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<?xml version="1.0" encoding="UTF-8"?>
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<!DOCTYPE why3session PUBLIC "-//Why3//proof session v5//EN"
"http://why3.lri.fr/why3session.dtd">
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<why3session shape_version="4">
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<prover id="0" name="Coq" version="8.5" timelimit="5" steplimit="1" memlimit="4000"/>
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<prover id="2" name="CVC3" version="2.4.1" timelimit="15" steplimit="1" memlimit="1000"/>
<prover id="4" name="Spass" version="3.7" timelimit="5" steplimit="1" memlimit="1000"/>
<prover id="5" name="Z3" version="3.2" timelimit="15" steplimit="1" memlimit="0"/>
<prover id="7" name="Vampire" version="0.6" timelimit="5" steplimit="1" memlimit="1000"/>
<prover id="8" name="Alt-Ergo" version="0.99.1" timelimit="5" steplimit="1" memlimit="1000"/>
<prover id="9" name="CVC4" version="1.4" timelimit="5" steplimit="1" memlimit="1000"/>
<prover id="10" name="Z3" version="4.3.2" timelimit="5" steplimit="1" memlimit="1000"/>
<prover id="11" name="Eprover" version="1.8-001" timelimit="5" steplimit="1" memlimit="1000"/>
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<file name="../bellman_ford.mlw" expanded="true">
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<theory name="Graph" sum="a445091f6d55fb03cbfd6988f8e23631" expanded="true">
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 <goal name="vertices_cardinal_pos">
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 <proof prover="8" timelimit="10" memlimit="0"><result status="valid" time="0.01" steps="5"/></proof>
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 </goal>
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 <goal name="path_in_vertices">
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 <proof prover="0" timelimit="10" memlimit="0" edited="bf_Graph_path_in_vertices_2.v"><result status="valid" time="0.56"/></proof>
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 </goal>
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 <goal name="long_path_decomposition_pigeon1">
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 <proof prover="0" timelimit="8" memlimit="1000" edited="bellman_ford_Graph_long_path_decomposition_pigeon1_1.v"><result status="valid" time="1.81"/></proof>
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 </goal>
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 <goal name="long_path_decomposition_pigeon2">
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 <proof prover="8"><result status="valid" time="0.06" steps="42"/></proof>
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 </goal>
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 <goal name="long_path_decomposition_pigeon3">
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 <proof prover="0" timelimit="7" memlimit="1000" edited="bellman_ford_Graph_long_path_decomposition_pigeon3_1.v"><result status="valid" time="2.41"/></proof>
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 </goal>
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 <goal name="long_path_decomposition">
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 <proof prover="0" memlimit="1000" edited="bellman_ford_Graph_long_path_decomposition_1.v"><result status="valid" time="1.94"/></proof>
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 </goal>
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 <goal name="simple_path">
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 <proof prover="0" timelimit="10" memlimit="0" edited="bf_Graph_simple_path_1.v"><result status="valid" time="2.71"/></proof>
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 </goal>
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 <goal name="key_lemma_1">
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 <proof prover="0" timelimit="10" memlimit="0" edited="bf_Graph_key_lemma_1_1.v"><result status="valid" time="4.10"/></proof>
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 </goal>
</theory>
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<theory name="BellmanFord" sum="4adb2a5542cdf0a204b760c08e69bfb1" expanded="true">
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 <goal name="key_lemma_2">
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 <proof prover="0" edited="bf_WP_BellmanFord_key_lemma_2_1.v"><result status="valid" time="6.78"/></proof>
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 </goal>
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 <goal name="WP_parameter relax" expl="VC for relax">
 <transf name="split_goal_wp">
  <goal name="WP_parameter relax.1" expl="1. postcondition">
  <transf name="inline_goal">
   <goal name="WP_parameter relax.1.1" expl="1. postcondition">
   <transf name="split_goal_wp">
    <goal name="WP_parameter relax.1.1.1" expl="1. postcondition">
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    <proof prover="0" timelimit="10" memlimit="0" edited="bf_WP_BellmanFord_WP_parameter_relax_7.v"><result status="valid" time="2.26"/></proof>
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    </goal>
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    <goal name="WP_parameter relax.1.1.2" expl="2. postcondition">
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    <proof prover="2" memlimit="0"><result status="valid" time="0.22"/></proof>
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    </goal>
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    <goal name="WP_parameter relax.1.1.3" expl="3. postcondition">
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    <proof prover="2" memlimit="0"><result status="valid" time="0.59"/></proof>
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    </goal>
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    <goal name="WP_parameter relax.1.1.4" expl="4. postcondition">
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    <proof prover="2" memlimit="0"><result status="valid" time="0.23"/></proof>
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    <proof prover="8" timelimit="18" memlimit="0"><result status="valid" time="0.14" steps="227"/></proof>
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    </goal>
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    <goal name="WP_parameter relax.1.1.5" expl="5. postcondition">
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    <proof prover="2" memlimit="0"><result status="valid" time="0.44"/></proof>
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    </goal>
   </transf>
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   </goal>
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  </transf>
  </goal>
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  <goal name="WP_parameter relax.2" expl="2. postcondition">
  <transf name="inline_goal">
   <goal name="WP_parameter relax.2.1" expl="1. postcondition">
   <transf name="split_goal_wp">
    <goal name="WP_parameter relax.2.1.1" expl="1. postcondition">
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    <proof prover="2" memlimit="0"><result status="valid" time="0.12"/></proof>
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    <proof prover="8" timelimit="15" memlimit="0"><result status="valid" time="0.04" steps="110"/></proof>
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    </goal>
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    <goal name="WP_parameter relax.2.1.2" expl="2. postcondition">
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    <proof prover="2" memlimit="0"><result status="valid" time="0.37"/></proof>
    <proof prover="8" timelimit="15" memlimit="0"><result status="valid" time="0.65" steps="809"/></proof>
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    </goal>
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    <goal name="WP_parameter relax.2.1.3" expl="3. postcondition">
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    <proof prover="2" memlimit="0"><result status="valid" time="0.31"/></proof>
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    </goal>
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    <goal name="WP_parameter relax.2.1.4" expl="4. postcondition">
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    <proof prover="2" memlimit="0"><result status="valid" time="0.08"/></proof>
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    <proof prover="8" timelimit="15" memlimit="0"><result status="valid" time="0.07" steps="189"/></proof>
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    </goal>
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    <goal name="WP_parameter relax.2.1.5" expl="5. postcondition">
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    <proof prover="2" memlimit="0"><result status="valid" time="0.13"/></proof>
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    </goal>
   </transf>
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   </goal>
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  </transf>
  </goal>
 </transf>
 </goal>
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 <goal name="WP_parameter bellman_ford" expl="VC for bellman_ford">
 <transf name="split_goal_wp">
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  <goal name="WP_parameter bellman_ford.1" expl="1. assertion">
  <transf name="inline_goal">
   <goal name="WP_parameter bellman_ford.1.1" expl="1. assertion">
   <transf name="split_goal_wp">
    <goal name="WP_parameter bellman_ford.1.1.1" expl="1. assertion">
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    <proof prover="4"><result status="valid" time="4.02"/></proof>
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    <proof prover="7"><result status="valid" time="0.16"/></proof>
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    <proof prover="11"><result status="valid" time="0.11"/></proof>
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    </goal>
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    <goal name="WP_parameter bellman_ford.1.1.2" expl="2. assertion">
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    <proof prover="8"><result status="valid" time="0.03" steps="48"/></proof>
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    </goal>
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    <goal name="WP_parameter bellman_ford.1.1.3" expl="3. assertion">
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    <proof prover="8"><result status="valid" time="0.03" steps="30"/></proof>
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    </goal>
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    <goal name="WP_parameter bellman_ford.1.1.4" expl="4. assertion">
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    <proof prover="8"><result status="valid" time="0.03" steps="53"/></proof>
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    </goal>
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    <goal name="WP_parameter bellman_ford.1.1.5" expl="5. assertion">
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    <proof prover="8"><result status="valid" time="0.03" steps="35"/></proof>
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    </goal>
   </transf>
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   </goal>
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  </transf>
  </goal>
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  <goal name="WP_parameter bellman_ford.2" expl="2. loop invariant init">
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  <proof prover="8"><result status="valid" time="0.03" steps="41"/></proof>
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  </goal>
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  <goal name="WP_parameter bellman_ford.3" expl="3. precondition">
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  <proof prover="8" timelimit="10" memlimit="0"><result status="valid" time="0.01" steps="12"/></proof>
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  </goal>
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  <goal name="WP_parameter bellman_ford.4" expl="4. exceptional postcondition">
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  <proof prover="0" memlimit="1000" edited="bf_WP_BellmanFord_WP_parameter_bellman_ford_20.v"><result status="valid" time="2.17"/></proof>
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  </goal>
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  <goal name="WP_parameter bellman_ford.5" expl="5. loop invariant preservation">
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  <proof prover="8"><result status="valid" time="0.96" steps="676"/></proof>
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  </goal>
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  <goal name="WP_parameter bellman_ford.6" expl="6. loop variant decrease">
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  <proof prover="8" timelimit="10" memlimit="0"><result status="valid" time="0.02" steps="18"/></proof>
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  </goal>
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  <goal name="WP_parameter bellman_ford.7" expl="7. assertion">
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  <proof prover="8"><result status="valid" time="0.06" steps="86"/></proof>
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  </goal>
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  <goal name="WP_parameter bellman_ford.8" expl="8. postcondition">
  <transf name="split_goal_wp">
   <goal name="WP_parameter bellman_ford.8.1" expl="1. postcondition">
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   <proof prover="2" memlimit="0"><result status="valid" time="0.10"/></proof>
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   <proof prover="8"><result status="valid" time="0.06" steps="83"/></proof>
   <proof prover="10" timelimit="10"><result status="valid" time="0.03"/></proof>
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   </goal>
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   <goal name="WP_parameter bellman_ford.8.2" expl="2. postcondition">
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   <proof prover="2" timelimit="30" memlimit="0"><result status="valid" time="0.10"/></proof>
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   </goal>
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   <goal name="WP_parameter bellman_ford.8.3" expl="3. postcondition">
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   <proof prover="2" timelimit="10" memlimit="0"><result status="valid" time="0.04"/></proof>
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   <proof prover="10" timelimit="10"><result status="valid" time="0.04"/></proof>
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   </goal>
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  </transf>
  </goal>
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  <goal name="WP_parameter bellman_ford.9" expl="9. loop invariant init">
  <transf name="inline_goal">
   <goal name="WP_parameter bellman_ford.9.1" expl="1. loop invariant init">
   <transf name="split_goal_wp">
    <goal name="WP_parameter bellman_ford.9.1.1" expl="1. loop invariant init">
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    <proof prover="7"><result status="valid" time="0.16"/></proof>
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    <proof prover="11"><result status="valid" time="0.12"/></proof>
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    </goal>
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    <goal name="WP_parameter bellman_ford.9.1.2" expl="2. loop invariant init">
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    <proof prover="8"><result status="valid" time="0.03" steps="43"/></proof>
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    </goal>
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    <goal name="WP_parameter bellman_ford.9.1.3" expl="3. loop invariant init">
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    <proof prover="8"><result status="valid" time="0.03" steps="30"/></proof>
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    </goal>
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    <goal name="WP_parameter bellman_ford.9.1.4" expl="4. loop invariant init">
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    <proof prover="8"><result status="valid" time="0.03" steps="60"/></proof>
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    </goal>
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    <goal name="WP_parameter bellman_ford.9.1.5" expl="5. loop invariant init">
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    <proof prover="8"><result status="valid" time="0.04" steps="59"/></proof>
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    </goal>
   </transf>
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   </goal>
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  </transf>
  </goal>
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  <goal name="WP_parameter bellman_ford.10" expl="10. loop invariant init">
  <transf name="split_goal_wp">
   <goal name="WP_parameter bellman_ford.10.1" expl="1. VC for bellman_ford">
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   <proof prover="8" timelimit="10"><result status="valid" time="0.02" steps="12"/></proof>
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   </goal>
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   <goal name="WP_parameter bellman_ford.10.2" expl="2. VC for bellman_ford">
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   <proof prover="8" timelimit="10"><result status="valid" time="0.13" steps="140"/></proof>
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   </goal>
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  </transf>
  </goal>
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  <goal name="WP_parameter bellman_ford.11" expl="11. precondition">
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  <proof prover="8" timelimit="10" memlimit="0"><result status="valid" time="0.01" steps="14"/></proof>
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  </goal>
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  <goal name="WP_parameter bellman_ford.12" expl="12. precondition">
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  <proof prover="8"><result status="valid" time="0.02" steps="19"/></proof>
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  </goal>
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  <goal name="WP_parameter bellman_ford.13" expl="13. precondition">
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  <proof prover="8" timelimit="15"><result status="valid" time="0.01" steps="16"/></proof>
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  </goal>
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  <goal name="WP_parameter bellman_ford.14" expl="14. loop invariant preservation">
  <transf name="split_goal_wp">
   <goal name="WP_parameter bellman_ford.14.1" expl="1. VC for bellman_ford">
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   <proof prover="8"><result status="valid" time="0.02" steps="25"/></proof>
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   </goal>
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   <goal name="WP_parameter bellman_ford.14.2" expl="2. VC for bellman_ford">
   <transf name="inline_goal">
    <goal name="WP_parameter bellman_ford.14.2.1" expl="1. VC for bellman_ford">
    <transf name="split_goal_wp">
     <goal name="WP_parameter bellman_ford.14.2.1.1" expl="1. VC for bellman_ford">
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     <proof prover="2" timelimit="30"><result status="valid" time="0.51"/></proof>
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     <proof prover="8" timelimit="10"><result status="valid" time="0.48" steps="858"/></proof>
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     </goal>
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     <goal name="WP_parameter bellman_ford.14.2.1.2" expl="2. VC for bellman_ford">
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     <proof prover="2" timelimit="5"><result status="valid" time="0.55"/></proof>
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     <proof prover="8" timelimit="10"><result status="valid" time="4.24" steps="4955"/></proof>
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     </goal>
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     <goal name="WP_parameter bellman_ford.14.2.1.3" expl="3. VC for bellman_ford">
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     <proof prover="2"><result status="valid" time="20.28"/></proof>
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     <metas>
      <ts_pos name="real" arity="0" id="2"
       ip_theory="BuiltIn">
       <ip_library name="why3"/>
       <ip_library name="BuiltIn"/>
       <ip_qualid name="real"/>
      </ts_pos>
      <ts_pos name="tuple0" arity="0" id="20"
       ip_theory="Tuple0">
       <ip_library name="why3"/>
       <ip_library name="Tuple0"/>
       <ip_qualid name="tuple0"/>
      </ts_pos>
      <ts_pos name="unit" arity="0" id="21"
       ip_theory="Unit">
       <ip_library name="why3"/>
       <ip_library name="Unit"/>
       <ip_qualid name="unit"/>
      </ts_pos>
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      <ts_pos name="&apos;mark" arity="0" id="68"
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       ip_theory="Mark">
       <ip_library name="why3"/>
       <ip_library name="Mark"/>
       <ip_qualid name="&apos;mark"/>
      </ts_pos>
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      <ts_pos name="ref" arity="1" id="5105"
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       ip_theory="Ref">
       <ip_library name="ref"/>
       <ip_qualid name="ref"/>
      </ts_pos>
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      <ts_pos name="t" arity="1" id="5255"
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       ip_theory="Impset">
       <ip_library name="impset"/>
       <ip_qualid name="t"/>
      </ts_pos>
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      <ts_pos name="distmap" arity="0" id="5347"
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       ip_theory="BellmanFord">
       <ip_qualid name="distmap"/>
      </ts_pos>
      <ls_pos name="infix =" id="10"
       ip_theory="BuiltIn">
       <ip_library name="why3"/>
       <ip_library name="BuiltIn"/>
       <ip_qualid name="infix ="/>
      </ls_pos>
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      <ls_pos name="zero" id="449"
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       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="zero"/>
      </ls_pos>
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      <ls_pos name="one" id="450"
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       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="one"/>
      </ls_pos>
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      <ls_pos name="infix &lt;" id="451"
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       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="infix &lt;"/>
      </ls_pos>
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      <ls_pos name="infix &gt;" id="454"
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       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="infix &gt;"/>
      </ls_pos>
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      <ls_pos name="infix +" id="1620"
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       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="infix +"/>
      </ls_pos>
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      <ls_pos name="prefix -" id="1621"
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       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="prefix -"/>
      </ls_pos>
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      <ls_pos name="infix *" id="1622"
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       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="infix *"/>
      </ls_pos>
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      <ls_pos name="add" id="2380"
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       ip_theory="IntInf">
       <ip_library name="int"/>
       <ip_qualid name="add"/>
      </ls_pos>
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      <ls_pos name="lt" id="2399"
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       ip_theory="IntInf">
       <ip_library name="int"/>
       <ip_qualid name="lt"/>
      </ls_pos>
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      <ls_pos name="le" id="2418"
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       ip_theory="IntInf">
       <ip_library name="int"/>
       <ip_qualid name="le"/>
      </ls_pos>
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      <ls_pos name="mem" id="2503"
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       ip_theory="Mem">
       <ip_library name="list"/>
       <ip_qualid name="mem"/>
      </ls_pos>
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      <ls_pos name="mem" id="2830"
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       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="mem"/>
      </ls_pos>
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      <ls_pos name="infix ==" id="2833"
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       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="infix =="/>
      </ls_pos>
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      <ls_pos name="subset" id="2851"
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       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="subset"/>
      </ls_pos>
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      <ls_pos name="empty" id="2872"
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       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="empty"/>
      </ls_pos>
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      <ls_pos name="is_empty" id="2874"
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       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="is_empty"/>
      </ls_pos>
347
      <ls_pos name="add" id="2884"
348 349 350 351
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="add"/>
      </ls_pos>
352
      <ls_pos name="singleton" id="2890"
353 354 355 356
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="singleton"/>
      </ls_pos>
357
      <ls_pos name="remove" id="2895"
358 359 360 361
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="remove"/>
      </ls_pos>
362
      <ls_pos name="union" id="2913"
363 364 365 366
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="union"/>
      </ls_pos>
367
      <ls_pos name="inter" id="2920"
368 369 370 371
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="inter"/>
      </ls_pos>
372
      <ls_pos name="diff" id="2927"
373 374 375 376
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="diff"/>
      </ls_pos>
377
      <ls_pos name="choose" id="2939"
378 379 380 381
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="choose"/>
      </ls_pos>
382
      <ls_pos name="cardinal" id="2943"
383 384 385 386
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="cardinal"/>
      </ls_pos>
387
      <ls_pos name="get" id="3153"
388 389 390 391
       ip_theory="Map">
       <ip_library name="map"/>
       <ip_qualid name="get"/>
      </ls_pos>
392
      <ls_pos name="set" id="3156"
393 394 395 396
       ip_theory="Map">
       <ip_library name="map"/>
       <ip_qualid name="set"/>
      </ls_pos>
397
      <ls_pos name="mixfix [&lt;-]" id="3176"
398 399 400 401
       ip_theory="Map">
       <ip_library name="map"/>
       <ip_qualid name="mixfix [&lt;-]"/>
      </ls_pos>
402
      <ls_pos name="const" id="3208"
403 404 405 406
       ip_theory="Const">
       <ip_library name="map"/>
       <ip_qualid name="const"/>
      </ls_pos>
407
      <ls_pos name="infix ++" id="3865"
408 409 410 411
       ip_theory="Append">
       <ip_library name="list"/>
       <ip_qualid name="infix ++"/>
      </ls_pos>
412
      <ls_pos name="vertices" id="4535"
413 414 415
       ip_theory="Graph">
       <ip_qualid name="vertices"/>
      </ls_pos>
416
      <ls_pos name="edges" id="4536"
417 418 419
       ip_theory="Graph">
       <ip_qualid name="edges"/>
      </ls_pos>
420
      <ls_pos name="s" id="4551" ip_theory="Graph">
421 422
       <ip_qualid name="s"/>
      </ls_pos>
423
      <ls_pos name="weight" id="4805"
424 425 426
       ip_theory="Graph">
       <ip_qualid name="weight"/>
      </ls_pos>
427
      <ls_pos name="pigeon_set" id="4942"
428 429 430 431
       ip_theory="Graph">
       <ip_qualid name="Pigeonhole"/>
       <ip_qualid name="pigeon_set"/>
      </ls_pos>
432
      <ls_pos name="negative_cycle" id="5080"
433 434 435
       ip_theory="Graph">
       <ip_qualid name="negative_cycle"/>
      </ls_pos>
436
      <ls_pos name="prefix !" id="5111"
437 438 439 440
       ip_theory="Ref">
       <ip_library name="ref"/>
       <ip_qualid name="prefix !"/>
      </ls_pos>
441
      <ls_pos name="initialize_single_source" id="5348"
442 443 444
       ip_theory="BellmanFord">
       <ip_qualid name="initialize_single_source"/>
      </ls_pos>
445
      <ls_pos name="inv1" id="5354"
446 447 448
       ip_theory="BellmanFord">
       <ip_qualid name="inv1"/>
      </ls_pos>
449
      <ls_pos name="inv2" id="5410"
450 451 452
       ip_theory="BellmanFord">
       <ip_qualid name="inv2"/>
      </ls_pos>
453
      <pr_pos name="Assoc" id="1623"
454 455 456 457 458
       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="CommutativeGroup"/>
       <ip_qualid name="Assoc"/>
      </pr_pos>
459
      <pr_pos name="Unit_def_l" id="1630"
460 461 462 463 464
       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="CommutativeGroup"/>
       <ip_qualid name="Unit_def_l"/>
      </pr_pos>
465
      <pr_pos name="Unit_def_r" id="1633"
466 467 468 469 470
       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="CommutativeGroup"/>
       <ip_qualid name="Unit_def_r"/>
      </pr_pos>
471
      <pr_pos name="Inv_def_l" id="1636"
472 473 474 475 476
       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="CommutativeGroup"/>
       <ip_qualid name="Inv_def_l"/>
      </pr_pos>
477
      <pr_pos name="Inv_def_r" id="1639"
478 479 480 481 482
       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="CommutativeGroup"/>
       <ip_qualid name="Inv_def_r"/>
      </pr_pos>
483
      <pr_pos name="Comm" id="1642"
484 485 486 487 488 489
       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="CommutativeGroup"/>
       <ip_qualid name="Comm"/>
       <ip_qualid name="Comm"/>
      </pr_pos>
490
      <pr_pos name="Assoc" id="1647"
491 492 493 494 495
       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="Assoc"/>
       <ip_qualid name="Assoc"/>
      </pr_pos>
496
      <pr_pos name="Mul_distr_l" id="1654"
497 498 499 500
       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="Mul_distr_l"/>
      </pr_pos>
501
      <pr_pos name="Mul_distr_r" id="1661"
502 503 504 505
       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="Mul_distr_r"/>
      </pr_pos>
506
      <pr_pos name="Comm" id="1679"
507 508 509 510 511
       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="Comm"/>
       <ip_qualid name="Comm"/>
      </pr_pos>
512
      <pr_pos name="Unitary" id="1684"
513 514 515 516
       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="Unitary"/>
      </pr_pos>
517
      <pr_pos name="NonTrivialRing" id="1687"
518 519 520 521
       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="NonTrivialRing"/>
      </pr_pos>
522
      <pr_pos name="Refl" id="1699"
523 524 525 526
       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="Refl"/>
      </pr_pos>
527
      <pr_pos name="Trans" id="1702"
528 529 530 531
       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="Trans"/>
      </pr_pos>
532
      <pr_pos name="Antisymm" id="1709"
533 534 535 536
       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="Antisymm"/>
      </pr_pos>
537
      <pr_pos name="Total" id="1714"
538 539 540 541
       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="Total"/>
      </pr_pos>
542
      <pr_pos name="ZeroLessOne" id="1719"
543 544 545 546
       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="ZeroLessOne"/>
      </pr_pos>
547
      <pr_pos name="CompatOrderAdd" id="1720"
548 549 550 551
       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="CompatOrderAdd"/>
      </pr_pos>
552
      <pr_pos name="CompatOrderMult" id="1727"
553 554 555 556
       ip_theory="Int">
       <ip_library name="int"/>
       <ip_qualid name="CompatOrderMult"/>
      </pr_pos>
557
      <pr_pos name="Refl" id="2427"
558 559 560 561
       ip_theory="IntInf">
       <ip_library name="int"/>
       <ip_qualid name="Refl"/>
      </pr_pos>
562
      <pr_pos name="Trans" id="2430"
563 564 565 566
       ip_theory="IntInf">
       <ip_library name="int"/>
       <ip_qualid name="Trans"/>
      </pr_pos>
567
      <pr_pos name="Antisymm" id="2437"
568 569 570 571
       ip_theory="IntInf">
       <ip_library name="int"/>
       <ip_qualid name="Antisymm"/>
      </pr_pos>
572
      <pr_pos name="Total" id="2442"
573 574 575 576
       ip_theory="IntInf">
       <ip_library name="int"/>
       <ip_qualid name="Total"/>
      </pr_pos>
577
      <pr_pos name="Length_nonnegative" id="2496"
578 579 580 581
       ip_theory="Length">
       <ip_library name="list"/>
       <ip_qualid name="Length_nonnegative"/>
      </pr_pos>
582
      <pr_pos name="Length_nil" id="2499"
583 584 585 586
       ip_theory="Length">
       <ip_library name="list"/>
       <ip_qualid name="Length_nil"/>
      </pr_pos>
587
      <pr_pos name="extensionality" id="2844"
588 589 590 591
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="extensionality"/>
      </pr_pos>
592
      <pr_pos name="subset_refl" id="2862"
593 594 595 596
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="subset_refl"/>
      </pr_pos>
597
      <pr_pos name="subset_trans" id="2865"
598 599 600 601
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="subset_trans"/>
      </pr_pos>
602
      <pr_pos name="empty_def1" id="2881"
603 604 605 606
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="empty_def1"/>
      </pr_pos>
607
      <pr_pos name="mem_empty" id="2882"
608 609 610 611
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="mem_empty"/>
      </pr_pos>
612
      <pr_pos name="add_remove" id="2901"
613 614 615 616
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="add_remove"/>
      </pr_pos>
617
      <pr_pos name="remove_add" id="2905"
618 619 620 621
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="remove_add"/>
      </pr_pos>
622
      <pr_pos name="subset_remove" id="2909"
623 624 625 626
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="subset_remove"/>
      </pr_pos>
627
      <pr_pos name="union_def1" id="2914"
628 629 630 631
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="union_def1"/>
      </pr_pos>
632
      <pr_pos name="inter_def1" id="2921"
633 634 635 636
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="inter_def1"/>
      </pr_pos>
637
      <pr_pos name="subset_diff" id="2934"
638 639 640 641
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="subset_diff"/>
      </pr_pos>
642
      <pr_pos name="choose_def" id="2940"
643 644 645 646
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="choose_def"/>
      </pr_pos>
647
      <pr_pos name="cardinal_nonneg" id="2945"
648 649 650 651
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="cardinal_nonneg"/>
      </pr_pos>
652
      <pr_pos name="cardinal_empty" id="2948"
653 654 655 656
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="cardinal_empty"/>
      </pr_pos>
657
      <pr_pos name="cardinal_add" id="2951"
658 659 660 661
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="cardinal_add"/>
      </pr_pos>
662
      <pr_pos name="cardinal_remove" id="2956"
663 664 665 666
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="cardinal_remove"/>
      </pr_pos>
667
      <pr_pos name="cardinal_subset" id="2961"
668 669 670 671
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="cardinal_subset"/>
      </pr_pos>
672
      <pr_pos name="subset_eq" id="2966"
673 674 675 676
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="subset_eq"/>
      </pr_pos>
677
      <pr_pos name="cardinal1" id="2971"
678 679 680 681
       ip_theory="Fset">
       <ip_library name="set"/>
       <ip_qualid name="cardinal1"/>
      </pr_pos>
682
      <pr_pos name="Select_eq" id="3189"
683 684 685 686
       ip_theory="Map">
       <ip_library name="map"/>
       <ip_qualid name="Select_eq"/>
      </pr_pos>
687
      <pr_pos name="Select_neq" id="3198"
688 689 690 691
       ip_theory="Map">
       <ip_library name="map"/>
       <ip_qualid name="Select_neq"/>
      </pr_pos>
692
      <pr_pos name="Const" id="3210"
693 694 695 696
       ip_theory="Const">
       <ip_library name="map"/>
       <ip_qualid name="Const"/>
      </pr_pos>
697
      <pr_pos name="Append_assoc" id="3882"
698 699 700 701
       ip_theory="Append">
       <ip_library name="list"/>
       <ip_qualid name="Append_assoc"/>
      </pr_pos>
702
      <pr_pos name="Append_l_nil" id="3889"
703 704 705 706
       ip_theory="Append">
       <ip_library name="list"/>
       <ip_qualid name="Append_l_nil"/>
      </pr_pos>
707
      <pr_pos name="Append_length" id="3892"
708 709 710 711
       ip_theory="Append">
       <ip_library name="list"/>
       <ip_qualid name="Append_length"/>
      </pr_pos>
712
      <pr_pos name="mem_append" id="3897"
713 714 715 716
       ip_theory="Append">
       <ip_library name="list"/>
       <ip_qualid name="mem_append"/>
      </pr_pos>
717
      <pr_pos name="mem_decomp" id="3904"
718 719 720 721
       ip_theory="Append">
       <ip_library name="list"/>
       <ip_qualid name="mem_decomp"/>
      </pr_pos>
722
      <pr_pos name="edges_def" id="4546"
723 724 725
       ip_theory="Graph">
       <ip_qualid name="edges_def"/>
      </pr_pos>
726
      <pr_pos name="s_in_graph" id="4552"
727 728 729
       ip_theory="Graph">
       <ip_qualid name="s_in_graph"/>
      </pr_pos>
730
      <pr_pos name="vertices_cardinal_pos" id="4553"
731 732 733
       ip_theory="Graph">
       <ip_qualid name="vertices_cardinal_pos"/>
      </pr_pos>
734
      <pr_pos name="path_right_extension" id="4758"
735 736 737
       ip_theory="Graph">
       <ip_qualid name="path_right_extension"/>
      </pr_pos>
738
      <pr_pos name="path_right_inversion" id="4767"
739 740 741
       ip_theory="Graph">
       <ip_qualid name="path_right_inversion"/>
      </pr_pos>
742
      <pr_pos name="path_trans" id="4778"
743 744 745
       ip_theory="Graph">
       <ip_qualid name="path_trans"/>
      </pr_pos>
746
      <pr_pos name="empty_path" id="4789"
747 748 749
       ip_theory="Graph">
       <ip_qualid name="empty_path"/>
      </pr_pos>
750
      <pr_pos name="path_decomposition" id="4794"
751 752 753
       ip_theory="Graph">
       <ip_qualid name="path_decomposition"/>
      </pr_pos>
754
      <pr_pos name="path_weight_right_extension" id="4837"
755 756 757
       ip_theory="Graph">
       <ip_qualid name="path_weight_right_extension"/>
      </pr_pos>
758
      <pr_pos name="path_weight_decomposition" id="4844"
759 760 761
       ip_theory="Graph">
       <ip_qualid name="path_weight_decomposition"/>
      </pr_pos>
762
      <pr_pos name="path_in_vertices" id="4853"
763 764 765
       ip_theory="Graph">
       <ip_qualid name="path_in_vertices"/>
      </pr_pos>
766
      <pr_pos name="Induction" id="4959"
767 768 769 770 771
       ip_theory="Graph">
       <ip_qualid name="Pigeonhole"/>
       <ip_qualid name="FsetInduction"/>
       <ip_qualid name="Induction"/>
      </pr_pos>
772
      <pr_pos name="corner" id="4968"
773 774 775 776
       ip_theory="Graph">
       <ip_qualid name="Pigeonhole"/>
       <ip_qualid name="corner"/>
      </pr_pos>
777
      <pr_pos name="pigeon_0" id="4985"
778 779 780 781
       ip_theory="Graph">
       <ip_qualid name="Pigeonhole"/>
       <ip_qualid name="pigeon_0"/>
      </pr_pos>
782
      <pr_pos name="pigeon_1" id="4986"
783 784 785 786
       ip_theory="Graph">
       <ip_qualid name="Pigeonhole"/>
       <ip_qualid name="pigeon_1"/>
      </pr_pos>
787
      <pr_pos name="pigeon_2" id="4991"
788 789 790 791
       ip_theory="Graph">
       <ip_qualid name="Pigeonhole"/>
       <ip_qualid name="pigeon_2"/>
      </pr_pos>
792
      <pr_pos name="pigeonhole" id="4994"
793 794 795 796
       ip_theory="Graph">
       <ip_qualid name="Pigeonhole"/>
       <ip_qualid name="pigeonhole"/>
      </pr_pos>
797
      <pr_pos name="long_path_decomposition_pigeon1" id="5009"
798 799 800
       ip_theory="Graph">
       <ip_qualid name="long_path_decomposition_pigeon1"/>
      </pr_pos>
801
      <pr_pos name="long_path_decomposition_pigeon2" id="5016"
802 803 804
       ip_theory="Graph">
       <ip_qualid name="long_path_decomposition_pigeon2"/>
      </pr_pos>
805
      <pr_pos name="long_path_decomposition_pigeon3" id="5031"
806 807 808
       ip_theory="Graph">
       <ip_qualid name="long_path_decomposition_pigeon3"/>
      </pr_pos>
809
      <pr_pos name="long_path_decomposition" id="5056"
810 811 812
       ip_theory="Graph">
       <ip_qualid name="long_path_decomposition"/>
      </pr_pos>
813
      <pr_pos name="simple_path" id="5073"
814 815 816
       ip_theory="Graph">
       <ip_qualid name="simple_path"/>
      </pr_pos>
817
      <pr_pos name="key_lemma_1" id="5091"
818 819 820
       ip_theory="Graph">
       <ip_qualid name="key_lemma_1"/>
      </pr_pos>