:: The Underlying Principle of {D}ijkstra's Shortest Path Algorithm
:: by Jingchao Chen and Yatsuka Nakamura
::
:: Received January 7, 2003
:: Copyright (c) 2003-2016 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, SUBSET_1, FUNCT_1, RELAT_1, TARSKI, CARD_1, FINSET_1,
XXREAL_0, FINSEQ_1, FINSEQ_2, CARD_3, NAT_1, XBOOLE_0, ARYTM_3, ARYTM_1,
ORDINAL4, GRAPH_1, STRUCT_0, TREES_2, GRAPH_4, PARTFUN1, GLIB_000,
REAL_1, GRAPH_5, ARYTM_2;
notations TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, CARD_1, NUMBERS, XCMPLX_0,
XXREAL_0, XREAL_0, RELAT_1, FUNCT_1, PARTFUN1, FINSEQ_1, FINSEQ_2,
FINSEQ_4, FINSET_1, ARYTM_2, NAT_1, NAT_D, STRUCT_0, GRAPH_1, FUNCT_2,
GRAPH_4, CARD_3, RVSUM_1;
constructors FINSEQ_4, FINSOP_1, BINARITH, GRAPH_4, NAT_D, RVSUM_1, ARYTM_2,
RELSET_1;
registrations XBOOLE_0, RELAT_1, FUNCT_1, FINSET_1, NUMBERS, XXREAL_0,
XREAL_0, NAT_1, FINSEQ_1, GRAPH_1, GRAPH_4, VALUED_0, CARD_1, STRUCT_0,
ARYTM_2, FINSEQ_3, ORDINAL1;
requirements REAL, BOOLE, SUBSET, NUMERALS, ARITHM;
definitions TARSKI, FUNCT_1;
expansions TARSKI, FUNCT_1;
theorems FUNCT_2, XBOOLE_1, GRAPH_1, FINSEQ_1, GRAPH_4, FUNCT_1, FINSEQ_3,
FINSEQ_4, NAT_1, TARSKI, CARD_5, FINSEQ_2, XBOOLE_0, CARD_2, RVSUM_1,
SUBSET_1, ZFMISC_1, XREAL_1, XXREAL_0, ORDINAL1, PARTFUN1, ARYTM_0,
REAL_1, XREAL_0;
schemes FUNCT_1, NAT_1, CLASSES1, PRE_CIRC;
begin :: Preliminaries
reserve n,m,i,j,k for Nat,
x,y,e,X,V,U for set,
W,f,g for Function;
theorem
rng f c= rng g & x in dom f implies
ex y being object st y in dom g & f.x = g. y
proof
assume that
A1: rng f c= rng g and
A2: x in dom f;
f.x in rng f by A2,FUNCT_1:3;
hence thesis by A1,FUNCT_1:def 3;
end;
theorem Th2:
for D being finite set, n being Element of NAT, X being set st X
= {x where x is Element of D* : 1 <= len x & len x <= n } holds X is finite
proof
let D be finite set,n be Element of NAT, X be set;
deffunc F(Element of NAT) = $1-tuples_on D;
consider f being Function such that
A1: dom f = Seg n & for x being Element of NAT st x in Seg n holds f.x =
F(x) from CLASSES1:sch 2;
assume
A2: X = {x where x is Element of D* : 1 <= len x & len x <= n };
A3: X c= Union f
proof
let x be object;
assume x in X;
then consider y being Element of D* such that
A4: x=y and
A5: 1 <= len y & len y <= n by A2;
consider m being Nat such that
A6: dom y = Seg m by FINSEQ_1:def 2;
m in NAT by ORDINAL1:def 12;
then
A7: len y = m by A6,FINSEQ_1:def 3;
then
A8: m in dom f by A1,A5,FINSEQ_1:1;
y in { s where s is Element of D*: len s = m } by A7;
then y in m-tuples_on D by FINSEQ_2:def 4;
then y in f.m by A1,A8;
hence thesis by A4,A8,CARD_5:2;
end;
now
let x be object;
assume
A9: x in dom f;
then reconsider z=x as Element of NAT by A1;
f.z = F(z) by A1,A9;
hence f.x is finite;
end;
then Union f is finite by A1,CARD_2:88;
hence thesis by A3;
end;
theorem Th3:
for D being finite set, n being Element of NAT, X being set st X
= {x where x is Element of D* : len x <= n } holds X is finite
proof
let D be finite set,n be Element of NAT, X be set;
set B = {x where x is Element of D* : 1 <= len x & len x <= n };
assume
A1: X = {x where x is Element of D* : len x <= n };
A2: X c= { {} } \/ B
proof
let y be object;
assume y in X;
then consider x being Element of D* such that
A3: y=x and
A4: len x <= n by A1;
per cases;
suppose
len x < 0+1;
then x = {} by NAT_1:13;
then x in { {} } by TARSKI:def 1;
hence thesis by A3,XBOOLE_0:def 3;
end;
suppose
len x >= 0+1;
then x in B by A4;
hence thesis by A3,XBOOLE_0:def 3;
end;
end;
B is finite by Th2;
hence thesis by A2;
end;
::$CT 2
definition
let D be set, X be non empty Subset of D*;
redefine mode Element of X -> FinSequence of D;
coherence
proof
let x be Element of X;
reconsider y=x as Element of D*;
y is FinSequence of D;
hence thesis;
end;
end;
begin :: Additional Properties of Finite Sequences
reserve p,q for FinSequence;
Lm1: 1 <= n & n <= len p implies p.n = (p^q).n
proof
assume 1 <= n & n <= len p;
then n in dom p by FINSEQ_3:25;
hence thesis by FINSEQ_1:def 7;
end;
Lm2: 1 <= n & n <= len q implies q.n = (p^q).(len p+n)
proof
assume 1 <= n & n <= len q;
then n in dom q by FINSEQ_3:25;
hence thesis by FINSEQ_1:def 7;
end;
theorem Th4:
(for n,m st 1<=n & n p.m) iff p is one-to-one
proof
hereby
assume
A1: for n,m st 1<=n & n p.m;
thus p is one-to-one
proof
let x1,x2 be object;
assume that
A2: x1 in dom p & x2 in dom p and
A3: p.x1 = p.x2;
reconsider n=x1, m=x2 as Element of NAT by A2;
A4: n <= len p & 1 <= m by A2,FINSEQ_3:25;
A5: 1 <= n & m <= len p by A2,FINSEQ_3:25;
assume
A6: x1 <> x2;
per cases;
suppose
m <= n;
then m < n by A6,XXREAL_0:1;
hence contradiction by A1,A3,A4;
end;
suppose
m > n;
hence contradiction by A1,A3,A5;
end;
end;
end;
assume
A7: p is one-to-one;
let n,m;
assume that
A8: 1<=n and
A9: n p.m) iff card rng p = len p
by Th4,FINSEQ_4:62;
reserve G for Graph,
pe,qe for FinSequence of the carrier' of G;
theorem
i in dom pe implies (the Source of G).(pe.i) in the carrier of G & (
the Target of G).(pe.i) in the carrier of G by FINSEQ_2:11,FUNCT_2:5;
theorem Th7:
for x being object holds
q^<*x*> is one-to-one & rng (q^<*x*>) c= rng p implies ex p1,p2
being FinSequence st p=p1^<*x*>^p2 & rng q c= rng (p1^p2)
proof let x be object;
set r=q^<*x*>, i=len q +1;
assume that
A1: r is one-to-one and
A2: rng r c= rng p;
A3: r.i=x by FINSEQ_1:42;
rng q c= rng r by FINSEQ_1:29;
then
A4: rng q c= rng p by A2;
len r =i & 1 <= i by FINSEQ_2:16,NAT_1:11;
then
A5: i in dom r by FINSEQ_3:25;
then consider y being object such that
A6: y in dom p and
A7: r.i = p.y by A2,FUNCT_1:114;
reconsider j=y as Element of NAT by A6;
j <= len p by A6,FINSEQ_3:25;
then consider k being Nat such that
A8: len p = j+k by NAT_1:10;
reconsider k as Element of NAT by ORDINAL1:def 12;
consider s, p2 being FinSequence such that
A9: len s = j and
len p2 = k and
A10: p = s^p2 by A8,FINSEQ_2:22;
A11: 1 <= j by A6,FINSEQ_3:25;
then ex m being Nat st j = 1 + m by NAT_1:10;
then consider p1 being FinSequence, d being object such that
A12: s = p1^<*d*> by A9,FINSEQ_2:18;
j in dom s by A9,A11,FINSEQ_3:25;
then
A13: s.j=x by A7,A10,A3,FINSEQ_1:def 7;
take p1,p2;
A14: j = len p1 +1 by A9,A12,FINSEQ_2:16;
hence p=p1^<*x*>^p2 by A10,A12,A13,FINSEQ_1:42;
let y be object;
assume
A15: y in rng q;
now
let y be set;
assume
A16: y in rng q;
assume y = x;
then consider z being object such that
A17: z in dom q and
A18: x = q.z by A16,FUNCT_1:def 3;
reconsider n=z as Element of NAT by A17;
n <= len q by A17,FINSEQ_3:25;
then
A19: n <> i by XREAL_1:29;
n in dom r & r.n=r.i by A3,A17,A18,FINSEQ_1:def 7,FINSEQ_2:15;
hence contradiction by A1,A5,A19;
end;
then
A20: y <> x by A15;
s = p1^<*x*> by A12,A14,A13,FINSEQ_1:42;
then rng p = rng (p1^<*x*>) \/ rng p2 by A10,FINSEQ_1:31
.= rng p1 \/ rng <*x*> \/ rng p2 by FINSEQ_1:31
.= rng p1 \/ rng p2 \/ rng <*x*> by XBOOLE_1:4
.= rng (p1^p2) \/ rng <*x*> by FINSEQ_1:31
.= rng (p1^p2) \/ {x} by FINSEQ_1:38;
then y in rng (p1^p2) or y in {x} by A15,A4,XBOOLE_0:def 3;
hence thesis by A20,TARSKI:def 1;
end;
begin :: Additional Properties of Chains and Oriented Paths
theorem Th8:
p^q is Chain of G implies p is Chain of G & q is Chain of G
proof
set r=p^q, D=the carrier' of G, V=the carrier of G;
assume
A1: p^q is Chain of G;
then consider f being FinSequence such that
A2: len f = len r + 1 and
A3: for n st 1 <= n & n <= len f holds f.n in V and
A4: for n st 1 <= n & n <= len r ex x, y being Element of V st x = f.n &
y = f.(n+1) & r.n joins x, y by GRAPH_1:def 14;
A5: len r = len p + len q by FINSEQ_1:22;
then len f = len p + (len q + 1) by A2;
then consider h1,h2 being FinSequence such that
A6: len h1 = len p and
A7: len h2 = len q +1 and
A8: f = h1^h2 by FINSEQ_2:22;
A9: now
take h2;
thus len h2 = len q + 1 by A7;
hereby
let n;
assume that
A10: 1 <= n and
A11: n <= len h2;
n <= len h1+n by NAT_1:11;
then
A12: 1 <= len h1 +n by A10,XXREAL_0:2;
len h1+n <= len h1 + len h2 by A11,XREAL_1:7;
then
A13: len h1+n <= len f by A8,FINSEQ_1:22;
n in dom h2 by A10,A11,FINSEQ_3:25;
then h2.n=f.(len h1+n) by A8,FINSEQ_1:def 7;
hence h2.n in V by A3,A13,A12;
end;
hereby
let n;
assume that
A14: 1 <= n and
A15: n <= len q;
set m=len p+n;
n <= m by NAT_1:11;
then 1 <= m by A14,XXREAL_0:2;
then consider x, y being Element of V such that
A16: x = f.m and
A17: y = f.(m+1) and
A18: r.m joins x, y by A4,A5,A15,XREAL_1:7;
take x,y;
len q <= len h2 by A7,NAT_1:11;
then n <= len h2 by A15,XXREAL_0:2;
hence x = h2.n by A6,A8,A14,A16,Lm2;
1 <= n+1 by NAT_1:11;
hence h2.(n+1)=f.(len h1+(n+1)) by A7,A8,A15,Lm2,XREAL_1:7
.=y by A6,A17;
thus q.n joins x, y by A14,A15,A18,Lm2;
end;
end;
A19: len f = len p + 1 +len q by A2,A5;
then consider f1,f2 being FinSequence such that
A20: len f1 = len p+1 and
len f2 = len q and
A21: f = f1^f2 by FINSEQ_2:22;
A22: now
take f1;
thus len f1 = len p + 1 by A20;
hereby
let n;
assume that
A23: 1 <= n and
A24: n <= len f1;
len f1 <= len f by A19,A20,NAT_1:11;
then n <= len f by A24,XXREAL_0:2;
then
A25: f.n in V by A3,A23;
n in dom f1 by A23,A24,FINSEQ_3:25;
hence f1.n in V by A21,A25,FINSEQ_1:def 7;
end;
hereby
let n;
assume that
A26: 1 <= n and
A27: n <= len p;
len p <= len r by A5,NAT_1:11;
then n <= len r by A27,XXREAL_0:2;
then consider x, y being Element of V such that
A28: x = f.n and
A29: y = f.(n+1) and
A30: r.n joins x, y by A4,A26;
take x,y;
len p <= len f1 by A20,NAT_1:11;
then n <= len f1 by A27,XXREAL_0:2;
hence x = f1.n by A21,A26,A28,Lm1;
1 <= n+1 & n+1 <= len f1 by A20,A27,NAT_1:11,XREAL_1:7;
then n+1 in dom f1 by FINSEQ_3:25;
hence y = f1.(n+1) by A21,A29,FINSEQ_1:def 7;
thus p.n joins x, y by A26,A27,A30,Lm1;
end;
end;
A31: p is FinSequence of D by A1,FINSEQ_1:36;
now
let n;
assume 1 <= n & n <= len p;
then n in dom p by FINSEQ_3:25;
hence p.n in D by A31,FINSEQ_2:11;
end;
hence p is Chain of G by A22,GRAPH_1:def 14;
A32: q is FinSequence of D by A1,FINSEQ_1:36;
now
let n;
assume 1 <= n & n <= len q;
then n in dom q by FINSEQ_3:25;
hence q.n in D by A32,FINSEQ_2:11;
end;
hence thesis by A9,GRAPH_1:def 14;
end;
theorem Th9:
p^q is oriented Chain of G implies p is oriented Chain of G & q
is oriented Chain of G
proof
assume
A1: p^q is oriented Chain of G;
set r=p^q;
A2: len r = len p + len q by FINSEQ_1:22;
A3: now
let n;
assume that
A4: 1 <= n and
A5: n < len q;
set m=len p + n;
n <= m by NAT_1:11;
then
A6: 1 <= m by A4,XXREAL_0:2;
n+1 <= len q by A5,NAT_1:13;
then
A7: q.(n+1)=r.(len p+(n+1)) by Lm2,NAT_1:11
.=r.(m+1);
m < len r by A2,A5,XREAL_1:8;
then (the Source of G).(r.(m+1))=(the Target of G).(r.m) by A1,A6,
GRAPH_1:def 15;
hence (the Source of G).(q.(n+1))=(the Target of G).(q.n) by A4,A5,A7,Lm2;
end;
now
let n;
assume that
A8: 1 <= n and
A9: n < len p;
n+1 <= len p by A9,NAT_1:13;
then
A10: p.(n+1)=r.(n+1) by Lm1,NAT_1:11;
len p <= len r by A2,NAT_1:11;
then n < len r by A9,XXREAL_0:2;
then (the Source of G).(r.(n+1))=(the Target of G).(r.n) by A1,A8,
GRAPH_1:def 15;
hence (the Source of G).(p.(n+1))=(the Target of G).(p.n) by A8,A9,A10,Lm1;
end;
hence thesis by A1,A3,Th8,GRAPH_1:def 15;
end;
theorem Th10:
for p,q being oriented Chain of G st (the Target of G).(p.len p)
=(the Source of G).(q.1) holds p^q is oriented Chain of G
proof
let p,q be oriented Chain of G;
assume
A1: (the Target of G).(p.len p)=(the Source of G).(q.1);
per cases;
suppose
A2: p={} or q={};
hereby
per cases by A2;
suppose
p={};
hence thesis by FINSEQ_1:34;
end;
suppose
q={};
hence thesis by FINSEQ_1:34;
end;
end;
end;
suppose
A3: not (p={} or q={});
consider vs2 being FinSequence of the carrier of G such that
A4: vs2 is_oriented_vertex_seq_of q by GRAPH_4:9;
len vs2 = len q + 1 by A4,GRAPH_4:def 5;
then
A5: len vs2 >= 1 by NAT_1:12;
len q >= 1 by A3,FINSEQ_1:20;
then q.1 orientedly_joins vs2/.1, vs2/.(1+1) by A4,GRAPH_4:def 5;
then
A6: (the Source of G).(q.1)=vs2/.1 by GRAPH_4:def 1
.=vs2.1 by A5,FINSEQ_4:15;
consider vs1 being FinSequence of the carrier of G such that
A7: vs1 is_oriented_vertex_seq_of p by GRAPH_4:9;
A8: len vs1 = len p + 1 by A7,GRAPH_4:def 5;
then
A9: len vs1 >= 1 by NAT_1:12;
len p >= 1 by A3,FINSEQ_1:20;
then p.(len p) orientedly_joins vs1/.(len p), vs1/.(len p+1) by A7,
GRAPH_4:def 5;
then (the Target of G).(p.(len p))=vs1/.(len vs1) by A8,GRAPH_4:def 1
.=vs1.len vs1 by A9,FINSEQ_4:15;
hence thesis by A1,A7,A4,A6,GRAPH_4:14;
end;
end;
begin :: Additional Properties of Acyclic Oriented Paths
theorem Th11:
{} is Simple oriented Chain of G
proof
set v = the Element of G;
set vs=<*v*>;
A1: now
let n,m;
assume that
A2: 1<=n & n;
A3: len vs = len p +1 by A2,FINSEQ_1:44;
A4: now
let n;
assume 1 <= n & n <= len p;
then
A5: n=1 by A2,XXREAL_0:1;
take v1,v2;
thus v1= vs.n & v2 = vs.(n+1) by A5,FINSEQ_1:44;
thus p.n joins v1, v2 by A5,GRAPH_1:def 12;
end;
A6: len vs = 2 by FINSEQ_1:44;
A7: now
let n,m;
assume that
A8: 1<=n and
A9: n= 2;
then n=2 by A6,A13,XXREAL_0:1;
then vs.n=v2 by FINSEQ_1:44;
hence vs.n in the carrier of G;
end;
end;
A14: for n st 1 <= n & n < len p & (the Source of G).(p.(n+1)) <> (the
Target of G).(p.n) holds contradiction by A2;
now
let n;
assume 1<=n & n<=len p;
then
A15: n=1 by A2,XXREAL_0:1;
vs/.1= v1 & vs/.(1+1)= v2 by FINSEQ_4:17;
hence p.n orientedly_joins vs/.n, vs/.(n+1) by A15,GRAPH_4:def 1;
end;
then vs is_oriented_vertex_seq_of p by A3,GRAPH_4:def 5;
hence thesis by A3,A7,A1,A11,A4,A14,GRAPH_1:def 14,def 15,GRAPH_4:def 7;
end;
theorem Th14:
for p being Simple oriented Chain of G, q being FinSequence of
the carrier' of G st len p >=1 & len q=1 & (the Source of G).(q.1)=(the Target
of G).(p.(len p)) & (the Source of G).(p.1) <> (the Target of G).(p.(len p)) &
not ex k st 1<=k & k <= len p & (the Target of G).(p.k) =(the Target of G).(q.1
) holds p^q is Simple oriented Chain of G
proof
let p be Simple oriented Chain of G, q be FinSequence of the carrier' of G;
set FS=the Source of G, FT=the Target of G, v1=FS.(q.1), v2=FT.(q.1), vp=(
the Target of G).(p.len p), E=the carrier' of G, V=the carrier of G;
assume that
A1: len p >=1 and
A2: len q=1 and
A3: v1=vp and
A4: FS.(p.1) <> vp and
A5: not ex k st 1<=k & k <= len p & FT.(p.k) = v2;
set lp=len p;
1 in dom q by A2,FINSEQ_3:25;
then reconsider v1, v2 as Element of V by FINSEQ_2:11,FUNCT_2:5;
consider r being FinSequence of V such that
A6: r is_oriented_vertex_seq_of p and
A7: for n,m st 1<=n & n;
A8: len r = len p + 1 by A6,GRAPH_4:def 5;
A9: for n st 1<=n & n<=len p holds p.n joins r/.n, r/.(n+1)
by A6,GRAPH_4:1,def 5;
A10: now
let n;
assume that
A11: 1 <= n and
A12: n <= len pq;
per cases;
suppose
A13: n=len pq;
set m=len p;
take v1,v2;
p.m orientedly_joins r/.m, r/.(m+1) by A1,A6,GRAPH_4:def 5;
then
A14: vp = r/.(m+1) by GRAPH_4:def 1;
A15: n = len r by A2,A8,A13,FINSEQ_1:22;
then n in dom r by A11,FINSEQ_3:25;
hence v1 = r.n by A3,A8,A15,A14,PARTFUN1:def 6
.=rv.n by A11,A15,Lm1;
thus v2 = rv.(n+1) by A15,FINSEQ_1:42;
q.1 joins v1, v2 by GRAPH_1:def 12;
hence pq.n joins v1, v2 by A2,A8,A15,Lm2;
end;
suppose
A16: n<>len pq;
take x=r/.n;
take y=r/.(n+1);
n < len pq by A12,A16,XXREAL_0:1;
then
A17: n < len p +1 by A2,FINSEQ_1:22;
then
A18: n+1 <= len r by A8,NAT_1:13;
n in dom r by A8,A11,A17,FINSEQ_3:25;
hence x = r.n by PARTFUN1:def 6
.= rv.n by A8,A11,A17,Lm1;
1 <= n+1 by NAT_1:12;
then n+1 in dom r by A18,FINSEQ_3:25;
hence y = r.(n+1) by PARTFUN1:def 6
.=rv.(n+1) by A18,Lm1,NAT_1:12;
A19: n <= len p by A17,NAT_1:13;
then p.n joins r/.n, r/.(n+1) by A9,A11;
hence pq.n joins x, y by A11,A19,Lm1;
end;
end;
A20: now
let n;
assume 1 <= n & n <= len pq;
then n in dom pq by FINSEQ_3:25;
then pq.n in rng pq by FUNCT_1:def 3;
then
A21: pq.n in rng p \/ rng q by FINSEQ_1:31;
rng p c= E & rng q c= E by FINSEQ_1:def 4;
then rng p \/ rng q c= E by XBOOLE_1:8;
hence pq.n in E by A21;
end;
A22: len rv = len r + 1 by FINSEQ_2:16;
then
A23: len rv = len pq +1 by A2,A8,FINSEQ_1:22;
p.lp orientedly_joins r/.lp, r/.(lp+1) by A1,A6,GRAPH_4:def 5;
then
A24: vp = r/.(lp+1) by GRAPH_4:def 1;
A25: now
let n,m;
assume that
A26: 1<=n and
A27: n= len rv;
then m=len rv by A28,XXREAL_0:1;
then
A39: v2 = rv.m by A22,FINSEQ_1:42;
consider k being Nat such that
A40: n=k+1 by A26,NAT_1:6;
reconsider k as Element of NAT by ORDINAL1:def 12;
1 < n by A26,A28,A30,A38,XXREAL_0:1;
then
A41: 1 <= k by A40,NAT_1:13;
k+1 < len r+1 by A22,A27,A28,A40,XXREAL_0:2;
then
A42: k+1 <= len r by NAT_1:13;
then
A43: k+1 in dom r by A26,A40,FINSEQ_3:25;
A44: k <= len p by A8,A42,XREAL_1:6;
then p.k orientedly_joins r/.k, r/.(k+1) by A6,A41,GRAPH_4:def 5;
then FT.(p.k) = r/.(k+1) by GRAPH_4:def 1
.=r.(k+1) by A43,PARTFUN1:def 6
.=v2 by A26,A29,A39,A40,A42,Lm1;
hence contradiction by A5,A41,A44;
end;
end;
A45: now
let n;
assume that
A46: 1 <= n and
A47: n < len pq;
per cases;
suppose
A48: n < len p;
then
A49: n+1 <= len p by NAT_1:13;
FS.(p.(n+1)) = FT.(p.n) & p.n = pq.n by A46,A48,Lm1,GRAPH_1:def 15;
hence FS.(pq.(n+1)) = FT.(pq.n) by A49,Lm1,NAT_1:12;
end;
suppose
A50: n >= len p;
n < len p + 1 by A2,A47,FINSEQ_1:22;
then n <= len p by NAT_1:13;
then
A51: n = len p by A50,XXREAL_0:1;
then pq.n=p.(len p) by A1,Lm1;
hence FS.(pq.(n+1)) = FT.(pq.n) by A2,A3,A51,Lm2;
end;
end;
A52: now
let n;
assume that
A53: 1 <= n and
A54: n <= len rv;
per cases;
suppose
n=len rv;
then rv.n=v2 by A22,FINSEQ_1:42;
hence rv.n in V;
end;
suppose
n<>len rv;
then n < len rv by A54,XXREAL_0:1;
then
A55: n <= len r by A22,NAT_1:13;
then n in dom r by A53,FINSEQ_3:25;
then r.n in V by FINSEQ_2:11;
hence rv.n in V by A53,A55,Lm1;
end;
end;
now
A56: dom r c= dom rv by FINSEQ_1:26;
let n;
assume that
A57: 1<=n and
A58: n<=len pq;
per cases;
suppose
A59: n <= len p;
then
A60: n+1 <= len r by A8,XREAL_1:7;
1 <= n+1 by NAT_1:12;
then
A61: n+1 in dom r by A60,FINSEQ_3:25;
then
A62: r/.(n+1)= r.(n+1) by PARTFUN1:def 6
.=rv.(n+1) by A60,Lm1,NAT_1:12
.=rv/.(n+1) by A56,A61,PARTFUN1:def 6;
A63: p.n orientedly_joins r/.n, r/.(n+1) by A6,A57,A59,GRAPH_4:def 5;
A64: n <= len r by A8,A59,NAT_1:12;
then
A65: n in dom r by A57,FINSEQ_3:25;
then r/.n= r.n by PARTFUN1:def 6
.=rv.n by A57,A64,Lm1
.=rv/.n by A56,A65,PARTFUN1:def 6;
hence pq.n orientedly_joins rv/.n, rv/.(n+1) by A57,A59,A63,A62,Lm1;
end;
suppose
A66: n > len p;
A67: len p+1 >= n by A2,A58,FINSEQ_1:22;
len p +1 <= n by A66,NAT_1:13;
then
A68: n = len r by A8,A67,XXREAL_0:1;
1 <= n+1 by NAT_1:12;
then
A69: n+1 in dom rv by A22,A68,FINSEQ_3:25;
A70: v2 = rv.(n+1) by A68,FINSEQ_1:42
.=rv/.(n+1) by A69,PARTFUN1:def 6;
A71: q.1 orientedly_joins v1, v2 by GRAPH_4:def 1;
A72: n in dom r by A8,A57,A67,FINSEQ_3:25;
then v1 = r.n by A3,A8,A24,A68,PARTFUN1:def 6
.=rv.n by A8,A57,A67,Lm1
.=rv/.n by A56,A72,PARTFUN1:def 6;
hence pq.n orientedly_joins rv/.n, rv/.(n+1) by A2,A8,A68,A70,A71,Lm2;
end;
end;
then rv is_oriented_vertex_seq_of pq by A23,GRAPH_4:def 5;
hence thesis by A20,A23,A52,A10,A45,A25,GRAPH_1:def 14,def 15,GRAPH_4:def 7;
end;
theorem Th15:
for p being Simple oriented Chain of G holds p is one-to-one
proof
let p be Simple oriented Chain of G;
set VV=the carrier of G;
consider vs being FinSequence of VV such that
A1: vs is_oriented_vertex_seq_of p and
A2: for n,m st 1<=n & n set equals
{(the Source of G).e, (the Target of G).e};
coherence;
end;
definition
let G,pe;
func vertices pe -> Subset of the carrier of G equals
{v where v is Vertex
of G : ex i st i in dom pe & v in vertices(pe/.i)};
coherence
proof
set VT={v where v is Vertex of G : ex i st i in dom pe & v in vertices(pe
/.i) };
VT c= the carrier of G
proof
let x be object;
assume x in VT;
then
ex v being Vertex of G st x=v & ex i st i in dom pe & v in vertices(
pe/.i);
hence thesis;
end;
hence thesis;
end;
end;
theorem Th16:
for p being Simple oriented Chain of G st p=pe^qe & len pe >=
1 & len qe >= 1 & (the Source of G).(p.1) <> (the Target of G).(p.len p) holds
not (the Source of G).(p.1) in vertices qe & not (the Target of G).(p.len p) in
vertices pe
proof
let p be Simple oriented Chain of G;
set FT=the Target of G, FS=the Source of G;
assume that
A1: p=pe^qe and
A2: len pe >= 1 and
A3: len qe >= 1 and
A4: FS.(p.1) <> FT.(p.len p);
consider vs being FinSequence of the carrier of G such that
A5: vs is_oriented_vertex_seq_of p and
A6: for n,m st 1<=n & n= 1 by A2,NAT_1:12;
then p.1 orientedly_joins vs/.1, vs/.(1+1) by A5,GRAPH_4:def 5;
then
A11: FS.(p.1)=vs/.1 by GRAPH_4:def 1
.=vs.1 by A8,FINSEQ_4:15;
p.len p orientedly_joins vs/.len p, vs/.(len p+1) by A5,A10,GRAPH_4:def 5;
then
A12: FT.(p.len p)=vs/.(len p+1) by GRAPH_4:def 1
.=vs.len vs by A7,A8,FINSEQ_4:15;
hereby
assume FS.(p.1) in vertices(qe);
then consider x being Vertex of G such that
A13: FS.(p.1)=x and
A14: ex i st i in dom qe & x in vertices(qe/.i);
consider i such that
A15: i in dom qe and
A16: x in vertices(qe/.i) by A14;
set k=len pe + i;
A17: qe/.i=qe.i by A15,PARTFUN1:def 6
.=p.k by A1,A15,FINSEQ_1:def 7;
1 <= i by A15,FINSEQ_3:25;
then
A18: 1 < i+1 by NAT_1:13;
i <= len qe by A15,FINSEQ_3:25;
then
A19: k <= len p by A9,XREAL_1:7;
then
A20: k <= len vs by A7,NAT_1:13;
1+i <= k by A2,XREAL_1:7;
then
A21: 1 < k by A18,XXREAL_0:2;
then
A22: p.k orientedly_joins vs/.k, vs/.(k+1) by A5,A19,GRAPH_4:def 5;
per cases by A16,A17,TARSKI:def 2;
suppose
A23: x=FS.(p.k);
FS.(p.k)=vs/.k by A22,GRAPH_4:def 1
.=vs.k by A21,A20,FINSEQ_4:15;
hence contradiction by A4,A6,A11,A12,A13,A21,A20,A23;
end;
suppose
A24: x=FT.(p.k);
A25: 1 < k+1 by A21,NAT_1:13;
A26: k+1 <= len vs by A7,A19,XREAL_1:7;
FT.(p.k)=vs/.(k+1) by A22,GRAPH_4:def 1
.=vs.(k+1) by A26,FINSEQ_4:15,NAT_1:12;
hence contradiction by A4,A6,A11,A12,A13,A24,A26,A25;
end;
end;
hereby
assume FT.(p.len p) in vertices(pe);
then consider x being Vertex of G such that
A27: FT.(p.len p)=x and
A28: ex i st i in dom pe & x in vertices(pe/.i);
consider i such that
A29: i in dom pe and
A30: x in vertices(pe/.i) by A28;
A31: pe/.i=pe.i by A29,PARTFUN1:def 6
.=p.i by A1,A29,FINSEQ_1:def 7;
A32: i <= len pe by A29,FINSEQ_3:25;
then
A33: i <= len p by A9,NAT_1:12;
then
A34: i < len vs by A7,NAT_1:13;
A35: 1 <= i by A29,FINSEQ_3:25;
then
A36: p.i orientedly_joins vs/.i, vs/.(i+1) by A5,A33,GRAPH_4:def 5;
per cases by A30,A31,TARSKI:def 2;
suppose
A37: x=FS.(p.i);
FS.(p.i)=vs/.i by A36,GRAPH_4:def 1
.=vs.i by A35,A34,FINSEQ_4:15;
hence contradiction by A4,A6,A12,A27,A35,A34,A37;
end;
suppose
A38: x=FT.(p.i);
A39: i+1 <= len vs by A7,A33,XREAL_1:7;
len pe+1 <= len pe + len qe & i+1 <= len pe+1 by A3,A32,XREAL_1:7;
then i+1 <= len p by A9,XXREAL_0:2;
then
A40: 1 <= i+1 & i+1 < len vs by A7,NAT_1:12,13;
FT.(p.i)=vs/.(i+1) by A36,GRAPH_4:def 1
.=vs.(i+1) by A39,FINSEQ_4:15,NAT_1:12;
hence contradiction by A4,A6,A11,A12,A27,A38,A40;
end;
end;
end;
theorem Th17:
vertices pe c= V iff for i be Nat st i in dom pe holds vertices( pe/.i) c= V
proof
set FS=the Source of G, FT=the Target of G;
hereby
assume
A1: vertices pe c= V;
hereby
let i be Nat;
assume
A2: i in dom pe;
then
A3: 1<=i & i <= len pe by FINSEQ_3:25;
thus vertices(pe/.i) c= V
proof
let x be object;
assume
A4: x in vertices(pe/.i);
then x = FS.(pe/.i) or x=FT.(pe/.i) by TARSKI:def 2;
then x = FS.(pe.i) or x=FT.(pe.i) by A3,FINSEQ_4:15;
then reconsider y=x as Vertex of G by A2,FINSEQ_2:11,FUNCT_2:5;
y in {v where v is Vertex of G : ex i st i in dom pe & v in
vertices(pe/.i)} by A2,A4;
hence thesis by A1;
end;
end;
end;
assume
A5: for i be Nat st i in dom pe holds vertices(pe/.i) c= V;
let x be object;
assume x in vertices pe;
then consider y being Vertex of G such that
A6: y=x and
A7: ex i st i in dom pe & y in vertices(pe/.i);
consider i such that
A8: i in dom pe and
A9: y in vertices(pe/.i) by A7;
vertices(pe/.i) c= V by A5,A8;
hence thesis by A6,A9;
end;
theorem Th18:
not vertices pe c= V implies ex i being Element of NAT, q,r
being FinSequence of the carrier' of G st i+1 <= len pe & not vertices(pe/.(i+1
)) c= V & len q=i & pe=q^r & vertices q c= V
proof
defpred P[Nat] means $1 in dom pe & not vertices(pe/.$1) c= V;
assume not vertices pe c= V;
then
A1: ex i be Nat st P[i] by Th17;
consider k be Nat such that
A2: P[k] & for n be Nat st P[n] holds k <= n from NAT_1:sch 5(A1);
k <= len pe by A2,FINSEQ_3:25;
then consider j being Nat such that
A3: len pe=k+j by NAT_1:10;
reconsider j as Element of NAT by ORDINAL1:def 12;
1 <= k by A2,FINSEQ_3:25;
then consider i being Nat such that
A4: k=1+i by NAT_1:10;
reconsider i as Element of NAT by ORDINAL1:def 12;
len pe=i+(1+j) by A4,A3;
then consider q,r being FinSequence such that
A5: len q = i and
len r = 1+j and
A6: pe = q^r by FINSEQ_2:22;
reconsider q,r as FinSequence of the carrier' of G by A6,FINSEQ_1:36;
take i,q,r;
thus i+1 <= len pe & not vertices(pe/.(i+1)) c= V by A2,A4,FINSEQ_3:25;
thus len q=i & pe=q^r by A5,A6;
now
let m be Nat;
assume
A7: m in dom q;
then
A8: m <= len q by FINSEQ_3:25;
A9: dom q c= dom pe by A6,FINSEQ_1:26;
assume
A10: not vertices(q/.m) c= V;
q/.m=q.m by A7,PARTFUN1:def 6
.=pe.m by A6,A7,FINSEQ_1:def 7
.=pe/.m by A7,A9,PARTFUN1:def 6;
then k <= m by A2,A7,A10,A9;
hence contradiction by A4,A5,A8,NAT_1:13;
end;
hence thesis by Th17;
end;
theorem Th19:
rng qe c= rng pe implies vertices qe c= vertices pe
proof
assume
A1: rng qe c= rng pe;
let x be object;
assume x in vertices qe;
then consider v being Vertex of G such that
A2: x=v and
A3: ex i st i in dom qe & v in vertices(qe/.i);
consider i such that
A4: i in dom qe and
A5: v in vertices(qe/.i) by A3;
qe.i in rng qe by A4,FUNCT_1:def 3;
then consider j being object such that
A6: j in dom pe and
A7: qe.i = pe.j by A1,FUNCT_1:def 3;
reconsider j as Element of NAT by A6;
qe/.i=qe.i by A4,PARTFUN1:def 6;
then pe/.j=qe/.i by A6,A7,PARTFUN1:def 6;
hence thesis by A2,A5,A6;
end;
theorem Th20:
rng qe c= rng pe & vertices(pe) \ X c= V implies vertices(qe) \ X c= V
proof
assume that
A1: rng qe c= rng pe and
A2: vertices(pe) \ X c= V;
vertices qe c= vertices pe by A1,Th19;
then vertices qe \ X c= vertices(pe) \ X by XBOOLE_1:35;
hence thesis by A2;
end;
theorem Th21:
vertices(pe^qe) \ X c= V implies vertices(pe) \ X c= V &
vertices(qe) \ X c= V
proof
A1: rng pe c= rng(pe^qe) & rng qe c= rng(pe^qe) by FINSEQ_1:29,30;
assume vertices(pe^qe) \ X c= V;
hence thesis by A1,Th20;
end;
reserve v,v1,v2,v3 for Element of G;
theorem Th22:
i in dom pe & (v=(the Source of G).(pe.i) or v=(the Target of G)
.(pe.i)) implies v in vertices pe
proof
set FS=the Source of G, FT=the Target of G;
assume that
A1: i in dom pe and
A2: v=FS.(pe.i) or v=FT.(pe.i);
v=FS.(pe/.i) or v=FT.(pe/.i) by A1,A2,PARTFUN1:def 6;
then v in vertices(pe/.i) by TARSKI:def 2;
hence thesis by A1;
end;
theorem Th23:
len pe = 1 implies vertices(pe) = vertices(pe/.1)
proof
set FS=the Source of G, FT=the Target of G;
assume
A1: len pe = 1;
then
A2: 1 in dom pe by FINSEQ_3:25;
now
let x be object;
hereby
assume x in vertices(pe);
then consider y being Vertex of G such that
A3: y=x and
A4: ex i st i in dom pe & y in vertices(pe/.i);
consider i such that
A5: i in dom pe and
A6: y in vertices(pe/.i) by A4;
1<=i & i <= len pe by A5,FINSEQ_3:25;
hence x in vertices(pe/.1) by A1,A3,A6,XXREAL_0:1;
end;
assume
A7: x in vertices(pe/.1);
then x=FS.(pe/.1) or x=FT.(pe/.1) by TARSKI:def 2;
then x=FS.(pe.1) or x=FT.(pe.1) by A1,FINSEQ_4:15;
then reconsider y=x as Vertex of G by A2,FINSEQ_2:11,FUNCT_2:5;
y in {v where v is Vertex of G : ex i st i in dom pe & v in vertices(
pe/.i)} by A2,A7;
hence x in vertices(pe);
end;
hence thesis by TARSKI:2;
end;
theorem Th24:
vertices pe c= vertices(pe^qe) & vertices qe c= vertices(pe^qe)
proof
rng pe c= rng (pe^qe) & rng qe c= rng (pe^qe) by FINSEQ_1:29,30;
hence thesis by Th19;
end;
reserve p,q for oriented Chain of G;
theorem Th25:
p = q^pe & len q >= 1 & len pe = 1 implies vertices(p) =
vertices(q) \/ {(the Target of G).(pe.1)}
proof
assume that
A1: p = q^pe and
A2: len q >= 1 and
A3: len pe = 1;
set FS=the Source of G, FT=the Target of G, V3={FT.(pe.1)};
A4: len p = len q + 1 by A1,A3,FINSEQ_1:22;
now
let x be object;
hereby
assume x in vertices p;
then consider y being Vertex of G such that
A5: y=x and
A6: ex i st i in dom p & y in vertices(p/.i);
consider i such that
A7: i in dom p and
A8: y in vertices(p/.i) by A6;
A9: 1<=i by A7,FINSEQ_3:25;
A10: i <= len p by A7,FINSEQ_3:25;
per cases;
suppose
A11: i <= len q;
then
A12: i in dom q by A9,FINSEQ_3:25;
p/.i=p.i by A9,A10,FINSEQ_4:15
.=q.i by A1,A9,A11,Lm1
.=q/.i by A9,A11,FINSEQ_4:15;
then y in {v where v is Vertex of G : ex j st j in dom q & v in
vertices(q/.j)} by A8,A12;
hence x in vertices(q) \/ V3 by A5,XBOOLE_0:def 3;
end;
suppose
A13: i > len q;
reconsider z=y as Vertex of G;
i >= len q+1 by A13,NAT_1:13;
then
A14: i = len q+1 by A4,A10,XXREAL_0:1;
A15: y=FS.(p/.i) or y=FT.(p/.i) by A8,TARSKI:def 2;
hereby
per cases by A9,A10,A15,FINSEQ_4:15;
suppose
A16: z=FS.(p.i);
len q < len p by A4,NAT_1:13;
then z=FT.(p.(len q)) by A2,A14,A16,GRAPH_1:def 15
.=FT.(q.(len q)) by A1,A2,Lm1
.=FT.(q/.(len q)) by A2,FINSEQ_4:15;
then
A17: z in vertices(q/.(len q)) by TARSKI:def 2;
len q in dom q by A2,FINSEQ_3:25;
then z in {v where v is Vertex of G : ex j st j in dom q & v in
vertices(q/.j)} by A17;
hence x in vertices(q) \/ V3 by A5,XBOOLE_0:def 3;
end;
suppose
z=FT.(p.i);
then z=FT.(pe.1) by A1,A3,A14,Lm2;
then z in V3 by TARSKI:def 1;
hence x in vertices(q) \/ V3 by A5,XBOOLE_0:def 3;
end;
end;
end;
end;
assume
A18: x in vertices(q) \/ V3;
per cases by A18,XBOOLE_0:def 3;
suppose
A19: x in vertices q;
vertices q c= vertices p by A1,Th24;
hence x in vertices p by A19;
end;
suppose
A20: x in V3;
1 in dom pe by A3,FINSEQ_3:25;
then reconsider y=FT.(pe.1) as Vertex of G by FINSEQ_2:11,FUNCT_2:5;
A21: 1 <= len p by A4,NAT_1:12;
then
A22: len p in dom p by FINSEQ_3:25;
y=FT.(p.(len p)) by A1,A3,A4,Lm2
.=FT.(p/.(len p)) by A21,FINSEQ_4:15;
then
A23: y in vertices(p/.(len p)) by TARSKI:def 2;
x = FT.(pe.1) by A20,TARSKI:def 1;
hence x in vertices(p) by A23,A22;
end;
end;
hence thesis by TARSKI:2;
end;
theorem Th26:
v <> (the Source of G).(p.1) & v in vertices(p) implies ex i st
1<=i & i <= len p & v = (the Target of G).(p.i)
proof
set FT=the Target of G, FS=the Source of G;
assume that
A1: v <> FS.(p.1) and
A2: v in vertices p;
consider u being Vertex of G such that
A3: v=u and
A4: ex i st i in dom p & u in vertices(p/.i) by A2;
consider i such that
A5: i in dom p and
A6: u in vertices(p/.i) by A4;
A7: u=FS.(p/.i) or u=FT.(p/.i) by A6,TARSKI:def 2;
A8: 1<=i by A5,FINSEQ_3:25;
A9: i <= len p by A5,FINSEQ_3:25;
per cases by A3,A7,A8,A9,FINSEQ_4:15;
suppose
A10: v=FT.(p.i);
take i;
thus thesis by A5,A10,FINSEQ_3:25;
end;
suppose
A11: v=FS.(p.i);
consider j being Nat such that
A12: i=1+j by A8,NAT_1:10;
reconsider j as Element of NAT by ORDINAL1:def 12;
A13: j < len p by A9,A12,NAT_1:13;
take j;
i > 1 by A1,A8,A11,XXREAL_0:1;
then j >= 1 by A12,NAT_1:13;
hence thesis by A11,A12,A13,GRAPH_1:def 15;
end;
end;
begin :: Directed Paths between Two vertices
definition
let G, p, v1,v2;
pred p is_orientedpath_of v1,v2 means
p <> {} & (the Source of G).(p. 1) = v1 & (the Target of G).(p.(len p))= v2;
end;
definition
let G, v1,v2, p,V;
pred p is_orientedpath_of v1,v2,V means
p is_orientedpath_of v1,v2 & vertices(p) \ {v2} c= V;
end;
definition
let G be Graph, v1,v2 be Element of G;
func OrientedPaths(v1,v2) -> Subset of ((the carrier' of G)*) equals
{p
where p is oriented Chain of G : p is_orientedpath_of v1,v2};
coherence
proof
set PT={p where p is oriented Chain of G: p is_orientedpath_of v1,v2};
PT c= ((the carrier' of G)*)
proof
let e be object;
assume e in PT;
then
ex p being oriented Chain of G st e=p & p is_orientedpath_of v1,v2;
hence thesis by FINSEQ_1:def 11;
end;
hence thesis;
end;
end;
theorem Th27:
p is_orientedpath_of v1,v2 implies v1 in vertices p & v2 in vertices p
proof
assume
A1: p is_orientedpath_of v1,v2;
then p <> {};
then 1 <= len p by FINSEQ_1:20;
then
A2: 1 in dom p & len p in dom p by FINSEQ_3:25;
(the Source of G).(p.1) = v1 & (the Target of G).(p.(len p))= v2 by A1;
hence thesis by A2,Th22;
end;
theorem
x in OrientedPaths(v1,v2) iff ex p st p=x & p is_orientedpath_of v1,v2;
theorem Th29:
p is_orientedpath_of v1,v2,V & v1 <> v2 implies v1 in V
proof
assume that
A1: p is_orientedpath_of v1,v2,V and
A2: v1 <> v2;
p is_orientedpath_of v1,v2 by A1;
then
A3: v1 in vertices p by Th27;
not v1 in {v2} by A2,TARSKI:def 1;
then
A4: v1 in vertices(p) \ {v2} by A3,XBOOLE_0:def 5;
vertices(p) \ {v2} c= V by A1;
hence thesis by A4;
end;
theorem Th30:
p is_orientedpath_of v1,v2,V & V c= U implies p is_orientedpath_of v1,v2,U
proof
assume that
A1: p is_orientedpath_of v1,v2,V and
A2: V c= U;
vertices(p) \ {v2} c= V by A1;
then
A3: vertices(p) \ {v2} c= U by A2;
p is_orientedpath_of v1,v2 by A1;
hence thesis by A3;
end;
theorem Th31:
len p >= 1 & p is_orientedpath_of v1,v2 & pe.1 orientedly_joins
v2,v3 & len pe=1 implies ex q st q=p^pe & q is_orientedpath_of v1,v3
proof
assume that
A1: len p >= 1 and
A2: p is_orientedpath_of v1,v2 and
A3: pe.1 orientedly_joins v2,v3 and
A4: len pe=1;
set FT=the Target of G, FS=the Source of G;
A5: pe is oriented Chain of G by A4,Th13;
FT.(p.len p)=v2 & FS.(pe.1)=v2 by A2,A3,GRAPH_4:def 1;
then reconsider q=p^pe as oriented Chain of G by A5,Th10;
A6: len q=len p +1 by A4,FINSEQ_1:22;
FT.(pe.1)=v3 by A3,GRAPH_4:def 1;
then
A7: FT.(q.(len q))=v3 by A4,A6,Lm2;
FS.(p.1)=v1 by A2;
then
A8: FS.(q.1)=v1 by A1,Lm1;
take q;
q <> {} by A6;
hence thesis by A8,A7;
end;
theorem Th32:
q=p^pe & len p >= 1 & len pe=1 & p is_orientedpath_of v1,v2,V &
pe.1 orientedly_joins v2,v3 implies q is_orientedpath_of v1,v3,V \/{v2}
proof
assume that
A1: q=p^pe and
A2: len p >= 1 & len pe=1 and
A3: p is_orientedpath_of v1,v2,V and
A4: pe.1 orientedly_joins v2,v3;
p is_orientedpath_of v1,v2 by A3;
then
A5: ex r being oriented Chain of G st r=p^pe & r is_orientedpath_of v1,v3 by A2
,A4,Th31;
set FT=the Target of G;
FT.(pe.1) = v3 by A4,GRAPH_4:def 1;
then vertices(q) \ {v3} = vertices(p) \/ {v3} \ {v3} by A1,A2,Th25
.=vertices(p) \ {v3} by XBOOLE_1:40;
then
A6: vertices(q) \ {v3} c= vertices(p) by XBOOLE_1:36;
vertices(p) \ {v2} c= V by A3;
then vertices p c= V \/ {v2} by XBOOLE_1:44;
then vertices(q) \ {v3} c= V \/ {v2} by A6;
hence thesis by A1,A5;
end;
begin :: Acyclic (or Simple) Paths
definition
let G be Graph, p be oriented Chain of G, v1,v2 be Element of
G;
pred p is_acyclicpath_of v1,v2 means
p is Simple & p is_orientedpath_of v1,v2;
end;
definition
let G be Graph, p be oriented Chain of G, v1,v2 be Element of
G,V be set;
pred p is_acyclicpath_of v1,v2,V means
p is Simple & p is_orientedpath_of v1,v2,V;
end;
definition
let G be Graph, v1,v2 be Element of G;
func AcyclicPaths(v1,v2) -> Subset of ((the carrier' of G)*) equals
{p where
p is Simple oriented Chain of G: p is_acyclicpath_of v1,v2};
coherence
proof
set PT={p where p is Simple oriented Chain of G: p is_acyclicpath_of v1,
v2};
PT c= (the carrier' of G)*
proof
let e be object;
assume e in PT;
then ex p being Simple oriented Chain of G st ( e=p)&( p
is_acyclicpath_of v1,v2);
hence thesis by FINSEQ_1:def 11;
end;
hence thesis;
end;
end;
definition
let G be Graph, v1,v2 be Element of G,V be set;
func AcyclicPaths(v1,v2,V) -> Subset of ((the carrier' of G)*) equals
{p
where p is Simple oriented Chain of G : p is_acyclicpath_of v1,v2,V};
coherence
proof
set PT={p where p is Simple oriented Chain of G: p is_acyclicpath_of v1,
v2,V};
PT c= (the carrier' of G)*
proof
let e be object;
assume e in PT;
then ex p being Simple oriented Chain of G st ( e=p)&( p
is_acyclicpath_of v1,v2,V);
hence thesis by FINSEQ_1:def 11;
end;
hence thesis;
end;
end;
definition
let G be Graph, p be oriented Chain of G;
func AcyclicPaths(p) -> Subset of ((the carrier' of G)*) equals
{q where q
is Simple oriented Chain of G : q <> {} & (the Source of G).(q.1) = (the
Source of G).(p.1) & (the Target of G).(q.(len q)) = (the Target of G).(p.(len
p)) & rng q c= rng p};
coherence
proof
set PT={q where q is Simple oriented Chain of G: q <> {} & (the Source
of G).(q.1) = (the Source of G).(p.1) & (the Target of G).(q.(len q)) = (the
Target of G).(p.(len p)) & rng q c= rng p};
PT c= (the carrier' of G)*
proof
let e be object;
assume e in PT;
then
ex q being Simple oriented Chain of G st ( e=q)&( q <> {} )&( (the
Source of G).(q.1) = (the Source of G).(p.1))&( (the Target of G).(q.( len q))
= (the Target of G).(p.(len p)))&( rng q c= rng p);
hence thesis by FINSEQ_1:def 11;
end;
hence thesis;
end;
end;
definition
let G be Graph;
func AcyclicPaths(G) -> Subset of (the carrier' of G)* equals
the set of all q where q is
Simple oriented Chain of G ;
coherence
proof
set PT=the set of all q where q is Simple oriented Chain of G;
PT c= (the carrier' of G)*
proof
let e be object;
assume e in PT;
then ex q being Simple oriented Chain of G st ( e=q);
hence thesis by FINSEQ_1:def 11;
end;
hence thesis;
end;
end;
theorem
p={} implies not p is_acyclicpath_of v1,v2
proof
assume p={};
then not p is_orientedpath_of v1,v2;
hence thesis;
end;
theorem
p is_acyclicpath_of v1,v2 implies p is_orientedpath_of v1,v2;
theorem
AcyclicPaths(v1,v2) c= OrientedPaths(v1,v2)
proof
let x be object;
assume x in AcyclicPaths(v1,v2);
then consider p being Simple oriented Chain of G such that
A1: x=p and
A2: p is_acyclicpath_of v1,v2;
p is_orientedpath_of v1,v2 by A2;
hence thesis by A1;
end;
theorem Th36:
AcyclicPaths(p) c= AcyclicPaths(G)
proof
let e be object;
assume e in AcyclicPaths(p);
then ex q being Simple oriented Chain of G st ( e=q)&( q <> {} )&( (the
Source of G).(q.1) = (the Source of G).(p.1))&( (the Target of G).(q.( len q))
= (the Target of G).(p.(len p)))&( rng q c= rng p);
hence thesis;
end;
theorem Th37:
AcyclicPaths(v1,v2) c= AcyclicPaths(G)
proof
let e be object;
assume e in AcyclicPaths(v1,v2);
then ex q being Simple oriented Chain of G st ( e=q)&( q is_acyclicpath_of
v1,v2);
hence thesis;
end;
theorem Th38:
p is_orientedpath_of v1,v2 implies AcyclicPaths(p) c= AcyclicPaths(v1,v2)
proof
assume p is_orientedpath_of v1,v2;
then
A1: (the Source of G).(p.1) = v1 & (the Target of G).(p.(len p)) = v2;
let x be object;
assume x in AcyclicPaths(p);
then consider q being Simple oriented Chain of G such that
A2: x=q and
A3: q <> {} & (the Source of G).(q.1) = (the Source of G).(p.1) & (the
Target of G).(q.(len q)) = (the Target of G).(p.(len p)) and
rng q c= rng p;
q is_orientedpath_of v1,v2 by A1,A3;
then q is_acyclicpath_of v1,v2;
hence thesis by A2;
end;
theorem Th39:
p is_orientedpath_of v1,v2,V implies AcyclicPaths(p) c= AcyclicPaths(v1,v2,V)
proof
assume
A1: p is_orientedpath_of v1,v2,V;
let x be object;
assume x in AcyclicPaths(p);
then consider q being Simple oriented Chain of G such that
A2: x=q and
A3: q <> {} & (the Source of G).(q.1) = (the Source of G).(p.1) & (the
Target of G).(q.(len q)) = (the Target of G).(p.(len p)) and
A4: rng q c= rng p;
vertices(p) \ {v2} c= V by A1;
then
A5: vertices(q) \ {v2} c= V by A4,Th20;
p is_orientedpath_of v1,v2 by A1;
then (the Source of G).(p.1) = v1 & (the Target of G).(p.(len p)) = v2;
then q is_orientedpath_of v1,v2 by A3;
then q is_orientedpath_of v1,v2,V by A5;
then q is_acyclicpath_of v1,v2,V;
hence thesis by A2;
end;
theorem
q in AcyclicPaths(p) implies len q <= len p
proof
A1: card (rng p) <= card (dom p) by CARD_2:47;
A2: card (dom p) = card (Seg len p) by FINSEQ_1:def 3
.= len p by FINSEQ_1:57;
assume q in AcyclicPaths(p);
then consider x being Simple oriented Chain of G such that
A3: q=x and
x <> {} and
(the Source of G).(x.1) = (the Source of G).(p.1) and
(the Target of G).(x.(len x)) = (the Target of G).(p.(len p)) and
A4: rng (x) c= rng p;
x is one-to-one by Th15;
then
A5: card(rng x)=len x by FINSEQ_4:62;
card (rng x) <= card (rng p) by A4,NAT_1:43;
hence thesis by A3,A5,A1,A2,XXREAL_0:2;
end;
theorem Th41:
p is_orientedpath_of v1,v2 implies AcyclicPaths(p) <> {} &
AcyclicPaths(v1,v2) <> {}
proof
defpred P[Nat] means for p,v1,v2 st $1+1 = len p & p
is_orientedpath_of v1,v2 holds AcyclicPaths(p) <> {};
set FS=the Source of G, FT=the Target of G;
A1: for k st P[k] holds P[k+1]
proof
let k;
assume
A2: P[k];
now
let p,v1,v2;
assume that
A3: k+1+1 = len p and
p is_orientedpath_of v1,v2;
consider p1,p2 being FinSequence such that
A4: len p1 = k+1 and
A5: len p2 = 1 and
A6: p = p1^p2 by A3,FINSEQ_2:22;
reconsider p1 as oriented Chain of G by A6,Th9;
A7: 1 <= len p1 by A4,NAT_1:11;
then 1 in dom p1 by FINSEQ_3:25;
then reconsider x=FS.(p1.1) as Element of G by FINSEQ_2:11,FUNCT_2:5;
A8: p1.1=p.1 by A6,A7,Lm1;
len p1 in dom p1 by A7,FINSEQ_3:25;
then reconsider y=FT.(p1.(len p1)) as Element of G by FINSEQ_2:11
,FUNCT_2:5;
p1 <> {} by A4;
then p1 is_orientedpath_of x,y;
then AcyclicPaths(p1) <> {} by A2,A4;
then consider r being object such that
A9: r in AcyclicPaths(p1) by XBOOLE_0:def 1;
A10: rng p1 c= rng p by A6,FINSEQ_1:29;
A11: rng p2 c= rng p by A6,FINSEQ_1:30;
A12: 1 in dom p2 by A5,FINSEQ_3:25;
then
A13: p.(len p1+1)=p2.1 by A6,FINSEQ_1:def 7;
consider q being Simple oriented Chain of G such that
r=q and
A14: q <> {} and
A15: FS.(q.1) = x and
A16: FT.(q.(len q)) = y and
A17: rng q c= rng p1 by A9;
len p1 < len p by A3,A4,NAT_1:13;
then
A18: FS.(p.(len p1+1)) = FT.(p.(len p1)) by A7,GRAPH_1:def 15
.=FT.(q.(len q)) by A6,A7,A16,Lm1;
then
A19: FS.(p2.1)=FT.(q.(len q)) by A6,A12,FINSEQ_1:def 7;
per cases;
suppose
ex k st 1<=k & k <= len q & FT.(q.k)=FT.(p2.1);
then consider k such that
A20: 1<=k and
A21: k <= len q and
A22: FT.(q.k)=FT.(p2.1);
consider i being Nat such that
A23: len q = k + i by A21,NAT_1:10;
consider q1,q2 being FinSequence such that
A24: len q1 = k and
len q2 = i and
A25: q = q1^q2 by A23,FINSEQ_2:22;
reconsider q1 as Simple oriented Chain of G by A25,Th12;
A26: q1 <> {} & FS.(q1.1) = FS.(p.1) by A8,A15,A20,A24,A25,Lm1;
rng q1 c= rng q by A25,FINSEQ_1:29;
then rng q1 c= rng p1 by A17;
then
A27: rng q1 c= rng p by A10;
FT.(q1.(len q1)) = FT.(p.(len p)) by A3,A4,A13,A20,A22,A24,A25,Lm1;
then
q1 in {w where w is Simple oriented Chain of G : w <> {} & FS.(
w.1) = FS.(p.1) & FT.(w.(len w)) = FT.(p.(len p)) & rng (w) c= rng p} by A27
,A26;
hence AcyclicPaths(p) <> {};
end;
suppose
A28: not ex k st 1<=k & k <= len q & FT.(q.k)=FT.(p2.1);
reconsider p2 as oriented Chain of G by A6,Th9;
hereby
per cases;
suppose
A29: FS.(q.1) <> FT.(q.(len q));
set qp=q^p2;
rng qp = rng q \/ rng p2 & rng q c= rng p by A17,A10,FINSEQ_1:31;
then
A30: rng qp c= rng p \/ rng p by A11,XBOOLE_1:13;
A31: len q >=1 by A14,FINSEQ_1:20;
then
A32: FS.(qp.1) = FS.(p.1) by A8,A15,Lm1;
len qp = len q + 1 by A5,FINSEQ_1:22;
then
A33: qp <> {} & FT.(qp.(len qp)) = FT.(p.(len p)) by A3,A4,A5,A13,Lm2;
qp is Simple oriented Chain of G by A5,A13,A18,A28,A29,A31,Th14;
then qp in {w where w is Simple oriented Chain of G : w <> {} &
FS.(w.1) = FS.(p.1) & FT.(w.(len w)) = FT.(p.(len p)) & rng (w) c= rng p} by
A32,A33,A30;
hence AcyclicPaths(p) <> {};
end;
suppose
A34: FS.(q.1) = FT.(q.(len q));
reconsider p2 as Simple oriented Chain of G by A5,Th13;
p2 <> {} & FT.(p2.(len p2)) =FT.(p.(len p)) by A3,A4,A5,A6,Lm2;
then p2 in {w where w is Simple oriented Chain of G : w <> {} &
FS.(w.1) = FS.(p.1) & FT.(w.(len w)) = FT.(p.(len p)) & rng (w) c= rng p} by A8
,A15,A11,A19,A34;
hence AcyclicPaths(p) <> {};
end;
end;
end;
end;
hence thesis;
end;
A35: P[0]
proof
let p,v1,v2;
assume that
A36: 0+1 = len p and
p is_orientedpath_of v1,v2;
reconsider r=p as Simple oriented Chain of G by A36,Th13;
r <> {} by A36;
then
r in {q where q is Simple oriented Chain of G : q <> {} & FS.(q.1) =
FS.(p.1) & FT.(q.(len q)) = FT.(p.(len p)) & rng q c= rng p};
hence thesis;
end;
A37: for k holds P[k] from NAT_1:sch 2(A35,A1);
assume
A38: p is_orientedpath_of v1,v2;
then p <> {};
then len p >=1 by FINSEQ_1:20;
then len p -' 1 + 1= len p by XREAL_1:235;
hence AcyclicPaths(p) <> {} by A38,A37;
then
A39: ex x being object st x in AcyclicPaths(p) by XBOOLE_0:def 1;
AcyclicPaths(p) c= AcyclicPaths(v1,v2) by A38,Th38;
hence thesis by A39;
end;
theorem Th42:
p is_orientedpath_of v1,v2,V implies AcyclicPaths(p) <> {} &
AcyclicPaths(v1,v2,V) <> {}
proof
defpred P[Nat] means for p,v1,v2 st len p <= $1+1 & p
is_orientedpath_of v1,v2,V holds AcyclicPaths(p) <> {};
set FS=the Source of G, FT=the Target of G;
assume
A1: p is_orientedpath_of v1,v2,V;
then p is_orientedpath_of v1,v2;
then p <> {};
then len p >=1 by FINSEQ_1:20;
then
A2: len p -' 1 + 1= len p by XREAL_1:235;
A3: for k st P[k] holds P[k+1]
proof
let k;
assume
A4: P[k];
now
let p,v1,v2;
assume that
A5: len p <= k+1+1 and
A6: p is_orientedpath_of v1,v2,V;
consider vs being FinSequence of the carrier of G such that
A7: vs is_oriented_vertex_seq_of p by GRAPH_4:9;
A8: vertices(p) \ {v2} c= V by A6;
A9: len vs = len p + 1 by A7,GRAPH_4:def 5;
then
A10: len vs >= 1 by NAT_1:12;
A11: p is_orientedpath_of v1,v2 by A6;
then
A12: p <> {};
then
A13: len p >= 1 by FINSEQ_1:20;
then p.(len p) orientedly_joins vs/.(len p), vs/.(len p+1) by A7,
GRAPH_4:def 5;
then FT.(p.(len p)) = vs/.(len p+1) by GRAPH_4:def 1;
then
A14: FT.(p.(len p)) = vs.(len vs) by A9,A10,FINSEQ_4:15;
per cases;
suppose
for n,m st 1<=n & n {} & FS.(w
.1) = FS.(p.1) & FT.(w.(len w)) = FT.(p.(len p)) & rng (w) c= rng p} by A12;
hence AcyclicPaths(p) <> {};
end;
suppose
not for n,m st 1<=n & n= 1 by A15,A16,XXREAL_0:2;
hereby
per cases;
suppose
A22: n=1;
then m < len vs by A17,A19,XXREAL_0:1;
then
A23: m <= len p by A9,NAT_1:13;
then consider j being Nat such that
A24: len p=m+j by NAT_1:10;
A25: p.1 orientedly_joins vs/.1, vs/.(1+1) by A7,A13,GRAPH_4:def 5;
A26: n <= len vs by A16,A17,XXREAL_0:2;
p.m orientedly_joins vs/.m, vs/.(m+1) by A7,A16,A22,A23,
GRAPH_4:def 5;
then
A27: FS.(p.m) = vs/.m by GRAPH_4:def 1
.=vs.m by A16,A17,A22,FINSEQ_4:15
.=vs/.n by A15,A18,A26,FINSEQ_4:15
.=FS.(p.1) by A22,A25,GRAPH_4:def 1
.=v1 by A11;
reconsider j as Element of NAT by ORDINAL1:def 12;
A28: len p=i+(1+j) by A20,A24;
then consider p1,p2 being FinSequence such that
A29: len p1 = i and
A30: len p2 = 1+j and
A31: p = p1^p2 by FINSEQ_2:22;
A32: 1 <= len p2 by A30,NAT_1:11;
then
A33: p2.1=p.m by A20,A29,A31,Lm2;
p2.(len p2)=p.(len p) by A28,A29,A30,A31,A32,Lm2;
then
A34: FT.(p2.(len p2))=v2 by A11;
reconsider p1,p2 as oriented Chain of G by A31,Th9;
i+1+-1 > 1+-1 by A16,A20,A22,XREAL_1:8;
then len p2 < len p by A28,A30,XREAL_1:29;
then len p2 < k+1+1 by A5,XXREAL_0:2;
then
A35: len p2 <= k+1 by NAT_1:13;
vertices(p1^p2) \ {v2} c= V by A6,A31;
then
A36: vertices(p2) \ {v2} c= V by Th21;
p2 <> {} by A30;
then p2 is_orientedpath_of v1,v2 by A27,A33,A34;
then p2 is_orientedpath_of v1,v2,V by A36;
then AcyclicPaths(p2) <> {} by A4,A35;
then consider r being object such that
A37: r in AcyclicPaths(p2) by XBOOLE_0:def 1;
consider q being Simple oriented Chain of G such that
r=q and
A38: q <> {} and
A39: FS.(q.1) = v1 & FT.(q.(len q)) = v2 and
A40: rng q c= rng p2 by A27,A33,A34,A37;
rng p2 c= rng p by A31,FINSEQ_1:30;
then
A41: rng q c= rng p by A40;
FS.(q.1) = FS.(p.1) & FT.(q.(len q)) = FT.(p.(len p)) by A11,A39;
then q in {w where w is Simple oriented Chain of G : w <> {} &
FS.(w.1) = FS.(p.1) & FT.(w.(len w)) = FT.(p.(len p)) & rng w c= rng p} by A38
,A41;
hence AcyclicPaths(p) <> {};
end;
suppose
A42: n<>1;
consider n1 being Nat such that
A43: n = 1+n1 by A15,NAT_1:10;
reconsider n1 as Element of NAT by ORDINAL1:def 12;
A44: n < len vs by A16,A17,XXREAL_0:2;
then
A45: n1 < len p by A9,A43,XREAL_1:7;
then consider j being Nat such that
A46: len p=n1+j by NAT_1:10;
reconsider j as Element of NAT by ORDINAL1:def 12;
consider p1,p2 being FinSequence such that
A47: len p1 = n1 and
len p2 = j and
A48: p = p1^p2 by A46,FINSEQ_2:22;
A49: n > 1 by A15,A42,XXREAL_0:1;
then
A50: n1 >= 1 by A43,NAT_1:13;
then
A51: p1.(len p1)=p.n1 by A47,A48,Lm1;
p1.1=p.1 by A47,A48,A50,Lm1;
then
A52: FS.(p1.1)=v1 by A11;
A53: rng p1 c= rng p by A48,FINSEQ_1:29;
reconsider p1,p2 as oriented Chain of G by A48,Th9;
p.n1 orientedly_joins vs/.n1,vs/.(n1+1)by A7,A45,A50,GRAPH_4:def 5;
then
A54: FT.(p.n1) = vs/.n by A43,GRAPH_4:def 1
.=vs.n by A15,A44,FINSEQ_4:15;
hereby
per cases;
suppose
m=len vs;
then
A55: FT.(p1.(len p1))=v2 by A11,A14,A18,A54,A51;
vertices(p1^p2) \ {v2} c= V by A6,A48;
then
A56: vertices(p1) \ {v2} c= V by Th21;
n1 < k+1+1 by A5,A45,XXREAL_0:2;
then
A57: len p1 <= k+1 by A47,NAT_1:13;
p1 <> {} by A49,A43,A47;
then p1 is_orientedpath_of v1,v2 by A52,A55;
then p1 is_orientedpath_of v1,v2,V by A56;
then AcyclicPaths(p1) <> {} by A4,A57;
then consider r being object such that
A58: r in AcyclicPaths(p1) by XBOOLE_0:def 1;
consider q being Simple oriented Chain of G such that
r=q and
A59: q <> {} and
A60: FS.(q.1) = v1 and
A61: FT.(q.(len q))= v2 and
A62: rng q c= rng p1 by A52,A55,A58;
A63: FT.(q.(len q)) = FT.(p.(len p)) by A11,A61;
rng q c= rng p & FS.(q.1) = FS.(p.1) by A11,A53,A60,A62;
then q in {w where w is Simple oriented Chain of G: w <> {}
& FS.(w.1) = FS.(p.1) & FT.(w.(len w))= FT.(p.(len p)) & rng w c= rng p} by A59
,A63;
hence AcyclicPaths(p) <> {};
end;
suppose
m<>len vs;
then m < len vs by A17,XXREAL_0:1;
then
A64: m <= len p by A9,NAT_1:13;
then consider h being Nat such that
A65: len p=m+h by NAT_1:10;
p.m orientedly_joins vs/.m,vs/.(m+1)by A7,A21,A64,GRAPH_4:def 5
;
then
A66: FS.(p.m) = vs/.m by GRAPH_4:def 1
.=FT.(p1.len p1) by A17,A18,A21,A54,A51,FINSEQ_4:15;
reconsider h as Element of NAT by ORDINAL1:def 12;
A67: len p=i+(1+h) by A20,A65;
then consider q1,q2 being FinSequence such that
A68: len q1 = i and
A69: len q2 = 1+h and
A70: p = q1^q2 by FINSEQ_2:22;
reconsider q2 as oriented Chain of G by A70,Th9;
A71: 1 <= len q2 by A69,NAT_1:12;
then q2.1=p.m by A20,A68,A70,Lm2;
then reconsider pq=p1^q2 as oriented Chain of G by A66,Th10;
pq.(len pq)=pq.(len p1+len q2) by FINSEQ_1:22
.=q2.(len q2) by A71,Lm2
.=p.(len p) by A67,A68,A69,A70,A71,Lm2;
then
A72: FT.(pq.(len pq))=v2 by A11;
A73: rng pq = rng p1 \/ rng q2 by FINSEQ_1:31;
rng q2 c= rng p by A70,FINSEQ_1:30;
then
A74: rng pq c= rng p by A53,A73,XBOOLE_1:8;
then
A75: vertices(pq) \ {v2} c= V by A8,Th20;
A76: len pq = n1 + (1+h) by A47,A69,FINSEQ_1:22;
m+h > n+h by A16,XREAL_1:8;
then len pq < k+1+1 by A5,A43,A65,A76,XXREAL_0:2;
then
A77: len pq <= k+1 by NAT_1:13;
A78: FS.(pq.1)=v1 by A47,A50,A52,Lm1;
pq <> {} by A76;
then pq is_orientedpath_of v1,v2 by A78,A72;
then pq is_orientedpath_of v1,v2,V by A75;
then AcyclicPaths(pq) <> {} by A4,A77;
then consider r being object such that
A79: r in AcyclicPaths(pq) by XBOOLE_0:def 1;
consider q being Simple oriented Chain of G such that
r=q and
A80: q <> {} and
A81: FS.(q.1) = v1 & FT.(q.(len q))= v2 and
A82: rng q c= rng pq by A78,A72,A79;
A83: rng q c= rng p by A74,A82;
FS.(q.1) = FS.(p.1) & FT.(q.(len q)) = FT.(p.(len p ) )
by A11,A81;
then q in {w where w is Simple oriented Chain of G : w <>
{} & FS.(w.1) = FS.(p.1) & FT.(w.(len w))= FT.(p.(len p)) & rng w c= rng p} by
A80,A83;
hence AcyclicPaths(p) <> {};
end;
end;
end;
end;
end;
end;
hence thesis;
end;
A84: P[0]
proof
let p,v1,v2;
assume that
A85: len p <= 0+1 and
A86: p is_orientedpath_of v1,v2,V;
p is_orientedpath_of v1,v2 by A86;
then
A87: p <> {};
then len p >= 1 by FINSEQ_1:20;
then reconsider r=p as Simple oriented Chain of G by A85,Th13,XXREAL_0:1;
r in {q where q is Simple oriented Chain of G : q <> {} & FS.(q.1) =
FS.(p.1) & FT.(q.(len q)) = FT.(p.(len p)) & rng q c= rng p} by A87;
hence thesis;
end;
for k holds P[k] from NAT_1:sch 2(A84,A3);
hence AcyclicPaths(p) <> {} by A1,A2;
then
A88: ex x being object st x in AcyclicPaths(p) by XBOOLE_0:def 1;
AcyclicPaths(p) c= AcyclicPaths(v1,v2,V) by A1,Th39;
hence thesis by A88;
end;
theorem Th43:
AcyclicPaths(v1,v2,V) c= AcyclicPaths(G)
proof
let e be object;
assume e in AcyclicPaths(v1,v2,V);
then ex q being Simple oriented Chain of G st ( e=q)&( q is_acyclicpath_of
v1,v2,V);
hence thesis;
end;
begin :: Weight Graphs and Their Basic Properties
notation
synonym Real>=0 for REAL+;
end;
definition
redefine func Real>=0 -> Subset of REAL equals
{ r where r is Real : r >=0 };
compatibility by REAL_1:1;
coherence by ARYTM_0:1;
end;
registration
cluster Real>=0 -> non empty;
coherence;
end;
definition
let G be Graph, W be Function;
pred W is_weight>=0of G means
W is Function of the carrier' of G, Real>=0;
end;
definition
let G be Graph, W be Function;
pred W is_weight_of G means
W is Function of (the carrier' of G), REAL;
end;
definition
let G be Graph, p be FinSequence of the carrier' of G, W be Function;
assume
A1: W is_weight_of G;
func RealSequence(p,W) -> FinSequence of REAL means
:Def15:
dom p = dom it & for i be Nat st i in dom p holds it.i=W.(p.i);
existence
proof
deffunc F(object) = W.(p.$1);
consider f such that
A2: dom f = dom p &
for x being object st x in dom p holds f.x = F(x) from FUNCT_1:
sch 3;
A3: now
let i be Nat;
A4: W is Function of (the carrier' of G), REAL by A1;
assume
A5: i in dom f;
then f.i=W.(p.i) by A2;
hence f.i in REAL by A2,A5,A4,FINSEQ_2:11,FUNCT_2:5;
end;
dom f = Seg len p by A2,FINSEQ_1:def 3;
then f is FinSequence by FINSEQ_1:def 2;
then reconsider g=f as FinSequence of REAL by A3,FINSEQ_2:12;
take g;
thus dom p = dom g by A2;
let i;
assume i in dom p;
hence thesis by A2;
end;
uniqueness
proof
let f1, f2 be FinSequence of REAL;
assume that
A6: dom p = dom f1 and
A7: for i be Nat st i in dom p holds f1.i=W.(p.i);
assume that
A8: dom p = dom f2 and
A9: for i be Nat st i in dom p holds f2.i=W.(p.i);
now
let i be Nat;
assume
A10: i in dom f1;
hence f1.i=W.(p.i) by A6,A7
.= f2.i by A6,A9,A10;
end;
hence thesis by A6,A8;
end;
end;
definition
let G be Graph, p be FinSequence of the carrier' of G,W be Function;
func cost(p,W) -> Real equals
Sum RealSequence(p,W);
coherence;
end;
theorem Th44:
W is_weight>=0of G implies W is_weight_of G
by FUNCT_2:7;
theorem Th45:
for f being FinSequence of REAL st W is_weight>=0of G & f =
RealSequence(pe,W) holds for i st i in dom f holds f.i >= 0
proof
let f be FinSequence of REAL;
assume that
A1: W is_weight>=0of G and
A2: f = RealSequence(pe,W);
A3: W is Function of the carrier' of G, Real>=0 by A1;
let i;
assume
A4: i in dom f;
A5: W is_weight_of G by A1,Th44;
then
A6: dom pe = dom f by A2,Def15;
then f.i=W.(pe.i) by A2,A5,A4,Def15;
then f.i in Real>=0 by A6,A3,A4,FINSEQ_2:11,FUNCT_2:5;
then ex r being Real st f.i=r & r >= 0;
hence thesis;
end;
theorem Th46:
rng qe c= rng pe & W is_weight_of G & i in dom qe implies ex j
st j in dom pe & RealSequence(pe,W).j = RealSequence(qe,W).i
proof
assume that
A1: rng qe c= rng pe and
A2: W is_weight_of G and
A3: i in dom qe;
set g=RealSequence(qe,W);
consider y being object such that
A4: y in dom pe and
A5: qe.i = pe.y by A1,A3,FUNCT_1:114;
reconsider j=y as Element of NAT by A4;
take j;
thus j in dom pe by A4;
g.i=W.(qe.i) by A2,A3,Def15;
hence thesis by A2,A4,A5,Def15;
end;
Lm3: for f being FinSequence of REAL holds (for y being Real st y in rng f
holds y >= 0) iff for i be Nat st i in dom f holds f.i >= 0
proof
let f be FinSequence of REAL;
hereby
assume
A1: for y being Real st y in rng f holds y >= 0;
hereby
let i be Nat;
assume i in dom f;
then f.i in rng f by FUNCT_1:def 3;
hence f.i >= 0 by A1;
end;
end;
assume
A2: for i be Nat st i in dom f holds f.i >= 0;
hereby
let x be Real;
assume x in rng f;
then consider y being object such that
A3: y in dom f and
A4: x = f.y by FUNCT_1:def 3;
thus x >= 0 by A2,A3,A4;
end;
end;
Lm4: for p,q,r being FinSequence of REAL st r=p^q & (for x being Real st x in
rng r holds x >= 0) holds (for i st i in dom p holds p.i >= 0) & for i st i in
dom q holds q.i >= 0
proof
let p, q,r be FinSequence of REAL;
assume that
A1: r=p^q and
A2: for x being Real st x in rng r holds x >= 0;
rng p c= rng r by A1,FINSEQ_1:29;
then for y be Real st y in rng p holds y >= 0 by A2;
hence for i st i in dom p holds p.i >= 0 by Lm3;
rng q c= rng r by A1,FINSEQ_1:30;
then for y be Real st y in rng q holds y >= 0 by A2;
hence thesis by Lm3;
end;
theorem
len qe = 1 & rng qe c= rng pe & W is_weight>=0of G implies cost(qe,W)
<= cost(pe,W)
proof
assume that
A1: len qe = 1 and
A2: rng qe c= rng pe and
A3: W is_weight>=0of G;
set f = RealSequence(pe,W), g = RealSequence(qe,W);
1 in dom qe by A1,FINSEQ_3:25;
then consider j such that
A4: j in dom pe and
A5: f.j = g.1 by A2,A3,Th44,Th46;
A6: W is_weight_of G by A3,Th44;
then dom pe = dom f by Def15;
then
A7: len pe = len f by FINSEQ_3:29;
reconsider g1 = g.1 as Element of REAL by XREAL_0:def 1;
dom g = dom qe by A6,Def15;
then len g = len qe by FINSEQ_3:29;
then g = <*g1*> by A1,FINSEQ_1:40;
then Sum g = g1 by RVSUM_1:73;
then
A8: cost(qe,W) = g1;
j <= len pe by A4,FINSEQ_3:25;
then consider m being Nat such that
A9: len f = j+m by A7,NAT_1:10;
reconsider m as Element of NAT by ORDINAL1:def 12;
consider f1,f2 being FinSequence of REAL such that
A10: len f1 = j and
len f2 = m and
A11: f = f1^f2 by A9,FINSEQ_2:23;
A12: 1 <= j by A4,FINSEQ_3:25;
then consider h being FinSequence of REAL,d being Element of REAL such that
A13: f1= h^<*d*> by A10,FINSEQ_2:19;
j in dom f1 by A12,A10,FINSEQ_3:25;
then
A14: f1.j = g.1 by A5,A11,FINSEQ_1:def 7;
j = len h +1 by A10,A13,FINSEQ_2:16;
then
A15: d=g.1 by A13,A14,FINSEQ_1:42;
for i be Nat st i in dom f holds f.i >= 0 by A3,Th45;
then
A16: for y being Real st y in rng f holds y >= 0 by Lm3;
then for i be Nat st i in dom f2 holds f2.i >= 0 by A11,Lm4;
then
A17: Sum f2 >= 0 by RVSUM_1:84;
for i be Nat st i in dom f1 holds f1.i >= 0 by A11,A16,Lm4;
then for y being Real st y in rng f1 holds y >= 0 by Lm3;
then for i be Nat st i in dom h holds h.i >= 0 by A13,Lm4;
then
A18: Sum h >= 0 by RVSUM_1:84;
reconsider d as Element of REAL;
reconsider dd = <*d*> as FinSequence of REAL;
Sum f1 = Sum h + Sum dd by A13,RVSUM_1:75
.= Sum h + d by RVSUM_1:73
.=Sum h + g.1 by A15;
then
A19: Sum f1 >= 0 qua Nat+ g.1 by A18,XREAL_1:7;
Sum f= Sum f1 + Sum f2 by A11,RVSUM_1:75;
hence thesis by A8,A17,A19,XREAL_1:7;
end;
theorem Th48:
W is_weight>=0of G implies cost(pe,W) >= 0
proof
set f = RealSequence(pe,W);
assume W is_weight>=0of G;
then for i be Nat st i in dom f holds f.i >= 0 by Th45;
hence thesis by RVSUM_1:84;
end;
theorem Th49:
pe = {} & W is_weight_of G implies cost(pe,W) = 0
proof
assume that
A1: pe = {} and
A2: W is_weight_of G;
set f=RealSequence(pe,W);
dom f = dom pe by A2,Def15;
then len f = len pe by FINSEQ_3:29
.= 0 by A1;
then f = <*>REAL;
hence thesis by RVSUM_1:72;
end;
theorem Th50:
for D being non empty finite Subset of (the carrier' of G)* st D
= AcyclicPaths(v1,v2) ex pe st pe in D & for qe st qe in D holds cost(pe,W) <=
cost(qe,W)
proof
let D be non empty finite Subset of (the carrier' of G)*;
deffunc F(Element of D) = cost($1,W);
consider x being Element of D such that
A1: for y being Element of D holds F(x) <= F(y) from PRE_CIRC:sch 5;
assume D = AcyclicPaths(v1,v2);
then x in AcyclicPaths(v1,v2);
then consider p being Simple oriented Chain of G such that
A2: x=p and
p is_acyclicpath_of v1,v2;
take p;
thus p in D by A2;
let pe be FinSequence of the carrier' of G;
assume pe in D;
then reconsider y=pe as Element of D;
F(x) <= F(y) by A1;
hence thesis by A2;
end;
theorem Th51:
for D being non empty finite Subset of (the carrier' of G)* st D
= AcyclicPaths(v1,v2,V) holds ex pe st pe in D & for qe st qe in D holds cost(
pe,W) <= cost(qe,W)
proof
let D be non empty finite Subset of ((the carrier' of G)*);
deffunc F(Element of D) = cost($1,W);
consider x being Element of D such that
A1: for y being Element of D holds F(x) <= F(y) from PRE_CIRC:sch 5;
assume D = AcyclicPaths(v1,v2,V);
then x in AcyclicPaths(v1,v2,V);
then consider p being Simple oriented Chain of G such that
A2: x=p and
p is_acyclicpath_of v1,v2,V;
take p;
thus p in D by A2;
let pe;
assume pe in D;
then reconsider y=pe as Element of D;
F(x) <= F(y) by A1;
hence thesis by A2;
end;
theorem Th52:
W is_weight_of G implies cost(pe^qe,W) = cost(pe,W) + cost(qe,W)
proof
set r=pe^qe, f=RealSequence(pe^qe,W), g=RealSequence(pe,W), h=RealSequence(
qe,W);
assume
A1: W is_weight_of G;
then
A2: dom pe = dom g by Def15;
then
A3: len pe = len g by FINSEQ_3:29;
A4: dom qe = dom h by A1,Def15;
then
A5: len qe = len h by FINSEQ_3:29;
A6: dom r = dom f by A1,Def15;
then len f = len r by FINSEQ_3:29;
then
A7: len f =len g + len h by A3,A5,FINSEQ_1:22;
A8: now
let i be Nat;
assume
A9: i in dom h;
then 1 <= i by FINSEQ_3:25;
then
A10: 1 <= len g +i by NAT_1:12;
i <= len h by A9,FINSEQ_3:25;
then len g +i <= len f by A7,XREAL_1:7;
then
A11: len g + i in dom f by A10,FINSEQ_3:25;
h.i=W.(qe.i) & r.(len g +i) = qe.i by A1,A4,A3,A9,Def15,FINSEQ_1:def 7;
hence f.(len g +i) = h.i by A1,A6,A11,Def15;
end;
now
let i be Nat;
assume
A12: i in dom g;
then i <= len g by FINSEQ_3:25;
then
A13: i <= len f by A7,NAT_1:12;
1 <= i by A12,FINSEQ_3:25;
then
A14: i in dom f by A13,FINSEQ_3:25;
g.i=W.(pe.i) & r.i = pe.i by A1,A2,A12,Def15,FINSEQ_1:def 7;
hence f.i = g.i by A1,A6,A14,Def15;
end;
then f=g^h by A7,A8,FINSEQ_3:38;
hence thesis by RVSUM_1:75;
end;
theorem Th53:
qe is one-to-one & rng qe c= rng pe & W is_weight>=0of G implies
cost(qe,W) <= cost(pe,W)
proof
set D=the carrier' of G;
assume that
A1: qe is one-to-one & rng qe c= rng pe and
A2: W is_weight>=0of G;
defpred P[Nat] means for p, q being FinSequence of D st q is one-to-one &
rng q c= rng p & len q = $1 holds cost(q,W) <= cost(p,W);
A3: for k being Nat st P[k] holds P[k + 1]
proof
let k be Nat;
assume
A4: P[k];
now
let p, q be FinSequence of D;
assume that
A5: q is one-to-one and
A6: rng q c= rng p and
A7: len q = k+1;
consider q1 being FinSequence, x being object such that
A8: q = q1^<*x*> by A7,FINSEQ_2:18;
A9: k+1= len q1 +1 by A7,A8,FINSEQ_2:16;
consider p1,p2 being FinSequence such that
A10: p=p1^<*x*>^p2 and
A11: rng q1 c= rng (p1^p2) by A5,A6,A8,Th7;
reconsider q1 as FinSequence of D by A8,FINSEQ_1:36;
A12: p1^<*x*> is FinSequence of D by A10,FINSEQ_1:36;
then reconsider y=<*x*> as FinSequence of D by FINSEQ_1:36;
reconsider p2 as FinSequence of D by A10,FINSEQ_1:36;
reconsider p1 as FinSequence of D by A12,FINSEQ_1:36;
q1 is one-to-one by A5,A8,FINSEQ_3:91;
then
A13: cost(q1,W) <= cost(p1^p2,W) by A4,A11,A9;
A14: cost(q,W)=cost(q1,W)+cost(y,W) by A2,A8,Th44,Th52;
cost(p,W)=cost(p1^y,W)+cost(p2,W) by A2,A10,Th44,Th52
.=cost(p1,W)+cost(y,W)+cost(p2,W) by A2,Th44,Th52
.=cost(p1,W)+cost(p2,W)+cost(y,W)
.=cost(p1^p2,W)+cost(y,W) by A2,Th44,Th52;
hence cost(q,W) <= cost(p,W) by A14,A13,XREAL_1:7;
end;
hence thesis;
end;
A15: P[0]
proof
let p, q be FinSequence of D;
assume that
q is one-to-one and
rng q c= rng p and
A16: len q = 0;
q = {} by A16;
then cost(q,W) = 0 by A2,Th44,Th49;
hence thesis by A2,Th48;
end;
for k being Nat holds P[k] from NAT_1:sch 2(A15,A3);
then P[len qe];
hence thesis by A1;
end;
theorem Th54:
pe in AcyclicPaths(p) & W is_weight>=0of G implies cost(pe,W) <= cost(p,W)
proof
assume that
A1: pe in AcyclicPaths(p) and
A2: W is_weight>=0of G;
A3: ex r being Simple oriented Chain of G st ( r=pe)&( r <> {})&( (the
Source of G).(r.1) = (the Source of G).(p.1))&( (the Target of G).(r .(len r))
= (the Target of G).(p.(len p)))&( rng r c= rng p) by A1;
then pe is one-to-one by Th15;
hence thesis by A2,A3,Th53;
end;
begin :: Shortest Paths and Their Basic Properties
definition
let G be Graph, v1,v2 be Vertex of G, p be oriented Chain of G, W be
Function;
pred p is_shortestpath_of v1,v2,W means
p is_orientedpath_of v1,v2 &
for q being oriented Chain of G st q is_orientedpath_of v1,v2 holds cost(p,W)
<= cost(q,W);
end;
definition
let G be Graph, v1,v2 be Vertex of G, p be oriented Chain of G, V be set, W
be Function;
pred p is_shortestpath_of v1,v2,V,W means
p is_orientedpath_of v1,v2
,V & for q being oriented Chain of G st q is_orientedpath_of v1,v2,V holds cost
(p,W) <= cost(q,W);
end;
begin :: Basic Properties of a Graph with Finite carrier
reserve G for finite Graph,
ps for Simple oriented Chain of G,
P,Q for oriented Chain of G,
v1,v2,v3 for Element of G,
pe,qe for FinSequence of the carrier' of G;
theorem
len ps <= VerticesCount G
proof
set VV=the carrier of G;
consider vs being FinSequence of VV such that
A1: vs is_oriented_vertex_seq_of ps and
A2: for n,m st 1<=n & n {};
then consider q being FinSequence, x be object such that
A4: vs = q^<*x*> by FINSEQ_1:46;
A5: len ps + 1 = len q + len <*x*> by A3,A4,FINSEQ_1:22
.=len q + 1 by FINSEQ_1:39;
now
let n,m;
assume that
A6: 1<=n and
A7: n finite;
coherence by Lm5;
end;
Lm6: AcyclicPaths(P) is finite
proof
AcyclicPaths(P) c= AcyclicPaths(G) by Th36;
hence thesis;
end;
Lm7: AcyclicPaths(v1,v2) is finite
proof
AcyclicPaths(v1,v2) c= AcyclicPaths(G) by Th37;
hence thesis;
end;
Lm8: AcyclicPaths(v1,v2,V) is finite
proof
AcyclicPaths(v1,v2,V) c= AcyclicPaths(G) by Th43;
hence thesis;
end;
registration
let G, P;
cluster AcyclicPaths P -> finite;
coherence by Lm6;
end;
registration
let G, v1, v2;
cluster AcyclicPaths(v1,v2) -> finite;
coherence by Lm7;
end;
registration
let G, v1, v2, V;
cluster AcyclicPaths(v1,v2,V) -> finite;
coherence by Lm8;
end;
theorem
AcyclicPaths(v1,v2) <> {} implies ex pe st pe in AcyclicPaths(v1,v2) &
for qe st qe in AcyclicPaths(v1,v2) holds cost(pe,W) <= cost(qe,W) by Th50;
theorem
AcyclicPaths(v1,v2,V) <> {} implies ex pe st pe in AcyclicPaths(v1,v2,
V) & for qe st qe in AcyclicPaths(v1,v2,V) holds cost(pe,W) <= cost(qe,W) by
Th51;
theorem
P is_orientedpath_of v1,v2 & W is_weight>=0of G implies ex q being
Simple oriented Chain of G st q is_shortestpath_of v1,v2,W
proof
assume that
A1: P is_orientedpath_of v1,v2 and
A2: W is_weight>=0of G;
AcyclicPaths(v1,v2) <> {} by A1,Th41;
then consider r being FinSequence of the carrier' of G such that
A3: r in AcyclicPaths(v1,v2) and
A4: for s being FinSequence of the carrier' of G st s in AcyclicPaths(v1
,v2) holds cost(r,W) <= cost(s,W) by Th50;
consider t being Simple oriented Chain of G such that
A5: r=t and
A6: t is_acyclicpath_of v1,v2 by A3;
take t;
thus t is_orientedpath_of v1,v2 by A6;
hereby
let s be oriented Chain of G;
assume
A7: s is_orientedpath_of v1,v2;
then consider x being Element of ((the carrier' of G)*) such that
A8: x in AcyclicPaths(s) by Th41,SUBSET_1:4;
AcyclicPaths(s) c= AcyclicPaths(v1,v2) by A7,Th38;
then
A9: cost(r,W) <= cost(x,W) by A4,A8;
cost(x,W) <= cost(s,W) by A2,A8,Th54;
hence cost(t,W) <= cost(s,W) by A5,A9,XXREAL_0:2;
end;
end;
theorem Th60:
P is_orientedpath_of v1,v2,V & W is_weight>=0of G implies ex q
being Simple oriented Chain of G st q is_shortestpath_of v1,v2,V,W
proof
assume that
A1: P is_orientedpath_of v1,v2,V and
A2: W is_weight>=0of G;
AcyclicPaths(v1,v2,V) <> {} by A1,Th42;
then consider r being FinSequence of the carrier' of G such that
A3: r in AcyclicPaths(v1,v2,V) and
A4: for s being FinSequence of the carrier' of G st s in AcyclicPaths(v1
,v2,V) holds cost(r,W) <= cost(s,W) by Th51;
consider t being Simple oriented Chain of G such that
A5: r=t and
A6: t is_acyclicpath_of v1,v2,V by A3;
take t;
thus t is_orientedpath_of v1,v2,V by A6;
hereby
let s be oriented Chain of G;
assume
A7: s is_orientedpath_of v1,v2,V;
then consider x being Element of ((the carrier' of G)*) such that
A8: x in AcyclicPaths(s) by Th42,SUBSET_1:4;
AcyclicPaths(s) c= AcyclicPaths(v1,v2,V) by A7,Th39;
then
A9: cost(r,W) <= cost(x,W) by A4,A8;
cost(x,W) <= cost(s,W) by A2,A8,Th54;
hence cost(t,W) <= cost(s,W) by A5,A9,XXREAL_0:2;
end;
end;
begin :: Three Basic Theorems for Dijkstra's Shortest Path Algorithm
theorem Th61:
W is_weight>=0of G & P is_shortestpath_of v1,v2,V,W & v1 <> v2 &
(for Q, v3 st not v3 in V & Q is_shortestpath_of v1,v3,V,W holds cost(P,W) <=
cost(Q,W)) implies P is_shortestpath_of v1,v2,W
proof
assume that
A1: W is_weight>=0of G and
A2: P is_shortestpath_of v1,v2,V,W and
A3: v1 <> v2 and
A4: for Q, v3 st not v3 in V & Q is_shortestpath_of v1,v3,V,W holds cost
(P,W) <= cost(Q,W);
A5: P is_orientedpath_of v1,v2,V by A2;
then
A6: v1 in V by A3,Th29;
A7: now
let r be oriented Chain of G;
assume
A8: r is_orientedpath_of v1,v2;
per cases;
suppose
A9: not vertices r c= V;
set FT=the Target of G, FS=the Source of G;
consider i being Element of NAT, s,t being FinSequence of the carrier'
of G such that
A10: i+1 <= len r and
A11: not vertices(r/.(i+1)) c= V and
A12: len s=i and
A13: r=s^t and
A14: vertices(s) c= V by A9,Th18;
i+1 <= len s+len t by A10,A13,FINSEQ_1:22;
then
A15: 1 <= len t by A12,XREAL_1:6;
then consider j being Nat such that
A16: len t = 1+j by NAT_1:10;
reconsider s,t as oriented Chain of G by A13,Th9;
reconsider j as Element of NAT by ORDINAL1:def 12;
consider t1,t2 being FinSequence such that
A17: len t1 = 1 and
len t2 = j and
A18: t = t1^t2 by A16,FINSEQ_2:22;
reconsider t1,t2 as oriented Chain of G by A18,Th9;
A19: r.(i+1)=t.1 by A12,A13,A15,Lm2
.=t1.1 by A17,A18,Lm1;
A20: cost(t2,W) >= 0 by A1,Th48;
A21: r=s^t1^t2 by A13,A18,FINSEQ_1:32;
then cost(r,W)=cost(s^t1,W)+cost(t2,W) by A1,Th44,Th52;
then
A22: cost(r,W) >= cost(s^t1,W)+(0 qua Nat) by A20,XREAL_1:7;
set e = r/.(i+1);
A23: e = r.(i+1) by A10,FINSEQ_4:15,NAT_1:11;
consider x being object such that
A24: x in vertices e and
A25: not x in V by A11;
A26: x=FS.e or x=FT.e by A24,TARSKI:def 2;
1 in dom t1 by A17,FINSEQ_3:25;
then reconsider v3=x as Vertex of G by A23,A19,A26,FINSEQ_2:11,FUNCT_2:5;
A27: v1=FS.(r.1) by A8;
hereby
per cases;
suppose
A28: i=0;
then
A29: FS.(t1.1) = v1 by A8,A19;
A30: vertices(t1) \ {v3} c= V
proof
let x be object;
assume
A31: x in vertices(t1) \ {v3};
then
A32: not x in {v3} by XBOOLE_0:def 5;
x in vertices(t1) by A31,XBOOLE_0:def 5;
then x in vertices(t1/.1) by A17,Th23;
then
A33: x=FS.(t1/.1) or x=FT.(t1/.1) by TARSKI:def 2;
FT.(t1/.1) = v3 by A3,A5,A17,A23,A19,A25,A26,A27,A28,Th29,
FINSEQ_4:15;
hence thesis by A6,A17,A29,A32,A33,FINSEQ_4:15,TARSKI:def 1;
end;
v1=FS.e by A8,A23,A28;
then t1 is_orientedpath_of v1,v3 by A3,A5,A17,A23,A19,A25,A26,Th29;
then
A34: t1 is_orientedpath_of v1,v3,V by A30;
then consider q being Simple oriented Chain of G such that
A35: q is_shortestpath_of v1,v3,V,W by A1,Th60;
s = {} by A12,A28;
then
A36: s^t1 = t1 by FINSEQ_1:34;
A37: cost(q,W) <= cost(t1,W) by A34,A35;
cost(P,W) <= cost(q,W) by A4,A25,A35;
then cost(P,W) <= cost(t1,W) by A37,XXREAL_0:2;
hence cost(P,W) <= cost(r,W) by A22,A36,XXREAL_0:2;
end;
suppose
A38: i<>0;
reconsider u=s^t1 as oriented Chain of G by A21,Th9;
A39: i < len r by A10,NAT_1:13;
i+1 > 0+1 by A38,XREAL_1:8;
then
A40: i >= 1 by NAT_1:13;
then
A41: i in dom s by A12,FINSEQ_3:25;
A42: now
assume FS.(r.(i+1))=v3;
then
A43: v3=FT.(r.i) by A40,A39,GRAPH_1:def 15;
r.i=s.i by A12,A13,A40,Lm1;
then v3 in vertices s by A41,A43,Th22;
hence contradiction by A14,A25;
end;
A44: vertices(u) \ {v3} c= V
proof
set V3={FT.(t1.1)};
let x be object;
assume
A45: x in vertices(u) \ {v3};
then
A46: x in vertices(u) by XBOOLE_0:def 5;
vertices(u) = vertices(s) \/ V3 by A12,A17,A40,Th25;
then
A47: x in vertices(s) or x in V3 by A46,XBOOLE_0:def 3;
not x in {v3} by A45,XBOOLE_0:def 5;
hence thesis by A10,A14,A19,A26,A42,A47,FINSEQ_4:15,NAT_1:11;
end;
len u = len s + len t1 by FINSEQ_1:22;
then
A48: u <> {} & u.len u =t1.1 by A17,Lm2;
u.1 =s.1 by A12,A40,Lm1
.=r.1 by A12,A13,A40,Lm1;
then FS.(u.1) = v1 by A8;
then u is_orientedpath_of v1,v3 by A23,A19,A26,A42,A48;
then
A49: u is_orientedpath_of v1,v3,V by A44;
then consider q being Simple oriented Chain of G such that
A50: q is_shortestpath_of v1,v3,V,W by A1,Th60;
A51: cost(q,W) <= cost(u,W) by A49,A50;
cost(P,W) <= cost(q,W) by A4,A25,A50;
then cost(P,W) <= cost(u,W) by A51,XXREAL_0:2;
hence cost(P,W) <= cost(r,W) by A22,XXREAL_0:2;
end;
end;
end;
suppose
vertices r c= V;
then vertices(r) \ {v2} c= V \ {v2} by XBOOLE_1:33;
then vertices(r) \ {v2} c= V by XBOOLE_1:1;
then r is_orientedpath_of v1,v2,V by A8;
hence cost(P,W) <= cost(r,W) by A2;
end;
end;
P is_orientedpath_of v1,v2 by A5;
hence thesis by A7;
end;
theorem
W is_weight>=0of G & P is_shortestpath_of v1,v2,V,W & v1 <> v2 & V c=
U & (for Q, v3 st not v3 in V & Q is_shortestpath_of v1,v3,V,W holds cost(P,W)
<= cost(Q,W)) implies P is_shortestpath_of v1,v2,U,W
proof
assume that
A1: W is_weight>=0of G and
A2: P is_shortestpath_of v1,v2,V,W and
A3: v1 <> v2 and
A4: V c= U and
A5: for Q, v3 st not v3 in V & Q is_shortestpath_of v1,v3,V,W holds cost
(P,W) <= cost(Q,W);
P is_shortestpath_of v1,v2,W by A1,A2,A3,A5,Th61;
then
A6: for q be oriented Chain of G st q is_orientedpath_of v1,v2,U holds
cost(P,W) <= cost(q,W);
P is_orientedpath_of v1,v2,V by A2;
then P is_orientedpath_of v1,v2,U by A4,Th30;
hence thesis by A6;
end;
definition
let G be Graph, pe be FinSequence of the carrier' of G, V be set, v1 be
Vertex of G, W be Function;
pred pe islongestInShortestpath V,v1,W means
for v being Vertex of G
st v in V & v <> v1 ex q being oriented Chain of G st q is_shortestpath_of v1,v
,V,W & cost(q,W) <= cost(pe,W);
end;
::$N Dijkstra's shortest path algorithm
theorem
for G being finite oriented Graph, P,Q,R being oriented Chain of G, v1
,v2,v3 being Element of G st e in the carrier' of G & W
is_weight>=0of G & len P >= 1 & P is_shortestpath_of v1,v2,V,W & v1 <> v2 & v1
<> v3 & R=P^<*e*> & Q is_shortestpath_of v1,v3,V,W & e orientedly_joins v2,v3 &
P islongestInShortestpath V,v1,W holds (cost(Q,W) <= cost(R,W) implies Q
is_shortestpath_of v1,v3,V \/{v2},W) & (cost(Q,W) >= cost(R,W) implies R
is_shortestpath_of v1,v3,V \/{v2},W)
proof
let G be finite oriented Graph, P,Q,R be oriented Chain of G, v1,v2,v3 be
Element of G;
assume that
A1: e in the carrier' of G and
A2: W is_weight>=0of G and
A3: len P >= 1 and
A4: P is_shortestpath_of v1,v2,V,W and
A5: v1 <> v2 and
A6: v1 <> v3 and
A7: R=P^<*e*> and
A8: Q is_shortestpath_of v1,v3,V,W and
A9: e orientedly_joins v2,v3 and
A10: P islongestInShortestpath V,v1,W;
A11: P is_orientedpath_of v1,v2,V by A4;
then
A12: v1 in V by A5,Th29;
set Eg=the carrier' of G;
reconsider pe=<*e*> as FinSequence of Eg by A1,FINSEQ_1:74;
set V9=V \/ {v2};
Q is_orientedpath_of v1,v3,V by A8;
then
A13: Q is_orientedpath_of v1,v3,V9 by Th30,XBOOLE_1:7;
A14: len pe = 1 & pe.1=e by FINSEQ_1:40;
then consider s being Simple oriented Chain of G such that
A15: s is_shortestpath_of v1,v3,V9,W by A2,A3,A7,A9,A11,Th32,Th60;
A16: R is_orientedpath_of v1,v3,V9 by A3,A7,A9,A11,A14,Th32;
A17: now
assume
A18: cost(Q,W) <= cost(s,W);
hereby
assume cost(Q,W)<= cost(R,W);
now
let u be oriented Chain of G;
assume u is_orientedpath_of v1,v3,V9;
then cost(s,W) <= cost(u,W) by A15;
hence cost(Q,W) <= cost (u,W) by A18,XXREAL_0:2;
end;
hence Q is_shortestpath_of v1,v3,V9,W by A13;
end;
hereby
assume
A19: cost(Q,W)>= cost(R,W);
now
let u be oriented Chain of G;
assume u is_orientedpath_of v1,v3,V9;
then cost(s,W) <= cost(u,W) by A15;
then cost(Q,W) <= cost (u,W) by A18,XXREAL_0:2;
hence cost(R,W) <= cost (u,W) by A19,XXREAL_0:2;
end;
hence R is_shortestpath_of v1,v3,V9,W by A16;
end;
end;
set FT=the Target of G, FS=the Source of G;
A20: FS.e=v2 & FT.e=v3 by A9,GRAPH_4:def 1;
A21: s is_orientedpath_of v1,v3,V9 by A15;
then
A22: s is_orientedpath_of v1,v3;
then
A23: FT.(s.len s)=v3;
s <> {} by A22;
then
A24: len s >= 1 by FINSEQ_1:20;
per cases;
suppose
len s >= 2;
then consider k being Nat such that
A25: len s = 1+1 + k by NAT_1:10;
A26: V9 \ {v2} = V \ {v2} by XBOOLE_1:40;
reconsider k as Element of NAT by ORDINAL1:def 12;
A27: len s = 1 + (1+k) by A25;
then consider s1,s2 being FinSequence such that
A28: len s1 = 1+k and
A29: len s2 = 1 and
A30: s = s1^s2 by FINSEQ_2:22;
reconsider s1,s2 as Simple oriented Chain of G by A30,Th12;
A31: len s1 >= 1 by A28,NAT_1:12;
A32: FS.(s.1)=v1 by A22;
then
A33: FS.(s1.1)=v1 by A30,A31,Lm1;
len s2 = 1 by A29;
then
A34: not v3 in vertices(s1) by A6,A23,A30,A31,A32,Th16;
A35: s2.1 = s.len s by A27,A28,A29,A30,Lm2;
then
A36: vertices(s) \ {v3} = vertices(s1) \/ {v3} \ {v3} by A23,A28,A29,A30,Th25,
NAT_1:12
.=vertices(s1) \ {v3} by XBOOLE_1:40
.=vertices(s1) by A34,ZFMISC_1:57;
then vertices(s1) c= V9 by A21;
then vertices(s1) \ {v2} c= V9 \ {v2} by XBOOLE_1:33;
then
A37: vertices(s1) \ {v2} c= V by A26,XBOOLE_1:1;
A38: len s1 < len s by A27,A28,NAT_1:13;
then
A39: FS.(s.(len s1+1))=FT.(s.(len s1)) by A31,GRAPH_1:def 15;
A40: s1.len s1=s.len s1 by A30,A31,Lm1;
then
A41: FS.(s2.1)=FT.(s1.(len s1)) by A27,A28,A35,A31,A38,GRAPH_1:def 15;
A42: cost(s,W) = cost(s1,W)+cost(s2,W) by A2,A30,Th44,Th52;
A43: not v1 in vertices(s2) by A6,A23,A29,A30,A31,A32,Th16;
hereby
per cases;
suppose
v2 in vertices(s1);
then consider i such that
A44: 1<=i and
A45: i <= len s1 and
A46: v2 = FT.(s1.i) by A5,A33,Th26;
s2/.1 in Eg by A1;
then
A47: s2.1 in Eg by A29,FINSEQ_4:15;
hereby
per cases;
suppose
A48: FS.(s2.1)=v2;
s1 <> {} by A28;
then s1 is_orientedpath_of v1,v2 by A33,A41,A48;
then s1 is_orientedpath_of v1,v2,V by A37;
then
A49: cost(P,W) <= cost(s1,W) by A4;
s2.1=e by A1,A23,A20,A35,A47,A48,GRAPH_1:def 7;
then s2=pe by A29,FINSEQ_1:40;
then cost(R,W) = cost(P,W)+cost(s2,W) by A2,A7,Th44,Th52;
then
A50: cost(R,W) <= cost(s,W) by A42,A49,XREAL_1:7;
hereby
assume cost(Q,W) <= cost(R,W);
then
A51: cost(Q,W) <= cost(s,W) by A50,XXREAL_0:2;
now
let u be oriented Chain of G;
assume u is_orientedpath_of v1,v3,V9;
then cost(s,W) <= cost(u,W) by A15;
hence cost(Q,W) <= cost (u,W) by A51,XXREAL_0:2;
end;
hence Q is_shortestpath_of v1,v3,V9,W by A13;
end;
hereby
assume cost(Q,W)>= cost(R,W);
now
let u be oriented Chain of G;
assume u is_orientedpath_of v1,v3,V9;
then cost(s,W) <= cost(u,W) by A15;
hence cost(R,W) <= cost (u,W) by A50,XXREAL_0:2;
end;
hence R is_shortestpath_of v1,v3,V9,W by A16;
end;
end;
suppose
A52: FS.(s2.1)<>v2;
len s1 in dom s1 by A31,FINSEQ_3:25;
then reconsider vx=FT.(s1.len s1) as Vertex of G by FINSEQ_2:11
,FUNCT_2:5;
A53: not vx in {v2} by A41,A52,TARSKI:def 1;
len s1 in dom s1 by A31,FINSEQ_3:25;
then
A54: vx in vertices s1 by Th22;
vertices s1 c= V9 by A21,A36;
then
A55: vx in V by A54,A53,XBOOLE_0:def 3;
{vx} c= V
by A55,TARSKI:def 1;
then
A56: V \/ {vx} c= V by XBOOLE_1:8;
1 in dom s2 by A29,FINSEQ_3:25;
then vx <> v1 by A25,A28,A35,A40,A43,A39,Th22;
then consider q9 being oriented Chain of G such that
A57: q9 is_shortestpath_of v1,vx,V,W and
A58: cost(q9,W) <= cost(P,W) by A10,A55;
consider j being Nat such that
A59: len s1=i+j by A45,NAT_1:10;
A60: q9 is_orientedpath_of v1,vx,V by A57;
then
A61: q9 is_orientedpath_of v1,vx;
then q9<>{};
then
A62: len q9 >= 1 by FINSEQ_1:20;
FT.(q9.len q9)=vx by A61;
then reconsider
qx=q9^s2 as oriented Chain of G by A25,A28,A35,A40,A39,Th10;
len qx = len q9 + 1 by A29,FINSEQ_1:22;
then
A63: qx <> {} & FT.(qx.len qx)=v3 by A23,A29,A35,Lm2;
FS.(q9.1)=v1 by A61;
then FS.(qx.1)=v1 by A62,Lm1;
then
A64: qx is_orientedpath_of v1,v3 by A63;
vertices(q9) \ {vx} c= V by A60;
then vertices(q9) c= V \/ {vx} by XBOOLE_1:44;
then vertices(q9) c= V by A56;
then
A65: vertices(q9) \ {v3} c= V \ {v3} by XBOOLE_1:33;
vertices(qx)= vertices(q9) \/ {v3} by A23,A29,A35,A62,Th25;
then vertices(qx) \ {v3} = vertices(q9) \ {v3} by XBOOLE_1:40;
then vertices(qx) \ {v3} c= V by A65,XBOOLE_1:1;
then qx is_orientedpath_of v1,v3,V by A64;
then
A66: cost(Q,W) <= cost(qx,W) by A8;
reconsider j as Element of NAT by ORDINAL1:def 12;
consider t1,t2 being FinSequence such that
A67: len t1 = i and
len t2 = j and
A68: s1 = t1^t2 by A59,FINSEQ_2:22;
reconsider t1,t2 as Simple oriented Chain of G by A68,Th12;
A69: FT.(t1.len t1)=v2 by A44,A46,A67,A68,Lm1;
vertices(t1) c= vertices(t1^t2) by Th24;
then vertices(t1) \ {v2} c= vertices(s1) \ {v2} by A68,XBOOLE_1:33;
then
A70: vertices(t1) \ {v2} c= V by A37;
t1 <> {} & FS.(t1.1)=v1 by A33,A44,A67,A68,Lm1;
then t1 is_orientedpath_of v1,v2 by A69;
then t1 is_orientedpath_of v1,v2,V by A70;
then
A71: cost(P,W) <= cost(t1,W) by A4;
A72: cost(t2,W) >= 0 by A2,Th48;
cost(s1,W)=cost(t1,W)+cost(t2,W) by A2,A68,Th44,Th52;
then cost(t1,W) + (0 qua Nat) <= cost(s1,W) by A72,XREAL_1:7;
then cost(P,W) <= cost(s1,W) by A71,XXREAL_0:2;
then
A73: cost(q9,W) <= cost(s1,W) by A58,XXREAL_0:2;
cost(qx,W) = cost(q9,W)+cost(s2,W) by A2,Th44,Th52;
then cost(qx,W) <= cost(s,W) by A42,A73,XREAL_1:7;
then
A74: cost(Q,W) <= cost(s,W) by A66,XXREAL_0:2;
hereby
assume cost(Q,W) <= cost(R,W);
now
let u be oriented Chain of G;
assume u is_orientedpath_of v1,v3,V9;
then cost(s,W) <= cost(u,W) by A15;
hence cost(Q,W) <= cost (u,W) by A74,XXREAL_0:2;
end;
hence Q is_shortestpath_of v1,v3,V9,W by A13;
end;
hereby
assume cost(Q,W)>= cost(R,W);
then
A75: cost(R,W) <= cost(s,W) by A74,XXREAL_0:2;
now
let u be oriented Chain of G;
assume u is_orientedpath_of v1,v3,V9;
then cost(s,W) <= cost(u,W) by A15;
hence cost(R,W) <= cost (u,W) by A75,XXREAL_0:2;
end;
hence R is_shortestpath_of v1,v3,V9,W by A16;
end;
end;
end;
end;
suppose
not v2 in vertices(s1);
then
A76: vertices(s1) \ {v2} = vertices(s1) by ZFMISC_1:57;
then vertices(s1) \ {v2} c= V9 by A21,A36;
then vertices(s) \ {v3} c= V by A36,A76,XBOOLE_1:43;
then s is_orientedpath_of v1,v3,V by A22;
hence thesis by A8,A17;
end;
end;
end;
suppose
len s < 1+1;
then
A77: len s <= 1 by NAT_1:13;
then
A78: len s=1 by A24,XXREAL_0:1;
A79: vertices s=vertices(s/.1) by A24,A77,Th23,XXREAL_0:1;
vertices(s) \ {v3} c= V
proof
let x be object;
assume
A80: x in vertices(s) \ {v3};
then
A81: not x in {v3} by XBOOLE_0:def 5;
x in vertices(s/.1) by A79,A80,XBOOLE_0:def 5;
then
A82: x=FS.(s/.1) or x=FT.(s/.1) by TARSKI:def 2;
A83: s/.1=s.1 by A24,FINSEQ_4:15;
v3=FT.(s.len s) by A22
.=FT.(s/.1) by A78,FINSEQ_4:15;
hence thesis by A22,A12,A81,A82,A83,TARSKI:def 1;
end;
then s is_orientedpath_of v1,v3,V by A22;
hence thesis by A8,A17;
end;
end;