<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE why3session PUBLIC "-//Why3//proof session v5//EN"
"http://why3.lri.fr/why3session.dtd">
<why3session shape_version="6">
<prover id="0" name="Alt-Ergo" version="2.4.0" timelimit="1" steplimit="0" memlimit="1000"/>
<prover id="1" name="CVC3" version="2.4.1" timelimit="5" steplimit="0" memlimit="1000"/>
<prover id="2" name="Alt-Ergo" version="2.0.0" timelimit="5" steplimit="0" memlimit="1000"/>
<prover id="3" name="CVC4" version="1.4" timelimit="5" steplimit="0" memlimit="1000"/>
<prover id="4" name="MetiTarski" version="2.4" timelimit="5" steplimit="0" memlimit="1000"/>
<prover id="5" name="Coq" version="8.11.2" timelimit="5" steplimit="0" memlimit="1000"/>
<prover id="6" name="Z3" version="3.2" timelimit="3" steplimit="0" memlimit="1000"/>
<prover id="7" name="Z3" version="4.3.2" timelimit="5" steplimit="0" memlimit="1000"/>
<prover id="9" name="CVC5" version="1.0.0" timelimit="1" steplimit="0" memlimit="1000"/>
<prover id="10" name="Alt-Ergo" version="2.4.1" timelimit="1" steplimit="0" memlimit="1000"/>
<file format="whyml" proved="true">
<path name=".."/><path name="triangle_inequality.why"/>
<theory name="CauchySchwarzInequality" proved="true">
 <goal name="norm2_pos" proved="true">
 <proof prover="4"><result status="valid" time="0.04"/></proof>
 <proof prover="6"><result status="valid" time="0.01"/></proof>
 <proof prover="7"><result status="valid" time="0.00"/></proof>
 </goal>
 <goal name="p_expr" proved="true">
 <proof prover="4"><result status="valid" time="0.03"/></proof>
 <proof prover="6"><result status="valid" time="0.00"/></proof>
 <proof prover="7"><result status="valid" time="0.00"/></proof>
 </goal>
 <goal name="p_pos" proved="true">
 <proof prover="1"><result status="valid" time="0.01"/></proof>
 <proof prover="2"><result status="valid" time="0.02" steps="4"/></proof>
 <proof prover="3"><result status="valid" time="0.00"/></proof>
 <proof prover="4"><result status="valid" time="0.04"/></proof>
 </goal>
 <goal name="mul_div_simpl" proved="true">
 <proof prover="4"><result status="valid" time="0.06"/></proof>
 <proof prover="6"><result status="valid" time="0.02"/></proof>
 <proof prover="7"><result status="valid" time="0.00"/></proof>
 </goal>
 <goal name="p_val_part" proved="true">
 <transf name="unfold" proved="true" arg1="p">
  <goal name="p_val_part.0" proved="true">
  <transf name="unfold" proved="true" arg1="norm2">
   <goal name="p_val_part.0.0" proved="true">
   <transf name="unfold" proved="true" arg1="dot">
    <goal name="p_val_part.0.0.0" proved="true">
    <transf name="unfold" proved="true" arg1="sqr">
     <goal name="p_val_part.0.0.0.0" proved="true">
     <proof prover="2"><result status="valid" time="0.08" steps="4"/></proof>
     </goal>
    </transf>
    </goal>
   </transf>
   </goal>
  </transf>
  </goal>
 </transf>
 </goal>
 <goal name="p_val_part_pos" proved="true">
 <proof prover="2"><result status="valid" time="0.40" steps="10"/></proof>
 <proof prover="3"><result status="valid" time="0.00"/></proof>
 <proof prover="4"><result status="valid" time="0.02"/></proof>
 </goal>
 <goal name="p_val_part_pos_aux" proved="true">
 <proof prover="2"><result status="valid" time="0.01" steps="3"/></proof>
 <proof prover="4"><result status="valid" time="0.03"/></proof>
 </goal>
 <goal name="CauchySchwarz_aux_non_null" proved="true">
 <proof prover="4"><result status="valid" time="0.03"/></proof>
 <proof prover="5"><path name="triangle_inequality_CauchySchwarzInequality_CauchySchwarz_aux_non_null_1.v"/><result status="valid" time="0.32"/></proof>
 </goal>
 <goal name="norm_null" proved="true">
 <proof prover="4"><result status="valid" time="0.06"/></proof>
 </goal>
 <goal name="CauchySchwarz_aux_null" proved="true">
 <proof prover="1"><result status="valid" time="0.01"/></proof>
 <proof prover="4"><result status="valid" time="0.06"/></proof>
 <proof prover="6"><result status="valid" time="0.28"/></proof>
 <proof prover="7"><result status="valid" time="0.00"/></proof>
 </goal>
 <goal name="CauchySchwarz_aux" proved="true">
 <proof prover="1"><result status="valid" time="0.00"/></proof>
 <proof prover="2"><result status="valid" time="0.02" steps="8"/></proof>
 <proof prover="3"><result status="valid" time="0.01"/></proof>
 <proof prover="4"><result status="valid" time="0.05"/></proof>
 </goal>
 <goal name="norm_pos" proved="true">
 <proof prover="1"><result status="valid" time="0.00"/></proof>
 <proof prover="2"><result status="valid" time="0.01" steps="5"/></proof>
 <proof prover="3"><result status="valid" time="0.00"/></proof>
 <proof prover="4"><result status="valid" time="0.02"/></proof>
 <proof prover="6"><result status="valid" time="0.02"/></proof>
 <proof prover="7"><result status="valid" time="0.01"/></proof>
 </goal>
 <goal name="sqr_le_sqrt" proved="true">
 <proof prover="2"><result status="valid" time="0.03" steps="18"/></proof>
 <proof prover="4"><result status="valid" time="0.07"/></proof>
 </goal>
 <goal name="CauchySchwarz" proved="true">
 <proof prover="5"><path name="triangle_inequality_CauchySchwarzInequality_CauchySchwarz_1.v"/><result status="valid" time="0.42"/></proof>
 </goal>
</theory>
<theory name="TriangleInequality" proved="true">
 <goal name="triangle_aux" proved="true">
 <transf name="split_vc" proved="true" >
  <goal name="triangle_aux.0" proved="true">
  <proof prover="0"><result status="valid" time="0.01" steps="44"/></proof>
  </goal>
  <goal name="triangle_aux.1" proved="true">
  <proof prover="9"><result status="valid" time="0.03" steps="3360"/></proof>
  </goal>
  <goal name="triangle_aux.2" proved="true">
  <proof prover="9"><result status="valid" time="0.02" steps="3346"/></proof>
  </goal>
  <goal name="triangle_aux.3" proved="true">
  <transf name="unfold" proved="true" arg1="norm">
   <goal name="triangle_aux.3.0" proved="true">
   <proof prover="10"><result status="valid" time="0.03" steps="204"/></proof>
   </goal>
  </transf>
  </goal>
  <goal name="triangle_aux.4" proved="true">
  <proof prover="9"><result status="valid" time="0.04" steps="6849"/></proof>
  </goal>
 </transf>
 </goal>
 <goal name="sqr_sqrt_le" proved="true">
 <proof prover="1"><result status="valid" time="0.01"/></proof>
 <proof prover="2"><result status="valid" time="0.04" steps="17"/></proof>
 <proof prover="4"><result status="valid" time="0.04"/></proof>
 </goal>
 <goal name="triangle" proved="true">
 <proof prover="5"><path name="triangle_inequality_TriangleInequality_triangle_1.v"/><result status="valid" time="0.33"/></proof>
 </goal>
</theory>
</file>
</why3session>
