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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<title>Module Qcert.Data.Operators.BinaryOperatorsSem</title>
<meta name="description" content="Documentation of Coq module Qcert.Data.Operators.BinaryOperatorsSem" />
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<body onload="hideAll('proofscript')">
<h1 class="title">Module Qcert.Data.Operators.BinaryOperatorsSem</h1>
<div class="coq">
<br/>
<span class="kwd">Require</span> <span class="kwd">Import</span> <span class="id"><a href="http://coq.inria.fr/library/Coq.Strings.String.html">String</a></span>.<br/>
<span class="kwd">Require</span> <span class="kwd">Import</span> <span class="id"><a href="http://coq.inria.fr/library/Coq.Lists.List.html">List</a></span>.<br/>
<span class="kwd">Require</span> <span class="kwd">Import</span> <span class="id"><a href="http://coq.inria.fr/library/Coq.Arith.Compare_dec.html">Compare_dec</a></span>.<br/>
<span class="kwd">Require</span> <span class="kwd">Import</span> <span class="id"><a href="http://coq.inria.fr/library/Coq.ZArith.ZArith.html">ZArith</a></span>.<br/>
<span class="kwd">Require</span> <span class="kwd">Import</span> <span class="id"><a href="Qcert.Utils.Utils.html">Utils</a></span>.<br/>
<span class="kwd">Require</span> <span class="kwd">Import</span> <span class="id"><a href="Qcert.Brands.BrandRelation.html">BrandRelation</a></span>.<br/>
<span class="kwd">Require</span> <span class="kwd">Import</span> <span class="id"><a href="Qcert.Data.Model.ForeignData.html">ForeignData</a></span>.<br/>
<span class="kwd">Require</span> <span class="kwd">Import</span> <span class="id"><a href="Qcert.Data.Model.Data.html">Data</a></span>.<br/>
<span class="kwd">Require</span> <span class="kwd">Import</span> <span class="id"><a href="Qcert.Data.Model.DataLift.html">DataLift</a></span>.<br/>
<span class="kwd">Require</span> <span class="kwd">Import</span> <span class="id"><a href="Qcert.Data.Model.DataNorm.html">DataNorm</a></span>.<br/>
<span class="kwd">Require</span> <span class="kwd">Import</span> <span class="id"><a href="Qcert.Data.Operators.ForeignOperators.html">ForeignOperators</a></span>.<br/>
<span class="kwd">Require</span> <span class="kwd">Import</span> <span class="id"><a href="Qcert.Data.Operators.OperatorsUtils.html">OperatorsUtils</a></span>.<br/>
<span class="kwd">Require</span> <span class="kwd">Import</span> <span class="id"><a href="Qcert.Data.Operators.Iterators.html">Iterators</a></span>.<br/>
<span class="kwd">Require</span> <span class="kwd">Export</span> <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html">BinaryOperators</a></span>.<br/>
<br/>
<span class="kwd">Section</span> <span class="id"><a name="BinaryOperatorsSem">BinaryOperatorsSem</a></span>.<br/>
<span class="kwd">Definition</span> <span class="id"><a name="nat_arith_binary_op_eval">nat_arith_binary_op_eval</a></span> (<span class="id">op</span>:<span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#nat_arith_binary_op">nat_arith_binary_op</a></span>) (<span class="id">z1</span> <span class="id">z2</span>:<span class="id"><a href="http://coq.inria.fr/library/Coq.Numbers.BinNums.html#Z">Z</a></span>) : <span class="id"><a href="http://coq.inria.fr/library/Coq.Numbers.BinNums.html#Z">Z</a></span> :=<br/>
<span class="kwd">match</span> <span class="id">op</span> <span class="kwd">with</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#NatPlus">NatPlus</a></span> => <span class="id"><a href="http://coq.inria.fr/library/Coq.ZArith.BinInt.html#Z.add">Z.add</a></span> <span class="id">z1</span> <span class="id">z2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#NatMinus">NatMinus</a></span>=> <span class="id"><a href="http://coq.inria.fr/library/Coq.ZArith.BinInt.html#Z.sub">Z.sub</a></span> <span class="id">z1</span> <span class="id">z2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#NatMult">NatMult</a></span> => <span class="id"><a href="http://coq.inria.fr/library/Coq.ZArith.BinInt.html#Z.mul">Z.mul</a></span> <span class="id">z1</span> <span class="id">z2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#NatMin">NatMin</a></span> => <span class="id"><a href="http://coq.inria.fr/library/Coq.ZArith.BinInt.html#Z.min">Z.min</a></span> <span class="id">z1</span> <span class="id">z2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#NatMax">NatMax</a></span> => <span class="id"><a href="http://coq.inria.fr/library/Coq.ZArith.BinInt.html#Z.max">Z.max</a></span> <span class="id">z1</span> <span class="id">z2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#NatDiv">NatDiv</a></span>=> <span class="id"><a href="http://coq.inria.fr/library/Coq.ZArith.BinInt.html#Z.quot">Z.quot</a></span> <span class="id">z1</span> <span class="id">z2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#NatRem">NatRem</a></span> => <span class="id"><a href="http://coq.inria.fr/library/Coq.ZArith.BinInt.html#Z.rem">Z.rem</a></span> <span class="id">z1</span> <span class="id">z2</span><br/>
<span class="kwd">end</span>.<br/>
<br/>
<span class="kwd">Definition</span> <span class="id"><a name="float_arith_binary_op_eval">float_arith_binary_op_eval</a></span> (<span class="id">op</span>:<span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#float_arith_binary_op">float_arith_binary_op</a></span>) (<span class="id">f1</span> <span class="id">f2</span>:<span class="id"><a href="Qcert.Utils.FloatAdd.html#float">float</a></span>) : <span class="id"><a href="Qcert.Utils.FloatAdd.html#float">float</a></span> :=<br/>
<span class="kwd">match</span> <span class="id">op</span> <span class="kwd">with</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#FloatPlus">FloatPlus</a></span> => <span class="id"><a href="Qcert.Utils.FloatAdd.html#float_add">float_add</a></span> <span class="id">f1</span> <span class="id">f2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#FloatMinus">FloatMinus</a></span> => <span class="id"><a href="Qcert.Utils.FloatAdd.html#float_sub">float_sub</a></span> <span class="id">f1</span> <span class="id">f2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#FloatMult">FloatMult</a></span> => <span class="id"><a href="Qcert.Utils.FloatAdd.html#float_mult">float_mult</a></span> <span class="id">f1</span> <span class="id">f2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#FloatDiv">FloatDiv</a></span> => <span class="id"><a href="Qcert.Utils.FloatAdd.html#float_div">float_div</a></span> <span class="id">f1</span> <span class="id">f2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#FloatPow">FloatPow</a></span> => <span class="id"><a href="Qcert.Utils.FloatAdd.html#float_pow">float_pow</a></span> <span class="id">f1</span> <span class="id">f2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#FloatMin">FloatMin</a></span> => <span class="id"><a href="Qcert.Utils.FloatAdd.html#float_min">float_min</a></span> <span class="id">f1</span> <span class="id">f2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#FloatMax">FloatMax</a></span> => <span class="id"><a href="Qcert.Utils.FloatAdd.html#float_max">float_max</a></span> <span class="id">f1</span> <span class="id">f2</span><br/>
<span class="kwd">end</span>.<br/>
<br/>
<span class="kwd">Definition</span> <span class="id"><a name="float_compare_binary_op_eval">float_compare_binary_op_eval</a></span> (<span class="id">op</span>:<span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#float_compare_binary_op">float_compare_binary_op</a></span>) (<span class="id">f1</span> <span class="id">f2</span>:<span class="id"><a href="Qcert.Utils.FloatAdd.html#float">float</a></span>) : <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#bool">bool</a></span> :=<br/>
<span class="kwd">match</span> <span class="id">op</span> <span class="kwd">with</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#FloatLt">FloatLt</a></span> => <span class="id"><a href="Qcert.Utils.FloatAdd.html#float_lt">float_lt</a></span> <span class="id">f1</span> <span class="id">f2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#FloatLe">FloatLe</a></span> => <span class="id"><a href="Qcert.Utils.FloatAdd.html#float_le">float_le</a></span> <span class="id">f1</span> <span class="id">f2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#FloatGt">FloatGt</a></span> => <span class="id"><a href="Qcert.Utils.FloatAdd.html#float_gt">float_gt</a></span> <span class="id">f1</span> <span class="id">f2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#FloatGe">FloatGe</a></span> => <span class="id"><a href="Qcert.Utils.FloatAdd.html#float_ge">float_ge</a></span> <span class="id">f1</span> <span class="id">f2</span><br/>
<span class="kwd">end</span>.<br/>
<br/>
<span class="kwd">Context</span> (<span class="id">h</span>:<span class="id"><a href="Qcert.Brands.BrandRelation.html#brand_relation_t">brand_relation_t</a></span>).<br/>
<span class="kwd">Context</span> {<span class="id">fdata</span>:<span class="id"><a href="Qcert.Data.Model.ForeignData.html#foreign_data">foreign_data</a></span>}.<br/>
<span class="kwd">Context</span> {<span class="id">foperators</span>:<span class="id"><a href="Qcert.Data.Operators.ForeignOperators.html#foreign_operators">foreign_operators</a></span>}.<br/>
<br/>
<span class="kwd">Definition</span> <span class="id"><a name="binary_op_eval">binary_op_eval</a></span> (<span class="id">bop</span>:<span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#binary_op">binary_op</a></span>) (<span class="id">d1</span> <span class="id">d2</span>:<span class="id"><a href="Qcert.Data.Model.Data.html#data">data</a></span>) : <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#option">option</a></span> <span class="id"><a href="Qcert.Data.Model.Data.html#data">data</a></span> :=<br/>
<span class="kwd">match</span> <span class="id">bop</span> <span class="kwd">with</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#OpEqual">OpEqual</a></span> => <span class="id"><a href="Qcert.Data.Model.DataLift.html#unbdata">unbdata</a></span> (<span class="kwd">fun</span> <span class="id">x</span> <span class="id">y</span> => <span class="kwd">if</span> <span class="id"><a href="Qcert.Data.Model.Data.html#data_eq_dec">data_eq_dec</a></span> <span class="id">x</span> <span class="id">y</span> <span class="kwd">then</span> <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#true">true</a></span> <span class="kwd">else</span> <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#false">false</a></span>) <span class="id">d1</span> <span class="id">d2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#OpRecConcat">OpRecConcat</a></span> =><br/>
<span class="kwd">match</span> <span class="id">d1</span>, <span class="id">d2</span> <span class="kwd">with</span><br/>
| (<span class="id"><a href="Qcert.Data.Model.Data.html#drec">drec</a></span> <span class="id">r1</span>), (<span class="id"><a href="Qcert.Data.Model.Data.html#drec">drec</a></span> <span class="id">r2</span>) => <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some">Some</a></span> (<span class="id"><a href="Qcert.Data.Model.Data.html#drec">drec</a></span> (<span class="id"><a href="Qcert.Utils.Bindings.html#rec_sort">rec_sort</a></span> (<span class="id">r1</span>++<span class="id">r2</span>)))<br/>
| <span class="id">_</span>, <span class="id">_</span> => <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None">None</a></span><br/>
<span class="kwd">end</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#OpRecMerge">OpRecMerge</a></span> =><br/>
<span class="kwd">match</span> <span class="id">d1</span>, <span class="id">d2</span> <span class="kwd">with</span><br/>
| (<span class="id"><a href="Qcert.Data.Model.Data.html#drec">drec</a></span> <span class="id">r1</span>), (<span class="id"><a href="Qcert.Data.Model.Data.html#drec">drec</a></span> <span class="id">r2</span>) =><br/>
<span class="kwd">match</span> <span class="id"><a href="Qcert.Utils.Bindings.html#merge_bindings">merge_bindings</a></span> <span class="id">r1</span> <span class="id">r2</span> <span class="kwd">with</span><br/>
| <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some">Some</a></span> <span class="id">x</span> => <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some">Some</a></span> (<span class="id"><a href="Qcert.Data.Model.Data.html#dcoll">dcoll</a></span> ((<span class="id"><a href="Qcert.Data.Model.Data.html#drec">drec</a></span> <span class="id">x</span>) :: <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nil">nil</a></span>))<br/>
| <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None">None</a></span> => <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some">Some</a></span> (<span class="id"><a href="Qcert.Data.Model.Data.html#dcoll">dcoll</a></span> <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nil">nil</a></span>)<br/>
<span class="kwd">end</span><br/>
| <span class="id">_</span>, <span class="id">_</span> => <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None">None</a></span><br/>
<span class="kwd">end</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#OpAnd">OpAnd</a></span> => <span class="id"><a href="Qcert.Data.Model.DataLift.html#unbdbool">unbdbool</a></span> <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#andb">andb</a></span> <span class="id">d1</span> <span class="id">d2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#OpOr">OpOr</a></span> => <span class="id"><a href="Qcert.Data.Model.DataLift.html#unbdbool">unbdbool</a></span> <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#orb">orb</a></span> <span class="id">d1</span> <span class="id">d2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#OpLt">OpLt</a></span> => <span class="id"><a href="Qcert.Data.Model.DataLift.html#unbdnat">unbdnat</a></span> (<span class="kwd">fun</span> <span class="id">x</span> <span class="id">y</span> => <span class="kwd">if</span> <span class="id"><a href="http://coq.inria.fr/library/Coq.ZArith.ZArith_dec.html#Z_lt_dec">Z_lt_dec</a></span> <span class="id">x</span> <span class="id">y</span> <span class="kwd">then</span> <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#true">true</a></span> <span class="kwd">else</span> <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#false">false</a></span>) <span class="id">d1</span> <span class="id">d2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#OpLe">OpLe</a></span> => <span class="id"><a href="Qcert.Data.Model.DataLift.html#unbdnat">unbdnat</a></span> (<span class="kwd">fun</span> <span class="id">x</span> <span class="id">y</span> => <span class="kwd">if</span> <span class="id"><a href="http://coq.inria.fr/library/Coq.ZArith.ZArith_dec.html#Z_le_dec">Z_le_dec</a></span> <span class="id">x</span> <span class="id">y</span> <span class="kwd">then</span> <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#true">true</a></span> <span class="kwd">else</span> <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#false">false</a></span>) <span class="id">d1</span> <span class="id">d2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#OpBagUnion">OpBagUnion</a></span> => <span class="id"><a href="Qcert.Data.Model.DataLift.html#rondcoll2">rondcoll2</a></span> <span class="id"><a href="Qcert.Utils.Bag.html#bunion">bunion</a></span> <span class="id">d1</span> <span class="id">d2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#OpBagDiff">OpBagDiff</a></span> => <span class="id"><a href="Qcert.Data.Model.DataLift.html#rondcoll2">rondcoll2</a></span> (@<span class="id"><a href="Qcert.Utils.Bag.html#bminus">bminus</a></span> <span class="id"><a href="Qcert.Data.Model.Data.html#data">data</a></span> <span class="id"><a href="Qcert.Data.Model.Data.html#data_eq_dec">data_eq_dec</a></span>) <span class="id">d2</span> <span class="id">d1</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#OpBagMin">OpBagMin</a></span> => <span class="id"><a href="Qcert.Data.Model.DataLift.html#rondcoll2">rondcoll2</a></span> (@<span class="id"><a href="Qcert.Utils.Bag.html#bmin">bmin</a></span> <span class="id"><a href="Qcert.Data.Model.Data.html#data">data</a></span> <span class="id"><a href="Qcert.Data.Model.Data.html#data_eq_dec">data_eq_dec</a></span>) <span class="id">d1</span> <span class="id">d2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#OpBagMax">OpBagMax</a></span> => <span class="id"><a href="Qcert.Data.Model.DataLift.html#rondcoll2">rondcoll2</a></span> (@<span class="id"><a href="Qcert.Utils.Bag.html#bmax">bmax</a></span> <span class="id"><a href="Qcert.Data.Model.Data.html#data">data</a></span> <span class="id"><a href="Qcert.Data.Model.Data.html#data_eq_dec">data_eq_dec</a></span>) <span class="id">d1</span> <span class="id">d2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#OpBagNth">OpBagNth</a></span> =><br/>
<span class="kwd">match</span> <span class="id">d1</span>, <span class="id">d2</span> <span class="kwd">with</span><br/>
| (<span class="id"><a href="Qcert.Data.Model.Data.html#dcoll">dcoll</a></span> <span class="id">c</span>), (<span class="id"><a href="Qcert.Data.Model.Data.html#dnat">dnat</a></span> <span class="id">n</span>) =><br/>
<span class="kwd">let</span> <span class="id">natish</span> := <span class="id"><a href="Qcert.Utils.CoqLibAdd.html#ZToSignedNat">ZToSignedNat</a></span> <span class="id">n</span> <span class="kwd">in</span><br/>
<span class="kwd">if</span> (<span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#fst">fst</a></span> <span class="id">natish</span>) <span class="kwd">then</span><br/>
<span class="kwd">match</span> <span class="id"><a href="http://coq.inria.fr/library/Coq.Lists.List.html#nth_error">List.nth_error</a></span> <span class="id">c</span> (<span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#snd">snd</a></span> <span class="id">natish</span>) <span class="kwd">with</span><br/>
| <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some">Some</a></span> <span class="id">d</span> => <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some">Some</a></span> (<span class="id"><a href="Qcert.Data.Model.Data.html#dsome">dsome</a></span> <span class="id">d</span>)<br/>
| <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None">None</a></span> => <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some">Some</a></span> <span class="id"><a href="Qcert.Data.Model.Data.html#dnone">dnone</a></span><br/>
<span class="kwd">end</span><br/>
<span class="kwd">else</span> <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some">Some</a></span> <span class="id"><a href="Qcert.Data.Model.Data.html#dnone">dnone</a></span><br/>
| <span class="id">_</span>, <span class="id">_</span> => <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None">None</a></span><br/>
<span class="kwd">end</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#OpContains">OpContains</a></span> =><br/>
<span class="id"><a href="Qcert.Data.Model.DataLift.html#ondcoll">ondcoll</a></span> (<span class="kwd">fun</span> <span class="id">l</span> =><br/>
<span class="kwd">if</span> <span class="id"><a href="http://coq.inria.fr/library/Coq.Lists.List.html#in_dec">in_dec</a></span> <span class="id"><a href="Qcert.Data.Model.Data.html#data_eq_dec">data_eq_dec</a></span> <span class="id">d1</span> <span class="id">l</span><br/>
<span class="kwd">then</span> <span class="id"><a href="Qcert.Data.Model.Data.html#dbool">dbool</a></span> <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#true">true</a></span> <span class="kwd">else</span> <span class="id"><a href="Qcert.Data.Model.Data.html#dbool">dbool</a></span> <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#false">false</a></span>) <span class="id">d2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#OpStringConcat">OpStringConcat</a></span> => <span class="id"><a href="Qcert.Data.Model.DataLift.html#unsdstring">unsdstring</a></span> <span class="id"><a href="http://coq.inria.fr/library/Coq.Strings.String.html#append">append</a></span> <span class="id">d1</span> <span class="id">d2</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#OpStringJoin">OpStringJoin</a></span> =><br/>
<span class="kwd">match</span> <span class="id">d1</span>, <span class="id">d2</span> <span class="kwd">with</span><br/>
| (<span class="id"><a href="Qcert.Data.Model.Data.html#dstring">dstring</a></span> <span class="id">sep</span>), (<span class="id"><a href="Qcert.Data.Model.Data.html#dcoll">dcoll</a></span> <span class="id">c</span>) =><br/>
<span class="id"><a href="Qcert.Data.Operators.OperatorsUtils.html#lifted_join">lifted_join</a></span> <span class="id">sep</span> <span class="id">c</span><br/>
| <span class="id">_</span>, <span class="id">_</span> => <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None">None</a></span><br/>
<span class="kwd">end</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#OpNatBinary">OpNatBinary</a></span> <span class="id">op</span> =><br/>
<span class="kwd">match</span> <span class="id">d1</span>, <span class="id">d2</span> <span class="kwd">with</span><br/>
| <span class="id"><a href="Qcert.Data.Model.Data.html#dnat">dnat</a></span> <span class="id">n1</span>, <span class="id"><a href="Qcert.Data.Model.Data.html#dnat">dnat</a></span> <span class="id">n2</span> => <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some">Some</a></span> (<span class="id"><a href="Qcert.Data.Model.Data.html#dnat">dnat</a></span> (<span class="id"><a href="Qcert.Data.Operators.BinaryOperatorsSem.html#nat_arith_binary_op_eval">nat_arith_binary_op_eval</a></span> <span class="id">op</span> <span class="id">n1</span> <span class="id">n2</span>))<br/>
| <span class="id">_</span>, <span class="id">_</span> => <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None">None</a></span><br/>
<span class="kwd">end</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#OpFloatBinary">OpFloatBinary</a></span> <span class="id">op</span> =><br/>
<span class="kwd">match</span> <span class="id">d1</span>, <span class="id">d2</span> <span class="kwd">with</span><br/>
| <span class="id"><a href="Qcert.Data.Model.Data.html#dfloat">dfloat</a></span> <span class="id">f1</span>, <span class="id"><a href="Qcert.Data.Model.Data.html#dfloat">dfloat</a></span> <span class="id">f2</span> => <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some">Some</a></span> (<span class="id"><a href="Qcert.Data.Model.Data.html#dfloat">dfloat</a></span> (<span class="id"><a href="Qcert.Data.Operators.BinaryOperatorsSem.html#float_arith_binary_op_eval">float_arith_binary_op_eval</a></span> <span class="id">op</span> <span class="id">f1</span> <span class="id">f2</span>))<br/>
| <span class="id">_</span>, <span class="id">_</span> => <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None">None</a></span><br/>
<span class="kwd">end</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#OpFloatCompare">OpFloatCompare</a></span> <span class="id">op</span> =><br/>
<span class="kwd">match</span> <span class="id">d1</span>, <span class="id">d2</span> <span class="kwd">with</span><br/>
| <span class="id"><a href="Qcert.Data.Model.Data.html#dfloat">dfloat</a></span> <span class="id">f1</span>, <span class="id"><a href="Qcert.Data.Model.Data.html#dfloat">dfloat</a></span> <span class="id">f2</span> => <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some">Some</a></span> (<span class="id"><a href="Qcert.Data.Model.Data.html#dbool">dbool</a></span> (<span class="id"><a href="Qcert.Data.Operators.BinaryOperatorsSem.html#float_compare_binary_op_eval">float_compare_binary_op_eval</a></span> <span class="id">op</span> <span class="id">f1</span> <span class="id">f2</span>))<br/>
| <span class="id">_</span>, <span class="id">_</span> => <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#None">None</a></span><br/>
<span class="kwd">end</span><br/>
| <span class="id"><a href="Qcert.Data.Operators.BinaryOperators.html#OpForeignBinary">OpForeignBinary</a></span> <span class="id">fb</span> => <span class="id"><a href="Qcert.Data.Operators.ForeignOperators.html#foreign_operators_binary_interp">foreign_operators_binary_interp</a></span> <span class="id"><a href="Qcert.Data.Operators.BinaryOperatorsSem.html#BinaryOperatorsSem.h">h</a></span> <span class="id">fb</span> <span class="id">d1</span> <span class="id">d2</span><br/>
<span class="kwd">end</span>.<br/>
<br/>
<span class="kwd">Local</span> <span class="kwd">Hint</span> <span class="id">Constructors</span> <span class="id"><a href="Qcert.Data.Operators.BinaryOperatorsSem.html#data_normalized">data_normalized</a></span> <span class="id"><a href="Qcert.Data.Operators.BinaryOperatorsSem.html#Forall">Forall</a></span> : <span class="id">qcert</span>.<br/>
<br/>
<span class="kwd">Lemma</span> <span class="id"><a name="binary_op_eval_normalized">binary_op_eval_normalized</a></span> {<span class="id">b</span> <span class="id">d1</span> <span class="id">d2</span> <span class="id">o</span>} :<br/>
<span class="id"><a href="Qcert.Data.Operators.BinaryOperatorsSem.html#binary_op_eval">binary_op_eval</a></span> <span class="id">b</span> <span class="id">d1</span> <span class="id">d2</span> = <span class="id"><a href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#Some">Some</a></span> <span class="id">o</span> -><br/>
<span class="id"><a href="Qcert.Data.Model.DataNorm.html#data_normalized">data_normalized</a></span> <span class="id"><a href="Qcert.Data.Operators.BinaryOperatorsSem.html#BinaryOperatorsSem.h">h</a></span> <span class="id">d1</span> -> <span class="id"><a href="Qcert.Data.Model.DataNorm.html#data_normalized">data_normalized</a></span> <span class="id"><a href="Qcert.Data.Operators.BinaryOperatorsSem.html#BinaryOperatorsSem.h">h</a></span> <span class="id">d2</span> -><br/>
<span class="id"><a href="Qcert.Data.Model.DataNorm.html#data_normalized">data_normalized</a></span> <span class="id"><a href="Qcert.Data.Operators.BinaryOperatorsSem.html#BinaryOperatorsSem.h">h</a></span> <span class="id">o</span>.<br/>
<div class="toggleproof" onclick="toggleDisplay('proof1767')">Proof.</div>
<div class="proofscript" id="proof1767">
<span class="tactic">destruct</span> <span class="id">b</span>; <span class="tactic">simpl</span>; <span class="tactic">intros</span>;<br/>
<span class="tactic">try</span> <span class="id">solve</span> [<br/>
<span class="tactic">unfold</span> <span class="id"><a href="Qcert.Data.Model.DataLift.html#rondcoll2">rondcoll2</a></span> <span class="kwd">in</span> *;<br/>
<span class="tactic">destruct</span> <span class="id">d1</span>; <span class="tactic">simpl</span> <span class="kwd">in</span> *; <span class="tactic">try</span> <span class="tactic">discriminate</span>;<br/>
<span class="tactic">destruct</span> <span class="id">d2</span>; <span class="tactic">simpl</span> <span class="kwd">in</span> *; <span class="tactic">try</span> <span class="tactic">discriminate</span>;<br/>
<span class="tactic">inversion</span> <span class="id">H</span>; <span class="tactic">subst</span>;<br/>
<span class="tactic">eauto</span> 1;<br/>
<span class="tactic">inversion</span> <span class="id">H</span>; <span class="tactic">subst</span>;<br/>
<span class="id">constructor</span>;<br/>
<span class="tactic">inversion</span> <span class="id">H0</span>; <span class="tactic">inversion</span> <span class="id">H1</span>; <span class="tactic">subst</span>;<br/>
<span class="id">solve</span> [<span class="tactic">apply</span> <span class="id"><a href="Qcert.Utils.Bag.html#bunion_Forall">bunion_Forall</a></span>; <span class="tactic">trivial</span><br/>
| <span class="tactic">apply</span> <span class="id"><a href="Qcert.Utils.Bag.html#bminus_Forall">bminus_Forall</a></span>; <span class="tactic">trivial</span><br/>
| <span class="tactic">apply</span> <span class="id"><a href="Qcert.Utils.Bag.html#bmin_Forall">bmin_Forall</a></span>; <span class="tactic">trivial</span><br/>
| <span class="tactic">apply</span> <span class="id"><a href="Qcert.Utils.Bag.html#bmax_Forall">bmax_Forall</a></span>; <span class="tactic">trivial</span>]<br/>
].<br/>
- <span class="tactic">do</span> 2 <span class="id">match_destr_in</span> <span class="id">H</span>.<br/>
<span class="tactic">inversion</span> <span class="id">H</span>; <span class="tactic">clear</span> <span class="id">H</span>; <span class="tactic">subst</span>.<br/>
<span class="tactic">apply</span> <span class="id"><a href="Qcert.Data.Model.DataNorm.html#data_normalized_rec_sort_app">data_normalized_rec_sort_app</a></span>; <span class="tactic">trivial</span>.<br/>
- <span class="tactic">do</span> 2 <span class="id">match_destr_in</span> <span class="id">H</span>.<br/>
<span class="tactic">unfold</span> <span class="id"><a href="Qcert.Utils.Bindings.html#merge_bindings">merge_bindings</a></span> <span class="kwd">in</span> <span class="id">H</span>.<br/>
<span class="tactic">destruct</span> (<span class="id"><a href="Qcert.Utils.Compat.html#compatible">Compat.compatible</a></span> <span class="id">l</span> <span class="id">l0</span>); <span class="tactic">inversion</span> <span class="id">H</span>; <span class="id">qeauto</span>.<br/>
<span class="id">constructor</span>. <span class="id">constructor</span>; <span class="tactic">trivial</span>.<br/>
<span class="tactic">apply</span> <span class="id"><a href="Qcert.Data.Model.DataNorm.html#data_normalized_rec_concat_sort">data_normalized_rec_concat_sort</a></span>; <span class="id">qtrivial</span>.<br/>
- <span class="tactic">do</span> 2 <span class="id">match_destr_in</span> <span class="id">H</span>.<br/>
<span class="tactic">destruct</span> <span class="id">z</span>; <span class="tactic">simpl</span> <span class="kwd">in</span> *; <span class="tactic">try</span> <span class="tactic">discriminate</span>.<br/>
+ <span class="tactic">destruct</span> <span class="id">l</span>; <span class="tactic">simpl</span> <span class="kwd">in</span> *.<br/>
<span class="tactic">inversion</span> <span class="id">H</span>; <span class="tactic">subst</span>; <span class="tactic">repeat</span> <span class="id">constructor</span>.<br/>
<span class="tactic">inversion</span> <span class="id">H</span>; <span class="tactic">subst</span>. <span class="tactic">inversion</span> <span class="id">H0</span>; <span class="tactic">simpl</span> <span class="kwd">in</span> *.<br/>
<span class="tactic">rewrite</span> <span class="id"><a href="http://coq.inria.fr/library/Coq.Lists.List.html#Forall_forall">Forall_forall</a></span> <span class="kwd">in</span> <span class="id">H3</span>; <span class="tactic">simpl</span> <span class="kwd">in</span> <span class="id">H3</span>.<br/>
<span class="id">specialize</span> (<span class="id">H3</span> <span class="id">d</span>).<br/>
<span class="id">constructor</span>.<br/>
<span class="tactic">apply</span> <span class="id">H3</span>; <span class="tactic">auto</span>.<br/>
+ <span class="id">case_eq</span> (<span class="id"><a href="http://coq.inria.fr/library/Coq.Lists.List.html#nth_error">nth_error</a></span> <span class="id">l</span> (<span class="id"><a href="http://coq.inria.fr/library/Coq.PArith.BinPos.html#Pos.to_nat">Pos.to_nat</a></span> <span class="id">p</span>)); <span class="tactic">intros</span>; <span class="tactic">rewrite</span> <span class="id">H2</span> <span class="kwd">in</span> <span class="id">H</span>.<br/>
* <span class="tactic">inversion</span> <span class="id">H</span>; <span class="tactic">clear</span> <span class="id">H</span>; <span class="tactic">subst</span>.<br/>
<span class="tactic">inversion</span> <span class="id">H0</span>; <span class="tactic">subst</span>.<br/>
<span class="tactic">apply</span> <span class="id"><a href="http://coq.inria.fr/library/Coq.Lists.List.html#nth_error_In">nth_error_In</a></span> <span class="kwd">in</span> <span class="id">H2</span>.<br/>
<span class="tactic">rewrite</span> <span class="id"><a href="http://coq.inria.fr/library/Coq.Lists.List.html#Forall_forall">Forall_forall</a></span> <span class="kwd">in</span> <span class="id">H3</span>.<br/>
<span class="id">specialize</span> (<span class="id">H3</span> <span class="id">d</span> <span class="id">H2</span>).<br/>
<span class="id">constructor</span>; <span class="tactic">eauto</span>.<br/>
* <span class="tactic">inversion</span> <span class="id">H</span>; <span class="tactic">subst</span>; <span class="tactic">repeat</span> <span class="id">constructor</span>.<br/>
+ <span class="tactic">inversion</span> <span class="id">H</span>; <span class="tactic">subst</span>; <span class="tactic">repeat</span> <span class="id">constructor</span>.<br/>
- <span class="tactic">destruct</span> <span class="id">d2</span>; <span class="tactic">simpl</span> <span class="kwd">in</span> *; <span class="tactic">try</span> <span class="tactic">discriminate</span>.<br/>
<span class="id">match_destr_in</span> <span class="id">H</span>; <span class="tactic">inversion</span> <span class="id">H</span>; <span class="id">qeauto</span>.<br/>
- <span class="tactic">destruct</span> <span class="id">d1</span>; <span class="tactic">destruct</span> <span class="id">d2</span>; <span class="tactic">simpl</span> <span class="kwd">in</span> *; <span class="tactic">try</span> <span class="tactic">discriminate</span>.<br/>
<span class="tactic">unfold</span> <span class="id"><a href="Qcert.Data.Operators.OperatorsUtils.html#lifted_join">lifted_join</a></span> <span class="kwd">in</span> <span class="id">H</span>.<br/>
<span class="tactic">apply</span> <span class="id"><a href="Qcert.Utils.Lift.html#some_lift">some_lift</a></span> <span class="kwd">in</span> <span class="id">H</span>; <span class="tactic">destruct</span> <span class="id">H</span>; <span class="tactic">subst</span>.<br/>
<span class="id">qeauto</span>.<br/>
- <span class="tactic">eapply</span> <span class="id"><a href="Qcert.Data.Operators.ForeignOperators.html#foreign_operators_binary_normalized">foreign_operators_binary_normalized</a></span>; <span class="tactic">eauto</span>.<br/>
Qed.</div>
<br/>
<span class="kwd">End</span> <span class="id"><a href="Qcert.Data.Operators.BinaryOperatorsSem.html#BinaryOperatorsSem">BinaryOperatorsSem</a></span>.<br/>
<br/>
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