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| <!DOCTYPE html> | |
| <html xmlns="http://www.w3.org/1999/xhtml"> | |
| <head> | |
| <meta http-equiv="Content-Type" content="text/html; charset=utf-8"/> | |
| <title>Theory Pumping (Isabelle2020: April 2020)</title> | |
| <link media="all" rel="stylesheet" type="text/css" href="isabelle.css"/> | |
| </head> | |
| <body> | |
| <div class="head"><h1>Theory Pumping</h1> | |
| <span class="command">theory</span> <span class="name">Pumping</span><br/> | |
| <span class="keyword">imports</span> <a href="../../HOL/HOL/Main.html"><span class="name">Main</span></a><br/> | |
| </div> | |
| <div class="source"> | |
| <pre class="source"><span class="keyword1"><span class="command">theory</span></span><span> </span><span>Pumping</span><span> | |
| </span><span> </span><span class="keyword2"><span class="keyword">imports</span></span><span> </span><span>Main</span><span> | |
| </span><span class="keyword2"><span class="keyword">begin</span></span><span> | |
| </span><span> | |
| </span><span class="keyword1"><span class="command">datatype</span></span><span> </span><span class="tfree">'t</span><span> </span><span>regexp</span><span> </span><span class="delimiter">=</span><span> | |
| </span><span> </span><span>EmptySet</span><span> | |
| </span><span> </span><span class="delimiter">|</span><span> </span><span>EmptyStr</span><span> | |
| </span><span> </span><span class="delimiter">|</span><span> </span><span>Char</span><span> </span><span class="tfree">'t</span><span> | |
| </span><span> </span><span class="delimiter">|</span><span> </span><span>App</span><span> </span><span class="string"><span class="delete"><span class="delete">"'t regexp"</span></span></span><span> </span><span class="string"><span class="delete"><span class="delete">"'t regexp"</span></span></span><span> | |
| </span><span> </span><span class="delimiter">|</span><span> </span><span>Union</span><span> </span><span class="string"><span class="delete"><span class="delete">"'t regexp"</span></span></span><span> </span><span class="string"><span class="delete"><span class="delete">"'t regexp"</span></span></span><span> | |
| </span><span> </span><span class="delimiter">|</span><span> </span><span>Star</span><span> </span><span class="string"><span class="delete"><span class="delete">"'t regexp"</span></span></span><span> | |
| </span><span> | |
| </span><span class="keyword1"><span class="command">inductive</span></span><span> </span><span>match_regexp</span><span> | |
| </span><span> </span><span class="delimiter">::</span><span> </span><span class="string"><span class="delete"><span class="delete">"'t list ⇒ 't regexp ⇒ bool"</span></span></span><span> | |
| </span><span> </span><span class="delimiter">(</span><span class="keyword2"><span class="keyword">infix</span></span><span> </span><span class="string"><span class="delete"><span class="delete">"=~"</span></span></span><span> </span><span>60</span><span class="delimiter">)</span><span> | |
| </span><span> </span><span class="keyword2"><span class="keyword">where</span></span><span> | |
| </span><span> </span><span>MEmpty</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"[] =~ EmptyStr"</span></span></span><span> </span><span class="delimiter">|</span><span> | |
| </span><span> </span><span>MChar</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"[x] =~ Char x"</span></span></span><span> </span><span class="delimiter">|</span><span> | |
| </span><span> </span><span>MApp</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"⟦x1 =~ e1; x2 =~ e2⟧ ⟹ x1 @ x2 =~ App e1 e2"</span></span></span><span> </span><span class="delimiter">|</span><span> | |
| </span><span> </span><span>MUnionL</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"x =~ e1 ⟹ x =~ Union e1 e2"</span></span></span><span> </span><span class="delimiter">|</span><span> | |
| </span><span> </span><span>MUnionR</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"x =~ e2 ⟹ x =~ Union e1 e2"</span></span></span><span> </span><span class="delimiter">|</span><span> | |
| </span><span> </span><span>MStar0</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"[] =~ Star e"</span></span></span><span> </span><span class="delimiter">|</span><span> | |
| </span><span> </span><span>MStarApp</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"⟦x1 =~ e; x2 =~ Star e⟧ ⟹ x1 @ x2 =~ Star e"</span></span></span><span> | |
| </span><span> | |
| </span><span class="keyword1"><span class="command">inductive_cases</span></span><span> </span><span>matchE</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"s =~ e"</span></span></span><span> | |
| </span><span class="keyword1"><span class="command">inductive_simps</span></span><span> </span><span>matchS</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"s =~ e"</span></span></span><span> | |
| </span><span> | |
| </span><span class="keyword1"><span class="command">fun</span></span><span> </span><span>pumping_constant</span><span> </span><span class="delimiter">::</span><span> </span><span class="string"><span class="delete"><span class="delete">"'t regexp ⇒ nat"</span></span></span><span> </span><span class="keyword2"><span class="keyword">where</span></span><span> | |
| </span><span> </span><span class="string"><span class="delete"><span class="delete">"pumping_constant EmptySet = 0"</span></span></span><span> </span><span class="delimiter">|</span><span> | |
| </span><span> </span><span class="string"><span class="delete"><span class="delete">"pumping_constant EmptyStr = 1"</span></span></span><span> </span><span class="delimiter">|</span><span> | |
| </span><span> </span><span class="string"><span class="delete"><span class="delete">"pumping_constant (Char _) = 2"</span></span></span><span> </span><span class="delimiter">|</span><span> | |
| </span><span> </span><span class="string"><span class="delete"><span class="delete">"pumping_constant (App e1 e2) = pumping_constant e1 + pumping_constant e2"</span></span></span><span> </span><span class="delimiter">|</span><span> | |
| </span><span> </span><span class="string"><span class="delete"><span class="delete">"pumping_constant (Union e1 e2) = pumping_constant e1 + pumping_constant e2"</span></span></span><span> </span><span class="delimiter">|</span><span> | |
| </span><span> </span><span class="string"><span class="delete"><span class="delete">"pumping_constant (Star _) = 1"</span></span></span><span> | |
| </span><span> | |
| </span><span class="keyword1"><span class="command">fun</span></span><span> </span><span>napp</span><span> </span><span class="delimiter">::</span><span> </span><span class="string"><span class="delete"><span class="delete">"nat ⇒ 't list ⇒ 't list"</span></span></span><span> </span><span class="keyword2"><span class="keyword">where</span></span><span> | |
| </span><span> </span><span class="string"><span class="delete"><span class="delete">"napp n s = concat (replicate n s)"</span></span></span><span> | |
| </span><span> | |
| </span><span class="keyword1"><span class="command">lemma</span></span><span> </span><span>napp_star</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"s =~ e ⟹ napp n s =~ Star e"</span></span></span><span> | |
| </span><span class="keyword1"><span class="command">proof</span></span><span> </span><span class="delimiter">(</span><span>induction</span><span> </span><span>n</span><span class="delimiter">)</span><span> | |
| </span><span> </span><span class="keyword3"><span class="command">case</span></span><span> </span><span>0</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">then</span></span><span> </span><span class="keyword3"><span class="command">show</span></span><span> </span><span class="var">?case</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span class="delimiter">(</span><span>simp</span><span> </span><span>add</span><span class="delimiter">:</span><span> </span><span>MStar0</span><span class="delimiter">)</span><span> | |
| </span><span class="keyword1"><span class="command">next</span></span><span> | |
| </span><span> </span><span class="keyword3"><span class="command">case</span></span><span> </span><span class="delimiter">(</span><span>Suc</span><span> </span><span>n</span><span class="delimiter">)</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">then</span></span><span> </span><span class="keyword3"><span class="command">show</span></span><span> </span><span class="var">?case</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span class="delimiter">(</span><span>simp</span><span> </span><span>add</span><span class="delimiter">:</span><span> </span><span>MStarApp</span><span class="delimiter">)</span><span> | |
| </span><span class="keyword1"><span class="command">qed</span></span><span> | |
| </span><span> | |
| </span><span class="keyword1"><span class="command">lemma</span></span><span> </span><span>pumping</span><span class="delimiter">:</span><span> | |
| </span><span> </span><span class="string"><span class="delete"><span class="delete">"⟦s =~ re; pumping_constant re ≤ length s⟧ ⟹ | |
| ∃s1 s2 s3. s = s1 @ s2 @ s3 ∧ s2 ≠ [] ∧ (∀m. s1 @ napp m s2 @ s3 =~ re)"</span></span></span><span> | |
| </span><span class="keyword1"><span class="command">proof</span></span><span> </span><span class="delimiter">(</span><span>induction</span><span> </span><span>rule</span><span class="delimiter">:</span><span> </span><span>match_regexp.induct</span><span class="delimiter">)</span><span> | |
| </span><span> </span><span class="keyword3"><span class="command">case</span></span><span> </span><span>MEmpty</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">then</span></span><span> </span><span class="keyword3"><span class="command">show</span></span><span> </span><span class="var">?case</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>simp</span><span> | |
| </span><span class="keyword1"><span class="command">next</span></span><span> | |
| </span><span> </span><span class="keyword3"><span class="command">case</span></span><span> </span><span class="delimiter">(</span><span>MChar</span><span> </span><span>x</span><span class="delimiter">)</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">then</span></span><span> </span><span class="keyword3"><span class="command">show</span></span><span> </span><span class="var">?case</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>simp</span><span> | |
| </span><span class="keyword1"><span class="command">next</span></span><span> | |
| </span><span> </span><span class="keyword3"><span class="command">case</span></span><span> </span><span class="delimiter">(</span><span>MApp</span><span> </span><span>x1</span><span> </span><span>e1</span><span> </span><span>x2</span><span> </span><span>e2</span><span class="delimiter">)</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">moreover</span></span><span> </span><span class="keyword3"><span class="command">show</span></span><span> </span><span class="var">?case</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">proof</span></span><span> </span><span class="delimiter">(</span><span>rule</span><span> </span><span>disjE</span><span class="delimiter">)</span><span> | |
| </span><span> </span><span class="keyword3"><span class="command">show</span></span><span> </span><span class="string"><span class="delete"><span class="delete">"pumping_constant e1 ≤ length x1 ∨ pumping_constant e2 ≤ length x2"</span></span></span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>calculation</span><span class="delimiter">(</span><span>5</span><span class="delimiter">)</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>auto</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">next</span></span><span> | |
| </span><span> </span><span class="keyword3"><span class="command">assume</span></span><span> </span><span class="string"><span class="delete"><span class="delete">"pumping_constant e1 ≤ length x1"</span></span></span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">then</span></span><span> </span><span class="keyword3"><span class="command">obtain</span></span><span> </span><span>s1</span><span> </span><span>s2</span><span> </span><span>s3</span><span> </span><span class="keyword2"><span class="keyword">where</span></span><span> | |
| </span><span> </span><span>pump</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"x1 = s1 @ s2 @ s3 ∧ s2 ≠ [] ∧ (∀m. s1 @ napp m s2 @ s3 =~ e1)"</span></span></span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>calculation</span><span class="delimiter">(</span><span>3</span><span class="delimiter">)</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>auto</span><span> | |
| </span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">have</span></span><span> </span><span>p1</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"x1 @ x2 = s1 @ s2 @ (s3 @ x2)"</span></span></span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>pump</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>auto</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">have</span></span><span> </span><span>p2</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"s2 ≠ []"</span></span></span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>pump</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>auto</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">have</span></span><span> </span><span>p3</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"∀m. s1 @ napp m s2 @ (s3 @ x2) =~ App e1 e2"</span></span></span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>MApp</span><span class="delimiter">(</span><span>2</span><span class="delimiter">)</span><span> </span><span>match_regexp.MApp</span><span> </span><span>pump</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>fastforce</span><span> | |
| </span><span> | |
| </span><span> </span><span class="keyword3"><span class="command">show</span></span><span> </span><span class="var">?thesis</span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>p1</span><span> </span><span>p2</span><span> </span><span>p3</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>blast</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">next</span></span><span> | |
| </span><span> </span><span class="keyword3"><span class="command">assume</span></span><span> </span><span class="string"><span class="delete"><span class="delete">"pumping_constant e2 ≤ length x2"</span></span></span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">then</span></span><span> </span><span class="keyword3"><span class="command">obtain</span></span><span> </span><span>s1</span><span> </span><span>s2</span><span> </span><span>s3</span><span> </span><span class="keyword2"><span class="keyword">where</span></span><span> | |
| </span><span> </span><span>pump</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"x2 = s1 @ s2 @ s3 ∧ s2 ≠ [] ∧ (∀m. s1 @ napp m s2 @ s3 =~ e2)"</span></span></span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>calculation</span><span class="delimiter">(</span><span>4</span><span class="delimiter">)</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>auto</span><span> | |
| </span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">have</span></span><span> </span><span>p1</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"x1 @ x2 = (x1 @ s1) @ s2 @ s3"</span></span></span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>pump</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>auto</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">have</span></span><span> </span><span>p2</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"s2 ≠ []"</span></span></span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>pump</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>auto</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">have</span></span><span> </span><span>p3</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"∀m. (x1 @ s1) @ napp m s2 @ s3 =~ App e1 e2"</span></span></span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>calculation</span><span class="delimiter">(</span><span>1</span><span class="delimiter">)</span><span> </span><span>match_regexp.MApp</span><span> </span><span>pump</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>fastforce</span><span> | |
| </span><span> | |
| </span><span> </span><span class="keyword3"><span class="command">show</span></span><span> </span><span class="var">?thesis</span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>p1</span><span> </span><span>p2</span><span> </span><span>p3</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>blast</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">qed</span></span><span> | |
| </span><span class="keyword1"><span class="command">next</span></span><span> | |
| </span><span> </span><span class="keyword3"><span class="command">case</span></span><span> </span><span class="delimiter">(</span><span>MUnionL</span><span> </span><span>x</span><span> </span><span>e1</span><span> </span><span>e2</span><span class="delimiter">)</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">moreover</span></span><span> </span><span class="keyword1"><span class="command">have</span></span><span> </span><span class="string"><span class="delete"><span class="delete">"pumping_constant e1 ≤ length x"</span></span></span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>calculation</span><span class="delimiter">(</span><span>3</span><span class="delimiter">)</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>auto</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">then</span></span><span> </span><span class="keyword3"><span class="command">obtain</span></span><span> </span><span>s1</span><span> </span><span>s2</span><span> </span><span>s3</span><span> </span><span class="keyword2"><span class="keyword">where</span></span><span> | |
| </span><span> </span><span>pump</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"x = s1 @ s2 @ s3 ∧ s2 ≠ [] ∧ (∀m. s1 @ napp m s2 @ s3 =~ e1)"</span></span></span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>calculation</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>auto</span><span> | |
| </span><span> </span><span class="keyword3"><span class="command">show</span></span><span> </span><span class="var">?case</span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>match_regexp.MUnionL</span><span> </span><span>pump</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>blast</span><span> | |
| </span><span class="keyword1"><span class="command">next</span></span><span> | |
| </span><span> </span><span class="keyword3"><span class="command">case</span></span><span> </span><span class="delimiter">(</span><span>MUnionR</span><span> </span><span>x</span><span> </span><span>e2</span><span> </span><span>e1</span><span class="delimiter">)</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">moreover</span></span><span> </span><span class="keyword1"><span class="command">have</span></span><span> </span><span class="string"><span class="delete"><span class="delete">"pumping_constant e2 ≤ length x"</span></span></span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>calculation</span><span class="delimiter">(</span><span>3</span><span class="delimiter">)</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>auto</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">then</span></span><span> </span><span class="keyword3"><span class="command">obtain</span></span><span> </span><span>s1</span><span> </span><span>s2</span><span> </span><span>s3</span><span> </span><span class="keyword2"><span class="keyword">where</span></span><span> | |
| </span><span> </span><span>pump</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"x = s1 @ s2 @ s3 ∧ s2 ≠ [] ∧ (∀m. s1 @ napp m s2 @ s3 =~ e2)"</span></span></span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>calculation</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>auto</span><span> | |
| </span><span> </span><span class="keyword3"><span class="command">show</span></span><span> </span><span class="var">?case</span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>match_regexp.MUnionR</span><span> </span><span>pump</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>blast</span><span> | |
| </span><span class="keyword1"><span class="command">next</span></span><span> | |
| </span><span> </span><span class="keyword3"><span class="command">case</span></span><span> </span><span class="delimiter">(</span><span>MStar0</span><span> </span><span>e</span><span class="delimiter">)</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">then</span></span><span> </span><span class="keyword3"><span class="command">show</span></span><span> </span><span class="var">?case</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>simp</span><span> | |
| </span><span class="keyword1"><span class="command">next</span></span><span> | |
| </span><span> </span><span class="keyword3"><span class="command">case</span></span><span> </span><span class="delimiter">(</span><span>MStarApp</span><span> </span><span>x1</span><span> </span><span>e</span><span> </span><span>x2</span><span class="delimiter">)</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">then</span></span><span> </span><span class="keyword3"><span class="command">show</span></span><span> </span><span class="var">?case</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">proof</span></span><span> </span><span class="delimiter">(</span><span>cases</span><span> </span><span>x2</span><span class="delimiter">)</span><span> | |
| </span><span> </span><span class="keyword3"><span class="command">case</span></span><span> </span><span>Nil</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">moreover</span></span><span> </span><span class="keyword1"><span class="command">have</span></span><span> </span><span class="string"><span class="delete"><span class="delete">"x1 ≠ []"</span></span></span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>calculation</span><span class="delimiter">(</span><span>1</span><span class="delimiter">)</span><span> </span><span>MStarApp.prems</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>auto</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">moreover</span></span><span> </span><span class="keyword1"><span class="command">have</span></span><span> </span><span class="string"><span class="delete"><span class="delete">"x1 @ x2 = [] @ x1 @ []"</span></span></span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>calculation</span><span class="delimiter">(</span><span>1</span><span class="delimiter">)</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>auto</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">moreover</span></span><span> </span><span class="keyword1"><span class="command">have</span></span><span> </span><span class="string"><span class="delete"><span class="delete">"∀m. [] @ napp m x1 @ [] =~ Star e"</span></span></span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>MStarApp</span><span class="delimiter">(</span><span>1</span><span class="delimiter">)</span><span> </span><span>napp_star</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>auto</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">ultimately</span></span><span> </span><span class="keyword3"><span class="command">show</span></span><span> </span><span class="var">?thesis</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>blast</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">next</span></span><span> | |
| </span><span> </span><span class="keyword3"><span class="command">case</span></span><span> </span><span class="delimiter">(</span><span>Cons</span><span> </span><span>a</span><span> </span><span>xs</span><span class="delimiter">)</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">then</span></span><span> </span><span class="keyword1"><span class="command">have</span></span><span> </span><span class="string"><span class="delete"><span class="delete">"pumping_constant (Star e) ≤ length x2"</span></span></span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>simp</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">then</span></span><span> </span><span class="keyword3"><span class="command">obtain</span></span><span> </span><span>s1</span><span> </span><span>s2</span><span> </span><span>s3</span><span> </span><span class="keyword2"><span class="keyword">where</span></span><span> | |
| </span><span> </span><span>pump</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"x2 = s1 @ s2 @ s3 ∧ s2 ≠ [] ∧ (∀m. s1 @ napp m s2 @ s3 =~ Star e)"</span></span></span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>MStarApp</span><span class="delimiter">(</span><span>4</span><span class="delimiter">)</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>auto</span><span> | |
| </span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">have</span></span><span> </span><span>p1</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"x1 @ x2 = (x1 @ s1) @ s2 @ s3"</span></span></span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>pump</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>auto</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">have</span></span><span> </span><span>p2</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"s2 ≠ []"</span></span></span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>pump</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>auto</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">have</span></span><span> </span><span>p3</span><span class="delimiter">:</span><span> </span><span class="string"><span class="delete"><span class="delete">"∀m. (x1 @ s1) @ napp m s2 @ s3 =~ Star e"</span></span></span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>MStarApp</span><span class="delimiter">(</span><span>1</span><span class="delimiter">)</span><span> </span><span>pump</span><span> </span><span>match_regexp.MStarApp</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>auto</span><span> | |
| </span><span> | |
| </span><span> </span><span class="keyword3"><span class="command">show</span></span><span> </span><span class="var">?thesis</span><span> </span><span class="keyword1"><span class="command">using</span></span><span> </span><span>p1</span><span> </span><span>p2</span><span> </span><span>p3</span><span> </span><span class="keyword1"><span class="command">by</span></span><span> </span><span>blast</span><span> | |
| </span><span> </span><span class="keyword1"><span class="command">qed</span></span><span> | |
| </span><span class="keyword1"><span class="command">qed</span></span><span> | |
| </span></pre> | |
| </div> | |
| </body> | |
| </html> |
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