:: GRAPHSP semantic presentation

begin


theorem Th1: :: GRAPHSP:1
for p being FinSequence
for x being set holds
( ( not x in rng p & p is one-to-one ) iff p ^ <*x*> is one-to-one )
proof
let p be FinSequence; ::_thesis: for x being set holds
( ( not x in rng p & p is one-to-one ) iff p ^ <*x*> is one-to-one )
let x be set ; ::_thesis: ( ( not x in rng p & p is one-to-one ) iff p ^ <*x*> is one-to-one )
A1: rng <*x*> = {x} by FINSEQ_1:38;
 ( ( rng p misses rng <*x*> & p is one-to-one & <*x*> is one-to-one ) iff p ^ <*x*> is one-to-one ) by FINSEQ_3:91;
hence  ( ( not x in rng p & p is one-to-one ) iff p ^ <*x*> is one-to-one ) by A1, FINSEQ_3:93, ZFMISC_1:48, ZFMISC_1:50; ::_thesis: verum
end;

theorem Th2: :: GRAPHSP:2
for X being set
for p being FinSequence of X
for ii being Integer st 1 <= ii & ii <= len p holds
p . ii in X
proof
let X be set ; ::_thesis: for p being FinSequence of X
for ii being Integer st 1 <= ii & ii <= len p holds
p . ii in X
let p be FinSequence of X; ::_thesis: for ii being Integer st 1 <= ii & ii <= len p holds
p . ii in X
let ii be Integer; ::_thesis: ( 1 <= ii & ii <= len p implies p . ii in X )
assume that
A1: 1 <= ii and
A2: ii <= len p ; ::_thesis: p . ii in X
reconsider ii = ii as Element of NAT by A1, INT_1:3;
 ii in dom p by A1, A2, FINSEQ_3:25;
hence  p . ii in X by PARTFUN1:4; ::_thesis: verum
end;

theorem Th3: :: GRAPHSP:3
for X being set
for p being FinSequence of X
for ii being Integer st 1 <= ii & ii <= len p holds
p /. ii = p . ii
proof
let X be set ; ::_thesis: for p being FinSequence of X
for ii being Integer st 1 <= ii & ii <= len p holds
p /. ii = p . ii
let p be FinSequence of X; ::_thesis: for ii being Integer st 1 <= ii & ii <= len p holds
p /. ii = p . ii
let ii be Integer; ::_thesis: ( 1 <= ii & ii <= len p implies p /. ii = p . ii )
assume that
A1: 1 <= ii and
A2: ii <= len p ; ::_thesis: p /. ii = p . ii
reconsider ii = ii as Element of NAT by A1, INT_1:3;
 ii in dom p by A1, A2, FINSEQ_3:25;
hence  p /. ii = p . ii by PARTFUN1:def_6; ::_thesis: verum
end;


theorem Th4: :: GRAPHSP:4
for G being Graph
for pe being FinSequence of the carrier' of G
for W being Function st W is_weight_of G & len pe = 1 holds
cost (pe,W) = W . (pe . 1)
proof
let G be Graph; ::_thesis: for pe being FinSequence of the carrier' of G
for W being Function st W is_weight_of G & len pe = 1 holds
cost (pe,W) = W . (pe . 1)
let pe be FinSequence of the carrier' of G; ::_thesis: for W being Function st W is_weight_of G & len pe = 1 holds
cost (pe,W) = W . (pe . 1)
let W be Function; ::_thesis: ( W is_weight_of G & len pe = 1 implies cost (pe,W) = W . (pe . 1) )
assume that
A1: W is_weight_of G and
A2: len pe = 1 ; ::_thesis: cost (pe,W) = W . (pe . 1)
A3: 1 in dom pe by A2, FINSEQ_3:25;
set f = RealSequence (pe,W);
 dom (RealSequence (pe,W)) = dom pe by A1, GRAPH_5:def_15;
then  len (RealSequence (pe,W)) = 1 by A2, FINSEQ_3:29;
then A4: RealSequence (pe,W) = <*((RealSequence (pe,W)) . 1)*> by FINSEQ_1:40;
thus  cost (pe,W) = Sum (RealSequence (pe,W)) by GRAPH_5:def_16
.= (RealSequence (pe,W)) . 1 by A4, FINSOP_1:11
.= W . (pe . 1) by A1, A3, GRAPH_5:def_15 ; ::_thesis: verum
end;

theorem Th5: :: GRAPHSP:5
for G being Graph
for e being set st e in the carrier' of G holds
<*e*> is oriented Simple Chain of G
proof
let G be Graph; ::_thesis: for e being set st e in the carrier' of G holds
<*e*> is oriented Simple Chain of G
let e be set ; ::_thesis: ( e in the carrier' of G implies <*e*> is oriented Simple Chain of G )
assume  e in the carrier' of G ; ::_thesis: <*e*> is oriented Simple Chain of G
then A1: <*e*> is FinSequence of the carrier' of G by FINSEQ_1:74;
 len <*e*> = 1 by FINSEQ_1:40;
hence  <*e*> is oriented Simple Chain of G by A1, GRAPH_5:15; ::_thesis: verum
end;

Lm1:  for n being Nat
for p, q being FinSequence st 1 <= n & n <= len p holds
p . n = (p ^ q) . n
proof
let n be Nat; ::_thesis: for p, q being FinSequence st 1 <= n & n <= len p holds
p . n = (p ^ q) . n
let p, q be FinSequence; ::_thesis: ( 1 <= n & n <= len p implies p . n = (p ^ q) . n )
assume  ( 1 <= n & n <= len p ) ; ::_thesis: p . n = (p ^ q) . n
then  n in dom p by FINSEQ_3:25;
hence  p . n = (p ^ q) . n by FINSEQ_1:def_7; ::_thesis: verum
end;

Lm2:  for n being Element of NAT
for p, q being FinSequence st 1 <= n & n <= len q holds
q . n = (p ^ q) . ((len p) + n)
proof
let n be Element of NAT ; ::_thesis: for p, q being FinSequence st 1 <= n & n <= len q holds
q . n = (p ^ q) . ((len p) + n)
let p, q be FinSequence; ::_thesis: ( 1 <= n & n <= len q implies q . n = (p ^ q) . ((len p) + n) )
assume  ( 1 <= n & n <= len q ) ; ::_thesis: q . n = (p ^ q) . ((len p) + n)
then  n in dom q by FINSEQ_3:25;
hence  q . n = (p ^ q) . ((len p) + n) by FINSEQ_1:def_7; ::_thesis: verum
end;

theorem Th6: :: GRAPHSP:6
for G being Graph
for pe, qe being FinSequence of the carrier' of G
for p being oriented Simple Chain of G st p = pe ^ qe & len pe >= 1 & len qe >= 1 holds
( the Target of G . (p . (len p)) <> the Target of G . (pe . (len pe)) & the Source of G . (p . 1) <> the Source of G . (qe . 1) )
proof
let G be Graph; ::_thesis: for pe, qe being FinSequence of the carrier' of G
for p being oriented Simple Chain of G st p = pe ^ qe & len pe >= 1 & len qe >= 1 holds
( the Target of G . (p . (len p)) <> the Target of G . (pe . (len pe)) & the Source of G . (p . 1) <> the Source of G . (qe . 1) )
let pe, qe be FinSequence of the carrier' of G; ::_thesis: for p being oriented Simple Chain of G st p = pe ^ qe & len pe >= 1 & len qe >= 1 holds
( the Target of G . (p . (len p)) <> the Target of G . (pe . (len pe)) & the Source of G . (p . 1) <> the Source of G . (qe . 1) )
let p be oriented Simple Chain of G; ::_thesis: ( p = pe ^ qe & len pe >= 1 & len qe >= 1 implies ( the Target of G . (p . (len p)) <> the Target of G . (pe . (len pe)) & the Source of G . (p . 1) <> the Source of G . (qe . 1) ) )
set FT = the Target of G;
set FS = the Source of G;
assume that
A1: p = pe ^ qe and
A2: len pe >= 1 and
A3: len qe >= 1 ; ::_thesis: ( the Target of G . (p . (len p)) <> the Target of G . (pe . (len pe)) & the Source of G . (p . 1) <> the Source of G . (qe . 1) )
consider vs being FinSequence of the carrier of G such that
A4: vs is_oriented_vertex_seq_of p and
A5: for n, m being Element of NAT st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs ) by GRAPH_4:def_7;
A6: len vs = (len p) + 1 by A4, GRAPH_4:def_5;
then A7: 1 <= len vs by NAT_1:12;
 len p = (len pe) + (len qe) by A1, FINSEQ_1:22;
then A8: len p >= (len pe) + 1 by A3, XREAL_1:7;
then A9: (len pe) + 1 < len vs by A6, NAT_1:13;
A10: len p > len pe by A8, NAT_1:13;
then A11: len p >= 1 by A2, XXREAL_0:2;
then  p . 1 orientedly_joins vs /. 1,vs /. (1 + 1) by A4, GRAPH_4:def_5;
then A12: the Source of G . (p . 1) = vs /. 1 by GRAPH_4:def_1
.= vs . 1 by A7, FINSEQ_4:15 ;
A13: p . (len pe) orientedly_joins vs /. (len pe),vs /. ((len pe) + 1) by A2, A4, A10, GRAPH_4:def_5;
 p . (len p) orientedly_joins vs /. (len p),vs /. ((len p) + 1) by A4, A11, GRAPH_4:def_5;
then A14: the Target of G . (p . (len p)) = vs /. ((len p) + 1) by GRAPH_4:def_1
.= vs . (len vs) by A6, A7, FINSEQ_4:15 ;
A15: 1 < (len pe) + 1 by A2, NAT_1:13;
then A16: p . ((len pe) + 1) orientedly_joins vs /. ((len pe) + 1),vs /. (((len pe) + 1) + 1) by A4, A8, GRAPH_4:def_5;
the Target of G . (pe . (len pe)) = the Target of G . (p . (len pe)) by A1, A2, Lm1
.= vs /. ((len pe) + 1) by A13, GRAPH_4:def_1
.= vs . ((len pe) + 1) by A15, A9, FINSEQ_4:15 ;
hence  the Target of G . (p . (len p)) <> the Target of G . (pe . (len pe)) by A5, A14, A15, A9; ::_thesis: the Source of G . (p . 1) <> the Source of G . (qe . 1)
assume A17: the Source of G . (p . 1) = the Source of G . (qe . 1) ; ::_thesis: contradiction
the Source of G . (qe . 1) = the Source of G . (p . ((len pe) + 1)) by A1, A3, Lm2
.= vs /. ((len pe) + 1) by A16, GRAPH_4:def_1
.= vs . ((len pe) + 1) by A15, A9, FINSEQ_4:15 ;
hence  contradiction by A5, A15, A9, A12, A17; ::_thesis: verum
end;

begin

theorem Th7: :: GRAPHSP:7
for G being Graph
for p being oriented Chain of G
for V being set
for v1, v2 being Vertex of G holds
( p is_orientedpath_of v1,v2,V iff p is_orientedpath_of v1,v2,V \/ {v2} )
proof
let G be Graph; ::_thesis: for p being oriented Chain of G
for V being set
for v1, v2 being Vertex of G holds
( p is_orientedpath_of v1,v2,V iff p is_orientedpath_of v1,v2,V \/ {v2} )
let p be oriented Chain of G; ::_thesis: for V being set
for v1, v2 being Vertex of G holds
( p is_orientedpath_of v1,v2,V iff p is_orientedpath_of v1,v2,V \/ {v2} )
let V be set ; ::_thesis: for v1, v2 being Vertex of G holds
( p is_orientedpath_of v1,v2,V iff p is_orientedpath_of v1,v2,V \/ {v2} )
let v1, v2 be Vertex of G; ::_thesis: ( p is_orientedpath_of v1,v2,V iff p is_orientedpath_of v1,v2,V \/ {v2} )
set V9 = V \/ {v2};
thus  ( p is_orientedpath_of v1,v2,V implies p is_orientedpath_of v1,v2,V \/ {v2} ) by GRAPH_5:32, XBOOLE_1:7; ::_thesis: ( p is_orientedpath_of v1,v2,V \/ {v2} implies p is_orientedpath_of v1,v2,V )
assume A1: p is_orientedpath_of v1,v2,V \/ {v2} ; ::_thesis: p is_orientedpath_of v1,v2,V
then  (vertices p) \ {v2} c= V \/ {v2} by GRAPH_5:def_4;
then  ((vertices p) \ {v2}) \ {v2} c= V by XBOOLE_1:43;
then A2: (vertices p) \ ({v2} \/ {v2}) c= V by XBOOLE_1:41;
 p is_orientedpath_of v1,v2 by A1, GRAPH_5:def_4;
hence  p is_orientedpath_of v1,v2,V by A2, GRAPH_5:def_4; ::_thesis: verum
end;

theorem Th8: :: GRAPHSP:8
for G being Graph
for p being oriented Chain of G
for W being Function
for V being set
for v1, v2 being Vertex of G holds
( p is_shortestpath_of v1,v2,V,W iff p is_shortestpath_of v1,v2,V \/ {v2},W )
proof
let G be Graph; ::_thesis: for p being oriented Chain of G
for W being Function
for V being set
for v1, v2 being Vertex of G holds
( p is_shortestpath_of v1,v2,V,W iff p is_shortestpath_of v1,v2,V \/ {v2},W )
let p be oriented Chain of G; ::_thesis: for W being Function
for V being set
for v1, v2 being Vertex of G holds
( p is_shortestpath_of v1,v2,V,W iff p is_shortestpath_of v1,v2,V \/ {v2},W )
let W be Function; ::_thesis: for V being set
for v1, v2 being Vertex of G holds
( p is_shortestpath_of v1,v2,V,W iff p is_shortestpath_of v1,v2,V \/ {v2},W )
let V be set ; ::_thesis: for v1, v2 being Vertex of G holds
( p is_shortestpath_of v1,v2,V,W iff p is_shortestpath_of v1,v2,V \/ {v2},W )
let v1, v2 be Vertex of G; ::_thesis: ( p is_shortestpath_of v1,v2,V,W iff p is_shortestpath_of v1,v2,V \/ {v2},W )
set V9 = V \/ {v2};
hereby  ::_thesis: ( p is_shortestpath_of v1,v2,V \/ {v2},W implies p is_shortestpath_of v1,v2,V,W )
assume A1: p is_shortestpath_of v1,v2,V,W ; ::_thesis: p is_shortestpath_of v1,v2,V \/ {v2},W
A2: now__::_thesis:_for_q_being_oriented_Chain_of_G_st_q_is_orientedpath_of_v1,v2,V_\/_{v2}_holds_
cost_(p,W)_<=_cost_(q,W)
let q be oriented Chain of G; ::_thesis: ( q is_orientedpath_of v1,v2,V \/ {v2} implies cost (p,W) <= cost (q,W) )
assume  q is_orientedpath_of v1,v2,V \/ {v2} ; ::_thesis: cost (p,W) <= cost (q,W)
then  q is_orientedpath_of v1,v2,V by Th7;
hence  cost (p,W) <= cost (q,W) by A1, GRAPH_5:def_18; ::_thesis: verum
end;
 p is_orientedpath_of v1,v2,V by A1, GRAPH_5:def_18;
then  p is_orientedpath_of v1,v2,V \/ {v2} by Th7;
hence  p is_shortestpath_of v1,v2,V \/ {v2},W by A2, GRAPH_5:def_18; ::_thesis: verum
end;
assume A3: p is_shortestpath_of v1,v2,V \/ {v2},W ; ::_thesis: p is_shortestpath_of v1,v2,V,W
A4: now__::_thesis:_for_q_being_oriented_Chain_of_G_st_q_is_orientedpath_of_v1,v2,V_holds_
cost_(p,W)_<=_cost_(q,W)
let q be oriented Chain of G; ::_thesis: ( q is_orientedpath_of v1,v2,V implies cost (p,W) <= cost (q,W) )
assume  q is_orientedpath_of v1,v2,V ; ::_thesis: cost (p,W) <= cost (q,W)
then  q is_orientedpath_of v1,v2,V \/ {v2} by Th7;
hence  cost (p,W) <= cost (q,W) by A3, GRAPH_5:def_18; ::_thesis: verum
end;
 p is_orientedpath_of v1,v2,V \/ {v2} by A3, GRAPH_5:def_18;
then  p is_orientedpath_of v1,v2,V by Th7;
hence  p is_shortestpath_of v1,v2,V,W by A4, GRAPH_5:def_18; ::_thesis: verum
end;

theorem Th9: :: GRAPHSP:9
for G being Graph
for p, q being oriented Chain of G
for W being Function
for V being set
for v1, v2 being Vertex of G st p is_shortestpath_of v1,v2,V,W & q is_shortestpath_of v1,v2,V,W holds
cost (p,W) = cost (q,W)
proof
let G be Graph; ::_thesis: for p, q being oriented Chain of G
for W being Function
for V being set
for v1, v2 being Vertex of G st p is_shortestpath_of v1,v2,V,W & q is_shortestpath_of v1,v2,V,W holds
cost (p,W) = cost (q,W)
let p, q be oriented Chain of G; ::_thesis: for W being Function
for V being set
for v1, v2 being Vertex of G st p is_shortestpath_of v1,v2,V,W & q is_shortestpath_of v1,v2,V,W holds
cost (p,W) = cost (q,W)
let W be Function; ::_thesis: for V being set
for v1, v2 being Vertex of G st p is_shortestpath_of v1,v2,V,W & q is_shortestpath_of v1,v2,V,W holds
cost (p,W) = cost (q,W)
let V be set ; ::_thesis: for v1, v2 being Vertex of G st p is_shortestpath_of v1,v2,V,W & q is_shortestpath_of v1,v2,V,W holds
cost (p,W) = cost (q,W)
let v1, v2 be Vertex of G; ::_thesis: ( p is_shortestpath_of v1,v2,V,W & q is_shortestpath_of v1,v2,V,W implies cost (p,W) = cost (q,W) )
assume that
A1: p is_shortestpath_of v1,v2,V,W and
A2: q is_shortestpath_of v1,v2,V,W ; ::_thesis: cost (p,W) = cost (q,W)
 q is_orientedpath_of v1,v2,V by A2, GRAPH_5:def_18;
then A3: cost (p,W) <= cost (q,W) by A1, GRAPH_5:def_18;
 p is_orientedpath_of v1,v2,V by A1, GRAPH_5:def_18;
then  cost (q,W) <= cost (p,W) by A2, GRAPH_5:def_18;
hence  cost (p,W) = cost (q,W) by A3, XXREAL_0:1; ::_thesis: verum
end;

theorem Th10: :: GRAPHSP:10
for G being oriented Graph
for v1, v2 being Vertex of G
for e1, e2 being set st e1 in the carrier' of G & e2 in the carrier' of G & e1 orientedly_joins v1,v2 & e2 orientedly_joins v1,v2 holds
e1 = e2
proof
let G be oriented Graph; ::_thesis: for v1, v2 being Vertex of G
for e1, e2 being set st e1 in the carrier' of G & e2 in the carrier' of G & e1 orientedly_joins v1,v2 & e2 orientedly_joins v1,v2 holds
e1 = e2
let v1, v2 be Vertex of G; ::_thesis: for e1, e2 being set st e1 in the carrier' of G & e2 in the carrier' of G & e1 orientedly_joins v1,v2 & e2 orientedly_joins v1,v2 holds
e1 = e2
let e1, e2 be set ; ::_thesis: ( e1 in the carrier' of G & e2 in the carrier' of G & e1 orientedly_joins v1,v2 & e2 orientedly_joins v1,v2 implies e1 = e2 )
assume that
A1: ( e1 in the carrier' of G & e2 in the carrier' of G ) and
A2: e1 orientedly_joins v1,v2 and
A3: e2 orientedly_joins v1,v2 ; ::_thesis: e1 = e2
A4: ( the Source of G . e2 = v1 & the Target of G . e2 = v2 ) by A3, GRAPH_4:def_1;
 ( the Source of G . e1 = v1 & the Target of G . e1 = v2 ) by A2, GRAPH_4:def_1;
hence  e1 = e2 by A1, A4, GRAPH_1:def_7; ::_thesis: verum
end;

Lm3:  for i being Element of NAT
for G being Graph
for pe being FinSequence of the carrier' of G st 1 <= i & i <= len pe holds
( the Source of G . (pe . i) in the carrier of G & the Target of G . (pe . i) in the carrier of G )
proof
let i be Element of NAT ; ::_thesis: for G being Graph
for pe being FinSequence of the carrier' of G st 1 <= i & i <= len pe holds
( the Source of G . (pe . i) in the carrier of G & the Target of G . (pe . i) in the carrier of G )
let G be Graph; ::_thesis: for pe being FinSequence of the carrier' of G st 1 <= i & i <= len pe holds
( the Source of G . (pe . i) in the carrier of G & the Target of G . (pe . i) in the carrier of G )
let pe be FinSequence of the carrier' of G; ::_thesis: ( 1 <= i & i <= len pe implies ( the Source of G . (pe . i) in the carrier of G & the Target of G . (pe . i) in the carrier of G ) )
assume  ( 1 <= i & i <= len pe ) ; ::_thesis: ( the Source of G . (pe . i) in the carrier of G & the Target of G . (pe . i) in the carrier of G )
then  i in dom pe by FINSEQ_3:25;
hence  ( the Source of G . (pe . i) in the carrier of G & the Target of G . (pe . i) in the carrier of G ) by GRAPH_5:8; ::_thesis: verum
end;

theorem Th11: :: GRAPHSP:11
for G being Graph
for U, V being set
for v1, v2 being Vertex of G st the carrier of G = U \/ V & v1 in U & v2 in V & ( for v3, v4 being Vertex of G st v3 in U & v4 in V holds
for e being set holds
( not e in the carrier' of G or not e orientedly_joins v3,v4 ) ) holds
for p being oriented Chain of G holds not p is_orientedpath_of v1,v2
proof
let G be Graph; ::_thesis: for U, V being set
for v1, v2 being Vertex of G st the carrier of G = U \/ V & v1 in U & v2 in V & ( for v3, v4 being Vertex of G st v3 in U & v4 in V holds
for e being set holds
( not e in the carrier' of G or not e orientedly_joins v3,v4 ) ) holds
for p being oriented Chain of G holds not p is_orientedpath_of v1,v2
let U, V be set ; ::_thesis: for v1, v2 being Vertex of G st the carrier of G = U \/ V & v1 in U & v2 in V & ( for v3, v4 being Vertex of G st v3 in U & v4 in V holds
for e being set holds
( not e in the carrier' of G or not e orientedly_joins v3,v4 ) ) holds
for p being oriented Chain of G holds not p is_orientedpath_of v1,v2
let v1, v2 be Vertex of G; ::_thesis: ( the carrier of G = U \/ V & v1 in U & v2 in V & ( for v3, v4 being Vertex of G st v3 in U & v4 in V holds
for e being set holds
( not e in the carrier' of G or not e orientedly_joins v3,v4 ) ) implies for p being oriented Chain of G holds not p is_orientedpath_of v1,v2 )
assume that
A1: the carrier of G = U \/ V and
A2: v1 in U and
A3: v2 in V and
A4: for v3, v4 being Vertex of G st v3 in U & v4 in V holds
for e being set holds
( not e in the carrier' of G or not e orientedly_joins v3,v4 ) ; ::_thesis: for p being oriented Chain of G holds not p is_orientedpath_of v1,v2
set FS = the Source of G;
set FT = the Target of G;
given p being oriented Chain of G such that A5: p is_orientedpath_of v1,v2 ; ::_thesis: contradiction
 p <> {} by A5, GRAPH_5:def_3;
then A6: len p >= 1 by FINSEQ_1:20;
defpred S1[ Nat] means ( $1 >= 1 & $1 <= len p & the Source of G . (p . $1) in U );
A7: for k being Nat st S1[k] holds
k <= len p ;
 the Source of G . (p . 1) = v1 by A5, GRAPH_5:def_3;
then A8: ex k being Nat st S1[k] by A2, A6;
consider k being Nat such that
A9: ( S1[k] & ( for n being Nat st S1[n] holds
n <= k ) ) from NAT_1:sch_6(A7, A8);
reconsider k = k as Element of NAT by ORDINAL1:def_12;
reconsider vx = the Source of G . (p . k) as Vertex of G by A9, Lm3;
A10: p . k in the carrier' of G by A9, Th2;
A11: the Target of G . (p . (len p)) = v2 by A5, GRAPH_5:def_3;
percases ( k = len p or k <> len p ) ;
suppose k = len p ; ::_thesis: contradiction
then  p . k orientedly_joins vx,v2 by A11, GRAPH_4:def_1;
hence  contradiction by A3, A4, A9, A10; ::_thesis: verum
end;
suppose k <> len p ; ::_thesis: contradiction
then A12: k < len p by A9, XXREAL_0:1;
A13: k < k + 1 by NAT_1:13;
A14: now__::_thesis:_not_the_Source_of_G_._(p_._(k_+_1))_in_U
assume A15: the Source of G . (p . (k + 1)) in U ; ::_thesis: contradiction
 ( k + 1 <= len p & 1 <= k + 1 ) by A12, INT_1:7, NAT_1:12;
hence  contradiction by A9, A13, A15; ::_thesis: verum
end;
reconsider vy = the Target of G . (p . k) as Vertex of G by A9, Lm3;
A16: p . k orientedly_joins vx,vy by GRAPH_4:def_1;
 the Source of G . (p . (k + 1)) = the Target of G . (p . k) by A9, A12, GRAPH_1:def_15;
then  vy in V by A1, A2, A14, XBOOLE_0:def_3;
hence  contradiction by A4, A9, A10, A16; ::_thesis: verum
end;
end;
end;

Lm4:  for i being Element of NAT
for G being Graph
for pe being FinSequence of the carrier' of G
for v1 being Vertex of G st 1 <= i & i <= len pe & ( v1 = the Source of G . (pe . i) or v1 = the Target of G . (pe . i) ) holds
v1 in vertices pe
proof
let i be Element of NAT ; ::_thesis: for G being Graph
for pe being FinSequence of the carrier' of G
for v1 being Vertex of G st 1 <= i & i <= len pe & ( v1 = the Source of G . (pe . i) or v1 = the Target of G . (pe . i) ) holds
v1 in vertices pe
let G be Graph; ::_thesis: for pe being FinSequence of the carrier' of G
for v1 being Vertex of G st 1 <= i & i <= len pe & ( v1 = the Source of G . (pe . i) or v1 = the Target of G . (pe . i) ) holds
v1 in vertices pe
let pe be FinSequence of the carrier' of G; ::_thesis: for v1 being Vertex of G st 1 <= i & i <= len pe & ( v1 = the Source of G . (pe . i) or v1 = the Target of G . (pe . i) ) holds
v1 in vertices pe
let v1 be Vertex of G; ::_thesis: ( 1 <= i & i <= len pe & ( v1 = the Source of G . (pe . i) or v1 = the Target of G . (pe . i) ) implies v1 in vertices pe )
assume that
A1: ( 1 <= i & i <= len pe ) and
A2: ( v1 = the Source of G . (pe . i) or v1 = the Target of G . (pe . i) ) ; ::_thesis: v1 in vertices pe
 i in dom pe by A1, FINSEQ_3:25;
hence  v1 in vertices pe by A2, GRAPH_5:24; ::_thesis: verum
end;

theorem Th12: :: GRAPHSP:12
for G being Graph
for p being oriented Chain of G
for U, V being set
for v1, v2 being Vertex of G st the carrier of G = U \/ V & v1 in U & ( for v3, v4 being Vertex of G st v3 in U & v4 in V holds
for e being set holds
( not e in the carrier' of G or not e orientedly_joins v3,v4 ) ) & p is_orientedpath_of v1,v2 holds
p is_orientedpath_of v1,v2,U
proof
let G be Graph; ::_thesis: for p being oriented Chain of G
for U, V being set
for v1, v2 being Vertex of G st the carrier of G = U \/ V & v1 in U & ( for v3, v4 being Vertex of G st v3 in U & v4 in V holds
for e being set holds
( not e in the carrier' of G or not e orientedly_joins v3,v4 ) ) & p is_orientedpath_of v1,v2 holds
p is_orientedpath_of v1,v2,U
let p be oriented Chain of G; ::_thesis: for U, V being set
for v1, v2 being Vertex of G st the carrier of G = U \/ V & v1 in U & ( for v3, v4 being Vertex of G st v3 in U & v4 in V holds
for e being set holds
( not e in the carrier' of G or not e orientedly_joins v3,v4 ) ) & p is_orientedpath_of v1,v2 holds
p is_orientedpath_of v1,v2,U
let U, V be set ; ::_thesis: for v1, v2 being Vertex of G st the carrier of G = U \/ V & v1 in U & ( for v3, v4 being Vertex of G st v3 in U & v4 in V holds
for e being set holds
( not e in the carrier' of G or not e orientedly_joins v3,v4 ) ) & p is_orientedpath_of v1,v2 holds
p is_orientedpath_of v1,v2,U
let v1, v2 be Vertex of G; ::_thesis: ( the carrier of G = U \/ V & v1 in U & ( for v3, v4 being Vertex of G st v3 in U & v4 in V holds
for e being set holds
( not e in the carrier' of G or not e orientedly_joins v3,v4 ) ) & p is_orientedpath_of v1,v2 implies p is_orientedpath_of v1,v2,U )
assume that
A1: the carrier of G = U \/ V and
A2: v1 in U and
A3: for v3, v4 being Vertex of G st v3 in U & v4 in V holds
for e being set holds
( not e in the carrier' of G or not e orientedly_joins v3,v4 ) and
A4: p is_orientedpath_of v1,v2 ; ::_thesis: p is_orientedpath_of v1,v2,U
set FS = the Source of G;
set FT = the Target of G;
A5: now__::_thesis:_vertices_p_c=_U
assume  not vertices p c= U ; ::_thesis: contradiction
then consider i being Element of NAT , q, r being FinSequence of the carrier' of G such that
A6: i + 1 <= len p and
A7: not vertices (p /. (i + 1)) c= U and
A8: len q = i and
A9: p = q ^ r and
A10: vertices q c= U by GRAPH_5:20;
A11: p . (i + 1) in the carrier' of G by A6, Th2, NAT_1:12;
 p /. (i + 1) = p . (i + 1) by A6, FINSEQ_4:15, NAT_1:12;
then A12: vertices (p /. (i + 1)) = {( the Source of G . (p . (i + 1))),( the Target of G . (p . (i + 1)))} by GRAPH_5:def_1;
A13: now__::_thesis:_(_the_Source_of_G_._(p_._(i_+_1))_in_U_implies_not_the_Target_of_G_._(p_._(i_+_1))_in_U_)
assume  ( the Source of G . (p . (i + 1)) in U & the Target of G . (p . (i + 1)) in U ) ; ::_thesis: contradiction
then  (vertices (p /. (i + 1))) \/ U = U by A12, ZFMISC_1:42;
hence  contradiction by A7, XBOOLE_1:7; ::_thesis: verum
end;
A14: 1 <= i + 1 by NAT_1:12;
then reconsider vy = the Target of G . (p . (i + 1)) as Vertex of G by A6, Lm3;
A15: ( vy in U or vy in V ) by A1, A2, XBOOLE_0:def_3;
percases ( i = 0 or i <> 0 ) ;
supposeA16: i = 0 ; ::_thesis: contradiction
then  the Source of G . (p . (i + 1)) = v1 by A4, GRAPH_5:def_3;
then  p . (i + 1) orientedly_joins v1,vy by GRAPH_4:def_1;
hence  contradiction by A2, A3, A4, A13, A11, A15, A16, GRAPH_5:def_3; ::_thesis: verum
end;
supposeA17: i <> 0 ; ::_thesis: contradiction
reconsider vx = the Source of G . (p . (i + 1)) as Vertex of G by A6, A14, Lm3;
hereby  ::_thesis: verum
percases ( vx in U or not vx in U ) ;
supposeA18: vx in U ; ::_thesis: contradiction
 p . (i + 1) orientedly_joins vx,vy by GRAPH_4:def_1;
hence  contradiction by A3, A6, A14, A13, A15, A18, Th2; ::_thesis: verum
end;
supposeA19: not vx in U ; ::_thesis: contradiction
A20: i < len p by A6, NAT_1:13;
A21: i >= 1 + 0 by A17, INT_1:7;
then reconsider vq = the Target of G . (q . i) as Vertex of G by A8, Lm3;
A22: vq in vertices q by A8, A21, Lm4;
the Target of G . (q . i) = the Target of G . (p . i) by A8, A9, A21, Lm1
.= the Source of G . (p . (i + 1)) by A21, A20, GRAPH_1:def_15 ;
hence  contradiction by A10, A19, A22; ::_thesis: verum
end;
end;
end;
end;
end;
end;
 (vertices p) \ {v2} c= vertices p by XBOOLE_1:36;
then  (vertices p) \ {v2} c= U by A5, XBOOLE_1:1;
hence  p is_orientedpath_of v1,v2,U by A4, GRAPH_5:def_4; ::_thesis: verum
end;

begin


theorem Th13: :: GRAPHSP:13
for W being Function
for V being set
for G being finite Graph
for P, Q being oriented Chain of G
for v1, v2, v3 being Vertex of G st W is_weight>=0of G & P is_shortestpath_of v1,v2,V,W & v1 <> v2 & v1 <> v3 & Q is_shortestpath_of v1,v3,V,W & ( for e being set holds
( not e in the carrier' of G or not e orientedly_joins v2,v3 ) ) & P islongestInShortestpath V,v1,W holds
Q is_shortestpath_of v1,v3,V \/ {v2},W
proof
let W be Function; ::_thesis: for V being set
for G being finite Graph
for P, Q being oriented Chain of G
for v1, v2, v3 being Vertex of G st W is_weight>=0of G & P is_shortestpath_of v1,v2,V,W & v1 <> v2 & v1 <> v3 & Q is_shortestpath_of v1,v3,V,W & ( for e being set holds
( not e in the carrier' of G or not e orientedly_joins v2,v3 ) ) & P islongestInShortestpath V,v1,W holds
Q is_shortestpath_of v1,v3,V \/ {v2},W
let V be set ; ::_thesis: for G being finite Graph
for P, Q being oriented Chain of G
for v1, v2, v3 being Vertex of G st W is_weight>=0of G & P is_shortestpath_of v1,v2,V,W & v1 <> v2 & v1 <> v3 & Q is_shortestpath_of v1,v3,V,W & ( for e being set holds
( not e in the carrier' of G or not e orientedly_joins v2,v3 ) ) & P islongestInShortestpath V,v1,W holds
Q is_shortestpath_of v1,v3,V \/ {v2},W
let G be finite Graph; ::_thesis: for P, Q being oriented Chain of G
for v1, v2, v3 being Vertex of G st W is_weight>=0of G & P is_shortestpath_of v1,v2,V,W & v1 <> v2 & v1 <> v3 & Q is_shortestpath_of v1,v3,V,W & ( for e being set holds
( not e in the carrier' of G or not e orientedly_joins v2,v3 ) ) & P islongestInShortestpath V,v1,W holds
Q is_shortestpath_of v1,v3,V \/ {v2},W
let P, Q be oriented Chain of G; ::_thesis: for v1, v2, v3 being Vertex of G st W is_weight>=0of G & P is_shortestpath_of v1,v2,V,W & v1 <> v2 & v1 <> v3 & Q is_shortestpath_of v1,v3,V,W & ( for e being set holds
( not e in the carrier' of G or not e orientedly_joins v2,v3 ) ) & P islongestInShortestpath V,v1,W holds
Q is_shortestpath_of v1,v3,V \/ {v2},W
let v1, v2, v3 be Vertex of G; ::_thesis: ( W is_weight>=0of G & P is_shortestpath_of v1,v2,V,W & v1 <> v2 & v1 <> v3 & Q is_shortestpath_of v1,v3,V,W & ( for e being set holds
( not e in the carrier' of G or not e orientedly_joins v2,v3 ) ) & P islongestInShortestpath V,v1,W implies Q is_shortestpath_of v1,v3,V \/ {v2},W )
assume that
A1: W is_weight>=0of G and
A2: P is_shortestpath_of v1,v2,V,W and
A3: v1 <> v2 and
A4: v1 <> v3 and
A5: Q is_shortestpath_of v1,v3,V,W and
A6: for e being set holds
( not e in the carrier' of G or not e orientedly_joins v2,v3 ) and
A7: P islongestInShortestpath V,v1,W ; ::_thesis: Q is_shortestpath_of v1,v3,V \/ {v2},W
set V9 = V \/ {v2};
set FS = the Source of G;
set FT = the Target of G;
A8: now__::_thesis:_for_S_being_oriented_Chain_of_G_st_S_is_orientedpath_of_v1,v3,V_\/_{v2}_holds_
cost_(Q,W)_<=_cost_(S,W)
let S be oriented Chain of G; ::_thesis: ( S is_orientedpath_of v1,v3,V \/ {v2} implies cost (Q,W) <= cost (b1,W) )
assume A9: S is_orientedpath_of v1,v3,V \/ {v2} ; ::_thesis: cost (Q,W) <= cost (b1,W)
then consider s being oriented Simple Chain of G such that
A10: s is_shortestpath_of v1,v3,V \/ {v2},W by A1, GRAPH_5:62;
A11: cost (s,W) <= cost (S,W) by A9, A10, GRAPH_5:def_18;
A12: s is_orientedpath_of v1,v3,V \/ {v2} by A10, GRAPH_5:def_18;
then A13: s is_orientedpath_of v1,v3 by GRAPH_5:def_4;
then A14: the Source of G . (s . 1) = v1 by GRAPH_5:def_3;
 s <> {} by A13, GRAPH_5:def_3;
then  len s >= 1 by FINSEQ_1:20;
then consider i being Nat such that
A15: len s = 1 + i by NAT_1:10;
A16: (vertices s) \ {v3} c= V \/ {v2} by A12, GRAPH_5:def_4;
A17: the Target of G . (s . (len s)) = v3 by A13, GRAPH_5:def_3;
consider s1, s2 being FinSequence such that
A18: len s1 = i and
A19: len s2 = 1 and
A20: s = s1 ^ s2 by A15, FINSEQ_2:22;
reconsider s1 = s1, s2 = s2 as oriented Simple Chain of G by A20, GRAPH_5:14;
reconsider vx = the Source of G . (s2 . 1) as Vertex of G by A19, Lm3;
A21: s2 . 1 = s . (len s) by A15, A18, A19, A20, Lm2;
A22: now__::_thesis:_not_vx_=_v2
assume  vx = v2 ; ::_thesis: contradiction
then A23: s2 . 1 orientedly_joins v2,v3 by A21, A17, GRAPH_4:def_1;
 1 in dom s2 by A19, FINSEQ_3:25;
hence  contradiction by A6, A23, FINSEQ_2:11; ::_thesis: verum
end;
percases ( not v2 in vertices s or v2 = v3 or ( v2 in vertices s & v2 <> v3 ) ) ;
supposeA24: ( not v2 in vertices s or v2 = v3 ) ; ::_thesis: cost (Q,W) <= cost (b1,W)
set Vs = vertices s;
 ((vertices s) \ {v3}) \ {v2} c= V by A16, XBOOLE_1:43;
then A25: (vertices s) \ ({v3} \/ {v2}) c= V by XBOOLE_1:41;
now__::_thesis:_(vertices_s)_\_{v3}_c=_V
percases ( not v2 in vertices s or v2 = v3 ) by A24;
supposeA26: not v2 in vertices s ; ::_thesis: (vertices s) \ {v3} c= V
 ((vertices s) \ {v2}) \ {v3} c= V by A25, XBOOLE_1:41;
hence  (vertices s) \ {v3} c= V by A26, ZFMISC_1:57; ::_thesis: verum
end;
suppose v2 = v3 ; ::_thesis: (vertices s) \ {v3} c= V
hence  (vertices s) \ {v3} c= V by A25; ::_thesis: verum
end;
end;
end;
then  s is_orientedpath_of v1,v3,V by A13, GRAPH_5:def_4;
then  cost (Q,W) <= cost (s,W) by A5, GRAPH_5:def_18;
hence  cost (Q,W) <= cost (S,W) by A11, XXREAL_0:2; ::_thesis: verum
end;
supposeA27: ( v2 in vertices s & v2 <> v3 ) ; ::_thesis: cost (Q,W) <= cost (b1,W)
A28: len s1 < len s by A15, A18, NAT_1:13;
consider j being Element of NAT  such that
A29: 1 <= j and
A30: j <= len s and
A31: v2 = the Target of G . (s . j) by A3, A14, A27, GRAPH_5:28;
 len s1 <> 0 by A15, A18, A17, A27, A29, A30, A31, XXREAL_0:1;
then A32: len s1 >= 0 + 1 by INT_1:7;
vx = the Source of G . (s . ((len s1) + 1)) by A19, A20, Lm2
.= the Target of G . (s . (len s1)) by A32, A28, GRAPH_1:def_15 ;
then  vx = the Target of G . (s1 . (len s1)) by A20, A32, Lm1;
then A33: vx in vertices s1 by A32, Lm4;
 not vx in {v2} by A22, TARSKI:def_1;
then A34: vx in (vertices s1) \ {v2} by A33, XBOOLE_0:def_5;
A35: now__::_thesis:_not_j_>_len_s1
assume  j > len s1 ; ::_thesis: contradiction
then  j >= (len s1) + 1 by INT_1:7;
hence  contradiction by A15, A18, A17, A27, A30, A31, XXREAL_0:1; ::_thesis: verum
end;
then consider k being Nat such that
A36: len s1 = j + k by NAT_1:10;
consider t1, t2 being FinSequence such that
A37: len t1 = j and
 len t2 = k and
A38: s1 = t1 ^ t2 by A36, FINSEQ_2:22;
reconsider t1 = t1, t2 = t2 as oriented Simple Chain of G by A38, GRAPH_5:14;
A39: t1 <> {} by A29, A37;
set Vt = vertices t1;
 (vertices (s1 ^ s2)) \ {v3} c= V \/ {v2} by A12, A20, GRAPH_5:def_4;
then A40: (vertices (t1 ^ t2)) \ {v3} c= V \/ {v2} by A38, GRAPH_5:23;
then A41: (vertices t1) \ {v3} c= V \/ {v2} by GRAPH_5:23;
A42: len s2 >= 1 by A19;
then  not v3 in vertices s1 by A4, A20, A14, A17, A32, GRAPH_5:18;
then  vertices s1 c= V \/ {v2} by A38, A40, ZFMISC_1:57;
then A43: (vertices s1) \ {v2} c= V by XBOOLE_1:43;
 not v1 in vertices s2 by A4, A19, A20, A14, A17, A32, GRAPH_5:18;
then  vx <> v1 by A19, Lm4;
then consider q being oriented Chain of G such that
A44: q is_shortestpath_of v1,vx,V,W and
A45: cost (q,W) <= cost (P,W) by A7, A34, A43, GRAPH_5:def_19;
A46: 0 <= cost (t2,W) by A1, GRAPH_5:50;
 vertices t1 c= vertices (t1 ^ t2) by GRAPH_5:26;
then  not v3 in vertices t1 by A4, A20, A14, A17, A32, A38, A42, GRAPH_5:18;
then  vertices t1 c= V \/ {v2} by A41, ZFMISC_1:57;
then A47: (vertices t1) \ {v2} c= V by XBOOLE_1:43;
A48: the Source of G . (t1 . 1) = the Source of G . (s1 . 1) by A29, A37, A38, Lm1
.= v1 by A20, A14, A32, Lm1 ;
 cost (s1,W) = (cost (t1,W)) + (cost (t2,W)) by A1, A38, GRAPH_5:46, GRAPH_5:54;
then A49: 0 + (cost (t1,W)) <= cost (s1,W) by A46, XREAL_1:7;
the Target of G . (t1 . (len t1)) = the Target of G . (s1 . j) by A29, A37, A38, Lm1
.= v2 by A20, A29, A31, A35, Lm1 ;
then  t1 is_orientedpath_of v1,v2 by A48, A39, GRAPH_5:def_3;
then  t1 is_orientedpath_of v1,v2,V by A47, GRAPH_5:def_4;
then  cost (P,W) <= cost (t1,W) by A2, GRAPH_5:def_18;
then  cost (q,W) <= cost (t1,W) by A45, XXREAL_0:2;
then A50: cost (q,W) <= cost (s1,W) by A49, XXREAL_0:2;
A51: s2 . 1 orientedly_joins vx,v3 by A21, A17, GRAPH_4:def_1;
A52: q is_orientedpath_of v1,vx,V by A44, GRAPH_5:def_18;
then A53: q is_orientedpath_of v1,vx by GRAPH_5:def_4;
then  q <> {} by GRAPH_5:def_3;
then A54: len q >= 1 by FINSEQ_1:20;
then consider p being oriented Chain of G such that
A55: p = q ^ s2 and
 p is_orientedpath_of v1,v3 by A19, A51, A53, GRAPH_5:33;
 p is_orientedpath_of v1,v3,V \/ {vx} by A19, A52, A51, A54, A55, GRAPH_5:34;
then  p is_orientedpath_of v1,v3,V by A34, A43, ZFMISC_1:40;
then  cost (Q,W) <= cost (p,W) by A5, GRAPH_5:def_18;
then A56: cost (Q,W) <= (cost (q,W)) + (cost (s2,W)) by A1, A55, GRAPH_5:46, GRAPH_5:54;
 cost (s,W) = (cost (s1,W)) + (cost (s2,W)) by A1, A20, GRAPH_5:46, GRAPH_5:54;
then  (cost (q,W)) + (cost (s2,W)) <= cost (s,W) by A50, XREAL_1:7;
then  cost (Q,W) <= cost (s,W) by A56, XXREAL_0:2;
hence  cost (Q,W) <= cost (S,W) by A11, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
 Q is_orientedpath_of v1,v3,V by A5, GRAPH_5:def_18;
then  Q is_orientedpath_of v1,v3,V \/ {v2} by GRAPH_5:32, XBOOLE_1:7;
hence  Q is_shortestpath_of v1,v3,V \/ {v2},W by A8, GRAPH_5:def_18; ::_thesis: verum
end;


theorem Th14: :: GRAPHSP:14
for e being set
for G being oriented finite Graph
for P being oriented Chain of G
for W being Function of the carrier' of G,Real>=0
for v1, v2 being Vertex of G st e in the carrier' of G & P = <*e*> & e orientedly_joins v1,v2 holds
P is_shortestpath_of v1,v2,{v1},W
proof
let e be set ; ::_thesis: for G being oriented finite Graph
for P being oriented Chain of G
for W being Function of the carrier' of G,Real>=0
for v1, v2 being Vertex of G st e in the carrier' of G & P = <*e*> & e orientedly_joins v1,v2 holds
P is_shortestpath_of v1,v2,{v1},W
let G be oriented finite Graph; ::_thesis: for P being oriented Chain of G
for W being Function of the carrier' of G,Real>=0
for v1, v2 being Vertex of G st e in the carrier' of G & P = <*e*> & e orientedly_joins v1,v2 holds
P is_shortestpath_of v1,v2,{v1},W
let P be oriented Chain of G; ::_thesis: for W being Function of the carrier' of G,Real>=0
for v1, v2 being Vertex of G st e in the carrier' of G & P = <*e*> & e orientedly_joins v1,v2 holds
P is_shortestpath_of v1,v2,{v1},W
let W be Function of the carrier' of G,Real>=0; ::_thesis: for v1, v2 being Vertex of G st e in the carrier' of G & P = <*e*> & e orientedly_joins v1,v2 holds
P is_shortestpath_of v1,v2,{v1},W
let v1, v2 be Vertex of G; ::_thesis: ( e in the carrier' of G & P = <*e*> & e orientedly_joins v1,v2 implies P is_shortestpath_of v1,v2,{v1},W )
assume that
A1: e in the carrier' of G and
A2: P = <*e*> and
A3: e orientedly_joins v1,v2 ; ::_thesis: P is_shortestpath_of v1,v2,{v1},W
A4: len P = 1 by A2, FINSEQ_1:40;
then A5: vertices P = vertices (P /. 1) by GRAPH_5:25;
set FS = the Source of G;
set FT = the Target of G;
set Eg = the carrier' of G;
A6: ( the Source of G . e = v1 & the Target of G . e = v2 ) by A3, GRAPH_4:def_1;
A7: now__::_thesis:_for_S_being_oriented_Chain_of_G_st_S_is_orientedpath_of_v1,v2,{v1}_holds_
cost_(P,W)_<=_cost_(S,W)
let S be oriented Chain of G; ::_thesis: ( S is_orientedpath_of v1,v2,{v1} implies cost (P,W) <= cost (S,W) )
assume A8: S is_orientedpath_of v1,v2,{v1} ; ::_thesis: cost (P,W) <= cost (S,W)
 W is_weight>=0of G by GRAPH_5:def_13;
then consider s being oriented Simple Chain of G such that
A9: s is_shortestpath_of v1,v2,{v1},W by A8, GRAPH_5:62;
set Vs = vertices s;
A10: s is_orientedpath_of v1,v2,{v1} by A9, GRAPH_5:def_18;
then A11: s is_orientedpath_of v1,v2 by GRAPH_5:def_4;
then  s <> {} by GRAPH_5:def_3;
then A12: len s >= 1 by FINSEQ_1:20;
then consider i being Nat such that
A13: len s = 1 + i by NAT_1:10;
A14: the Target of G . (s . (len s)) = v2 by A11, GRAPH_5:def_3;
then  v2 in vertices s by A12, Lm4;
then A15: {v2} c= vertices s by ZFMISC_1:31;
A16: the Source of G . (s . 1) = v1 by A11, GRAPH_5:def_3;
then  v1 in vertices s by A12, Lm4;
then  {v1} c= vertices s by ZFMISC_1:31;
then  {v1} \/ {v2} c= vertices s by A15, XBOOLE_1:8;
then A17: {v1,v2} c= vertices s by ENUMSET1:1;
consider s1, s2 being FinSequence such that
A18: len s1 = i and
A19: len s2 = 1 and
A20: s = s1 ^ s2 by A13, FINSEQ_2:22;
reconsider s1 = s1, s2 = s2 as oriented Simple Chain of G by A20, GRAPH_5:14;
 (vertices s) \ {v2} c= {v1} by A10, GRAPH_5:def_4;
then  vertices s c= {v2} \/ {v1} by XBOOLE_1:44;
then A21: vertices s c= {v1,v2} by ENUMSET1:1;
now__::_thesis:_not_s1_<>_{}
reconsider vx = the Source of G . (s2 . 1) as Vertex of G by A19, Lm3;
A22: len s1 < len s by A13, A18, NAT_1:13;
A23: vx in vertices s2 by A19, Lm4;
 vertices s2 c= vertices (s1 ^ s2) by GRAPH_5:26;
then A24: vertices s2 c= {v1,v2} by A21, A17, A20, XBOOLE_0:def_10;
assume  s1 <> {} ; ::_thesis: contradiction
then A25: len s1 >= 1 by FINSEQ_1:20;
then A26: the Source of G . (s . 1) <> the Source of G . (s2 . 1) by A19, A20, Th6;
 len s2 = 1 by A19;
then A27: the Target of G . (s . (len s)) <> the Target of G . (s1 . (len s1)) by A20, A25, Th6;
the Source of G . (s2 . 1) = the Source of G . (s . ((len s1) + 1)) by A19, A20, Lm2
.= the Target of G . (s . (len s1)) by A25, A22, GRAPH_1:def_15
.= the Target of G . (s1 . (len s1)) by A20, A25, Lm1 ;
hence  contradiction by A16, A14, A27, A26, A24, A23, TARSKI:def_2; ::_thesis: verum
end;
then A28: len s = 0 + 1 by A13, A18;
 s /. 1 in the carrier' of G by A1;
then  s . 1 in the carrier' of G by A28, FINSEQ_4:15;
then  e = s . 1 by A1, A6, A16, A14, A28, GRAPH_1:def_7;
then  s = P by A2, A28, FINSEQ_1:40;
hence  cost (P,W) <= cost (S,W) by A8, A9, GRAPH_5:def_18; ::_thesis: verum
end;
A29: P . 1 = e by A2, FINSEQ_1:40;
then  P /. 1 = e by A4, FINSEQ_4:15;
then  vertices P = {v1,v2} by A6, A5, GRAPH_5:def_1;
then A30: (vertices P) \ {v2} = ({v1} \/ {v2}) \ {v2} by ENUMSET1:1
.= {v1} \ {v2} by XBOOLE_1:40 ;
 P is_orientedpath_of v1,v2 by A2, A6, A4, A29, GRAPH_5:def_3;
then  P is_orientedpath_of v1,v2,{v1} by A30, GRAPH_5:def_4;
hence  P is_shortestpath_of v1,v2,{v1},W by A7, GRAPH_5:def_18; ::_thesis: verum
end;

theorem Th15: :: GRAPHSP:15
for e, V being set
for G being oriented finite Graph
for P, Q being oriented Chain of G
for W being Function of the carrier' of G,Real>=0
for v1, v2, v3 being Vertex of G st e in the carrier' of G & P is_shortestpath_of v1,v2,V,W & v1 <> v3 & Q = P ^ <*e*> & e orientedly_joins v2,v3 & v1 in V & ( for v4 being Vertex of G st v4 in V holds
for ee being set holds
( not ee in the carrier' of G or not ee orientedly_joins v4,v3 ) ) holds
Q is_shortestpath_of v1,v3,V \/ {v2},W
proof
let e, V be set ; ::_thesis: for G being oriented finite Graph
for P, Q being oriented Chain of G
for W being Function of the carrier' of G,Real>=0
for v1, v2, v3 being Vertex of G st e in the carrier' of G & P is_shortestpath_of v1,v2,V,W & v1 <> v3 & Q = P ^ <*e*> & e orientedly_joins v2,v3 & v1 in V & ( for v4 being Vertex of G st v4 in V holds
for ee being set holds
( not ee in the carrier' of G or not ee orientedly_joins v4,v3 ) ) holds
Q is_shortestpath_of v1,v3,V \/ {v2},W
let G be oriented finite Graph; ::_thesis: for P, Q being oriented Chain of G
for W being Function of the carrier' of G,Real>=0
for v1, v2, v3 being Vertex of G st e in the carrier' of G & P is_shortestpath_of v1,v2,V,W & v1 <> v3 & Q = P ^ <*e*> & e orientedly_joins v2,v3 & v1 in V & ( for v4 being Vertex of G st v4 in V holds
for ee being set holds
( not ee in the carrier' of G or not ee orientedly_joins v4,v3 ) ) holds
Q is_shortestpath_of v1,v3,V \/ {v2},W
let P, Q be oriented Chain of G; ::_thesis: for W being Function of the carrier' of G,Real>=0
for v1, v2, v3 being Vertex of G st e in the carrier' of G & P is_shortestpath_of v1,v2,V,W & v1 <> v3 & Q = P ^ <*e*> & e orientedly_joins v2,v3 & v1 in V & ( for v4 being Vertex of G st v4 in V holds
for ee being set holds
( not ee in the carrier' of G or not ee orientedly_joins v4,v3 ) ) holds
Q is_shortestpath_of v1,v3,V \/ {v2},W
let W be Function of the carrier' of G,Real>=0; ::_thesis: for v1, v2, v3 being Vertex of G st e in the carrier' of G & P is_shortestpath_of v1,v2,V,W & v1 <> v3 & Q = P ^ <*e*> & e orientedly_joins v2,v3 & v1 in V & ( for v4 being Vertex of G st v4 in V holds
for ee being set holds
( not ee in the carrier' of G or not ee orientedly_joins v4,v3 ) ) holds
Q is_shortestpath_of v1,v3,V \/ {v2},W
let v1, v2, v3 be Vertex of G; ::_thesis: ( e in the carrier' of G & P is_shortestpath_of v1,v2,V,W & v1 <> v3 & Q = P ^ <*e*> & e orientedly_joins v2,v3 & v1 in V & ( for v4 being Vertex of G st v4 in V holds
for ee being set holds
( not ee in the carrier' of G or not ee orientedly_joins v4,v3 ) ) implies Q is_shortestpath_of v1,v3,V \/ {v2},W )
assume that
A1: e in the carrier' of G and
A2: P is_shortestpath_of v1,v2,V,W and
A3: v1 <> v3 and
A4: Q = P ^ <*e*> and
A5: e orientedly_joins v2,v3 and
A6: v1 in V and
A7: for v4 being Vertex of G st v4 in V holds
for ee being set holds
( not ee in the carrier' of G or not ee orientedly_joins v4,v3 ) ; ::_thesis: Q is_shortestpath_of v1,v3,V \/ {v2},W
set Eg = the carrier' of G;
reconsider pe = <*e*> as FinSequence of the carrier' of G by A1, FINSEQ_1:74;
A8: P is_orientedpath_of v1,v2,V by A2, GRAPH_5:def_18;
then  P is_orientedpath_of v1,v2 by GRAPH_5:def_4;
then  P <> {} by GRAPH_5:def_3;
then A9: len P >= 1 by FINSEQ_1:20;
set V9 = V \/ {v2};
set FS = the Source of G;
set FT = the Target of G;
A10: W is_weight>=0of G by GRAPH_5:def_13;
A11: now__::_thesis:_for_S_being_oriented_Chain_of_G_st_S_is_orientedpath_of_v1,v3,V_\/_{v2}_holds_
cost_(Q,W)_<=_cost_(S,W)
let S be oriented Chain of G; ::_thesis: ( S is_orientedpath_of v1,v3,V \/ {v2} implies cost (Q,W) <= cost (S,W) )
assume A12: S is_orientedpath_of v1,v3,V \/ {v2} ; ::_thesis: cost (Q,W) <= cost (S,W)
then consider s being oriented Simple Chain of G such that
A13: s is_shortestpath_of v1,v3,V \/ {v2},W by A10, GRAPH_5:62;
set Vs = vertices s;
A14: s is_orientedpath_of v1,v3,V \/ {v2} by A13, GRAPH_5:def_18;
then A15: s is_orientedpath_of v1,v3 by GRAPH_5:def_4;
then A16: the Target of G . (s . (len s)) = v3 by GRAPH_5:def_3;
A17: (vertices s) \ {v3} c= V \/ {v2} by A14, GRAPH_5:def_4;
 s <> {} by A15, GRAPH_5:def_3;
then A18: len s >= 1 by FINSEQ_1:20;
then consider i being Nat such that
A19: len s = 1 + i by NAT_1:10;
consider s1, s2 being FinSequence such that
A20: len s1 = i and
A21: len s2 = 1 and
A22: s = s1 ^ s2 by A19, FINSEQ_2:22;
reconsider s1 = s1, s2 = s2 as oriented Simple Chain of G by A22, GRAPH_5:14;
reconsider vx = the Source of G . (s2 . 1) as Vertex of G by A21, Lm3;
A23: s2 . 1 = s . (len s) by A19, A20, A21, A22, Lm2;
A24: the Source of G . (s . 1) = v1 by A15, GRAPH_5:def_3;
A25: s1 <> {}
proof
assume  s1 = {} ; ::_thesis: contradiction
then A26: len s = 1 + 0 by A19, A20;
then  s . 1 orientedly_joins v1,v3 by A24, A16, GRAPH_4:def_1;
hence  contradiction by A6, A7, A26, Th2; ::_thesis: verum
end;
then A27: len s1 >= 1 by FINSEQ_1:20;
 len s1 < len s by A19, A20, NAT_1:13;
then A28: the Source of G . (s . (len s)) = the Target of G . (s . (len s1)) by A19, A20, A27, GRAPH_1:def_15
.= the Target of G . (s1 . (len s1)) by A22, A27, Lm1 ;
A29: now__::_thesis:_not_vx_<>_v2
 vx <> the Target of G . (s . (len s)) by A21, A22, A23, A27, A28, Th6;
then A30: not vx in {v3} by A16, TARSKI:def_1;
 vx in vertices s by A18, A23, Lm4;
then  vx in (vertices s) \ {v3} by A30, XBOOLE_0:def_5;
then A31: ( vx in V or vx in {v2} ) by A17, XBOOLE_0:def_3;
assume A32: vx <> v2 ; ::_thesis: contradiction
 ( s2 . 1 orientedly_joins vx,v3 & s2 . 1 in the carrier' of G ) by A21, A23, A16, Th2, GRAPH_4:def_1;
hence  contradiction by A7, A32, A31, TARSKI:def_1; ::_thesis: verum
end;
 len s2 = 1 by A21;
then  not the Target of G . (s . (len s)) in vertices s1 by A3, A22, A24, A16, A27, GRAPH_5:18;
then A33: (vertices s1) \ {v3} = vertices s1 by A16, ZFMISC_1:57;
 vertices s1 c= vertices (s1 ^ s2) by GRAPH_5:26;
then  vertices s1 c= (vertices s) \ {v3} by A22, A33, XBOOLE_1:33;
then  vertices s1 c= V \/ {v2} by A17, XBOOLE_1:1;
then  (vertices s1) \ {v2} c= (V \/ {v2}) \ {v2} by XBOOLE_1:33;
then  (vertices s1) \ {v2} c= V \ {v2} by XBOOLE_1:40;
then A34: (vertices s1) \ {v2} c= V by XBOOLE_1:1;
 s2 /. 1 in the carrier' of G by A1;
then A35: s2 . 1 in the carrier' of G by A21, FINSEQ_4:15;
 ( the Source of G . e = v2 & the Target of G . e = v3 ) by A5, GRAPH_4:def_1;
then  e = s2 . 1 by A1, A23, A16, A29, A35, GRAPH_1:def_7;
then  s2 = <*e*> by A21, FINSEQ_1:40;
then A36: cost (Q,W) = (cost (P,W)) + (cost (s2,W)) by A4, A10, GRAPH_5:46, GRAPH_5:54;
 the Source of G . (s1 . 1) = v1 by A22, A24, A27, Lm1;
then  s1 is_orientedpath_of v1,v2 by A23, A25, A28, A29, GRAPH_5:def_3;
then  s1 is_orientedpath_of v1,v2,V by A34, GRAPH_5:def_4;
then A37: cost (P,W) <= cost (s1,W) by A2, GRAPH_5:def_18;
A38: cost (s,W) <= cost (S,W) by A12, A13, GRAPH_5:def_18;
 cost (s,W) = (cost (s1,W)) + (cost (s2,W)) by A10, A22, GRAPH_5:46, GRAPH_5:54;
then  cost (Q,W) <= cost (s,W) by A37, A36, XREAL_1:7;
hence  cost (Q,W) <= cost (S,W) by A38, XXREAL_0:2; ::_thesis: verum
end;
 ( len pe = 1 & pe . 1 = e ) by FINSEQ_1:40;
then  Q is_orientedpath_of v1,v3,V \/ {v2} by A4, A5, A8, A9, GRAPH_5:34;
hence  Q is_shortestpath_of v1,v3,V \/ {v2},W by A11, GRAPH_5:def_18; ::_thesis: verum
end;

theorem Th16: :: GRAPHSP:16
for U, V being set
for G being oriented finite Graph
for P being oriented Chain of G
for W being Function of the carrier' of G,Real>=0
for v1, v2 being Vertex of G st the carrier of G = U \/ V & v1 in U & ( for v3, v4 being Vertex of G st v3 in U & v4 in V holds
for e being set holds
( not e in the carrier' of G or not e orientedly_joins v3,v4 ) ) holds
( P is_shortestpath_of v1,v2,U,W iff P is_shortestpath_of v1,v2,W )
proof
let U, V be set ; ::_thesis: for G being oriented finite Graph
for P being oriented Chain of G
for W being Function of the carrier' of G,Real>=0
for v1, v2 being Vertex of G st the carrier of G = U \/ V & v1 in U & ( for v3, v4 being Vertex of G st v3 in U & v4 in V holds
for e being set holds
( not e in the carrier' of G or not e orientedly_joins v3,v4 ) ) holds
( P is_shortestpath_of v1,v2,U,W iff P is_shortestpath_of v1,v2,W )
let G be oriented finite Graph; ::_thesis: for P being oriented Chain of G
for W being Function of the carrier' of G,Real>=0
for v1, v2 being Vertex of G st the carrier of G = U \/ V & v1 in U & ( for v3, v4 being Vertex of G st v3 in U & v4 in V holds
for e being set holds
( not e in the carrier' of G or not e orientedly_joins v3,v4 ) ) holds
( P is_shortestpath_of v1,v2,U,W iff P is_shortestpath_of v1,v2,W )
let P be oriented Chain of G; ::_thesis: for W being Function of the carrier' of G,Real>=0
for v1, v2 being Vertex of G st the carrier of G = U \/ V & v1 in U & ( for v3, v4 being Vertex of G st v3 in U & v4 in V holds
for e being set holds
( not e in the carrier' of G or not e orientedly_joins v3,v4 ) ) holds
( P is_shortestpath_of v1,v2,U,W iff P is_shortestpath_of v1,v2,W )
let W be Function of the carrier' of G,Real>=0; ::_thesis: for v1, v2 being Vertex of G st the carrier of G = U \/ V & v1 in U & ( for v3, v4 being Vertex of G st v3 in U & v4 in V holds
for e being set holds
( not e in the carrier' of G or not e orientedly_joins v3,v4 ) ) holds
( P is_shortestpath_of v1,v2,U,W iff P is_shortestpath_of v1,v2,W )
let v1, v2 be Vertex of G; ::_thesis: ( the carrier of G = U \/ V & v1 in U & ( for v3, v4 being Vertex of G st v3 in U & v4 in V holds
for e being set holds
( not e in the carrier' of G or not e orientedly_joins v3,v4 ) ) implies ( P is_shortestpath_of v1,v2,U,W iff P is_shortestpath_of v1,v2,W ) )
assume A1: ( the carrier of G = U \/ V & v1 in U & ( for v3, v4 being Vertex of G st v3 in U & v4 in V holds
for e being set holds
( not e in the carrier' of G or not e orientedly_joins v3,v4 ) ) ) ; ::_thesis: ( P is_shortestpath_of v1,v2,U,W iff P is_shortestpath_of v1,v2,W )
hereby  ::_thesis: ( P is_shortestpath_of v1,v2,W implies P is_shortestpath_of v1,v2,U,W )
assume A2: P is_shortestpath_of v1,v2,U,W ; ::_thesis: P is_shortestpath_of v1,v2,W
A3: now__::_thesis:_for_Q_being_oriented_Chain_of_G_st_Q_is_orientedpath_of_v1,v2_holds_
cost_(P,W)_<=_cost_(Q,W)
let Q be oriented Chain of G; ::_thesis: ( Q is_orientedpath_of v1,v2 implies cost (P,W) <= cost (Q,W) )
assume  Q is_orientedpath_of v1,v2 ; ::_thesis: cost (P,W) <= cost (Q,W)
then  Q is_orientedpath_of v1,v2,U by A1, Th12;
hence  cost (P,W) <= cost (Q,W) by A2, GRAPH_5:def_18; ::_thesis: verum
end;
 P is_orientedpath_of v1,v2,U by A2, GRAPH_5:def_18;
then  P is_orientedpath_of v1,v2 by GRAPH_5:def_4;
hence  P is_shortestpath_of v1,v2,W by A3, GRAPH_5:def_17; ::_thesis: verum
end;
hereby  ::_thesis: verum
assume A4: P is_shortestpath_of v1,v2,W ; ::_thesis: P is_shortestpath_of v1,v2,U,W
A5: now__::_thesis:_for_Q_being_oriented_Chain_of_G_st_Q_is_orientedpath_of_v1,v2,U_holds_
cost_(P,W)_<=_cost_(Q,W)
let Q be oriented Chain of G; ::_thesis: ( Q is_orientedpath_of v1,v2,U implies cost (P,W) <= cost (Q,W) )
assume  Q is_orientedpath_of v1,v2,U ; ::_thesis: cost (P,W) <= cost (Q,W)
then  Q is_orientedpath_of v1,v2 by GRAPH_5:def_4;
hence  cost (P,W) <= cost (Q,W) by A4, GRAPH_5:def_17; ::_thesis: verum
end;
 P is_orientedpath_of v1,v2 by A4, GRAPH_5:def_17;
then  P is_orientedpath_of v1,v2,U by A1, Th12;
hence  P is_shortestpath_of v1,v2,U,W by A5, GRAPH_5:def_18; ::_thesis: verum
end;
end;

begin

notation
let f be Function;
let i, x be set ;
synonym (f,i) := x for f +* (i,x);
end;

definition
let f be FinSequence of REAL ;
let x be set ;
let r be Real;
:: original: :=
redefine func(f,x) := r ->  FinSequence of REAL ;
coherence
 (f,x) := r is FinSequence of REAL
proof
A1: rng ((f,x) := r) c= REAL
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((f,x) := r) or y in REAL )
A2: rng ((f,x) := r) c= (rng f) \/ {r} by FUNCT_7:100;
assume  y in rng ((f,x) := r) ; ::_thesis: y in REAL
then  ( y in rng f or y in {r} ) by A2, XBOOLE_0:def_3;
hence  y in REAL ; ::_thesis: verum
end;
dom ((f,x) := r) = dom f by FUNCT_7:30
.= Seg (len f) by FINSEQ_1:def_3 ;
then  (f,x) := r is FinSequence by FINSEQ_1:def_2;
hence  (f,x) := r is FinSequence of REAL by A1, FINSEQ_1:def_4; ::_thesis: verum
end;
end;

definition
let i, k be Element of NAT ;
let f be FinSequence of REAL ;
let r be Real;
func(f,i) := (k,r) ->  FinSequence of REAL  equals :: GRAPHSP:def 1
(((f,i) := k),k) := r;
coherence
 (((f,i) := k),k) := r is FinSequence of REAL ;
end;

:: deftheorem  defines := GRAPHSP:def_1_:_
 for i, k being Element of NAT
for f being FinSequence of REAL
for r being Real holds (f,i) := (k,r) = (((f,i) := k),k) := r;


theorem Th17: :: GRAPHSP:17
for i, k being Element of NAT
for f being Element of REAL *
for r being Real st i <> k & i in dom f holds
((f,i) := (k,r)) . i = k
proof
let i, k be Element of NAT ; ::_thesis: for f being Element of REAL *
for r being Real st i <> k & i in dom f holds
((f,i) := (k,r)) . i = k
let f be Element of REAL * ; ::_thesis: for r being Real st i <> k & i in dom f holds
((f,i) := (k,r)) . i = k
let r be Real; ::_thesis: ( i <> k & i in dom f implies ((f,i) := (k,r)) . i = k )
assume that
A1: i <> k and
A2: i in dom f ; ::_thesis: ((f,i) := (k,r)) . i = k
set fik = (f,i) := k;
thus ((f,i) := (k,r)) . i = ((f,i) := k) . i by A1, FUNCT_7:32
.= k by A2, FUNCT_7:31 ; ::_thesis: verum
end;

theorem Th18: :: GRAPHSP:18
for m, i, k being Element of NAT
for f being Element of REAL *
for r being Real st m <> i & m <> k holds
((f,i) := (k,r)) . m = f . m
proof
let m, i, k be Element of NAT ; ::_thesis: for f being Element of REAL *
for r being Real st m <> i & m <> k holds
((f,i) := (k,r)) . m = f . m
let f be Element of REAL * ; ::_thesis: for r being Real st m <> i & m <> k holds
((f,i) := (k,r)) . m = f . m
let r be Real; ::_thesis: ( m <> i & m <> k implies ((f,i) := (k,r)) . m = f . m )
assume that
A1: m <> i and
A2: m <> k ; ::_thesis: ((f,i) := (k,r)) . m = f . m
set fik = (f,i) := k;
thus ((f,i) := (k,r)) . m = ((f,i) := k) . m by A2, FUNCT_7:32
.= f . m by A1, FUNCT_7:32 ; ::_thesis: verum
end;

theorem Th19: :: GRAPHSP:19
for k, i being Element of NAT
for f being Element of REAL *
for r being Real st k in dom f holds
((f,i) := (k,r)) . k = r
proof
let k, i be Element of NAT ; ::_thesis: for f being Element of REAL *
for r being Real st k in dom f holds
((f,i) := (k,r)) . k = r
let f be Element of REAL * ; ::_thesis: for r being Real st k in dom f holds
((f,i) := (k,r)) . k = r
let r be Real; ::_thesis: ( k in dom f implies ((f,i) := (k,r)) . k = r )
set fik = (f,i) := k;
A1: dom ((f,i) := k) = dom f by FUNCT_7:30;
assume  k in dom f ; ::_thesis: ((f,i) := (k,r)) . k = r
hence  ((f,i) := (k,r)) . k = r by A1, FUNCT_7:31; ::_thesis: verum
end;

theorem Th20: :: GRAPHSP:20
for i, k being Element of NAT
for f being Element of REAL *
for r being Real holds dom ((f,i) := (k,r)) = dom f
proof
let i, k be Element of NAT ; ::_thesis: for f being Element of REAL *
for r being Real holds dom ((f,i) := (k,r)) = dom f
let f be Element of REAL * ; ::_thesis: for r being Real holds dom ((f,i) := (k,r)) = dom f
let r be Real; ::_thesis: dom ((f,i) := (k,r)) = dom f
set fik = (f,i) := k;
thus  dom ((f,i) := (k,r)) = dom ((f,i) := k) by FUNCT_7:30
.= dom f by FUNCT_7:30 ; ::_thesis: verum
end;

begin

definition
let X be set ;
let f, g be Element of Funcs (X,X);
:: original: *
redefine funcg * f ->  Element of Funcs (X,X);
coherence
 f * g is Element of Funcs (X,X)
proof
reconsider f = f, g = g as Function of X,X by FUNCT_2:66;
 g * f in Funcs (X,X) by FUNCT_2:9;
hence  f * g is Element of Funcs (X,X) ; ::_thesis: verum
end;
end;

definition
let X be set ;
let f be Element of Funcs (X,X);
let g be Element of X;
:: original: .
redefine funcf . g ->  Element of X;
coherence
 f . g is Element of X
proof
A1: f is Function of X,X by FUNCT_2:66;
percases ( X = {} or X <> {} ) ;
supposeA2: X = {} ; ::_thesis: f . g is Element of X
then  {} = dom f by A1;
then  f . g = {} by FUNCT_1:def_2;
hence  f . g is Element of X by A2, SUBSET_1:def_1; ::_thesis: verum
end;
suppose X <> {} ; ::_thesis: f . g is Element of X
hence  f . g is Element of X by A1, FUNCT_2:5; ::_thesis: verum
end;
end;
end;
end;

definition
let X be set ;
let f be Element of Funcs (X,X);
func repeat f ->  Function of NAT,(Funcs (X,X)) means :Def2: :: GRAPHSP:def 2
( it . 0 = id X & ( for i being Nat holds it . (i + 1) = f * (it . i) ) );
existence
 ex b1 being Function of NAT,(Funcs (X,X)) st
( b1 . 0 = id X & ( for i being Nat holds b1 . (i + 1) = f * (b1 . i) ) )
proof
deffunc H1( Nat, Element of Funcs (X,X)) ->  Element of Funcs (X,X) = f * $2;
 ex F being Function of NAT,(Funcs (X,X)) st
( F . 0 = id X & ( for n being Nat holds F . (n + 1) = H1(n,F . n) ) ) from NAT_1:sch_12();
hence  ex b1 being Function of NAT,(Funcs (X,X)) st
( b1 . 0 = id X & ( for i being Nat holds b1 . (i + 1) = f * (b1 . i) ) ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function of NAT,(Funcs (X,X)) st b1 . 0 = id X & ( for i being Nat holds b1 . (i + 1) = f * (b1 . i) ) & b2 . 0 = id X & ( for i being Nat holds b2 . (i + 1) = f * (b2 . i) ) holds
b1 = b2
proof
deffunc H1( Nat, Element of Funcs (X,X)) ->  Element of Funcs (X,X) = f * $2;
let F1, F2 be Function of NAT,(Funcs (X,X)); ::_thesis: ( F1 . 0 = id X & ( for i being Nat holds F1 . (i + 1) = f * (F1 . i) ) & F2 . 0 = id X & ( for i being Nat holds F2 . (i + 1) = f * (F2 . i) ) implies F1 = F2 )
assume that
A1: F1 . 0 = id X and
A2: for i being Nat holds F1 . (i + 1) = H1(i,F1 . i) and
A3: F2 . 0 = id X and
A4: for i being Nat holds F2 . (i + 1) = H1(i,F2 . i) ; ::_thesis: F1 = F2
thus  F1 = F2 from NAT_1:sch_16(A1, A2, A3, A4); ::_thesis: verum
end;
end;

:: deftheorem Def2 defines repeat GRAPHSP:def_2_:_
 for X being set
for f being Element of Funcs (X,X)
for b3 being Function of NAT,(Funcs (X,X)) holds
( b3 = repeat f iff ( b3 . 0 = id X & ( for i being Nat holds b3 . (i + 1) = f * (b3 . i) ) ) );

theorem Th21: :: GRAPHSP:21
for F being Element of Funcs ((REAL *),(REAL *))
for f being Element of REAL *
for n, i being Element of NAT holds ((repeat F) . 0) . f = f
proof
let F be Element of Funcs ((REAL *),(REAL *)); ::_thesis: for f being Element of REAL *
for n, i being Element of NAT holds ((repeat F) . 0) . f = f
let f be Element of REAL * ; ::_thesis: for n, i being Element of NAT holds ((repeat F) . 0) . f = f
let n, i be Element of NAT ; ::_thesis: ((repeat F) . 0) . f = f
thus ((repeat F) . 0) . f = (id (REAL *)) . f by Def2
.= f by FUNCT_1:18 ; ::_thesis: verum
end;

Lm5:  for X being set
for f being Element of Funcs (X,X) holds dom f = X
proof
let X be set ; ::_thesis: for f being Element of Funcs (X,X) holds dom f = X
let f be Element of Funcs (X,X); ::_thesis: dom f = X
 ex ff being Function st
( f = ff & dom ff = X & rng ff c= X ) by FUNCT_2:def_2;
hence  dom f = X ; ::_thesis: verum
end;

theorem Th22: :: GRAPHSP:22
for F, G being Element of Funcs ((REAL *),(REAL *))
for f being Element of REAL *
for i being Element of NAT holds ((repeat (F * G)) . (i + 1)) . f = F . (G . (((repeat (F * G)) . i) . f))
proof
let F, G be Element of Funcs ((REAL *),(REAL *)); ::_thesis: for f being Element of REAL *
for i being Element of NAT holds ((repeat (F * G)) . (i + 1)) . f = F . (G . (((repeat (F * G)) . i) . f))
let f be Element of REAL * ; ::_thesis: for i being Element of NAT holds ((repeat (F * G)) . (i + 1)) . f = F . (G . (((repeat (F * G)) . i) . f))
let i be Element of NAT ; ::_thesis: ((repeat (F * G)) . (i + 1)) . f = F . (G . (((repeat (F * G)) . i) . f))
set Fi = (repeat (F * G)) . i;
set ff = ((repeat (F * G)) . i) . f;
set FFi = (F * G) * ((repeat (F * G)) . i);
A1: dom (F * G) = REAL * by Lm5;
A2: dom ((F * G) * ((repeat (F * G)) . i)) = REAL * by Lm5;
thus ((repeat (F * G)) . (i + 1)) . f = ((F * G) * ((repeat (F * G)) . i)) . f by Def2
.= (F * G) . (((repeat (F * G)) . i) . f) by A2, FUNCT_1:12
.= F . (G . (((repeat (F * G)) . i) . f)) by A1, FUNCT_1:12 ; ::_thesis: verum
end;

definition
let g be Element of Funcs ((REAL *),(REAL *));
let f be Element of REAL * ;
:: original: .
redefine funcg . f ->  Element of REAL * ;
coherence
 g . f is Element of REAL *
proof
 g . f in REAL * ;
hence  g . f is Element of REAL * ; ::_thesis: verum
end;
end;

definition
let f be Element of REAL * ;
let n be Nat;
func OuterVx (f,n) ->  Subset of NAT equals :: GRAPHSP:def 3
 {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i <> - 1 & f . (n + i) <> - 1 )  }  ;
coherence
  {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i <> - 1 & f . (n + i) <> - 1 )  }  is Subset of NAT
proof
set NS =  {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i <> - 1 & f . (n + i) <> - 1 )  }  ;
  {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i <> - 1 & f . (n + i) <> - 1 )  }  c= NAT
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i <> - 1 & f . (n + i) <> - 1 )  }  or x in NAT )
assume  x in  {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i <> - 1 & f . (n + i) <> - 1 )  }  ; ::_thesis: x in NAT
then  ex k being Element of NAT st
( x = k & k in dom f & 1 <= k & k <= n & f . k <> - 1 & f . (n + k) <> - 1 ) ;
hence  x in NAT ; ::_thesis: verum
end;
hence   {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i <> - 1 & f . (n + i) <> - 1 )  }  is Subset of NAT ; ::_thesis: verum
end;
end;

:: deftheorem  defines OuterVx GRAPHSP:def_3_:_
 for f being Element of REAL *
for n being Nat holds OuterVx (f,n) =  {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i <> - 1 & f . (n + i) <> - 1 )  }  ;

definition
let f be Element of Funcs ((REAL *),(REAL *));
let g be Element of REAL * ;
let n be Nat;
assume A1: ex i being Element of NAT st OuterVx ((((repeat f) . i) . g),n) = {} ;
func LifeSpan (f,g,n) ->  Element of NAT  means :Def4: :: GRAPHSP:def 4
( OuterVx ((((repeat f) . it) . g),n) = {} & ( for k being Nat st OuterVx ((((repeat f) . k) . g),n) = {} holds
it <= k ) );
existence
 ex b1 being Element of NAT st
( OuterVx ((((repeat f) . b1) . g),n) = {} & ( for k being Nat st OuterVx ((((repeat f) . k) . g),n) = {} holds
b1 <= k ) )
proof
defpred S1[ Nat] means  OuterVx ((((repeat f) . $1) . g),n) = {} ;
A2: ex k being Nat st S1[k] by A1;
 ex k being Nat st
( S1[k] & ( for n being Nat st S1[n] holds
k <= n ) ) from NAT_1:sch_5(A2);
then consider k being Nat such that
A3: ( S1[k] & ( for n being Nat st S1[n] holds
k <= n ) ) ;
 k in NAT by ORDINAL1:def_12;
hence  ex b1 being Element of NAT st
( OuterVx ((((repeat f) . b1) . g),n) = {} & ( for k being Nat st OuterVx ((((repeat f) . k) . g),n) = {} holds
b1 <= k ) ) by A3; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Element of NAT st OuterVx ((((repeat f) . b1) . g),n) = {} & ( for k being Nat st OuterVx ((((repeat f) . k) . g),n) = {} holds
b1 <= k ) & OuterVx ((((repeat f) . b2) . g),n) = {} & ( for k being Nat st OuterVx ((((repeat f) . k) . g),n) = {} holds
b2 <= k ) holds
b1 = b2
proof
let it1, it2 be Element of NAT ; ::_thesis: ( OuterVx ((((repeat f) . it1) . g),n) = {} & ( for k being Nat st OuterVx ((((repeat f) . k) . g),n) = {} holds
it1 <= k ) & OuterVx ((((repeat f) . it2) . g),n) = {} & ( for k being Nat st OuterVx ((((repeat f) . k) . g),n) = {} holds
it2 <= k ) implies it1 = it2 )
assume A4: ( OuterVx ((((repeat f) . it1) . g),n) = {} & ( for k being Nat st OuterVx ((((repeat f) . k) . g),n) = {} holds
it1 <= k ) & OuterVx ((((repeat f) . it2) . g),n) = {} & ( for k being Nat st OuterVx ((((repeat f) . k) . g),n) = {} holds
it2 <= k ) & not it1 = it2 ) ; ::_thesis: contradiction
then  ( it1 <= it2 & it2 <= it1 ) ;
hence  contradiction by A4, XXREAL_0:1; ::_thesis: verum
end;
end;

:: deftheorem Def4 defines LifeSpan GRAPHSP:def_4_:_
 for f being Element of Funcs ((REAL *),(REAL *))
for g being Element of REAL *
for n being Nat st ex i being Element of NAT st OuterVx ((((repeat f) . i) . g),n) = {} holds
for b4 being Element of NAT holds
( b4 = LifeSpan (f,g,n) iff ( OuterVx ((((repeat f) . b4) . g),n) = {} & ( for k being Nat st OuterVx ((((repeat f) . k) . g),n) = {} holds
b4 <= k ) ) );

definition
let f be Element of Funcs ((REAL *),(REAL *));
let n be Element of NAT ;
func while_do (f,n) ->  Element of Funcs ((REAL *),(REAL *)) means :Def5: :: GRAPHSP:def 5
( dom it = REAL * & ( for h being Element of REAL * holds it . h = ((repeat f) . (LifeSpan (f,h,n))) . h ) );
existence
 ex b1 being Element of Funcs ((REAL *),(REAL *)) st
( dom b1 = REAL * & ( for h being Element of REAL * holds b1 . h = ((repeat f) . (LifeSpan (f,h,n))) . h ) )
proof
set X = REAL * ;
defpred S1[ set , set ] means  for h being Element of REAL * st $1 = h holds
$2 = ((repeat f) . (LifeSpan (f,h,n))) . h;
A1: ex ff being Function st
( f = ff & dom ff = REAL * & rng ff c= REAL * ) by FUNCT_2:def_2;
A2: now__::_thesis:_for_xx_being_set_st_xx_in_dom_f_holds_
ex_y1_being_set_st_S1[xx,y1]
let xx be set ; ::_thesis: ( xx in dom f implies ex y1 being set st S1[xx,y1] )
assume  xx in dom f ; ::_thesis: ex y1 being set st S1[xx,y1]
then reconsider h9 = xx as Element of REAL * by A1;
now__::_thesis:_ex_yy_being_Element_of_REAL_*_st_
for_h_being_Element_of_REAL_*_st_xx_=_h_holds_
yy_=_((repeat_f)_._(LifeSpan_(f,h,n)))_._h
take yy = ((repeat f) . (LifeSpan (f,h9,n))) . h9; ::_thesis: for h being Element of REAL * st xx = h holds
yy = ((repeat f) . (LifeSpan (f,h,n))) . h
let h be Element of REAL * ; ::_thesis: ( xx = h implies yy = ((repeat f) . (LifeSpan (f,h,n))) . h )
assume  xx = h ; ::_thesis: yy = ((repeat f) . (LifeSpan (f,h,n))) . h
hence  yy = ((repeat f) . (LifeSpan (f,h,n))) . h ; ::_thesis: verum
end;
hence  ex y1 being set st S1[xx,y1] ; ::_thesis: verum
end;
consider f9 being Function such that
A3: ( dom f9 = dom f & ( for xx being set st xx in dom f holds
S1[xx,f9 . xx] ) ) from CLASSES1:sch_1(A2);
 rng f9 c= REAL *
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng f9 or y in REAL * )
assume  y in rng f9 ; ::_thesis: y in REAL *
then consider xx being set  such that
A4: xx in dom f9 and
A5: y = f9 . xx by FUNCT_1:def_3;
reconsider h9 = xx as Element of REAL * by A1, A3, A4;
 y = ((repeat f) . (LifeSpan (f,h9,n))) . h9 by A3, A4, A5;
hence  y in REAL * ; ::_thesis: verum
end;
then reconsider f9 = f9 as Element of Funcs ((REAL *),(REAL *)) by A1, A3, FUNCT_2:def_2;
take  f9 ; ::_thesis: ( dom f9 = REAL * & ( for h being Element of REAL * holds f9 . h = ((repeat f) . (LifeSpan (f,h,n))) . h ) )
thus  dom f9 = REAL * by A1, A3; ::_thesis: for h being Element of REAL * holds f9 . h = ((repeat f) . (LifeSpan (f,h,n))) . h
let h be Element of REAL * ; ::_thesis: f9 . h = ((repeat f) . (LifeSpan (f,h,n))) . h
thus  f9 . h = ((repeat f) . (LifeSpan (f,h,n))) . h by A1, A3; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Element of Funcs ((REAL *),(REAL *)) st dom b1 = REAL * & ( for h being Element of REAL * holds b1 . h = ((repeat f) . (LifeSpan (f,h,n))) . h ) & dom b2 = REAL * & ( for h being Element of REAL * holds b2 . h = ((repeat f) . (LifeSpan (f,h,n))) . h ) holds
b1 = b2
proof
set X = REAL * ;
let g1, g2 be Element of Funcs ((REAL *),(REAL *)); ::_thesis: ( dom g1 = REAL * & ( for h being Element of REAL * holds g1 . h = ((repeat f) . (LifeSpan (f,h,n))) . h ) & dom g2 = REAL * & ( for h being Element of REAL * holds g2 . h = ((repeat f) . (LifeSpan (f,h,n))) . h ) implies g1 = g2 )
assume that
A6: dom g1 = REAL * and
A7: for h being Element of REAL * holds g1 . h = ((repeat f) . (LifeSpan (f,h,n))) . h and
A8: dom g2 = REAL * and
A9: for h being Element of REAL * holds g2 . h = ((repeat f) . (LifeSpan (f,h,n))) . h ; ::_thesis: g1 = g2
now__::_thesis:_for_xx_being_set_st_xx_in_dom_g1_holds_
g1_._xx_=_g2_._xx
let xx be set ; ::_thesis: ( xx in dom g1 implies g1 . xx = g2 . xx )
assume  xx in dom g1 ; ::_thesis: g1 . xx = g2 . xx
then reconsider h = xx as Element of REAL * by A6;
thus g1 . xx = ((repeat f) . (LifeSpan (f,h,n))) . h by A7
.= g2 . xx by A9 ; ::_thesis: verum
end;
hence  g1 = g2 by A6, A8, FUNCT_1:2; ::_thesis: verum
end;
end;

:: deftheorem Def5 defines while_do GRAPHSP:def_5_:_
 for f being Element of Funcs ((REAL *),(REAL *))
for n being Element of NAT
for b3 being Element of Funcs ((REAL *),(REAL *)) holds
( b3 = while_do (f,n) iff ( dom b3 = REAL * & ( for h being Element of REAL * holds b3 . h = ((repeat f) . (LifeSpan (f,h,n))) . h ) ) );

begin

definition
let G be oriented Graph;
let v1, v2 be Vertex of G;
assume A1: ex e being set st
( e in the carrier' of G & e orientedly_joins v1,v2 ) ;
func XEdge (v1,v2) ->  set  means :Def6: :: GRAPHSP:def 6
 ex e being set st
( it = e & e in the carrier' of G & e orientedly_joins v1,v2 );
existence
 ex b1 being set ex e being set st
( b1 = e & e in the carrier' of G & e orientedly_joins v1,v2 ) by A1;
uniqueness
 for b1, b2 being set st ex e being set st
( b1 = e & e in the carrier' of G & e orientedly_joins v1,v2 ) & ex e being set st
( b2 = e & e in the carrier' of G & e orientedly_joins v1,v2 ) holds
b1 = b2 by Th10;
end;

:: deftheorem Def6 defines XEdge GRAPHSP:def_6_:_
 for G being oriented Graph
for v1, v2 being Vertex of G st ex e being set st
( e in the carrier' of G & e orientedly_joins v1,v2 ) holds
for b4 being set holds
( b4 = XEdge (v1,v2) iff ex e being set st
( b4 = e & e in the carrier' of G & e orientedly_joins v1,v2 ) );

definition
let G be oriented Graph;
let v1, v2 be Vertex of G;
let W be Function;
func Weight (v1,v2,W) ->  set  equals :Def7: :: GRAPHSP:def 7
W . (XEdge (v1,v2)) if  ex e being set st
( e in the carrier' of G & e orientedly_joins v1,v2 )
otherwise  - 1;
correctness
coherence
( ( ex e being set st
( e in the carrier' of G & e orientedly_joins v1,v2 ) implies W . (XEdge (v1,v2)) is set ) & ( ( for e being set holds
( not e in the carrier' of G or not e orientedly_joins v1,v2 ) ) implies - 1 is set ) );
consistency
for b1 being set holds verum;
;
end;

:: deftheorem Def7 defines Weight GRAPHSP:def_7_:_
 for G being oriented Graph
for v1, v2 being Vertex of G
for W being Function holds
( ( ex e being set st
( e in the carrier' of G & e orientedly_joins v1,v2 ) implies Weight (v1,v2,W) = W . (XEdge (v1,v2)) ) & ( ( for e being set holds
( not e in the carrier' of G or not e orientedly_joins v1,v2 ) ) implies Weight (v1,v2,W) = - 1 ) );

definition
let G be oriented Graph;
let v1, v2 be Vertex of G;
let W be Function of the carrier' of G,Real>=0;
:: original: Weight
redefine func Weight (v1,v2,W) ->  Real;
coherence
 Weight (v1,v2,W) is Real
proof
percases ( ex e being set st
( e in the carrier' of G & e orientedly_joins v1,v2 ) or for e being set holds
( not e in the carrier' of G or not e orientedly_joins v1,v2 ) ) ;
supposeA1: ex e being set st
( e in the carrier' of G & e orientedly_joins v1,v2 ) ; ::_thesis: Weight (v1,v2,W) is Real
then consider e being set  such that
A2: XEdge (v1,v2) = e and
A3: e in the carrier' of G and
 e orientedly_joins v1,v2 by Def6;
 e in dom W by A3, FUNCT_2:def_1;
then  W . (XEdge (v1,v2)) in Real>=0 by A2, PARTFUN1:4;
hence  Weight (v1,v2,W) is Real by A1, Def7; ::_thesis: verum
end;
suppose for e being set holds
( not e in the carrier' of G or not e orientedly_joins v1,v2 ) ; ::_thesis: Weight (v1,v2,W) is Real
hence  Weight (v1,v2,W) is Real by Def7; ::_thesis: verum
end;
end;
end;
end;


theorem Th23: :: GRAPHSP:23
for G being oriented Graph
for v1, v2 being Vertex of G
for W being Function of the carrier' of G,Real>=0 holds
( Weight (v1,v2,W) >= 0 iff ex e being set st
( e in the carrier' of G & e orientedly_joins v1,v2 ) )
proof
let G be oriented Graph; ::_thesis: for v1, v2 being Vertex of G
for W being Function of the carrier' of G,Real>=0 holds
( Weight (v1,v2,W) >= 0 iff ex e being set st
( e in the carrier' of G & e orientedly_joins v1,v2 ) )
let v1, v2 be Vertex of G; ::_thesis: for W being Function of the carrier' of G,Real>=0 holds
( Weight (v1,v2,W) >= 0 iff ex e being set st
( e in the carrier' of G & e orientedly_joins v1,v2 ) )
let W be Function of the carrier' of G,Real>=0; ::_thesis: ( Weight (v1,v2,W) >= 0 iff ex e being set st
( e in the carrier' of G & e orientedly_joins v1,v2 ) )
set EG = the carrier' of G;
hereby  ::_thesis: ( ex e being set st
( e in the carrier' of G & e orientedly_joins v1,v2 ) implies Weight (v1,v2,W) >= 0 )
assume A1: Weight (v1,v2,W) >= 0 ; ::_thesis: ex e being set st
( e in the carrier' of G & e orientedly_joins v1,v2 )
assume  for e being set holds
( not e in the carrier' of G or not e orientedly_joins v1,v2 ) ; ::_thesis: contradiction
then  Weight (v1,v2,W) = - 1 by Def7;
hence  contradiction by A1; ::_thesis: verum
end;
assume  ex e being set st
( e in the carrier' of G & e orientedly_joins v1,v2 ) ; ::_thesis: Weight (v1,v2,W) >= 0
then consider e being set  such that
A2: XEdge (v1,v2) = e and
A3: e in the carrier' of G and
A4: e orientedly_joins v1,v2 by Def6;
 e in dom W by A3, FUNCT_2:def_1;
then  W . e in Real>=0 by PARTFUN1:4;
then  ex r being Real st
( W . e = r & r >= 0 ) by GRAPH_5:def_12;
hence  Weight (v1,v2,W) >= 0 by A2, A3, A4, Def7; ::_thesis: verum
end;

theorem :: GRAPHSP:24
for G being oriented Graph
for v1, v2 being Vertex of G
for W being Function of the carrier' of G,Real>=0 holds
( Weight (v1,v2,W) = - 1 iff for e being set holds
( not e in the carrier' of G or not e orientedly_joins v1,v2 ) ) by Def7, Th23;

theorem Th25: :: GRAPHSP:25
for e being set
for G being oriented Graph
for v1, v2 being Vertex of G
for W being Function of the carrier' of G,Real>=0 st e in the carrier' of G & e orientedly_joins v1,v2 holds
Weight (v1,v2,W) = W . e
proof
let e be set ; ::_thesis: for G being oriented Graph
for v1, v2 being Vertex of G
for W being Function of the carrier' of G,Real>=0 st e in the carrier' of G & e orientedly_joins v1,v2 holds
Weight (v1,v2,W) = W . e
let G be oriented Graph; ::_thesis: for v1, v2 being Vertex of G
for W being Function of the carrier' of G,Real>=0 st e in the carrier' of G & e orientedly_joins v1,v2 holds
Weight (v1,v2,W) = W . e
let v1, v2 be Vertex of G; ::_thesis: for W being Function of the carrier' of G,Real>=0 st e in the carrier' of G & e orientedly_joins v1,v2 holds
Weight (v1,v2,W) = W . e
let W be Function of the carrier' of G,Real>=0; ::_thesis: ( e in the carrier' of G & e orientedly_joins v1,v2 implies Weight (v1,v2,W) = W . e )
set EG = the carrier' of G;
assume A1: ( e in the carrier' of G & e orientedly_joins v1,v2 ) ; ::_thesis: Weight (v1,v2,W) = W . e
then consider e1 being set  such that
A2: XEdge (v1,v2) = e1 and
A3: ( e1 in the carrier' of G & e1 orientedly_joins v1,v2 ) by Def6;
 e = e1 by A1, A3, Th10;
hence  Weight (v1,v2,W) = W . e by A2, A3, Def7; ::_thesis: verum
end;

begin

definition
let f be Element of REAL * ;
let n be Element of NAT ;
func UnusedVx (f,n) ->  Subset of NAT equals :: GRAPHSP:def 8
 {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i <> - 1 )  }  ;
coherence
  {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i <> - 1 )  }  is Subset of NAT
proof
set NS =  {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i <> - 1 )  }  ;
  {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i <> - 1 )  }  c= NAT
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i <> - 1 )  }  or x in NAT )
assume  x in  {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i <> - 1 )  }  ; ::_thesis: x in NAT
then  ex k being Element of NAT st
( x = k & k in dom f & 1 <= k & k <= n & f . k <> - 1 ) ;
hence  x in NAT ; ::_thesis: verum
end;
hence   {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i <> - 1 )  }  is Subset of NAT ; ::_thesis: verum
end;
end;

:: deftheorem  defines UnusedVx GRAPHSP:def_8_:_
 for f being Element of REAL *
for n being Element of NAT holds UnusedVx (f,n) =  {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i <> - 1 )  }  ;

definition
let f be Element of REAL * ;
let n be Element of NAT ;
func UsedVx (f,n) ->  Subset of NAT equals :: GRAPHSP:def 9
 {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i = - 1 )  }  ;
coherence
  {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i = - 1 )  }  is Subset of NAT
proof
set NS =  {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i = - 1 )  }  ;
  {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i = - 1 )  }  c= NAT
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i = - 1 )  }  or x in NAT )
assume  x in  {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i = - 1 )  }  ; ::_thesis: x in NAT
then  ex k being Element of NAT st
( x = k & k in dom f & 1 <= k & k <= n & f . k = - 1 ) ;
hence  x in NAT ; ::_thesis: verum
end;
hence   {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i = - 1 )  }  is Subset of NAT ; ::_thesis: verum
end;
end;

:: deftheorem  defines UsedVx GRAPHSP:def_9_:_
 for f being Element of REAL *
for n being Element of NAT holds UsedVx (f,n) =  {  i where i is Element of NAT : ( i in dom f & 1 <= i & i <= n & f . i = - 1 )  }  ;

theorem Th26: :: GRAPHSP:26
for n being Element of NAT
for f being Element of REAL * holds UnusedVx (f,n) c= Seg n
proof
let n be Element of NAT ; ::_thesis: for f being Element of REAL * holds UnusedVx (f,n) c= Seg n
let f be Element of REAL * ; ::_thesis: UnusedVx (f,n) c= Seg n
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in UnusedVx (f,n) or x in Seg n )
assume  x in UnusedVx (f,n) ; ::_thesis: x in Seg n
then  ex i being Element of NAT st
( x = i & i in dom f & 1 <= i & i <= n & f . i <> - 1 ) ;
then  x in  {  k where k is Element of NAT : ( 1 <= k & k <= n )  }  ;
hence  x in Seg n by FINSEQ_1:def_1; ::_thesis: verum
end;

registration
let f be Element of REAL * ;
let n be Element of NAT ;
cluster UnusedVx (f,n) ->  finite ;
coherence
 UnusedVx (f,n) is finite
proof
 UnusedVx (f,n) c= Seg n by Th26;
hence  UnusedVx (f,n) is finite ; ::_thesis: verum
end;
end;

theorem Th27: :: GRAPHSP:27
for n being Element of NAT
for f being Element of REAL * holds OuterVx (f,n) c= UnusedVx (f,n)
proof
let n be Element of NAT ; ::_thesis: for f being Element of REAL * holds OuterVx (f,n) c= UnusedVx (f,n)
let f be Element of REAL * ; ::_thesis: OuterVx (f,n) c= UnusedVx (f,n)
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in OuterVx (f,n) or x in UnusedVx (f,n) )
assume  x in OuterVx (f,n) ; ::_thesis: x in UnusedVx (f,n)
then  ex k being Element of NAT st
( x = k & k in dom f & 1 <= k & k <= n & f . k <> - 1 & f . (n + k) <> - 1 ) ;
hence  x in UnusedVx (f,n) ; ::_thesis: verum
end;

theorem Th28: :: GRAPHSP:28
for n being Element of NAT
for f being Element of REAL * holds OuterVx (f,n) c= Seg n
proof
let n be Element of NAT ; ::_thesis: for f being Element of REAL * holds OuterVx (f,n) c= Seg n
let f be Element of REAL * ; ::_thesis: OuterVx (f,n) c= Seg n
 ( OuterVx (f,n) c= UnusedVx (f,n) & UnusedVx (f,n) c= Seg n ) by Th26, Th27;
hence  OuterVx (f,n) c= Seg n by XBOOLE_1:1; ::_thesis: verum
end;

registration
let f be Element of REAL * ;
let n be Element of NAT ;
cluster OuterVx (f,n) ->  finite ;
coherence
 OuterVx (f,n) is finite
proof
 OuterVx (f,n) c= Seg n by Th28;
hence  OuterVx (f,n) is finite ; ::_thesis: verum
end;
end;

definition
let X be finite Subset of NAT;
let f be Element of REAL * ;
let n be Element of NAT ;
func Argmin (X,f,n) ->  Element of NAT  means :Def10: :: GRAPHSP:def 10
( ( X <> {} implies ex i being Element of NAT st
( i = it & i in X & ( for k being Element of NAT st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being Element of NAT st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k ) ) ) & ( X = {} implies it = 0 ) );
existence
 ex b1 being Element of NAT st
( ( X <> {} implies ex i being Element of NAT st
( i = b1 & i in X & ( for k being Element of NAT st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being Element of NAT st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k ) ) ) & ( X = {} implies b1 = 0 ) )
proof
percases ( X = {} or X <> {} ) ;
supposeA1: X = {} ; ::_thesis: ex b1 being Element of NAT st
( ( X <> {} implies ex i being Element of NAT st
( i = b1 & i in X & ( for k being Element of NAT st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being Element of NAT st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k ) ) ) & ( X = {} implies b1 = 0 ) )
take  0 ; ::_thesis: ( ( X <> {} implies ex i being Element of NAT st
( i = 0 & i in X & ( for k being Element of NAT st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being Element of NAT st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k ) ) ) & ( X = {} implies 0 = 0 ) )
thus  ( ( X <> {} implies ex i being Element of NAT st
( i = 0 & i in X & ( for k being Element of NAT st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being Element of NAT st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k ) ) ) & ( X = {} implies 0 = 0 ) ) by A1; ::_thesis: verum
end;
supposeA2: X <> {} ; ::_thesis: ex b1 being Element of NAT st
( ( X <> {} implies ex i being Element of NAT st
( i = b1 & i in X & ( for k being Element of NAT st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being Element of NAT st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k ) ) ) & ( X = {} implies b1 = 0 ) )
then reconsider X9 = X as non empty finite Subset of NAT ;
deffunc H1( Element of X9) ->  Element of REAL = f /. ((2 * n) + $1);
consider x being Element of X9 such that
A3: for y being Element of X9 holds H1(x) <= H1(y) from GRAPH_5:sch_2();
reconsider x = x as Element of NAT ;
defpred S1[ Nat] means ( $1 in X & f /. ((2 * n) + x) = f /. ((2 * n) + $1) );
A4: ex i being Nat st S1[i] ;
consider i being Nat such that
A5: ( S1[i] & ( for k being Nat st S1[k] holds
i <= k ) ) from NAT_1:sch_5(A4);
reconsider i = i as Element of NAT by ORDINAL1:def_12;
take F = i; ::_thesis: ( ( X <> {} implies ex i being Element of NAT st
( i = F & i in X & ( for k being Element of NAT st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being Element of NAT st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k ) ) ) & ( X = {} implies F = 0 ) )
hereby  ::_thesis: ( X = {} implies F = 0 )
assume  X <> {} ; ::_thesis: ex i being Element of NAT st
( i = F & i in X & ( for k being Element of NAT st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being Element of NAT st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k ) )
take i = F; ::_thesis: ( i = F & i in X & ( for k being Element of NAT st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being Element of NAT st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k ) )
thus  ( i = F & i in X ) by A5; ::_thesis: ( ( for k being Element of NAT st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being Element of NAT st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k ) )
thus  for k being Element of NAT st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) by A3, A5; ::_thesis: for k being Element of NAT st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k
thus  for k being Element of NAT st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k by A5; ::_thesis: verum
end;
thus  ( X = {} implies F = 0 ) by A2; ::_thesis: verum
end;
end;
end;
uniqueness
 for b1, b2 being Element of NAT st ( X <> {} implies ex i being Element of NAT st
( i = b1 & i in X & ( for k being Element of NAT st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being Element of NAT st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k ) ) ) & ( X = {} implies b1 = 0 ) & ( X <> {} implies ex i being Element of NAT st
( i = b2 & i in X & ( for k being Element of NAT st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being Element of NAT st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k ) ) ) & ( X = {} implies b2 = 0 ) holds
b1 = b2
proof
let F1, F2 be Element of NAT ; ::_thesis: ( ( X <> {} implies ex i being Element of NAT st
( i = F1 & i in X & ( for k being Element of NAT st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being Element of NAT st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k ) ) ) & ( X = {} implies F1 = 0 ) & ( X <> {} implies ex i being Element of NAT st
( i = F2 & i in X & ( for k being Element of NAT st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being Element of NAT st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k ) ) ) & ( X = {} implies F2 = 0 ) implies F1 = F2 )
assume that
A6: ( X <> {} implies ex i being Element of NAT st
( i = F1 & i in X & ( for k being Element of NAT st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being Element of NAT st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k ) ) ) and
A7: ( X = {} implies F1 = 0 ) and
A8: ( X <> {} implies ex i being Element of NAT st
( i = F2 & i in X & ( for k being Element of NAT st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being Element of NAT st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k ) ) ) and
A9: ( X = {} implies F2 = 0 ) ; ::_thesis: F1 = F2
percases ( X <> {} or X = {} ) ;
supposeA10: X <> {} ; ::_thesis: F1 = F2
then consider j being Element of NAT  such that
A11: j = F2 and
A12: j in X and
A13: for k being Element of NAT st k in X holds
f /. ((2 * n) + j) <= f /. ((2 * n) + k) and
A14: for k being Element of NAT st k in X & f /. ((2 * n) + j) = f /. ((2 * n) + k) holds
j <= k by A8;
consider i being Element of NAT  such that
A15: i = F1 and
A16: i in X and
A17: for k being Element of NAT st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) and
A18: for k being Element of NAT st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k by A6, A10;
 ( f /. ((2 * n) + i) <= f /. ((2 * n) + j) & f /. ((2 * n) + j) <= f /. ((2 * n) + i) ) by A16, A17, A12, A13;
then  f /. ((2 * n) + j) = f /. ((2 * n) + i) by XXREAL_0:1;
then  ( i <= j & j <= i ) by A16, A18, A12, A14;
hence  F1 = F2 by A15, A11, XXREAL_0:1; ::_thesis: verum
end;
suppose X = {} ; ::_thesis: F1 = F2
hence  F1 = F2 by A7, A9; ::_thesis: verum
end;
end;
end;
end;

:: deftheorem Def10 defines Argmin GRAPHSP:def_10_:_
 for X being finite Subset of NAT
for f being Element of REAL *
for n, b4 being Element of NAT holds
( b4 = Argmin (X,f,n) iff ( ( X <> {} implies ex i being Element of NAT st
( i = b4 & i in X & ( for k being Element of NAT st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being Element of NAT st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k ) ) ) & ( X = {} implies b4 = 0 ) ) );

theorem Th29: :: GRAPHSP:29
for n, j being Element of NAT
for f being Element of REAL * st OuterVx (f,n) <> {} & j = Argmin ((OuterVx (f,n)),f,n) holds
( j in dom f & 1 <= j & j <= n & f . j <> - 1 & f . (n + j) <> - 1 )
proof
let n, j be Element of NAT ; ::_thesis: for f being Element of REAL * st OuterVx (f,n) <> {} & j = Argmin ((OuterVx (f,n)),f,n) holds
( j in dom f & 1 <= j & j <= n & f . j <> - 1 & f . (n + j) <> - 1 )
let f be Element of REAL * ; ::_thesis: ( OuterVx (f,n) <> {} & j = Argmin ((OuterVx (f,n)),f,n) implies ( j in dom f & 1 <= j & j <= n & f . j <> - 1 & f . (n + j) <> - 1 ) )
set IN = OuterVx (f,n);
assume  ( OuterVx (f,n) <> {} & j = Argmin ((OuterVx (f,n)),f,n) ) ; ::_thesis: ( j in dom f & 1 <= j & j <= n & f . j <> - 1 & f . (n + j) <> - 1 )
then  ex i being Element of NAT st
( i = j & i in OuterVx (f,n) & ( for k being Element of NAT st k in OuterVx (f,n) holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being Element of NAT st k in OuterVx (f,n) & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k ) ) by Def10;
then  ex k being Element of NAT st
( j = k & k in dom f & 1 <= k & k <= n & f . k <> - 1 & f . (n + k) <> - 1 ) ;
hence  ( j in dom f & 1 <= j & j <= n & f . j <> - 1 & f . (n + j) <> - 1 ) ; ::_thesis: verum
end;

theorem Th30: :: GRAPHSP:30
for n being Element of NAT
for f being Element of REAL * holds Argmin ((OuterVx (f,n)),f,n) <= n
proof
let n be Element of NAT ; ::_thesis: for f being Element of REAL * holds Argmin ((OuterVx (f,n)),f,n) <= n
let f be Element of REAL * ; ::_thesis: Argmin ((OuterVx (f,n)),f,n) <= n
set IN = OuterVx (f,n);
percases ( OuterVx (f,n) <> {} or OuterVx (f,n) = {} ) ;
suppose OuterVx (f,n) <> {} ; ::_thesis: Argmin ((OuterVx (f,n)),f,n) <= n
hence  Argmin ((OuterVx (f,n)),f,n) <= n by Th29; ::_thesis: verum
end;
suppose OuterVx (f,n) = {} ; ::_thesis: Argmin ((OuterVx (f,n)),f,n) <= n
hence  Argmin ((OuterVx (f,n)),f,n) <= n by Def10; ::_thesis: verum
end;
end;
end;

definition
let n be Element of NAT ;
func findmin n ->  Element of Funcs ((REAL *),(REAL *)) means :Def11: :: GRAPHSP:def 11
( dom it = REAL * & ( for f being Element of REAL * holds it . f = (f,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (f,n)),f,n)),(- 1)) ) );
existence
 ex b1 being Element of Funcs ((REAL *),(REAL *)) st
( dom b1 = REAL * & ( for f being Element of REAL * holds b1 . f = (f,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (f,n)),f,n)),(- 1)) ) )
proof
set X = REAL * ;
set mi = ((n * n) + (3 * n)) + 1;
defpred S1[ set , set ] means  for f being Element of REAL * st $1 = f holds
$2 = (f,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (f,n)),f,n)),(- 1));
A1: now__::_thesis:_for_xx_being_set_st_xx_in_REAL_*_holds_
ex_y_being_set_st_S1[xx,y]
let xx be set ; ::_thesis: ( xx in REAL * implies ex y being set st S1[xx,y] )
assume  xx in REAL * ; ::_thesis: ex y being set st S1[xx,y]
then reconsider h = xx as Element of REAL * ;
reconsider y = (h,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (h,n)),h,n)),(- 1)) as set ;
take y = y; ::_thesis: S1[xx,y]
thus  S1[xx,y] ; ::_thesis: verum
end;
consider F being Function such that
A2: ( dom F = REAL * & ( for x being set st x in REAL * holds
S1[x,F . x] ) ) from CLASSES1:sch_1(A1);
 rng F c= REAL *
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng F or y in REAL * )
assume  y in rng F ; ::_thesis: y in REAL *
then consider xx being set  such that
A3: xx in dom F and
A4: y = F . xx by FUNCT_1:def_3;
reconsider h = xx as Element of REAL * by A2, A3;
 y = (h,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (h,n)),h,n)),(- 1)) by A2, A4;
hence  y in REAL * by FINSEQ_1:def_11; ::_thesis: verum
end;
then reconsider F = F as Element of Funcs ((REAL *),(REAL *)) by A2, FUNCT_2:def_2;
take  F ; ::_thesis: ( dom F = REAL * & ( for f being Element of REAL * holds F . f = (f,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (f,n)),f,n)),(- 1)) ) )
thus  dom F = REAL * by A2; ::_thesis: for f being Element of REAL * holds F . f = (f,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (f,n)),f,n)),(- 1))
let f be Element of REAL * ; ::_thesis: F . f = (f,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (f,n)),f,n)),(- 1))
thus  F . f = (f,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (f,n)),f,n)),(- 1)) by A2; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Element of Funcs ((REAL *),(REAL *)) st dom b1 = REAL * & ( for f being Element of REAL * holds b1 . f = (f,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (f,n)),f,n)),(- 1)) ) & dom b2 = REAL * & ( for f being Element of REAL * holds b2 . f = (f,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (f,n)),f,n)),(- 1)) ) holds
b1 = b2
proof
set X = REAL * ;
set mi = ((n * n) + (3 * n)) + 1;
let F1, F2 be Element of Funcs ((REAL *),(REAL *)); ::_thesis: ( dom F1 = REAL * & ( for f being Element of REAL * holds F1 . f = (f,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (f,n)),f,n)),(- 1)) ) & dom F2 = REAL * & ( for f being Element of REAL * holds F2 . f = (f,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (f,n)),f,n)),(- 1)) ) implies F1 = F2 )
assume that
A5: dom F1 = REAL * and
A6: for f being Element of REAL * holds F1 . f = (f,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (f,n)),f,n)),(- 1)) and
A7: dom F2 = REAL * and
A8: for f being Element of REAL * holds F2 . f = (f,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (f,n)),f,n)),(- 1)) ; ::_thesis: F1 = F2
now__::_thesis:_for_xx_being_set_st_xx_in_dom_F1_holds_
F1_._xx_=_F2_._xx
let xx be set ; ::_thesis: ( xx in dom F1 implies F1 . xx = F2 . xx )
assume  xx in dom F1 ; ::_thesis: F1 . xx = F2 . xx
then reconsider h = xx as Element of REAL * by A5;
thus F1 . xx = (h,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (h,n)),h,n)),(- 1)) by A6
.= F2 . xx by A8 ; ::_thesis: verum
end;
hence  F1 = F2 by A5, A7, FUNCT_1:2; ::_thesis: verum
end;
end;

:: deftheorem Def11 defines findmin GRAPHSP:def_11_:_
 for n being Element of NAT
for b2 being Element of Funcs ((REAL *),(REAL *)) holds
( b2 = findmin n iff ( dom b2 = REAL * & ( for f being Element of REAL * holds b2 . f = (f,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (f,n)),f,n)),(- 1)) ) ) );

theorem Th31: :: GRAPHSP:31
for i, n being Element of NAT
for f being Element of REAL * st i > n & i <> ((n * n) + (3 * n)) + 1 holds
((findmin n) . f) . i = f . i
proof
let i, n be Element of NAT ; ::_thesis: for f being Element of REAL * st i > n & i <> ((n * n) + (3 * n)) + 1 holds
((findmin n) . f) . i = f . i
let f be Element of REAL * ; ::_thesis: ( i > n & i <> ((n * n) + (3 * n)) + 1 implies ((findmin n) . f) . i = f . i )
set k = Argmin ((OuterVx (f,n)),f,n);
set mi = ((n * n) + (3 * n)) + 1;
assume A1: ( i > n & i <> ((n * n) + (3 * n)) + 1 ) ; ::_thesis: ((findmin n) . f) . i = f . i
 ( ((findmin n) . f) . i = ((f,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (f,n)),f,n)),(- 1))) . i & Argmin ((OuterVx (f,n)),f,n) <= n ) by Def11, Th30;
hence  ((findmin n) . f) . i = f . i by A1, Th18; ::_thesis: verum
end;

theorem Th32: :: GRAPHSP:32
for i, n being Element of NAT
for f being Element of REAL * st i in dom f & f . i = - 1 & i <> ((n * n) + (3 * n)) + 1 holds
((findmin n) . f) . i = - 1
proof
let i, n be Element of NAT ; ::_thesis: for f being Element of REAL * st i in dom f & f . i = - 1 & i <> ((n * n) + (3 * n)) + 1 holds
((findmin n) . f) . i = - 1
let f be Element of REAL * ; ::_thesis: ( i in dom f & f . i = - 1 & i <> ((n * n) + (3 * n)) + 1 implies ((findmin n) . f) . i = - 1 )
set k = Argmin ((OuterVx (f,n)),f,n);
set mi = ((n * n) + (3 * n)) + 1;
assume that
A1: i in dom f and
A2: ( f . i = - 1 & i <> ((n * n) + (3 * n)) + 1 ) ; ::_thesis: ((findmin n) . f) . i = - 1
A3: ((findmin n) . f) . i = ((f,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (f,n)),f,n)),(- 1))) . i by Def11;
percases ( i = Argmin ((OuterVx (f,n)),f,n) or i <> Argmin ((OuterVx (f,n)),f,n) ) ;
suppose i = Argmin ((OuterVx (f,n)),f,n) ; ::_thesis: ((findmin n) . f) . i = - 1
hence  ((findmin n) . f) . i = - 1 by A1, A3, Th19; ::_thesis: verum
end;
suppose i <> Argmin ((OuterVx (f,n)),f,n) ; ::_thesis: ((findmin n) . f) . i = - 1
hence  ((findmin n) . f) . i = - 1 by A2, A3, Th18; ::_thesis: verum
end;
end;
end;

theorem Th33: :: GRAPHSP:33
for n being Element of NAT
for f being Element of REAL * holds dom ((findmin n) . f) = dom f
proof
let n be Element of NAT ; ::_thesis: for f being Element of REAL * holds dom ((findmin n) . f) = dom f
let f be Element of REAL * ; ::_thesis: dom ((findmin n) . f) = dom f
 (findmin n) . f = (f,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (f,n)),f,n)),(- 1)) by Def11;
hence  dom ((findmin n) . f) = dom f by Th20; ::_thesis: verum
end;

Lm6:  for k, n being Element of NAT st k >= 1 holds
n <= k * n
proof
let k, n be Element of NAT ; ::_thesis: ( k >= 1 implies n <= k * n )
assume  k >= 1 ; ::_thesis: n <= k * n
then  1 * n <= k * n by NAT_1:4;
hence  n <= k * n ; ::_thesis: verum
end;

Lm7:  for n being Element of NAT holds
( 3 * n < ((n * n) + (3 * n)) + 1 & n < ((n * n) + (3 * n)) + 1 & 2 * n < ((n * n) + (3 * n)) + 1 )
proof
let n be Element of NAT ; ::_thesis: ( 3 * n < ((n * n) + (3 * n)) + 1 & n < ((n * n) + (3 * n)) + 1 & 2 * n < ((n * n) + (3 * n)) + 1 )
 3 * n <= (n * n) + (3 * n) by NAT_1:12;
hence A1: 3 * n < ((n * n) + (3 * n)) + 1 by NAT_1:13; ::_thesis: ( n < ((n * n) + (3 * n)) + 1 & 2 * n < ((n * n) + (3 * n)) + 1 )
 n <= 3 * n by Lm6;
hence  n < ((n * n) + (3 * n)) + 1 by A1, XXREAL_0:2; ::_thesis: 2 * n < ((n * n) + (3 * n)) + 1
 2 * n <= 3 * n by NAT_1:4;
hence  2 * n < ((n * n) + (3 * n)) + 1 by A1, XXREAL_0:2; ::_thesis: verum
end;

Lm8:  for n, k being Element of NAT holds
( ( n < k & k <= 2 * n implies ( ( not 2 * n < k or not k <= 3 * n ) & not k <= n & not k > 3 * n ) ) & ( ( k <= n or k > 3 * n ) implies ( ( not 2 * n < k or not k <= 3 * n ) & ( not n < k or not k <= 2 * n ) ) ) & ( 2 * n < k & k <= 3 * n implies ( ( not n < k or not k <= 2 * n ) & not k <= n & not k > 3 * n ) ) )
proof
let n, k be Element of NAT ; ::_thesis: ( ( n < k & k <= 2 * n implies ( ( not 2 * n < k or not k <= 3 * n ) & not k <= n & not k > 3 * n ) ) & ( ( k <= n or k > 3 * n ) implies ( ( not 2 * n < k or not k <= 3 * n ) & ( not n < k or not k <= 2 * n ) ) ) & ( 2 * n < k & k <= 3 * n implies ( ( not n < k or not k <= 2 * n ) & not k <= n & not k > 3 * n ) ) )
A1: 2 * n <= 3 * n by NAT_1:4;
hence  ( n < k & k <= 2 * n implies ( ( not 2 * n < k or not k <= 3 * n ) & not k <= n & not k > 3 * n ) ) by XXREAL_0:2; ::_thesis: ( ( ( k <= n or k > 3 * n ) implies ( ( not 2 * n < k or not k <= 3 * n ) & ( not n < k or not k <= 2 * n ) ) ) & ( 2 * n < k & k <= 3 * n implies ( ( not n < k or not k <= 2 * n ) & not k <= n & not k > 3 * n ) ) )
A2: 2 * n = n + n ;
hereby  ::_thesis: ( 2 * n < k & k <= 3 * n implies ( ( not n < k or not k <= 2 * n ) & not k <= n & not k > 3 * n ) )
assume A3: ( k <= n or k > 3 * n ) ; ::_thesis: ( ( not 2 * n < k or not k <= 3 * n ) & ( not n < k or not k <= 2 * n ) )
percases ( k <= n or k > 3 * n ) by A3;
supposeA4: k <= n ; ::_thesis: ( ( not 2 * n < k or not k <= 3 * n ) & ( not n < k or not k <= 2 * n ) )
hence  ( not 2 * n < k or not k <= 3 * n ) by A2, NAT_1:12; ::_thesis: ( not n < k or not k <= 2 * n )
thus  ( not n < k or not k <= 2 * n ) by A4; ::_thesis: verum
end;
supposeA5: k > 3 * n ; ::_thesis: ( ( not 2 * n < k or not k <= 3 * n ) & ( not n < k or not k <= 2 * n ) )
hence  ( not 2 * n < k or not k <= 3 * n ) ; ::_thesis: ( not n < k or not k <= 2 * n )
thus  ( not n < k or not k <= 2 * n ) by A1, A5, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
assume that
A6: 2 * n < k and
A7: k <= 3 * n ; ::_thesis: ( ( not n < k or not k <= 2 * n ) & not k <= n & not k > 3 * n )
thus  ( not n < k or not k <= 2 * n ) by A6; ::_thesis: ( not k <= n & not k > 3 * n )
thus  ( not k <= n & not k > 3 * n ) by A2, A6, A7, NAT_1:12; ::_thesis: verum
end;

theorem Th34: :: GRAPHSP:34
for n being Element of NAT
for f being Element of REAL * st OuterVx (f,n) <> {} holds
ex j being Element of NAT st
( j in OuterVx (f,n) & 1 <= j & j <= n & ((findmin n) . f) . j = - 1 )
proof
let n be Element of NAT ; ::_thesis: for f being Element of REAL * st OuterVx (f,n) <> {} holds
ex j being Element of NAT st
( j in OuterVx (f,n) & 1 <= j & j <= n & ((findmin n) . f) . j = - 1 )
let f be Element of REAL * ; ::_thesis: ( OuterVx (f,n) <> {} implies ex j being Element of NAT st
( j in OuterVx (f,n) & 1 <= j & j <= n & ((findmin n) . f) . j = - 1 ) )
set IX = OuterVx (f,n);
assume  OuterVx (f,n) <> {} ; ::_thesis: ex j being Element of NAT st
( j in OuterVx (f,n) & 1 <= j & j <= n & ((findmin n) . f) . j = - 1 )
then consider i being Element of NAT  such that
A1: i = Argmin ((OuterVx (f,n)),f,n) and
A2: i in OuterVx (f,n) and
 for k being Element of NAT st k in OuterVx (f,n) holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) and
 for k being Element of NAT st k in OuterVx (f,n) & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k by Def10;
take  i ; ::_thesis: ( i in OuterVx (f,n) & 1 <= i & i <= n & ((findmin n) . f) . i = - 1 )
thus  i in OuterVx (f,n) by A2; ::_thesis: ( 1 <= i & i <= n & ((findmin n) . f) . i = - 1 )
A3: ex k being Element of NAT st
( i = k & k in dom f & 1 <= k & k <= n & f . k <> - 1 & f . (n + k) <> - 1 ) by A2;
hence  ( 1 <= i & i <= n ) ; ::_thesis: ((findmin n) . f) . i = - 1
thus ((findmin n) . f) . i = ((f,(((n * n) + (3 * n)) + 1)) := (i,(- 1))) . i by A1, Def11
.= - 1 by A3, Th19 ; ::_thesis: verum
end;

definition
let f be Element of REAL * ;
let n, k be Element of NAT ;
func newpathcost (f,n,k) ->  Real equals :: GRAPHSP:def 12
(f /. ((2 * n) + (f /. (((n * n) + (3 * n)) + 1)))) + (f /. (((2 * n) + (n * (f /. (((n * n) + (3 * n)) + 1)))) + k));
correctness
coherence
(f /. ((2 * n) + (f /. (((n * n) + (3 * n)) + 1)))) + (f /. (((2 * n) + (n * (f /. (((n * n) + (3 * n)) + 1)))) + k)) is Real;
;
end;

:: deftheorem  defines newpathcost GRAPHSP:def_12_:_
 for f being Element of REAL *
for n, k being Element of NAT holds newpathcost (f,n,k) = (f /. ((2 * n) + (f /. (((n * n) + (3 * n)) + 1)))) + (f /. (((2 * n) + (n * (f /. (((n * n) + (3 * n)) + 1)))) + k));

definition
let n, k be Element of NAT ;
let f be Element of REAL * ;
predf hasBetterPathAt n,k means :Def13: :: GRAPHSP:def 13
( ( f . (n + k) = - 1 or f /. ((2 * n) + k) > newpathcost (f,n,k) ) & f /. (((2 * n) + (n * (f /. (((n * n) + (3 * n)) + 1)))) + k) >= 0 & f . k <> - 1 );
end;

:: deftheorem Def13 defines hasBetterPathAt GRAPHSP:def_13_:_
 for n, k being Element of NAT
for f being Element of REAL * holds
( f hasBetterPathAt n,k iff ( ( f . (n + k) = - 1 or f /. ((2 * n) + k) > newpathcost (f,n,k) ) & f /. (((2 * n) + (n * (f /. (((n * n) + (3 * n)) + 1)))) + k) >= 0 & f . k <> - 1 ) );

definition
let f be Element of REAL * ;
let n be Element of NAT ;
func Relax (f,n) ->  Element of REAL *  means :Def14: :: GRAPHSP:def 14
( dom it = dom f & ( for k being Element of NAT st k in dom f holds
( ( n < k & k <= 2 * n implies ( ( f hasBetterPathAt n,k -' n implies it . k = f /. (((n * n) + (3 * n)) + 1) ) & ( not f hasBetterPathAt n,k -' n implies it . k = f . k ) ) ) & ( 2 * n < k & k <= 3 * n implies ( ( f hasBetterPathAt n,k -' (2 * n) implies it . k = newpathcost (f,n,(k -' (2 * n))) ) & ( not f hasBetterPathAt n,k -' (2 * n) implies it . k = f . k ) ) ) & ( ( k <= n or k > 3 * n ) implies it . k = f . k ) ) ) );
existence
 ex b1 being Element of REAL * st
( dom b1 = dom f & ( for k being Element of NAT st k in dom f holds
( ( n < k & k <= 2 * n implies ( ( f hasBetterPathAt n,k -' n implies b1 . k = f /. (((n * n) + (3 * n)) + 1) ) & ( not f hasBetterPathAt n,k -' n implies b1 . k = f . k ) ) ) & ( 2 * n < k & k <= 3 * n implies ( ( f hasBetterPathAt n,k -' (2 * n) implies b1 . k = newpathcost (f,n,(k -' (2 * n))) ) & ( not f hasBetterPathAt n,k -' (2 * n) implies b1 . k = f . k ) ) ) & ( ( k <= n or k > 3 * n ) implies b1 . k = f . k ) ) ) )
proof
defpred S1[ Element of NAT ] means f hasBetterPathAt n,$1 -' (2 * n);
defpred S2[ Element of NAT ] means f hasBetterPathAt n,$1 -' n;
set X = dom f;
defpred S3[ set , set ] means  for k being Element of NAT st $1 = k & k in dom f holds
( ( n < k & k <= 2 * n implies ( ( S2[k] implies $2 = f /. (((n * n) + (3 * n)) + 1) ) & ( not S2[k] implies $2 = f . k ) ) ) & ( 2 * n < k & k <= 3 * n implies ( ( S1[k] implies $2 = newpathcost (f,n,(k -' (2 * n))) ) & ( not S1[k] implies $2 = f . k ) ) ) & ( ( k <= n or k > 3 * n ) implies $2 = f . k ) );
A1: now__::_thesis:_for_xx_being_set_st_xx_in_dom_f_holds_
ex_y1_being_set_st_S3[xx,y1]
let xx be set ; ::_thesis: ( xx in dom f implies ex y1 being set st S3[y1,b2] )
assume  xx in dom f ; ::_thesis: ex y1 being set st S3[y1,b2]
then reconsider k = xx as Element of NAT ;
percases ( ( n < k & k <= 2 * n ) or ( 2 * n < k & k <= 3 * n ) or k <= n or k > 3 * n ) ;
supposeA2: ( n < k & k <= 2 * n ) ; ::_thesis: ex y1 being set st S3[y1,b2]
thus  ex y1 being set st S3[xx,y1] ::_thesis: verum
proof
percases ( S2[k] or not S2[k] ) ;
supposeA3: S2[k] ; ::_thesis: ex y1 being set st S3[xx,y1]
take y1 = f /. (((n * n) + (3 * n)) + 1); ::_thesis: S3[xx,y1]
thus  S3[xx,y1] by A2, A3, Lm8; ::_thesis: verum
end;
supposeA4: not S2[k] ; ::_thesis: ex y1 being set st S3[xx,y1]
take y1 = f . k; ::_thesis: S3[xx,y1]
thus  S3[xx,y1] by A2, A4; ::_thesis: verum
end;
end;
end;
end;
supposeA5: ( 2 * n < k & k <= 3 * n ) ; ::_thesis: ex y1 being set st S3[y1,b2]
thus  ex y1 being set st S3[xx,y1] ::_thesis: verum
proof
percases ( S1[k] or not S1[k] ) ;
supposeA6: S1[k] ; ::_thesis: ex y1 being set st S3[xx,y1]
take y1 = newpathcost (f,n,(k -' (2 * n))); ::_thesis: S3[xx,y1]
thus  S3[xx,y1] by A5, A6, Lm8; ::_thesis: verum
end;
supposeA7: not S1[k] ; ::_thesis: ex y1 being set st S3[xx,y1]
take y1 = f . k; ::_thesis: S3[xx,y1]
thus  S3[xx,y1] by A5, A7; ::_thesis: verum
end;
end;
end;
end;
supposeA8: ( k <= n or k > 3 * n ) ; ::_thesis: ex y1 being set st S3[y1,b2]
thus  ex y1 being set st S3[xx,y1] ::_thesis: verum
proof
take y1 = f . k; ::_thesis: S3[xx,y1]
thus  S3[xx,y1] by A8, Lm8; ::_thesis: verum
end;
end;
end;
end;
consider F being Function such that
A9: ( dom F = dom f & ( for x being set st x in dom f holds
S3[x,F . x] ) ) from CLASSES1:sch_1(A1);
A10: rng F c= REAL
proof
let y1 be set ; :: according to TARSKI:def_3 ::_thesis: ( not y1 in rng F or y1 in REAL )
assume  y1 in rng F ; ::_thesis: y1 in REAL
then consider xx being set  such that
A11: xx in dom F and
A12: y1 = F . xx by FUNCT_1:def_3;
reconsider k = xx as Element of NAT by A9, A11;
percases ( ( n < k & k <= 2 * n ) or ( 2 * n < k & k <= 3 * n ) or k <= n or k > 3 * n ) ;
supposeA13: ( n < k & k <= 2 * n ) ; ::_thesis: y1 in REAL
hereby  ::_thesis: verum
percases ( S2[k] or not S2[k] ) ;
suppose S2[k] ; ::_thesis: y1 in REAL
then  y1 = f /. (((n * n) + (3 * n)) + 1) by A9, A11, A12, A13;
hence  y1 in REAL ; ::_thesis: verum
end;
suppose not S2[k] ; ::_thesis: y1 in REAL
then y1 = f . k by A9, A11, A12, A13
.= f /. k by A9, A11, PARTFUN1:def_6 ;
hence  y1 in REAL ; ::_thesis: verum
end;
end;
end;
end;
supposeA14: ( 2 * n < k & k <= 3 * n ) ; ::_thesis: y1 in REAL
hereby  ::_thesis: verum
percases ( S1[k] or not S1[k] ) ;
suppose S1[k] ; ::_thesis: y1 in REAL
then  y1 = newpathcost (f,n,(k -' (2 * n))) by A9, A11, A12, A14;
hence  y1 in REAL ; ::_thesis: verum
end;
suppose not S1[k] ; ::_thesis: y1 in REAL
then y1 = f . k by A9, A11, A12, A14
.= f /. k by A9, A11, PARTFUN1:def_6 ;
hence  y1 in REAL ; ::_thesis: verum
end;
end;
end;
end;
suppose ( k <= n or k > 3 * n ) ; ::_thesis: y1 in REAL
then y1 = f . k by A9, A11, A12
.= f /. k by A9, A11, PARTFUN1:def_6 ;
hence  y1 in REAL ; ::_thesis: verum
end;
end;
end;
 dom f = Seg (len f) by FINSEQ_1:def_3;
then  F is FinSequence by A9, FINSEQ_1:def_2;
then  F is FinSequence of REAL by A10, FINSEQ_1:def_4;
then reconsider F = F as Element of REAL * by FINSEQ_1:def_11;
take  F ; ::_thesis: ( dom F = dom f & ( for k being Element of NAT st k in dom f holds
( ( n < k & k <= 2 * n implies ( ( f hasBetterPathAt n,k -' n implies F . k = f /. (((n * n) + (3 * n)) + 1) ) & ( not f hasBetterPathAt n,k -' n implies F . k = f . k ) ) ) & ( 2 * n < k & k <= 3 * n implies ( ( f hasBetterPathAt n,k -' (2 * n) implies F . k = newpathcost (f,n,(k -' (2 * n))) ) & ( not f hasBetterPathAt n,k -' (2 * n) implies F . k = f . k ) ) ) & ( ( k <= n or k > 3 * n ) implies F . k = f . k ) ) ) )
thus  dom F = dom f by A9; ::_thesis: for k being Element of NAT st k in dom f holds
( ( n < k & k <= 2 * n implies ( ( f hasBetterPathAt n,k -' n implies F . k = f /. (((n * n) + (3 * n)) + 1) ) & ( not f hasBetterPathAt n,k -' n implies F . k = f . k ) ) ) & ( 2 * n < k & k <= 3 * n implies ( ( f hasBetterPathAt n,k -' (2 * n) implies F . k = newpathcost (f,n,(k -' (2 * n))) ) & ( not f hasBetterPathAt n,k -' (2 * n) implies F . k = f . k ) ) ) & ( ( k <= n or k > 3 * n ) implies F . k = f . k ) )
let k be Element of NAT ; ::_thesis: ( k in dom f implies ( ( n < k & k <= 2 * n implies ( ( f hasBetterPathAt n,k -' n implies F . k = f /. (((n * n) + (3 * n)) + 1) ) & ( not f hasBetterPathAt n,k -' n implies F . k = f . k ) ) ) & ( 2 * n < k & k <= 3 * n implies ( ( f hasBetterPathAt n,k -' (2 * n) implies F . k = newpathcost (f,n,(k -' (2 * n))) ) & ( not f hasBetterPathAt n,k -' (2 * n) implies F . k = f . k ) ) ) & ( ( k <= n or k > 3 * n ) implies F . k = f . k ) ) )
assume  k in dom f ; ::_thesis: ( ( n < k & k <= 2 * n implies ( ( f hasBetterPathAt n,k -' n implies F . k = f /. (((n * n) + (3 * n)) + 1) ) & ( not f hasBetterPathAt n,k -' n implies F . k = f . k ) ) ) & ( 2 * n < k & k <= 3 * n implies ( ( f hasBetterPathAt n,k -' (2 * n) implies F . k = newpathcost (f,n,(k -' (2 * n))) ) & ( not f hasBetterPathAt n,k -' (2 * n) implies F . k = f . k ) ) ) & ( ( k <= n or k > 3 * n ) implies F . k = f . k ) )
hence  ( ( n < k & k <= 2 * n implies ( ( f hasBetterPathAt n,k -' n implies F . k = f /. (((n * n) + (3 * n)) + 1) ) & ( not f hasBetterPathAt n,k -' n implies F . k = f . k ) ) ) & ( 2 * n < k & k <= 3 * n implies ( ( f hasBetterPathAt n,k -' (2 * n) implies F . k = newpathcost (f,n,(k -' (2 * n))) ) & ( not f hasBetterPathAt n,k -' (2 * n) implies F . k = f . k ) ) ) & ( ( k <= n or k > 3 * n ) implies F . k = f . k ) ) by A9; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Element of REAL * st dom b1 = dom f & ( for k being Element of NAT st k in dom f holds
( ( n < k & k <= 2 * n implies ( ( f hasBetterPathAt n,k -' n implies b1 . k = f /. (((n * n) + (3 * n)) + 1) ) & ( not f hasBetterPathAt n,k -' n implies b1 . k = f . k ) ) ) & ( 2 * n < k & k <= 3 * n implies ( ( f hasBetterPathAt n,k -' (2 * n) implies b1 . k = newpathcost (f,n,(k -' (2 * n))) ) & ( not f hasBetterPathAt n,k -' (2 * n) implies b1 . k = f . k ) ) ) & ( ( k <= n or k > 3 * n ) implies b1 . k = f . k ) ) ) & dom b2 = dom f & ( for k being Element of NAT st k in dom f holds
( ( n < k & k <= 2 * n implies ( ( f hasBetterPathAt n,k -' n implies b2 . k = f /. (((n * n) + (3 * n)) + 1) ) & ( not f hasBetterPathAt n,k -' n implies b2 . k = f . k ) ) ) & ( 2 * n < k & k <= 3 * n implies ( ( f hasBetterPathAt n,k -' (2 * n) implies b2 . k = newpathcost (f,n,(k -' (2 * n))) ) & ( not f hasBetterPathAt n,k -' (2 * n) implies b2 . k = f . k ) ) ) & ( ( k <= n or k > 3 * n ) implies b2 . k = f . k ) ) ) holds
b1 = b2
proof
let F1, F2 be Element of REAL * ; ::_thesis: ( dom F1 = dom f & ( for k being Element of NAT st k in dom f holds
( ( n < k & k <= 2 * n implies ( ( f hasBetterPathAt n,k -' n implies F1 . k = f /. (((n * n) + (3 * n)) + 1) ) & ( not f hasBetterPathAt n,k -' n implies F1 . k = f . k ) ) ) & ( 2 * n < k & k <= 3 * n implies ( ( f hasBetterPathAt n,k -' (2 * n) implies F1 . k = newpathcost (f,n,(k -' (2 * n))) ) & ( not f hasBetterPathAt n,k -' (2 * n) implies F1 . k = f . k ) ) ) & ( ( k <= n or k > 3 * n ) implies F1 . k = f . k ) ) ) & dom F2 = dom f & ( for k being Element of NAT st k in dom f holds
( ( n < k & k <= 2 * n implies ( ( f hasBetterPathAt n,k -' n implies F2 . k = f /. (((n * n) + (3 * n)) + 1) ) & ( not f hasBetterPathAt n,k -' n implies F2 . k = f . k ) ) ) & ( 2 * n < k & k <= 3 * n implies ( ( f hasBetterPathAt n,k -' (2 * n) implies F2 . k = newpathcost (f,n,(k -' (2 * n))) ) & ( not f hasBetterPathAt n,k -' (2 * n) implies F2 . k = f . k ) ) ) & ( ( k <= n or k > 3 * n ) implies F2 . k = f . k ) ) ) implies F1 = F2 )
assume that
A15: dom F1 = dom f and
A16: for k being Element of NAT st k in dom f holds
( ( n < k & k <= 2 * n implies ( ( f hasBetterPathAt n,k -' n implies F1 . k = f /. (((n * n) + (3 * n)) + 1) ) & ( not f hasBetterPathAt n,k -' n implies F1 . k = f . k ) ) ) & ( 2 * n < k & k <= 3 * n implies ( ( f hasBetterPathAt n,k -' (2 * n) implies F1 . k = newpathcost (f,n,(k -' (2 * n))) ) & ( not f hasBetterPathAt n,k -' (2 * n) implies F1 . k = f . k ) ) ) & ( ( k <= n or k > 3 * n ) implies F1 . k = f . k ) ) and
A17: dom F2 = dom f and
A18: for k being Element of NAT st k in dom f holds
( ( n < k & k <= 2 * n implies ( ( f hasBetterPathAt n,k -' n implies F2 . k = f /. (((n * n) + (3 * n)) + 1) ) & ( not f hasBetterPathAt n,k -' n implies F2 . k = f . k ) ) ) & ( 2 * n < k & k <= 3 * n implies ( ( f hasBetterPathAt n,k -' (2 * n) implies F2 . k = newpathcost (f,n,(k -' (2 * n))) ) & ( not f hasBetterPathAt n,k -' (2 * n) implies F2 . k = f . k ) ) ) & ( ( k <= n or k > 3 * n ) implies F2 . k = f . k ) ) ; ::_thesis: F1 = F2
now__::_thesis:_for_xx_being_set_st_xx_in_dom_F1_holds_
F1_._xx_=_F2_._xx
let xx be set ; ::_thesis: ( xx in dom F1 implies F1 . b1 = F2 . b1 )
assume A19: xx in dom F1 ; ::_thesis: F1 . b1 = F2 . b1
then reconsider k = xx as Element of NAT ;
defpred S1[] means f hasBetterPathAt n,k -' (2 * n);
defpred S2[] means f hasBetterPathAt n,k -' n;
percases ( ( n < k & k <= 2 * n ) or ( 2 * n < k & k <= 3 * n ) or k <= n or k > 3 * n ) ;
supposeA20: ( n < k & k <= 2 * n ) ; ::_thesis: F1 . b1 = F2 . b1
hereby  ::_thesis: verum
percases ( S2[] or not S2[] ) ;
supposeA21: S2[] ; ::_thesis: F1 . xx = F2 . xx
hence F1 . xx = f /. (((n * n) + (3 * n)) + 1) by A15, A16, A19, A20
.= F2 . xx by A15, A18, A19, A20, A21 ;
::_thesis: verum
end;
supposeA22: not S2[] ; ::_thesis: F1 . xx = F2 . xx
hence F1 . xx = f . k by A15, A16, A19, A20
.= F2 . xx by A15, A18, A19, A20, A22 ;
::_thesis: verum
end;
end;
end;
end;
supposeA23: ( 2 * n < k & k <= 3 * n ) ; ::_thesis: F1 . b1 = F2 . b1
hereby  ::_thesis: verum
percases ( S1[] or not S1[] ) ;
supposeA24: S1[] ; ::_thesis: F1 . xx = F2 . xx
hence F1 . xx = newpathcost (f,n,(k -' (2 * n))) by A15, A16, A19, A23
.= F2 . xx by A15, A18, A19, A23, A24 ;
::_thesis: verum
end;
supposeA25: not S1[] ; ::_thesis: F1 . xx = F2 . xx
hence F1 . xx = f . k by A15, A16, A19, A23
.= F2 . xx by A15, A18, A19, A23, A25 ;
::_thesis: verum
end;
end;
end;
end;
supposeA26: ( k <= n or k > 3 * n ) ; ::_thesis: F1 . b1 = F2 . b1
hence F1 . xx = f . k by A15, A16, A19
.= F2 . xx by A15, A18, A19, A26 ;
::_thesis: verum
end;
end;
end;
hence  F1 = F2 by A15, A17, FUNCT_1:2; ::_thesis: verum
end;
end;

:: deftheorem Def14 defines Relax GRAPHSP:def_14_:_
 for f being Element of REAL *
for n being Element of NAT
for b3 being Element of REAL * holds
( b3 = Relax (f,n) iff ( dom b3 = dom f & ( for k being Element of NAT st k in dom f holds
( ( n < k & k <= 2 * n implies ( ( f hasBetterPathAt n,k -' n implies b3 . k = f /. (((n * n) + (3 * n)) + 1) ) & ( not f hasBetterPathAt n,k -' n implies b3 . k = f . k ) ) ) & ( 2 * n < k & k <= 3 * n implies ( ( f hasBetterPathAt n,k -' (2 * n) implies b3 . k = newpathcost (f,n,(k -' (2 * n))) ) & ( not f hasBetterPathAt n,k -' (2 * n) implies b3 . k = f . k ) ) ) & ( ( k <= n or k > 3 * n ) implies b3 . k = f . k ) ) ) ) );

definition
let n be Element of NAT ;
func Relax n ->  Element of Funcs ((REAL *),(REAL *)) means :Def15: :: GRAPHSP:def 15
( dom it = REAL * & ( for f being Element of REAL * holds it . f = Relax (f,n) ) );
existence
 ex b1 being Element of Funcs ((REAL *),(REAL *)) st
( dom b1 = REAL * & ( for f being Element of REAL * holds b1 . f = Relax (f,n) ) )
proof
defpred S1[ set , set ] means  for f being Element of REAL * st $1 = f holds
$2 = Relax (f,n);
set X = REAL * ;
A1: now__::_thesis:_for_xx_being_set_st_xx_in_REAL_*_holds_
ex_y1_being_set_st_S1[xx,y1]
let xx be set ; ::_thesis: ( xx in REAL * implies ex y1 being set st S1[xx,y1] )
assume  xx in REAL * ; ::_thesis: ex y1 being set st S1[xx,y1]
then reconsider h = xx as Element of REAL * ;
thus  ex y1 being set st S1[xx,y1] ::_thesis: verum
proof
take y1 = Relax (h,n); ::_thesis: S1[xx,y1]
thus  S1[xx,y1] ; ::_thesis: verum
end;
end;
consider F being Function such that
A2: ( dom F = REAL * & ( for x being set st x in REAL * holds
S1[x,F . x] ) ) from CLASSES1:sch_1(A1);
now__::_thesis:_for_y1_being_set_st_y1_in_rng_F_holds_
y1_in_REAL_*
let y1 be set ; ::_thesis: ( y1 in rng F implies y1 in REAL * )
assume  y1 in rng F ; ::_thesis: y1 in REAL *
then consider xx being set  such that
A3: xx in dom F and
A4: y1 = F . xx by FUNCT_1:def_3;
reconsider h = xx as Element of REAL * by A2, A3;
 y1 = Relax (h,n) by A2, A4;
hence  y1 in REAL * ; ::_thesis: verum
end;
then  rng F c= REAL * by TARSKI:def_3;
then reconsider F = F as Element of Funcs ((REAL *),(REAL *)) by A2, FUNCT_2:def_2;
take  F ; ::_thesis: ( dom F = REAL * & ( for f being Element of REAL * holds F . f = Relax (f,n) ) )
thus  dom F = REAL * by A2; ::_thesis: for f being Element of REAL * holds F . f = Relax (f,n)
let f be Element of REAL * ; ::_thesis: F . f = Relax (f,n)
thus  F . f = Relax (f,n) by A2; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Element of Funcs ((REAL *),(REAL *)) st dom b1 = REAL * & ( for f being Element of REAL * holds b1 . f = Relax (f,n) ) & dom b2 = REAL * & ( for f being Element of REAL * holds b2 . f = Relax (f,n) ) holds
b1 = b2
proof
set X = REAL * ;
let F1, F2 be Element of Funcs ((REAL *),(REAL *)); ::_thesis: ( dom F1 = REAL * & ( for f being Element of REAL * holds F1 . f = Relax (f,n) ) & dom F2 = REAL * & ( for f being Element of REAL * holds F2 . f = Relax (f,n) ) implies F1 = F2 )
assume that
A5: dom F1 = REAL * and
A6: for f being Element of REAL * holds F1 . f = Relax (f,n) and
A7: dom F2 = REAL * and
A8: for f being Element of REAL * holds F2 . f = Relax (f,n) ; ::_thesis: F1 = F2
now__::_thesis:_for_xx_being_set_st_xx_in_dom_F1_holds_
F1_._xx_=_F2_._xx
let xx be set ; ::_thesis: ( xx in dom F1 implies F1 . xx = F2 . xx )
assume  xx in dom F1 ; ::_thesis: F1 . xx = F2 . xx
then reconsider h = xx as Element of REAL * by A5;
thus F1 . xx = Relax (h,n) by A6
.= F2 . xx by A8 ; ::_thesis: verum
end;
hence  F1 = F2 by A5, A7, FUNCT_1:2; ::_thesis: verum
end;
end;

:: deftheorem Def15 defines Relax GRAPHSP:def_15_:_
 for n being Element of NAT
for b2 being Element of Funcs ((REAL *),(REAL *)) holds
( b2 = Relax n iff ( dom b2 = REAL * & ( for f being Element of REAL * holds b2 . f = Relax (f,n) ) ) );

theorem Th35: :: GRAPHSP:35
for n being Element of NAT
for f being Element of REAL * holds dom ((Relax n) . f) = dom f
proof
let n be Element of NAT ; ::_thesis: for f being Element of REAL * holds dom ((Relax n) . f) = dom f
let f be Element of REAL * ; ::_thesis: dom ((Relax n) . f) = dom f
thus  dom ((Relax n) . f) = dom (Relax (f,n)) by Def15
.= dom f by Def14 ; ::_thesis: verum
end;

theorem Th36: :: GRAPHSP:36
for i, n being Element of NAT
for f being Element of REAL * st ( i <= n or i > 3 * n ) & i in dom f holds
((Relax n) . f) . i = f . i
proof
let i, n be Element of NAT ; ::_thesis: for f being Element of REAL * st ( i <= n or i > 3 * n ) & i in dom f holds
((Relax n) . f) . i = f . i
let f be Element of REAL * ; ::_thesis: ( ( i <= n or i > 3 * n ) & i in dom f implies ((Relax n) . f) . i = f . i )
assume A1: ( ( i <= n or i > 3 * n ) & i in dom f ) ; ::_thesis: ((Relax n) . f) . i = f . i
thus ((Relax n) . f) . i = (Relax (f,n)) . i by Def15
.= f . i by A1, Def14 ; ::_thesis: verum
end;

theorem Th37: :: GRAPHSP:37
for n, i being Element of NAT
for f being Element of REAL * holds dom (((repeat ((Relax n) * (findmin n))) . i) . f) = dom (((repeat ((Relax n) * (findmin n))) . (i + 1)) . f)
proof
let n, i be Element of NAT ; ::_thesis: for f being Element of REAL * holds dom (((repeat ((Relax n) * (findmin n))) . i) . f) = dom (((repeat ((Relax n) * (findmin n))) . (i + 1)) . f)
let f be Element of REAL * ; ::_thesis: dom (((repeat ((Relax n) * (findmin n))) . i) . f) = dom (((repeat ((Relax n) * (findmin n))) . (i + 1)) . f)
set R = Relax n;
set M = findmin n;
set ff = ((repeat ((Relax n) * (findmin n))) . i) . f;
thus  dom (((repeat ((Relax n) * (findmin n))) . (i + 1)) . f) = dom ((Relax n) . ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f))) by Th22
.= dom ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f)) by Th35
.= dom (((repeat ((Relax n) * (findmin n))) . i) . f) by Th33 ; ::_thesis: verum
end;

theorem Th38: :: GRAPHSP:38
for n, i being Element of NAT
for f being Element of REAL * st OuterVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n) <> {} holds
UnusedVx ((((repeat ((Relax n) * (findmin n))) . (i + 1)) . f),n) c< UnusedVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n)
proof
let n, i be Element of NAT ; ::_thesis: for f being Element of REAL * st OuterVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n) <> {} holds
UnusedVx ((((repeat ((Relax n) * (findmin n))) . (i + 1)) . f),n) c< UnusedVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n)
let f be Element of REAL * ; ::_thesis: ( OuterVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n) <> {} implies UnusedVx ((((repeat ((Relax n) * (findmin n))) . (i + 1)) . f),n) c< UnusedVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n) )
set R = Relax n;
set M = findmin n;
set ff = ((repeat ((Relax n) * (findmin n))) . i) . f;
set Fi1 = ((repeat ((Relax n) * (findmin n))) . (i + 1)) . f;
assume  OuterVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n) <> {} ; ::_thesis: UnusedVx ((((repeat ((Relax n) * (findmin n))) . (i + 1)) . f),n) c< UnusedVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n)
then A1: ex j being Element of NAT st
( j in OuterVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n) & 1 <= j & j <= n & ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f)) . j = - 1 ) by Th34;
A2: dom (((repeat ((Relax n) * (findmin n))) . (i + 1)) . f) = dom (((repeat ((Relax n) * (findmin n))) . i) . f) by Th37
.= dom ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f)) by Th33 ;
A3: now__::_thesis:_for_x_being_set_st_x_in_UnusedVx_((((repeat_((Relax_n)_*_(findmin_n)))_._(i_+_1))_._f),n)_holds_
((findmin_n)_._(((repeat_((Relax_n)_*_(findmin_n)))_._i)_._f))_._x_<>_-_1
let x be set ; ::_thesis: ( x in UnusedVx ((((repeat ((Relax n) * (findmin n))) . (i + 1)) . f),n) implies ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f)) . x <> - 1 )
assume  x in UnusedVx ((((repeat ((Relax n) * (findmin n))) . (i + 1)) . f),n) ; ::_thesis: ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f)) . x <> - 1
then consider k being Element of NAT  such that
A4: x = k and
A5: k in dom (((repeat ((Relax n) * (findmin n))) . (i + 1)) . f) and
 1 <= k and
A6: k <= n and
A7: (((repeat ((Relax n) * (findmin n))) . (i + 1)) . f) . k <> - 1 ;
(((repeat ((Relax n) * (findmin n))) . (i + 1)) . f) . k = ((Relax n) . ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f))) . k by Th22
.= ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f)) . k by A2, A5, A6, Th36 ;
hence  ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f)) . x <> - 1 by A4, A7; ::_thesis: verum
end;
A8: UnusedVx ((((repeat ((Relax n) * (findmin n))) . (i + 1)) . f),n) c= UnusedVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n)
proof
A9: n < ((n * n) + (3 * n)) + 1 by Lm7;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in UnusedVx ((((repeat ((Relax n) * (findmin n))) . (i + 1)) . f),n) or x in UnusedVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n) )
assume  x in UnusedVx ((((repeat ((Relax n) * (findmin n))) . (i + 1)) . f),n) ; ::_thesis: x in UnusedVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n)
then consider k being Element of NAT  such that
A10: x = k and
A11: k in dom (((repeat ((Relax n) * (findmin n))) . (i + 1)) . f) and
A12: 1 <= k and
A13: k <= n and
A14: (((repeat ((Relax n) * (findmin n))) . (i + 1)) . f) . k <> - 1 ;
A15: k in dom (((repeat ((Relax n) * (findmin n))) . i) . f) by A11, Th37;
(((repeat ((Relax n) * (findmin n))) . (i + 1)) . f) . k = ((Relax n) . ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f))) . k by Th22
.= ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f)) . k by A2, A11, A13, Th36 ;
then  (((repeat ((Relax n) * (findmin n))) . i) . f) . k <> - 1 by A13, A14, A15, A9, Th32;
hence  x in UnusedVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n) by A10, A12, A13, A15; ::_thesis: verum
end;
 OuterVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n) c= UnusedVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n) by Th27;
then  UnusedVx ((((repeat ((Relax n) * (findmin n))) . (i + 1)) . f),n) <> UnusedVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n) by A1, A3;
hence  UnusedVx ((((repeat ((Relax n) * (findmin n))) . (i + 1)) . f),n) c< UnusedVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n) by A8, XBOOLE_0:def_8; ::_thesis: verum
end;

theorem Th39: :: GRAPHSP:39
for n, i, k being Element of NAT
for g, f, h being Element of REAL * st g = ((repeat ((Relax n) * (findmin n))) . i) . f & h = ((repeat ((Relax n) * (findmin n))) . (i + 1)) . f & k = Argmin ((OuterVx (g,n)),g,n) & OuterVx (g,n) <> {} holds
( UsedVx (h,n) = (UsedVx (g,n)) \/ {k} & not k in UsedVx (g,n) )
proof
let n, i, k be Element of NAT ; ::_thesis: for g, f, h being Element of REAL * st g = ((repeat ((Relax n) * (findmin n))) . i) . f & h = ((repeat ((Relax n) * (findmin n))) . (i + 1)) . f & k = Argmin ((OuterVx (g,n)),g,n) & OuterVx (g,n) <> {} holds
( UsedVx (h,n) = (UsedVx (g,n)) \/ {k} & not k in UsedVx (g,n) )
let g, f, h be Element of REAL * ; ::_thesis: ( g = ((repeat ((Relax n) * (findmin n))) . i) . f & h = ((repeat ((Relax n) * (findmin n))) . (i + 1)) . f & k = Argmin ((OuterVx (g,n)),g,n) & OuterVx (g,n) <> {} implies ( UsedVx (h,n) = (UsedVx (g,n)) \/ {k} & not k in UsedVx (g,n) ) )
set R = Relax n;
set M = findmin n;
set ff = ((repeat ((Relax n) * (findmin n))) . i) . f;
set Fi1 = ((repeat ((Relax n) * (findmin n))) . (i + 1)) . f;
set mi = ((n * n) + (3 * n)) + 1;
assume that
A1: g = ((repeat ((Relax n) * (findmin n))) . i) . f and
A2: h = ((repeat ((Relax n) * (findmin n))) . (i + 1)) . f and
A3: k = Argmin ((OuterVx (g,n)),g,n) and
A4: OuterVx (g,n) <> {} ; ::_thesis: ( UsedVx (h,n) = (UsedVx (g,n)) \/ {k} & not k in UsedVx (g,n) )
A5: (findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f) = ((((repeat ((Relax n) * (findmin n))) . i) . f),(((n * n) + (3 * n)) + 1)) := (k,(- 1)) by A1, A3, Def11;
A6: dom h = dom (((repeat ((Relax n) * (findmin n))) . i) . f) by A2, Th37;
A7: dom g = dom ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f)) by A1, Th33;
A8: now__::_thesis:_for_x_being_set_st_x_in_(UsedVx_(g,n))_\/_{k}_holds_
x_in_UsedVx_(h,n)
let x be set ; ::_thesis: ( x in (UsedVx (g,n)) \/ {k} implies b1 in UsedVx (h,n) )
assume A9: x in (UsedVx (g,n)) \/ {k} ; ::_thesis: b1 in UsedVx (h,n)
percases ( x in UsedVx (g,n) or x in {k} ) by A9, XBOOLE_0:def_3;
supposeA10: x in UsedVx (g,n) ; ::_thesis: b1 in UsedVx (h,n)
A11: n < ((n * n) + (3 * n)) + 1 by Lm7;
consider m being Element of NAT  such that
A12: x = m and
A13: m in dom g and
A14: 1 <= m and
A15: m <= n and
A16: g . m = - 1 by A10;
h . m = ((Relax n) . ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f))) . m by A2, Th22
.= ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f)) . m by A7, A13, A15, Th36
.= - 1 by A1, A13, A15, A16, A11, Th32 ;
hence  x in UsedVx (h,n) by A1, A6, A12, A13, A14, A15; ::_thesis: verum
end;
suppose x in {k} ; ::_thesis: b1 in UsedVx (h,n)
then A17: x = k by TARSKI:def_1;
A18: k in dom g by A3, A4, Th29;
A19: 1 <= k by A3, A4, Th29;
A20: k <= n by A3, A4, Th29;
h . k = ((Relax n) . ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f))) . k by A2, Th22
.= ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f)) . k by A7, A18, A20, Th36
.= - 1 by A1, A5, A18, Th19 ;
hence  x in UsedVx (h,n) by A1, A6, A17, A18, A19, A20; ::_thesis: verum
end;
end;
end;
A21: dom h = dom ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f)) by A6, Th33;
now__::_thesis:_for_x_being_set_st_x_in_UsedVx_(h,n)_holds_
x_in_(UsedVx_(g,n))_\/_{k}
let x be set ; ::_thesis: ( x in UsedVx (h,n) implies b1 in (UsedVx (g,n)) \/ {k} )
assume  x in UsedVx (h,n) ; ::_thesis: b1 in (UsedVx (g,n)) \/ {k}
then consider m being Element of NAT  such that
A22: x = m and
A23: m in dom h and
A24: 1 <= m and
A25: m <= n and
A26: h . m = - 1 ;
percases ( m = k or m <> k ) ;
suppose m = k ; ::_thesis: b1 in (UsedVx (g,n)) \/ {k}
then  x in {k} by A22, TARSKI:def_1;
hence  x in (UsedVx (g,n)) \/ {k} by XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA27: m <> k ; ::_thesis: b1 in (UsedVx (g,n)) \/ {k}
A28: n < ((n * n) + (3 * n)) + 1 by Lm7;
- 1 = ((Relax n) . ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f))) . m by A2, A26, Th22
.= ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f)) . m by A21, A23, A25, Th36
.= (((repeat ((Relax n) * (findmin n))) . i) . f) . m by A5, A25, A27, A28, Th18 ;
then  m in  {  j where j is Element of NAT : ( j in dom (((repeat ((Relax n) * (findmin n))) . i) . f) & 1 <= j & j <= n & (((repeat ((Relax n) * (findmin n))) . i) . f) . j = - 1 )  }  by A6, A23, A24, A25;
hence  x in (UsedVx (g,n)) \/ {k} by A1, A22, XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
hence  UsedVx (h,n) = (UsedVx (g,n)) \/ {k} by A8, TARSKI:1; ::_thesis: not k in UsedVx (g,n)
assume  k in UsedVx (g,n) ; ::_thesis: contradiction
then  ex j being Element of NAT st
( j = k & j in dom g & 1 <= j & j <= n & g . j = - 1 ) ;
hence  contradiction by A3, A4, Th29; ::_thesis: verum
end;

theorem Th40: :: GRAPHSP:40
for n being Element of NAT
for f being Element of REAL * ex i being Element of NAT st
( i <= n & OuterVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n) = {} )
proof
let n be Element of NAT ; ::_thesis: for f being Element of REAL * ex i being Element of NAT st
( i <= n & OuterVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n) = {} )
let f be Element of REAL * ; ::_thesis: ex i being Element of NAT st
( i <= n & OuterVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n) = {} )
set R = Relax n;
set M = findmin n;
defpred S1[ Element of NAT ] means ( $1 <= n implies card (UnusedVx ((((repeat ((Relax n) * (findmin n))) . $1) . f),n)) <= n - $1 );
set nf = ((repeat ((Relax n) * (findmin n))) . n) . f;
assume A1: for i being Element of NAT holds
( not i <= n or not OuterVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n) = {} ) ; ::_thesis: contradiction
A2: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A3: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_(_k_+_1_<=_n_implies_card_(UnusedVx_((((repeat_((Relax_n)_*_(findmin_n)))_._(k_+_1))_._f),n))_<=_n_-_(k_+_1)_)
set fk = UnusedVx ((((repeat ((Relax n) * (findmin n))) . k) . f),n);
set fk1 = UnusedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n);
A4: k <= k + 1 by NAT_1:11;
assume A5: k + 1 <= n ; ::_thesis: card (UnusedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n)) <= n - (k + 1)
then  OuterVx ((((repeat ((Relax n) * (findmin n))) . k) . f),n) <> {} by A1, A4, XXREAL_0:2;
then  UnusedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) c< UnusedVx ((((repeat ((Relax n) * (findmin n))) . k) . f),n) by Th38;
then  card (UnusedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n)) < n - k by A3, A5, A4, CARD_2:48, XXREAL_0:2;
then  (card (UnusedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n))) + 1 <= n - k by INT_1:7;
then  card (UnusedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n)) <= (n - k) - 1 by XREAL_1:19;
hence  card (UnusedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n)) <= n - (k + 1) ; ::_thesis: verum
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
A6: S1[ 0 ]
proof
set f0 = ((repeat ((Relax n) * (findmin n))) . 0) . f;
assume  0 <= n ; ::_thesis: card (UnusedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n)) <= n - 0
 card (UnusedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n)) <= card (Seg n) by Th26, NAT_1:43;
hence  card (UnusedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n)) <= n - 0 by FINSEQ_1:57; ::_thesis: verum
end;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A6, A2);
then  S1[n] ;
then A7: UnusedVx ((((repeat ((Relax n) * (findmin n))) . n) . f),n) = {} ;
 OuterVx ((((repeat ((Relax n) * (findmin n))) . n) . f),n) c= UnusedVx ((((repeat ((Relax n) * (findmin n))) . n) . f),n) by Th27;
hence  contradiction by A1, A7, XBOOLE_1:3; ::_thesis: verum
end;

Lm9:  for n, k being Element of NAT holds n - k <= n
proof
let n, k be Element of NAT ; ::_thesis: n - k <= n
 n <= n + k by NAT_1:11;
hence  n - k <= n by XREAL_1:20; ::_thesis: verum
end;

Lm10:  for p, q being FinSequence of NAT
for f being Element of REAL *
for i, n being Element of NAT st ( for k being Element of NAT st 1 <= k & k < len p holds
p . ((len p) - k) = f . ((p /. (((len p) - k) + 1)) + n) ) & ( for k being Element of NAT st 1 <= k & k < len q holds
q . ((len q) - k) = f . ((q /. (((len q) - k) + 1)) + n) ) & len p <= len q & p . (len p) = q . (len q) holds
for k being Element of NAT st 1 <= k & k < len p holds
p . ((len p) - k) = q . ((len q) - k)
proof
let p, q be FinSequence of NAT ; ::_thesis: for f being Element of REAL *
for i, n being Element of NAT st ( for k being Element of NAT st 1 <= k & k < len p holds
p . ((len p) - k) = f . ((p /. (((len p) - k) + 1)) + n) ) & ( for k being Element of NAT st 1 <= k & k < len q holds
q . ((len q) - k) = f . ((q /. (((len q) - k) + 1)) + n) ) & len p <= len q & p . (len p) = q . (len q) holds
for k being Element of NAT st 1 <= k & k < len p holds
p . ((len p) - k) = q . ((len q) - k)
let f be Element of REAL * ; ::_thesis: for i, n being Element of NAT st ( for k being Element of NAT st 1 <= k & k < len p holds
p . ((len p) - k) = f . ((p /. (((len p) - k) + 1)) + n) ) & ( for k being Element of NAT st 1 <= k & k < len q holds
q . ((len q) - k) = f . ((q /. (((len q) - k) + 1)) + n) ) & len p <= len q & p . (len p) = q . (len q) holds
for k being Element of NAT st 1 <= k & k < len p holds
p . ((len p) - k) = q . ((len q) - k)
let i, n be Element of NAT ; ::_thesis: ( ( for k being Element of NAT st 1 <= k & k < len p holds
p . ((len p) - k) = f . ((p /. (((len p) - k) + 1)) + n) ) & ( for k being Element of NAT st 1 <= k & k < len q holds
q . ((len q) - k) = f . ((q /. (((len q) - k) + 1)) + n) ) & len p <= len q & p . (len p) = q . (len q) implies for k being Element of NAT st 1 <= k & k < len p holds
p . ((len p) - k) = q . ((len q) - k) )
assume that
A1: for k being Element of NAT st 1 <= k & k < len p holds
p . ((len p) - k) = f . ((p /. (((len p) - k) + 1)) + n) and
A2: for k being Element of NAT st 1 <= k & k < len q holds
q . ((len q) - k) = f . ((q /. (((len q) - k) + 1)) + n) and
A3: len p <= len q and
A4: p . (len p) = q . (len q) ; ::_thesis: for k being Element of NAT st 1 <= k & k < len p holds
p . ((len p) - k) = q . ((len q) - k)
defpred S1[ Element of NAT ] means ( $1 < len p implies p . ((len p) - $1) = q . ((len q) - $1) );
A5: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A6: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_(_k_+_1_<_len_p_implies_p_._((len_p)_-_(k_+_1))_=_q_._((len_q)_-_(k_+_1))_)
A7: ( k <= k + 1 & (len q) - k <= len q ) by Lm9, NAT_1:11;
assume A8: k + 1 < len p ; ::_thesis: p . ((len p) - (k + 1)) = q . ((len q) - (k + 1))
then A9: 1 <= (len p) - k by XREAL_1:19;
 (len p) - k <= (len q) - k by A3, XREAL_1:9;
then A10: 1 <= (len q) - k by A9, XXREAL_0:2;
A11: k + 1 < len q by A3, A8, XXREAL_0:2;
A12: p /. ((len p) - k) = p . ((len p) - k) by A9, Lm9, Th3
.= q /. ((len q) - k) by A6, A8, A7, A10, Th3, XXREAL_0:2 ;
thus p . ((len p) - (k + 1)) = f . ((p /. (((len p) - (k + 1)) + 1)) + n) by A1, A8, NAT_1:11
.= f . ((q /. (((len q) - (k + 1)) + 1)) + n) by A12
.= q . ((len q) - (k + 1)) by A2, A11, NAT_1:11 ; ::_thesis: verum
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
A13: S1[ 0 ] by A4;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A13, A5);
hence  for k being Element of NAT st 1 <= k & k < len p holds
p . ((len p) - k) = q . ((len q) - k) ; ::_thesis: verum
end;

theorem Th41: :: GRAPHSP:41
for n, i being Element of NAT
for f being Element of REAL * holds dom f = dom (((repeat ((Relax n) * (findmin n))) . i) . f)
proof
let n, i be Element of NAT ; ::_thesis: for f being Element of REAL * holds dom f = dom (((repeat ((Relax n) * (findmin n))) . i) . f)
let f be Element of REAL * ; ::_thesis: dom f = dom (((repeat ((Relax n) * (findmin n))) . i) . f)
set R = Relax n;
set M = findmin n;
defpred S1[ Element of NAT ] means  dom f = dom (((repeat ((Relax n) * (findmin n))) . $1) . f);
dom (((repeat ((Relax n) * (findmin n))) . 0) . f) = dom ((id (REAL *)) . f) by Def2
.= dom f by FUNCT_1:18 ;
then A1: S1[ 0 ] ;
A2: for k being Element of NAT st S1[k] holds
S1[k + 1] by Th37;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A1, A2);
hence  dom f = dom (((repeat ((Relax n) * (findmin n))) . i) . f) ; ::_thesis: verum
end;

definition
let f, g be Element of REAL * ;
let m, n be Element of NAT ;
predf,g equal_at m,n means :Def16: :: GRAPHSP:def 16
( dom f = dom g & ( for k being Element of NAT st k in dom f & m <= k & k <= n holds
f . k = g . k ) );
end;

:: deftheorem Def16 defines equal_at GRAPHSP:def_16_:_
 for f, g being Element of REAL *
for m, n being Element of NAT holds
( f,g equal_at m,n iff ( dom f = dom g & ( for k being Element of NAT st k in dom f & m <= k & k <= n holds
f . k = g . k ) ) );

theorem Th42: :: GRAPHSP:42
for m, n being Element of NAT
for f being Element of REAL * holds f,f equal_at m,n
proof
let m, n be Element of NAT ; ::_thesis: for f being Element of REAL * holds f,f equal_at m,n
let f be Element of REAL * ; ::_thesis: f,f equal_at m,n
 for k being Element of NAT st k in dom f & m <= k & k <= n holds
f . k = f . k ;
hence  f,f equal_at m,n by Def16; ::_thesis: verum
end;

theorem Th43: :: GRAPHSP:43
for m, n being Element of NAT
for f, g, h being Element of REAL * st f,g equal_at m,n & g,h equal_at m,n holds
f,h equal_at m,n
proof
let m, n be Element of NAT ; ::_thesis: for f, g, h being Element of REAL * st f,g equal_at m,n & g,h equal_at m,n holds
f,h equal_at m,n
let f, g, h be Element of REAL * ; ::_thesis: ( f,g equal_at m,n & g,h equal_at m,n implies f,h equal_at m,n )
assume that
A1: f,g equal_at m,n and
A2: g,h equal_at m,n ; ::_thesis: f,h equal_at m,n
A3: dom f = dom g by A1, Def16;
A4: now__::_thesis:_for_k_being_Element_of_NAT_st_k_in_dom_f_&_m_<=_k_&_k_<=_n_holds_
f_._k_=_h_._k
let k be Element of NAT ; ::_thesis: ( k in dom f & m <= k & k <= n implies f . k = h . k )
assume A5: ( k in dom f & m <= k & k <= n ) ; ::_thesis: f . k = h . k
hence f . k = g . k by A1, Def16
.= h . k by A2, A3, A5, Def16 ;
::_thesis: verum
end;
 dom g = dom h by A2, Def16;
hence  f,h equal_at m,n by A3, A4, Def16; ::_thesis: verum
end;

theorem Th44: :: GRAPHSP:44
for n, i being Element of NAT
for f being Element of REAL * holds ((repeat ((Relax n) * (findmin n))) . i) . f,((repeat ((Relax n) * (findmin n))) . (i + 1)) . f equal_at (3 * n) + 1,(n * n) + (3 * n)
proof
let n, i be Element of NAT ; ::_thesis: for f being Element of REAL * holds ((repeat ((Relax n) * (findmin n))) . i) . f,((repeat ((Relax n) * (findmin n))) . (i + 1)) . f equal_at (3 * n) + 1,(n * n) + (3 * n)
let f be Element of REAL * ; ::_thesis: ((repeat ((Relax n) * (findmin n))) . i) . f,((repeat ((Relax n) * (findmin n))) . (i + 1)) . f equal_at (3 * n) + 1,(n * n) + (3 * n)
set R = Relax n;
set M = findmin n;
set ff = ((repeat ((Relax n) * (findmin n))) . i) . f;
set Fi1 = ((repeat ((Relax n) * (findmin n))) . (i + 1)) . f;
A1: now__::_thesis:_for_k_being_Element_of_NAT_st_k_in_dom_(((repeat_((Relax_n)_*_(findmin_n)))_._i)_._f)_&_(3_*_n)_+_1_<=_k_&_k_<=_(n_*_n)_+_(3_*_n)_holds_
(((repeat_((Relax_n)_*_(findmin_n)))_._(i_+_1))_._f)_._k_=_(((repeat_((Relax_n)_*_(findmin_n)))_._i)_._f)_._k
let k be Element of NAT ; ::_thesis: ( k in dom (((repeat ((Relax n) * (findmin n))) . i) . f) & (3 * n) + 1 <= k & k <= (n * n) + (3 * n) implies (((repeat ((Relax n) * (findmin n))) . (i + 1)) . f) . k = (((repeat ((Relax n) * (findmin n))) . i) . f) . k )
assume that
A2: k in dom (((repeat ((Relax n) * (findmin n))) . i) . f) and
A3: (3 * n) + 1 <= k and
A4: k <= (n * n) + (3 * n) ; ::_thesis: (((repeat ((Relax n) * (findmin n))) . (i + 1)) . f) . k = (((repeat ((Relax n) * (findmin n))) . i) . f) . k
A5: k > 3 * n by A3, NAT_1:13;
A6: k in dom ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f)) by A2, Th33;
A7: k < ((n * n) + (3 * n)) + 1 by A4, NAT_1:13;
 3 * n >= n by Lm6;
then A8: k > n by A5, XXREAL_0:2;
thus (((repeat ((Relax n) * (findmin n))) . (i + 1)) . f) . k = ((Relax n) . ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f))) . k by Th22
.= ((findmin n) . (((repeat ((Relax n) * (findmin n))) . i) . f)) . k by A5, A6, Th36
.= (((repeat ((Relax n) * (findmin n))) . i) . f) . k by A7, A8, Th31 ; ::_thesis: verum
end;
 dom (((repeat ((Relax n) * (findmin n))) . (i + 1)) . f) = dom (((repeat ((Relax n) * (findmin n))) . i) . f) by Th37;
hence  ((repeat ((Relax n) * (findmin n))) . i) . f,((repeat ((Relax n) * (findmin n))) . (i + 1)) . f equal_at (3 * n) + 1,(n * n) + (3 * n) by A1, Def16; ::_thesis: verum
end;

theorem :: GRAPHSP:45
for F being Element of Funcs ((REAL *),(REAL *))
for f being Element of REAL *
for n, i being Element of NAT st i < LifeSpan (F,f,n) holds
OuterVx ((((repeat F) . i) . f),n) <> {} by Def4;

theorem Th46: :: GRAPHSP:46
for n, i being Element of NAT
for f being Element of REAL * holds f,((repeat ((Relax n) * (findmin n))) . i) . f equal_at (3 * n) + 1,(n * n) + (3 * n)
proof
let n, i be Element of NAT ; ::_thesis: for f being Element of REAL * holds f,((repeat ((Relax n) * (findmin n))) . i) . f equal_at (3 * n) + 1,(n * n) + (3 * n)
let f be Element of REAL * ; ::_thesis: f,((repeat ((Relax n) * (findmin n))) . i) . f equal_at (3 * n) + 1,(n * n) + (3 * n)
set R = Relax n;
set M = findmin n;
set m = (3 * n) + 1;
set mm = (n * n) + (3 * n);
defpred S1[ Element of NAT ] means f,((repeat ((Relax n) * (findmin n))) . $1) . f equal_at (3 * n) + 1,(n * n) + (3 * n);
A1: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A2: S1[k] ; ::_thesis: S1[k + 1]
 ((repeat ((Relax n) * (findmin n))) . k) . f,((repeat ((Relax n) * (findmin n))) . (k + 1)) . f equal_at (3 * n) + 1,(n * n) + (3 * n) by Th44;
hence  S1[k + 1] by A2, Th43; ::_thesis: verum
end;
 ((repeat ((Relax n) * (findmin n))) . 0) . f = f by Th21;
then A3: S1[ 0 ] by Th42;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A3, A1);
hence  f,((repeat ((Relax n) * (findmin n))) . i) . f equal_at (3 * n) + 1,(n * n) + (3 * n) ; ::_thesis: verum
end;

theorem Th47: :: GRAPHSP:47
for n being Element of NAT
for f being Element of REAL * st 1 <= n & 1 in dom f & f . (n + 1) <> - 1 & ( for i being Element of NAT st 1 <= i & i <= n holds
f . i = 1 ) & ( for i being Element of NAT st 2 <= i & i <= n holds
f . (n + i) = - 1 ) holds
( 1 = Argmin ((OuterVx (f,n)),f,n) & UsedVx (f,n) = {} & {1} = UsedVx ((((repeat ((Relax n) * (findmin n))) . 1) . f),n) )
proof
let n be Element of NAT ; ::_thesis: for f being Element of REAL * st 1 <= n & 1 in dom f & f . (n + 1) <> - 1 & ( for i being Element of NAT st 1 <= i & i <= n holds
f . i = 1 ) & ( for i being Element of NAT st 2 <= i & i <= n holds
f . (n + i) = - 1 ) holds
( 1 = Argmin ((OuterVx (f,n)),f,n) & UsedVx (f,n) = {} & {1} = UsedVx ((((repeat ((Relax n) * (findmin n))) . 1) . f),n) )
let f be Element of REAL * ; ::_thesis: ( 1 <= n & 1 in dom f & f . (n + 1) <> - 1 & ( for i being Element of NAT st 1 <= i & i <= n holds
f . i = 1 ) & ( for i being Element of NAT st 2 <= i & i <= n holds
f . (n + i) = - 1 ) implies ( 1 = Argmin ((OuterVx (f,n)),f,n) & UsedVx (f,n) = {} & {1} = UsedVx ((((repeat ((Relax n) * (findmin n))) . 1) . f),n) ) )
set R = Relax n;
set M = findmin n;
set f0 = ((repeat ((Relax n) * (findmin n))) . 0) . f;
set RT = repeat ((Relax n) * (findmin n));
assume that
A1: 1 <= n and
A2: ( 1 in dom f & f . (n + 1) <> - 1 ) and
A3: for i being Element of NAT st 1 <= i & i <= n holds
f . i = 1 and
A4: for i being Element of NAT st 2 <= i & i <= n holds
f . (n + i) = - 1 ; ::_thesis: ( 1 = Argmin ((OuterVx (f,n)),f,n) & UsedVx (f,n) = {} & {1} = UsedVx ((((repeat ((Relax n) * (findmin n))) . 1) . f),n) )
set k = Argmin ((OuterVx (f,n)),f,n);
 f . 1 = 1 by A1, A3;
then A5: 1 in  {  j where j is Element of NAT : ( j in dom f & 1 <= j & j <= n & f . j <> - 1 & f . (n + j) <> - 1 )  }  by A1, A2;
thus A6: Argmin ((OuterVx (f,n)),f,n) = 1 ::_thesis: ( UsedVx (f,n) = {} & {1} = UsedVx ((((repeat ((Relax n) * (findmin n))) . 1) . f),n) )
proof
assume A7: Argmin ((OuterVx (f,n)),f,n) <> 1 ; ::_thesis: contradiction
 1 <= Argmin ((OuterVx (f,n)),f,n) by A5, Th29;
then  1 < Argmin ((OuterVx (f,n)),f,n) by A7, XXREAL_0:1;
then A8: 1 + 1 <= Argmin ((OuterVx (f,n)),f,n) by INT_1:7;
 ( Argmin ((OuterVx (f,n)),f,n) <= n & f . (n + (Argmin ((OuterVx (f,n)),f,n))) <> - 1 ) by A5, Th29;
hence  contradiction by A4, A8; ::_thesis: verum
end;
thus A9: UsedVx (f,n) = {} ::_thesis: {1} = UsedVx ((((repeat ((Relax n) * (findmin n))) . 1) . f),n)
proof
assume  UsedVx (f,n) <> {} ; ::_thesis: contradiction
then consider x being set  such that
A10: x in UsedVx (f,n) by XBOOLE_0:def_1;
 ex j being Element of NAT st
( x = j & j in dom f & 1 <= j & j <= n & f . j = - 1 ) by A10;
hence  contradiction by A3; ::_thesis: verum
end;
 OuterVx (f,n) <> {} by A5;
then A11: OuterVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n) <> {} by Th21;
A12: Argmin ((OuterVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n)),(((repeat ((Relax n) * (findmin n))) . 0) . f),n) = Argmin ((OuterVx (f,n)),(((repeat ((Relax n) * (findmin n))) . 0) . f),n) by Th21
.= 1 by A6, Th21 ;
thus  UsedVx ((((repeat ((Relax n) * (findmin n))) . 1) . f),n) = UsedVx ((((repeat ((Relax n) * (findmin n))) . (0 + 1)) . f),n)
.= (UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n)) \/ {1} by A11, A12, Th39
.= (UsedVx (f,n)) \/ {1} by Th21
.= {1} by A9 ; ::_thesis: verum
end;

theorem Th48: :: GRAPHSP:48
for n, i, m being Element of NAT
for g, f, h being Element of REAL * st g = ((repeat ((Relax n) * (findmin n))) . 1) . f & h = ((repeat ((Relax n) * (findmin n))) . i) . f & 1 <= i & i <= LifeSpan (((Relax n) * (findmin n)),f,n) & m in UsedVx (g,n) holds
m in UsedVx (h,n)
proof
let n, i, m be Element of NAT ; ::_thesis: for g, f, h being Element of REAL * st g = ((repeat ((Relax n) * (findmin n))) . 1) . f & h = ((repeat ((Relax n) * (findmin n))) . i) . f & 1 <= i & i <= LifeSpan (((Relax n) * (findmin n)),f,n) & m in UsedVx (g,n) holds
m in UsedVx (h,n)
let g, f, h be Element of REAL * ; ::_thesis: ( g = ((repeat ((Relax n) * (findmin n))) . 1) . f & h = ((repeat ((Relax n) * (findmin n))) . i) . f & 1 <= i & i <= LifeSpan (((Relax n) * (findmin n)),f,n) & m in UsedVx (g,n) implies m in UsedVx (h,n) )
set RF = (Relax n) * (findmin n);
set RT = repeat ((Relax n) * (findmin n));
set cn = LifeSpan (((Relax n) * (findmin n)),f,n);
assume that
A1: g = ((repeat ((Relax n) * (findmin n))) . 1) . f and
A2: ( h = ((repeat ((Relax n) * (findmin n))) . i) . f & 1 <= i & i <= LifeSpan (((Relax n) * (findmin n)),f,n) ) and
A3: m in UsedVx (g,n) ; ::_thesis: m in UsedVx (h,n)
defpred S1[ Element of NAT ] means ( 1 <= $1 & $1 <= LifeSpan (((Relax n) * (findmin n)),f,n) implies m in UsedVx ((((repeat ((Relax n) * (findmin n))) . $1) . f),n) );
A4: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A5: S1[k] ; ::_thesis: S1[k + 1]
hereby  ::_thesis: verum
assume that
 1 <= k + 1 and
A6: k + 1 <= LifeSpan (((Relax n) * (findmin n)),f,n) ; ::_thesis: m in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n)
percases ( k = 0 or k <> 0 ) ;
suppose k = 0 ; ::_thesis: m in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n)
hence  m in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) by A1, A3; ::_thesis: verum
end;
supposeA7: k <> 0 ; ::_thesis: m in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n)
 k < LifeSpan (((Relax n) * (findmin n)),f,n) by A6, NAT_1:13;
then  OuterVx ((((repeat ((Relax n) * (findmin n))) . k) . f),n) <> {} by Def4;
then  UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) = (UsedVx ((((repeat ((Relax n) * (findmin n))) . k) . f),n)) \/ {(Argmin ((OuterVx ((((repeat ((Relax n) * (findmin n))) . k) . f),n)),(((repeat ((Relax n) * (findmin n))) . k) . f),n))} by Th39;
then A8: UsedVx ((((repeat ((Relax n) * (findmin n))) . k) . f),n) c= UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) by XBOOLE_1:7;
 k >= 1 + 0 by A7, INT_1:7;
hence  m in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) by A5, A6, A8, NAT_1:13; ::_thesis: verum
end;
end;
end;
end;
A9: S1[ 0 ] ;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A9, A4);
hence  m in UsedVx (h,n) by A2; ::_thesis: verum
end;

definition
let p be FinSequence of NAT ;
let f be Element of REAL * ;
let i, n be Element of NAT ;
predp is_vertex_seq_at f,i,n means :Def17: :: GRAPHSP:def 17
( p . (len p) = i & ( for k being Element of NAT st 1 <= k & k < len p holds
p . ((len p) - k) = f . (n + (p /. (((len p) - k) + 1))) ) );
end;

:: deftheorem Def17 defines is_vertex_seq_at GRAPHSP:def_17_:_
 for p being FinSequence of NAT
for f being Element of REAL *
for i, n being Element of NAT holds
( p is_vertex_seq_at f,i,n iff ( p . (len p) = i & ( for k being Element of NAT st 1 <= k & k < len p holds
p . ((len p) - k) = f . (n + (p /. (((len p) - k) + 1))) ) ) );

definition
let p be FinSequence of NAT ;
let f be Element of REAL * ;
let i, n be Element of NAT ;
predp is_simple_vertex_seq_at f,i,n means :Def18: :: GRAPHSP:def 18
( p . 1 = 1 & len p > 1 & p is_vertex_seq_at f,i,n & p is one-to-one );
end;

:: deftheorem Def18 defines is_simple_vertex_seq_at GRAPHSP:def_18_:_
 for p being FinSequence of NAT
for f being Element of REAL *
for i, n being Element of NAT holds
( p is_simple_vertex_seq_at f,i,n iff ( p . 1 = 1 & len p > 1 & p is_vertex_seq_at f,i,n & p is one-to-one ) );

theorem :: GRAPHSP:49
for p, q being FinSequence of NAT
for f being Element of REAL *
for i, n being Element of NAT st p is_simple_vertex_seq_at f,i,n & q is_simple_vertex_seq_at f,i,n holds
p = q
proof
let p, q be FinSequence of NAT ; ::_thesis: for f being Element of REAL *
for i, n being Element of NAT st p is_simple_vertex_seq_at f,i,n & q is_simple_vertex_seq_at f,i,n holds
p = q
let f be Element of REAL * ; ::_thesis: for i, n being Element of NAT st p is_simple_vertex_seq_at f,i,n & q is_simple_vertex_seq_at f,i,n holds
p = q
let i, n be Element of NAT ; ::_thesis: ( p is_simple_vertex_seq_at f,i,n & q is_simple_vertex_seq_at f,i,n implies p = q )
assume that
A1: p is_simple_vertex_seq_at f,i,n and
A2: q is_simple_vertex_seq_at f,i,n ; ::_thesis: p = q
A3: p . 1 = 1 by A1, Def18;
A4: q . 1 = 1 by A2, Def18;
A5: p is_vertex_seq_at f,i,n by A1, Def18;
then A6: ( p . (len p) = i & ( for k being Element of NAT st 1 <= k & k < len p holds
p . ((len p) - k) = f . ((p /. (((len p) - k) + 1)) + n) ) ) by Def17;
A7: q is_vertex_seq_at f,i,n by A2, Def18;
then A8: q . (len q) = i by Def17;
A9: for k being Element of NAT st 1 <= k & k < len q holds
q . ((len q) - k) = f . ((q /. (((len q) - k) + 1)) + n) by A7, Def17;
A10: len p > 1 by A1, Def18;
A11: now__::_thesis:_not_len_p_<>_len_q
assume A12: len p <> len q ; ::_thesis: contradiction
percases ( len p < len q or len p >= len q ) ;
supposeA13: len p < len q ; ::_thesis: contradiction
A14: (len p) - 1 > 1 - 1 by A10, XREAL_1:14;
then reconsider k = (len p) - 1 as Element of NAT by INT_1:3;
A15: (len p) - k = 0 + 1 ;
then A16: (len q) - k > 1 by A13, XREAL_1:14;
then reconsider m = (len q) - k as Element of NAT by INT_1:3;
A17: k < len p by XREAL_1:146;
 k >= 0 + 1 by A14, INT_1:7;
then A18: 1 = q . ((len q) - k) by A3, A6, A8, A9, A13, A15, A17, Lm10;
 ( q is one-to-one & m <= len q ) by A2, Def18, Lm9;
hence  contradiction by A4, A18, A16, GRAPH_5:6; ::_thesis: verum
end;
supposeA19: len p >= len q ; ::_thesis: contradiction
A20: p is one-to-one by A1, Def18;
A21: len p > len q by A12, A19, XXREAL_0:1;
hereby  ::_thesis: verum
percases ( len q <= 1 or len q > 1 ) ;
suppose len q <= 1 ; ::_thesis: contradiction
hence  contradiction by A2, Def18; ::_thesis: verum
end;
suppose len q > 1 ; ::_thesis: contradiction
then A22: (len q) - 1 > 1 - 1 by XREAL_1:14;
then reconsider k = (len q) - 1 as Element of NAT by INT_1:3;
A23: k < len q by XREAL_1:146;
A24: (len q) - k = 0 + 1 ;
then A25: (len p) - k > 1 by A21, XREAL_1:14;
then reconsider m = (len p) - k as Element of NAT by INT_1:3;
 k >= 0 + 1 by A22, INT_1:7;
then A26: 1 = p . ((len p) - k) by A6, A4, A8, A9, A19, A24, A23, Lm10;
 m <= len p by Lm9;
hence  contradiction by A3, A20, A26, A25, GRAPH_5:6; ::_thesis: verum
end;
end;
end;
end;
end;
end;
now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_len_p_holds_
p_._k_=_q_._k
let k be Nat; ::_thesis: ( 1 <= k & k <= len p implies p . b1 = q . b1 )
assume that
A27: 1 <= k and
A28: k <= len p ; ::_thesis: p . b1 = q . b1
percases ( k = len p or k <> len p ) ;
suppose k = len p ; ::_thesis: p . b1 = q . b1
hence  p . k = q . k by A5, A8, A11, Def17; ::_thesis: verum
end;
suppose k <> len p ; ::_thesis: p . b1 = q . b1
then  k < len p by A28, XXREAL_0:1;
then A29: (len p) - k > k - k by XREAL_1:14;
then reconsider m = (len p) - k as Element of NAT by INT_1:3;
A30: (len q) - m = 0 + k by A11;
 (len p) - k <= (len p) - 1 by A27, XREAL_1:13;
then A31: m < len p by XREAL_1:146, XXREAL_0:2;
 m >= 0 + 1 by A29, INT_1:7;
hence  p . k = q . k by A6, A8, A9, A30, A31, Lm10; ::_thesis: verum
end;
end;
end;
hence  p = q by A11, FINSEQ_1:14; ::_thesis: verum
end;

definition
let G be Graph;
let p be FinSequence of the carrier' of G;
let vs be FinSequence;
predp is_oriented_edge_seq_of vs means :Def19: :: GRAPHSP:def 19
( len vs = (len p) + 1 & ( for n being Nat st 1 <= n & n <= len p holds
( the Source of G . (p . n) = vs . n & the Target of G . (p . n) = vs . (n + 1) ) ) );
end;

:: deftheorem Def19 defines is_oriented_edge_seq_of GRAPHSP:def_19_:_
 for G being Graph
for p being FinSequence of the carrier' of G
for vs being FinSequence holds
( p is_oriented_edge_seq_of vs iff ( len vs = (len p) + 1 & ( for n being Nat st 1 <= n & n <= len p holds
( the Source of G . (p . n) = vs . n & the Target of G . (p . n) = vs . (n + 1) ) ) ) );

theorem :: GRAPHSP:50
for G being oriented Graph
for vs being FinSequence
for p, q being oriented Chain of G st p is_oriented_edge_seq_of vs & q is_oriented_edge_seq_of vs holds
p = q
proof
let G be oriented Graph; ::_thesis: for vs being FinSequence
for p, q being oriented Chain of G st p is_oriented_edge_seq_of vs & q is_oriented_edge_seq_of vs holds
p = q
let vs be FinSequence; ::_thesis: for p, q being oriented Chain of G st p is_oriented_edge_seq_of vs & q is_oriented_edge_seq_of vs holds
p = q
let p, q be oriented Chain of G; ::_thesis: ( p is_oriented_edge_seq_of vs & q is_oriented_edge_seq_of vs implies p = q )
assume that
A1: p is_oriented_edge_seq_of vs and
A2: q is_oriented_edge_seq_of vs ; ::_thesis: p = q
A3: (len p) + 1 = len vs by A1, Def19
.= (len q) + 1 by A2, Def19 ;
now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_len_p_holds_
p_._k_=_q_._k
let k be Nat; ::_thesis: ( 1 <= k & k <= len p implies p . k = q . k )
assume A4: ( 1 <= k & k <= len p ) ; ::_thesis: p . k = q . k
then A5: the Target of G . (p . k) = vs . (k + 1) by A1, Def19
.= the Target of G . (q . k) by A2, A3, A4, Def19 ;
A6: ( p . k in the carrier' of G & q . k in the carrier' of G ) by A3, A4, Th2;
the Source of G . (p . k) = vs . k by A1, A4, Def19
.= the Source of G . (q . k) by A2, A3, A4, Def19 ;
hence  p . k = q . k by A5, A6, GRAPH_1:def_7; ::_thesis: verum
end;
hence  p = q by A3, FINSEQ_1:14; ::_thesis: verum
end;

theorem :: GRAPHSP:51
for G being Graph
for vs1, vs2 being FinSequence
for p being oriented Chain of G st p is_oriented_edge_seq_of vs1 & p is_oriented_edge_seq_of vs2 & len p >= 1 holds
vs1 = vs2
proof
let G be Graph; ::_thesis: for vs1, vs2 being FinSequence
for p being oriented Chain of G st p is_oriented_edge_seq_of vs1 & p is_oriented_edge_seq_of vs2 & len p >= 1 holds
vs1 = vs2
let vs1, vs2 be FinSequence; ::_thesis: for p being oriented Chain of G st p is_oriented_edge_seq_of vs1 & p is_oriented_edge_seq_of vs2 & len p >= 1 holds
vs1 = vs2
let p be oriented Chain of G; ::_thesis: ( p is_oriented_edge_seq_of vs1 & p is_oriented_edge_seq_of vs2 & len p >= 1 implies vs1 = vs2 )
assume that
A1: p is_oriented_edge_seq_of vs1 and
A2: p is_oriented_edge_seq_of vs2 and
A3: len p >= 1 ; ::_thesis: vs1 = vs2
A4: now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_len_vs1_holds_
vs1_._k_=_vs2_._k
let k be Nat; ::_thesis: ( 1 <= k & k <= len vs1 implies vs1 . b1 = vs2 . b1 )
assume that
A5: 1 <= k and
A6: k <= len vs1 ; ::_thesis: vs1 . b1 = vs2 . b1
percases ( k = len vs1 or k <> len vs1 ) ;
suppose k = len vs1 ; ::_thesis: vs1 . b1 = vs2 . b1
then A7: k = (len p) + 1 by A1, Def19;
hence vs1 . k = the Target of G . (p . (len p)) by A1, A3, Def19
.= vs2 . k by A2, A3, A7, Def19 ;
::_thesis: verum
end;
suppose k <> len vs1 ; ::_thesis: vs1 . b1 = vs2 . b1
then  k < len vs1 by A6, XXREAL_0:1;
then  k + 1 <= len vs1 by INT_1:7;
then  k + 1 <= (len p) + 1 by A1, Def19;
then A8: k <= len p by XREAL_1:6;
hence vs1 . k = the Source of G . (p . k) by A1, A5, Def19
.= vs2 . k by A2, A5, A8, Def19 ;
::_thesis: verum
end;
end;
end;
len vs1 = (len p) + 1 by A1, Def19
.= len vs2 by A2, Def19 ;
hence  vs1 = vs2 by A4, FINSEQ_1:14; ::_thesis: verum
end;

Lm11:  for i, n being Element of NAT st 1 <= i & i <= n holds
( 1 < (2 * n) + i & (2 * n) + i < ((n * n) + (3 * n)) + 1 & i < (2 * n) + i )
proof
let i, n be Element of NAT ; ::_thesis: ( 1 <= i & i <= n implies ( 1 < (2 * n) + i & (2 * n) + i < ((n * n) + (3 * n)) + 1 & i < (2 * n) + i ) )
assume that
A1: 1 <= i and
A2: i <= n ; ::_thesis: ( 1 < (2 * n) + i & (2 * n) + i < ((n * n) + (3 * n)) + 1 & i < (2 * n) + i )
set m2 = (2 * n) + i;
 (2 * n) + 1 <= (2 * n) + i by A1, XREAL_1:7;
then A3: 2 * n < (2 * n) + i by NAT_1:13;
 1 <= n by A1, A2, XXREAL_0:2;
then  2 * 1 <= 2 * n by XREAL_1:64;
then  2 < (2 * n) + i by A3, XXREAL_0:2;
hence  1 < (2 * n) + i by XXREAL_0:2; ::_thesis: ( (2 * n) + i < ((n * n) + (3 * n)) + 1 & i < (2 * n) + i )
A4: 3 * n < ((n * n) + (3 * n)) + 1 by Lm7;
 (2 * n) + n = (2 + 1) * n ;
then  (2 * n) + i <= 3 * n by A2, XREAL_1:7;
hence  (2 * n) + i < ((n * n) + (3 * n)) + 1 by A4, XXREAL_0:2; ::_thesis: i < (2 * n) + i
 n <= 2 * n by Lm6;
then  i <= 2 * n by A2, XXREAL_0:2;
hence  i < (2 * n) + i by A3, XXREAL_0:2; ::_thesis: verum
end;

Lm12:  for i, n being Element of NAT st 1 <= i & i <= n holds
( 1 < n + i & n + i <= 2 * n & n + i < ((n * n) + (3 * n)) + 1 )
proof
let i, n be Element of NAT ; ::_thesis: ( 1 <= i & i <= n implies ( 1 < n + i & n + i <= 2 * n & n + i < ((n * n) + (3 * n)) + 1 ) )
assume that
A1: 1 <= i and
A2: i <= n ; ::_thesis: ( 1 < n + i & n + i <= 2 * n & n + i < ((n * n) + (3 * n)) + 1 )
set ni = n + i;
 1 <= n by A1, A2, XXREAL_0:2;
then  1 + 1 <= n + i by A1, XREAL_1:7;
hence  1 < n + i by NAT_1:13; ::_thesis: ( n + i <= 2 * n & n + i < ((n * n) + (3 * n)) + 1 )
A3: n + i <= n + n by A2, XREAL_1:7;
hence  n + i <= 2 * n ; ::_thesis: n + i < ((n * n) + (3 * n)) + 1
 2 * n < ((n * n) + (3 * n)) + 1 by Lm7;
hence  n + i < ((n * n) + (3 * n)) + 1 by A3, XXREAL_0:2; ::_thesis: verum
end;

Lm13:  for i, n, j being Element of NAT st 1 <= i & i <= n & j <= n holds
( 1 < ((2 * n) + (n * i)) + j & i < ((2 * n) + (n * i)) + j & ((2 * n) + (n * i)) + j < ((n * n) + (3 * n)) + 1 )
proof
let i, n, j be Element of NAT ; ::_thesis: ( 1 <= i & i <= n & j <= n implies ( 1 < ((2 * n) + (n * i)) + j & i < ((2 * n) + (n * i)) + j & ((2 * n) + (n * i)) + j < ((n * n) + (3 * n)) + 1 ) )
assume that
A1: 1 <= i and
A2: i <= n and
A3: j <= n ; ::_thesis: ( 1 < ((2 * n) + (n * i)) + j & i < ((2 * n) + (n * i)) + j & ((2 * n) + (n * i)) + j < ((n * n) + (3 * n)) + 1 )
A4: 1 <= n by A1, A2, XXREAL_0:2;
then  2 * 1 <= 2 * n by XREAL_1:64;
then A5: 1 < 2 * n by XXREAL_0:2;
set m3 = ((2 * n) + (n * i)) + j;
A6: ( ((2 * n) + (n * i)) + j = ((2 * n) + j) + (n * i) & n * i <= n * n ) by A2, XREAL_1:64;
 ((2 * n) + (n * i)) + j = (2 * n) + ((n * i) + j) ;
then  2 * n <= ((2 * n) + (n * i)) + j by NAT_1:12;
hence  1 < ((2 * n) + (n * i)) + j by A5, XXREAL_0:2; ::_thesis: ( i < ((2 * n) + (n * i)) + j & ((2 * n) + (n * i)) + j < ((n * n) + (3 * n)) + 1 )
A7: (2 * n) + (n * i) <= ((2 * n) + (n * i)) + j by NAT_1:12;
 1 * i <= n * i by A4, XREAL_1:64;
then  1 + i < (2 * n) + (n * i) by A5, XREAL_1:8;
then  1 + i < ((2 * n) + (n * i)) + j by A7, XXREAL_0:2;
hence  i < ((2 * n) + (n * i)) + j by NAT_1:13; ::_thesis: ((2 * n) + (n * i)) + j < ((n * n) + (3 * n)) + 1
 (2 * n) + n = (2 + 1) * n ;
then  (2 * n) + j <= 3 * n by A3, XREAL_1:7;
then  ((2 * n) + (n * i)) + j <= (n * n) + (3 * n) by A6, XREAL_1:7;
hence  ((2 * n) + (n * i)) + j < ((n * n) + (3 * n)) + 1 by NAT_1:13; ::_thesis: verum
end;

Lm14:  for i, n, j being Element of NAT st 1 <= i & i <= n & 1 <= j & j <= n holds
( (3 * n) + 1 <= ((2 * n) + (n * i)) + j & ((2 * n) + (n * i)) + j <= (n * n) + (3 * n) )
proof
let i, n, j be Element of NAT ; ::_thesis: ( 1 <= i & i <= n & 1 <= j & j <= n implies ( (3 * n) + 1 <= ((2 * n) + (n * i)) + j & ((2 * n) + (n * i)) + j <= (n * n) + (3 * n) ) )
assume that
A1: 1 <= i and
A2: i <= n and
A3: 1 <= j and
A4: j <= n ; ::_thesis: ( (3 * n) + 1 <= ((2 * n) + (n * i)) + j & ((2 * n) + (n * i)) + j <= (n * n) + (3 * n) )
set m3 = ((2 * n) + (n * i)) + j;
A5: ( ((2 * n) + (n * i)) + j = ((2 * n) + j) + (n * i) & n * i <= n * n ) by A2, XREAL_1:64;
 n * 1 <= n * i by A1, XREAL_1:64;
then  (2 * n) + n <= (2 * n) + (n * i) by XREAL_1:7;
hence  (3 * n) + 1 <= ((2 * n) + (n * i)) + j by A3, XREAL_1:7; ::_thesis: ((2 * n) + (n * i)) + j <= (n * n) + (3 * n)
 (2 * n) + n = (2 + 1) * n ;
then  (2 * n) + j <= 3 * n by A4, XREAL_1:7;
hence  ((2 * n) + (n * i)) + j <= (n * n) + (3 * n) by A5, XREAL_1:7; ::_thesis: verum
end;

begin

definition
let f be Element of REAL * ;
let G be oriented Graph;
let n be Element of NAT ;
let W be Function of the carrier' of G,Real>=0;
predf is_Input_of_Dijkstra_Alg G,n,W means :Def20: :: GRAPHSP:def 20
( len f = ((n * n) + (3 * n)) + 1 & Seg n = the carrier of G & ( for i being Element of NAT st 1 <= i & i <= n holds
( f . i = 1 & f . ((2 * n) + i) = 0 ) ) & f . (n + 1) = 0 & ( for i being Element of NAT st 2 <= i & i <= n holds
f . (n + i) = - 1 ) & ( for i, j being Vertex of G
for k, m being Element of NAT st k = i & m = j holds
f . (((2 * n) + (n * k)) + m) = Weight (i,j,W) ) );
end;

:: deftheorem Def20 defines is_Input_of_Dijkstra_Alg GRAPHSP:def_20_:_
 for f being Element of REAL *
for G being oriented Graph
for n being Element of NAT
for W being Function of the carrier' of G,Real>=0 holds
( f is_Input_of_Dijkstra_Alg G,n,W iff ( len f = ((n * n) + (3 * n)) + 1 & Seg n = the carrier of G & ( for i being Element of NAT st 1 <= i & i <= n holds
( f . i = 1 & f . ((2 * n) + i) = 0 ) ) & f . (n + 1) = 0 & ( for i being Element of NAT st 2 <= i & i <= n holds
f . (n + i) = - 1 ) & ( for i, j being Vertex of G
for k, m being Element of NAT st k = i & m = j holds
f . (((2 * n) + (n * k)) + m) = Weight (i,j,W) ) ) );

begin

definition
let n be Element of NAT ;
func DijkstraAlgorithm n ->  Element of Funcs ((REAL *),(REAL *)) equals :: GRAPHSP:def 21
 while_do (((Relax n) * (findmin n)),n);
coherence
 while_do (((Relax n) * (findmin n)),n) is Element of Funcs ((REAL *),(REAL *)) ;
end;

:: deftheorem  defines DijkstraAlgorithm GRAPHSP:def_21_:_
 for n being Element of NAT holds DijkstraAlgorithm n = while_do (((Relax n) * (findmin n)),n);

begin

           Lm15:  for n being Element of NAT
for f, h being Element of REAL *
for G being oriented finite Graph
for W being Function of the carrier' of G,Real>=0
for v1 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & v1 = 1 & n >= 1 & h = ((repeat ((Relax n) * (findmin n))) . 1) . f holds
( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & h . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) ) ) & ( for m, j being Element of NAT st h . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (h,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (h,n) holds
h . (n + m) <> - 1 ) )
proof
let n be Element of NAT ; ::_thesis: for f, h being Element of REAL *
for G being oriented finite Graph
for W being Function of the carrier' of G,Real>=0
for v1 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & v1 = 1 & n >= 1 & h = ((repeat ((Relax n) * (findmin n))) . 1) . f holds
( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & h . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) ) ) & ( for m, j being Element of NAT st h . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (h,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (h,n) holds
h . (n + m) <> - 1 ) )
let f, h be Element of REAL * ; ::_thesis: for G being oriented finite Graph
for W being Function of the carrier' of G,Real>=0
for v1 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & v1 = 1 & n >= 1 & h = ((repeat ((Relax n) * (findmin n))) . 1) . f holds
( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & h . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) ) ) & ( for m, j being Element of NAT st h . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (h,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (h,n) holds
h . (n + m) <> - 1 ) )
let G be oriented finite Graph; ::_thesis: for W being Function of the carrier' of G,Real>=0
for v1 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & v1 = 1 & n >= 1 & h = ((repeat ((Relax n) * (findmin n))) . 1) . f holds
( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & h . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) ) ) & ( for m, j being Element of NAT st h . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (h,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (h,n) holds
h . (n + m) <> - 1 ) )
let W be Function of the carrier' of G,Real>=0; ::_thesis: for v1 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & v1 = 1 & n >= 1 & h = ((repeat ((Relax n) * (findmin n))) . 1) . f holds
( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & h . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) ) ) & ( for m, j being Element of NAT st h . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (h,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (h,n) holds
h . (n + m) <> - 1 ) )
let v1 be Vertex of G; ::_thesis: ( f is_Input_of_Dijkstra_Alg G,n,W & v1 = 1 & n >= 1 & h = ((repeat ((Relax n) * (findmin n))) . 1) . f implies ( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & h . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) ) ) & ( for m, j being Element of NAT st h . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (h,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (h,n) holds
h . (n + m) <> - 1 ) ) )
set R = Relax n;
set M = findmin n;
set f0 = ((repeat ((Relax n) * (findmin n))) . 0) . f;
set mi = ((n * n) + (3 * n)) + 1;
assume that
A1: f is_Input_of_Dijkstra_Alg G,n,W and
A2: v1 = 1 and
A3: n >= 1 and
A4: h = ((repeat ((Relax n) * (findmin n))) . 1) . f ; ::_thesis: ( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & h . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) ) ) & ( for m, j being Element of NAT st h . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (h,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (h,n) holds
h . (n + m) <> - 1 ) )
A5: len f = ((n * n) + (3 * n)) + 1 by A1, Def20;
set nk = n + 1;
A6: (2 * n) + n = (2 + 1) * n ;
A7: f . (n + 1) = 0 by A1, Def20;
A8: 1 <= ((n * n) + (3 * n)) + 1 by NAT_1:12;
then A9: 1 in dom f by A5, FINSEQ_3:25;
A10: ( ( for j being Element of NAT st 1 <= j & j <= n holds
f . j = 1 ) & ( for j being Element of NAT st 2 <= j & j <= n holds
f . (n + j) = - 1 ) ) by A1, Def20;
then A11: {1} = UsedVx (h,n) by A3, A4, A7, A9, Th47;
 h = ((repeat ((Relax n) * (findmin n))) . (0 + 1)) . f by A4;
then  h = (Relax n) . ((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) by Th22;
then A12: h = Relax (((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)),n) by Def15;
A13: Seg n = the carrier of G by A1, Def20;
then reconsider VG = the carrier of G as non empty Subset of NAT by A3;
set Ak = Argmin ((OuterVx (f,n)),f,n);
A14: dom ((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) = dom (((repeat ((Relax n) * (findmin n))) . 0) . f) by Th33
.= dom f by Th21 ;
A15: 1 = Argmin ((OuterVx (f,n)),f,n) by A3, A7, A10, A9, Th47;
A16: (findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f) = (findmin n) . f by Th21
.= (f,(((n * n) + (3 * n)) + 1)) := (1,(- 1)) by A15, Def11 ;
then  ((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) . 1 = - 1 by A9, Th19;
then  not (findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f) hasBetterPathAt n,1 by Def13;
then A17: not (findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f) hasBetterPathAt n,(n + 1) -' n by NAT_D:34;
A18: n + 1 < ((n * n) + (3 * n)) + 1 by A3, Lm12;
A19: W is_weight>=0of G by GRAPH_5:def_13;
hereby  ::_thesis: ( ( for m, j being Element of NAT st h . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (h,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (h,n) holds
h . (n + m) <> - 1 ) )
set w1 = (2 * n) + 1;
let v3 be Vertex of G; ::_thesis: for j being Element of NAT st v3 <> v1 & v3 = j & h . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex pe being oriented Chain of G st
( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & pe is_oriented_edge_seq_of p & pe is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (pe,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies pe islongestInShortestpath UsedVx (h,n),v1,W ) )
let j be Element of NAT ; ::_thesis: ( v3 <> v1 & v3 = j & h . (n + j) <> - 1 implies ex p being FinSequence of NAT ex pe being oriented Chain of G st
( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & pe is_oriented_edge_seq_of p & pe is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (pe,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies pe islongestInShortestpath UsedVx (h,n),v1,W ) ) )
set nj = n + j;
assume that
A20: v3 <> v1 and
A21: v3 = j and
A22: h . (n + j) <> - 1 ; ::_thesis: ex p being FinSequence of NAT ex pe being oriented Chain of G st
( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & pe is_oriented_edge_seq_of p & pe is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (pe,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies pe islongestInShortestpath UsedVx (h,n),v1,W ) )
set m2 = (2 * n) + j;
A23: j in VG by A21;
then A24: 1 <= j by A13, FINSEQ_1:1;
then  n + 1 <= n + j by XREAL_1:7;
then A25: n < n + j by NAT_1:13;
A26: j <= n by A13, A23, FINSEQ_1:1;
then A27: ( 1 < (2 * n) + j & (2 * n) + j < ((n * n) + (3 * n)) + 1 ) by A24, Lm11;
 j > 1 by A2, A20, A21, A24, XXREAL_0:1;
then  j >= 1 + 1 by INT_1:7;
then A28: f . (n + j) = - 1 by A1, A26, Def20;
A29: n + j <= 2 * n by A24, A26, Lm12;
A30: ((n * n) + (3 * n)) + 1 in dom f by A8, A5, FINSEQ_3:25;
then A31: ((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) /. (((n * n) + (3 * n)) + 1) = ((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) . (((n * n) + (3 * n)) + 1) by A14, PARTFUN1:def_6
.= 1 by A16, A27, A30, Th17 ;
A32: ( 1 < (2 * n) + 1 & (2 * n) + 1 < ((n * n) + (3 * n)) + 1 ) by A3, Lm11;
then  (2 * n) + 1 in dom f by A5, FINSEQ_3:25;
then A33: ((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) /. ((2 * n) + (((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) /. (((n * n) + (3 * n)) + 1))) = ((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) . ((2 * n) + 1) by A14, A31, PARTFUN1:def_6
.= f . ((2 * n) + 1) by A16, A32, Th18
.= 0 by A1, A3, Def20 ;
A34: (2 * n) + j <= 3 * n by A6, A26, XREAL_1:7;
A35: ( 1 < n + j & n + j < ((n * n) + (3 * n)) + 1 ) by A24, A26, Lm12;
then A36: n + j in dom f by A5, FINSEQ_3:25;
 ((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) . (n + j) = f . (n + j) by A16, A35, Th18;
then A37: (findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f) hasBetterPathAt n,(n + j) -' n by A14, A12, A22, A29, A36, A25, A28, Def14;
 (2 * n) + 1 <= (2 * n) + j by A24, XREAL_1:7;
then A38: 2 * n < (2 * n) + j by NAT_1:13;
set m3 = ((2 * n) + (n * 1)) + j;
reconsider p = <*1,j*> as FinSequence of NAT ;
A39: p . 1 = 1 by FINSEQ_1:44;
A40: ( 1 < ((2 * n) + (n * 1)) + j & ((2 * n) + (n * 1)) + j < ((n * n) + (3 * n)) + 1 ) by A3, A26, Lm13;
then  ((2 * n) + (n * 1)) + j in dom f by A5, FINSEQ_3:25;
then A41: ((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) /. (((2 * n) + (n * (((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) /. (((n * n) + (3 * n)) + 1)))) + j) = ((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) . (((2 * n) + (n * 1)) + j) by A14, A31, PARTFUN1:def_6
.= f . (((2 * n) + (n * 1)) + j) by A16, A40, Th18
.= Weight (v1,v3,W) by A1, A2, A21, Def20 ;
A42: (n + j) -' n = j by NAT_D:34;
then  ((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) /. (((2 * n) + (n * (((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) /. (((n * n) + (3 * n)) + 1)))) + j) >= 0 by A37, Def13;
then consider e being set  such that
A43: e in the carrier' of G and
A44: e orientedly_joins v1,v3 by A41, Th23;
reconsider pe = <*e*> as oriented Chain of G by A43, Th5;
A45: len pe = 1 by FINSEQ_1:40;
A46: len p = 2 by FINSEQ_1:44;
then A47: p . (len p) = j by FINSEQ_1:44;
A48: now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_len_pe_holds_
(_the_Source_of_G_._(pe_._k)_=_p_._k_&_the_Target_of_G_._(pe_._k)_=_p_._(k_+_1)_)
let k be Nat; ::_thesis: ( 1 <= k & k <= len pe implies ( the Source of G . (pe . k) = p . k & the Target of G . (pe . k) = p . (k + 1) ) )
assume  ( 1 <= k & k <= len pe ) ; ::_thesis: ( the Source of G . (pe . k) = p . k & the Target of G . (pe . k) = p . (k + 1) )
then A49: k = 1 by A45, XXREAL_0:1;
hence  the Source of G . (pe . k) = the Source of G . e by FINSEQ_1:40
.= p . k by A2, A44, A39, A49, GRAPH_4:def_1 ;
::_thesis: the Target of G . (pe . k) = p . (k + 1)
thus  the Target of G . (pe . k) = the Target of G . e by A49, FINSEQ_1:40
.= p . (k + 1) by A21, A44, A46, A47, A49, GRAPH_4:def_1 ; ::_thesis: verum
end;
A50: h . (n + j) = 1 by A14, A12, A29, A36, A25, A37, A31, Def14;
now__::_thesis:_for_k_being_Element_of_NAT_st_1_<=_k_&_k_<_len_p_holds_
p_._((len_p)_-_k)_=_h_._(n_+_(p_/._(((len_p)_-_k)_+_1)))
let k be Element of NAT ; ::_thesis: ( 1 <= k & k < len p implies p . ((len p) - k) = h . (n + (p /. (((len p) - k) + 1))) )
assume that
A51: 1 <= k and
A52: k < len p ; ::_thesis: p . ((len p) - k) = h . (n + (p /. (((len p) - k) + 1)))
 k < 1 + 1 by A52, FINSEQ_1:44;
then  k <= 1 by NAT_1:13;
then  k = 1 by A51, XXREAL_0:1;
hence  p . ((len p) - k) = h . (n + (p /. (((len p) - k) + 1))) by A50, A39, A46, A47, FINSEQ_4:15; ::_thesis: verum
end;
then A53: p is_vertex_seq_at h,j,n by A47, Def17;
 ( ((2 * n) + j) -' (2 * n) = j & (2 * n) + j in dom ((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) ) by A5, A14, A27, FINSEQ_3:25, NAT_D:34;
then A54: (Relax (((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)),n)) . ((2 * n) + j) = newpathcost (((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)),n,(((2 * n) + j) -' (2 * n))) by A42, A37, A34, A38, Def14
.= 0 + (((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) /. (((2 * n) + (n * (((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) /. (((n * n) + (3 * n)) + 1)))) + j)) by A33, NAT_D:34 ;
take p = p; ::_thesis: ex pe being oriented Chain of G st
( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & pe is_oriented_edge_seq_of p & pe is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (pe,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies pe islongestInShortestpath UsedVx (h,n),v1,W ) )
take pe = pe; ::_thesis: ( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & pe is_oriented_edge_seq_of p & pe is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (pe,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies pe islongestInShortestpath UsedVx (h,n),v1,W ) )
 p is one-to-one by A2, A20, A21, FINSEQ_3:94;
hence  p is_simple_vertex_seq_at h,j,n by A39, A46, A53, Def18; ::_thesis: ( ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & pe is_oriented_edge_seq_of p & pe is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (pe,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies pe islongestInShortestpath UsedVx (h,n),v1,W ) )
hereby  ::_thesis: ( pe is_oriented_edge_seq_of p & pe is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (pe,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies pe islongestInShortestpath UsedVx (h,n),v1,W ) )
let i be Element of NAT ; ::_thesis: ( 1 <= i & i < len p implies p . i in UsedVx (h,n) )
assume that
A55: 1 <= i and
A56: i < len p ; ::_thesis: p . i in UsedVx (h,n)
 i + 1 <= 1 + 1 by A46, A56, INT_1:7;
then  i <= 1 by XREAL_1:6;
then  i = 1 by A55, XXREAL_0:1;
hence  p . i in UsedVx (h,n) by A11, A39, TARSKI:def_1; ::_thesis: verum
end;
 len p = (len pe) + 1 by A45, FINSEQ_1:44;
hence  pe is_oriented_edge_seq_of p by A48, Def19; ::_thesis: ( pe is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (pe,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies pe islongestInShortestpath UsedVx (h,n),v1,W ) )
thus  pe is_shortestpath_of v1,v3, UsedVx (h,n),W by A2, A11, A43, A44, Th14; ::_thesis: ( cost (pe,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies pe islongestInShortestpath UsedVx (h,n),v1,W ) )
thus  cost (pe,W) = W . (pe . 1) by A19, A45, Th4, GRAPH_5:46
.= W . e by FINSEQ_1:40
.= h . ((2 * n) + j) by A12, A41, A54, A43, A44, Th25 ; ::_thesis: ( not v3 in UsedVx (h,n) implies pe islongestInShortestpath UsedVx (h,n),v1,W )
assume  not v3 in UsedVx (h,n) ; ::_thesis: pe islongestInShortestpath UsedVx (h,n),v1,W
 for v2 being Vertex of G st v2 in UsedVx (h,n) & v2 <> v1 holds
ex Q being oriented Chain of G st
( Q is_shortestpath_of v1,v2, UsedVx (h,n),W & cost (Q,W) <= cost (pe,W) ) by A2, A11, TARSKI:def_1;
hence  pe islongestInShortestpath UsedVx (h,n),v1,W by GRAPH_5:def_19; ::_thesis: verum
end;
A57: 2 * n < ((n * n) + (3 * n)) + 1 by Lm7;
A58: 2 * 1 <= 2 * n by A3, XREAL_1:64;
 1 < n + 1 by A3, Lm12;
then A59: n + 1 in dom f by A5, A18, FINSEQ_3:25;
A60: n < n + 1 by NAT_1:13;
 n + 1 <= 2 * n by A3, Lm12;
then A61: h . (n + 1) = ((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) . (n + 1) by A14, A12, A60, A59, A17, Def14
.= (((repeat ((Relax n) * (findmin n))) . 0) . f) . (n + 1) by A18, A60, Th31
.= 0 by A7, Th21 ;
A62: n * 1 <= n * n by A3, XREAL_1:64;
hereby  ::_thesis: for m being Element of NAT st m in UsedVx (h,n) holds
h . (n + m) <> - 1
A63: n < ((n * n) + (3 * n)) + 1 by Lm7;
let m, j be Element of NAT ; ::_thesis: ( h . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (h,n) implies f . (((2 * n) + (n * m)) + j) = - 1 )
assume that
A64: h . (n + j) = - 1 and
A65: 1 <= j and
A66: j <= n and
A67: m in UsedVx (h,n) ; ::_thesis: f . (((2 * n) + (n * m)) + j) = - 1
reconsider v2 = j as Vertex of G by A13, A65, A66, FINSEQ_1:1;
A68: 3 * n < ((n * n) + (3 * n)) + 1 by Lm7;
set m2 = (2 * n) + j;
 (2 * n) + j <= 3 * n by A6, A66, XREAL_1:7;
then A69: (2 * n) + j < ((n * n) + (3 * n)) + 1 by A68, XXREAL_0:2;
 (2 * n) + 1 <= (2 * n) + j by A65, XREAL_1:7;
then  2 * n < (2 * n) + j by NAT_1:13;
then  2 < (2 * n) + j by A58, XXREAL_0:2;
then A70: 1 < (2 * n) + j by XXREAL_0:2;
 j <> 1 by A61, A64;
then A71: ((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) . j = f . j by A16, A66, A63, Th18
.= 1 by A1, A65, A66, Def20 ;
A72: ((n * n) + (3 * n)) + 1 in dom f by A8, A5, FINSEQ_3:25;
then A73: ((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) /. (((n * n) + (3 * n)) + 1) = ((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) . (((n * n) + (3 * n)) + 1) by A14, PARTFUN1:def_6
.= 1 by A16, A69, A70, A72, Th17 ;
set nj = n + j;
 1 + 1 <= n + j by A3, A65, XREAL_1:7;
then A74: 1 < n + j by NAT_1:13;
A75: n + j <= n + n by A66, XREAL_1:7;
then  n + j < ((n * n) + (3 * n)) + 1 by A57, XXREAL_0:2;
then A76: n + j in dom f by A5, A74, FINSEQ_3:25;
 n + 1 <= n + j by A65, XREAL_1:7;
then A77: n < n + j by NAT_1:13;
A78: n + j <= 2 * n by A75;
then A79: not (findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f) hasBetterPathAt n,(n + j) -' n by A14, A12, A64, A76, A77, A73, Def14;
then  ( (n + j) -' n = j & ((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) . (n + j) = - 1 ) by A14, A12, A64, A78, A76, A77, Def14, NAT_D:34;
then A80: not ((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) /. (((2 * n) + (n * (((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) /. (((n * n) + (3 * n)) + 1)))) + j) >= 0 by A79, A71, Def13;
set m3 = ((2 * n) + (n * 1)) + j;
 ((2 * n) + (n * 1)) + j = (2 * n) + ((n * 1) + j) ;
then  2 <= ((2 * n) + (n * 1)) + j by A58, NAT_1:12;
then A81: 1 < ((2 * n) + (n * 1)) + j by XXREAL_0:2;
 j <= n * n by A62, A66, XXREAL_0:2;
then  ((2 * n) + (n * 1)) + j <= (3 * n) + (n * n) by XREAL_1:7;
then A82: ((2 * n) + (n * 1)) + j < ((n * n) + (3 * n)) + 1 by NAT_1:13;
then  ((2 * n) + (n * 1)) + j in dom f by A5, A81, FINSEQ_3:25;
then A83: ((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) /. (((2 * n) + (n * (((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) /. (((n * n) + (3 * n)) + 1)))) + j) = ((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) . (((2 * n) + (n * 1)) + j) by A14, A73, PARTFUN1:def_6;
((findmin n) . (((repeat ((Relax n) * (findmin n))) . 0) . f)) . (((2 * n) + (n * 1)) + j) = f . (((2 * n) + (n * 1)) + j) by A16, A81, A82, Th18
.= Weight (v1,v2,W) by A1, A2, Def20 ;
then  for e being set holds
( not e in the carrier' of G or not e orientedly_joins v1,v2 ) by A80, A83, Th23;
then A84: Weight (v1,v2,W) = - 1 by Def7;
 m = 1 by A11, A67, TARSKI:def_1;
hence  f . (((2 * n) + (n * m)) + j) = - 1 by A1, A2, A84, Def20; ::_thesis: verum
end;
let m be Element of NAT ; ::_thesis: ( m in UsedVx (h,n) implies h . (n + m) <> - 1 )
assume  m in UsedVx (h,n) ; ::_thesis: h . (n + m) <> - 1
then  m = 1 by A11, TARSKI:def_1;
hence  h . (n + m) <> - 1 by A61; ::_thesis: verum
end;

Lm16:  for n, k, i being Element of NAT
for g, f, h being Element of REAL * st g = ((repeat ((Relax n) * (findmin n))) . k) . f & h = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f & OuterVx (g,n) <> {} & i in UsedVx (g,n) & len f = ((n * n) + (3 * n)) + 1 holds
h . (n + i) = g . (n + i)
proof
let n, k, i be Element of NAT ; ::_thesis: for g, f, h being Element of REAL * st g = ((repeat ((Relax n) * (findmin n))) . k) . f & h = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f & OuterVx (g,n) <> {} & i in UsedVx (g,n) & len f = ((n * n) + (3 * n)) + 1 holds
h . (n + i) = g . (n + i)
let g, f, h be Element of REAL * ; ::_thesis: ( g = ((repeat ((Relax n) * (findmin n))) . k) . f & h = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f & OuterVx (g,n) <> {} & i in UsedVx (g,n) & len f = ((n * n) + (3 * n)) + 1 implies h . (n + i) = g . (n + i) )
set R = Relax n;
set M = findmin n;
set RF = repeat ((Relax n) * (findmin n));
set mi = ((n * n) + (3 * n)) + 1;
set Ak = Argmin ((OuterVx (g,n)),g,n);
assume that
A1: g = ((repeat ((Relax n) * (findmin n))) . k) . f and
A2: h = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f and
A3: OuterVx (g,n) <> {} and
A4: i in UsedVx (g,n) and
A5: len f = ((n * n) + (3 * n)) + 1 ; ::_thesis: h . (n + i) = g . (n + i)
A6: h = (Relax n) . ((findmin n) . g) by A1, A2, Th22
.= Relax (((findmin n) . g),n) by Def15 ;
set ni = n + i;
A7: ex j being Element of NAT st
( i = j & j in dom g & 1 <= j & j <= n & g . j = - 1 ) by A4;
then A8: n + i <= n + n by XREAL_1:7;
A9: 1 <= n + i by A7, NAT_1:12;
A10: 2 * n < ((n * n) + (3 * n)) + 1 by Lm7;
then  n + i < ((n * n) + (3 * n)) + 1 by A8, XXREAL_0:2;
then  n + i in dom f by A5, A9, FINSEQ_3:25;
then  n + i in dom g by A1, Th41;
then A11: ( (n + i) -' n = i & n + i in dom ((findmin n) . g) ) by Th33, NAT_D:34;
A12: Argmin ((OuterVx (g,n)),g,n) <= n by A3, Th29;
A13: n < ((n * n) + (3 * n)) + 1 by Lm7;
A14: (findmin n) . g = (g,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (g,n)),g,n)),(- 1)) by Def11;
A15: g . (Argmin ((OuterVx (g,n)),g,n)) <> - 1 by A3, Th29;
A16: now__::_thesis:_not_(findmin_n)_._g_hasBetterPathAt_n,i
assume  (findmin n) . g hasBetterPathAt n,i ; ::_thesis: contradiction
then  ((findmin n) . g) . i <> - 1 by Def13;
hence  contradiction by A14, A15, A7, A13, Th18; ::_thesis: verum
end;
 n + 1 <= n + i by A7, XREAL_1:7;
then A17: n < n + i by NAT_1:13;
 n + i <= 2 * n by A8;
hence h . (n + i) = ((findmin n) . g) . (n + i) by A6, A17, A11, A16, Def14
.= g . (n + i) by A14, A12, A8, A17, A10, Th18 ;
::_thesis: verum
end;

Lm17:  for n, k, j being Element of NAT
for g, f, h being Element of REAL *
for p being FinSequence of NAT st g = ((repeat ((Relax n) * (findmin n))) . k) . f & h = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f & OuterVx (g,n) <> {} & len f = ((n * n) + (3 * n)) + 1 & p is_simple_vertex_seq_at g,j,n & g . (n + j) = h . (n + j) & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (g,n) ) holds
p is_simple_vertex_seq_at h,j,n
proof
let n, k, j be Element of NAT ; ::_thesis: for g, f, h being Element of REAL *
for p being FinSequence of NAT st g = ((repeat ((Relax n) * (findmin n))) . k) . f & h = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f & OuterVx (g,n) <> {} & len f = ((n * n) + (3 * n)) + 1 & p is_simple_vertex_seq_at g,j,n & g . (n + j) = h . (n + j) & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (g,n) ) holds
p is_simple_vertex_seq_at h,j,n
let g, f, h be Element of REAL * ; ::_thesis: for p being FinSequence of NAT st g = ((repeat ((Relax n) * (findmin n))) . k) . f & h = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f & OuterVx (g,n) <> {} & len f = ((n * n) + (3 * n)) + 1 & p is_simple_vertex_seq_at g,j,n & g . (n + j) = h . (n + j) & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (g,n) ) holds
p is_simple_vertex_seq_at h,j,n
let p be FinSequence of NAT ; ::_thesis: ( g = ((repeat ((Relax n) * (findmin n))) . k) . f & h = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f & OuterVx (g,n) <> {} & len f = ((n * n) + (3 * n)) + 1 & p is_simple_vertex_seq_at g,j,n & g . (n + j) = h . (n + j) & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (g,n) ) implies p is_simple_vertex_seq_at h,j,n )
set RT = repeat ((Relax n) * (findmin n));
assume that
A1: ( g = ((repeat ((Relax n) * (findmin n))) . k) . f & h = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f & OuterVx (g,n) <> {} & len f = ((n * n) + (3 * n)) + 1 ) and
A2: p is_simple_vertex_seq_at g,j,n and
A3: g . (n + j) = h . (n + j) and
A4: for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (g,n) ; ::_thesis: p is_simple_vertex_seq_at h,j,n
A5: len p > 1 by A2, Def18;
A6: p is_vertex_seq_at g,j,n by A2, Def18;
then A7: p . (len p) = j by Def17;
now__::_thesis:_for_k_being_Element_of_NAT_st_1_<=_k_&_k_<_len_p_holds_
p_._((len_p)_-_k)_=_h_._(n_+_(p_/._(((len_p)_-_k)_+_1)))
let k be Element of NAT ; ::_thesis: ( 1 <= k & k < len p implies p . ((len p) - b1) = h . (n + (p /. (((len p) - b1) + 1))) )
assume that
A8: 1 <= k and
A9: k < len p ; ::_thesis: p . ((len p) - b1) = h . (n + (p /. (((len p) - b1) + 1)))
A10: k - k < (len p) - k by A9, XREAL_1:14;
then reconsider m = (len p) - k as Element of NAT by INT_1:3;
 m <= (len p) - 1 by A8, XREAL_1:13;
then A11: m + 1 <= ((len p) - 1) + 1 by XREAL_1:7;
A12: 1 + 0 < m + 1 by A10, XREAL_1:8;
then A13: p /. (m + 1) = p . (m + 1) by A11, FINSEQ_4:15;
percases ( m + 1 = len p or m + 1 <> len p ) ;
suppose m + 1 = len p ; ::_thesis: p . ((len p) - b1) = h . (n + (p /. (((len p) - b1) + 1)))
hence  p . ((len p) - k) = h . (n + (p /. (((len p) - k) + 1))) by A3, A5, A6, A7, A13, Def17; ::_thesis: verum
end;
suppose m + 1 <> len p ; ::_thesis: p . ((len p) - b1) = h . (n + (p /. (((len p) - b1) + 1)))
then A14: m + 1 < len p by A11, XXREAL_0:1;
thus p . ((len p) - k) = g . (n + (p /. (((len p) - k) + 1))) by A6, A8, A9, Def17
.= h . (n + (p /. (((len p) - k) + 1))) by A1, A4, A12, A13, A14, Lm16 ; ::_thesis: verum
end;
end;
end;
then A15: p is_vertex_seq_at h,j,n by A7, Def17;
 ( p . 1 = 1 & p is one-to-one ) by A2, Def18;
hence  p is_simple_vertex_seq_at h,j,n by A5, A15, Def18; ::_thesis: verum
end;

Lm18:  for n, k, m, j being Element of NAT
for g, f, h being Element of REAL *
for p being FinSequence of NAT st g = ((repeat ((Relax n) * (findmin n))) . k) . f & h = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f & OuterVx (g,n) <> {} & len f = ((n * n) + (3 * n)) + 1 & p is_simple_vertex_seq_at g,m,n & m = h . (n + j) & g . (n + m) = h . (n + m) & m <> j & not j in UsedVx (g,n) & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (g,n) ) holds
p ^ <*j*> is_simple_vertex_seq_at h,j,n
proof
let n, k, m, j be Element of NAT ; ::_thesis: for g, f, h being Element of REAL *
for p being FinSequence of NAT st g = ((repeat ((Relax n) * (findmin n))) . k) . f & h = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f & OuterVx (g,n) <> {} & len f = ((n * n) + (3 * n)) + 1 & p is_simple_vertex_seq_at g,m,n & m = h . (n + j) & g . (n + m) = h . (n + m) & m <> j & not j in UsedVx (g,n) & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (g,n) ) holds
p ^ <*j*> is_simple_vertex_seq_at h,j,n
let g, f, h be Element of REAL * ; ::_thesis: for p being FinSequence of NAT st g = ((repeat ((Relax n) * (findmin n))) . k) . f & h = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f & OuterVx (g,n) <> {} & len f = ((n * n) + (3 * n)) + 1 & p is_simple_vertex_seq_at g,m,n & m = h . (n + j) & g . (n + m) = h . (n + m) & m <> j & not j in UsedVx (g,n) & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (g,n) ) holds
p ^ <*j*> is_simple_vertex_seq_at h,j,n
let p be FinSequence of NAT ; ::_thesis: ( g = ((repeat ((Relax n) * (findmin n))) . k) . f & h = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f & OuterVx (g,n) <> {} & len f = ((n * n) + (3 * n)) + 1 & p is_simple_vertex_seq_at g,m,n & m = h . (n + j) & g . (n + m) = h . (n + m) & m <> j & not j in UsedVx (g,n) & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (g,n) ) implies p ^ <*j*> is_simple_vertex_seq_at h,j,n )
set RT = repeat ((Relax n) * (findmin n));
assume that
A1: ( g = ((repeat ((Relax n) * (findmin n))) . k) . f & h = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f & OuterVx (g,n) <> {} & len f = ((n * n) + (3 * n)) + 1 ) and
A2: p is_simple_vertex_seq_at g,m,n and
A3: m = h . (n + j) and
A4: g . (n + m) = h . (n + m) and
A5: m <> j and
A6: not j in UsedVx (g,n) and
A7: for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (g,n) ; ::_thesis: p ^ <*j*> is_simple_vertex_seq_at h,j,n
A8: p is_vertex_seq_at g,m,n by A2, Def18;
then A9: p . (len p) = m by Def17;
set q = p ^ <*j*>;
A10: len p > 1 by A2, Def18;
A11: len (p ^ <*j*>) = (len p) + 1 by FINSEQ_2:16;
then A12: (p ^ <*j*>) . (len (p ^ <*j*>)) = j by FINSEQ_1:42;
now__::_thesis:_for_ii_being_Element_of_NAT_st_1_<=_ii_&_ii_<_len_(p_^_<*j*>)_holds_
(p_^_<*j*>)_._((len_(p_^_<*j*>))_-_ii)_=_h_._(n_+_((p_^_<*j*>)_/._(((len_(p_^_<*j*>))_-_ii)_+_1)))
let ii be Element of NAT ; ::_thesis: ( 1 <= ii & ii < len (p ^ <*j*>) implies (p ^ <*j*>) . ((len (p ^ <*j*>)) - b1) = h . (n + ((p ^ <*j*>) /. (((len (p ^ <*j*>)) - b1) + 1))) )
assume that
A13: 1 <= ii and
A14: ii < len (p ^ <*j*>) ; ::_thesis: (p ^ <*j*>) . ((len (p ^ <*j*>)) - b1) = h . (n + ((p ^ <*j*>) /. (((len (p ^ <*j*>)) - b1) + 1)))
A15: ii - ii < (len (p ^ <*j*>)) - ii by A14, XREAL_1:14;
then reconsider mm = (len (p ^ <*j*>)) - ii as Element of NAT by INT_1:3;
A16: 1 + 0 < mm + 1 by A15, XREAL_1:8;
 mm <= (len (p ^ <*j*>)) - 1 by A13, XREAL_1:13;
then A17: mm + 1 <= ((len (p ^ <*j*>)) - 1) + 1 by XREAL_1:7;
then A18: (p ^ <*j*>) /. (mm + 1) = (p ^ <*j*>) . (mm + 1) by A16, FINSEQ_4:15;
percases ( mm + 1 = len (p ^ <*j*>) or mm + 1 <> len (p ^ <*j*>) ) ;
suppose mm + 1 = len (p ^ <*j*>) ; ::_thesis: (p ^ <*j*>) . ((len (p ^ <*j*>)) - b1) = h . (n + ((p ^ <*j*>) /. (((len (p ^ <*j*>)) - b1) + 1)))
hence  (p ^ <*j*>) . ((len (p ^ <*j*>)) - ii) = h . (n + ((p ^ <*j*>) /. (((len (p ^ <*j*>)) - ii) + 1))) by A3, A10, A9, A11, A12, A18, Lm1; ::_thesis: verum
end;
suppose mm + 1 <> len (p ^ <*j*>) ; ::_thesis: (p ^ <*j*>) . ((len (p ^ <*j*>)) - b1) = h . (n + ((p ^ <*j*>) /. (((len (p ^ <*j*>)) - b1) + 1)))
then A19: mm + 1 < len (p ^ <*j*>) by A17, XXREAL_0:1;
then A20: mm + 1 <= len p by A11, INT_1:7;
A21: 1 + 0 <= mm by A15, INT_1:7;
 mm < len p by A11, A19, XREAL_1:7;
then A22: (p ^ <*j*>) . ((len (p ^ <*j*>)) - ii) = p . mm by A21, Lm1;
hereby  ::_thesis: verum
percases ( mm + 1 = len p or mm + 1 <> len p ) ;
supposeA23: mm + 1 = len p ; ::_thesis: (p ^ <*j*>) . ((len (p ^ <*j*>)) - ii) = h . (n + ((p ^ <*j*>) /. (((len (p ^ <*j*>)) - ii) + 1)))
A24: p /. (((len p) - 1) + 1) = m by A10, A9, FINSEQ_4:15;
 (p ^ <*j*>) /. (((len (p ^ <*j*>)) - ii) + 1) = m by A10, A9, A18, A23, Lm1;
hence  (p ^ <*j*>) . ((len (p ^ <*j*>)) - ii) = h . (n + ((p ^ <*j*>) /. (((len (p ^ <*j*>)) - ii) + 1))) by A4, A10, A8, A22, A23, A24, Def17; ::_thesis: verum
end;
supposeA25: mm + 1 <> len p ; ::_thesis: (p ^ <*j*>) . ((len (p ^ <*j*>)) - ii) = h . (n + ((p ^ <*j*>) /. (((len (p ^ <*j*>)) - ii) + 1)))
set i2 = ii - 1;
A26: mm + 1 < len p by A20, A25, XXREAL_0:1;
A27: now__::_thesis:_not_ii_-_1_<=_1
assume  ii - 1 <= 1 ; ::_thesis: contradiction
then  (len p) - 1 <= (len p) - (ii - 1) by XREAL_1:13;
hence  contradiction by A11, A26, XREAL_1:20; ::_thesis: verum
end;
then reconsider i3 = ii - 1 as Element of NAT by INT_1:3;
A28: p /. (mm + 1) = p . (mm + 1) by A16, A20, FINSEQ_4:15;
A29: (p ^ <*j*>) /. (((len (p ^ <*j*>)) - ii) + 1) = (p ^ <*j*>) . (mm + 1) by A16, A17, FINSEQ_4:15
.= p /. (mm + 1) by A16, A20, A28, Lm1 ;
 ii - 1 < (len (p ^ <*j*>)) - 1 by A14, XREAL_1:14;
hence (p ^ <*j*>) . ((len (p ^ <*j*>)) - ii) = g . (n + (p /. (((len p) - i3) + 1))) by A8, A11, A22, A27, Def17
.= h . (n + ((p ^ <*j*>) /. (((len (p ^ <*j*>)) - ii) + 1))) by A1, A7, A11, A16, A26, A28, A29, Lm16 ;
::_thesis: verum
end;
end;
end;
end;
end;
end;
then A30: p ^ <*j*> is_vertex_seq_at h,j,n by A12, Def17;
A31: now__::_thesis:_not_j_in_rng_p
assume  j in rng p ; ::_thesis: contradiction
then consider i being Nat such that
A32: i in dom p and
A33: j = p . i by FINSEQ_2:10;
A34: 1 <= i by A32, FINSEQ_3:25;
A35: i <= len p by A32, FINSEQ_3:25;
percases ( i = len p or i <> len p ) ;
suppose i = len p ; ::_thesis: contradiction
hence  contradiction by A5, A8, A33, Def17; ::_thesis: verum
end;
suppose i <> len p ; ::_thesis: contradiction
then  i < len p by A35, XXREAL_0:1;
hence  contradiction by A6, A7, A32, A33, A34; ::_thesis: verum
end;
end;
end;
 p is one-to-one by A2, Def18;
then A36: p ^ <*j*> is one-to-one by A31, Th1;
 p . 1 = 1 by A2, Def18;
then A37: (p ^ <*j*>) . 1 = 1 by A10, Lm1;
 len (p ^ <*j*>) > 1 by A10, A11, NAT_1:13;
hence  p ^ <*j*> is_simple_vertex_seq_at h,j,n by A37, A30, A36, Def18; ::_thesis: verum
end;

Lm19:  for n, i being Element of NAT
for V being set
for f, g being Element of REAL *
for G being oriented finite Graph
for P being oriented Chain of G
for W being Function of the carrier' of G,Real>=0
for v2, v1 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & W is_weight>=0of G & v2 = i & v1 <> v2 & 1 <= i & i <= n & P is_shortestpath_of v1,v2,V,W & ( for m, j being Element of NAT st g . (n + j) = - 1 & 1 <= j & j <= n & m in V holds
f . (((2 * n) + (n * m)) + j) = - 1 ) holds
g . (n + i) <> - 1
proof
let n, i be Element of NAT ; ::_thesis: for V being set
for f, g being Element of REAL *
for G being oriented finite Graph
for P being oriented Chain of G
for W being Function of the carrier' of G,Real>=0
for v2, v1 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & W is_weight>=0of G & v2 = i & v1 <> v2 & 1 <= i & i <= n & P is_shortestpath_of v1,v2,V,W & ( for m, j being Element of NAT st g . (n + j) = - 1 & 1 <= j & j <= n & m in V holds
f . (((2 * n) + (n * m)) + j) = - 1 ) holds
g . (n + i) <> - 1
let V be set ; ::_thesis: for f, g being Element of REAL *
for G being oriented finite Graph
for P being oriented Chain of G
for W being Function of the carrier' of G,Real>=0
for v2, v1 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & W is_weight>=0of G & v2 = i & v1 <> v2 & 1 <= i & i <= n & P is_shortestpath_of v1,v2,V,W & ( for m, j being Element of NAT st g . (n + j) = - 1 & 1 <= j & j <= n & m in V holds
f . (((2 * n) + (n * m)) + j) = - 1 ) holds
g . (n + i) <> - 1
let f, g be Element of REAL * ; ::_thesis: for G being oriented finite Graph
for P being oriented Chain of G
for W being Function of the carrier' of G,Real>=0
for v2, v1 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & W is_weight>=0of G & v2 = i & v1 <> v2 & 1 <= i & i <= n & P is_shortestpath_of v1,v2,V,W & ( for m, j being Element of NAT st g . (n + j) = - 1 & 1 <= j & j <= n & m in V holds
f . (((2 * n) + (n * m)) + j) = - 1 ) holds
g . (n + i) <> - 1
let G be oriented finite Graph; ::_thesis: for P being oriented Chain of G
for W being Function of the carrier' of G,Real>=0
for v2, v1 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & W is_weight>=0of G & v2 = i & v1 <> v2 & 1 <= i & i <= n & P is_shortestpath_of v1,v2,V,W & ( for m, j being Element of NAT st g . (n + j) = - 1 & 1 <= j & j <= n & m in V holds
f . (((2 * n) + (n * m)) + j) = - 1 ) holds
g . (n + i) <> - 1
let P be oriented Chain of G; ::_thesis: for W being Function of the carrier' of G,Real>=0
for v2, v1 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & W is_weight>=0of G & v2 = i & v1 <> v2 & 1 <= i & i <= n & P is_shortestpath_of v1,v2,V,W & ( for m, j being Element of NAT st g . (n + j) = - 1 & 1 <= j & j <= n & m in V holds
f . (((2 * n) + (n * m)) + j) = - 1 ) holds
g . (n + i) <> - 1
let W be Function of the carrier' of G,Real>=0; ::_thesis: for v2, v1 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & W is_weight>=0of G & v2 = i & v1 <> v2 & 1 <= i & i <= n & P is_shortestpath_of v1,v2,V,W & ( for m, j being Element of NAT st g . (n + j) = - 1 & 1 <= j & j <= n & m in V holds
f . (((2 * n) + (n * m)) + j) = - 1 ) holds
g . (n + i) <> - 1
let v2, v1 be Vertex of G; ::_thesis: ( f is_Input_of_Dijkstra_Alg G,n,W & W is_weight>=0of G & v2 = i & v1 <> v2 & 1 <= i & i <= n & P is_shortestpath_of v1,v2,V,W & ( for m, j being Element of NAT st g . (n + j) = - 1 & 1 <= j & j <= n & m in V holds
f . (((2 * n) + (n * m)) + j) = - 1 ) implies g . (n + i) <> - 1 )
assume that
A1: f is_Input_of_Dijkstra_Alg G,n,W and
A2: W is_weight>=0of G and
A3: v2 = i and
A4: v1 <> v2 and
A5: ( 1 <= i & i <= n ) and
A6: P is_shortestpath_of v1,v2,V,W and
A7: for m, j being Element of NAT st g . (n + j) = - 1 & 1 <= j & j <= n & m in V holds
f . (((2 * n) + (n * m)) + j) = - 1 ; ::_thesis: g . (n + i) <> - 1
 P is_orientedpath_of v1,v2,V by A6, GRAPH_5:def_18;
then consider q being oriented Simple Chain of G such that
A8: q is_shortestpath_of v1,v2,V,W by A2, GRAPH_5:62;
set FT = the Target of G;
assume A9: g . (n + i) = - 1 ; ::_thesis: contradiction
set e = q . (len q);
consider vs being FinSequence of the carrier of G such that
A10: vs is_oriented_vertex_seq_of q and
A11: for n1, m1 being Element of NAT st 1 <= n1 & n1 < m1 & m1 <= len vs & vs . n1 = vs . m1 holds
( n1 = 1 & m1 = len vs ) by GRAPH_4:def_7;
A12: q is_orientedpath_of v1,v2,V by A8, GRAPH_5:def_18;
then A13: q is_orientedpath_of v1,v2 by GRAPH_5:def_4;
then  q <> {} by GRAPH_5:def_3;
then A14: len q >= 1 by FINSEQ_1:20;
then A15: q . (len q) orientedly_joins vs /. (len q),vs /. ((len q) + 1) by A10, GRAPH_4:def_5;
 len q in dom q by A14, FINSEQ_3:25;
then A16: q . (len q) in the carrier' of G by FINSEQ_2:11;
A17: len vs = (len q) + 1 by A10, GRAPH_4:def_5;
then A18: len q < len vs by NAT_1:13;
then A19: len q in dom vs by A14, FINSEQ_3:25;
A20: vs /. (len q) = vs . (len q) by A14, A18, FINSEQ_4:15;
then reconsider v3 = vs . (len q) as Vertex of G ;
 the Target of G . (q . (len q)) = v2 by A13, GRAPH_5:def_3;
then A21: v2 = vs /. ((len q) + 1) by A15, GRAPH_4:def_1;
A22: 1 < (len q) + 1 by A14, NAT_1:13;
then A23: v2 = vs . (len vs) by A17, A21, FINSEQ_4:15;
now__::_thesis:_not_v3_=_v2
A24: q . 1 orientedly_joins vs /. 1,vs /. (1 + 1) by A10, A14, GRAPH_4:def_5;
assume A25: v3 = v2 ; ::_thesis: contradiction
v1 = the Source of G . (q . 1) by A13, GRAPH_5:def_3
.= vs /. 1 by A24, GRAPH_4:def_1
.= vs . 1 by A17, A22, FINSEQ_4:15
.= v3 by A11, A14, A18, A23, A25 ;
hence  contradiction by A4, A25; ::_thesis: verum
end;
then A26: not v3 in {v2} by TARSKI:def_1;
 Seg n = the carrier of G by A1, Def20;
then  v3 in Seg n by A19, FINSEQ_2:11;
then reconsider m = v3 as Element of NAT ;
A27: f . (((2 * n) + (n * m)) + i) = Weight (v3,v2,W) by A1, A3, Def20;
 v3 = the Source of G . (q . (len q)) by A15, A20, GRAPH_4:def_1;
then  v3 in vertices q by A14, Lm4;
then A28: m in (vertices q) \ {v2} by A26, XBOOLE_0:def_5;
 (vertices q) \ {v2} c= V by A12, GRAPH_5:def_4;
then  f . (((2 * n) + (n * m)) + i) = - 1 by A5, A7, A9, A28;
hence  contradiction by A15, A16, A20, A21, A27, Th23; ::_thesis: verum
end;

Lm20:  for n, k being Element of NAT
for f, g, h being Element of REAL *
for G being oriented finite Graph
for W being Function of the carrier' of G,Real>=0
for v1 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & v1 = 1 & n >= 1 & g = ((repeat ((Relax n) * (findmin n))) . k) . f & h = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f & OuterVx (g,n) <> {} & 1 in UsedVx (g,n) & ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & g . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at g,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (g,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (g,n),W & cost (P,W) = g . ((2 * n) + j) & ( not v3 in UsedVx (g,n) implies P islongestInShortestpath UsedVx (g,n),v1,W ) ) ) & ( for m, j being Element of NAT st g . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (g,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (g,n) holds
g . (n + m) <> - 1 ) holds
( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & h . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) ) ) & ( for m, j being Element of NAT st h . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (h,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (h,n) holds
h . (n + m) <> - 1 ) )
proof
let n, k be Element of NAT ; ::_thesis: for f, g, h being Element of REAL *
for G being oriented finite Graph
for W being Function of the carrier' of G,Real>=0
for v1 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & v1 = 1 & n >= 1 & g = ((repeat ((Relax n) * (findmin n))) . k) . f & h = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f & OuterVx (g,n) <> {} & 1 in UsedVx (g,n) & ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & g . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at g,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (g,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (g,n),W & cost (P,W) = g . ((2 * n) + j) & ( not v3 in UsedVx (g,n) implies P islongestInShortestpath UsedVx (g,n),v1,W ) ) ) & ( for m, j being Element of NAT st g . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (g,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (g,n) holds
g . (n + m) <> - 1 ) holds
( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & h . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) ) ) & ( for m, j being Element of NAT st h . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (h,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (h,n) holds
h . (n + m) <> - 1 ) )
let f, g, h be Element of REAL * ; ::_thesis: for G being oriented finite Graph
for W being Function of the carrier' of G,Real>=0
for v1 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & v1 = 1 & n >= 1 & g = ((repeat ((Relax n) * (findmin n))) . k) . f & h = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f & OuterVx (g,n) <> {} & 1 in UsedVx (g,n) & ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & g . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at g,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (g,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (g,n),W & cost (P,W) = g . ((2 * n) + j) & ( not v3 in UsedVx (g,n) implies P islongestInShortestpath UsedVx (g,n),v1,W ) ) ) & ( for m, j being Element of NAT st g . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (g,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (g,n) holds
g . (n + m) <> - 1 ) holds
( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & h . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) ) ) & ( for m, j being Element of NAT st h . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (h,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (h,n) holds
h . (n + m) <> - 1 ) )
let G be oriented finite Graph; ::_thesis: for W being Function of the carrier' of G,Real>=0
for v1 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & v1 = 1 & n >= 1 & g = ((repeat ((Relax n) * (findmin n))) . k) . f & h = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f & OuterVx (g,n) <> {} & 1 in UsedVx (g,n) & ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & g . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at g,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (g,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (g,n),W & cost (P,W) = g . ((2 * n) + j) & ( not v3 in UsedVx (g,n) implies P islongestInShortestpath UsedVx (g,n),v1,W ) ) ) & ( for m, j being Element of NAT st g . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (g,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (g,n) holds
g . (n + m) <> - 1 ) holds
( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & h . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) ) ) & ( for m, j being Element of NAT st h . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (h,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (h,n) holds
h . (n + m) <> - 1 ) )
let W be Function of the carrier' of G,Real>=0; ::_thesis: for v1 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & v1 = 1 & n >= 1 & g = ((repeat ((Relax n) * (findmin n))) . k) . f & h = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f & OuterVx (g,n) <> {} & 1 in UsedVx (g,n) & ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & g . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at g,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (g,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (g,n),W & cost (P,W) = g . ((2 * n) + j) & ( not v3 in UsedVx (g,n) implies P islongestInShortestpath UsedVx (g,n),v1,W ) ) ) & ( for m, j being Element of NAT st g . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (g,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (g,n) holds
g . (n + m) <> - 1 ) holds
( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & h . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) ) ) & ( for m, j being Element of NAT st h . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (h,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (h,n) holds
h . (n + m) <> - 1 ) )
let v1 be Vertex of G; ::_thesis: ( f is_Input_of_Dijkstra_Alg G,n,W & v1 = 1 & n >= 1 & g = ((repeat ((Relax n) * (findmin n))) . k) . f & h = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f & OuterVx (g,n) <> {} & 1 in UsedVx (g,n) & ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & g . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at g,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (g,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (g,n),W & cost (P,W) = g . ((2 * n) + j) & ( not v3 in UsedVx (g,n) implies P islongestInShortestpath UsedVx (g,n),v1,W ) ) ) & ( for m, j being Element of NAT st g . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (g,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (g,n) holds
g . (n + m) <> - 1 ) implies ( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & h . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) ) ) & ( for m, j being Element of NAT st h . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (h,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (h,n) holds
h . (n + m) <> - 1 ) ) )
set R = Relax n;
set M = findmin n;
set IN = OuterVx (g,n);
set Ug = UsedVx (g,n);
assume that
A1: f is_Input_of_Dijkstra_Alg G,n,W and
A2: v1 = 1 and
A3: n >= 1 and
A4: g = ((repeat ((Relax n) * (findmin n))) . k) . f and
A5: h = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f and
A6: OuterVx (g,n) <> {} and
A7: 1 in UsedVx (g,n) ; ::_thesis: ( ex v3 being Vertex of G ex j being Element of NAT st
( v3 <> v1 & v3 = j & g . (n + j) <> - 1 & ( for p being FinSequence of NAT
for P being oriented Chain of G holds
( not p is_simple_vertex_seq_at g,j,n or ex i being Element of NAT st
( 1 <= i & i < len p & not p . i in UsedVx (g,n) ) or not P is_oriented_edge_seq_of p or not P is_shortestpath_of v1,v3, UsedVx (g,n),W or not cost (P,W) = g . ((2 * n) + j) or ( not v3 in UsedVx (g,n) & not P islongestInShortestpath UsedVx (g,n),v1,W ) ) ) ) or ex m, j being Element of NAT st
( g . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (g,n) & not f . (((2 * n) + (n * m)) + j) = - 1 ) or ex m being Element of NAT st
( m in UsedVx (g,n) & not g . (n + m) <> - 1 ) or ( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & h . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) ) ) & ( for m, j being Element of NAT st h . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (h,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (h,n) holds
h . (n + m) <> - 1 ) ) )
assume A8: for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & g . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at g,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (g,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (g,n),W & cost (P,W) = g . ((2 * n) + j) & ( not v3 in UsedVx (g,n) implies P islongestInShortestpath UsedVx (g,n),v1,W ) ) ; ::_thesis: ( ex m, j being Element of NAT st
( g . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (g,n) & not f . (((2 * n) + (n * m)) + j) = - 1 ) or ex m being Element of NAT st
( m in UsedVx (g,n) & not g . (n + m) <> - 1 ) or ( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & h . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) ) ) & ( for m, j being Element of NAT st h . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (h,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (h,n) holds
h . (n + m) <> - 1 ) ) )
assume that
A9: for m, j being Element of NAT st g . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (g,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 and
A10: for m being Element of NAT st m in UsedVx (g,n) holds
g . (n + m) <> - 1 ; ::_thesis: ( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & h . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) ) ) & ( for m, j being Element of NAT st h . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (h,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (h,n) holds
h . (n + m) <> - 1 ) )
set mi = ((n * n) + (3 * n)) + 1;
set Ak = Argmin ((OuterVx (g,n)),g,n);
A11: 1 <= ((n * n) + (3 * n)) + 1 by NAT_1:12;
A12: len f = ((n * n) + (3 * n)) + 1 by A1, Def20;
A13: (findmin n) . g = (g,(((n * n) + (3 * n)) + 1)) := ((Argmin ((OuterVx (g,n)),g,n)),(- 1)) by Def11;
A14: dom ((findmin n) . g) = dom g by Th33;
 h = (Relax n) . ((findmin n) . g) by A4, A5, Th22;
then A15: h = Relax (((findmin n) . g),n) by Def15;
A16: Seg n = the carrier of G by A1, Def20;
then reconsider VG = the carrier of G as non empty Subset of NAT by A3;
A17: W is_weight>=0of G by GRAPH_5:def_13;
A18: (2 * n) + n = (2 + 1) * n ;
A19: dom f = dom g by A4, Th41;
A20: 1 <= Argmin ((OuterVx (g,n)),g,n) by A6, Th29;
A21: Argmin ((OuterVx (g,n)),g,n) <= n by A6, Th29;
A22: g . (n + (Argmin ((OuterVx (g,n)),g,n))) <> - 1 by A6, Th29;
set Uh = UsedVx (h,n);
A23: ( UsedVx (h,n) = (UsedVx (g,n)) \/ {(Argmin ((OuterVx (g,n)),g,n))} & not Argmin ((OuterVx (g,n)),g,n) in UsedVx (g,n) ) by A4, A5, A6, Th39;
then A24: UsedVx (g,n) c= UsedVx (h,n) by XBOOLE_1:7;
A25: n < ((n * n) + (3 * n)) + 1 by Lm7;
A26: dom f = dom h by A5, Th41;
reconsider vk = Argmin ((OuterVx (g,n)),g,n) as Vertex of G by A16, A20, A21, FINSEQ_1:1;
consider pk being FinSequence of NAT , PK being oriented Chain of G such that
A27: pk is_simple_vertex_seq_at g, Argmin ((OuterVx (g,n)),g,n),n and
A28: for i being Element of NAT st 1 <= i & i < len pk holds
pk . i in UsedVx (g,n) and
A29: PK is_oriented_edge_seq_of pk and
A30: PK is_shortestpath_of v1,vk, UsedVx (g,n),W and
A31: cost (PK,W) = g . ((2 * n) + (Argmin ((OuterVx (g,n)),g,n))) and
A32: ( not vk in UsedVx (g,n) implies PK islongestInShortestpath UsedVx (g,n),v1,W ) by A2, A7, A8, A22, A23;
A33: ex kk being Element of NAT st
( kk = Argmin ((OuterVx (g,n)),g,n) & kk in OuterVx (g,n) & ( for i being Element of NAT st i in OuterVx (g,n) holds
g /. ((2 * n) + kk) <= g /. ((2 * n) + i) ) & ( for i being Element of NAT st i in OuterVx (g,n) & g /. ((2 * n) + kk) = g /. ((2 * n) + i) holds
kk <= i ) ) by A6, Def10;
set nAk = (2 * n) + (Argmin ((OuterVx (g,n)),g,n));
A34: 1 < (2 * n) + (Argmin ((OuterVx (g,n)),g,n)) by A20, A21, Lm11;
A35: (2 * n) + (Argmin ((OuterVx (g,n)),g,n)) < ((n * n) + (3 * n)) + 1 by A20, A21, Lm11;
A36: Argmin ((OuterVx (g,n)),g,n) < (2 * n) + (Argmin ((OuterVx (g,n)),g,n)) by A20, A21, Lm11;
A37: (2 * n) + (Argmin ((OuterVx (g,n)),g,n)) in dom g by A12, A19, A34, A35, FINSEQ_3:25;
A38: f,g equal_at (3 * n) + 1,(n * n) + (3 * n) by A4, Th46;
 PK is_orientedpath_of v1,vk, UsedVx (g,n) by A30, GRAPH_5:def_18;
then A39: PK is_orientedpath_of v1,vk by GRAPH_5:def_4;
then  PK <> {} by GRAPH_5:def_3;
then A40: len PK >= 1 by FINSEQ_1:20;
A41: ((n * n) + (3 * n)) + 1 in dom g by A11, A12, A19, FINSEQ_3:25;
then A42: ((findmin n) . g) /. (((n * n) + (3 * n)) + 1) = ((findmin n) . g) . (((n * n) + (3 * n)) + 1) by A14, PARTFUN1:def_6
.= Argmin ((OuterVx (g,n)),g,n) by A13, A21, A25, A41, Th17 ;
A43: ((findmin n) . g) /. ((2 * n) + (Argmin ((OuterVx (g,n)),g,n))) = ((findmin n) . g) . ((2 * n) + (Argmin ((OuterVx (g,n)),g,n))) by A14, A37, PARTFUN1:def_6
.= cost (PK,W) by A13, A31, A35, A36, Th18 ;
set nk = n + (Argmin ((OuterVx (g,n)),g,n));
A44: 1 < n + (Argmin ((OuterVx (g,n)),g,n)) by A20, A21, Lm12;
A45: n + (Argmin ((OuterVx (g,n)),g,n)) <= 2 * n by A20, A21, Lm12;
A46: n + (Argmin ((OuterVx (g,n)),g,n)) < ((n * n) + (3 * n)) + 1 by A20, A21, Lm12;
 n + 1 <= n + (Argmin ((OuterVx (g,n)),g,n)) by A20, XREAL_1:7;
then A47: n < n + (Argmin ((OuterVx (g,n)),g,n)) by NAT_1:13;
A48: n + (Argmin ((OuterVx (g,n)),g,n)) in dom g by A12, A19, A44, A46, FINSEQ_3:25;
A49: ((findmin n) . g) . (n + (Argmin ((OuterVx (g,n)),g,n))) = g . (n + (Argmin ((OuterVx (g,n)),g,n))) by A46, A47, Th31;
now__::_thesis:_not_(findmin_n)_._g_hasBetterPathAt_n,_Argmin_((OuterVx_(g,n)),g,n)
set Wke = ((findmin n) . g) /. (((2 * n) + (n * (((findmin n) . g) /. (((n * n) + (3 * n)) + 1)))) + (Argmin ((OuterVx (g,n)),g,n)));
assume A50: (findmin n) . g hasBetterPathAt n, Argmin ((OuterVx (g,n)),g,n) ; ::_thesis: contradiction
then A51: ( ((findmin n) . g) . (n + (Argmin ((OuterVx (g,n)),g,n))) = - 1 or ((findmin n) . g) /. ((2 * n) + (Argmin ((OuterVx (g,n)),g,n))) > newpathcost (((findmin n) . g),n,(Argmin ((OuterVx (g,n)),g,n))) ) by Def13;
 ((findmin n) . g) /. (((2 * n) + (n * (((findmin n) . g) /. (((n * n) + (3 * n)) + 1)))) + (Argmin ((OuterVx (g,n)),g,n))) >= 0 by A50, Def13;
then  (((findmin n) . g) /. ((2 * n) + (Argmin ((OuterVx (g,n)),g,n)))) + (((findmin n) . g) /. (((2 * n) + (n * (((findmin n) . g) /. (((n * n) + (3 * n)) + 1)))) + (Argmin ((OuterVx (g,n)),g,n)))) >= (((findmin n) . g) /. ((2 * n) + (Argmin ((OuterVx (g,n)),g,n)))) + 0 by XREAL_1:7;
hence  contradiction by A6, A42, A49, A51, Th29; ::_thesis: verum
end;
then  not (findmin n) . g hasBetterPathAt n,(n + (Argmin ((OuterVx (g,n)),g,n))) -' n by NAT_D:34;
then A52: h . (n + (Argmin ((OuterVx (g,n)),g,n))) = g . (n + (Argmin ((OuterVx (g,n)),g,n))) by A14, A15, A45, A47, A48, A49, Def14;
hereby  ::_thesis: ( ( for m, j being Element of NAT st h . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (h,n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx (h,n) holds
h . (n + m) <> - 1 ) )
let v3 be Vertex of G; ::_thesis: for j being Element of NAT st v3 <> v1 & v3 = j & h . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( b4 is_simple_vertex_seq_at h,P,n & ( for i being Element of NAT st 1 <= b6 & b6 < len i holds
i . b6 in UsedVx (h,n) ) & b5 is_oriented_edge_seq_of b4 & b5 is_shortestpath_of v1,p, UsedVx (h,n),W & cost (b5,W) = h . ((2 * n) + P) & ( not p in UsedVx (h,n) implies b5 islongestInShortestpath UsedVx (h,n),v1,W ) )
let j be Element of NAT ; ::_thesis: ( v3 <> v1 & v3 = j & h . (n + j) <> - 1 implies ex p being FinSequence of NAT ex P being oriented Chain of G st
( b3 is_simple_vertex_seq_at h,P,n & ( for i being Element of NAT st 1 <= b5 & b5 < len i holds
i . b5 in UsedVx (h,n) ) & b4 is_oriented_edge_seq_of b3 & b4 is_shortestpath_of v1,p, UsedVx (h,n),W & cost (b4,W) = h . ((2 * n) + P) & ( not p in UsedVx (h,n) implies b4 islongestInShortestpath UsedVx (h,n),v1,W ) ) )
assume that
A53: v3 <> v1 and
A54: v3 = j and
A55: h . (n + j) <> - 1 ; ::_thesis: ex p being FinSequence of NAT ex P being oriented Chain of G st
( b3 is_simple_vertex_seq_at h,P,n & ( for i being Element of NAT st 1 <= b5 & b5 < len i holds
i . b5 in UsedVx (h,n) ) & b4 is_oriented_edge_seq_of b3 & b4 is_shortestpath_of v1,p, UsedVx (h,n),W & cost (b4,W) = h . ((2 * n) + P) & ( not p in UsedVx (h,n) implies b4 islongestInShortestpath UsedVx (h,n),v1,W ) )
set nj = n + j;
A56: j in VG by A54;
then A57: 1 <= j by A16, FINSEQ_1:1;
A58: j <= n by A16, A56, FINSEQ_1:1;
then A59: 1 < n + j by A57, Lm12;
A60: n + j <= 2 * n by A57, A58, Lm12;
A61: n + j < ((n * n) + (3 * n)) + 1 by A57, A58, Lm12;
then A62: n + j in dom g by A12, A19, A59, FINSEQ_3:25;
A63: (n + j) -' n = j by NAT_D:34;
 n + 1 <= n + j by A57, XREAL_1:7;
then A64: n < n + j by NAT_1:13;
set m2 = (2 * n) + j;
A65: (2 * n) + j <= 3 * n by A18, A58, XREAL_1:7;
 (2 * n) + 1 <= (2 * n) + j by A57, XREAL_1:7;
then A66: 2 * n < (2 * n) + j by NAT_1:13;
A67: ((2 * n) + j) -' (2 * n) = j by NAT_D:34;
A68: 1 < (2 * n) + j by A57, A58, Lm11;
A69: (2 * n) + j < ((n * n) + (3 * n)) + 1 by A57, A58, Lm11;
then A70: (2 * n) + j in dom g by A12, A19, A68, FINSEQ_3:25;
A71: (2 * n) + j in dom ((findmin n) . g) by A12, A14, A19, A68, A69, FINSEQ_3:25;
A72: ((findmin n) . g) . (n + j) = g . (n + j) by A13, A21, A61, A64, Th18;
 n <= 2 * n by Lm6;
then  n < (2 * n) + j by A66, XXREAL_0:2;
then A73: ((findmin n) . g) . ((2 * n) + j) = g . ((2 * n) + j) by A13, A21, A69, Th18;
A74: j < ((n * n) + (3 * n)) + 1 by A25, A58, XXREAL_0:2;
then A75: j in dom g by A12, A19, A57, FINSEQ_3:25;
A76: j in dom h by A12, A26, A57, A74, FINSEQ_3:25;
set Akj = ((2 * n) + (n * (Argmin ((OuterVx (g,n)),g,n)))) + j;
A77: 1 < ((2 * n) + (n * (Argmin ((OuterVx (g,n)),g,n)))) + j by A20, A21, A58, Lm13;
A78: Argmin ((OuterVx (g,n)),g,n) < ((2 * n) + (n * (Argmin ((OuterVx (g,n)),g,n)))) + j by A20, A21, A58, Lm13;
A79: ((2 * n) + (n * (Argmin ((OuterVx (g,n)),g,n)))) + j < ((n * n) + (3 * n)) + 1 by A20, A21, A58, Lm13;
then A80: ((2 * n) + (n * (Argmin ((OuterVx (g,n)),g,n)))) + j in dom g by A12, A19, A77, FINSEQ_3:25;
A81: (3 * n) + 1 <= ((2 * n) + (n * (Argmin ((OuterVx (g,n)),g,n)))) + j by A20, A21, A57, A58, Lm14;
A82: ((2 * n) + (n * (Argmin ((OuterVx (g,n)),g,n)))) + j <= (n * n) + (3 * n) by A20, A21, A57, A58, Lm14;
A83: ((findmin n) . g) /. (((2 * n) + (n * (((findmin n) . g) /. (((n * n) + (3 * n)) + 1)))) + j) = ((findmin n) . g) . (((2 * n) + (n * (Argmin ((OuterVx (g,n)),g,n)))) + j) by A14, A42, A80, PARTFUN1:def_6
.= g . (((2 * n) + (n * (Argmin ((OuterVx (g,n)),g,n)))) + j) by A13, A78, A79, Th18
.= f . (((2 * n) + (n * (Argmin ((OuterVx (g,n)),g,n)))) + j) by A19, A38, A80, A81, A82, Def16
.= Weight (vk,v3,W) by A1, A54, Def20 ;
A84: ((findmin n) . g) /. ((2 * n) + j) = g . ((2 * n) + j) by A14, A70, A73, PARTFUN1:def_6;
percases ( not (findmin n) . g hasBetterPathAt n,(n + j) -' n or (findmin n) . g hasBetterPathAt n,(n + j) -' n ) ;
supposeA85: not (findmin n) . g hasBetterPathAt n,(n + j) -' n ; ::_thesis: ex p being FinSequence of NAT ex P being oriented Chain of G st
( b3 is_simple_vertex_seq_at h,P,n & ( for i being Element of NAT st 1 <= b5 & b5 < len i holds
i . b5 in UsedVx (h,n) ) & b4 is_oriented_edge_seq_of b3 & b4 is_shortestpath_of v1,p, UsedVx (h,n),W & cost (b4,W) = h . ((2 * n) + P) & ( not p in UsedVx (h,n) implies b4 islongestInShortestpath UsedVx (h,n),v1,W ) )
then A86: h . (n + j) = g . (n + j) by A14, A15, A60, A62, A64, A72, Def14;
then consider p being FinSequence of NAT , P being oriented Chain of G such that
A87: p is_simple_vertex_seq_at g,j,n and
A88: for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (g,n) and
A89: P is_oriented_edge_seq_of p and
A90: P is_shortestpath_of v1,v3, UsedVx (g,n),W and
A91: cost (P,W) = g . ((2 * n) + j) and
A92: ( not v3 in UsedVx (g,n) implies P islongestInShortestpath UsedVx (g,n),v1,W ) by A8, A53, A54, A55;
take p = p; ::_thesis: ex P being oriented Chain of G st
( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) )
take P = P; ::_thesis: ( p is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) )
thus  p is_simple_vertex_seq_at h,j,n by A4, A5, A6, A12, A86, A87, A88, Lm17; ::_thesis: ( ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (h,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) )
hereby  ::_thesis: ( P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) )
let i be Element of NAT ; ::_thesis: ( 1 <= i & i < len p implies p . i in UsedVx (h,n) )
assume that
A93: 1 <= i and
A94: i < len p ; ::_thesis: p . i in UsedVx (h,n)
 p . i in UsedVx (g,n) by A88, A93, A94;
hence  p . i in UsedVx (h,n) by A24; ::_thesis: verum
end;
thus  P is_oriented_edge_seq_of p by A89; ::_thesis: ( P is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) )
hereby  ::_thesis: ( cost (P,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W ) )
percases ( ((findmin n) . g) . j = - 1 or ((findmin n) . g) . j <> - 1 ) ;
suppose ((findmin n) . g) . j = - 1 ; ::_thesis: P is_shortestpath_of v1,v3, UsedVx (h,n),W
then  (Relax (((findmin n) . g),n)) . j = - 1 by A14, A58, A75, Def14;
then  j in  {  i where i is Element of NAT : ( i in dom h & 1 <= i & i <= n & h . i = - 1 )  }  by A15, A57, A58, A76;
then A95: ( j in UsedVx (g,n) or j in {(Argmin ((OuterVx (g,n)),g,n))} ) by A23, XBOOLE_0:def_3;
now__::_thesis:_for_Q_being_oriented_Chain_of_G
for_v4_being_Vertex_of_G_st_not_v4_in_UsedVx_(g,n)_&_Q_is_shortestpath_of_v1,v4,_UsedVx_(g,n),W_holds_
cost_(P,W)_<=_cost_(Q,W)
let Q be oriented Chain of G; ::_thesis: for v4 being Vertex of G st not v4 in UsedVx (g,n) & Q is_shortestpath_of v1,v4, UsedVx (g,n),W holds
cost (P,W) <= cost (b2,W)
let v4 be Vertex of G; ::_thesis: ( not v4 in UsedVx (g,n) & Q is_shortestpath_of v1,v4, UsedVx (g,n),W implies cost (P,W) <= cost (b1,W) )
assume that
A96: not v4 in UsedVx (g,n) and
A97: Q is_shortestpath_of v1,v4, UsedVx (g,n),W ; ::_thesis: cost (P,W) <= cost (b1,W)
A98: v4 in VG ;
then reconsider j4 = v4 as Element of NAT ;
A99: 1 <= j4 by A16, A98, FINSEQ_1:1;
A100: j4 <= n by A16, A98, FINSEQ_1:1;
then A101: g . (n + j4) <> - 1 by A1, A2, A7, A9, A17, A96, A97, A99, Lm19;
then consider q being FinSequence of NAT , R being oriented Chain of G such that
 q is_simple_vertex_seq_at g,j4,n and
 for i being Element of NAT st 1 <= i & i < len q holds
q . i in UsedVx (g,n) and
 R is_oriented_edge_seq_of q and
A102: R is_shortestpath_of v1,v4, UsedVx (g,n),W and
A103: cost (R,W) = g . ((2 * n) + j4) and
A104: ( not v4 in UsedVx (g,n) implies R islongestInShortestpath UsedVx (g,n),v1,W ) by A2, A7, A8, A96;
A105: cost (R,W) = cost (Q,W) by A97, A102, Th9;
percases ( j in UsedVx (g,n) or j = Argmin ((OuterVx (g,n)),g,n) ) by A95, TARSKI:def_1;
suppose j in UsedVx (g,n) ; ::_thesis: cost (P,W) <= cost (b1,W)
then  ex PP being oriented Chain of G st
( PP is_shortestpath_of v1,v3, UsedVx (g,n),W & cost (PP,W) <= cost (R,W) ) by A53, A54, A96, A104, GRAPH_5:def_19;
hence  cost (P,W) <= cost (Q,W) by A90, A105, Th9; ::_thesis: verum
end;
supposeA106: j = Argmin ((OuterVx (g,n)),g,n) ; ::_thesis: cost (P,W) <= cost (b1,W)
 j4 <= ((n * n) + (3 * n)) + 1 by A25, A100, XXREAL_0:2;
then A107: j4 in dom g by A12, A19, A99, FINSEQ_3:25;
then  g . j4 <> - 1 by A96, A99, A100;
then  j4 in  {  i where i is Element of NAT : ( i in dom g & 1 <= i & i <= n & g . i <> - 1 & g . (n + i) <> - 1 )  }  by A99, A100, A101, A107;
then A108: g /. ((2 * n) + (Argmin ((OuterVx (g,n)),g,n))) <= g /. ((2 * n) + j4) by A33;
A109: g /. ((2 * n) + (Argmin ((OuterVx (g,n)),g,n))) = g . ((2 * n) + (Argmin ((OuterVx (g,n)),g,n))) by A37, PARTFUN1:def_6;
A110: 1 < (2 * n) + j4 by A99, A100, Lm11;
 (2 * n) + j4 < ((n * n) + (3 * n)) + 1 by A99, A100, Lm11;
then  (2 * n) + j4 in dom g by A12, A19, A110, FINSEQ_3:25;
hence  cost (P,W) <= cost (Q,W) by A91, A103, A105, A106, A108, A109, PARTFUN1:def_6; ::_thesis: verum
end;
end;
end;
hence  P is_shortestpath_of v1,v3, UsedVx (h,n),W by A17, A24, A53, A90, GRAPH_5:64; ::_thesis: verum
end;
supposeA111: ((findmin n) . g) . j <> - 1 ; ::_thesis: P is_shortestpath_of v1,v3, UsedVx (h,n),W
hereby  ::_thesis: verum
percases ( ((findmin n) . g) /. (((2 * n) + (n * (((findmin n) . g) /. (((n * n) + (3 * n)) + 1)))) + j) >= 0 or ((findmin n) . g) /. (((2 * n) + (n * (((findmin n) . g) /. (((n * n) + (3 * n)) + 1)))) + j) < 0 ) ;
supposeA112: ((findmin n) . g) /. (((2 * n) + (n * (((findmin n) . g) /. (((n * n) + (3 * n)) + 1)))) + j) >= 0 ; ::_thesis: P is_shortestpath_of v1,v3, UsedVx (h,n),W
then A113: ((findmin n) . g) /. ((2 * n) + j) <= newpathcost (((findmin n) . g),n,j) by A63, A85, A111, Def13;
A114: ((findmin n) . g) /. ((2 * n) + j) = cost (P,W) by A71, A73, A91, PARTFUN1:def_6;
consider e being set  such that
A115: e in the carrier' of G and
A116: e orientedly_joins vk,v3 by A83, A112, Th23;
reconsider pe = <*e*> as oriented Chain of G by A115, Th5;
A117: len pe = 1 by FINSEQ_1:40;
A118: pe . 1 = e by FINSEQ_1:40;
then consider Q being oriented Chain of G such that
A119: Q = PK ^ pe and
 Q is_orientedpath_of v1,v3 by A39, A40, A116, A117, GRAPH_5:33;
cost (pe,W) = W . (pe . 1) by A17, A117, Th4, GRAPH_5:46
.= Weight (vk,v3,W) by A115, A116, A118, Th25 ;
then  cost (Q,W) = newpathcost (((findmin n) . g),n,j) by A17, A42, A43, A83, A119, GRAPH_5:46, GRAPH_5:54;
hence  P is_shortestpath_of v1,v3, UsedVx (h,n),W by A2, A7, A17, A23, A30, A32, A40, A53, A90, A113, A114, A115, A116, A119, GRAPH_5:65; ::_thesis: verum
end;
suppose ((findmin n) . g) /. (((2 * n) + (n * (((findmin n) . g) /. (((n * n) + (3 * n)) + 1)))) + j) < 0 ; ::_thesis: P is_shortestpath_of v1,v3, UsedVx (h,n),W
then  for e being set holds
( not e in the carrier' of G or not e orientedly_joins vk,v3 ) by A83, Th23;
hence  P is_shortestpath_of v1,v3, UsedVx (h,n),W by A2, A7, A17, A23, A30, A32, A53, A90, Th13; ::_thesis: verum
end;
end;
end;
end;
end;
end;
thus  cost (P,W) = h . ((2 * n) + j) by A15, A63, A65, A66, A67, A71, A73, A85, A91, Def14; ::_thesis: ( not v3 in UsedVx (h,n) implies P islongestInShortestpath UsedVx (h,n),v1,W )
hereby  ::_thesis: verum
assume A120: not v3 in UsedVx (h,n) ; ::_thesis: P islongestInShortestpath UsedVx (h,n),v1,W
then A121: not v3 in UsedVx (g,n) by A23, XBOOLE_0:def_3;
now__::_thesis:_for_v2_being_Vertex_of_G_st_v2_in_UsedVx_(h,n)_&_v2_<>_v1_holds_
ex_PK_being_oriented_Chain_of_G_st_
(_PK_is_shortestpath_of_v1,v2,_UsedVx_(h,n),W_&_cost_(PK,W)_<=_cost_(P,W)_)
let v2 be Vertex of G; ::_thesis: ( v2 in UsedVx (h,n) & v2 <> v1 implies ex PK being oriented Chain of G st
( b2 is_shortestpath_of v1,PK, UsedVx (h,n),W & cost (b2,W) <= cost (P,W) ) )
assume that
A122: v2 in UsedVx (h,n) and
A123: v2 <> v1 ; ::_thesis: ex PK being oriented Chain of G st
( b2 is_shortestpath_of v1,PK, UsedVx (h,n),W & cost (b2,W) <= cost (P,W) )
percases ( v2 in {(Argmin ((OuterVx (g,n)),g,n))} or v2 in UsedVx (g,n) ) by A23, A122, XBOOLE_0:def_3;
suppose v2 in {(Argmin ((OuterVx (g,n)),g,n))} ; ::_thesis: ex PK being oriented Chain of G st
( b2 is_shortestpath_of v1,PK, UsedVx (h,n),W & cost (b2,W) <= cost (P,W) )
then A124: v2 = vk by TARSKI:def_1;
take PK = PK; ::_thesis: ( PK is_shortestpath_of v1,v2, UsedVx (h,n),W & cost (PK,W) <= cost (P,W) )
thus  PK is_shortestpath_of v1,v2, UsedVx (h,n),W by A23, A30, A124, Th8; ::_thesis: cost (PK,W) <= cost (P,W)
 g . j <> - 1 by A54, A57, A58, A75, A121;
then  j in  {  i where i is Element of NAT : ( i in dom g & 1 <= i & i <= n & g . i <> - 1 & g . (n + i) <> - 1 )  }  by A55, A57, A58, A75, A86;
then A125: g /. ((2 * n) + (Argmin ((OuterVx (g,n)),g,n))) <= g /. ((2 * n) + j) by A33;
 g /. ((2 * n) + (Argmin ((OuterVx (g,n)),g,n))) = cost (PK,W) by A31, A37, PARTFUN1:def_6;
hence  cost (PK,W) <= cost (P,W) by A70, A91, A125, PARTFUN1:def_6; ::_thesis: verum
end;
supposeA126: v2 in UsedVx (g,n) ; ::_thesis: ex Q being oriented Chain of G st
( b2 is_shortestpath_of v1,Q, UsedVx (h,n),W & cost (b2,W) <= cost (P,W) )
then consider Q being oriented Chain of G such that
A127: Q is_shortestpath_of v1,v2, UsedVx (g,n),W and
A128: cost (Q,W) <= cost (P,W) by A23, A92, A120, A123, GRAPH_5:def_19, XBOOLE_0:def_3;
A129: now__::_thesis:_for_R_being_oriented_Chain_of_G
for_v4_being_Vertex_of_G_st_not_v4_in_UsedVx_(g,n)_&_R_is_shortestpath_of_v1,v4,_UsedVx_(g,n),W_holds_
cost_(Q,W)_<=_cost_(R,W)
let R be oriented Chain of G; ::_thesis: for v4 being Vertex of G st not v4 in UsedVx (g,n) & R is_shortestpath_of v1,v4, UsedVx (g,n),W holds
cost (Q,W) <= cost (R,W)
let v4 be Vertex of G; ::_thesis: ( not v4 in UsedVx (g,n) & R is_shortestpath_of v1,v4, UsedVx (g,n),W implies cost (Q,W) <= cost (R,W) )
assume that
A130: not v4 in UsedVx (g,n) and
A131: R is_shortestpath_of v1,v4, UsedVx (g,n),W ; ::_thesis: cost (Q,W) <= cost (R,W)
A132: v4 in VG ;
then reconsider j4 = v4 as Element of NAT ;
A133: 1 <= j4 by A16, A132, FINSEQ_1:1;
 j4 <= n by A16, A132, FINSEQ_1:1;
then  g . (n + j4) <> - 1 by A1, A2, A7, A9, A17, A130, A131, A133, Lm19;
then consider rn being FinSequence of NAT , RR being oriented Chain of G such that
 rn is_simple_vertex_seq_at g,j4,n and
 for i being Element of NAT st 1 <= i & i < len rn holds
rn . i in UsedVx (g,n) and
 RR is_oriented_edge_seq_of rn and
A134: RR is_shortestpath_of v1,v4, UsedVx (g,n),W and
 cost (RR,W) = g . ((2 * n) + j4) and
A135: ( not v4 in UsedVx (g,n) implies RR islongestInShortestpath UsedVx (g,n),v1,W ) by A2, A7, A8, A130;
consider QQ being oriented Chain of G such that
A136: QQ is_shortestpath_of v1,v2, UsedVx (g,n),W and
A137: cost (QQ,W) <= cost (RR,W) by A123, A126, A130, A135, GRAPH_5:def_19;
 cost (QQ,W) = cost (Q,W) by A127, A136, Th9;
hence  cost (Q,W) <= cost (R,W) by A131, A134, A137, Th9; ::_thesis: verum
end;
take Q = Q; ::_thesis: ( Q is_shortestpath_of v1,v2, UsedVx (h,n),W & cost (Q,W) <= cost (P,W) )
thus  Q is_shortestpath_of v1,v2, UsedVx (h,n),W by A17, A24, A123, A127, A129, GRAPH_5:64; ::_thesis: cost (Q,W) <= cost (P,W)
thus  cost (Q,W) <= cost (P,W) by A128; ::_thesis: verum
end;
end;
end;
hence  P islongestInShortestpath UsedVx (h,n),v1,W by GRAPH_5:def_19; ::_thesis: verum
end;
end;
supposeA138: (findmin n) . g hasBetterPathAt n,(n + j) -' n ; ::_thesis: ex q being FinSequence of NAT ex Q being oriented Chain of G st
( b3 is_simple_vertex_seq_at h,Q,n & ( for i being Element of NAT st 1 <= b5 & b5 < len i holds
i . b5 in UsedVx (h,n) ) & b4 is_oriented_edge_seq_of b3 & b4 is_shortestpath_of v1,q, UsedVx (h,n),W & cost (b4,W) = h . ((2 * n) + Q) & ( not q in UsedVx (h,n) implies b4 islongestInShortestpath UsedVx (h,n),v1,W ) )
then A139: (Relax (((findmin n) . g),n)) . (n + j) = Argmin ((OuterVx (g,n)),g,n) by A14, A42, A60, A62, A64, Def14;
A140: ( ((findmin n) . g) . (n + j) = - 1 or ((findmin n) . g) /. ((2 * n) + j) > newpathcost (((findmin n) . g),n,j) ) by A63, A138, Def13;
A141: ((findmin n) . g) /. (((2 * n) + (n * (((findmin n) . g) /. (((n * n) + (3 * n)) + 1)))) + j) >= 0 by A63, A138, Def13;
A142: ((findmin n) . g) . j <> - 1 by A63, A138, Def13;
A143: newpathcost (((findmin n) . g),n,j) = (((findmin n) . g) /. ((2 * n) + (Argmin ((OuterVx (g,n)),g,n)))) + (Weight (vk,v3,W)) by A42, A83;
A144: now__::_thesis:_not_Argmin_((OuterVx_(g,n)),g,n)_=_j
assume A145: Argmin ((OuterVx (g,n)),g,n) = j ; ::_thesis: contradiction
then A146: ((findmin n) . g) . (n + j) <> - 1 by A13, A21, A22, A61, A64, Th18;
 (((findmin n) . g) /. ((2 * n) + j)) + (Weight (vk,v3,W)) >= (((findmin n) . g) /. ((2 * n) + j)) + 0 by A83, A141, XREAL_1:7;
hence  contradiction by A63, A138, A143, A145, A146, Def13; ::_thesis: verum
end;
A147: now__::_thesis:_not_j_in_UsedVx_(g,n)
assume  j in UsedVx (g,n) ; ::_thesis: contradiction
then  ex i being Element of NAT st
( j = i & i in dom g & 1 <= i & i <= n & g . i = - 1 ) ;
hence  contradiction by A25, A142, Th32; ::_thesis: verum
end;
consider e being set  such that
A148: ( e in the carrier' of G & e orientedly_joins vk,v3 ) by A83, A141, Th23;
reconsider pe = <*e*> as oriented Chain of G by A148, Th5;
A149: len pe = 1 by FINSEQ_1:40;
A150: pe . 1 = e by FINSEQ_1:40;
then consider Q being oriented Chain of G such that
A151: Q = PK ^ pe and
 Q is_orientedpath_of v1,v3 by A39, A40, A148, A149, GRAPH_5:33;
take q = pk ^ <*j*>; ::_thesis: ex Q being oriented Chain of G st
( q is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len q holds
q . i in UsedVx (h,n) ) & Q is_oriented_edge_seq_of q & Q is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (Q,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies Q islongestInShortestpath UsedVx (h,n),v1,W ) )
take Q = Q; ::_thesis: ( q is_simple_vertex_seq_at h,j,n & ( for i being Element of NAT st 1 <= i & i < len q holds
q . i in UsedVx (h,n) ) & Q is_oriented_edge_seq_of q & Q is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (Q,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies Q islongestInShortestpath UsedVx (h,n),v1,W ) )
thus  q is_simple_vertex_seq_at h,j,n by A4, A5, A6, A12, A15, A27, A28, A52, A139, A144, A147, Lm18; ::_thesis: ( ( for i being Element of NAT st 1 <= i & i < len q holds
q . i in UsedVx (h,n) ) & Q is_oriented_edge_seq_of q & Q is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (Q,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies Q islongestInShortestpath UsedVx (h,n),v1,W ) )
A152: len pk > 1 by A27, Def18;
A153: pk is_vertex_seq_at g, Argmin ((OuterVx (g,n)),g,n),n by A27, Def18;
A154: q . (len pk) = pk . (len pk) by A152, Lm1
.= Argmin ((OuterVx (g,n)),g,n) by A153, Def17 ;
hereby  ::_thesis: ( Q is_oriented_edge_seq_of q & Q is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (Q,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies Q islongestInShortestpath UsedVx (h,n),v1,W ) )
let i be Element of NAT ; ::_thesis: ( 1 <= i & i < len q implies q . i in UsedVx (h,n) )
assume that
A155: 1 <= i and
A156: i < len q ; ::_thesis: q . i in UsedVx (h,n)
 i < (len pk) + 1 by A156, FINSEQ_2:16;
then A157: i <= len pk by NAT_1:13;
now__::_thesis:_(_q_._i_in_{(Argmin_((OuterVx_(g,n)),g,n))}_or_q_._i_in_UsedVx_(g,n)_)
percases ( i = len pk or i <> len pk ) ;
suppose i = len pk ; ::_thesis: ( q . i in {(Argmin ((OuterVx (g,n)),g,n))} or q . i in UsedVx (g,n) )
hence  ( q . i in {(Argmin ((OuterVx (g,n)),g,n))} or q . i in UsedVx (g,n) ) by A154, TARSKI:def_1; ::_thesis: verum
end;
suppose i <> len pk ; ::_thesis: ( q . i in {(Argmin ((OuterVx (g,n)),g,n))} or q . i in UsedVx (g,n) )
then A158: i < len pk by A157, XXREAL_0:1;
 q . i = pk . i by A155, A157, Lm1;
hence  ( q . i in {(Argmin ((OuterVx (g,n)),g,n))} or q . i in UsedVx (g,n) ) by A28, A155, A158; ::_thesis: verum
end;
end;
end;
hence  q . i in UsedVx (h,n) by A23, XBOOLE_0:def_3; ::_thesis: verum
end;
A159: len Q = (len PK) + 1 by A149, A151, FINSEQ_1:22;
A160: len pk = (len PK) + 1 by A29, Def19;
then A161: len q = (len Q) + 1 by A159, FINSEQ_2:16;
set FS = the Source of G;
set FT = the Target of G;
now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_Q_holds_
(_the_Source_of_G_._(Q_._i)_=_q_._i_&_the_Target_of_G_._(Q_._i)_=_q_._(i_+_1)_)
let i be Nat; ::_thesis: ( 1 <= i & i <= len Q implies ( the Source of G . (Q . b1) = q . b1 & the Target of G . (Q . b1) = q . (b1 + 1) ) )
assume that
A162: 1 <= i and
A163: i <= len Q ; ::_thesis: ( the Source of G . (Q . b1) = q . b1 & the Target of G . (Q . b1) = q . (b1 + 1) )
percases ( i = len Q or i <> len Q ) ;
supposeA164: i = len Q ; ::_thesis: ( the Source of G . (Q . b1) = q . b1 & the Target of G . (Q . b1) = q . (b1 + 1) )
then A165: Q . i = e by A149, A150, A151, A159, Lm2;
then A166: the Target of G . (Q . i) = v3 by A148, GRAPH_4:def_1;
thus  the Source of G . (Q . i) = q . i by A148, A154, A159, A160, A164, A165, GRAPH_4:def_1; ::_thesis: the Target of G . (Q . i) = q . (i + 1)
thus  the Target of G . (Q . i) = q . (i + 1) by A54, A159, A160, A164, A166, FINSEQ_1:42; ::_thesis: verum
end;
suppose i <> len Q ; ::_thesis: ( the Source of G . (Q . b1) = q . b1 & the Target of G . (Q . b1) = q . (b1 + 1) )
then A167: i < len Q by A163, XXREAL_0:1;
then A168: i <= len PK by A159, NAT_1:13;
then A169: the Source of G . (PK . i) = pk . i by A29, A162, Def19;
A170: the Target of G . (PK . i) = pk . (i + 1) by A29, A162, A168, Def19;
A171: Q . i = PK . i by A151, A162, A168, Lm1;
A172: i + 1 <= len pk by A159, A160, A167, NAT_1:13;
thus  the Source of G . (Q . i) = q . i by A159, A160, A162, A163, A169, A171, Lm1; ::_thesis: the Target of G . (Q . i) = q . (i + 1)
thus  the Target of G . (Q . i) = q . (i + 1) by A170, A171, A172, Lm1, NAT_1:12; ::_thesis: verum
end;
end;
end;
hence  Q is_oriented_edge_seq_of q by A161, Def19; ::_thesis: ( Q is_shortestpath_of v1,v3, UsedVx (h,n),W & cost (Q,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies Q islongestInShortestpath UsedVx (h,n),v1,W ) )
A173: (cost (PK,W)) + (cost (pe,W)) = cost (Q,W) by A17, A151, GRAPH_5:46, GRAPH_5:54;
A174: cost (pe,W) = W . (pe . 1) by A17, A149, Th4, GRAPH_5:46
.= Weight (vk,v3,W) by A148, A150, Th25 ;
then A175: newpathcost (((findmin n) . g),n,j) = cost (Q,W) by A17, A42, A43, A83, A151, GRAPH_5:46, GRAPH_5:54;
hereby  ::_thesis: ( cost (Q,W) = h . ((2 * n) + j) & ( not v3 in UsedVx (h,n) implies Q islongestInShortestpath UsedVx (h,n),v1,W ) )
percases ( g . (n + j) = - 1 or g . (n + j) <> - 1 ) ;
supposeA176: g . (n + j) = - 1 ; ::_thesis: Q is_shortestpath_of v1,v3, UsedVx (h,n),W
now__::_thesis:_for_v2_being_Vertex_of_G_st_v2_in_UsedVx_(g,n)_holds_
for_e_being_set_holds_
(_not_e_in_the_carrier'_of_G_or_not_e_orientedly_joins_v2,v3_)
let v2 be Vertex of G; ::_thesis: ( v2 in UsedVx (g,n) implies for e being set holds
( not e in the carrier' of G or not e orientedly_joins v2,v3 ) )
assume A177: v2 in UsedVx (g,n) ; ::_thesis: for e being set holds
( not e in the carrier' of G or not e orientedly_joins v2,v3 )
then reconsider m = v2 as Element of NAT ;
- 1 = f . (((2 * n) + (n * m)) + j) by A9, A57, A58, A176, A177
.= Weight (v2,v3,W) by A1, A54, Def20 ;
hence  for e being set holds
( not e in the carrier' of G or not e orientedly_joins v2,v3 ) by Th23; ::_thesis: verum
end;
hence  Q is_shortestpath_of v1,v3, UsedVx (h,n),W by A2, A7, A23, A30, A53, A148, A151, Th15; ::_thesis: verum
end;
supposeA178: g . (n + j) <> - 1 ; ::_thesis: Q is_shortestpath_of v1,v3, UsedVx (h,n),W
then  ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at g,j,n & ( for i being Element of NAT st 1 <= i & i < len p holds
p . i in UsedVx (g,n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx (g,n),W & cost (P,W) = g . ((2 * n) + j) & ( not v3 in UsedVx (g,n) implies P islongestInShortestpath UsedVx (g,n),v1,W ) ) by A8, A53, A54;
hence  Q is_shortestpath_of v1,v3, UsedVx (h,n),W by A2, A7, A13, A17, A21, A23, A30, A32, A40, A42, A43, A53, A61, A64, A83, A84, A140, A148, A151, A173, A174, A178, Th18, GRAPH_5:65; ::_thesis: verum
end;
end;
end;
thus  cost (Q,W) = h . ((2 * n) + j) by A15, A63, A65, A66, A67, A71, A138, A175, Def14; ::_thesis: ( not v3 in UsedVx (h,n) implies Q islongestInShortestpath UsedVx (h,n),v1,W )
 0 <= cost (pe,W) by A17, GRAPH_5:50;
then A179: (cost (PK,W)) + 0 <= cost (Q,W) by A173, XREAL_1:7;
hereby  ::_thesis: verum
assume  not v3 in UsedVx (h,n) ; ::_thesis: Q islongestInShortestpath UsedVx (h,n),v1,W
now__::_thesis:_for_v2_being_Vertex_of_G_st_v2_in_UsedVx_(h,n)_&_v2_<>_v1_holds_
ex_PK_being_oriented_Chain_of_G_st_
(_PK_is_shortestpath_of_v1,v2,_UsedVx_(h,n),W_&_cost_(PK,W)_<=_cost_(Q,W)_)
let v2 be Vertex of G; ::_thesis: ( v2 in UsedVx (h,n) & v2 <> v1 implies ex PK being oriented Chain of G st
( b2 is_shortestpath_of v1,PK, UsedVx (h,n),W & cost (b2,W) <= cost (Q,W) ) )
assume that
A180: v2 in UsedVx (h,n) and
A181: v2 <> v1 ; ::_thesis: ex PK being oriented Chain of G st
( b2 is_shortestpath_of v1,PK, UsedVx (h,n),W & cost (b2,W) <= cost (Q,W) )
percases ( v2 in {(Argmin ((OuterVx (g,n)),g,n))} or v2 in UsedVx (g,n) ) by A23, A180, XBOOLE_0:def_3;
suppose v2 in {(Argmin ((OuterVx (g,n)),g,n))} ; ::_thesis: ex PK being oriented Chain of G st
( b2 is_shortestpath_of v1,PK, UsedVx (h,n),W & cost (b2,W) <= cost (Q,W) )
then A182: v2 = Argmin ((OuterVx (g,n)),g,n) by TARSKI:def_1;
take PK = PK; ::_thesis: ( PK is_shortestpath_of v1,v2, UsedVx (h,n),W & cost (PK,W) <= cost (Q,W) )
thus  PK is_shortestpath_of v1,v2, UsedVx (h,n),W by A23, A30, A182, Th8; ::_thesis: cost (PK,W) <= cost (Q,W)
thus  cost (PK,W) <= cost (Q,W) by A179; ::_thesis: verum
end;
supposeA183: v2 in UsedVx (g,n) ; ::_thesis: ex P being oriented Chain of G st
( b2 is_shortestpath_of v1,P, UsedVx (h,n),W & cost (b2,W) <= cost (Q,W) )
then consider P being oriented Chain of G such that
A184: P is_shortestpath_of v1,v2, UsedVx (g,n),W and
A185: cost (P,W) <= cost (PK,W) by A4, A5, A6, A32, A181, Th39, GRAPH_5:def_19;
A186: now__::_thesis:_for_R_being_oriented_Chain_of_G
for_v4_being_Vertex_of_G_st_not_v4_in_UsedVx_(g,n)_&_R_is_shortestpath_of_v1,v4,_UsedVx_(g,n),W_holds_
cost_(P,W)_<=_cost_(R,W)
let R be oriented Chain of G; ::_thesis: for v4 being Vertex of G st not v4 in UsedVx (g,n) & R is_shortestpath_of v1,v4, UsedVx (g,n),W holds
cost (P,W) <= cost (R,W)
let v4 be Vertex of G; ::_thesis: ( not v4 in UsedVx (g,n) & R is_shortestpath_of v1,v4, UsedVx (g,n),W implies cost (P,W) <= cost (R,W) )
assume that
A187: not v4 in UsedVx (g,n) and
A188: R is_shortestpath_of v1,v4, UsedVx (g,n),W ; ::_thesis: cost (P,W) <= cost (R,W)
A189: v4 in VG ;
then reconsider j4 = v4 as Element of NAT ;
A190: 1 <= j4 by A16, A189, FINSEQ_1:1;
 j4 <= n by A16, A189, FINSEQ_1:1;
then  g . (n + j4) <> - 1 by A1, A2, A7, A9, A17, A187, A188, A190, Lm19;
then consider rn being FinSequence of NAT , RR being oriented Chain of G such that
 rn is_simple_vertex_seq_at g,j4,n and
 for i being Element of NAT st 1 <= i & i < len rn holds
rn . i in UsedVx (g,n) and
 RR is_oriented_edge_seq_of rn and
A191: RR is_shortestpath_of v1,v4, UsedVx (g,n),W and
 cost (RR,W) = g . ((2 * n) + j4) and
A192: ( not v4 in UsedVx (g,n) implies RR islongestInShortestpath UsedVx (g,n),v1,W ) by A2, A7, A8, A187;
consider PP being oriented Chain of G such that
A193: PP is_shortestpath_of v1,v2, UsedVx (g,n),W and
A194: cost (PP,W) <= cost (RR,W) by A181, A183, A187, A192, GRAPH_5:def_19;
 cost (PP,W) = cost (P,W) by A184, A193, Th9;
hence  cost (P,W) <= cost (R,W) by A188, A191, A194, Th9; ::_thesis: verum
end;
take P = P; ::_thesis: ( P is_shortestpath_of v1,v2, UsedVx (h,n),W & cost (P,W) <= cost (Q,W) )
thus  P is_shortestpath_of v1,v2, UsedVx (h,n),W by A17, A24, A181, A184, A186, GRAPH_5:64; ::_thesis: cost (P,W) <= cost (Q,W)
thus  cost (P,W) <= cost (Q,W) by A179, A185, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence  Q islongestInShortestpath UsedVx (h,n),v1,W by GRAPH_5:def_19; ::_thesis: verum
end;
end;
end;
end;
hereby  ::_thesis: for m being Element of NAT st m in UsedVx (h,n) holds
h . (n + m) <> - 1
let m, j be Element of NAT ; ::_thesis: ( h . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx (h,n) implies f . (((2 * n) + (n * b1)) + b2) = - 1 )
assume that
A195: h . (n + j) = - 1 and
A196: 1 <= j and
A197: j <= n and
A198: m in UsedVx (h,n) ; ::_thesis: f . (((2 * n) + (n * b1)) + b2) = - 1
set nj = n + j;
A199: 1 < n + j by A196, A197, Lm12;
A200: n + j <= 2 * n by A196, A197, Lm12;
A201: n + j < ((n * n) + (3 * n)) + 1 by A196, A197, Lm12;
then A202: n + j in dom g by A12, A19, A199, FINSEQ_3:25;
 n + 1 <= n + j by A196, XREAL_1:7;
then A203: n < n + j by NAT_1:13;
A204: now__::_thesis:_not_(findmin_n)_._g_hasBetterPathAt_n,(n_+_j)_-'_n
assume  (findmin n) . g hasBetterPathAt n,(n + j) -' n ; ::_thesis: contradiction
then  h . (n + j) = Argmin ((OuterVx (g,n)),g,n) by A14, A15, A42, A200, A202, A203, Def14;
hence  contradiction by A195, NAT_1:2; ::_thesis: verum
end;
A205: ((findmin n) . g) . (n + j) = g . (n + j) by A13, A21, A201, A203, Th18;
then A206: g . (n + j) = - 1 by A14, A15, A195, A200, A202, A203, A204, Def14;
then A207: not j in UsedVx (g,n) by A10;
 j < ((n * n) + (3 * n)) + 1 by A25, A197, XXREAL_0:2;
then  j in dom g by A12, A19, A196, FINSEQ_3:25;
then  g . j <> - 1 by A196, A197, A207;
then A208: ((findmin n) . g) . j <> - 1 by A13, A22, A25, A197, A206, Th18;
 not (findmin n) . g hasBetterPathAt n,j by A204, NAT_D:34;
then A209: ((findmin n) . g) /. (((2 * n) + (n * (((findmin n) . g) /. (((n * n) + (3 * n)) + 1)))) + j) < 0 by A205, A206, A208, Def13;
reconsider v3 = j as Vertex of G by A16, A196, A197, FINSEQ_1:1;
set Akj = ((2 * n) + (n * (Argmin ((OuterVx (g,n)),g,n)))) + j;
A210: 1 < ((2 * n) + (n * (Argmin ((OuterVx (g,n)),g,n)))) + j by A20, A21, A197, Lm13;
A211: Argmin ((OuterVx (g,n)),g,n) < ((2 * n) + (n * (Argmin ((OuterVx (g,n)),g,n)))) + j by A20, A21, A197, Lm13;
A212: ((2 * n) + (n * (Argmin ((OuterVx (g,n)),g,n)))) + j < ((n * n) + (3 * n)) + 1 by A20, A21, A197, Lm13;
then A213: ((2 * n) + (n * (Argmin ((OuterVx (g,n)),g,n)))) + j in dom g by A12, A19, A210, FINSEQ_3:25;
A214: (3 * n) + 1 <= ((2 * n) + (n * (Argmin ((OuterVx (g,n)),g,n)))) + j by A20, A21, A196, A197, Lm14;
A215: ((2 * n) + (n * (Argmin ((OuterVx (g,n)),g,n)))) + j <= (n * n) + (3 * n) by A20, A21, A196, A197, Lm14;
A216: f . (((2 * n) + (n * (Argmin ((OuterVx (g,n)),g,n)))) + j) = Weight (vk,v3,W) by A1, Def20;
A217: ((findmin n) . g) /. (((2 * n) + (n * (((findmin n) . g) /. (((n * n) + (3 * n)) + 1)))) + j) = ((findmin n) . g) . (((2 * n) + (n * (Argmin ((OuterVx (g,n)),g,n)))) + j) by A14, A42, A213, PARTFUN1:def_6
.= g . (((2 * n) + (n * (Argmin ((OuterVx (g,n)),g,n)))) + j) by A13, A211, A212, Th18
.= Weight (vk,v3,W) by A19, A38, A213, A214, A215, A216, Def16 ;
percases ( m in {(Argmin ((OuterVx (g,n)),g,n))} or m in UsedVx (g,n) ) by A23, A198, XBOOLE_0:def_3;
suppose m in {(Argmin ((OuterVx (g,n)),g,n))} ; ::_thesis: f . (((2 * n) + (n * b1)) + b2) = - 1
then A218: m = Argmin ((OuterVx (g,n)),g,n) by TARSKI:def_1;
 for e being set holds
( not e in the carrier' of G or not e orientedly_joins vk,v3 ) by A209, A217, Th23;
then  Weight (vk,v3,W) = - 1 by Def7;
hence  f . (((2 * n) + (n * m)) + j) = - 1 by A1, A218, Def20; ::_thesis: verum
end;
suppose m in UsedVx (g,n) ; ::_thesis: f . (((2 * n) + (n * b1)) + b2) = - 1
hence  f . (((2 * n) + (n * m)) + j) = - 1 by A9, A196, A197, A206; ::_thesis: verum
end;
end;
end;
let m be Element of NAT ; ::_thesis: ( m in UsedVx (h,n) implies h . (n + m) <> - 1 )
assume A219: m in UsedVx (h,n) ; ::_thesis: h . (n + m) <> - 1
percases ( m in UsedVx (g,n) or m in {(Argmin ((OuterVx (g,n)),g,n))} ) by A23, A219, XBOOLE_0:def_3;
supposeA220: m in UsedVx (g,n) ; ::_thesis: h . (n + m) <> - 1
then  h . (n + m) = g . (n + m) by A4, A5, A6, A12, Lm16;
hence  h . (n + m) <> - 1 by A10, A220; ::_thesis: verum
end;
suppose m in {(Argmin ((OuterVx (g,n)),g,n))} ; ::_thesis: h . (n + m) <> - 1
hence  h . (n + m) <> - 1 by A22, A52, TARSKI:def_1; ::_thesis: verum
end;
end;
end;

theorem :: GRAPHSP:52
for n, i being Element of NAT
for f, g being Element of REAL *
for G being oriented finite Graph
for W being Function of the carrier' of G,Real>=0
for v1, v2 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & v1 = 1 & 1 <> v2 & v2 = i & n >= 1 & g = (DijkstraAlgorithm n) . f holds
( the carrier of G = (UsedVx (g,n)) \/ (UnusedVx (g,n)) & ( v2 in UsedVx (g,n) implies ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at g,i,n & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v2,W & cost (P,W) = g . ((2 * n) + i) ) ) & ( v2 in UnusedVx (g,n) implies for Q being oriented Chain of G holds not Q is_orientedpath_of v1,v2 ) )
proof
let n, i be Element of NAT ; ::_thesis: for f, g being Element of REAL *
for G being oriented finite Graph
for W being Function of the carrier' of G,Real>=0
for v1, v2 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & v1 = 1 & 1 <> v2 & v2 = i & n >= 1 & g = (DijkstraAlgorithm n) . f holds
( the carrier of G = (UsedVx (g,n)) \/ (UnusedVx (g,n)) & ( v2 in UsedVx (g,n) implies ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at g,i,n & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v2,W & cost (P,W) = g . ((2 * n) + i) ) ) & ( v2 in UnusedVx (g,n) implies for Q being oriented Chain of G holds not Q is_orientedpath_of v1,v2 ) )
let f, g be Element of REAL * ; ::_thesis: for G being oriented finite Graph
for W being Function of the carrier' of G,Real>=0
for v1, v2 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & v1 = 1 & 1 <> v2 & v2 = i & n >= 1 & g = (DijkstraAlgorithm n) . f holds
( the carrier of G = (UsedVx (g,n)) \/ (UnusedVx (g,n)) & ( v2 in UsedVx (g,n) implies ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at g,i,n & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v2,W & cost (P,W) = g . ((2 * n) + i) ) ) & ( v2 in UnusedVx (g,n) implies for Q being oriented Chain of G holds not Q is_orientedpath_of v1,v2 ) )
let G be oriented finite Graph; ::_thesis: for W being Function of the carrier' of G,Real>=0
for v1, v2 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & v1 = 1 & 1 <> v2 & v2 = i & n >= 1 & g = (DijkstraAlgorithm n) . f holds
( the carrier of G = (UsedVx (g,n)) \/ (UnusedVx (g,n)) & ( v2 in UsedVx (g,n) implies ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at g,i,n & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v2,W & cost (P,W) = g . ((2 * n) + i) ) ) & ( v2 in UnusedVx (g,n) implies for Q being oriented Chain of G holds not Q is_orientedpath_of v1,v2 ) )
let W be Function of the carrier' of G,Real>=0; ::_thesis: for v1, v2 being Vertex of G st f is_Input_of_Dijkstra_Alg G,n,W & v1 = 1 & 1 <> v2 & v2 = i & n >= 1 & g = (DijkstraAlgorithm n) . f holds
( the carrier of G = (UsedVx (g,n)) \/ (UnusedVx (g,n)) & ( v2 in UsedVx (g,n) implies ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at g,i,n & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v2,W & cost (P,W) = g . ((2 * n) + i) ) ) & ( v2 in UnusedVx (g,n) implies for Q being oriented Chain of G holds not Q is_orientedpath_of v1,v2 ) )
let v1, v2 be Vertex of G; ::_thesis: ( f is_Input_of_Dijkstra_Alg G,n,W & v1 = 1 & 1 <> v2 & v2 = i & n >= 1 & g = (DijkstraAlgorithm n) . f implies ( the carrier of G = (UsedVx (g,n)) \/ (UnusedVx (g,n)) & ( v2 in UsedVx (g,n) implies ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at g,i,n & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v2,W & cost (P,W) = g . ((2 * n) + i) ) ) & ( v2 in UnusedVx (g,n) implies for Q being oriented Chain of G holds not Q is_orientedpath_of v1,v2 ) ) )
assume that
A1: f is_Input_of_Dijkstra_Alg G,n,W and
A2: v1 = 1 and
A3: 1 <> v2 and
A4: v2 = i and
A5: n >= 1 and
A6: g = (DijkstraAlgorithm n) . f ; ::_thesis: ( the carrier of G = (UsedVx (g,n)) \/ (UnusedVx (g,n)) & ( v2 in UsedVx (g,n) implies ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at g,i,n & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v2,W & cost (P,W) = g . ((2 * n) + i) ) ) & ( v2 in UnusedVx (g,n) implies for Q being oriented Chain of G holds not Q is_orientedpath_of v1,v2 ) )
A7: Seg n = the carrier of G by A1, Def20;
then reconsider VG = the carrier of G as non empty Subset of NAT by A5;
A8: f . (n + 1) = 0 by A1, Def20;
set Ug = UsedVx (g,n);
set Vg = UnusedVx (g,n);
set R = Relax n;
set M = findmin n;
set RM = repeat ((Relax n) * (findmin n));
set cn = LifeSpan (((Relax n) * (findmin n)),f,n);
set mi = ((n * n) + (3 * n)) + 1;
A9: g = ((repeat ((Relax n) * (findmin n))) . (LifeSpan (((Relax n) * (findmin n)),f,n))) . f by A6, Def5;
A10: (UsedVx (g,n)) \/ (UnusedVx (g,n)) c= VG
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (UsedVx (g,n)) \/ (UnusedVx (g,n)) or x in VG )
assume A11: x in (UsedVx (g,n)) \/ (UnusedVx (g,n)) ; ::_thesis: x in VG
percases ( x in UsedVx (g,n) or x in UnusedVx (g,n) ) by A11, XBOOLE_0:def_3;
suppose x in UsedVx (g,n) ; ::_thesis: x in VG
then  ex k being Element of NAT st
( x = k & k in dom g & 1 <= k & k <= n & g . k = - 1 ) ;
hence  x in VG by A7, FINSEQ_1:1; ::_thesis: verum
end;
suppose x in UnusedVx (g,n) ; ::_thesis: x in VG
then  ex k being Element of NAT st
( x = k & k in dom g & 1 <= k & k <= n & g . k <> - 1 ) ;
hence  x in VG by A7, FINSEQ_1:1; ::_thesis: verum
end;
end;
end;
A12: len f = ((n * n) + (3 * n)) + 1 by A1, Def20;
 VG c= (UsedVx (g,n)) \/ (UnusedVx (g,n))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in VG or x in (UsedVx (g,n)) \/ (UnusedVx (g,n)) )
assume A13: x in VG ; ::_thesis: x in (UsedVx (g,n)) \/ (UnusedVx (g,n))
then reconsider j = x as Element of NAT ;
A14: 1 <= j by A7, A13, FINSEQ_1:1;
A15: j <= n by A7, A13, FINSEQ_1:1;
 n < ((n * n) + (3 * n)) + 1 by Lm7;
then  j < ((n * n) + (3 * n)) + 1 by A15, XXREAL_0:2;
then  j in dom f by A12, A14, FINSEQ_3:25;
then A16: j in dom g by A9, Th41;
percases ( g . j = - 1 or g . j <> - 1 ) ;
suppose g . j = - 1 ; ::_thesis: x in (UsedVx (g,n)) \/ (UnusedVx (g,n))
then  j in  {  k where k is Element of NAT : ( k in dom g & 1 <= k & k <= n & g . k = - 1 )  }  by A14, A15, A16;
hence  x in (UsedVx (g,n)) \/ (UnusedVx (g,n)) by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose g . j <> - 1 ; ::_thesis: x in (UsedVx (g,n)) \/ (UnusedVx (g,n))
then  j in  {  k where k is Element of NAT : ( k in dom g & 1 <= k & k <= n & g . k <> - 1 )  }  by A14, A15, A16;
hence  x in (UsedVx (g,n)) \/ (UnusedVx (g,n)) by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
hence A17: the carrier of G = (UsedVx (g,n)) \/ (UnusedVx (g,n)) by A10, XBOOLE_0:def_10; ::_thesis: ( ( v2 in UsedVx (g,n) implies ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at g,i,n & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v2,W & cost (P,W) = g . ((2 * n) + i) ) ) & ( v2 in UnusedVx (g,n) implies for Q being oriented Chain of G holds not Q is_orientedpath_of v1,v2 ) )
defpred S1[ Element of NAT ] means ( $1 <= LifeSpan (((Relax n) * (findmin n)),f,n) implies ( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & (((repeat ((Relax n) * (findmin n))) . $1) . f) . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at ((repeat ((Relax n) * (findmin n))) . $1) . f,j,n & ( for m being Element of NAT st 1 <= m & m < len p holds
p . m in UsedVx ((((repeat ((Relax n) * (findmin n))) . $1) . f),n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx ((((repeat ((Relax n) * (findmin n))) . $1) . f),n),W & cost (P,W) = (((repeat ((Relax n) * (findmin n))) . $1) . f) . ((2 * n) + j) & ( not v3 in UsedVx ((((repeat ((Relax n) * (findmin n))) . $1) . f),n) implies P islongestInShortestpath UsedVx ((((repeat ((Relax n) * (findmin n))) . $1) . f),n),v1,W ) ) ) & ( for m, j being Element of NAT st (((repeat ((Relax n) * (findmin n))) . $1) . f) . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx ((((repeat ((Relax n) * (findmin n))) . $1) . f),n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx ((((repeat ((Relax n) * (findmin n))) . $1) . f),n) holds
(((repeat ((Relax n) * (findmin n))) . $1) . f) . (n + m) <> - 1 ) ) );
 1 <= ((n * n) + (3 * n)) + 1 by NAT_1:12;
then A18: 1 in dom f by A12, FINSEQ_3:25;
A19: ( ( for m being Element of NAT st 1 <= m & m <= n holds
f . m = 1 ) & ( for m being Element of NAT st 2 <= m & m <= n holds
f . (n + m) = - 1 ) ) by A1, Def20;
then  {1} = UsedVx ((((repeat ((Relax n) * (findmin n))) . 1) . f),n) by A5, A8, A18, Th47;
then A20: 1 in UsedVx ((((repeat ((Relax n) * (findmin n))) . 1) . f),n) by TARSKI:def_1;
A21: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A22: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_(_k_+_1_<=_LifeSpan_(((Relax_n)_*_(findmin_n)),f,n)_implies_(_(_for_v3_being_Vertex_of_G
for_j_being_Element_of_NAT_st_v3_<>_v1_&_v3_=_j_&_(((repeat_((Relax_n)_*_(findmin_n)))_._(k_+_1))_._f)_._(n_+_j)_<>_-_1_holds_
ex_p_being_FinSequence_of_NAT_ex_P_being_oriented_Chain_of_G_st_
(_p_is_simple_vertex_seq_at_((repeat_((Relax_n)_*_(findmin_n)))_._(k_+_1))_._f,j,n_&_(_for_m_being_Element_of_NAT_st_1_<=_m_&_m_<_len_p_holds_
p_._m_in_UsedVx_((((repeat_((Relax_n)_*_(findmin_n)))_._(k_+_1))_._f),n)_)_&_P_is_oriented_edge_seq_of_p_&_P_is_shortestpath_of_v1,v3,_UsedVx_((((repeat_((Relax_n)_*_(findmin_n)))_._(k_+_1))_._f),n),W_&_cost_(P,W)_=_(((repeat_((Relax_n)_*_(findmin_n)))_._(k_+_1))_._f)_._((2_*_n)_+_j)_&_(_not_v3_in_UsedVx_((((repeat_((Relax_n)_*_(findmin_n)))_._(k_+_1))_._f),n)_implies_P_islongestInShortestpath_UsedVx_((((repeat_((Relax_n)_*_(findmin_n)))_._(k_+_1))_._f),n),v1,W_)_)_)_&_(_for_m,_j_being_Element_of_NAT_st_(((repeat_((Relax_n)_*_(findmin_n)))_._(k_+_1))_._f)_._(n_+_j)_=_-_1_&_1_<=_j_&_j_<=_n_&_m_in_UsedVx_((((repeat_((Relax_n)_*_(findmin_n)))_._(k_+_1))_._f),n)_holds_
f_._(((2_*_n)_+_(n_*_m))_+_j)_=_-_1_)_&_(_for_m_being_Element_of_NAT_st_m_in_UsedVx_((((repeat_((Relax_n)_*_(findmin_n)))_._(k_+_1))_._f),n)_holds_
(((repeat_((Relax_n)_*_(findmin_n)))_._(k_+_1))_._f)_._(n_+_m)_<>_-_1_)_)_)
set FK1 = ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f;
set UV1 = UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n);
set FK = ((repeat ((Relax n) * (findmin n))) . k) . f;
assume A23: k + 1 <= LifeSpan (((Relax n) * (findmin n)),f,n) ; ::_thesis: ( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & (((repeat ((Relax n) * (findmin n))) . (k + 1)) . f) . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f,j,n & ( for m being Element of NAT st 1 <= m & m < len p holds
p . m in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n),W & cost (P,W) = (((repeat ((Relax n) * (findmin n))) . (k + 1)) . f) . ((2 * n) + j) & ( not v3 in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) implies P islongestInShortestpath UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n),v1,W ) ) ) & ( for m, j being Element of NAT st (((repeat ((Relax n) * (findmin n))) . (k + 1)) . f) . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) holds
(((repeat ((Relax n) * (findmin n))) . (k + 1)) . f) . (n + m) <> - 1 ) )
then A24: k < LifeSpan (((Relax n) * (findmin n)),f,n) by NAT_1:13;
then A25: OuterVx ((((repeat ((Relax n) * (findmin n))) . k) . f),n) <> {} by Def4;
percases ( k = 0 or k <> 0 ) ;
suppose k = 0 ; ::_thesis: ( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & (((repeat ((Relax n) * (findmin n))) . (k + 1)) . f) . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f,j,n & ( for m being Element of NAT st 1 <= m & m < len p holds
p . m in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n),W & cost (P,W) = (((repeat ((Relax n) * (findmin n))) . (k + 1)) . f) . ((2 * n) + j) & ( not v3 in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) implies P islongestInShortestpath UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n),v1,W ) ) ) & ( for m, j being Element of NAT st (((repeat ((Relax n) * (findmin n))) . (k + 1)) . f) . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) holds
(((repeat ((Relax n) * (findmin n))) . (k + 1)) . f) . (n + m) <> - 1 ) )
hence  ( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & (((repeat ((Relax n) * (findmin n))) . (k + 1)) . f) . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f,j,n & ( for m being Element of NAT st 1 <= m & m < len p holds
p . m in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n),W & cost (P,W) = (((repeat ((Relax n) * (findmin n))) . (k + 1)) . f) . ((2 * n) + j) & ( not v3 in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) implies P islongestInShortestpath UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n),v1,W ) ) ) & ( for m, j being Element of NAT st (((repeat ((Relax n) * (findmin n))) . (k + 1)) . f) . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) holds
(((repeat ((Relax n) * (findmin n))) . (k + 1)) . f) . (n + m) <> - 1 ) ) by A1, A2, A5, Lm15; ::_thesis: verum
end;
suppose k <> 0 ; ::_thesis: ( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & (((repeat ((Relax n) * (findmin n))) . (k + 1)) . f) . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f,j,n & ( for m being Element of NAT st 1 <= m & m < len p holds
p . m in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n),W & cost (P,W) = (((repeat ((Relax n) * (findmin n))) . (k + 1)) . f) . ((2 * n) + j) & ( not v3 in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) implies P islongestInShortestpath UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n),v1,W ) ) ) & ( for m, j being Element of NAT st (((repeat ((Relax n) * (findmin n))) . (k + 1)) . f) . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) holds
(((repeat ((Relax n) * (findmin n))) . (k + 1)) . f) . (n + m) <> - 1 ) )
then  k >= 1 + 0 by INT_1:7;
then  1 in UsedVx ((((repeat ((Relax n) * (findmin n))) . k) . f),n) by A20, A24, Th48;
hence  ( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & (((repeat ((Relax n) * (findmin n))) . (k + 1)) . f) . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at ((repeat ((Relax n) * (findmin n))) . (k + 1)) . f,j,n & ( for m being Element of NAT st 1 <= m & m < len p holds
p . m in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n),W & cost (P,W) = (((repeat ((Relax n) * (findmin n))) . (k + 1)) . f) . ((2 * n) + j) & ( not v3 in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) implies P islongestInShortestpath UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n),v1,W ) ) ) & ( for m, j being Element of NAT st (((repeat ((Relax n) * (findmin n))) . (k + 1)) . f) . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx ((((repeat ((Relax n) * (findmin n))) . (k + 1)) . f),n) holds
(((repeat ((Relax n) * (findmin n))) . (k + 1)) . f) . (n + m) <> - 1 ) ) by A1, A2, A5, A22, A23, A25, Lm20, NAT_1:13; ::_thesis: verum
end;
end;
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
A26: ((repeat ((Relax n) * (findmin n))) . 0) . f = f by Th21;
A27: S1[ 0 ]
proof
set UV = UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n);
set h = ((repeat ((Relax n) * (findmin n))) . 0) . f;
assume  0 <= LifeSpan (((Relax n) * (findmin n)),f,n) ; ::_thesis: ( ( for v3 being Vertex of G
for j being Element of NAT st v3 <> v1 & v3 = j & (((repeat ((Relax n) * (findmin n))) . 0) . f) . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at ((repeat ((Relax n) * (findmin n))) . 0) . f,j,n & ( for m being Element of NAT st 1 <= m & m < len p holds
p . m in UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n),W & cost (P,W) = (((repeat ((Relax n) * (findmin n))) . 0) . f) . ((2 * n) + j) & ( not v3 in UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n) implies P islongestInShortestpath UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n),v1,W ) ) ) & ( for m, j being Element of NAT st (((repeat ((Relax n) * (findmin n))) . 0) . f) . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n) holds
(((repeat ((Relax n) * (findmin n))) . 0) . f) . (n + m) <> - 1 ) )
hereby  ::_thesis: ( ( for m, j being Element of NAT st (((repeat ((Relax n) * (findmin n))) . 0) . f) . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n) holds
f . (((2 * n) + (n * m)) + j) = - 1 ) & ( for m being Element of NAT st m in UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n) holds
(((repeat ((Relax n) * (findmin n))) . 0) . f) . (n + m) <> - 1 ) )
let v3 be Vertex of G; ::_thesis: for j being Element of NAT st v3 <> v1 & v3 = j & (((repeat ((Relax n) * (findmin n))) . 0) . f) . (n + j) <> - 1 holds
ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at ((repeat ((Relax n) * (findmin n))) . 0) . f,j,n & ( for m being Element of NAT st 1 <= m & m < len p holds
p . m in UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n),W & cost (P,W) = (((repeat ((Relax n) * (findmin n))) . 0) . f) . ((2 * n) + j) & ( not v3 in UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n) implies P islongestInShortestpath UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n),v1,W ) )
let j be Element of NAT ; ::_thesis: ( v3 <> v1 & v3 = j & (((repeat ((Relax n) * (findmin n))) . 0) . f) . (n + j) <> - 1 implies ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at ((repeat ((Relax n) * (findmin n))) . 0) . f,j,n & ( for m being Element of NAT st 1 <= m & m < len p holds
p . m in UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n),W & cost (P,W) = (((repeat ((Relax n) * (findmin n))) . 0) . f) . ((2 * n) + j) & ( not v3 in UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n) implies P islongestInShortestpath UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n),v1,W ) ) )
assume that
A28: v3 <> v1 and
A29: v3 = j and
A30: (((repeat ((Relax n) * (findmin n))) . 0) . f) . (n + j) <> - 1 ; ::_thesis: ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at ((repeat ((Relax n) * (findmin n))) . 0) . f,j,n & ( for m being Element of NAT st 1 <= m & m < len p holds
p . m in UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n) ) & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v3, UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n),W & cost (P,W) = (((repeat ((Relax n) * (findmin n))) . 0) . f) . ((2 * n) + j) & ( not v3 in UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n) implies P islongestInShortestpath UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n),v1,W ) )
A31: v3 in VG ;
then  1 <= j by A7, A29, FINSEQ_1:1;
then  1 < j by A2, A28, A29, XXREAL_0:1;
then A32: 1 + 1 <= j by INT_1:7;
assume  for p being FinSequence of NAT
for P being oriented Chain of G holds
( not p is_simple_vertex_seq_at ((repeat ((Relax n) * (findmin n))) . 0) . f,j,n or ex m being Element of NAT st
( 1 <= m & m < len p & not p . m in UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n) ) or not P is_oriented_edge_seq_of p or not P is_shortestpath_of v1,v3, UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n),W or not cost (P,W) = (((repeat ((Relax n) * (findmin n))) . 0) . f) . ((2 * n) + j) or ( not v3 in UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n) & not P islongestInShortestpath UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n),v1,W ) ) ; ::_thesis: contradiction
 j <= n by A7, A29, A31, FINSEQ_1:1;
hence  contradiction by A1, A26, A30, A32, Def20; ::_thesis: verum
end;
thus  for m, j being Element of NAT st (((repeat ((Relax n) * (findmin n))) . 0) . f) . (n + j) = - 1 & 1 <= j & j <= n & m in UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n) holds
f . (((2 * n) + (n * m)) + j) = - 1 by A5, A26, A8, A19, A18, Th47; ::_thesis: for m being Element of NAT st m in UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n) holds
(((repeat ((Relax n) * (findmin n))) . 0) . f) . (n + m) <> - 1
let m be Element of NAT ; ::_thesis: ( m in UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n) implies (((repeat ((Relax n) * (findmin n))) . 0) . f) . (n + m) <> - 1 )
assume A33: m in UsedVx ((((repeat ((Relax n) * (findmin n))) . 0) . f),n) ; ::_thesis: (((repeat ((Relax n) * (findmin n))) . 0) . f) . (n + m) <> - 1
assume  (((repeat ((Relax n) * (findmin n))) . 0) . f) . (n + m) = - 1 ; ::_thesis: contradiction
thus  contradiction by A5, A26, A8, A19, A18, A33, Th47; ::_thesis: verum
end;
A34: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A27, A21);
 ex ii being Element of NAT st
( ii <= n & OuterVx ((((repeat ((Relax n) * (findmin n))) . ii) . f),n) = {} ) by Th40;
then A35: OuterVx (g,n) = {} by A9, Def4;
A36: now__::_thesis:_for_v3,_v4_being_Vertex_of_G_st_v3_in_UsedVx_(g,n)_&_v4_in_UnusedVx_(g,n)_holds_
for_e_being_set_holds_
(_not_e_in_the_carrier'_of_G_or_not_e_orientedly_joins_v3,v4_)
let v3, v4 be Vertex of G; ::_thesis: ( v3 in UsedVx (g,n) & v4 in UnusedVx (g,n) implies for e being set holds
( not e in the carrier' of G or not e orientedly_joins v3,v4 ) )
assume that
A37: v3 in UsedVx (g,n) and
A38: v4 in UnusedVx (g,n) ; ::_thesis: for e being set holds
( not e in the carrier' of G or not e orientedly_joins v3,v4 )
 v3 in VG ;
then reconsider m = v3 as Element of NAT ;
consider j being Element of NAT  such that
A39: v4 = j and
A40: j in dom g and
A41: ( 1 <= j & j <= n ) and
A42: g . j <> - 1 by A38;
now__::_thesis:_not_g_._(n_+_j)_<>_-_1
assume  g . (n + j) <> - 1 ; ::_thesis: contradiction
then  j in  {  k where k is Element of NAT : ( k in dom g & 1 <= k & k <= n & g . k <> - 1 & g . (n + k) <> - 1 )  }  by A40, A41, A42;
hence  contradiction by A35; ::_thesis: verum
end;
then  - 1 = f . (((2 * n) + (n * m)) + j) by A9, A34, A37, A41
.= Weight (v3,v4,W) by A1, A39, Def20 ;
hence  for e being set holds
( not e in the carrier' of G or not e orientedly_joins v3,v4 ) by Th23; ::_thesis: verum
end;
A43: f . 1 = 1 by A1, A5, Def20;
now__::_thesis:_not_LifeSpan_(((Relax_n)_*_(findmin_n)),f,n)_=_0
assume A44: LifeSpan (((Relax n) * (findmin n)),f,n) = 0 ; ::_thesis: contradiction
 1 in  {  k where k is Element of NAT : ( k in dom f & 1 <= k & k <= n & f . k <> - 1 & f . (n + k) <> - 1 )  }  by A5, A43, A8, A18;
hence  contradiction by A9, A26, A35, A44; ::_thesis: verum
end;
then  LifeSpan (((Relax n) * (findmin n)),f,n) >= 1 + 0 by INT_1:7;
then A45: v1 in UsedVx (g,n) by A2, A9, A20, Th48;
hereby  ::_thesis: ( v2 in UnusedVx (g,n) implies for Q being oriented Chain of G holds not Q is_orientedpath_of v1,v2 )
assume  v2 in UsedVx (g,n) ; ::_thesis: ex p being FinSequence of NAT ex P being oriented Chain of G st
( p is_simple_vertex_seq_at g,i,n & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v2,W & cost (P,W) = g . ((2 * n) + i) )
then  g . (n + i) <> - 1 by A4, A9, A34;
then consider p being FinSequence of NAT , P being oriented Chain of G such that
A46: p is_simple_vertex_seq_at g,i,n and
 for m being Element of NAT st 1 <= m & m < len p holds
p . m in UsedVx (g,n) and
A47: P is_oriented_edge_seq_of p and
A48: P is_shortestpath_of v1,v2, UsedVx (g,n),W and
A49: cost (P,W) = g . ((2 * n) + i) and
 ( not v2 in UsedVx (g,n) implies P islongestInShortestpath UsedVx (g,n),v1,W ) by A2, A3, A4, A9, A34;
take p = p; ::_thesis: ex P being oriented Chain of G st
( p is_simple_vertex_seq_at g,i,n & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v2,W & cost (P,W) = g . ((2 * n) + i) )
take P = P; ::_thesis: ( p is_simple_vertex_seq_at g,i,n & P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v2,W & cost (P,W) = g . ((2 * n) + i) )
thus  p is_simple_vertex_seq_at g,i,n by A46; ::_thesis: ( P is_oriented_edge_seq_of p & P is_shortestpath_of v1,v2,W & cost (P,W) = g . ((2 * n) + i) )
thus  P is_oriented_edge_seq_of p by A47; ::_thesis: ( P is_shortestpath_of v1,v2,W & cost (P,W) = g . ((2 * n) + i) )
thus  P is_shortestpath_of v1,v2,W by A17, A36, A45, A48, Th16; ::_thesis: cost (P,W) = g . ((2 * n) + i)
thus  cost (P,W) = g . ((2 * n) + i) by A49; ::_thesis: verum
end;
thus  ( v2 in UnusedVx (g,n) implies for Q being oriented Chain of G holds not Q is_orientedpath_of v1,v2 ) by A17, A36, A45, Th11; ::_thesis: verum
end;