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<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <br/>  for <font color="Olive" title="b4">G</font> being   <a href="graph_1.html#V2" title="GRAPH_1:attr.2">oriented</a>   <a href="graph_1.html#V6" title="GRAPH_1:attr.6">finite</a>  <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a><br/>  for <font color="Olive" title="b5">W</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Olive" title="b4">G</font>,<a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a><br/>  for <font color="Olive" title="b6">v1</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Olive" title="b4">G</font>  st <font color="Olive" title="b1">f</font> <a href="graphsp.html#R6" title="GRAPHSP:pred.6">is_Input_of_Dijkstra_Alg</a> <font color="Olive" title="b4">G</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b5">W</font> &amp; <font color="Olive" title="b6">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Olive" title="b1">f</font> &amp;  <a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c1">n</font>) &amp; (  for <font color="Olive" title="b7">v3</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Olive" title="b4">G</font><br/>  for <font color="Olive" title="b8">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b7">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">v1</font> &amp; <font color="Olive" title="b7">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">j</font> &amp; <font color="Olive" title="b2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b8">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 holds <br/> ex <font color="Olive" title="b9">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b10">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Olive" title="b4">G</font> st <br/>( <font color="Olive" title="b9">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b8">j</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b11">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b11">i</font> &amp; <font color="Olive" title="b11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b9">p</font> holds <br/><font color="Olive" title="b9">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive" title="b10">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive" title="b9">p</font> &amp; <font color="Olive" title="b10">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b6">v1</font>,<font color="Olive" title="b7">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Olive" title="b5">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b10">P</font>,<font color="Olive" title="b5">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b8">j</font></span>)</span> &amp; (  not <font color="Olive" title="b7">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive" title="b10">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Olive" title="b6">v1</font>,<font color="Olive" title="b5">W</font> ) ) ) &amp; (  for <font color="Olive" title="b7">m</font>, <font color="Olive" title="b8">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b8">j</font> &amp; <font color="Olive" title="b8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Olive" title="b1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b7">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) &amp; (  for <font color="Olive" title="b7">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Olive" title="b2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b7">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) holds <br/>( (  for <font color="Olive" title="b7">v3</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Olive" title="b4">G</font><br/>  for <font color="Olive" title="b8">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b7">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">v1</font> &amp; <font color="Olive" title="b7">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">j</font> &amp; <font color="Olive" title="b3">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b8">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 holds <br/> ex <font color="Olive" title="b9">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b10">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Olive" title="b4">G</font> st <br/>( <font color="Olive" title="b9">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Olive" title="b3">h</font>,<font color="Olive" title="b8">j</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b11">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b11">i</font> &amp; <font color="Olive" title="b11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b9">p</font> holds <br/><font color="Olive" title="b9">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Olive" title="b3">h</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive" title="b10">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive" title="b9">p</font> &amp; <font color="Olive" title="b10">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b6">v1</font>,<font color="Olive" title="b7">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Olive" title="b3">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Olive" title="b5">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b10">P</font>,<font color="Olive" title="b5">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b8">j</font></span>)</span> &amp; (  not <font color="Olive" title="b7">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Olive" title="b3">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive" title="b10">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Olive" title="b3">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Olive" title="b6">v1</font>,<font color="Olive" title="b5">W</font> ) ) ) &amp; (  for <font color="Olive" title="b7">m</font>, <font color="Olive" title="b8">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b8">j</font> &amp; <font color="Olive" title="b8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Olive" title="b3">h</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Olive" title="b1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b7">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) &amp; (  for <font color="Olive" title="b7">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Olive" title="b3">h</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Olive" title="b3">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b7">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font>, <font color="Maroon" title="c5">h</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being   <a href="graph_1.html#V2" title="GRAPH_1:attr.2">oriented</a>   <a href="graph_1.html#V6" title="GRAPH_1:attr.6">finite</a>  <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a><br/>  for <font color="Olive" title="b2">W</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Olive" title="b1">G</font>,<a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a><br/>  for <font color="Olive" title="b3">v1</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Olive" title="b1">G</font>  st <font color="Maroon" title="c3">f</font> <a href="graphsp.html#R6" title="GRAPHSP:pred.6">is_Input_of_Dijkstra_Alg</a> <font color="Olive" title="b1">G</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c3">f</font> &amp;  <a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) &amp; (  for <font color="Olive" title="b4">v3</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Olive" title="b1">G</font><br/>  for <font color="Olive" title="b5">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b4">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">v1</font> &amp; <font color="Olive" title="b4">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j</font> &amp; <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b5">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 holds <br/> ex <font color="Olive" title="b6">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b7">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Olive" title="b1">G</font> st <br/>( <font color="Olive" title="b6">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c4">g</font>,<font color="Olive" title="b5">j</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b8">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b8">i</font> &amp; <font color="Olive" title="b8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b6">p</font> holds <br/><font color="Olive" title="b6">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive" title="b7">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive" title="b6">p</font> &amp; <font color="Olive" title="b7">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b4">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Olive" title="b2">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b7">P</font>,<font color="Olive" title="b2">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b5">j</font></span>)</span> &amp; (  not <font color="Olive" title="b4">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive" title="b7">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Olive" title="b3">v1</font>,<font color="Olive" title="b2">W</font> ) ) ) &amp; (  for <font color="Olive" title="b4">m</font>, <font color="Olive" title="b5">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b5">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">j</font> &amp; <font color="Olive" title="b5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b4">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b5">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) &amp; (  for <font color="Olive" title="b4">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b4">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) holds <br/>( (  for <font color="Olive" title="b4">v3</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Olive" title="b1">G</font><br/>  for <font color="Olive" title="b5">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b4">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">v1</font> &amp; <font color="Olive" title="b4">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j</font> &amp; <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b5">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 holds <br/> ex <font color="Olive" title="b6">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b7">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Olive" title="b1">G</font> st <br/>( <font color="Olive" title="b6">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c5">h</font>,<font color="Olive" title="b5">j</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b8">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b8">i</font> &amp; <font color="Olive" title="b8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b6">p</font> holds <br/><font color="Olive" title="b6">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive" title="b7">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive" title="b6">p</font> &amp; <font color="Olive" title="b7">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b4">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Olive" title="b2">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b7">P</font>,<font color="Olive" title="b2">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b5">j</font></span>)</span> &amp; (  not <font color="Olive" title="b4">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive" title="b7">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Olive" title="b3">v1</font>,<font color="Olive" title="b2">W</font> ) ) ) &amp; (  for <font color="Olive" title="b4">m</font>, <font color="Olive" title="b5">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b5">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">j</font> &amp; <font color="Olive" title="b5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b4">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b5">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) &amp; (  for <font color="Olive" title="b4">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b4">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">G</font> be   <a href="graph_1.html#V2" title="GRAPH_1:attr.2">oriented</a>   <a href="graph_1.html#V6" title="GRAPH_1:attr.6">finite</a>  <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">W</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c6">G</font>,<a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a><br/>  for <font color="Olive" title="b2">v1</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font>  st <font color="Maroon" title="c3">f</font> <a href="graphsp.html#R6" title="GRAPHSP:pred.6">is_Input_of_Dijkstra_Alg</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c3">f</font> &amp;  <a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) &amp; (  for <font color="Olive" title="b3">v3</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font><br/>  for <font color="Olive" title="b4">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v1</font> &amp; <font color="Olive" title="b3">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j</font> &amp; <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b4">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 holds <br/> ex <font color="Olive" title="b5">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b6">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive" title="b5">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c4">g</font>,<font color="Olive" title="b4">j</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b7">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b7">i</font> &amp; <font color="Olive" title="b7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">p</font> holds <br/><font color="Olive" title="b5">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive" title="b6">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive" title="b5">p</font> &amp; <font color="Olive" title="b6">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b3">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Olive" title="b1">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b6">P</font>,<font color="Olive" title="b1">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b4">j</font></span>)</span> &amp; (  not <font color="Olive" title="b3">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive" title="b6">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Olive" title="b2">v1</font>,<font color="Olive" title="b1">W</font> ) ) ) &amp; (  for <font color="Olive" title="b3">m</font>, <font color="Olive" title="b4">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b4">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">j</font> &amp; <font color="Olive" title="b4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b3">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b4">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) &amp; (  for <font color="Olive" title="b3">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) holds <br/>( (  for <font color="Olive" title="b3">v3</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font><br/>  for <font color="Olive" title="b4">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v1</font> &amp; <font color="Olive" title="b3">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j</font> &amp; <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b4">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 holds <br/> ex <font color="Olive" title="b5">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b6">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive" title="b5">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c5">h</font>,<font color="Olive" title="b4">j</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b7">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b7">i</font> &amp; <font color="Olive" title="b7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">p</font> holds <br/><font color="Olive" title="b5">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive" title="b6">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive" title="b5">p</font> &amp; <font color="Olive" title="b6">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b3">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Olive" title="b1">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b6">P</font>,<font color="Olive" title="b1">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b4">j</font></span>)</span> &amp; (  not <font color="Olive" title="b3">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive" title="b6">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Olive" title="b2">v1</font>,<font color="Olive" title="b1">W</font> ) ) ) &amp; (  for <font color="Olive" title="b3">m</font>, <font color="Olive" title="b4">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b4">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">j</font> &amp; <font color="Olive" title="b4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b3">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b4">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) &amp; (  for <font color="Olive" title="b3">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">W</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c6">G</font>,<a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v1</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font>  st <font color="Maroon" title="c3">f</font> <a href="graphsp.html#R6" title="GRAPHSP:pred.6">is_Input_of_Dijkstra_Alg</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font> &amp; <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c3">f</font> &amp;  <a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) &amp; (  for <font color="Olive" title="b2">v3</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font><br/>  for <font color="Olive" title="b3">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">v1</font> &amp; <font color="Olive" title="b2">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">j</font> &amp; <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 holds <br/> ex <font color="Olive" title="b4">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b5">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive" title="b4">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c4">g</font>,<font color="Olive" title="b3">j</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b6">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">p</font> holds <br/><font color="Olive" title="b4">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive" title="b5">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive" title="b4">p</font> &amp; <font color="Olive" title="b5">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b5">P</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span> &amp; (  not <font color="Olive" title="b2">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive" title="b5">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Olive" title="b1">v1</font>,<font color="Maroon" title="c7">W</font> ) ) ) &amp; (  for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) &amp; (  for <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) holds <br/>( (  for <font color="Olive" title="b2">v3</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font><br/>  for <font color="Olive" title="b3">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">v1</font> &amp; <font color="Olive" title="b2">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">j</font> &amp; <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 holds <br/> ex <font color="Olive" title="b4">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b5">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive" title="b4">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c5">h</font>,<font color="Olive" title="b3">j</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b6">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">p</font> holds <br/><font color="Olive" title="b4">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive" title="b5">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive" title="b4">p</font> &amp; <font color="Olive" title="b5">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b5">P</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span> &amp; (  not <font color="Olive" title="b2">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive" title="b5">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Olive" title="b1">v1</font>,<font color="Maroon" title="c7">W</font> ) ) ) &amp; (  for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) &amp; (  for <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">v1</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="graphsp.html#R6" title="GRAPHSP:pred.6">is_Input_of_Dijkstra_Alg</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font> &amp; <font color="Maroon" title="c8">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c3">f</font> &amp;  <a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) &amp; (  for <font color="Olive" title="b1">v3</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font><br/>  for <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font> &amp; <font color="Olive" title="b1">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 holds <br/> ex <font color="Olive" title="b3">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b4">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive" title="b3">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c4">g</font>,<font color="Olive" title="b2">j</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b3">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive" title="b4">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b4">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b4">P</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> &amp; (  not <font color="Olive" title="b1">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive" title="b4">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) ) ) &amp; (  for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) &amp; (  for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) implies ( (  for <font color="Olive" title="b1">v3</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font><br/>  for <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font> &amp; <font color="Olive" title="b1">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 holds <br/> ex <font color="Olive" title="b3">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b4">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive" title="b3">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c5">h</font>,<font color="Olive" title="b2">j</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b3">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive" title="b4">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b4">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b4">P</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> &amp; (  not <font color="Olive" title="b1">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive" title="b4">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) ) ) &amp; (  for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) &amp; (  for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) ) )</span><br/>















<span class="kw">set </span><font color="Maroon" title="c9">R</font> =  <a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">M</font> =  <a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">IN</font> =  <a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c12">Ug</font> =  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>);<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:98"/><span class="lab"><font color="Green" title="E82">A1</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="graphsp.html#R6" title="GRAPHSP:pred.6">is_Input_of_Dijkstra_Alg</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">W</font>
 <span class="kw">and </span><br/><a NAME="E2:98"/><span class="lab"><font color="Green" title="E83">A2</font></span>: 
<font color="Maroon" title="c8">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E3:98"/><span class="lab"><font color="Green" title="E84">A3</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">and </span><br/><a NAME="E4:98"/><span class="lab"><font color="Green" title="E85">A4</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E5:98"/><span class="lab"><font color="Green" title="E86">A5</font></span>: 
<font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K5" title="GRAPHSP:func.5">repeat</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K3" title="GRAPHSP:func.3">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E6:98"/><span class="lab"><font color="Green" title="E87">A6</font></span>: 
 <a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E7:98"/><span class="lab"><font color="Green" title="E88">A7</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">v3</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font> ex <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font> &amp; <font color="Olive" title="b1">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; (  for <font color="Olive" title="b3">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b4">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font>  holds <br/>(  not <font color="Olive" title="b3">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c4">g</font>,<font color="Olive" title="b2">j</font>,<font color="Maroon" title="c1">n</font> or  ex <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">p</font> &amp;  not <font color="Olive" title="b3">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) ) or  not <font color="Olive" title="b4">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive" title="b3">p</font> or  not <font color="Olive" title="b4">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> or  not  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b4">P</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> or (  not <font color="Olive" title="b1">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) &amp;  not <font color="Olive" title="b4">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) ) ) ) or  ex <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) &amp;  not <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) or  ex <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) &amp;  not <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) or ( (  for <font color="Olive" title="b1">v3</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font><br/>  for <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font> &amp; <font color="Olive" title="b1">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 holds <br/> ex <font color="Olive" title="b3">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b4">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive" title="b3">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c5">h</font>,<font color="Olive" title="b2">j</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b3">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive" title="b4">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b4">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b4">P</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> &amp; (  not <font color="Olive" title="b1">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive" title="b4">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) ) ) &amp; (  for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) &amp; (  for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E8:98"/><span class="lab"><font color="Green" title="E89">A8</font></span>: 
 for <font color="Olive" title="b1">v3</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font><br/>  for <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font> &amp; <font color="Olive" title="b1">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 holds <br/> ex <font color="Olive" title="b3">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b4">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive" title="b3">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c4">g</font>,<font color="Olive" title="b2">j</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b3">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive" title="b4">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b4">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b4">P</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> &amp; (  not <font color="Olive" title="b1">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive" title="b4">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) &amp;  not <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) or  ex <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) &amp;  not <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) or ( (  for <font color="Olive" title="b1">v3</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font><br/>  for <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font> &amp; <font color="Olive" title="b1">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 holds <br/> ex <font color="Olive" title="b3">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b4">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive" title="b3">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c5">h</font>,<font color="Olive" title="b2">j</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b3">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive" title="b4">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b4">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b4">P</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> &amp; (  not <font color="Olive" title="b1">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive" title="b4">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) ) ) &amp; (  for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) &amp; (  for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E9:98"/><span class="lab"><font color="Green" title="E90">A9</font></span>: 
 for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">and </span><br/><a NAME="E10:98"/><span class="lab"><font color="Green" title="E91">A10</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">v3</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font><br/>  for <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font> &amp; <font color="Olive" title="b1">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 holds <br/> ex <font color="Olive" title="b3">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b4">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive" title="b3">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c5">h</font>,<font color="Olive" title="b2">j</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b3">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive" title="b4">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b4">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b4">P</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> &amp; (  not <font color="Olive" title="b1">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive" title="b4">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) ) ) &amp; (  for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) &amp; (  for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c13">mi</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">Ak</font> =  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>);<br/>
<a NAME="E11:98"/><span class="lab"><font color="Green" title="E92">A11</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E12:98"/><span class="lab"><font color="Green" title="E93">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="graphsp.html#D20" target="_self" title="GRAPHSP:def.20">Def20</a></span>;<br/>
<a NAME="E13:98"/><span class="lab"><font color="Green" title="E94">A13</font></span>: 
<span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon" title="c4">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="graphsp.html#K2" title="GRAPHSP:func.2">:=</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graphsp.html#D11" target="_self" title="GRAPHSP:def.11">Def11</a></span>;<br/>
<a NAME="E14:98"/><span class="lab"><font color="Green" title="E95">A14</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graphsp.html#T33" target="_self" title="GRAPHSP:th.33">Th34</a></span>;<br/>
<a NAME="E15:98"/>
<font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graphsp.html#K19" title="GRAPHSP:func.19">Relax</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E5:98"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="ref" href="graphsp.html#T22" target="_self" title="GRAPHSP:th.22">Th23</a></span>;<br/>
<a NAME="E16:98"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A15</font></span>: 
<font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K18" title="GRAPHSP:func.18">Relax</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c1">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graphsp.html#D15" target="_self" title="GRAPHSP:def.15">Def15</a></span>;<br/>
<a NAME="E17:98"/><span class="lab"><font color="Green" title="E97">A16</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="graphsp.html#D20" target="_self" title="GRAPHSP:def.20">Def20</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">VG</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">G</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98"><span class="lab"><font color="Green" title="E84">A3</font></span></a></span>;<br/>
<a NAME="E19:98"/><span class="lab"><font color="Green" title="E98">A17</font></span>: 
<font color="Maroon" title="c7">W</font> <a href="graph_5.html#R5" title="GRAPH_5:pred.5">is_weight&gt;=0of</a> <font color="Maroon" title="c6">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#D13" title="GRAPH_5:def.13">GRAPH_5:def 13</a></span>;<br/>
<a NAME="E20:98"/><span class="lab"><font color="Green" title="E99">A18</font></span>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font>
 
;<br/>
<a NAME="E21:98"/><span class="lab"><font color="Green" title="E100">A19</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="ref" href="graphsp.html#T41" target="_self" title="GRAPHSP:th.41">Th42</a></span>;<br/>
<a NAME="E22:98"/><span class="lab"><font color="Green" title="E101">A20</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:98"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="ref" href="graphsp.html#T29" target="_self" title="GRAPHSP:th.29">Th30</a></span>;<br/>
<a NAME="E23:98"/><span class="lab"><font color="Green" title="E102">A21</font></span>: 
 <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:98"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="ref" href="graphsp.html#T29" target="_self" title="GRAPHSP:th.29">Th30</a></span>;<br/>
<a NAME="E24:98"/><span class="lab"><font color="Green" title="E103">A22</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:98"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="ref" href="graphsp.html#T29" target="_self" title="GRAPHSP:th.29">Th30</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">Uh</font> =  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>);<br/>
<a NAME="E25:98"/><span class="lab"><font color="Green" title="E104">A23</font></span>: 
(  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  not  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E5:98"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="txt" href="#E6:98"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="ref" href="graphsp.html#T39" target="_self" title="GRAPHSP:th.39">Th40</a></span>;<br/>
<a NAME="E26:98"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A24</font></span>: 
 <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E27:98"/><span class="lab"><font color="Green" title="E106">A25</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="graphsp.html#E51"><span class="lab"><font color="Green" title="E46">Lm7</font></span></a></span>;<br/>
<a NAME="E28:98"/><span class="lab"><font color="Green" title="E107">A26</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="ref" href="graphsp.html#T41" target="_self" title="GRAPHSP:th.41">Th42</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">vk</font> =  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) as   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:98"><span class="lab"><font color="Green" title="E97">A16</font></span></a>, <a class="txt" href="#E22:98"><span class="lab"><font color="Green" title="E101">A20</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">pk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c19">PK</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font><span class="kw"> such that </span><br/><a NAME="E31:98"/><span class="lab"><font color="Green" title="E108">A27</font></span>: 
<font color="Maroon" title="c18">pk</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c4">g</font>, <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E32:98"/><span class="lab"><font color="Green" title="E109">A28</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">pk</font> holds <br/><font color="Maroon" title="c18">pk</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">and </span><br/><a NAME="E33:98"/><span class="lab"><font color="Green" title="E110">A29</font></span>: 
<font color="Maroon" title="c19">PK</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c18">pk</font>
 <span class="kw">and </span><br/><a NAME="E34:98"/><span class="lab"><font color="Green" title="E111">A30</font></span>: 
<font color="Maroon" title="c19">PK</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c17">vk</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font>
 <span class="kw">and </span><br/><a NAME="E35:98"/><span class="lab"><font color="Green" title="E112">A31</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c19">PK</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E36:98"/><span class="lab"><font color="Green" title="E113">A32</font></span>: 
(  not <font color="Maroon" title="c17">vk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Maroon" title="c19">PK</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E7:98"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="txt" href="#E8:98"><span class="lab"><font color="Green" title="E89">A8</font></span></a>, <a class="txt" href="#E24:98"><span class="lab"><font color="Green" title="E103">A22</font></span></a>, <a class="txt" href="#E25:98"><span class="lab"><font color="Green" title="E104">A23</font></span></a></span>;<br/>
<a NAME="E37:98"/><span class="lab"><font color="Green" title="E114">A33</font></span>: 
 ex <font color="Olive" title="b1">kk</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b1">kk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) &amp; (  for <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">kk</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">i</font></span>)</span> ) &amp; (  for <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">kk</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">i</font></span>)</span> holds <br/><font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:98"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="ref" href="graphsp.html#D10" target="_self" title="GRAPHSP:def.10">Def10</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">nAk</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>;<br/>
<a NAME="E38:98"/><span class="lab"><font color="Green" title="E115">A34</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:98"><span class="lab"><font color="Green" title="E101">A20</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="graphsp.html#E81"><span class="lab"><font color="Green" title="E71">Lm11</font></span></a></span>;<br/>
<a NAME="E39:98"/><span class="lab"><font color="Green" title="E116">A35</font></span>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:98"><span class="lab"><font color="Green" title="E101">A20</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="graphsp.html#E81"><span class="lab"><font color="Green" title="E71">Lm11</font></span></a></span>;<br/>
<a NAME="E40:98"/><span class="lab"><font color="Green" title="E117">A36</font></span>: 
 <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:98"><span class="lab"><font color="Green" title="E101">A20</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="graphsp.html#E81"><span class="lab"><font color="Green" title="E71">Lm11</font></span></a></span>;<br/>
<a NAME="E41:98"/><span class="lab"><font color="Green" title="E118">A37</font></span>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:98"><span class="lab"><font color="Green" title="E93">A12</font></span></a>, <a class="txt" href="#E21:98"><span class="lab"><font color="Green" title="E100">A19</font></span></a>, <a class="txt" href="#E38:98"><span class="lab"><font color="Green" title="E115">A34</font></span></a>, <a class="txt" href="#E39:98"><span class="lab"><font color="Green" title="E116">A35</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E42:98"/><span class="lab"><font color="Green" title="E119">A38</font></span>: 
<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> <a href="graphsp.html#R2" title="GRAPHSP:pred.2">equal_at</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="ref" href="graphsp.html#T46" target="_self" title="GRAPHSP:th.46">Th47</a></span>;<br/>
<a NAME="E43:98"/>
<font color="Maroon" title="c19">PK</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c17">vk</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:98"><span class="lab"><font color="Green" title="E111">A30</font></span></a>, <a class="ref" href="graph_5.html#D18" title="GRAPH_5:def.18">GRAPH_5:def 18</a></span>;<br/>
<a NAME="E44:98"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A39</font></span>: 
<font color="Maroon" title="c19">PK</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c17">vk</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#D4" title="GRAPH_5:def.4">GRAPH_5:def 4</a></span>;<br/>
<a NAME="E45:98"/><span class="kw">then </span>
<font color="Maroon" title="c19">PK</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#D3" title="GRAPH_5:def.3">GRAPH_5:def 3</a></span>;<br/>
<a NAME="E46:98"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A40</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">PK</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T20" title="FINSEQ_1:th.20">FINSEQ_1:20</a></span>;<br/>
<a NAME="E47:98"/><span class="lab"><font color="Green" title="E122">A41</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:98"><span class="lab"><font color="Green" title="E92">A11</font></span></a>, <a class="txt" href="#E12:98"><span class="lab"><font color="Green" title="E93">A12</font></span></a>, <a class="txt" href="#E21:98"><span class="lab"><font color="Green" title="E100">A19</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E123">A42</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:98"><span class="lab"><font color="Green" title="E95">A14</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98"><span class="lab"><font color="Green" title="E94">A13</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="#E27:98"><span class="lab"><font color="Green" title="E106">A25</font></span></a>, <a class="txt" href="#E47:98"><span class="lab"><font color="Green" title="E122">A41</font></span></a>, <a class="ref" href="graphsp.html#T17" target="_self" title="GRAPHSP:th.17">Th18</a></span>
;<br/>
<span class="lab"><font color="Green" title="E124">A43</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:98"><span class="lab"><font color="Green" title="E95">A14</font></span></a>, <a class="txt" href="#E41:98"><span class="lab"><font color="Green" title="E118">A37</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c19">PK</font>,<font color="Maroon" title="c7">W</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98"><span class="lab"><font color="Green" title="E94">A13</font></span></a>, <a class="txt" href="#E35:98"><span class="lab"><font color="Green" title="E112">A31</font></span></a>, <a class="txt" href="#E39:98"><span class="lab"><font color="Green" title="E116">A35</font></span></a>, <a class="txt" href="#E40:98"><span class="lab"><font color="Green" title="E117">A36</font></span></a>, <a class="ref" href="graphsp.html#T18" target="_self" title="GRAPHSP:th.18">Th19</a></span>
;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">nk</font> = <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>;<br/>
<a NAME="E50:98"/><span class="lab"><font color="Green" title="E125">A44</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:98"><span class="lab"><font color="Green" title="E101">A20</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="graphsp.html#E82"><span class="lab"><font color="Green" title="E72">Lm12</font></span></a></span>;<br/>
<a NAME="E51:98"/><span class="lab"><font color="Green" title="E126">A45</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:98"><span class="lab"><font color="Green" title="E101">A20</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="graphsp.html#E82"><span class="lab"><font color="Green" title="E72">Lm12</font></span></a></span>;<br/>
<a NAME="E52:98"/><span class="lab"><font color="Green" title="E127">A46</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:98"><span class="lab"><font color="Green" title="E101">A20</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="graphsp.html#E82"><span class="lab"><font color="Green" title="E72">Lm12</font></span></a></span>;<br/>
<a NAME="E53:98"/>
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:98"><span class="lab"><font color="Green" title="E101">A20</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E54:98"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A47</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E55:98"/><span class="lab"><font color="Green" title="E129">A48</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:98"><span class="lab"><font color="Green" title="E93">A12</font></span></a>, <a class="txt" href="#E21:98"><span class="lab"><font color="Green" title="E100">A19</font></span></a>, <a class="txt" href="#E50:98"><span class="lab"><font color="Green" title="E125">A44</font></span></a>, <a class="txt" href="#E52:98"><span class="lab"><font color="Green" title="E127">A46</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E56:98"/><span class="lab"><font color="Green" title="E130">A49</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:98"><span class="lab"><font color="Green" title="E127">A46</font></span></a>, <a class="txt" href="#E54:98"><span class="lab"><font color="Green" title="E128">A47</font></span></a>, <a class="ref" href="graphsp.html#T31" target="_self" title="GRAPHSP:th.31">Th32</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_3"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c22">Wke</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>;<br/><span class="kw">assume </span><a NAME="E1:98_3"/><span class="lab"><font color="Green" title="E131">A50</font></span>: 
<span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c1">n</font>, <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:98_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A51</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="graphsp.html#K17" title="GRAPHSP:func.17">newpathcost</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p2">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graphsp.html#D13" target="_self" title="GRAPHSP:def.13">Def13</a></span>;<br/><a NAME="E3:98_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98_3"><span class="lab"><font color="Green" title="E131">A50</font></span></a>, <a class="ref" href="graphsp.html#D13" target="_self" title="GRAPHSP:def.13">Def13</a></span>;<br/><a NAME="E4:98_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E5:98_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:98"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="txt" href="#E48:98"><span class="lab"><font color="Green" title="E123">A42</font></span></a>, <a class="txt" href="#E56:98"><span class="lab"><font color="Green" title="E130">A49</font></span></a>, <a class="txt" href="#E2:98_3"><span class="lab"><font color="Green" title="E132">A51</font></span></a>, <a class="ref" href="graphsp.html#T29" target="_self" title="GRAPHSP:th.29">Th30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E58:98"/><span class="kw">then </span>
 not <span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/>
<a NAME="E59:98"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A52</font></span>: 
<font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:98"><span class="lab"><font color="Green" title="E95">A14</font></span></a>, <a class="txt" href="#E16:98"><span class="lab"><font color="Green" title="E96">A15</font></span></a>, <a class="txt" href="#E51:98"><span class="lab"><font color="Green" title="E126">A45</font></span></a>, <a class="txt" href="#E54:98"><span class="lab"><font color="Green" title="E128">A47</font></span></a>, <a class="txt" href="#E55:98"><span class="lab"><font color="Green" title="E129">A48</font></span></a>, <a class="txt" href="#E56:98"><span class="lab"><font color="Green" title="E130">A49</font></span></a>, <a class="ref" href="graphsp.html#D14" target="_self" title="GRAPHSP:def.14">Def14</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) &amp; (  for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c22">v3</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c22">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font> &amp; <font color="Maroon" title="c22">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 holds <br/> ex <font color="Olive" title="b2">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b3">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive">b<sub>4</sub></font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c5">h</font>,<font color="Olive" title="b3">P</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>6</sub></font> &amp; <font color="Olive">b<sub>6</sub></font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">i</font> holds <br/><font color="Olive" title="b4">i</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive">b<sub>5</sub></font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b2">p</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive">b<sub>5</sub></font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">P</font></span>)</span> &amp; (  not <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive">b<sub>5</sub></font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c23">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font> &amp; <font color="Maroon" title="c22">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j</font> &amp; <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 implies  ex <font color="Olive" title="b1">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b2">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c5">h</font>,<font color="Olive" title="b2">P</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">i</font> holds <br/><font color="Olive" title="b3">i</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive">b<sub>4</sub></font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">p</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive">b<sub>4</sub></font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">P</font></span>)</span> &amp; (  not <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive">b<sub>4</sub></font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:98_4"/><span class="lab"><font color="Green" title="E132">A53</font></span>: 
<font color="Maroon" title="c22">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font>
 <span class="kw">and </span><br/><a NAME="E2:98_4"/><span class="lab"><font color="Green" title="E133">A54</font></span>: 
<font color="Maroon" title="c22">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">and </span><br/><a NAME="E3:98_4"/><span class="lab"><font color="Green" title="E134">A55</font></span>: 
<font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b2">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c5">h</font>,<font color="Olive" title="b2">P</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">i</font> holds <br/><font color="Olive" title="b3">i</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive">b<sub>4</sub></font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">p</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive">b<sub>4</sub></font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">P</font></span>)</span> &amp; (  not <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive">b<sub>4</sub></font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )</span><br/><span class="kw">set </span><font color="Maroon" title="c24">nj</font> = <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font>;<br/><a NAME="E4:98_4"/><span class="lab"><font color="Green" title="E135">A56</font></span>: 
<font color="Maroon" title="c23">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">VG</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_4"><span class="lab"><font color="Green" title="E133">A54</font></span></a></span>;<br/><a NAME="E5:98_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E136">A57</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:98"><span class="lab"><font color="Green" title="E97">A16</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E6:98_4"/><span class="lab"><font color="Green" title="E137">A58</font></span>: 
<font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:98"><span class="lab"><font color="Green" title="E97">A16</font></span></a>, <a class="txt" href="#E4:98_4"><span class="lab"><font color="Green" title="E135">A56</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E7:98_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E138">A59</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_4"><span class="lab"><font color="Green" title="E136">A57</font></span></a>, <a class="txt" href="graphsp.html#E82"><span class="lab"><font color="Green" title="E72">Lm12</font></span></a></span>;<br/><a NAME="E8:98_4"/><span class="lab"><font color="Green" title="E139">A60</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_4"><span class="lab"><font color="Green" title="E136">A57</font></span></a>, <a class="txt" href="#E6:98_4"><span class="lab"><font color="Green" title="E137">A58</font></span></a>, <a class="txt" href="graphsp.html#E82"><span class="lab"><font color="Green" title="E72">Lm12</font></span></a></span>;<br/><a NAME="E9:98_4"/><span class="lab"><font color="Green" title="E140">A61</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_4"><span class="lab"><font color="Green" title="E136">A57</font></span></a>, <a class="txt" href="#E6:98_4"><span class="lab"><font color="Green" title="E137">A58</font></span></a>, <a class="txt" href="graphsp.html#E82"><span class="lab"><font color="Green" title="E72">Lm12</font></span></a></span>;<br/><a NAME="E10:98_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E141">A62</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:98"><span class="lab"><font color="Green" title="E93">A12</font></span></a>, <a class="txt" href="#E21:98"><span class="lab"><font color="Green" title="E100">A19</font></span></a>, <a class="txt" href="#E7:98_4"><span class="lab"><font color="Green" title="E138">A59</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E11:98_4"/><span class="lab"><font color="Green" title="E142">A63</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E12:98_4"/>
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_4"><span class="lab"><font color="Green" title="E136">A57</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><a NAME="E13:98_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A64</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c25">m2</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font>;<br/><a NAME="E14:98_4"/><span class="lab"><font color="Green" title="E144">A65</font></span>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:98"><span class="lab"><font color="Green" title="E99">A18</font></span></a>, <a class="txt" href="#E6:98_4"><span class="lab"><font color="Green" title="E137">A58</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><a NAME="E15:98_4"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_4"><span class="lab"><font color="Green" title="E136">A57</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><a NAME="E16:98_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E145">A66</font></span>: 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E17:98_4"/><span class="lab"><font color="Green" title="E146">A67</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E18:98_4"/><span class="lab"><font color="Green" title="E147">A68</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_4"><span class="lab"><font color="Green" title="E136">A57</font></span></a>, <a class="txt" href="#E6:98_4"><span class="lab"><font color="Green" title="E137">A58</font></span></a>, <a class="txt" href="graphsp.html#E81"><span class="lab"><font color="Green" title="E71">Lm11</font></span></a></span>;<br/><a NAME="E19:98_4"/><span class="lab"><font color="Green" title="E148">A69</font></span>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_4"><span class="lab"><font color="Green" title="E136">A57</font></span></a>, <a class="txt" href="#E6:98_4"><span class="lab"><font color="Green" title="E137">A58</font></span></a>, <a class="txt" href="graphsp.html#E81"><span class="lab"><font color="Green" title="E71">Lm11</font></span></a></span>;<br/><a NAME="E20:98_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E149">A70</font></span>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:98"><span class="lab"><font color="Green" title="E93">A12</font></span></a>, <a class="txt" href="#E21:98"><span class="lab"><font color="Green" title="E100">A19</font></span></a>, <a class="txt" href="#E18:98_4"><span class="lab"><font color="Green" title="E147">A68</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E21:98_4"/><span class="lab"><font color="Green" title="E150">A71</font></span>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:98"><span class="lab"><font color="Green" title="E93">A12</font></span></a>, <a class="txt" href="#E14:98"><span class="lab"><font color="Green" title="E95">A14</font></span></a>, <a class="txt" href="#E21:98"><span class="lab"><font color="Green" title="E100">A19</font></span></a>, <a class="txt" href="#E18:98_4"><span class="lab"><font color="Green" title="E147">A68</font></span></a>, <a class="txt" href="#E19:98_4"><span class="lab"><font color="Green" title="E148">A69</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E22:98_4"/><span class="lab"><font color="Green" title="E151">A72</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98"><span class="lab"><font color="Green" title="E94">A13</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="#E9:98_4"><span class="lab"><font color="Green" title="E140">A61</font></span></a>, <a class="txt" href="#E13:98_4"><span class="lab"><font color="Green" title="E143">A64</font></span></a>, <a class="ref" href="graphsp.html#T18" target="_self" title="GRAPHSP:th.18">Th19</a></span>;<br/><a NAME="E23:98_4"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="graphsp.html#E50"><span class="lab"><font color="Green" title="E45">Lm6</font></span></a></span>;<br/><a NAME="E24:98_4"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:98_4"><span class="lab"><font color="Green" title="E145">A66</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E25:98_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E152">A73</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98"><span class="lab"><font color="Green" title="E94">A13</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="#E19:98_4"><span class="lab"><font color="Green" title="E148">A69</font></span></a>, <a class="ref" href="graphsp.html#T18" target="_self" title="GRAPHSP:th.18">Th19</a></span>;<br/><a NAME="E26:98_4"/><span class="lab"><font color="Green" title="E153">A74</font></span>: 
<font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:98"><span class="lab"><font color="Green" title="E106">A25</font></span></a>, <a class="txt" href="#E6:98_4"><span class="lab"><font color="Green" title="E137">A58</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E27:98_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E154">A75</font></span>: 
<font color="Maroon" title="c23">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:98"><span class="lab"><font color="Green" title="E93">A12</font></span></a>, <a class="txt" href="#E21:98"><span class="lab"><font color="Green" title="E100">A19</font></span></a>, <a class="txt" href="#E5:98_4"><span class="lab"><font color="Green" title="E136">A57</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E28:98_4"/><span class="lab"><font color="Green" title="E155">A76</font></span>: 
<font color="Maroon" title="c23">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:98"><span class="lab"><font color="Green" title="E93">A12</font></span></a>, <a class="txt" href="#E28:98"><span class="lab"><font color="Green" title="E107">A26</font></span></a>, <a class="txt" href="#E5:98_4"><span class="lab"><font color="Green" title="E136">A57</font></span></a>, <a class="txt" href="#E26:98_4"><span class="lab"><font color="Green" title="E153">A74</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c26">Akj</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font>;<br/><a NAME="E29:98_4"/><span class="lab"><font color="Green" title="E156">A77</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:98"><span class="lab"><font color="Green" title="E101">A20</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="#E6:98_4"><span class="lab"><font color="Green" title="E137">A58</font></span></a>, <a class="txt" href="graphsp.html#E83"><span class="lab"><font color="Green" title="E73">Lm13</font></span></a></span>;<br/><a NAME="E30:98_4"/><span class="lab"><font color="Green" title="E157">A78</font></span>: 
 <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:98"><span class="lab"><font color="Green" title="E101">A20</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="#E6:98_4"><span class="lab"><font color="Green" title="E137">A58</font></span></a>, <a class="txt" href="graphsp.html#E83"><span class="lab"><font color="Green" title="E73">Lm13</font></span></a></span>;<br/><a NAME="E31:98_4"/><span class="lab"><font color="Green" title="E158">A79</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:98"><span class="lab"><font color="Green" title="E101">A20</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="#E6:98_4"><span class="lab"><font color="Green" title="E137">A58</font></span></a>, <a class="txt" href="graphsp.html#E83"><span class="lab"><font color="Green" title="E73">Lm13</font></span></a></span>;<br/><a NAME="E32:98_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E159">A80</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:98"><span class="lab"><font color="Green" title="E93">A12</font></span></a>, <a class="txt" href="#E21:98"><span class="lab"><font color="Green" title="E100">A19</font></span></a>, <a class="txt" href="#E29:98_4"><span class="lab"><font color="Green" title="E156">A77</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E33:98_4"/><span class="lab"><font color="Green" title="E160">A81</font></span>: 
<span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:98"><span class="lab"><font color="Green" title="E101">A20</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="#E5:98_4"><span class="lab"><font color="Green" title="E136">A57</font></span></a>, <a class="txt" href="#E6:98_4"><span class="lab"><font color="Green" title="E137">A58</font></span></a>, <a class="txt" href="graphsp.html#E84"><span class="lab"><font color="Green" title="E74">Lm14</font></span></a></span>;<br/><a NAME="E34:98_4"/><span class="lab"><font color="Green" title="E161">A82</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:98"><span class="lab"><font color="Green" title="E101">A20</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="#E5:98_4"><span class="lab"><font color="Green" title="E136">A57</font></span></a>, <a class="txt" href="#E6:98_4"><span class="lab"><font color="Green" title="E137">A58</font></span></a>, <a class="txt" href="graphsp.html#E84"><span class="lab"><font color="Green" title="E74">Lm14</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E162">A83</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">j</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p5">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:98"><span class="lab"><font color="Green" title="E95">A14</font></span></a>, <a class="txt" href="#E48:98"><span class="lab"><font color="Green" title="E123">A42</font></span></a>, <a class="txt" href="#E32:98_4"><span class="lab"><font color="Green" title="E159">A80</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p5">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98"><span class="lab"><font color="Green" title="E94">A13</font></span></a>, <a class="txt" href="#E30:98_4"><span class="lab"><font color="Green" title="E157">A78</font></span></a>, <a class="txt" href="#E31:98_4"><span class="lab"><font color="Green" title="E158">A79</font></span></a>, <a class="ref" href="graphsp.html#T18" target="_self" title="GRAPHSP:th.18">Th19</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p5">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:98"><span class="lab"><font color="Green" title="E100">A19</font></span></a>, <a class="txt" href="#E42:98"><span class="lab"><font color="Green" title="E119">A38</font></span></a>, <a class="txt" href="#E32:98_4"><span class="lab"><font color="Green" title="E159">A80</font></span></a>, <a class="txt" href="#E33:98_4"><span class="lab"><font color="Green" title="E160">A81</font></span></a>, <a class="txt" href="#E34:98_4"><span class="lab"><font color="Green" title="E161">A82</font></span></a>, <a class="ref" href="graphsp.html#D16" target="_self" title="GRAPHSP:def.16">Def16</a></span>
<br/>.= 
 <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> (<font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c22">v3</font>,<font color="Maroon" title="c7">W</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:98_4"><span class="lab"><font color="Green" title="E133">A54</font></span></a>, <a class="ref" href="graphsp.html#D20" target="_self" title="GRAPHSP:def.20">Def20</a></span>
;<br/><a NAME="E36:98_4"/><span class="lab"><font color="Green" title="E163">A84</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:98"><span class="lab"><font color="Green" title="E95">A14</font></span></a>, <a class="txt" href="#E20:98_4"><span class="lab"><font color="Green" title="E149">A70</font></span></a>, <a class="txt" href="#E25:98_4"><span class="lab"><font color="Green" title="E152">A73</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:98_4_2"/>
(  not <span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font> or <span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1"><span class="kw">suppose </span></a><a NAME="E1:98_4_2_1"/><span class="lab"><font color="Green" title="E164">A85</font></span>: 
 not <span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b2">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c5">h</font>,<font color="Olive" title="b2">P</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">i</font> holds <br/><font color="Olive" title="b3">i</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive">b<sub>4</sub></font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">p</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive">b<sub>4</sub></font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">P</font></span>)</span> &amp; (  not <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive">b<sub>4</sub></font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )</span><br/><div class="add"><a NAME="E2:98_4_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E165">A86</font></span>: 
<font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:98"><span class="lab"><font color="Green" title="E95">A14</font></span></a>, <a class="txt" href="#E16:98"><span class="lab"><font color="Green" title="E96">A15</font></span></a>, <a class="txt" href="#E8:98_4"><span class="lab"><font color="Green" title="E139">A60</font></span></a>, <a class="txt" href="#E10:98_4"><span class="lab"><font color="Green" title="E141">A62</font></span></a>, <a class="txt" href="#E13:98_4"><span class="lab"><font color="Green" title="E143">A64</font></span></a>, <a class="txt" href="#E22:98_4"><span class="lab"><font color="Green" title="E151">A72</font></span></a>, <a class="ref" href="graphsp.html#D14" target="_self" title="GRAPHSP:def.14">Def14</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c27">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c28">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font><span class="kw"> such that </span><br/><a NAME="E4:98_4_2_1"/><span class="lab"><font color="Green" title="E166">A87</font></span>: 
<font color="Maroon" title="c27">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c23">j</font>,<font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E5:98_4_2_1"/><span class="lab"><font color="Green" title="E167">A88</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">p</font> holds <br/><font color="Maroon" title="c27">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">and </span><br/><a NAME="E6:98_4_2_1"/><span class="lab"><font color="Green" title="E168">A89</font></span>: 
<font color="Maroon" title="c28">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c27">p</font>
 <span class="kw">and </span><br/><a NAME="E7:98_4_2_1"/><span class="lab"><font color="Green" title="E169">A90</font></span>: 
<font color="Maroon" title="c28">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font>
 <span class="kw">and </span><br/><a NAME="E8:98_4_2_1"/><span class="lab"><font color="Green" title="E170">A91</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c28">P</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:98_4_2_1"/><span class="lab"><font color="Green" title="E171">A92</font></span>: 
(  not <font color="Maroon" title="c22">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Maroon" title="c28">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:98"><span class="lab"><font color="Green" title="E89">A8</font></span></a>, <a class="txt" href="#E1:98_4"><span class="lab"><font color="Green" title="E132">A53</font></span></a>, <a class="txt" href="#E2:98_4"><span class="lab"><font color="Green" title="E133">A54</font></span></a>, <a class="txt" href="#E3:98_4"><span class="lab"><font color="Green" title="E134">A55</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c29">p</font> = <font color="Maroon" title="c27">p</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Maroon" title="c29">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c23">j</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">p</font> holds <br/><font color="Maroon" title="c29">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive" title="b1">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c29">p</font> &amp; <font color="Olive" title="b1">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b1">P</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> &amp; (  not <font color="Maroon" title="c22">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive" title="b1">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )</span><br/><span class="kw">take </span><font color="Maroon" title="c30">P</font> = <font color="Maroon" title="c28">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c29">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c23">j</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">p</font> holds <br/><font color="Maroon" title="c29">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Maroon" title="c30">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c29">p</font> &amp; <font color="Maroon" title="c30">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> &amp; (  not <font color="Maroon" title="c22">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Maroon" title="c30">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )</span><br/><span class="kw">thus </span><a NAME="E10:98_4_2_1"/>
<font color="Maroon" title="c29">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c23">j</font>,<font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E5:98"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="txt" href="#E6:98"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="txt" href="#E12:98"><span class="lab"><font color="Green" title="E93">A12</font></span></a>, <a class="txt" href="#E2:98_4_2_1"><span class="lab"><font color="Green" title="E165">A86</font></span></a>, <a class="txt" href="#E4:98_4_2_1"><span class="lab"><font color="Green" title="E166">A87</font></span></a>, <a class="txt" href="#E5:98_4_2_1"><span class="lab"><font color="Green" title="E167">A88</font></span></a>, <a class="txt" href="graphsp.html#E89"><span class="lab"><font color="Green" title="E78">Lm17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">p</font> holds <br/><font color="Maroon" title="c29">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Maroon" title="c30">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c29">p</font> &amp; <font color="Maroon" title="c30">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> &amp; (  not <font color="Maroon" title="c22">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Maroon" title="c30">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c30">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c29">p</font> &amp; <font color="Maroon" title="c30">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> &amp; (  not <font color="Maroon" title="c22">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Maroon" title="c30">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c31">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">i</font> &amp; <font color="Maroon" title="c31">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">p</font> implies <font color="Maroon" title="c29">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:98_4_2_1_1"/><span class="lab"><font color="Green" title="E172">A93</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">i</font>
 <span class="kw">and </span><br/><a NAME="E2:98_4_2_1_1"/><span class="lab"><font color="Green" title="E173">A94</font></span>: 
<font color="Maroon" title="c31">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>)</span><br/><a NAME="E3:98_4_2_1_1"/>
<font color="Maroon" title="c29">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_4_2_1"><span class="lab"><font color="Green" title="E167">A88</font></span></a>, <a class="txt" href="#E1:98_4_2_1_1"><span class="lab"><font color="Green" title="E172">A93</font></span></a>, <a class="txt" href="#E2:98_4_2_1_1"><span class="lab"><font color="Green" title="E173">A94</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:98_4_2_1_1"/>
<font color="Maroon" title="c29">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:98"><span class="lab"><font color="Green" title="E105">A24</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E12:98_4_2_1"/>
<font color="Maroon" title="c30">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c29">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:98_4_2_1"><span class="lab"><font color="Green" title="E168">A89</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c30">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> &amp; (  not <font color="Maroon" title="c22">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Maroon" title="c30">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> &amp; (  not <font color="Maroon" title="c22">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Maroon" title="c30">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )</span>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:98_4_2_1_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:98_4_2_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c30">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font></span><br/><div class="add"><a NAME="E2:98_4_2_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="graphsp.html#K18" title="GRAPHSP:func.18">Relax</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:98"><span class="lab"><font color="Green" title="E95">A14</font></span></a>, <a class="txt" href="#E6:98_4"><span class="lab"><font color="Green" title="E137">A58</font></span></a>, <a class="txt" href="#E27:98_4"><span class="lab"><font color="Green" title="E154">A75</font></span></a>, <a class="ref" href="graphsp.html#D14" target="_self" title="GRAPHSP:def.14">Def14</a></span>;<br/><a NAME="E3:98_4_2_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">h</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:98"><span class="lab"><font color="Green" title="E96">A15</font></span></a>, <a class="txt" href="#E5:98_4"><span class="lab"><font color="Green" title="E136">A57</font></span></a>, <a class="txt" href="#E6:98_4"><span class="lab"><font color="Green" title="E137">A58</font></span></a>, <a class="txt" href="#E28:98_4"><span class="lab"><font color="Green" title="E155">A76</font></span></a></span>;<br/><a NAME="E4:98_4_2_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E172">A95</font></span>: 
( <font color="Maroon" title="c23">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) or <font color="Maroon" title="c23">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:98"><span class="lab"><font color="Green" title="E104">A23</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_4_2_1_2_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c31">Q</font> be   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v4</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font>  st  not <font color="Olive" title="b1">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c31">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">v4</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c7">W</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c32">v4</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c32">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c31">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c32">v4</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> implies  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c7">W</font>) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:98_4_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E173">A96</font></span>: 
 not <font color="Maroon" title="c32">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">and </span><br/><a NAME="E2:98_4_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E174">A97</font></span>: 
<font color="Maroon" title="c31">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c32">v4</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c7">W</font>)</span><br/><a NAME="E3:98_4_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E175">A98</font></span>: 
<font color="Maroon" title="c32">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">VG</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c33">j4</font> = <font color="Maroon" title="c32">v4</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E5:98_4_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E176">A99</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c33">j4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:98"><span class="lab"><font color="Green" title="E97">A16</font></span></a>, <a class="txt" href="#E3:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E175">A98</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E6:98_4_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E177">A100</font></span>: 
<font color="Maroon" title="c33">j4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:98"><span class="lab"><font color="Green" title="E97">A16</font></span></a>, <a class="txt" href="#E3:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E175">A98</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E7:98_4_2_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E178">A101</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c33">j4</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E7:98"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="txt" href="#E9:98"><span class="lab"><font color="Green" title="E90">A9</font></span></a>, <a class="txt" href="#E19:98"><span class="lab"><font color="Green" title="E98">A17</font></span></a>, <a class="txt" href="#E1:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E173">A96</font></span></a>, <a class="txt" href="#E2:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E174">A97</font></span></a>, <a class="txt" href="#E5:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E176">A99</font></span></a>, <a class="txt" href="graphsp.html#E91"><span class="lab"><font color="Green" title="E80">Lm19</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c34">q</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c35">R</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font><span class="kw"> such that </span><br/><a NAME="E9:98_4_2_1_2_1_1_1"/>
<font color="Maroon" title="c34">q</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c33">j4</font>,<font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E10:98_4_2_1_2_1_1_1"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">q</font> holds <br/><font color="Maroon" title="c34">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">and </span><br/><a NAME="E11:98_4_2_1_2_1_1_1"/>
<font color="Maroon" title="c35">R</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c34">q</font>
 <span class="kw">and </span><br/><a NAME="E12:98_4_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E179">A102</font></span>: 
<font color="Maroon" title="c35">R</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c32">v4</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font>
 <span class="kw">and </span><br/><a NAME="E13:98_4_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E180">A103</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c35">R</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c33">j4</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E14:98_4_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E181">A104</font></span>: 
(  not <font color="Maroon" title="c32">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Maroon" title="c35">R</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E7:98"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="txt" href="#E8:98"><span class="lab"><font color="Green" title="E89">A8</font></span></a>, <a class="txt" href="#E1:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E173">A96</font></span></a></span>;<br/><a NAME="E15:98_4_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E182">A105</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c35">R</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E174">A97</font></span></a>, <a class="txt" href="#E12:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E179">A102</font></span></a>, <a class="ref" href="graphsp.html#T9" target="_self" title="GRAPHSP:th.9">Th9</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1_2_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:98_4_2_1_2_1_1_1_1"/>
( <font color="Maroon" title="c23">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) or <font color="Maroon" title="c23">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98_4_2_1_2_1_1"><span class="lab"><font color="Green" title="E172">A95</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1_2_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:98_4_2_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c23">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c7">W</font>)</span><br/><div class="add"><a NAME="E2:98_4_2_1_2_1_1_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">PP</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive" title="b1">PP</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b1">PP</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c35">R</font>,<font color="Maroon" title="c7">W</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98_4"><span class="lab"><font color="Green" title="E132">A53</font></span></a>, <a class="txt" href="#E2:98_4"><span class="lab"><font color="Green" title="E133">A54</font></span></a>, <a class="txt" href="#E1:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E173">A96</font></span></a>, <a class="txt" href="#E14:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E181">A104</font></span></a>, <a class="ref" href="graph_5.html#D19" title="GRAPH_5:def.19">GRAPH_5:def 19</a></span>;<br/><span class="kw">hence </span><a NAME="E3:98_4_2_1_2_1_1_1_1_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:98_4_2_1"><span class="lab"><font color="Green" title="E169">A90</font></span></a>, <a class="txt" href="#E15:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E182">A105</font></span></a>, <a class="ref" href="graphsp.html#T9" target="_self" title="GRAPHSP:th.9">Th9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1_2_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:98_4_2_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E183">A106</font></span>: 
<font color="Maroon" title="c23">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c7">W</font>)</span><br/><div class="add"><a NAME="E2:98_4_2_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c33">j4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:98"><span class="lab"><font color="Green" title="E106">A25</font></span></a>, <a class="txt" href="#E6:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E177">A100</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:98_4_2_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E184">A107</font></span>: 
<font color="Maroon" title="c33">j4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:98"><span class="lab"><font color="Green" title="E93">A12</font></span></a>, <a class="txt" href="#E21:98"><span class="lab"><font color="Green" title="E100">A19</font></span></a>, <a class="txt" href="#E5:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E176">A99</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E4:98_4_2_1_2_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c33">j4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E173">A96</font></span></a>, <a class="txt" href="#E5:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E176">A99</font></span></a>, <a class="txt" href="#E6:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E177">A100</font></span></a></span>;<br/><a NAME="E5:98_4_2_1_2_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c33">j4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E176">A99</font></span></a>, <a class="txt" href="#E6:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E177">A100</font></span></a>, <a class="txt" href="#E7:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E178">A101</font></span></a>, <a class="txt" href="#E3:98_4_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E184">A107</font></span></a></span>;<br/><a NAME="E6:98_4_2_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E185">A108</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c33">j4</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:98"><span class="lab"><font color="Green" title="E114">A33</font></span></a></span>;<br/><a NAME="E7:98_4_2_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E186">A109</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:98"><span class="lab"><font color="Green" title="E118">A37</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E8:98_4_2_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E187">A110</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c33">j4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E176">A99</font></span></a>, <a class="txt" href="#E6:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E177">A100</font></span></a>, <a class="txt" href="graphsp.html#E81"><span class="lab"><font color="Green" title="E71">Lm11</font></span></a></span>;<br/><a NAME="E9:98_4_2_1_2_1_1_1_1_2"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c33">j4</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E176">A99</font></span></a>, <a class="txt" href="#E6:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E177">A100</font></span></a>, <a class="txt" href="graphsp.html#E81"><span class="lab"><font color="Green" title="E71">Lm11</font></span></a></span>;<br/><a NAME="E10:98_4_2_1_2_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c33">j4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:98"><span class="lab"><font color="Green" title="E93">A12</font></span></a>, <a class="txt" href="#E21:98"><span class="lab"><font color="Green" title="E100">A19</font></span></a>, <a class="txt" href="#E8:98_4_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E187">A110</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">hence </span><a NAME="E11:98_4_2_1_2_1_1_1_1_2"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:98_4_2_1"><span class="lab"><font color="Green" title="E170">A91</font></span></a>, <a class="txt" href="#E13:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E180">A103</font></span></a>, <a class="txt" href="#E15:98_4_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E182">A105</font></span></a>, <a class="txt" href="#E1:98_4_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E183">A106</font></span></a>, <a class="txt" href="#E6:98_4_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E185">A108</font></span></a>, <a class="txt" href="#E7:98_4_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E186">A109</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:98_4_2_1_2_1_1"/>
<font color="Maroon" title="c30">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:98"><span class="lab"><font color="Green" title="E98">A17</font></span></a>, <a class="txt" href="#E26:98"><span class="lab"><font color="Green" title="E105">A24</font></span></a>, <a class="txt" href="#E1:98_4"><span class="lab"><font color="Green" title="E132">A53</font></span></a>, <a class="txt" href="#E7:98_4_2_1"><span class="lab"><font color="Green" title="E169">A90</font></span></a>, <a class="ref" href="graph_5.html#T64" title="GRAPH_5:th.64">GRAPH_5:64</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:98_4_2_1_2_1_2"/><span class="lab"><font color="Green" title="E172">A111</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c30">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font></span><br/><div class="add"><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1_2_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:98_4_2_1_2_1_2_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:98_4_2_1_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E173">A112</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c30">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font></span><br/><div class="add"><a NAME="E2:98_4_2_1_2_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E174">A113</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graphsp.html#K17" title="GRAPHSP:func.17">newpathcost</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c23">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:98_4"><span class="lab"><font color="Green" title="E142">A63</font></span></a>, <a class="txt" href="#E1:98_4_2_1"><span class="lab"><font color="Green" title="E164">A85</font></span></a>, <a class="txt" href="#E1:98_4_2_1_2_1_2"><span class="lab"><font color="Green" title="E172">A111</font></span></a>, <a class="ref" href="graphsp.html#D13" target="_self" title="GRAPHSP:def.13">Def13</a></span>;<br/><a NAME="E3:98_4_2_1_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E175">A114</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:98_4"><span class="lab"><font color="Green" title="E150">A71</font></span></a>, <a class="txt" href="#E25:98_4"><span class="lab"><font color="Green" title="E152">A73</font></span></a>, <a class="txt" href="#E8:98_4_2_1"><span class="lab"><font color="Green" title="E170">A91</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c31">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:98_4_2_1_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E176">A115</font></span>: 
<font color="Maroon" title="c31">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c6">G</font>
 <span class="kw">and </span><br/><a NAME="E6:98_4_2_1_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E177">A116</font></span>: 
<font color="Maroon" title="c31">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c22">v3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:98_4"><span class="lab"><font color="Green" title="E162">A83</font></span></a>, <a class="txt" href="#E1:98_4_2_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E173">A112</font></span></a>, <a class="ref" href="graphsp.html#T23" target="_self" title="GRAPHSP:th.23">Th24</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c32">pe</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c31">e</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_4_2_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E176">A115</font></span></a>, <a class="ref" href="graphsp.html#T5" target="_self" title="GRAPHSP:th.5">Th5</a></span>;<br/><a NAME="E8:98_4_2_1_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E178">A117</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">pe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/><a NAME="E9:98_4_2_1_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E179">A118</font></span>: 
<font color="Maroon" title="c32">pe</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c33">Q</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font><span class="kw"> such that </span><br/><a NAME="E11:98_4_2_1_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E180">A119</font></span>: 
<font color="Maroon" title="c33">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">PK</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c32">pe</font>
 <span class="kw">and </span><br/><a NAME="E12:98_4_2_1_2_1_2_1_1_1"/>
<font color="Maroon" title="c33">Q</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:98"><span class="lab"><font color="Green" title="E120">A39</font></span></a>, <a class="txt" href="#E46:98"><span class="lab"><font color="Green" title="E121">A40</font></span></a>, <a class="txt" href="#E6:98_4_2_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E177">A116</font></span></a>, <a class="txt" href="#E8:98_4_2_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E178">A117</font></span></a>, <a class="ref" href="graph_5.html#T33" title="GRAPH_5:th.33">GRAPH_5:33</a></span>;<br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c32">pe</font>,<font color="Maroon" title="c7">W</font>) = 
<font color="Maroon" title="c7">W</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">pe</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:98"><span class="lab"><font color="Green" title="E98">A17</font></span></a>, <a class="txt" href="#E8:98_4_2_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E178">A117</font></span></a>, <a class="ref" href="graphsp.html#T4" target="_self" title="GRAPHSP:th.4">Th4</a>, <a class="ref" href="graph_5.html#T46" title="GRAPH_5:th.46">GRAPH_5:46</a></span>
<br/>.= 
 <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> (<font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c22">v3</font>,<font color="Maroon" title="c7">W</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_4_2_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E176">A115</font></span></a>, <a class="txt" href="#E6:98_4_2_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E177">A116</font></span></a>, <a class="txt" href="#E9:98_4_2_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E179">A118</font></span></a>, <a class="ref" href="graphsp.html#T25" target="_self" title="GRAPHSP:th.25">Th26</a></span>
;<br/><a NAME="E14:98_4_2_1_2_1_2_1_1_1"/><span class="kw">then </span>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c33">Q</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K17" title="GRAPHSP:func.17">newpathcost</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c23">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:98"><span class="lab"><font color="Green" title="E98">A17</font></span></a>, <a class="txt" href="#E48:98"><span class="lab"><font color="Green" title="E123">A42</font></span></a>, <a class="txt" href="#E49:98"><span class="lab"><font color="Green" title="E124">A43</font></span></a>, <a class="txt" href="#E35:98_4"><span class="lab"><font color="Green" title="E162">A83</font></span></a>, <a class="txt" href="#E11:98_4_2_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E180">A119</font></span></a>, <a class="ref" href="graph_5.html#T46" title="GRAPH_5:th.46">GRAPH_5:46</a>, <a class="ref" href="graph_5.html#T54" title="GRAPH_5:th.54">GRAPH_5:54</a></span>;<br/><span class="kw">hence </span><a NAME="E15:98_4_2_1_2_1_2_1_1_1"/>
<font color="Maroon" title="c30">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E7:98"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="txt" href="#E19:98"><span class="lab"><font color="Green" title="E98">A17</font></span></a>, <a class="txt" href="#E25:98"><span class="lab"><font color="Green" title="E104">A23</font></span></a>, <a class="txt" href="#E34:98"><span class="lab"><font color="Green" title="E111">A30</font></span></a>, <a class="txt" href="#E36:98"><span class="lab"><font color="Green" title="E113">A32</font></span></a>, <a class="txt" href="#E46:98"><span class="lab"><font color="Green" title="E121">A40</font></span></a>, <a class="txt" href="#E1:98_4"><span class="lab"><font color="Green" title="E132">A53</font></span></a>, <a class="txt" href="#E7:98_4_2_1"><span class="lab"><font color="Green" title="E169">A90</font></span></a>, <a class="txt" href="#E2:98_4_2_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E174">A113</font></span></a>, <a class="txt" href="#E3:98_4_2_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E175">A114</font></span></a>, <a class="txt" href="#E5:98_4_2_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E176">A115</font></span></a>, <a class="txt" href="#E6:98_4_2_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E177">A116</font></span></a>, <a class="txt" href="#E11:98_4_2_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E180">A119</font></span></a>, <a class="ref" href="graph_5.html#T65" title="GRAPH_5:th.65">GRAPH_5:65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:98_4_2_1_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c30">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font></span><br/><div class="add"><a NAME="E2:98_4_2_1_2_1_2_1_1_2"/><span class="kw">then </span>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c6">G</font> or  not <font color="Olive" title="b1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c22">v3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:98_4"><span class="lab"><font color="Green" title="E162">A83</font></span></a>, <a class="ref" href="graphsp.html#T23" target="_self" title="GRAPHSP:th.23">Th24</a></span>;<br/><span class="kw">hence </span><a NAME="E3:98_4_2_1_2_1_2_1_1_2"/>
<font color="Maroon" title="c30">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E7:98"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="txt" href="#E19:98"><span class="lab"><font color="Green" title="E98">A17</font></span></a>, <a class="txt" href="#E25:98"><span class="lab"><font color="Green" title="E104">A23</font></span></a>, <a class="txt" href="#E34:98"><span class="lab"><font color="Green" title="E111">A30</font></span></a>, <a class="txt" href="#E36:98"><span class="lab"><font color="Green" title="E113">A32</font></span></a>, <a class="txt" href="#E1:98_4"><span class="lab"><font color="Green" title="E132">A53</font></span></a>, <a class="txt" href="#E7:98_4_2_1"><span class="lab"><font color="Green" title="E169">A90</font></span></a>, <a class="ref" href="graphsp.html#T13" target="_self" title="GRAPHSP:th.13">Th13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E14:98_4_2_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:98"><span class="lab"><font color="Green" title="E96">A15</font></span></a>, <a class="txt" href="#E11:98_4"><span class="lab"><font color="Green" title="E142">A63</font></span></a>, <a class="txt" href="#E14:98_4"><span class="lab"><font color="Green" title="E144">A65</font></span></a>, <a class="txt" href="#E16:98_4"><span class="lab"><font color="Green" title="E145">A66</font></span></a>, <a class="txt" href="#E17:98_4"><span class="lab"><font color="Green" title="E146">A67</font></span></a>, <a class="txt" href="#E21:98_4"><span class="lab"><font color="Green" title="E150">A71</font></span></a>, <a class="txt" href="#E25:98_4"><span class="lab"><font color="Green" title="E152">A73</font></span></a>, <a class="txt" href="#E1:98_4_2_1"><span class="lab"><font color="Green" title="E164">A85</font></span></a>, <a class="txt" href="#E8:98_4_2_1"><span class="lab"><font color="Green" title="E170">A91</font></span></a>, <a class="ref" href="graphsp.html#D14" target="_self" title="GRAPHSP:def.14">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c22">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Maroon" title="c30">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:98_4_2_1_3"/><span class="lab"><font color="Green" title="E172">A120</font></span>: 
 not <font color="Maroon" title="c22">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c30">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font></span><br/><a NAME="E2:98_4_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E173">A121</font></span>: 
 not <font color="Maroon" title="c22">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:98"><span class="lab"><font color="Green" title="E104">A23</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_4_2_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c31">v2</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c31">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c31">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font> implies  ex <font color="Olive" title="b1">PK</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">PK</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:98_4_2_1_3_1"/><span class="lab"><font color="Green" title="E174">A122</font></span>: 
<font color="Maroon" title="c31">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">and </span><br/><a NAME="E2:98_4_2_1_3_1"/><span class="lab"><font color="Green" title="E175">A123</font></span>: 
<font color="Maroon" title="c31">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">PK</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">PK</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>) )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:98_4_2_1_3_1_1"/>
( <font color="Maroon" title="c31">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c31">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:98"><span class="lab"><font color="Green" title="E104">A23</font></span></a>, <a class="txt" href="#E1:98_4_2_1_3_1"><span class="lab"><font color="Green" title="E174">A122</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:98_4_2_1_3_1_1_1"/>
<font color="Maroon" title="c31">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">PK</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">PK</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>) )</span><br/><div class="add"><a NAME="E2:98_4_2_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E176">A124</font></span>: 
<font color="Maroon" title="c31">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">vk</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c32">PK</font> = <font color="Maroon" title="c19">PK</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c32">PK</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c31">v2</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c32">PK</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>) )</span><br/><span class="kw">thus </span><a NAME="E3:98_4_2_1_3_1_1_1"/>
<font color="Maroon" title="c32">PK</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c31">v2</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:98"><span class="lab"><font color="Green" title="E104">A23</font></span></a>, <a class="txt" href="#E34:98"><span class="lab"><font color="Green" title="E111">A30</font></span></a>, <a class="txt" href="#E2:98_4_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E176">A124</font></span></a>, <a class="ref" href="graphsp.html#T8" target="_self" title="GRAPHSP:th.8">Th8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c32">PK</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>)</span><br/><a NAME="E4:98_4_2_1_3_1_1_1"/>
<font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_4"><span class="lab"><font color="Green" title="E133">A54</font></span></a>, <a class="txt" href="#E5:98_4"><span class="lab"><font color="Green" title="E136">A57</font></span></a>, <a class="txt" href="#E6:98_4"><span class="lab"><font color="Green" title="E137">A58</font></span></a>, <a class="txt" href="#E27:98_4"><span class="lab"><font color="Green" title="E154">A75</font></span></a>, <a class="txt" href="#E2:98_4_2_1_3"><span class="lab"><font color="Green" title="E173">A121</font></span></a></span>;<br/><a NAME="E5:98_4_2_1_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98_4"><span class="lab"><font color="Green" title="E134">A55</font></span></a>, <a class="txt" href="#E5:98_4"><span class="lab"><font color="Green" title="E136">A57</font></span></a>, <a class="txt" href="#E6:98_4"><span class="lab"><font color="Green" title="E137">A58</font></span></a>, <a class="txt" href="#E27:98_4"><span class="lab"><font color="Green" title="E154">A75</font></span></a>, <a class="txt" href="#E2:98_4_2_1"><span class="lab"><font color="Green" title="E165">A86</font></span></a></span>;<br/><a NAME="E6:98_4_2_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E177">A125</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:98"><span class="lab"><font color="Green" title="E114">A33</font></span></a></span>;<br/><a NAME="E7:98_4_2_1_3_1_1_1"/>
<font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c32">PK</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:98"><span class="lab"><font color="Green" title="E112">A31</font></span></a>, <a class="txt" href="#E41:98"><span class="lab"><font color="Green" title="E118">A37</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E8:98_4_2_1_3_1_1_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c32">PK</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:98_4"><span class="lab"><font color="Green" title="E149">A70</font></span></a>, <a class="txt" href="#E8:98_4_2_1"><span class="lab"><font color="Green" title="E170">A91</font></span></a>, <a class="txt" href="#E6:98_4_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E177">A125</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:98_4_2_1_3_1_1_2"/><span class="lab"><font color="Green" title="E176">A126</font></span>: 
<font color="Maroon" title="c31">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">Q</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">Q</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c32">Q</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font><span class="kw"> such that </span><br/><a NAME="E3:98_4_2_1_3_1_1_2"/><span class="lab"><font color="Green" title="E177">A127</font></span>: 
<font color="Maroon" title="c32">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c31">v2</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font>
 <span class="kw">and </span><br/><a NAME="E4:98_4_2_1_3_1_1_2"/><span class="lab"><font color="Green" title="E178">A128</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c32">Q</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:98"><span class="lab"><font color="Green" title="E104">A23</font></span></a>, <a class="txt" href="#E9:98_4_2_1"><span class="lab"><font color="Green" title="E171">A92</font></span></a>, <a class="txt" href="#E1:98_4_2_1_3"><span class="lab"><font color="Green" title="E172">A120</font></span></a>, <a class="txt" href="#E2:98_4_2_1_3_1"><span class="lab"><font color="Green" title="E175">A123</font></span></a>, <a class="ref" href="graph_5.html#D19" title="GRAPH_5:def.19">GRAPH_5:def 19</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div><span class="lab"><font color="Green" title="E179">A129</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="98_4_2_1_3_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c33">R</font> be   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v4</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font>  st  not <font color="Olive" title="b1">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c33">R</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">v4</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c32">Q</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c33">R</font>,<font color="Maroon" title="c7">W</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c34">v4</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c34">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c33">R</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c34">v4</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> implies  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c32">Q</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c33">R</font>,<font color="Maroon" title="c7">W</font>) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:98_4_2_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E180">A130</font></span>: 
 not <font color="Maroon" title="c34">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">and </span><br/><a NAME="E2:98_4_2_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E181">A131</font></span>: 
<font color="Maroon" title="c33">R</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c34">v4</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c32">Q</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c33">R</font>,<font color="Maroon" title="c7">W</font>)</span><br/><a NAME="E3:98_4_2_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E182">A132</font></span>: 
<font color="Maroon" title="c34">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">VG</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c35">j4</font> = <font color="Maroon" title="c34">v4</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E5:98_4_2_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E183">A133</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">j4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:98"><span class="lab"><font color="Green" title="E97">A16</font></span></a>, <a class="txt" href="#E3:98_4_2_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E182">A132</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E6:98_4_2_1_3_1_1_2_1"/>
<font color="Maroon" title="c35">j4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:98"><span class="lab"><font color="Green" title="E97">A16</font></span></a>, <a class="txt" href="#E3:98_4_2_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E182">A132</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E7:98_4_2_1_3_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">j4</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E7:98"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="txt" href="#E9:98"><span class="lab"><font color="Green" title="E90">A9</font></span></a>, <a class="txt" href="#E19:98"><span class="lab"><font color="Green" title="E98">A17</font></span></a>, <a class="txt" href="#E1:98_4_2_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E180">A130</font></span></a>, <a class="txt" href="#E2:98_4_2_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E181">A131</font></span></a>, <a class="txt" href="#E5:98_4_2_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E183">A133</font></span></a>, <a class="txt" href="graphsp.html#E91"><span class="lab"><font color="Green" title="E80">Lm19</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c36">rn</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c37">RR</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font><span class="kw"> such that </span><br/><a NAME="E9:98_4_2_1_3_1_1_2_1"/>
<font color="Maroon" title="c36">rn</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c35">j4</font>,<font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E10:98_4_2_1_3_1_1_2_1"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c36">rn</font> holds <br/><font color="Maroon" title="c36">rn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">and </span><br/><a NAME="E11:98_4_2_1_3_1_1_2_1"/>
<font color="Maroon" title="c37">RR</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c36">rn</font>
 <span class="kw">and </span><br/><a NAME="E12:98_4_2_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E184">A134</font></span>: 
<font color="Maroon" title="c37">RR</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c34">v4</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font>
 <span class="kw">and </span><br/><a NAME="E13:98_4_2_1_3_1_1_2_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c37">RR</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">j4</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E14:98_4_2_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E185">A135</font></span>: 
(  not <font color="Maroon" title="c34">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Maroon" title="c37">RR</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E7:98"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="txt" href="#E8:98"><span class="lab"><font color="Green" title="E89">A8</font></span></a>, <a class="txt" href="#E1:98_4_2_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E180">A130</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c38">QQ</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font><span class="kw"> such that </span><br/><a NAME="E16:98_4_2_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E186">A136</font></span>: 
<font color="Maroon" title="c38">QQ</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c31">v2</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font>
 <span class="kw">and </span><br/><a NAME="E17:98_4_2_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E187">A137</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c38">QQ</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c37">RR</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_4_2_1_3_1"><span class="lab"><font color="Green" title="E175">A123</font></span></a>, <a class="txt" href="#E1:98_4_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E176">A126</font></span></a>, <a class="txt" href="#E1:98_4_2_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E180">A130</font></span></a>, <a class="txt" href="#E14:98_4_2_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E185">A135</font></span></a>, <a class="ref" href="graph_5.html#D19" title="GRAPH_5:def.19">GRAPH_5:def 19</a></span>;<br/><a NAME="E18:98_4_2_1_3_1_1_2_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c38">QQ</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c32">Q</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98_4_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E177">A127</font></span></a>, <a class="txt" href="#E16:98_4_2_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E186">A136</font></span></a>, <a class="ref" href="graphsp.html#T9" target="_self" title="GRAPHSP:th.9">Th9</a></span>;<br/><span class="kw">hence </span><a NAME="E19:98_4_2_1_3_1_1_2_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c32">Q</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c33">R</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_4_2_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E181">A131</font></span></a>, <a class="txt" href="#E12:98_4_2_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E184">A134</font></span></a>, <a class="txt" href="#E17:98_4_2_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E187">A137</font></span></a>, <a class="ref" href="graphsp.html#T9" target="_self" title="GRAPHSP:th.9">Th9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">take </span><font color="Maroon" title="c33">Q</font> = <font color="Maroon" title="c32">Q</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c33">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c31">v2</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c33">Q</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>) )</span><br/><span class="kw">thus </span><a NAME="E6:98_4_2_1_3_1_1_2"/>
<font color="Maroon" title="c33">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c31">v2</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:98"><span class="lab"><font color="Green" title="E98">A17</font></span></a>, <a class="txt" href="#E26:98"><span class="lab"><font color="Green" title="E105">A24</font></span></a>, <a class="txt" href="#E2:98_4_2_1_3_1"><span class="lab"><font color="Green" title="E175">A123</font></span></a>, <a class="txt" href="#E3:98_4_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E177">A127</font></span></a>, <a class="txt" href="#E5:98_4_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E179">A129</font></span></a>, <a class="ref" href="graph_5.html#T64" title="GRAPH_5:th.64">GRAPH_5:64</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c33">Q</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>)</span><br/><span class="kw">thus </span><a NAME="E7:98_4_2_1_3_1_1_2"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c33">Q</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c30">P</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98_4_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E178">A128</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:98_4_2_1_3"/>
<font color="Maroon" title="c30">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#D19" title="GRAPH_5:def.19">GRAPH_5:def 19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_2"><span class="kw">suppose </span></a><a NAME="E1:98_4_2_2"/><span class="lab"><font color="Green" title="E164">A138</font></span>: 
<span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">q</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b2">Q</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c5">h</font>,<font color="Olive" title="b2">Q</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">i</font> holds <br/><font color="Olive" title="b3">i</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive">b<sub>4</sub></font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">q</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive">b<sub>4</sub></font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">Q</font></span>)</span> &amp; (  not <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive">b<sub>4</sub></font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )</span><br/><div class="add"><a NAME="E2:98_4_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E165">A139</font></span>: 
<span class="p1">(<span class="default"><a href="graphsp.html#K18" title="GRAPHSP:func.18">Relax</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:98"><span class="lab"><font color="Green" title="E95">A14</font></span></a>, <a class="txt" href="#E48:98"><span class="lab"><font color="Green" title="E123">A42</font></span></a>, <a class="txt" href="#E8:98_4"><span class="lab"><font color="Green" title="E139">A60</font></span></a>, <a class="txt" href="#E10:98_4"><span class="lab"><font color="Green" title="E141">A62</font></span></a>, <a class="txt" href="#E13:98_4"><span class="lab"><font color="Green" title="E143">A64</font></span></a>, <a class="ref" href="graphsp.html#D14" target="_self" title="GRAPHSP:def.14">Def14</a></span>;<br/><a NAME="E3:98_4_2_2"/><span class="lab"><font color="Green" title="E166">A140</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="graphsp.html#K17" title="GRAPHSP:func.17">newpathcost</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c23">j</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:98_4"><span class="lab"><font color="Green" title="E142">A63</font></span></a>, <a class="txt" href="#E1:98_4_2_2"><span class="lab"><font color="Green" title="E164">A138</font></span></a>, <a class="ref" href="graphsp.html#D13" target="_self" title="GRAPHSP:def.13">Def13</a></span>;<br/><a NAME="E4:98_4_2_2"/><span class="lab"><font color="Green" title="E167">A141</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:98_4"><span class="lab"><font color="Green" title="E142">A63</font></span></a>, <a class="txt" href="#E1:98_4_2_2"><span class="lab"><font color="Green" title="E164">A138</font></span></a>, <a class="ref" href="graphsp.html#D13" target="_self" title="GRAPHSP:def.13">Def13</a></span>;<br/><a NAME="E5:98_4_2_2"/><span class="lab"><font color="Green" title="E168">A142</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:98_4"><span class="lab"><font color="Green" title="E142">A63</font></span></a>, <a class="txt" href="#E1:98_4_2_2"><span class="lab"><font color="Green" title="E164">A138</font></span></a>, <a class="ref" href="graphsp.html#D13" target="_self" title="GRAPHSP:def.13">Def13</a></span>;<br/><a NAME="E6:98_4_2_2"/><span class="lab"><font color="Green" title="E169">A143</font></span>: 
 <a href="graphsp.html#K17" title="GRAPHSP:func.17">newpathcost</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c23">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> (<font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c22">v3</font>,<font color="Maroon" title="c7">W</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:98"><span class="lab"><font color="Green" title="E123">A42</font></span></a>, <a class="txt" href="#E35:98_4"><span class="lab"><font color="Green" title="E162">A83</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E170">A144</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="98_4_2_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:98_4_2_2_1"/><span class="lab"><font color="Green" title="E171">A145</font></span>: 
 <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:98_4_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E172">A146</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98"><span class="lab"><font color="Green" title="E94">A13</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="#E24:98"><span class="lab"><font color="Green" title="E103">A22</font></span></a>, <a class="txt" href="#E9:98_4"><span class="lab"><font color="Green" title="E140">A61</font></span></a>, <a class="txt" href="#E13:98_4"><span class="lab"><font color="Green" title="E143">A64</font></span></a>, <a class="ref" href="graphsp.html#T18" target="_self" title="GRAPHSP:th.18">Th19</a></span>;<br/><a NAME="E3:98_4_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> (<font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c22">v3</font>,<font color="Maroon" title="c7">W</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:98_4"><span class="lab"><font color="Green" title="E162">A83</font></span></a>, <a class="txt" href="#E4:98_4_2_2"><span class="lab"><font color="Green" title="E167">A141</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E4:98_4_2_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:98_4"><span class="lab"><font color="Green" title="E142">A63</font></span></a>, <a class="txt" href="#E1:98_4_2_2"><span class="lab"><font color="Green" title="E164">A138</font></span></a>, <a class="txt" href="#E6:98_4_2_2"><span class="lab"><font color="Green" title="E169">A143</font></span></a>, <a class="txt" href="#E1:98_4_2_2_1"><span class="lab"><font color="Green" title="E171">A145</font></span></a>, <a class="txt" href="#E2:98_4_2_2_1"><span class="lab"><font color="Green" title="E172">A146</font></span></a>, <a class="ref" href="graphsp.html#D13" target="_self" title="GRAPHSP:def.13">Def13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E171">A147</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="98_4_2_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:98_4_2_2_2"/>
<font color="Maroon" title="c23">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:98_4_2_2_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c23">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 ;<br/><span class="kw">hence </span><a NAME="E3:98_4_2_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:98"><span class="lab"><font color="Green" title="E106">A25</font></span></a>, <a class="txt" href="#E5:98_4_2_2"><span class="lab"><font color="Green" title="E168">A142</font></span></a>, <a class="ref" href="graphsp.html#T32" target="_self" title="GRAPHSP:th.32">Th33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c27">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:98_4_2_2"/><span class="lab"><font color="Green" title="E172">A148</font></span>: 
( <font color="Maroon" title="c27">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c6">G</font> &amp; <font color="Maroon" title="c27">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c22">v3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:98_4"><span class="lab"><font color="Green" title="E162">A83</font></span></a>, <a class="txt" href="#E4:98_4_2_2"><span class="lab"><font color="Green" title="E167">A141</font></span></a>, <a class="ref" href="graphsp.html#T23" target="_self" title="GRAPHSP:th.23">Th24</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c28">pe</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c27">e</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:98_4_2_2"><span class="lab"><font color="Green" title="E172">A148</font></span></a>, <a class="ref" href="graphsp.html#T5" target="_self" title="GRAPHSP:th.5">Th5</a></span>;<br/><a NAME="E12:98_4_2_2"/><span class="lab"><font color="Green" title="E173">A149</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">pe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/><a NAME="E13:98_4_2_2"/><span class="lab"><font color="Green" title="E174">A150</font></span>: 
<font color="Maroon" title="c28">pe</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">Q</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font><span class="kw"> such that </span><br/><a NAME="E15:98_4_2_2"/><span class="lab"><font color="Green" title="E175">A151</font></span>: 
<font color="Maroon" title="c29">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">PK</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c28">pe</font>
 <span class="kw">and </span><br/><a NAME="E16:98_4_2_2"/>
<font color="Maroon" title="c29">Q</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:98"><span class="lab"><font color="Green" title="E120">A39</font></span></a>, <a class="txt" href="#E46:98"><span class="lab"><font color="Green" title="E121">A40</font></span></a>, <a class="txt" href="#E10:98_4_2_2"><span class="lab"><font color="Green" title="E172">A148</font></span></a>, <a class="txt" href="#E12:98_4_2_2"><span class="lab"><font color="Green" title="E173">A149</font></span></a>, <a class="ref" href="graph_5.html#T33" title="GRAPH_5:th.33">GRAPH_5:33</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c30">q</font> = <font color="Maroon" title="c18">pk</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c23">j</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">Q</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Maroon" title="c30">q</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c23">j</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">q</font> holds <br/><font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive" title="b1">Q</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c30">q</font> &amp; <font color="Olive" title="b1">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b1">Q</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> &amp; (  not <font color="Maroon" title="c22">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive" title="b1">Q</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )</span><br/><span class="kw">take </span><font color="Maroon" title="c31">Q</font> = <font color="Maroon" title="c29">Q</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c30">q</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c23">j</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">q</font> holds <br/><font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Maroon" title="c31">Q</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c30">q</font> &amp; <font color="Maroon" title="c31">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> &amp; (  not <font color="Maroon" title="c22">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Maroon" title="c31">Q</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )</span><br/><span class="kw">thus </span><a NAME="E17:98_4_2_2"/>
<font color="Maroon" title="c30">q</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c23">j</font>,<font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E5:98"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="txt" href="#E6:98"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="txt" href="#E12:98"><span class="lab"><font color="Green" title="E93">A12</font></span></a>, <a class="txt" href="#E16:98"><span class="lab"><font color="Green" title="E96">A15</font></span></a>, <a class="txt" href="#E31:98"><span class="lab"><font color="Green" title="E108">A27</font></span></a>, <a class="txt" href="#E32:98"><span class="lab"><font color="Green" title="E109">A28</font></span></a>, <a class="txt" href="#E59:98"><span class="lab"><font color="Green" title="E131">A52</font></span></a>, <a class="txt" href="#E2:98_4_2_2"><span class="lab"><font color="Green" title="E165">A139</font></span></a>, <a class="txt" href="#E7:98_4_2_2"><span class="lab"><font color="Green" title="E170">A144</font></span></a>, <a class="txt" href="#E8:98_4_2_2"><span class="lab"><font color="Green" title="E171">A147</font></span></a>, <a class="txt" href="graphsp.html#E90"><span class="lab"><font color="Green" title="E79">Lm18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">q</font> holds <br/><font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Maroon" title="c31">Q</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c30">q</font> &amp; <font color="Maroon" title="c31">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> &amp; (  not <font color="Maroon" title="c22">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Maroon" title="c31">Q</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )</span><br/><a NAME="E18:98_4_2_2"/><span class="lab"><font color="Green" title="E176">A152</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">pk</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:98"><span class="lab"><font color="Green" title="E108">A27</font></span></a>, <a class="ref" href="graphsp.html#D18" target="_self" title="GRAPHSP:def.18">Def18</a></span>;<br/><a NAME="E19:98_4_2_2"/><span class="lab"><font color="Green" title="E177">A153</font></span>: 
<font color="Maroon" title="c18">pk</font> <a href="graphsp.html#R3" title="GRAPHSP:pred.3">is_vertex_seq_at</a> <font color="Maroon" title="c4">g</font>, <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:98"><span class="lab"><font color="Green" title="E108">A27</font></span></a>, <a class="ref" href="graphsp.html#D18" target="_self" title="GRAPHSP:def.18">Def18</a></span>;<br/><span class="lab"><font color="Green" title="E178">A154</font></span>: <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">pk</font></span>)</span> = 
<font color="Maroon" title="c18">pk</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">pk</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:98_4_2_2"><span class="lab"><font color="Green" title="E176">A152</font></span></a>, <a class="txt" href="graphsp.html#E6"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>
<br/>.= 
 <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:98_4_2_2"><span class="lab"><font color="Green" title="E177">A153</font></span></a>, <a class="ref" href="graphsp.html#D17" target="_self" title="GRAPHSP:def.17">Def17</a></span>
;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c31">Q</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c30">q</font> &amp; <font color="Maroon" title="c31">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> &amp; (  not <font color="Maroon" title="c22">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Maroon" title="c31">Q</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c32">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">i</font> &amp; <font color="Maroon" title="c32">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">q</font> implies <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:98_4_2_2_4"/><span class="lab"><font color="Green" title="E179">A155</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">i</font>
 <span class="kw">and </span><br/><a NAME="E2:98_4_2_2_4"/><span class="lab"><font color="Green" title="E180">A156</font></span>: 
<font color="Maroon" title="c32">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>)</span><br/><a NAME="E3:98_4_2_2_4"/>
<font color="Maroon" title="c32">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">pk</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_4_2_2_4"><span class="lab"><font color="Green" title="E180">A156</font></span></a>, <a class="ref" href="finseq_2.html#T16" title="FINSEQ_2:th.16">FINSEQ_2:16</a></span>;<br/><a NAME="E4:98_4_2_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E181">A157</font></span>: 
<font color="Maroon" title="c32">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">pk</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_4_2_2_4_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_2_4_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:98_4_2_2_4_1_1"/>
( <font color="Maroon" title="c32">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">pk</font> or <font color="Maroon" title="c32">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">pk</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_2_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:98_4_2_2_4_1_1_1"/>
<font color="Maroon" title="c32">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">pk</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:98_4_2_2_4_1_1_1"/>
( <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:98_4_2_2"><span class="lab"><font color="Green" title="E178">A154</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_2_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:98_4_2_2_4_1_1_2"/>
<font color="Maroon" title="c32">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">pk</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) )</span><br/><div class="add"><a NAME="E2:98_4_2_2_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E182">A158</font></span>: 
<font color="Maroon" title="c32">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">pk</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98_4_2_2_4"><span class="lab"><font color="Green" title="E181">A157</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:98_4_2_2_4_1_1_2"/>
<font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">pk</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c32">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98_4_2_2_4"><span class="lab"><font color="Green" title="E179">A155</font></span></a>, <a class="txt" href="#E4:98_4_2_2_4"><span class="lab"><font color="Green" title="E181">A157</font></span></a>, <a class="txt" href="graphsp.html#E6"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:98_4_2_2_4_1_1_2"/>
( <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:98"><span class="lab"><font color="Green" title="E109">A28</font></span></a>, <a class="txt" href="#E1:98_4_2_2_4"><span class="lab"><font color="Green" title="E179">A155</font></span></a>, <a class="txt" href="#E2:98_4_2_2_4_1_1_2"><span class="lab"><font color="Green" title="E182">A158</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:98_4_2_2_4"/>
<font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:98"><span class="lab"><font color="Green" title="E104">A23</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><a NAME="E22:98_4_2_2"/><span class="lab"><font color="Green" title="E179">A159</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c31">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">PK</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:98_4_2_2"><span class="lab"><font color="Green" title="E173">A149</font></span></a>, <a class="txt" href="#E15:98_4_2_2"><span class="lab"><font color="Green" title="E175">A151</font></span></a>, <a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/><a NAME="E23:98_4_2_2"/><span class="lab"><font color="Green" title="E180">A160</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">pk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">PK</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:98"><span class="lab"><font color="Green" title="E110">A29</font></span></a>, <a class="ref" href="graphsp.html#D19" target="_self" title="GRAPHSP:def.19">Def19</a></span>;<br/><a NAME="E24:98_4_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E181">A161</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c31">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:98_4_2_2"><span class="lab"><font color="Green" title="E179">A159</font></span></a>, <a class="ref" href="finseq_2.html#T16" title="FINSEQ_2:th.16">FINSEQ_2:16</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c32">FS</font> =  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c6">G</font>;<br/><span class="kw">set </span><font color="Maroon" title="c33">FT</font> =  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c6">G</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_4_2_2_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c34">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c34">i</font> &amp; <font color="Maroon" title="c34">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c31">Q</font> implies (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> &amp;  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:98_4_2_2_5"/><span class="lab"><font color="Green" title="E182">A162</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c34">i</font>
 <span class="kw">and </span><br/><a NAME="E2:98_4_2_2_5"/><span class="lab"><font color="Green" title="E183">A163</font></span>: 
<font color="Maroon" title="c34">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c31">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> &amp;  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_2_5_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:98_4_2_2_5_1"/>
( <font color="Maroon" title="c34">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c31">Q</font> or <font color="Maroon" title="c34">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c31">Q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_2_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:98_4_2_2_5_1_1"/><span class="lab"><font color="Green" title="E184">A164</font></span>: 
<font color="Maroon" title="c34">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c31">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> &amp;  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )</span><br/><div class="add"><a NAME="E2:98_4_2_2_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E185">A165</font></span>: 
<font color="Maroon" title="c31">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:98_4_2_2"><span class="lab"><font color="Green" title="E173">A149</font></span></a>, <a class="txt" href="#E13:98_4_2_2"><span class="lab"><font color="Green" title="E174">A150</font></span></a>, <a class="txt" href="#E15:98_4_2_2"><span class="lab"><font color="Green" title="E175">A151</font></span></a>, <a class="txt" href="#E22:98_4_2_2"><span class="lab"><font color="Green" title="E179">A159</font></span></a>, <a class="txt" href="graphsp.html#E7"><span class="lab"><font color="Green" title="E7">Lm2</font></span></a></span>;<br/><a NAME="E3:98_4_2_2_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E186">A166</font></span>: 
 the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">v3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:98_4_2_2"><span class="lab"><font color="Green" title="E172">A148</font></span></a>, <a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></span>;<br/><span class="kw">thus </span><a NAME="E4:98_4_2_2_5_1_1"/>
 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c34">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:98_4_2_2"><span class="lab"><font color="Green" title="E172">A148</font></span></a>, <a class="txt" href="#E20:98_4_2_2"><span class="lab"><font color="Green" title="E178">A154</font></span></a>, <a class="txt" href="#E22:98_4_2_2"><span class="lab"><font color="Green" title="E179">A159</font></span></a>, <a class="txt" href="#E23:98_4_2_2"><span class="lab"><font color="Green" title="E180">A160</font></span></a>, <a class="txt" href="#E1:98_4_2_2_5_1_1"><span class="lab"><font color="Green" title="E184">A164</font></span></a>, <a class="txt" href="#E2:98_4_2_2_5_1_1"><span class="lab"><font color="Green" title="E185">A165</font></span></a>, <a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><span class="kw">thus </span><a NAME="E5:98_4_2_2_5_1_1"/>
 the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_4"><span class="lab"><font color="Green" title="E133">A54</font></span></a>, <a class="txt" href="#E22:98_4_2_2"><span class="lab"><font color="Green" title="E179">A159</font></span></a>, <a class="txt" href="#E23:98_4_2_2"><span class="lab"><font color="Green" title="E180">A160</font></span></a>, <a class="txt" href="#E1:98_4_2_2_5_1_1"><span class="lab"><font color="Green" title="E184">A164</font></span></a>, <a class="txt" href="#E3:98_4_2_2_5_1_1"><span class="lab"><font color="Green" title="E186">A166</font></span></a>, <a class="ref" href="finseq_1.html#T42" title="FINSEQ_1:th.42">FINSEQ_1:42</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_2_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:98_4_2_2_5_1_2"/>
<font color="Maroon" title="c34">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c31">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> &amp;  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )</span><br/><div class="add"><a NAME="E2:98_4_2_2_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E184">A167</font></span>: 
<font color="Maroon" title="c34">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c31">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_4_2_2_5"><span class="lab"><font color="Green" title="E183">A163</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:98_4_2_2_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E185">A168</font></span>: 
<font color="Maroon" title="c34">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">PK</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:98_4_2_2"><span class="lab"><font color="Green" title="E179">A159</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:98_4_2_2_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E186">A169</font></span>: 
 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">PK</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">pk</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c34">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:98"><span class="lab"><font color="Green" title="E110">A29</font></span></a>, <a class="txt" href="#E1:98_4_2_2_5"><span class="lab"><font color="Green" title="E182">A162</font></span></a>, <a class="ref" href="graphsp.html#D19" target="_self" title="GRAPHSP:def.19">Def19</a></span>;<br/><a NAME="E5:98_4_2_2_5_1_2"/><span class="lab"><font color="Green" title="E187">A170</font></span>: 
 the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">PK</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">pk</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:98"><span class="lab"><font color="Green" title="E110">A29</font></span></a>, <a class="txt" href="#E1:98_4_2_2_5"><span class="lab"><font color="Green" title="E182">A162</font></span></a>, <a class="txt" href="#E3:98_4_2_2_5_1_2"><span class="lab"><font color="Green" title="E185">A168</font></span></a>, <a class="ref" href="graphsp.html#D19" target="_self" title="GRAPHSP:def.19">Def19</a></span>;<br/><a NAME="E6:98_4_2_2_5_1_2"/><span class="lab"><font color="Green" title="E188">A171</font></span>: 
<font color="Maroon" title="c31">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">PK</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:98_4_2_2"><span class="lab"><font color="Green" title="E175">A151</font></span></a>, <a class="txt" href="#E1:98_4_2_2_5"><span class="lab"><font color="Green" title="E182">A162</font></span></a>, <a class="txt" href="#E3:98_4_2_2_5_1_2"><span class="lab"><font color="Green" title="E185">A168</font></span></a>, <a class="txt" href="graphsp.html#E6"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/><a NAME="E7:98_4_2_2_5_1_2"/><span class="lab"><font color="Green" title="E189">A172</font></span>: 
<font color="Maroon" title="c34">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">pk</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:98_4_2_2"><span class="lab"><font color="Green" title="E179">A159</font></span></a>, <a class="txt" href="#E23:98_4_2_2"><span class="lab"><font color="Green" title="E180">A160</font></span></a>, <a class="txt" href="#E2:98_4_2_2_5_1_2"><span class="lab"><font color="Green" title="E184">A167</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">thus </span><a NAME="E8:98_4_2_2_5_1_2"/>
 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c34">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:98_4_2_2"><span class="lab"><font color="Green" title="E179">A159</font></span></a>, <a class="txt" href="#E23:98_4_2_2"><span class="lab"><font color="Green" title="E180">A160</font></span></a>, <a class="txt" href="#E1:98_4_2_2_5"><span class="lab"><font color="Green" title="E182">A162</font></span></a>, <a class="txt" href="#E2:98_4_2_2_5"><span class="lab"><font color="Green" title="E183">A163</font></span></a>, <a class="txt" href="#E4:98_4_2_2_5_1_2"><span class="lab"><font color="Green" title="E186">A169</font></span></a>, <a class="txt" href="#E6:98_4_2_2_5_1_2"><span class="lab"><font color="Green" title="E188">A171</font></span></a>, <a class="txt" href="graphsp.html#E6"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><span class="kw">thus </span><a NAME="E9:98_4_2_2_5_1_2"/>
 the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">q</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:98_4_2_2_5_1_2"><span class="lab"><font color="Green" title="E187">A170</font></span></a>, <a class="txt" href="#E6:98_4_2_2_5_1_2"><span class="lab"><font color="Green" title="E188">A171</font></span></a>, <a class="txt" href="#E7:98_4_2_2_5_1_2"><span class="lab"><font color="Green" title="E189">A172</font></span></a>, <a class="txt" href="graphsp.html#E6"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E26:98_4_2_2"/>
<font color="Maroon" title="c31">Q</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c30">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:98_4_2_2"><span class="lab"><font color="Green" title="E181">A161</font></span></a>, <a class="ref" href="graphsp.html#D19" target="_self" title="GRAPHSP:def.19">Def19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c31">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> &amp; (  not <font color="Maroon" title="c22">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Maroon" title="c31">Q</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )</span><br/><a NAME="E27:98_4_2_2"/><span class="lab"><font color="Green" title="E182">A173</font></span>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c19">PK</font>,<font color="Maroon" title="c7">W</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c28">pe</font>,<font color="Maroon" title="c7">W</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:98"><span class="lab"><font color="Green" title="E98">A17</font></span></a>, <a class="txt" href="#E15:98_4_2_2"><span class="lab"><font color="Green" title="E175">A151</font></span></a>, <a class="ref" href="graph_5.html#T46" title="GRAPH_5:th.46">GRAPH_5:46</a>, <a class="ref" href="graph_5.html#T54" title="GRAPH_5:th.54">GRAPH_5:54</a></span>;<br/><span class="lab"><font color="Green" title="E183">A174</font></span>:  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c28">pe</font>,<font color="Maroon" title="c7">W</font>) = 
<font color="Maroon" title="c7">W</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">pe</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:98"><span class="lab"><font color="Green" title="E98">A17</font></span></a>, <a class="txt" href="#E12:98_4_2_2"><span class="lab"><font color="Green" title="E173">A149</font></span></a>, <a class="ref" href="graphsp.html#T4" target="_self" title="GRAPHSP:th.4">Th4</a>, <a class="ref" href="graph_5.html#T46" title="GRAPH_5:th.46">GRAPH_5:46</a></span>
<br/>.= 
 <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> (<font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c22">v3</font>,<font color="Maroon" title="c7">W</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:98_4_2_2"><span class="lab"><font color="Green" title="E172">A148</font></span></a>, <a class="txt" href="#E13:98_4_2_2"><span class="lab"><font color="Green" title="E174">A150</font></span></a>, <a class="ref" href="graphsp.html#T25" target="_self" title="GRAPHSP:th.25">Th26</a></span>
;<br/><a NAME="E29:98_4_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E184">A175</font></span>: 
 <a href="graphsp.html#K17" title="GRAPHSP:func.17">newpathcost</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c23">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:98"><span class="lab"><font color="Green" title="E98">A17</font></span></a>, <a class="txt" href="#E48:98"><span class="lab"><font color="Green" title="E123">A42</font></span></a>, <a class="txt" href="#E49:98"><span class="lab"><font color="Green" title="E124">A43</font></span></a>, <a class="txt" href="#E35:98_4"><span class="lab"><font color="Green" title="E162">A83</font></span></a>, <a class="txt" href="#E15:98_4_2_2"><span class="lab"><font color="Green" title="E175">A151</font></span></a>, <a class="ref" href="graph_5.html#T46" title="GRAPH_5:th.46">GRAPH_5:46</a>, <a class="ref" href="graph_5.html#T54" title="GRAPH_5:th.54">GRAPH_5:54</a></span>;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> &amp; (  not <font color="Maroon" title="c22">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Maroon" title="c31">Q</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )</span>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_2_7_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:98_4_2_2_7_1"/>
( <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 or <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_2_7_1_1"><span class="kw">suppose </span></a><a NAME="E1:98_4_2_2_7_1_1"/><span class="lab"><font color="Green" title="E185">A176</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c31">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_4_2_2_7_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c34">v2</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c34">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) implies  for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c6">G</font> or  not <font color="Olive" title="b1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c34">v2</font>,<font color="Maroon" title="c22">v3</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:98_4_2_2_7_1_1_1"/><span class="lab"><font color="Green" title="E186">A177</font></span>: 
<font color="Maroon" title="c34">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c6">G</font> or  not <font color="Olive" title="b1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c34">v2</font>,<font color="Maroon" title="c22">v3</font> )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c35">m</font> = <font color="Maroon" title="c34">v2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/> <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 = 
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c35">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:98"><span class="lab"><font color="Green" title="E90">A9</font></span></a>, <a class="txt" href="#E5:98_4"><span class="lab"><font color="Green" title="E136">A57</font></span></a>, <a class="txt" href="#E6:98_4"><span class="lab"><font color="Green" title="E137">A58</font></span></a>, <a class="txt" href="#E1:98_4_2_2_7_1_1"><span class="lab"><font color="Green" title="E185">A176</font></span></a>, <a class="txt" href="#E1:98_4_2_2_7_1_1_1"><span class="lab"><font color="Green" title="E186">A177</font></span></a></span>
<br/>.= 
 <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> (<font color="Maroon" title="c34">v2</font>,<font color="Maroon" title="c22">v3</font>,<font color="Maroon" title="c7">W</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:98_4"><span class="lab"><font color="Green" title="E133">A54</font></span></a>, <a class="ref" href="graphsp.html#D20" target="_self" title="GRAPHSP:def.20">Def20</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:98_4_2_2_7_1_1_1"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c6">G</font> or  not <font color="Olive" title="b1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c34">v2</font>,<font color="Maroon" title="c22">v3</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graphsp.html#T23" target="_self" title="GRAPHSP:th.23">Th24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:98_4_2_2_7_1_1"/>
<font color="Maroon" title="c31">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E7:98"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="txt" href="#E25:98"><span class="lab"><font color="Green" title="E104">A23</font></span></a>, <a class="txt" href="#E34:98"><span class="lab"><font color="Green" title="E111">A30</font></span></a>, <a class="txt" href="#E1:98_4"><span class="lab"><font color="Green" title="E132">A53</font></span></a>, <a class="txt" href="#E10:98_4_2_2"><span class="lab"><font color="Green" title="E172">A148</font></span></a>, <a class="txt" href="#E15:98_4_2_2"><span class="lab"><font color="Green" title="E175">A151</font></span></a>, <a class="ref" href="graphsp.html#T15" target="_self" title="GRAPHSP:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_2_7_1_2"><span class="kw">suppose </span></a><a NAME="E1:98_4_2_2_7_1_2"/><span class="lab"><font color="Green" title="E185">A178</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c31">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font></span><br/><div class="add"><a NAME="E2:98_4_2_2_7_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b2">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c23">j</font>,<font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Olive" title="b1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) ) &amp; <font color="Olive" title="b2">P</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b2">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive" title="b2">P</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> &amp; (  not <font color="Maroon" title="c22">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Olive" title="b2">P</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:98"><span class="lab"><font color="Green" title="E89">A8</font></span></a>, <a class="txt" href="#E1:98_4"><span class="lab"><font color="Green" title="E132">A53</font></span></a>, <a class="txt" href="#E2:98_4"><span class="lab"><font color="Green" title="E133">A54</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:98_4_2_2_7_1_2"/>
<font color="Maroon" title="c31">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c22">v3</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E7:98"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="txt" href="#E13:98"><span class="lab"><font color="Green" title="E94">A13</font></span></a>, <a class="txt" href="#E19:98"><span class="lab"><font color="Green" title="E98">A17</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="#E25:98"><span class="lab"><font color="Green" title="E104">A23</font></span></a>, <a class="txt" href="#E34:98"><span class="lab"><font color="Green" title="E111">A30</font></span></a>, <a class="txt" href="#E36:98"><span class="lab"><font color="Green" title="E113">A32</font></span></a>, <a class="txt" href="#E46:98"><span class="lab"><font color="Green" title="E121">A40</font></span></a>, <a class="txt" href="#E48:98"><span class="lab"><font color="Green" title="E123">A42</font></span></a>, <a class="txt" href="#E49:98"><span class="lab"><font color="Green" title="E124">A43</font></span></a>, <a class="txt" href="#E1:98_4"><span class="lab"><font color="Green" title="E132">A53</font></span></a>, <a class="txt" href="#E9:98_4"><span class="lab"><font color="Green" title="E140">A61</font></span></a>, <a class="txt" href="#E13:98_4"><span class="lab"><font color="Green" title="E143">A64</font></span></a>, <a class="txt" href="#E35:98_4"><span class="lab"><font color="Green" title="E162">A83</font></span></a>, <a class="txt" href="#E36:98_4"><span class="lab"><font color="Green" title="E163">A84</font></span></a>, <a class="txt" href="#E3:98_4_2_2"><span class="lab"><font color="Green" title="E166">A140</font></span></a>, <a class="txt" href="#E10:98_4_2_2"><span class="lab"><font color="Green" title="E172">A148</font></span></a>, <a class="txt" href="#E15:98_4_2_2"><span class="lab"><font color="Green" title="E175">A151</font></span></a>, <a class="txt" href="#E27:98_4_2_2"><span class="lab"><font color="Green" title="E182">A173</font></span></a>, <a class="txt" href="#E28:98_4_2_2"><span class="lab"><font color="Green" title="E183">A174</font></span></a>, <a class="txt" href="#E1:98_4_2_2_7_1_2"><span class="lab"><font color="Green" title="E185">A178</font></span></a>, <a class="ref" href="graphsp.html#T18" target="_self" title="GRAPHSP:th.18">Th19</a>, <a class="ref" href="graph_5.html#T65" title="GRAPH_5:th.65">GRAPH_5:65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E31:98_4_2_2"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:98"><span class="lab"><font color="Green" title="E96">A15</font></span></a>, <a class="txt" href="#E11:98_4"><span class="lab"><font color="Green" title="E142">A63</font></span></a>, <a class="txt" href="#E14:98_4"><span class="lab"><font color="Green" title="E144">A65</font></span></a>, <a class="txt" href="#E16:98_4"><span class="lab"><font color="Green" title="E145">A66</font></span></a>, <a class="txt" href="#E17:98_4"><span class="lab"><font color="Green" title="E146">A67</font></span></a>, <a class="txt" href="#E21:98_4"><span class="lab"><font color="Green" title="E150">A71</font></span></a>, <a class="txt" href="#E1:98_4_2_2"><span class="lab"><font color="Green" title="E164">A138</font></span></a>, <a class="txt" href="#E29:98_4_2_2"><span class="lab"><font color="Green" title="E184">A175</font></span></a>, <a class="ref" href="graphsp.html#D14" target="_self" title="GRAPHSP:def.14">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c22">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Maroon" title="c31">Q</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> )</span><br/><a NAME="E32:98_4_2_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c28">pe</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:98"><span class="lab"><font color="Green" title="E98">A17</font></span></a>, <a class="ref" href="graph_5.html#T50" title="GRAPH_5:th.50">GRAPH_5:50</a></span>;<br/><a NAME="E33:98_4_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E185">A179</font></span>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c19">PK</font>,<font color="Maroon" title="c7">W</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:98_4_2_2"><span class="lab"><font color="Green" title="E182">A173</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:98_4_2_2_8"/>
 not <font color="Maroon" title="c22">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c31">Q</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_4_2_2_8_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c34">v2</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c34">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c34">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font> implies  ex <font color="Olive" title="b1">PK</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">PK</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:98_4_2_2_8_1"/><span class="lab"><font color="Green" title="E186">A180</font></span>: 
<font color="Maroon" title="c34">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">and </span><br/><a NAME="E2:98_4_2_2_8_1"/><span class="lab"><font color="Green" title="E187">A181</font></span>: 
<font color="Maroon" title="c34">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">PK</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">PK</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>) )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_2_8_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:98_4_2_2_8_1_1"/>
( <font color="Maroon" title="c34">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c34">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:98"><span class="lab"><font color="Green" title="E104">A23</font></span></a>, <a class="txt" href="#E1:98_4_2_2_8_1"><span class="lab"><font color="Green" title="E186">A180</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_2_8_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:98_4_2_2_8_1_1_1"/>
<font color="Maroon" title="c34">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">PK</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">PK</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>) )</span><br/><div class="add"><a NAME="E2:98_4_2_2_8_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E188">A182</font></span>: 
<font color="Maroon" title="c34">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c35">PK</font> = <font color="Maroon" title="c19">PK</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c35">PK</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c34">v2</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c35">PK</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>) )</span><br/><span class="kw">thus </span><a NAME="E3:98_4_2_2_8_1_1_1"/>
<font color="Maroon" title="c35">PK</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c34">v2</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:98"><span class="lab"><font color="Green" title="E104">A23</font></span></a>, <a class="txt" href="#E34:98"><span class="lab"><font color="Green" title="E111">A30</font></span></a>, <a class="txt" href="#E2:98_4_2_2_8_1_1_1"><span class="lab"><font color="Green" title="E188">A182</font></span></a>, <a class="ref" href="graphsp.html#T8" target="_self" title="GRAPHSP:th.8">Th8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c35">PK</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>)</span><br/><span class="kw">thus </span><a NAME="E4:98_4_2_2_8_1_1_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c35">PK</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:98_4_2_2"><span class="lab"><font color="Green" title="E185">A179</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_2_2_8_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:98_4_2_2_8_1_1_2"/><span class="lab"><font color="Green" title="E188">A183</font></span>: 
<font color="Maroon" title="c34">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">P</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c35">P</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font><span class="kw"> such that </span><br/><a NAME="E3:98_4_2_2_8_1_1_2"/><span class="lab"><font color="Green" title="E189">A184</font></span>: 
<font color="Maroon" title="c35">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c34">v2</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font>
 <span class="kw">and </span><br/><a NAME="E4:98_4_2_2_8_1_1_2"/><span class="lab"><font color="Green" title="E190">A185</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c35">P</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c19">PK</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E5:98"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="txt" href="#E6:98"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="txt" href="#E36:98"><span class="lab"><font color="Green" title="E113">A32</font></span></a>, <a class="txt" href="#E2:98_4_2_2_8_1"><span class="lab"><font color="Green" title="E187">A181</font></span></a>, <a class="ref" href="graphsp.html#T39" target="_self" title="GRAPHSP:th.39">Th40</a>, <a class="ref" href="graph_5.html#D19" title="GRAPH_5:def.19">GRAPH_5:def 19</a></span>;<br/><div><span class="lab"><font color="Green" title="E191">A186</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="98_4_2_2_8_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c36">R</font> be   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v4</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font>  st  not <font color="Olive" title="b1">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c36">R</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Olive" title="b1">v4</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c35">P</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c36">R</font>,<font color="Maroon" title="c7">W</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c37">v4</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c37">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c36">R</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c37">v4</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> implies  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c35">P</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c36">R</font>,<font color="Maroon" title="c7">W</font>) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:98_4_2_2_8_1_1_2_1"/><span class="lab"><font color="Green" title="E192">A187</font></span>: 
 not <font color="Maroon" title="c37">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">and </span><br/><a NAME="E2:98_4_2_2_8_1_1_2_1"/><span class="lab"><font color="Green" title="E193">A188</font></span>: 
<font color="Maroon" title="c36">R</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c37">v4</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c35">P</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c36">R</font>,<font color="Maroon" title="c7">W</font>)</span><br/><a NAME="E3:98_4_2_2_8_1_1_2_1"/><span class="lab"><font color="Green" title="E194">A189</font></span>: 
<font color="Maroon" title="c37">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">VG</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c38">j4</font> = <font color="Maroon" title="c37">v4</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E5:98_4_2_2_8_1_1_2_1"/><span class="lab"><font color="Green" title="E195">A190</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">j4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:98"><span class="lab"><font color="Green" title="E97">A16</font></span></a>, <a class="txt" href="#E3:98_4_2_2_8_1_1_2_1"><span class="lab"><font color="Green" title="E194">A189</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E6:98_4_2_2_8_1_1_2_1"/>
<font color="Maroon" title="c38">j4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:98"><span class="lab"><font color="Green" title="E97">A16</font></span></a>, <a class="txt" href="#E3:98_4_2_2_8_1_1_2_1"><span class="lab"><font color="Green" title="E194">A189</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E7:98_4_2_2_8_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c38">j4</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E7:98"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="txt" href="#E9:98"><span class="lab"><font color="Green" title="E90">A9</font></span></a>, <a class="txt" href="#E19:98"><span class="lab"><font color="Green" title="E98">A17</font></span></a>, <a class="txt" href="#E1:98_4_2_2_8_1_1_2_1"><span class="lab"><font color="Green" title="E192">A187</font></span></a>, <a class="txt" href="#E2:98_4_2_2_8_1_1_2_1"><span class="lab"><font color="Green" title="E193">A188</font></span></a>, <a class="txt" href="#E5:98_4_2_2_8_1_1_2_1"><span class="lab"><font color="Green" title="E195">A190</font></span></a>, <a class="txt" href="graphsp.html#E91"><span class="lab"><font color="Green" title="E80">Lm19</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c39">rn</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c40">RR</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font><span class="kw"> such that </span><br/><a NAME="E9:98_4_2_2_8_1_1_2_1"/>
<font color="Maroon" title="c39">rn</font> <a href="graphsp.html#R4" title="GRAPHSP:pred.4">is_simple_vertex_seq_at</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c38">j4</font>,<font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E10:98_4_2_2_8_1_1_2_1"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">rn</font> holds <br/><font color="Maroon" title="c39">rn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">and </span><br/><a NAME="E11:98_4_2_2_8_1_1_2_1"/>
<font color="Maroon" title="c40">RR</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c39">rn</font>
 <span class="kw">and </span><br/><a NAME="E12:98_4_2_2_8_1_1_2_1"/><span class="lab"><font color="Green" title="E196">A191</font></span>: 
<font color="Maroon" title="c40">RR</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c37">v4</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font>
 <span class="kw">and </span><br/><a NAME="E13:98_4_2_2_8_1_1_2_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c40">RR</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c38">j4</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E14:98_4_2_2_8_1_1_2_1"/><span class="lab"><font color="Green" title="E197">A192</font></span>: 
(  not <font color="Maroon" title="c37">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Maroon" title="c40">RR</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E7:98"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="txt" href="#E8:98"><span class="lab"><font color="Green" title="E89">A8</font></span></a>, <a class="txt" href="#E1:98_4_2_2_8_1_1_2_1"><span class="lab"><font color="Green" title="E192">A187</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c41">PP</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c6">G</font><span class="kw"> such that </span><br/><a NAME="E16:98_4_2_2_8_1_1_2_1"/><span class="lab"><font color="Green" title="E198">A193</font></span>: 
<font color="Maroon" title="c41">PP</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c34">v2</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font>
 <span class="kw">and </span><br/><a NAME="E17:98_4_2_2_8_1_1_2_1"/><span class="lab"><font color="Green" title="E199">A194</font></span>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c41">PP</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c40">RR</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_4_2_2_8_1"><span class="lab"><font color="Green" title="E187">A181</font></span></a>, <a class="txt" href="#E1:98_4_2_2_8_1_1_2"><span class="lab"><font color="Green" title="E188">A183</font></span></a>, <a class="txt" href="#E1:98_4_2_2_8_1_1_2_1"><span class="lab"><font color="Green" title="E192">A187</font></span></a>, <a class="txt" href="#E14:98_4_2_2_8_1_1_2_1"><span class="lab"><font color="Green" title="E197">A192</font></span></a>, <a class="ref" href="graph_5.html#D19" title="GRAPH_5:def.19">GRAPH_5:def 19</a></span>;<br/><a NAME="E18:98_4_2_2_8_1_1_2_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c41">PP</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c35">P</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98_4_2_2_8_1_1_2"><span class="lab"><font color="Green" title="E189">A184</font></span></a>, <a class="txt" href="#E16:98_4_2_2_8_1_1_2_1"><span class="lab"><font color="Green" title="E198">A193</font></span></a>, <a class="ref" href="graphsp.html#T9" target="_self" title="GRAPHSP:th.9">Th9</a></span>;<br/><span class="kw">hence </span><a NAME="E19:98_4_2_2_8_1_1_2_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c35">P</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c36">R</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_4_2_2_8_1_1_2_1"><span class="lab"><font color="Green" title="E193">A188</font></span></a>, <a class="txt" href="#E12:98_4_2_2_8_1_1_2_1"><span class="lab"><font color="Green" title="E196">A191</font></span></a>, <a class="txt" href="#E17:98_4_2_2_8_1_1_2_1"><span class="lab"><font color="Green" title="E199">A194</font></span></a>, <a class="ref" href="graphsp.html#T9" target="_self" title="GRAPHSP:th.9">Th9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">take </span><font color="Maroon" title="c36">P</font> = <font color="Maroon" title="c35">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c36">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c34">v2</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font> &amp;  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c36">P</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>) )</span><br/><span class="kw">thus </span><a NAME="E6:98_4_2_2_8_1_1_2"/>
<font color="Maroon" title="c36">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c34">v2</font>, <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c7">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:98"><span class="lab"><font color="Green" title="E98">A17</font></span></a>, <a class="txt" href="#E26:98"><span class="lab"><font color="Green" title="E105">A24</font></span></a>, <a class="txt" href="#E2:98_4_2_2_8_1"><span class="lab"><font color="Green" title="E187">A181</font></span></a>, <a class="txt" href="#E3:98_4_2_2_8_1_1_2"><span class="lab"><font color="Green" title="E189">A184</font></span></a>, <a class="txt" href="#E5:98_4_2_2_8_1_1_2"><span class="lab"><font color="Green" title="E191">A186</font></span></a>, <a class="ref" href="graph_5.html#T64" title="GRAPH_5:th.64">GRAPH_5:64</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c36">P</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>)</span><br/><span class="kw">thus </span><a NAME="E7:98_4_2_2_8_1_1_2"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c36">P</font>,<font color="Maroon" title="c7">W</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> (<font color="Maroon" title="c31">Q</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:98_4_2_2"><span class="lab"><font color="Green" title="E185">A179</font></span></a>, <a class="txt" href="#E4:98_4_2_2_8_1_1_2"><span class="lab"><font color="Green" title="E190">A185</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:98_4_2_2_8"/>
<font color="Maroon" title="c31">Q</font> <a href="graph_5.html#R9" title="GRAPH_5:pred.9">islongestInShortestpath</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>),<font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c7">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#D19" title="GRAPH_5:def.19">GRAPH_5:def 19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) holds <br/><font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c22">m</font>, <font color="Maroon" title="c23">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">j</font> &amp; <font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c22">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:98_5"/><span class="lab"><font color="Green" title="E132">A195</font></span>: 
<font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">and </span><br/><a NAME="E2:98_5"/><span class="lab"><font color="Green" title="E133">A196</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">and </span><br/><a NAME="E3:98_5"/><span class="lab"><font color="Green" title="E134">A197</font></span>: 
<font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E4:98_5"/><span class="lab"><font color="Green" title="E135">A198</font></span>: 
<font color="Maroon" title="c22">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span><br/><span class="kw">set </span><font color="Maroon" title="c24">nj</font> = <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font>;<br/><a NAME="E5:98_5"/><span class="lab"><font color="Green" title="E136">A199</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_5"><span class="lab"><font color="Green" title="E133">A196</font></span></a>, <a class="txt" href="#E3:98_5"><span class="lab"><font color="Green" title="E134">A197</font></span></a>, <a class="txt" href="graphsp.html#E82"><span class="lab"><font color="Green" title="E72">Lm12</font></span></a></span>;<br/><a NAME="E6:98_5"/><span class="lab"><font color="Green" title="E137">A200</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_5"><span class="lab"><font color="Green" title="E133">A196</font></span></a>, <a class="txt" href="#E3:98_5"><span class="lab"><font color="Green" title="E134">A197</font></span></a>, <a class="txt" href="graphsp.html#E82"><span class="lab"><font color="Green" title="E72">Lm12</font></span></a></span>;<br/><a NAME="E7:98_5"/><span class="lab"><font color="Green" title="E138">A201</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_5"><span class="lab"><font color="Green" title="E133">A196</font></span></a>, <a class="txt" href="#E3:98_5"><span class="lab"><font color="Green" title="E134">A197</font></span></a>, <a class="txt" href="graphsp.html#E82"><span class="lab"><font color="Green" title="E72">Lm12</font></span></a></span>;<br/><a NAME="E8:98_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E139">A202</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:98"><span class="lab"><font color="Green" title="E93">A12</font></span></a>, <a class="txt" href="#E21:98"><span class="lab"><font color="Green" title="E100">A19</font></span></a>, <a class="txt" href="#E5:98_5"><span class="lab"><font color="Green" title="E136">A199</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E9:98_5"/>
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_5"><span class="lab"><font color="Green" title="E133">A196</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><a NAME="E10:98_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A203</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><div><span class="lab"><font color="Green" title="E141">A204</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="98_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:98_5_1"/>
<span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:98_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:98"><span class="lab"><font color="Green" title="E95">A14</font></span></a>, <a class="txt" href="#E16:98"><span class="lab"><font color="Green" title="E96">A15</font></span></a>, <a class="txt" href="#E48:98"><span class="lab"><font color="Green" title="E123">A42</font></span></a>, <a class="txt" href="#E6:98_5"><span class="lab"><font color="Green" title="E137">A200</font></span></a>, <a class="txt" href="#E8:98_5"><span class="lab"><font color="Green" title="E139">A202</font></span></a>, <a class="txt" href="#E10:98_5"><span class="lab"><font color="Green" title="E140">A203</font></span></a>, <a class="ref" href="graphsp.html#D14" target="_self" title="GRAPHSP:def.14">Def14</a></span>;<br/><span class="kw">hence </span><a NAME="E3:98_5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98_5"><span class="lab"><font color="Green" title="E132">A195</font></span></a>, <a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E12:98_5"/><span class="lab"><font color="Green" title="E142">A205</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98"><span class="lab"><font color="Green" title="E94">A13</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="#E7:98_5"><span class="lab"><font color="Green" title="E138">A201</font></span></a>, <a class="txt" href="#E10:98_5"><span class="lab"><font color="Green" title="E140">A203</font></span></a>, <a class="ref" href="graphsp.html#T18" target="_self" title="GRAPHSP:th.18">Th19</a></span>;<br/><a NAME="E13:98_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A206</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:98"><span class="lab"><font color="Green" title="E95">A14</font></span></a>, <a class="txt" href="#E16:98"><span class="lab"><font color="Green" title="E96">A15</font></span></a>, <a class="txt" href="#E1:98_5"><span class="lab"><font color="Green" title="E132">A195</font></span></a>, <a class="txt" href="#E6:98_5"><span class="lab"><font color="Green" title="E137">A200</font></span></a>, <a class="txt" href="#E8:98_5"><span class="lab"><font color="Green" title="E139">A202</font></span></a>, <a class="txt" href="#E10:98_5"><span class="lab"><font color="Green" title="E140">A203</font></span></a>, <a class="txt" href="#E11:98_5"><span class="lab"><font color="Green" title="E141">A204</font></span></a>, <a class="ref" href="graphsp.html#D14" target="_self" title="GRAPHSP:def.14">Def14</a></span>;<br/><a NAME="E14:98_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E144">A207</font></span>: 
 not <font color="Maroon" title="c23">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:98"><span class="lab"><font color="Green" title="E91">A10</font></span></a></span>;<br/><a NAME="E15:98_5"/>
<font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:98"><span class="lab"><font color="Green" title="E106">A25</font></span></a>, <a class="txt" href="#E3:98_5"><span class="lab"><font color="Green" title="E134">A197</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E16:98_5"/><span class="kw">then </span>
<font color="Maroon" title="c23">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:98"><span class="lab"><font color="Green" title="E93">A12</font></span></a>, <a class="txt" href="#E21:98"><span class="lab"><font color="Green" title="E100">A19</font></span></a>, <a class="txt" href="#E2:98_5"><span class="lab"><font color="Green" title="E133">A196</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E17:98_5"/><span class="kw">then </span>
<font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98_5"><span class="lab"><font color="Green" title="E133">A196</font></span></a>, <a class="txt" href="#E3:98_5"><span class="lab"><font color="Green" title="E134">A197</font></span></a>, <a class="txt" href="#E14:98_5"><span class="lab"><font color="Green" title="E144">A207</font></span></a></span>;<br/><a NAME="E18:98_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E145">A208</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98"><span class="lab"><font color="Green" title="E94">A13</font></span></a>, <a class="txt" href="#E24:98"><span class="lab"><font color="Green" title="E103">A22</font></span></a>, <a class="txt" href="#E27:98"><span class="lab"><font color="Green" title="E106">A25</font></span></a>, <a class="txt" href="#E3:98_5"><span class="lab"><font color="Green" title="E134">A197</font></span></a>, <a class="txt" href="#E13:98_5"><span class="lab"><font color="Green" title="E143">A206</font></span></a>, <a class="ref" href="graphsp.html#T18" target="_self" title="GRAPHSP:th.18">Th19</a></span>;<br/><a NAME="E19:98_5"/>
 not <span class="p1">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:98_5"><span class="lab"><font color="Green" title="E141">A204</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E20:98_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E146">A209</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:98_5"><span class="lab"><font color="Green" title="E142">A205</font></span></a>, <a class="txt" href="#E13:98_5"><span class="lab"><font color="Green" title="E143">A206</font></span></a>, <a class="txt" href="#E18:98_5"><span class="lab"><font color="Green" title="E145">A208</font></span></a>, <a class="ref" href="graphsp.html#D13" target="_self" title="GRAPHSP:def.13">Def13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c25">v3</font> = <font color="Maroon" title="c23">j</font> as   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c6">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:98"><span class="lab"><font color="Green" title="E97">A16</font></span></a>, <a class="txt" href="#E2:98_5"><span class="lab"><font color="Green" title="E133">A196</font></span></a>, <a class="txt" href="#E3:98_5"><span class="lab"><font color="Green" title="E134">A197</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c26">Akj</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font>;<br/><a NAME="E22:98_5"/><span class="lab"><font color="Green" title="E147">A210</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:98"><span class="lab"><font color="Green" title="E101">A20</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="#E3:98_5"><span class="lab"><font color="Green" title="E134">A197</font></span></a>, <a class="txt" href="graphsp.html#E83"><span class="lab"><font color="Green" title="E73">Lm13</font></span></a></span>;<br/><a NAME="E23:98_5"/><span class="lab"><font color="Green" title="E148">A211</font></span>: 
 <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:98"><span class="lab"><font color="Green" title="E101">A20</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="#E3:98_5"><span class="lab"><font color="Green" title="E134">A197</font></span></a>, <a class="txt" href="graphsp.html#E83"><span class="lab"><font color="Green" title="E73">Lm13</font></span></a></span>;<br/><a NAME="E24:98_5"/><span class="lab"><font color="Green" title="E149">A212</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:98"><span class="lab"><font color="Green" title="E101">A20</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="#E3:98_5"><span class="lab"><font color="Green" title="E134">A197</font></span></a>, <a class="txt" href="graphsp.html#E83"><span class="lab"><font color="Green" title="E73">Lm13</font></span></a></span>;<br/><a NAME="E25:98_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E150">A213</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:98"><span class="lab"><font color="Green" title="E93">A12</font></span></a>, <a class="txt" href="#E21:98"><span class="lab"><font color="Green" title="E100">A19</font></span></a>, <a class="txt" href="#E22:98_5"><span class="lab"><font color="Green" title="E147">A210</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E26:98_5"/><span class="lab"><font color="Green" title="E151">A214</font></span>: 
<span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:98"><span class="lab"><font color="Green" title="E101">A20</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="#E2:98_5"><span class="lab"><font color="Green" title="E133">A196</font></span></a>, <a class="txt" href="#E3:98_5"><span class="lab"><font color="Green" title="E134">A197</font></span></a>, <a class="txt" href="graphsp.html#E84"><span class="lab"><font color="Green" title="E74">Lm14</font></span></a></span>;<br/><a NAME="E27:98_5"/><span class="lab"><font color="Green" title="E152">A215</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p4">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:98"><span class="lab"><font color="Green" title="E101">A20</font></span></a>, <a class="txt" href="#E23:98"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="txt" href="#E2:98_5"><span class="lab"><font color="Green" title="E133">A196</font></span></a>, <a class="txt" href="#E3:98_5"><span class="lab"><font color="Green" title="E134">A197</font></span></a>, <a class="txt" href="graphsp.html#E84"><span class="lab"><font color="Green" title="E74">Lm14</font></span></a></span>;<br/><a NAME="E28:98_5"/><span class="lab"><font color="Green" title="E153">A216</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p5">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> (<font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c25">v3</font>,<font color="Maroon" title="c7">W</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="graphsp.html#D20" target="_self" title="GRAPHSP:def.20">Def20</a></span>;<br/><span class="lab"><font color="Green" title="E154">A217</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">j</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K16" title="GRAPHSP:func.16">findmin</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="graphsp.html#K6" title="GRAPHSP:func.6">.</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p5">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:98"><span class="lab"><font color="Green" title="E95">A14</font></span></a>, <a class="txt" href="#E48:98"><span class="lab"><font color="Green" title="E123">A42</font></span></a>, <a class="txt" href="#E25:98_5"><span class="lab"><font color="Green" title="E150">A213</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p5">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:98"><span class="lab"><font color="Green" title="E94">A13</font></span></a>, <a class="txt" href="#E23:98_5"><span class="lab"><font color="Green" title="E148">A211</font></span></a>, <a class="txt" href="#E24:98_5"><span class="lab"><font color="Green" title="E149">A212</font></span></a>, <a class="ref" href="graphsp.html#T18" target="_self" title="GRAPHSP:th.18">Th19</a></span>
<br/>.= 
 <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> (<font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c25">v3</font>,<font color="Maroon" title="c7">W</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:98"><span class="lab"><font color="Green" title="E100">A19</font></span></a>, <a class="txt" href="#E42:98"><span class="lab"><font color="Green" title="E119">A38</font></span></a>, <a class="txt" href="#E25:98_5"><span class="lab"><font color="Green" title="E150">A213</font></span></a>, <a class="txt" href="#E26:98_5"><span class="lab"><font color="Green" title="E151">A214</font></span></a>, <a class="txt" href="#E27:98_5"><span class="lab"><font color="Green" title="E152">A215</font></span></a>, <a class="txt" href="#E28:98_5"><span class="lab"><font color="Green" title="E153">A216</font></span></a>, <a class="ref" href="graphsp.html#D16" target="_self" title="GRAPHSP:def.16">Def16</a></span>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_5_3"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:98_5_3"/>
( <font color="Maroon" title="c22">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c22">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:98"><span class="lab"><font color="Green" title="E104">A23</font></span></a>, <a class="txt" href="#E4:98_5"><span class="lab"><font color="Green" title="E135">A198</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_5_3_1"><span class="kw">suppose </span></a><a NAME="E1:98_5_3_1"/>
<font color="Maroon" title="c22">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span><br/><div class="add"><a NAME="E2:98_5_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E155">A218</font></span>: 
<font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p1">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:98_5_3_1"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c6">G</font> or  not <font color="Olive" title="b1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c25">v3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:98_5"><span class="lab"><font color="Green" title="E146">A209</font></span></a>, <a class="txt" href="#E29:98_5"><span class="lab"><font color="Green" title="E154">A217</font></span></a>, <a class="ref" href="graphsp.html#T23" target="_self" title="GRAPHSP:th.23">Th24</a></span>;<br/><a NAME="E4:98_5_3_1"/><span class="kw">then </span>
 <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> (<font color="Maroon" title="c17">vk</font>,<font color="Maroon" title="c25">v3</font>,<font color="Maroon" title="c7">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graphsp.html#D7" target="_self" title="GRAPHSP:def.7">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E5:98_5_3_1"/>
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c22">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E2:98_5_3_1"><span class="lab"><font color="Green" title="E155">A218</font></span></a>, <a class="ref" href="graphsp.html#D20" target="_self" title="GRAPHSP:def.20">Def20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_5_3_2"><span class="kw">suppose </span></a><a NAME="E1:98_5_3_2"/>
<font color="Maroon" title="c22">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:98_5_3_2"/>
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c22">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:98"><span class="lab"><font color="Green" title="E90">A9</font></span></a>, <a class="txt" href="#E2:98_5"><span class="lab"><font color="Green" title="E133">A196</font></span></a>, <a class="txt" href="#E3:98_5"><span class="lab"><font color="Green" title="E134">A197</font></span></a>, <a class="txt" href="#E13:98_5"><span class="lab"><font color="Green" title="E143">A206</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c22">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>) implies <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E62:98"/><span class="lab"><font color="Green" title="E132">A219</font></span>: 
<font color="Maroon" title="c22">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c1">n</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_6"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:98_6"/>
( <font color="Maroon" title="c22">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>) or <font color="Maroon" title="c22">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:98"><span class="lab"><font color="Green" title="E104">A23</font></span></a>, <a class="txt" href="#E62:98"><span class="lab"><font color="Green" title="E132">A219</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_6_1"><span class="kw">suppose </span></a><a NAME="E1:98_6_1"/><span class="lab"><font color="Green" title="E133">A220</font></span>: 
<font color="Maroon" title="c22">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graphsp.html#K14" title="GRAPHSP:func.14">UsedVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span><br/><div class="add"><a NAME="E2:98_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E5:98"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="txt" href="#E6:98"><span class="lab"><font color="Green" title="E87">A6</font></span></a>, <a class="txt" href="#E12:98"><span class="lab"><font color="Green" title="E93">A12</font></span></a>, <a class="txt" href="graphsp.html#E88"><span class="lab"><font color="Green" title="E77">Lm16</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:98_6_1"/>
<font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:98"><span class="lab"><font color="Green" title="E91">A10</font></span></a>, <a class="txt" href="#E1:98_6_1"><span class="lab"><font color="Green" title="E133">A220</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_6_2"><span class="kw">suppose </span></a><a NAME="E1:98_6_2"/>
<font color="Maroon" title="c22">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="graphsp.html#K15" title="GRAPHSP:func.15">Argmin</a> (<span class="p3">(<span class="default"><a href="graphsp.html#K7" title="GRAPHSP:func.7">OuterVx</a> (<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:98_6_2"/>
<font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:98"><span class="lab"><font color="Green" title="E103">A22</font></span></a>, <a class="txt" href="#E59:98"><span class="lab"><font color="Green" title="E131">A52</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
