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language_util.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1"/>
<link href="coqdoc.css" rel="stylesheet" type="text/css"/>
<title>language_util</title>
</head>
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<h1 class="libtitle">language_util</h1>
<div class="code">
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Export</span> <a class="idref" href="language_flagged.html#"><span class="id" type="library">language_flagged</span></a>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab90"></a><h1 class="section">Auxiliary Functions For Language</h1>
</div>
<div class="code">
<span class="id" type="keyword">Section</span> <a name="AuxiliaryFunctions_For_Language"><span class="id" type="section">AuxiliaryFunctions_For_Language</span></a>.<br/>
<br/>
<span class="id" type="keyword">End</span> <a class="idref" href="language_util.html#AuxiliaryFunctions_For_Language"><span class="id" type="section">AuxiliaryFunctions_For_Language</span></a>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab91"></a><h1 class="section">Auxiliary Functions For Language_Flagged</h1>
</div>
<div class="code">
<br/>
<span class="id" type="keyword">Section</span> <a name="AuxiliaryFunctions_For_Language_Flagged"><span class="id" type="section">AuxiliaryFunctions_For_Language_Flagged</span></a>.<br/>
<br/>
</div>
<div class="doc">
Check Flags Extraction Functions
</div>
<div class="code">
<br/>
<span class="id" type="keyword">Function</span> <span class="id" type="var">name_exterior_checks</span> (<span class="id" type="var">n</span>: <a class="idref" href="language_flagged.html#name_x"><span class="id" type="inductive">name_x</span></a>): <a class="idref" href="checks.html#exterior_checks"><span class="id" type="definition">exterior_checks</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="language_util.html#n"><span class="id" type="variable">n</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="language_flagged.html#E_Identifier_X"><span class="id" type="constructor">E_Identifier_X</span></a> <span class="id" type="var">ast_num</span> <span class="id" type="var">x</span> <span class="id" type="var">ex_cks</span> => <span class="id" type="var">ex_cks</span><br/>
| <a class="idref" href="language_flagged.html#E_Indexed_Component_X"><span class="id" type="constructor">E_Indexed_Component_X</span></a> <span class="id" type="var">ast_num</span> <span class="id" type="var">x</span> <span class="id" type="var">e</span> <span class="id" type="var">ex_cks</span> => <span class="id" type="var">ex_cks</span><br/>
| <a class="idref" href="language_flagged.html#E_Selected_Component_X"><span class="id" type="constructor">E_Selected_Component_X</span></a> <span class="id" type="var">ast_num</span> <span class="id" type="var">x</span> <span class="id" type="var">f</span> <span class="id" type="var">ex_cks</span> => <span class="id" type="var">ex_cks</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Function</span> <span class="id" type="var">exp_exterior_checks</span> (<span class="id" type="var">e</span>: <a class="idref" href="language_flagged.html#expression_x"><span class="id" type="inductive">expression_x</span></a>): <a class="idref" href="checks.html#exterior_checks"><span class="id" type="definition">exterior_checks</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="language_util.html#e"><span class="id" type="variable">e</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="language_flagged.html#E_Literal_X"><span class="id" type="constructor">E_Literal_X</span></a> <span class="id" type="var">ast_num</span> <span class="id" type="var">l</span> <span class="id" type="var">in_cks</span> <span class="id" type="var">ex_cks</span> => <span class="id" type="var">ex_cks</span><br/>
| <a class="idref" href="language_flagged.html#E_Name_X"><span class="id" type="constructor">E_Name_X</span></a> <span class="id" type="var">ast_num</span> <span class="id" type="var">n</span> => <a class="idref" href="language_util.html#name_exterior_checks"><span class="id" type="definition">name_exterior_checks</span></a> <span class="id" type="var">n</span><br/>
| <a class="idref" href="language_flagged.html#E_Binary_Operation_X"><span class="id" type="constructor">E_Binary_Operation_X</span></a> <span class="id" type="var">ast_num</span> <span class="id" type="var">op</span> <span class="id" type="var">e1</span> <span class="id" type="var">e2</span> <span class="id" type="var">in_cks</span> <span class="id" type="var">ex_cks</span> => <span class="id" type="var">ex_cks</span><br/>
| <a class="idref" href="language_flagged.html#E_Unary_Operation_X"><span class="id" type="constructor">E_Unary_Operation_X</span></a> <span class="id" type="var">ast_num</span> <span class="id" type="var">op</span> <span class="id" type="var">e</span> <span class="id" type="var">in_cks</span> <span class="id" type="var">ex_cks</span> => <span class="id" type="var">ex_cks</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <a name="update_exterior_checks_name"><span class="id" type="definition">update_exterior_checks_name</span></a> <span class="id" type="var">n</span> <span class="id" type="var">checks</span> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="language_util.html#n"><span class="id" type="variable">n</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="language_flagged.html#E_Identifier_X"><span class="id" type="constructor">E_Identifier_X</span></a> <span class="id" type="var">ast_num</span> <span class="id" type="var">x</span> <span class="id" type="var">ex_checks</span> => <a class="idref" href="language_flagged.html#E_Identifier_X"><span class="id" type="constructor">E_Identifier_X</span></a> <span class="id" type="var">ast_num</span> <span class="id" type="var">x</span> <a class="idref" href="language_util.html#checks"><span class="id" type="variable">checks</span></a><br/>
| <a class="idref" href="language_flagged.html#E_Indexed_Component_X"><span class="id" type="constructor">E_Indexed_Component_X</span></a> <span class="id" type="var">ast_num</span> <span class="id" type="var">x</span> <span class="id" type="var">e</span> <span class="id" type="var">ex_checks</span> => <a class="idref" href="language_flagged.html#E_Indexed_Component_X"><span class="id" type="constructor">E_Indexed_Component_X</span></a> <span class="id" type="var">ast_num</span> <span class="id" type="var">x</span> <span class="id" type="var">e</span> <a class="idref" href="language_util.html#checks"><span class="id" type="variable">checks</span></a><br/>
| <a class="idref" href="language_flagged.html#E_Selected_Component_X"><span class="id" type="constructor">E_Selected_Component_X</span></a> <span class="id" type="var">ast_num</span> <span class="id" type="var">x</span> <span class="id" type="var">f</span> <span class="id" type="var">ex_checks</span> => <a class="idref" href="language_flagged.html#E_Selected_Component_X"><span class="id" type="constructor">E_Selected_Component_X</span></a> <span class="id" type="var">ast_num</span> <span class="id" type="var">x</span> <span class="id" type="var">f</span> <a class="idref" href="language_util.html#checks"><span class="id" type="variable">checks</span></a><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <a name="update_exterior_checks_exp"><span class="id" type="definition">update_exterior_checks_exp</span></a> <span class="id" type="var">e</span> <span class="id" type="var">checks</span> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="language_util.html#e"><span class="id" type="variable">e</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="language_flagged.html#E_Literal_X"><span class="id" type="constructor">E_Literal_X</span></a> <span class="id" type="var">ast_num</span> <span class="id" type="var">l</span> <span class="id" type="var">in_checks</span> <span class="id" type="var">ex_checks</span> => <a class="idref" href="language_flagged.html#E_Literal_X"><span class="id" type="constructor">E_Literal_X</span></a> <span class="id" type="var">ast_num</span> <span class="id" type="var">l</span> <span class="id" type="var">in_checks</span> <a class="idref" href="language_util.html#checks"><span class="id" type="variable">checks</span></a><br/>
| <a class="idref" href="language_flagged.html#E_Name_X"><span class="id" type="constructor">E_Name_X</span></a> <span class="id" type="var">ast_num</span> <span class="id" type="var">n</span> => <br/>
<span class="id" type="keyword">let</span> <span class="id" type="var">n'</span> := <a class="idref" href="language_util.html#update_exterior_checks_name"><span class="id" type="definition">update_exterior_checks_name</span></a> <span class="id" type="var">n</span> <a class="idref" href="language_util.html#checks"><span class="id" type="variable">checks</span></a> <span class="id" type="keyword">in</span> <br/>
<a class="idref" href="language_flagged.html#E_Name_X"><span class="id" type="constructor">E_Name_X</span></a> <span class="id" type="var">ast_num</span> <a class="idref" href="language_util.html#n'"><span class="id" type="variable">n'</span></a><br/>
| <a class="idref" href="language_flagged.html#E_Binary_Operation_X"><span class="id" type="constructor">E_Binary_Operation_X</span></a> <span class="id" type="var">ast_num</span> <span class="id" type="var">bop</span> <span class="id" type="var">e1</span> <span class="id" type="var">e2</span> <span class="id" type="var">in_checks</span> <span class="id" type="var">ex_checks</span> => <a class="idref" href="language_flagged.html#E_Binary_Operation_X"><span class="id" type="constructor">E_Binary_Operation_X</span></a> <span class="id" type="var">ast_num</span> <span class="id" type="var">bop</span> <span class="id" type="var">e1</span> <span class="id" type="var">e2</span> <span class="id" type="var">in_checks</span> <a class="idref" href="language_util.html#checks"><span class="id" type="variable">checks</span></a><br/>
| <a class="idref" href="language_flagged.html#E_Unary_Operation_X"><span class="id" type="constructor">E_Unary_Operation_X</span></a> <span class="id" type="var">ast_num</span> <span class="id" type="var">uop</span> <span class="id" type="var">e</span> <span class="id" type="var">in_checks</span> <span class="id" type="var">ex_checks</span> => <a class="idref" href="language_flagged.html#E_Unary_Operation_X"><span class="id" type="constructor">E_Unary_Operation_X</span></a> <span class="id" type="var">ast_num</span> <span class="id" type="var">uop</span> <a class="idref" href="language_util.html#e"><span class="id" type="variable">e</span></a> <span class="id" type="var">in_checks</span> <a class="idref" href="language_util.html#checks"><span class="id" type="variable">checks</span></a><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="exp_updated_exterior_checks"><span class="id" type="lemma">exp_updated_exterior_checks</span></a>: <span class="id" type="keyword">forall</span> <span class="id" type="var">e</span> <span class="id" type="var">cks</span>,<br/>
<a class="idref" href="language_util.html#exp_exterior_checks"><span class="id" type="definition">exp_exterior_checks</span></a> (<a class="idref" href="language_util.html#update_exterior_checks_exp"><span class="id" type="definition">update_exterior_checks_exp</span></a> <a class="idref" href="language_util.html#e"><span class="id" type="variable">e</span></a> <a class="idref" href="language_util.html#cks"><span class="id" type="variable">cks</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="language_util.html#cks"><span class="id" type="variable">cks</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>; <span class="id" type="tactic">destruct</span> <span class="id" type="var">e</span>; <span class="id" type="var">smack</span>;<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">n</span>; <span class="id" type="var">smack</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="name_updated_exterior_checks"><span class="id" type="lemma">name_updated_exterior_checks</span></a>: <span class="id" type="keyword">forall</span> <span class="id" type="var">n</span> <span class="id" type="var">cks</span>,<br/>
<a class="idref" href="language_util.html#name_exterior_checks"><span class="id" type="definition">name_exterior_checks</span></a> (<a class="idref" href="language_util.html#update_exterior_checks_name"><span class="id" type="definition">update_exterior_checks_name</span></a> <a class="idref" href="language_util.html#n"><span class="id" type="variable">n</span></a> <a class="idref" href="language_util.html#cks"><span class="id" type="variable">cks</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="language_util.html#cks"><span class="id" type="variable">cks</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>; <span class="id" type="tactic">destruct</span> <span class="id" type="var">n</span>; <span class="id" type="var">smack</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="exp_exterior_checks_refl"><span class="id" type="lemma">exp_exterior_checks_refl</span></a>: <span class="id" type="keyword">forall</span> <span class="id" type="var">e</span>,<br/>
<a class="idref" href="language_util.html#update_exterior_checks_exp"><span class="id" type="definition">update_exterior_checks_exp</span></a> <a class="idref" href="language_util.html#e"><span class="id" type="variable">e</span></a> (<a class="idref" href="language_util.html#exp_exterior_checks"><span class="id" type="definition">exp_exterior_checks</span></a> <a class="idref" href="language_util.html#e"><span class="id" type="variable">e</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="language_util.html#e"><span class="id" type="variable">e</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">e</span>; <span class="id" type="var">smack</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">n</span>; <span class="id" type="tactic">simpl</span>; <span class="id" type="var">smack</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="name_exterior_checks_refl"><span class="id" type="lemma">name_exterior_checks_refl</span></a>: <span class="id" type="keyword">forall</span> <span class="id" type="var">n</span>,<br/>
<a class="idref" href="language_util.html#update_exterior_checks_name"><span class="id" type="definition">update_exterior_checks_name</span></a> <a class="idref" href="language_util.html#n"><span class="id" type="variable">n</span></a> (<a class="idref" href="language_util.html#name_exterior_checks"><span class="id" type="definition">name_exterior_checks</span></a> <a class="idref" href="language_util.html#n"><span class="id" type="variable">n</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="language_util.html#n"><span class="id" type="variable">n</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">n</span>; <span class="id" type="var">smack</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="update_exterior_checks_exp_astnum_eq"><span class="id" type="lemma">update_exterior_checks_exp_astnum_eq</span></a>: <span class="id" type="keyword">forall</span> <span class="id" type="var">e</span> <span class="id" type="var">cks</span>,<br/>
<a class="idref" href="language_flagged.html#expression_astnum_x"><span class="id" type="definition">expression_astnum_x</span></a> (<a class="idref" href="language_util.html#update_exterior_checks_exp"><span class="id" type="definition">update_exterior_checks_exp</span></a> <a class="idref" href="language_util.html#e"><span class="id" type="variable">e</span></a> <a class="idref" href="language_util.html#cks"><span class="id" type="variable">cks</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="language_flagged.html#expression_astnum_x"><span class="id" type="definition">expression_astnum_x</span></a> <a class="idref" href="language_util.html#e"><span class="id" type="variable">e</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>;<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">e</span>; <span class="id" type="var">smack</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <a name="update_exterior_checks_name_astnum_eq"><span class="id" type="lemma">update_exterior_checks_name_astnum_eq</span></a>: <span class="id" type="keyword">forall</span> <span class="id" type="var">n</span> <span class="id" type="var">cks</span>,<br/>
<a class="idref" href="language_flagged.html#name_astnum_x"><span class="id" type="definition">name_astnum_x</span></a> (<a class="idref" href="language_util.html#update_exterior_checks_name"><span class="id" type="definition">update_exterior_checks_name</span></a> <a class="idref" href="language_util.html#n"><span class="id" type="variable">n</span></a> <a class="idref" href="language_util.html#cks"><span class="id" type="variable">cks</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl4/stdlib/Coq.Init.Logic.html#:type_scope:x_'='_x"><span class="id" type="notation">=</span></a> <a class="idref" href="language_flagged.html#name_astnum_x"><span class="id" type="definition">name_astnum_x</span></a> <a class="idref" href="language_util.html#n"><span class="id" type="variable">n</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>;<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">n</span>; <span class="id" type="var">smack</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">End</span> <a class="idref" href="language_util.html#AuxiliaryFunctions_For_Language_Flagged"><span class="id" type="section">AuxiliaryFunctions_For_Language_Flagged</span></a>.<br/>
<br/>
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