:: Dijkstra's Shortest Path Algorithm
:: by Jing-Chao Chen
::
:: Received March 17, 2003
:: Copyright (c) 2003-2016 Association of Mizar Users

environ

vocabularies HIDDEN, NUMBERS, SUBSET_1, FINSEQ_1, INT_1, RELAT_1, FUNCT_1, ORDINAL4, XBOOLE_0, XXREAL_0, PARTFUN1, GRAPH_1, STRUCT_0, TREES_2, GLIB_000, GRAPH_5, CARD_3, GRAPH_4, NAT_1, ARYTM_3, TARSKI, CARD_1, FINSET_1, FUNCT_4, REAL_1, FUNCT_2, ARYTM_1, GRAPHSP, XREAL_0;
notations HIDDEN, XCMPLX_0, XXREAL_0, REAL_1, INT_1, TARSKI, XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, FINSEQ_1, FINSEQ_2, FINSEQ_4, CARD_1, FINSET_1, ORDINAL1, XREAL_0, STRUCT_0, GRAPH_1, PARTFUN1, FUNCT_2, CQC_SIM1, GRAPH_4, GRAPH_5, NAT_D, DOMAIN_1, RVSUM_1, NUMBERS, FUNCT_7, NAT_1;
definitions TARSKI;
theorems FUNCT_2, FUNCT_1, PARTFUN1, FINSEQ_1, NAT_1, ZFMISC_1, TARSKI, FINSEQ_3, XBOOLE_0, XBOOLE_1, SUBSET_1, GRAPH_1, FINSEQ_4, CARD_2, INT_1, GRAPH_5, GRAPH_4, FINSEQ_2, ENUMSET1, FUNCT_7, XREAL_1, XXREAL_0, FINSOP_1, ORDINAL1, NAT_D, XREAL_0;
schemes NAT_1, CLASSES1, PRE_CIRC;
registrations XBOOLE_0, ORDINAL1, RELSET_1, FUNCT_2, FINSET_1, NUMBERS, XXREAL_0, XREAL_0, NAT_1, INT_1, FINSEQ_1, GRAPH_1, GRAPH_4, GRAPH_5, VALUED_0, CARD_1, FUNCT_1, XCMPLX_0, MEMBERED;
constructors HIDDEN, DOMAIN_1, REAL_1, FINSEQ_4, FINSOP_1, NAT_D, FUNCT_7, CQC_SIM1, GRAPH_4, GRAPH_5, BINOP_2, RVSUM_1, RELSET_1;
requirements HIDDEN, NUMERALS, SUBSET, BOOLE, REAL, ARITHM;
equalities RVSUM_1;
expansions TARSKI;


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 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 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 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 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 end;

Lm1: for n being Nat
for p, q being FinSequence st 1 <= n & n <= len p holds
p . n = (p ^ q) . n
proof end;

Lm2: for n being Nat
for p, q being FinSequence st 1 <= n & n <= len q holds
q . n = (p ^ q) . ((len p) + n)
proof 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 end;

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 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 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 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 end;

Lm3: for i being 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 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 end;

Lm4: for i being 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 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 end;

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 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 end;

theorem Th15: :: GRAPHSP:15
for V, e 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 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 end;

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

definition
let f be FinSequence of REAL ;
let x be object ;
let r be Real;
:: original: :=
redefine func (f,x) := r -> FinSequence of REAL ;
coherence
(f,x) := r is FinSequence of REAL
proof end;
end;

definition
let i, k be 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 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 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 end;

theorem Th18: :: GRAPHSP:18
for i, k, m being 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 end;

theorem Th19: :: GRAPHSP:19
for i, k being Nat
for f being Element of REAL *
for r being Real st k in dom f holds
((f,i) := (k,r)) . k = r
proof end;

theorem Th20: :: GRAPHSP:20
for i, k being Nat
for f being Element of REAL *
for r being Real holds dom ((f,i) := (k,r)) = dom f
proof end;

definition
let X be set ;
let f, g be Element of Funcs (X,X);
:: original: *
redefine func g * f -> Element of Funcs (X,X);
coherence
f * g is Element of Funcs (X,X)
proof end;
end;

definition
let X be set ;
let f be Element of Funcs (X,X);
let g be Element of X;
:: original: .
redefine func f . g -> Element of X;
coherence
f . g is Element of X
proof end;
end;

definition
let X be set ;
let f be Element of Funcs (X,X);
func repeat f -> sequence of (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 sequence of (Funcs (X,X)) st
( b1 . 0 = id X & ( for i being Nat holds b1 . (i + 1) = f * (b1 . i) ) )
proof end;
uniqueness
for b1, b2 being sequence of (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 end;
end;

:: deftheorem Def2 defines repeat GRAPHSP:def 2 :
for X being set
for f being Element of Funcs (X,X)
for b3 being sequence of (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 end;

Lm5: for X being set
for f being Element of Funcs (X,X) holds dom f = X
proof end;

theorem Th22: :: GRAPHSP:22
for F, G being Element of Funcs ((REAL *),(REAL *))
for f being Element of REAL *
for i being Nat holds ((repeat (F * G)) . (i + 1)) . f = F . (G . (((repeat (F * G)) . i) . f))
proof end;

definition
let g be Element of Funcs ((REAL *),(REAL *));
let f be Element of REAL * ;
:: original: .
redefine func g . f -> Element of REAL * ;
coherence
g . f is Element of REAL *
proof 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 Nat : ( i in dom f & 1 <= i & i <= n & f . i <> - 1 & f . (n + i) <> - 1 ) } ;
coherence
{ i where i is Nat : ( i in dom f & 1 <= i & i <= n & f . i <> - 1 & f . (n + i) <> - 1 ) } is Subset of NAT
proof 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 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 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 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 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 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 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 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 end;
end;

:: deftheorem Def5 defines while_do GRAPHSP:def 5 :
for f being Element of Funcs ((REAL *),(REAL *))
for n being 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 ) ) );

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, 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;
by TARSKI:1;
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 ) );

registration
let G be oriented Graph;
let v1, v2 be Vertex of G;
let W be Function of the carrier' of G,Real>=0;
cluster Weight (v1,v2,W) -> real ;
coherence
Weight (v1,v2,W) is real
proof 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 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 end;

definition
let f be Element of REAL * ;
let n be Nat;
func UnusedVx (f,n) -> Subset of NAT equals :: GRAPHSP:def 8
 { i where i is Nat : ( i in dom f & 1 <= i & i <= n & f . i <> - 1 ) } ;
coherence
{ i where i is Nat : ( i in dom f & 1 <= i & i <= n & f . i <> - 1 ) } is Subset of NAT
proof end;
end;

:: deftheorem defines UnusedVx GRAPHSP:def 8 :
for f being Element of REAL *
for n being Nat holds UnusedVx (f,n) = { i where i is Nat : ( i in dom f & 1 <= i & i <= n & f . i <> - 1 ) } ;

definition
let f be Element of REAL * ;
let n be Nat;
func UsedVx (f,n) -> Subset of NAT equals :: GRAPHSP:def 9
 { i where i is Nat : ( i in dom f & 1 <= i & i <= n & f . i = - 1 ) } ;
coherence
{ i where i is Nat : ( i in dom f & 1 <= i & i <= n & f . i = - 1 ) } is Subset of NAT
proof end;
end;

:: deftheorem defines UsedVx GRAPHSP:def 9 :
for f being Element of REAL *
for n being Nat holds UsedVx (f,n) = { i where i is Nat : ( i in dom f & 1 <= i & i <= n & f . i = - 1 ) } ;

theorem Th26: :: GRAPHSP:26
for n being Nat
for f being Element of REAL * holds UnusedVx (f,n) c= Seg n
proof end;

registration
let f be Element of REAL * ;
let n be Nat;
cluster UnusedVx (f,n) -> finite ;
coherence
UnusedVx (f,n) is finite
proof end;
end;

theorem Th27: :: GRAPHSP:27
for n being Nat
for f being Element of REAL * holds OuterVx (f,n) c= UnusedVx (f,n)
proof end;

theorem Th28: :: GRAPHSP:28
for n being Nat
for f being Element of REAL * holds OuterVx (f,n) c= Seg n
proof end;

registration
let f be Element of REAL * ;
let n be Nat;
cluster OuterVx (f,n) -> finite ;
coherence
OuterVx (f,n) is finite
proof end;
end;

definition
let X be finite Subset of NAT;
let f be Element of REAL * ;
let n be Nat;
func Argmin (X,f,n) -> Nat means :Def10: :: GRAPHSP:def 10
( ( X <> {} implies ex i being Nat st
( i = it & i in X & ( for k being Nat st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being Nat st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k ) ) ) & ( X = {} implies it = 0 ) );
existence
ex b1 being Nat st
( ( X <> {} implies ex i being Nat st
( i = b1 & i in X & ( for k being Nat st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being Nat st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k ) ) ) & ( X = {} implies b1 = 0 ) )
proof end;
uniqueness
for b1, b2 being Nat st ( X <> {} implies ex i being Nat st
( i = b1 & i in X & ( for k being Nat st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being 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 Nat st
( i = b2 & i in X & ( for k being Nat st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being Nat st k in X & f /. ((2 * n) + i) = f /. ((2 * n) + k) holds
i <= k ) ) ) & ( X = {} implies b2 = 0 ) holds
b1 = b2
proof 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 Nat holds
( b4 = Argmin (X,f,n) iff ( ( X <> {} implies ex i being Nat st
( i = b4 & i in X & ( for k being Nat st k in X holds
f /. ((2 * n) + i) <= f /. ((2 * n) + k) ) & ( for k being 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 j, n being 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 end;

theorem Th30: :: GRAPHSP:30
for n being Nat
for f being Element of REAL * holds Argmin ((OuterVx (f,n)),f,n) <= n
proof end;

definition
let n be 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 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 end;
end;

:: deftheorem Def11 defines findmin GRAPHSP:def 11 :
for n being 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)) ) ) );

reconsider jj = 1 as Real ;

theorem Th31: :: GRAPHSP:31
for i, n being Nat
for f being Element of REAL * st i > n & i <> ((n * n) + (3 * n)) + 1 holds
((findmin n) . f) . i = f . i
proof end;

theorem Th32: :: GRAPHSP:32
for i, n being 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 end;

theorem Th33: :: GRAPHSP:33
for n being Nat
for f being Element of REAL * holds dom ((findmin n) . f) = dom f
proof end;

Lm6: for k, n being Nat st k >= 1 holds
n <= k * n
proof end;

Lm7: for n being Nat holds
( 3 * n < ((n * n) + (3 * n)) + 1 & n < ((n * n) + (3 * n)) + 1 & 2 * n < ((n * n) + (3 * n)) + 1 )
proof end;

Lm8: for k, n being 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 end;

theorem Th34: :: GRAPHSP:34
for n being Nat
for f being Element of REAL * st OuterVx (f,n) <> {} holds
ex j being Nat st
( j in OuterVx (f,n) & 1 <= j & j <= n & ((findmin n) . f) . j = - 1 )
proof end;

definition
let f be Element of REAL * ;
let n, k be 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 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 Nat;
let f be Element of REAL * ;
pred f hasBetterPathAt n,k means :: 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 defines hasBetterPathAt GRAPHSP:def 13 :
for n, k being 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 Nat;
func Relax (f,n) -> Element of REAL * means :Def14: :: GRAPHSP:def 14
( dom it = dom f & ( for k being 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 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 end;
uniqueness
for b1, b2 being Element of REAL * st dom b1 = dom f & ( for k being 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 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 end;
end;

:: deftheorem Def14 defines Relax GRAPHSP:def 14 :
for f being Element of REAL *
for n being Nat
for b3 being Element of REAL * holds
( b3 = Relax (f,n) iff ( dom b3 = dom f & ( for k being 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 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 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 end;
end;

:: deftheorem Def15 defines Relax GRAPHSP:def 15 :
for n being 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 Nat
for f being Element of REAL * holds dom ((Relax n) . f) = dom f
proof end;

theorem Th36: :: GRAPHSP:36
for i, n being 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 end;

theorem Th37: :: GRAPHSP:37
for i, n being 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 end;

theorem Th38: :: GRAPHSP:38
for i, n being 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 end;

theorem Th39: :: GRAPHSP:39
for i, k, n being Nat
for f, g, 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 end;

theorem Th40: :: GRAPHSP:40
for n being Nat
for f being Element of REAL * ex i being Nat st
( i <= n & OuterVx ((((repeat ((Relax n) * (findmin n))) . i) . f),n) = {} )
proof end;

Lm9: for k, n being Nat holds n - k <= n
proof 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 Nat st 1 <= k & k < len p holds
p . ((len p) - k) = f . ((p /. (((len p) - k) + 1)) + n) ) & ( for k being 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 Nat st 1 <= k & k < len p holds
p . ((len p) - k) = q . ((len q) - k)
proof end;

theorem Th41: :: GRAPHSP:41
for i, n being Nat
for f being Element of REAL * holds dom f = dom (((repeat ((Relax n) * (findmin n))) . i) . f)
proof end;

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

:: deftheorem defines equal_at GRAPHSP:def 16 :
for f, g being Element of REAL *
for m, n being Nat holds
( f,g equal_at m,n iff ( dom f = dom g & ( for k being Nat st k in dom f & m <= k & k <= n holds
f . k = g . k ) ) );

theorem Th42: :: GRAPHSP:42
for m, n being Nat
for f being Element of REAL * holds f,f equal_at m,n ;

theorem Th43: :: GRAPHSP:43
for m, n being 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 end;

theorem Th44: :: GRAPHSP:44
for i, n being 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 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 i, n being 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 end;

theorem Th47: :: GRAPHSP:47
for n being Nat
for f being Element of REAL * st 1 <= n & 1 in dom f & f . (n + 1) <> - 1 & ( for i being Nat st 1 <= i & i <= n holds
f . i = 1 ) & ( for i being 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 end;

theorem Th48: :: GRAPHSP:48
for i, m, n being Nat
for f, g, 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 end;

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

:: deftheorem defines is_vertex_seq_at GRAPHSP:def 17 :
for p being FinSequence of NAT
for f being Element of REAL *
for i, n being Nat holds
( p is_vertex_seq_at f,i,n iff ( p . (len p) = i & ( for k being 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 Nat;
pred p is_simple_vertex_seq_at f,i,n means :: GRAPHSP:def 18
( p . 1 = 1 & len p > 1 & p is_vertex_seq_at f,i,n & p is one-to-one );
end;

:: deftheorem 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 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 end;

definition
let G be Graph;
let p be FinSequence of the carrier' of G;
let vs be FinSequence;
pred p is_oriented_edge_seq_of vs means :: 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 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 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 end;

Lm11: for i, n being Nat st 1 <= i & i <= n holds
( 1 < (2 * n) + i & (2 * n) + i < ((n * n) + (3 * n)) + 1 & i < (2 * n) + i )
proof end;

Lm12: for i, n being Nat st 1 <= i & i <= n holds
( 1 < n + i & n + i <= 2 * n & n + i < ((n * n) + (3 * n)) + 1 )
proof end;

Lm13: for i, j, n being 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 end;

Lm14: for i, j, n being 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 end;

:: address possible value init. value comment
:: 1 1 or -1 1 -1 if node v1 is used
:: 2 1 or -1 1 -1 if node v2 is used
:: : : : :
:: n 1 or -1 1 -1 if node vn is used
:: n+1 0 0 preceding-node of v1 toward v1
:: n+2 -1 or Node No. -1 preceding-node of v2 toward v1
:: : : : :
:: 2*n -1 or Node No. -1 preceding-node of vn toward v1
:: 2*n+1 0 0 cost from v1 to v1
:: 2*n+2 >=0 0 minimum cost from v2 to v1
:: : : : :
:: 3*n >=0 0 minimum cost from vn to v1
:: 3*n+1 weight(v1,v1) the weight of edge(v1,v1)
:: 3*n+2 weight(v1,v2) the weight of edge(v1,v2)
:: : : :
:: 4*n weight(v1,vn) the weight of edge(v1,vn)
:: : : :
:: n*n+3*n weight(vn,vn) the weight of edge(vn,vn)
:: n*n+3n+1 Node No. current node with the shortest path
definition
let f be Element of REAL * ;
let G be oriented Graph;
let n be Nat;
let W be Function of the carrier' of G,Real>=0;
pred f is_Input_of_Dijkstra_Alg G,n,W means :: GRAPHSP:def 20
( len f = ((n * n) + (3 * n)) + 1 & Seg n = the carrier of G & ( for i being Nat st 1 <= i & i <= n holds
( f . i = 1 & f . ((2 * n) + i) = 0 ) ) & f . (n + 1) = 0 & ( for i being Nat st 2 <= i & i <= n holds
f . (n + i) = - 1 ) & ( for i, j being Vertex of G
for k, m being Nat st k = i & m = j holds
f . (((2 * n) + (n * k)) + m) = Weight (i,j,W) ) );
end;

:: deftheorem defines is_Input_of_Dijkstra_Alg GRAPHSP:def 20 :
for f being Element of REAL *
for G being oriented Graph
for n being 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 Nat st 1 <= i & i <= n holds
( f . i = 1 & f . ((2 * n) + i) = 0 ) ) & f . (n + 1) = 0 & ( for i being Nat st 2 <= i & i <= n holds
f . (n + i) = - 1 ) & ( for i, j being Vertex of G
for k, m being Nat st k = i & m = j holds
f . (((2 * n) + (n * k)) + m) = Weight (i,j,W) ) ) );

definition
let n be 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 Nat holds DijkstraAlgorithm n = while_do (((Relax n) * (findmin n)),n);

Lm15: for n being 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 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 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 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 Nat st m in UsedVx (h,n) holds
h . (n + m) <> - 1 ) )
proof end;

Lm16: for i, k, n being Nat
for f, g, 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 end;

Lm17: for j, k, n being Nat
for f, g, 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 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 end;

Lm18: for j, k, m, n being Nat
for f, g, 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 Nat st 1 <= i & i < len p holds
p . i in UsedVx (g,n) ) holds
for q being FinSequence of NAT st q = p ^ <*j*> holds
q is_simple_vertex_seq_at h,j,n
proof end;

Lm19: for i, n being 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 v1, v2 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 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 end;

Lm20: for k, n being 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 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 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 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 Nat st m in UsedVx (g,n) holds
g . (n + m) <> - 1 ) holds
( ( for v3 being Vertex of G
for j being 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 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 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 Nat st m in UsedVx (h,n) holds
h . (n + m) <> - 1 ) )
proof end;

theorem :: GRAPHSP:52
for i, n being 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 end;