color="#006699"><strong>have</strong></font> b3<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(step s b) ⊲ bs ≅ (step t b) ⊲ bs @ d"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 744 </font></span> <font color="#006699"><strong>using</strong></font> Cons.hyps b0 b2 reachableStep <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 745 </font></span> <font color="#006699"><strong>have</strong></font> b4<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"s ⊲ b # bs ≅ t ⊲ b # bs @ d"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 746 </font></span> <font color="#006699"><strong>using</strong></font> b3 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 747 </font></span> <font color="#006699"><strong>}</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 748 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 749 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 750 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 751 </font></span> <font color="#006699"><strong>}</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 752 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 753 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 754 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 755 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 756 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 757 </font></span> <font color="#006699"><strong>theorem</strong></font> sc_eq_nonlk<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"dynamic_step_consistent = nonleakage"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 758 </font></span> <font color="#006699"><strong>using</strong></font> nonlk_imp_sc sc_imp_nonlk <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 759 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 760 </font></span> <font color="#006699"><strong>lemma</strong></font> noninf_imp_dlr<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"noninfluence ⟹ dynamic_local_respect"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 761 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 762 </font></span> <font color="#0099ff"><strong>assume</strong></font> p0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"noninfluence"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 763 </font></span> <font color="#006699"><strong>have</strong></font> p1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀ d as s t. reachable0 s ∧ reachable0 t</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 764 </font></span><font color="#ff00cc"> ∧ (s ≈ (sources as d s) ≈ t)</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 765 </font></span><font color="#ff00cc"> ⟶ (s ⊲ as ≅ t ⊲ (ipurge as d t) @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 766 </font></span> <font color="#006699"><strong>using</strong></font> p0 noninfluence_def <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 767 </font></span> <font color="#006699"><strong>have</strong></font> <font color="#ff00cc">"∀a d s. reachable0 s ∧ ¬((the (domain a)) @ s ↝ d) </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 768 </font></span><font color="#ff00cc"> ⟶ (s ∼ d ∼ (step s a)) "</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 769 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 770 </font></span> <font color="#006699"><strong>{</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 771 </font></span> <font color="#0099ff"><strong>fix</strong></font> a d s
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 772 </font></span> <font color="#0099ff"><strong>assume</strong></font> a0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 s ∧ ¬((the (domain a)) @ s ↝ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 773 </font></span> <font color="#006699"><strong>have</strong></font> a1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"sources [a] d s = {d}"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 774 </font></span> <font color="#006699"><strong>using</strong></font> a0 sources_Cons sources_Nil <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 775 </font></span> <font color="#006699"><strong>have</strong></font> a2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(ipurge [a] d s) = []"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 776 </font></span> <font color="#006699"><strong>using</strong></font> a0 a1 interf_reflexive <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 777 </font></span> <font color="#006699"><strong>have</strong></font> a3<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"s ∼ d ∼ s"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 778 </font></span> <font color="#006699"><strong>using</strong></font> vpeq_reflexive_lemma <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 779 </font></span> <font color="#006699"><strong>have</strong></font> a4<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ≈ (sources [a] d s) ≈ s)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 780 </font></span> <font color="#006699"><strong>using</strong></font> a1 a3 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 781 </font></span> <font color="#006699"><strong>have</strong></font> a5<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ⊲ [a] ≅ s ⊲ (ipurge [a] d s) @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 782 </font></span> <font color="#006699"><strong>using</strong></font> a4 a0 p1 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 783 </font></span> <font color="#006699"><strong>have</strong></font> a6<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ⊲ [a] ≅ s ⊲ [] @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 784 </font></span> <font color="#006699"><strong>using</strong></font> a5 a2 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 785 </font></span> <font color="#006699"><strong>have</strong></font> a7<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ∼ d ∼ (step s a))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 786 </font></span> <font color="#006699"><strong>using</strong></font> a6 vpeq_symmetric_lemma <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 787 </font></span> <font color="#006699"><strong>}</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 788 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 789 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 790 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 791 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 792 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 793 </font></span> <font color="#006699"><strong>lemma</strong></font> noninf_imp_sc<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"noninfluence ⟹ dynamic_step_consistent"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 794 </font></span> <font color="#006699"><strong>using</strong></font> nonlk_imp_sc noninf_impl_nonlk <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 795 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 796 </font></span> <font color="#006699"><strong>theorem</strong></font> UnwindingTheorem <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"⟦dynamic_step_consistent;</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 797 </font></span><font color="#ff00cc"> dynamic_local_respect⟧ </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 798 </font></span><font color="#ff00cc"> ⟹ noninfluence"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 799 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 800 </font></span> <font color="#0099ff"><strong>assume</strong></font> p3<font color="#000000"><strong>:</strong></font> dynamic_step_consistent
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 801 </font></span> <font color="#0099ff"><strong>assume</strong></font> p4<font color="#000000"><strong>:</strong></font> dynamic_local_respect
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 802 </font></span> <font color="#006699"><strong>{</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 803 </font></span> <font color="#0099ff"><strong>fix</strong></font> as d
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 804 </font></span> <font color="#006699"><strong>have</strong></font> <font color="#ff00cc">"∀s t. reachable0 s ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 805 </font></span><font color="#ff00cc"> reachable0 t ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 806 </font></span><font color="#ff00cc"> (s ≈ (sources as d s) ≈ t)</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 807 </font></span><font color="#ff00cc"> ⟶ ((s ⊲ as ≅ t ⊲ (ipurge as d t) @ d))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 808 </font></span> <font color="#006699"><strong>proof</strong></font><font color="#000000"><strong>(</strong></font>induct as<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 809 </font></span> <font color="#0099ff"><strong>case</strong></font> Nil <font color="#0099ff"><strong>show</strong></font> ?case <font color="#006699"><strong>using</strong></font> sources_Nil <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 810 </font></span> <font color="#006699"><strong>next</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 811 </font></span> <font color="#0099ff"><strong>case</strong></font> <font color="#000000"><strong>(</strong></font>Cons b bs<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 812 </font></span> <font color="#0099ff"><strong>assume</strong></font> p0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀s t. reachable0 s ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 813 </font></span><font color="#ff00cc"> reachable0 t ∧</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 814 </font></span><font color="#ff00cc"> (s ≈ (sources bs d s) ≈ t)</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 815 </font></span><font color="#ff00cc"> ⟶ ((s ⊲ bs ≅ t ⊲ (ipurge bs d t) @ d))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 816 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?case
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 817 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 818 </font></span> <font color="#006699"><strong>{</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 819 </font></span> <font color="#0099ff"><strong>fix</strong></font> s t
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 820 </font></span> <font color="#0099ff"><strong>assume</strong></font> p1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 s"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 821 </font></span> <font color="#0099ff"><strong>assume</strong></font> p2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 t"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 822 </font></span> <font color="#0099ff"><strong>assume</strong></font> p8<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ≈ (sources (b # bs) d s) ≈ t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 823 </font></span> <font color="#006699"><strong>have</strong></font> a1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"((step s b) ≈ (sources bs d (step s b)) ≈ (step t b))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 824 </font></span> <font color="#006699"><strong>using</strong></font> lemma_1 p1 p2 p3 p4 p8 <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 825 </font></span> <font color="#006699"><strong>have</strong></font> a2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"s ⊲ b # bs ≅ t ⊲ ipurge (b # bs) d t @ d"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 826 </font></span> <font color="#006699"><strong>proof</strong></font> <font color="#000000"><strong>(</strong></font>cases <font color="#ff00cc">"the (domain b) ∈ sources (b # bs) d s"</font><font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 827 </font></span> <font color="#0099ff"><strong>assume</strong></font> b0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"the (domain b) ∈ sources (b # bs) d s"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 828 </font></span> <font color="#006699"><strong>have</strong></font> b1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"interferes (the (domain b)) s d = interferes (the (domain b)) t d "</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 829 </font></span> <font color="#006699"><strong>using</strong></font> p1 p2 policy_respect p8 sources_refl <font color="#006699"><strong>by</strong></font> fastforce
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 830 </font></span> <font color="#006699"><strong>have</strong></font> b2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀v. v∈sources bs d (step s b) </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 831 </font></span><font color="#ff00cc"> ⟶ interferes (the (domain b)) s v = interferes (the (domain b)) t v "</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 832 </font></span> <font color="#006699"><strong>using</strong></font> p1 p2 ivpeq_def policy_respect p8 sources_Cons <font color="#006699"><strong>by</strong></font> fastforce
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 833 </font></span> <font color="#006699"><strong>have</strong></font> b3<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"ipurge (b # bs) d t = b # (ipurge bs d (step t b))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 834 </font></span> <font color="#006699"><strong>by</strong></font> <font color="#000000"><strong>(</strong></font>metis b0 ipurge_Cons p1 p2 p3 p4 p8 sources_eq1<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 835 </font></span> <font color="#006699"><strong>have</strong></font> b4<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(((step s b) ⊲ bs ≅ (step t b) ⊲ (ipurge bs d (step t b)) @ d))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 836 </font></span> <font color="#006699"><strong>using</strong></font> a1 p0 p1 p2 p3 p4 reachableStep <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 837 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 838 </font></span> <font color="#006699"><strong>using</strong></font> b3 b4 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 839 </font></span> <font color="#006699"><strong>next</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 840 </font></span> <font color="#0099ff"><strong>assume</strong></font> b0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"the (domain b) ∉ sources (b # bs) d s"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 841 </font></span> <font color="#006699"><strong>have</strong></font> b1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"ipurge (b # bs) d t = (ipurge bs d (step t b))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 842 </font></span> <font color="#006699"><strong>by</strong></font> <font color="#000000"><strong>(</strong></font>metis a1 b0 ipurge_Cons ipurge_eq p1 p2 p3 p4 p8 reachableStep<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 843 </font></span> <font color="#006699"><strong>have</strong></font> b2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ≈ (sources bs d (step s b)) ≈ (step s b))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 844 </font></span> <font color="#006699"><strong>using</strong></font> b0 lemma_2 p1 p4 <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 845 </font></span> <font color="#006699"><strong>have</strong></font> b3<font color="#000000"><strong>:</strong></font><font color="#ff00cc">"(s ≈ (sources bs d (step s b)) ≈ t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 846 </font></span> <font color="#006699"><strong>using</strong></font> b0 lemma_1_sub_3 p8 <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 847 </font></span> <font color="#006699"><strong>have</strong></font> b4<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"((step s b) ≈ (sources bs d (step s b)) ≈ t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 848 </font></span> <font color="#006699"><strong>by</strong></font> <font color="#000000"><strong>(</strong></font>meson b3 b2 ivpeq_def vpeq_symmetric_lemma vpeq_transitive_lemma<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 849 </font></span> <font color="#006699"><strong>have</strong></font> b5<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(((step s b) ⊲ bs ≅ t ⊲ (ipurge bs d t) @ d))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 850 </font></span> <font color="#006699"><strong>using</strong></font> b4 p0 p1 p2 p3 p4 reachableStep <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 851 </font></span> <font color="#006699"><strong>have</strong></font> b6<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(t ≈ (sources bs d (step s b)) ≈ (step t b))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 852 </font></span> <font color="#006699"><strong>using</strong></font> p1 p2 b0 lemma_1_sub_2 p4 p8 <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 853 </font></span> <font color="#006699"><strong>have</strong></font> b7<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"ipurge bs d t = ipurge bs d (step t b)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 854 </font></span> <font color="#006699"><strong>by</strong></font> <font color="#000000"><strong>(</strong></font>metis a1 b4 ipurge_eq p1 p2 p3 p4 reachableStep<font color="#000000"><strong>)</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 855 </font></span> <font color="#006699"><strong>have</strong></font> b8<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(((step s b) ⊲ bs ≅ t ⊲ (ipurge bs d (step t b)) @ d))"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 856 </font></span> <font color="#006699"><strong>using</strong></font> b5 b7 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 857 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 858 </font></span> <font color="#006699"><strong>using</strong></font> b1 observ_equivalence_def run_Cons <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 859 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 860 </font></span> <font color="#006699"><strong>}</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 861 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 862 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 863 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 864 </font></span> <font color="#006699"><strong>}</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 865 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>using</strong></font> noninfluence_def <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 866 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 867 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 868 </font></span> <font color="#006699"><strong>theorem</strong></font> UnwindingTheorem1 <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"⟦dynamic_weakly_step_consistent;</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 869 </font></span><font color="#ff00cc"> dynamic_local_respect⟧ ⟹ noninfluence"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 870 </font></span> <font color="#006699"><strong>using</strong></font> UnwindingTheorem weak_with_step_cons <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 871 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 872 </font></span> <font color="#006699"><strong>theorem</strong></font> uc_eq_noninf <font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(dynamic_step_consistent ∧ dynamic_local_respect) = noninfluence"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 873 </font></span> <font color="#006699"><strong>using</strong></font> UnwindingTheorem1 step_cons_impl_weak noninf_imp_dlr noninf_imp_sc <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 874 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 875 </font></span> <font color="#cc0000">(*</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 876 </font></span><font color="#cc0000"> dipurge_eq required</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 877 </font></span><font color="#cc0000"> *)</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 878 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 879 </font></span> <font color="#006699"><strong>theorem</strong></font> noninf_impl_weak<font color="#000000"><strong>:</strong></font><font color="#ff00cc">"noninfluence ⟹ weak_noninfluence"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 880 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 881 </font></span> <font color="#0099ff"><strong>assume</strong></font> p0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"noninfluence"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 882 </font></span> <font color="#006699"><strong>have</strong></font> p1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀ d as s t. reachable0 s ∧ reachable0 t</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 883 </font></span><font color="#ff00cc"> ∧ (s ≈ (sources as d s) ≈ t) </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 884 </font></span><font color="#ff00cc"> ⟶ (s ⊲ as ≅ t ⊲ (ipurge as d t) @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 885 </font></span> <font color="#006699"><strong>using</strong></font> p0 noninfluence_def <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 886 </font></span> <font color="#006699"><strong>have</strong></font> p2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(dynamic_step_consistent ∧ dynamic_local_respect)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 887 </font></span> <font color="#006699"><strong>using</strong></font> p0 uc_eq_noninf <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 888 </font></span> <font color="#006699"><strong>have</strong></font> <font color="#ff00cc">"∀ d as bs s t . reachable0 s ∧ reachable0 t ∧ (s ≈ (sources as d s) ≈ t) </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 889 </font></span><font color="#ff00cc"> ∧ ipurge as d t = ipurge bs d t</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 890 </font></span><font color="#ff00cc"> ⟶ (s ⊲ as ≅ t ⊲ bs @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 891 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 892 </font></span> <font color="#006699"><strong>{</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 893 </font></span> <font color="#0099ff"><strong>fix</strong></font> d as bs s t
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 894 </font></span> <font color="#0099ff"><strong>assume</strong></font> a0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 s ∧ reachable0 t ∧ (s ≈ (sources as d s) ≈ t) </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 895 </font></span><font color="#ff00cc"> ∧ ipurge as d t = ipurge bs d t"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 896 </font></span> <font color="#006699"><strong>have</strong></font> a4<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"noninterference_r"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 897 </font></span> <font color="#006699"><strong>using</strong></font> noninf_impl_nonintf_r p0 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 898 </font></span> <font color="#006699"><strong>have</strong></font> a7<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"weak_noninterference_r"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 899 </font></span> <font color="#006699"><strong>using</strong></font> a4 nonintf_r_impl_wk_nonintf_r <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 900 </font></span> <font color="#006699"><strong>have</strong></font> a6<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"ipurge as d s = ipurge as d t"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 901 </font></span> <font color="#006699"><strong>using</strong></font> a0 p2 ipurge_eq <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 902 </font></span> <font color="#006699"><strong>have</strong></font> b1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ⊲ as ≅ t ⊲ (ipurge as d t) @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 903 </font></span> <font color="#006699"><strong>using</strong></font> a0 p1 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 904 </font></span> <font color="#006699"><strong>have</strong></font> b4<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ⊲ as ≅ t ⊲ as @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 905 </font></span> <font color="#006699"><strong>using</strong></font> a0 noninf_imp_sc nonleakage_def p0 sc_imp_nonlk <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 906 </font></span> <font color="#006699"><strong>have</strong></font> b5<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(t ⊲ bs ≅ t ⊲ (ipurge bs d t) @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 907 </font></span> <font color="#006699"><strong>using</strong></font> a0 a4 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 908 </font></span> <font color="#006699"><strong>have</strong></font> b6<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(t ⊲ bs ≅ t ⊲ (ipurge as d t) @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 909 </font></span> <font color="#006699"><strong>using</strong></font> b5 a0 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 910 </font></span> <font color="#006699"><strong>have</strong></font> b7<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ⊲ as ≅ t ⊲ bs @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 911 </font></span> <font color="#006699"><strong>using</strong></font> a0 b1 b6 observ_equiv_sym observ_equiv_trans <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 912 </font></span> <font color="#006699"><strong>}</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 913 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 914 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 915 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 916 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 917 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 918 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 919 </font></span> <font color="#006699"><strong>lemma</strong></font> wk_nonintf_r_and_nonlk_impl_noninfl<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"⟦weak_noninterference_r; nonleakage⟧ ⟹ weak_noninfluence"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 920 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 921 </font></span> <font color="#0099ff"><strong>assume</strong></font> p0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"weak_noninterference_r"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 922 </font></span> <font color="#0099ff"><strong>assume</strong></font> p1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"nonleakage"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 923 </font></span> <font color="#006699"><strong>have</strong></font> p2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀d as bs s. reachable0 s ∧ ipurge as d s = ipurge bs d s</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 924 </font></span><font color="#ff00cc"> ⟶ (s ⊲ as ≅ s ⊲ bs @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 925 </font></span> <font color="#006699"><strong>using</strong></font> weak_noninterference_r_def p0 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 926 </font></span> <font color="#006699"><strong>have</strong></font> p3<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀d as s t. reachable0 s ∧ reachable0 t</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 927 </font></span><font color="#ff00cc"> ∧ (s ≈ (sources as d s) ≈ t) ⟶ (s ⊲ as ≅ t ⊲ as @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 928 </font></span> <font color="#006699"><strong>using</strong></font> nonleakage_def p1 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 929 </font></span> <font color="#006699"><strong>have</strong></font> <font color="#ff00cc">"∀ d as bs s t . reachable0 s ∧ reachable0 t ∧ (s ≈ (sources as d s) ≈ t) </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 930 </font></span><font color="#ff00cc"> ∧ ipurge as d t = ipurge bs d t</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 931 </font></span><font color="#ff00cc"> ⟶ (s ⊲ as ≅ t ⊲ bs @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 932 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 933 </font></span> <font color="#006699"><strong>{</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 934 </font></span> <font color="#0099ff"><strong>fix</strong></font> d as bs s t
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 935 </font></span> <font color="#0099ff"><strong>assume</strong></font> a0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 s ∧ reachable0 t ∧ (s ≈ (sources as d s) ≈ t) </font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 936 </font></span><font color="#ff00cc"> ∧ ipurge as d t = ipurge bs d t"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 937 </font></span> <font color="#006699"><strong>have</strong></font> a1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"s ⊲ as ≅ t ⊲ as @ d"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 938 </font></span> <font color="#006699"><strong>using</strong></font> a0 p3 <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 939 </font></span> <font color="#006699"><strong>have</strong></font> a2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"t ⊲ as ≅ t ⊲ bs @ d"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 940 </font></span> <font color="#006699"><strong>using</strong></font> a0 p2 <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 941 </font></span> <font color="#006699"><strong>have</strong></font> a3<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(s ⊲ as ≅ t ⊲ bs @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 942 </font></span> <font color="#006699"><strong>using</strong></font> a0 a1 a2 observ_equiv_trans <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 943 </font></span> <font color="#006699"><strong>}</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 944 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 945 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 946 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 947 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 948 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 949 </font></span> <font color="#006699"><strong>lemma</strong></font> nonintf_r_and_nonlk_impl_noninfl<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"⟦noninterference_r; nonleakage⟧ ⟹ noninfluence"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 950 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 951 </font></span> <font color="#0099ff"><strong>assume</strong></font> p0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"noninterference_r"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 952 </font></span> <font color="#0099ff"><strong>assume</strong></font> p1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"nonleakage"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 953 </font></span> <font color="#006699"><strong>have</strong></font> p2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">" ∀d as s. reachable0 s ⟶ (s ⊲ as ≅ s ⊲ (ipurge as d s) @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 954 </font></span> <font color="#006699"><strong>using</strong></font> p0 noninterference_r_def <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 955 </font></span> <font color="#006699"><strong>have</strong></font> p3<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"∀d as s t. reachable0 s ∧ reachable0 t</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 956 </font></span><font color="#ff00cc"> ∧ (s ≈ (sources as d s) ≈ t) ⟶ (s ⊲ as ≅ t ⊲ as @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 957 </font></span> <font color="#006699"><strong>using</strong></font> p1 nonleakage_def <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 958 </font></span> <font color="#006699"><strong>have</strong></font> <font color="#ff00cc">"∀ d as s t. reachable0 s ∧ reachable0 t</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 959 </font></span><font color="#ff00cc"> ∧ (s ≈ (sources as d s) ≈ t)</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 960 </font></span><font color="#ff00cc"> ⟶ (s ⊲ as ≅ t ⊲ (ipurge as d t) @ d)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 961 </font></span> <font color="#006699"><strong>proof</strong></font> -
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 962 </font></span> <font color="#006699"><strong>{</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 963 </font></span> <font color="#0099ff"><strong>fix</strong></font> d as bs s t
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 964 </font></span> <font color="#0099ff"><strong>assume</strong></font> a0<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"reachable0 s ∧ reachable0 t</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 965 </font></span><font color="#ff00cc"> ∧ (s ≈ (sources as d s) ≈ t)"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 966 </font></span> <font color="#006699"><strong>have</strong></font> a1<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"s ⊲ as ≅ t ⊲ as @ d"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 967 </font></span> <font color="#006699"><strong>using</strong></font> p3 a0 <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 968 </font></span> <font color="#006699"><strong>have</strong></font> a2<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"s ⊲ as ≅ s ⊲ (ipurge as d s) @ d"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 969 </font></span> <font color="#006699"><strong>using</strong></font> a0 p2 <font color="#006699"><strong>by</strong></font> fast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 970 </font></span> <font color="#006699"><strong>have</strong></font> a3<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"t ⊲ as ≅ t ⊲ (ipurge as d t) @ d"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 971 </font></span> <font color="#006699"><strong>using</strong></font> a0 p2 <font color="#006699"><strong>by</strong></font> fast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 972 </font></span> <font color="#006699"><strong>have</strong></font> <font color="#ff00cc">"s ⊲ as ≅ t ⊲ (ipurge as d t) @ d"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 973 </font></span> <font color="#006699"><strong>using</strong></font> a0 a1 a3 observ_equiv_trans <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 974 </font></span> <font color="#006699"><strong>}</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 975 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>by</strong></font> auto
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 976 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 977 </font></span> <font color="#006699"><strong>then</strong></font> <font color="#0099ff"><strong>show</strong></font> ?thesis <font color="#006699"><strong>using</strong></font> noninfluence_def <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 978 </font></span> <font color="#006699"><strong>qed</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 979 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff990066"> 980 </font></span> <font color="#006699"><strong>theorem</strong></font> nonintf_r_and_nonlk_eq_strnoninfl<font color="#000000"><strong>:</strong></font> <font color="#ff00cc">"(noninterference_r ∧ nonleakage) = noninfluence"</font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 981 </font></span> <font color="#006699"><strong>using</strong></font> nonintf_r_and_nonlk_impl_noninfl noninf_impl_nonintf_r noninf_impl_nonlk <font color="#006699"><strong>by</strong></font> blast
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 982 </font></span>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 983 </font></span> <font color="#009966"><strong>end</strong></font>
<span style="background:#ffdbdbdb; border-right:solid 2px black; margin-right:5px; "><font color="#ff000000"> 984 </font></span><font color="#009966"><strong>end</strong></font>
</font></pre>
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