:: GRAPH_1 semantic presentation

definition

attr c₁ is strict;
struct MultiGraphStruct -> ;
aggr MultiGraphStruct(# Vertices, Edges, Source, Target #) -> MultiGraphStruct ;
sel Vertices c₁ -> set ;
sel Edges c₁ -> set ;
sel Source c₁ -> Function of the Edges of c₁,the Vertices of c₁;
sel Target c₁ -> Function of the Edges of c₁,the Vertices of c₁;

end;

set DomEx = {1,2};

reconsider empty1 = {} as Function of {} ,{1,2} by FUNCT_2:55, RELAT_1:60;

definition

let IT be MultiGraphStruct ;
attr IT is Graph-like means :Def1: :: GRAPH_1:def 1
the Vertices of IT is non empty set ;

end;

:: deftheorem Def1 defines Graph-like GRAPH_1:def 1 :

registration

cluster strict Graph-like MultiGraphStruct ;
existence

proof end;

end;

definition

mode Graph is Graph-like MultiGraphStruct ;

end;

Lm1: for G being Graph holds
( dom the Source of G = the Edges of G & dom the Target of G = the Edges of G & rng the Source of G c= the Vertices of G & rng the Target of G c= the Vertices of G )

proof end;

Lm2: for G being Graph
for x being Element of the Vertices of G holds x in the Vertices of G

proof end;

definition

let G1, G2 be Graph;
assume A1: ( the Source of G1 tolerates the Source of G2 & the Target of G1 tolerates the Target of G2 ) ;
func G1 \/ G2 -> strict Graph means :Def2: :: GRAPH_1:def 2
( the Vertices of it = the Vertices of G1 \/ the Vertices of G2 & the Edges of it = the Edges of G1 \/ the Edges of G2 & ( for v being set st v in the Edges of G1 holds
( the Source of it . v = the Source of G1 . v & the Target of it . v = the Target of G1 . v ) ) & ( for v being set st v in the Edges of G2 holds
( the Source of it . v = the Source of G2 . v & the Target of it . v = the Target of G2 . v ) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def2 defines \/ GRAPH_1:def 2 :

definition

let G, G1, G2 be Graph;
pred G is_sum_of G1,G2 means :Def3: :: GRAPH_1:def 3
( the Target of G1 tolerates the Target of G2 & the Source of G1 tolerates the Source of G2 & MultiGraphStruct(# the Vertices of G,the Edges of G,the Source of G,the Target of G #) = G1 \/ G2 );

end;

:: deftheorem Def3 defines is_sum_of GRAPH_1:def 3 :

definition

let IT be Graph;
attr IT is oriented means :Def4: :: GRAPH_1:def 4
for x, y being set st x in the Edges of IT & y in the Edges of IT & the Source of IT . x = the Source of IT . y & the Target of IT . x = the Target of IT . y holds
x = y;
attr IT is non-multi means :Def5: :: GRAPH_1:def 5
for x, y being set st x in the Edges of IT & y in the Edges of IT & ( ( the Source of IT . x = the Source of IT . y & the Target of IT . x = the Target of IT . y ) or ( the Source of IT . x = the Target of IT . y & the Source of IT . y = the Target of IT . x ) ) holds
x = y;
attr IT is simple means :Def6: :: GRAPH_1:def 6
for x being set holds
( not x in the Edges of IT or not the Source of IT . x = the Target of IT . x );
attr IT is connected means :Def7: :: GRAPH_1:def 7
for G1, G2 being Graph holds
( not the Vertices of G1 misses the Vertices of G2 or not IT is_sum_of G1,G2 );

end;

:: deftheorem Def4 defines oriented GRAPH_1:def 4 :

:: deftheorem Def5 defines non-multi GRAPH_1:def 5 :

:: deftheorem Def6 defines simple GRAPH_1:def 6 :

:: deftheorem Def7 defines connected GRAPH_1:def 7 :

definition

let IT be MultiGraphStruct ;
attr IT is finite means :Def8: :: GRAPH_1:def 8
( the Vertices of IT is finite & the Edges of IT is finite );

end;

:: deftheorem Def8 defines finite GRAPH_1:def 8 :

registration

cluster finite MultiGraphStruct ;
existence

proof end;

cluster oriented non-multi simple connected finite MultiGraphStruct ;
existence

proof end;

end;

definition

let G be Graph;
let x, y be Element of the Vertices of G;
let v be set ;
pred v joins x,y means :: GRAPH_1:def 9
( ( the Source of G . v = x & the Target of G . v = y ) or ( the Source of G . v = y & the Target of G . v = x ) );

end;

:: deftheorem defines joins GRAPH_1:def 9 :

definition

let G be Graph;
let x, y be Element of the Vertices of G;
pred x,y are_incydent means :: GRAPH_1:def 10
ex v being set st
( v in the Edges of G & v joins x,y );

end;

:: deftheorem defines are_incydent GRAPH_1:def 10 :

definition

let G be Graph;
mode Chain of G -> FinSequence means :Def11: :: GRAPH_1:def 11
( ( for n being Nat st 1 <= n & n <= len it holds
it . n in the Edges of G ) & ex p being FinSequence st
( len p = (len it) + 1 & ( for n being Nat st 1 <= n & n <= len p holds
p . n in the Vertices of G ) & ( for n being Nat st 1 <= n & n <= len it holds
ex x', y' being Element of the Vertices of G st
( x' = p . n & y' = p . (n + 1) & it . n joins x',y' ) ) ) );
existence

proof end;

end;

:: deftheorem Def11 defines Chain GRAPH_1:def 11 :

Lm3: for G being Graph holds {} is Chain of G

proof end;

definition

let G be Graph;
:: original: Chain
redefine mode Chain of G -> FinSequence of the Edges of G;
coherence

proof end;

end;

definition

let G be Graph;
let IT be Chain of G;
attr IT is oriented means :Def12: :: GRAPH_1:def 12
for n being Nat st 1 <= n & n < len IT holds
the Source of G . (IT . (n + 1)) = the Target of G . (IT . n);

end;

:: deftheorem Def12 defines oriented GRAPH_1:def 12 :

registration

let G be Graph;
cluster oriented Chain of G;
existence

proof end;

end;

Lm4: for G being Graph holds {} is oriented Chain of G

proof end;

definition

let G be Graph;
let IT be Chain of G;
:: original: one-to-one
redefine attr IT is one-to-one means :: GRAPH_1:def 13
for n, m being Nat st 1 <= n & n < m & m <= len IT holds
IT . n <> IT . m;
compatibility

proof end;

end;

:: deftheorem defines one-to-one GRAPH_1:def 13 :

registration

let G be Graph;
cluster V5 Chain of G;
existence

proof end;

end;

definition

let G be Graph;
mode Path of G is V5 Chain of G;

end;

registration

let G be Graph;
cluster V5 oriented Chain of G;
existence

proof end;

end;

definition

let G be Graph;
mode OrientedPath of G is V5 oriented Chain of G;

end;

definition

let G be Graph;
let IT be Path of G;
canceled;
attr IT is cyclic means :Def15: :: GRAPH_1:def 15
ex p being FinSequence st
( len p = (len IT) + 1 & ( for n being Nat st 1 <= n & n <= len p holds
p . n in the Vertices of G ) & ( for n being Nat st 1 <= n & n <= len IT holds
ex x', y' being Element of the Vertices of G st
( x' = p . n & y' = p . (n + 1) & IT . n joins x',y' ) ) & p . 1 = p . (len p) );

end;

:: deftheorem GRAPH_1:def 14 :

:: deftheorem Def15 defines cyclic GRAPH_1:def 15 :

registration

let G be Graph;
cluster cyclic Chain of G;
existence

proof end;

end;

definition

let G be Graph;
mode Cycle of G is cyclic Path of G;

end;

Lm5: for G being Graph holds {} is Cycle of G

proof end;

registration

let G be Graph;
cluster cyclic Chain of G;
existence

proof end;

end;

definition

let G be Graph;
mode OrientedCycle of G is cyclic OrientedPath of G;

end;

Lm6: for G being Graph
for v being set st v in the Edges of G holds
( the Source of G . v in the Vertices of G & the Target of G . v in the Vertices of G )

proof end;

definition

let G be Graph;
canceled;
mode Subgraph of G -> Graph means :Def17: :: GRAPH_1:def 17
( the Vertices of it c= the Vertices of G & the Edges of it c= the Edges of G & ( for v being set st v in the Edges of it holds
( the Source of it . v = the Source of G . v & the Target of it . v = the Target of G . v & the Source of G . v in the Vertices of it & the Target of G . v in the Vertices of it ) ) );
existence

proof end;

end;

:: deftheorem GRAPH_1:def 16 :

:: deftheorem Def17 defines Subgraph GRAPH_1:def 17 :

registration

let G be Graph;
cluster strict Subgraph of G;
existence

proof end;

end;

definition

let G be finite Graph;
func VerticesCount G -> Nat means :: GRAPH_1:def 18
ex B being finite set st
( B = the Vertices of G & it = card B );
existence

proof end;

uniqueness ;
func EdgesCount G -> Nat means :: GRAPH_1:def 19
ex B being finite set st
( B = the Edges of G & it = card B );
existence

proof end;

uniqueness ;

end;

:: deftheorem defines VerticesCount GRAPH_1:def 18 :

:: deftheorem defines EdgesCount GRAPH_1:def 19 :

definition

let G be finite Graph;
let x be Element of the Vertices of G;
func EdgesIn x -> Nat means :: GRAPH_1:def 20
ex X being finite set st
( ( for z being set holds
( z in X iff ( z in the Edges of G & the Target of G . z = x ) ) ) & it = card X );
existence

proof end;

uniqueness

proof end;

func EdgesOut x -> Nat means :: GRAPH_1:def 21
ex X being finite set st
( ( for z being set holds
( z in X iff ( z in the Edges of G & the Source of G . z = x ) ) ) & it = card X );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem defines EdgesIn GRAPH_1:def 20 :

:: deftheorem defines EdgesOut GRAPH_1:def 21 :

definition

let G be finite Graph;
let x be Element of the Vertices of G;
func Degree x -> Nat equals :: GRAPH_1:def 22
(EdgesIn x) + (EdgesOut x);
correctness ;

end;

:: deftheorem defines Degree GRAPH_1:def 22 :

Lm7: for n being Nat
for G being Graph
for p being Chain of G holds p | (Seg n) is Chain of G

proof end;

Lm8: for G being Graph
for H1, H2 being strict Subgraph of G st the Vertices of H1 = the Vertices of H2 & the Edges of H1 = the Edges of H2 holds
H1 = H2

proof end;

definition

let G1, G2 be Graph;
pred G1 c= G2 means :Def23: :: GRAPH_1:def 23
G1 is Subgraph of G2;
reflexivity

proof end;

end;

:: deftheorem Def23 defines c= GRAPH_1:def 23 :

Lm9: for G being Graph
for H being Subgraph of G holds
( the Source of H in PFuncs the Edges of G,the Vertices of G & the Target of H in PFuncs the Edges of G,the Vertices of G )

proof end;

definition

let G be Graph;
func bool G -> set means :Def24: :: GRAPH_1:def 24
for x being set holds
( x in it iff x is strict Subgraph of G );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def24 defines bool GRAPH_1:def 24 :

scheme :: GRAPH_1:sch 1
GraphSeparation{ F₁() -> Graph, P₁[ set ] } :

ex X being set st
for x being set holds
( x in X iff ( x is strict Subgraph of F₁() & P₁[x] ) )

proof end;

theorem :: GRAPH_1:1

for G being Graph holds
( dom the Source of G = the Edges of G & dom the Target of G = the Edges of G & rng the Source of G c= the Vertices of G & rng the Target of G c= the Vertices of G ) by Lm1;

theorem :: GRAPH_1:2

for G being Graph
for x being Element of the Vertices of G holds x in the Vertices of G by Lm2;

theorem :: GRAPH_1:3

for G being Graph
for v being set st v in the Edges of G holds
( the Source of G . v in the Vertices of G & the Target of G . v in the Vertices of G ) by Lm6;

theorem :: GRAPH_1:4

for n being Nat
for G being Graph
for p being Chain of G holds p | (Seg n) is Chain of G by Lm7;

theorem Th5: :: GRAPH_1:5

for G1, G being Graph st G1 c= G holds
( the Source of G1 c= the Source of G & the Target of G1 c= the Target of G )

proof end;

theorem :: GRAPH_1:6

for G1, G2 being Graph st the Source of G1 tolerates the Source of G2 & the Target of G1 tolerates the Target of G2 holds
( the Source of (G1 \/ G2) = the Source of G1 \/ the Source of G2 & the Target of (G1 \/ G2) = the Target of G1 \/ the Target of G2 )

proof end;

theorem Th7: :: GRAPH_1:7

for G being strict Graph holds G = G \/ G

proof end;

theorem Th8: :: GRAPH_1:8

for G1, G2 being Graph st the Source of G1 tolerates the Source of G2 & the Target of G1 tolerates the Target of G2 holds
G1 \/ G2 = G2 \/ G1

proof end;

theorem Th9: :: GRAPH_1:9

for G1, G2, G3 being Graph st the Source of G1 tolerates the Source of G2 & the Target of G1 tolerates the Target of G2 & the Source of G1 tolerates the Source of G3 & the Target of G1 tolerates the Target of G3 & the Source of G2 tolerates the Source of G3 & the Target of G2 tolerates the Target of G3 holds
(G1 \/ G2) \/ G3 = G1 \/ (G2 \/ G3)

proof end;

theorem Th10: :: GRAPH_1:10

for G, G1, G2 being Graph st G is_sum_of G1,G2 holds
G is_sum_of G2,G1

proof end;

theorem :: GRAPH_1:11

for G being strict Graph holds G is_sum_of G,G

proof end;

theorem Th12: :: GRAPH_1:12

for G1, G2 being Graph st ex G being Graph st
( G1 c= G & G2 c= G ) holds
G1 \/ G2 = G2 \/ G1

proof end;

theorem :: GRAPH_1:13

for G1, G2, G3 being Graph st ex G being Graph st
( G1 c= G & G2 c= G & G3 c= G ) holds
(G1 \/ G2) \/ G3 = G1 \/ (G2 \/ G3)

proof end;

theorem :: GRAPH_1:14

for G being Graph holds {} is oriented cyclic Path of G by Lm4, Lm5;

theorem :: GRAPH_1:15

for G being Graph
for H1, H2 being strict Subgraph of G st the Vertices of H1 = the Vertices of H2 & the Edges of H1 = the Edges of H2 holds
H1 = H2 by Lm8;

theorem Th16: :: GRAPH_1:16

for G1, G2 being strict Graph st G1 c= G2 & G2 c= G1 holds
G1 = G2

proof end;

theorem Th17: :: GRAPH_1:17

for G1, G2, G3 being Graph st G1 c= G2 & G2 c= G3 holds
G1 c= G3

proof end;

theorem Th18: :: GRAPH_1:18

for G, G1, G2 being Graph st G is_sum_of G1,G2 holds
( G1 c= G & G2 c= G )

proof end;

theorem Th19: :: GRAPH_1:19

for G1, G2 being Graph st the Source of G1 tolerates the Source of G2 & the Target of G1 tolerates the Target of G2 holds
( G1 c= G1 \/ G2 & G2 c= G1 \/ G2 )

proof end;

theorem Th20: :: GRAPH_1:20

for G1, G2 being Graph st ex G being Graph st
( G1 c= G & G2 c= G ) holds
( G1 c= G1 \/ G2 & G2 c= G1 \/ G2 )

proof end;

theorem Th21: :: GRAPH_1:21

for G1, G3, G2, G being Graph st G1 c= G3 & G2 c= G3 & G is_sum_of G1,G2 holds
G c= G3

proof end;

theorem Th22: :: GRAPH_1:22

for G1, G, G2 being Graph st G1 c= G & G2 c= G holds
G1 \/ G2 c= G

proof end;

theorem :: GRAPH_1:23

for G1, G2 being strict Graph st G1 c= G2 holds
( G1 \/ G2 = G2 & G2 \/ G1 = G2 )

proof end;

theorem Th24: :: GRAPH_1:24

for G1, G2 being Graph st the Source of G1 tolerates the Source of G2 & the Target of G1 tolerates the Target of G2 & ( G1 \/ G2 = G2 or G2 \/ G1 = G2 ) holds
G1 c= G2

proof end;

theorem :: GRAPH_1:25

canceled;

theorem :: GRAPH_1:26

canceled;

theorem :: GRAPH_1:27

for G1 being Graph
for G being oriented Graph st G1 c= G holds
G1 is oriented

proof end;

theorem :: GRAPH_1:28

for G1 being Graph
for G being non-multi Graph st G1 c= G holds
G1 is non-multi

proof end;

theorem :: GRAPH_1:29

for G1 being Graph
for G being simple Graph st G1 c= G holds
G1 is simple

proof end;

theorem :: GRAPH_1:30

for G being Graph
for G1 being strict Graph holds
( G1 in bool G iff G1 c= G )

proof end;

theorem Th31: :: GRAPH_1:31

for G being strict Graph holds G in bool G

proof end;

theorem :: GRAPH_1:32

for G2 being Graph
for G1 being strict Graph holds
( G1 c= G2 iff bool G1 c= bool G2 )

proof end;

theorem :: GRAPH_1:33

canceled;

theorem :: GRAPH_1:34

for G being strict Graph holds {G} c= bool G

proof end;

theorem :: GRAPH_1:35

for G1, G2 being strict Graph st the Source of G1 tolerates the Source of G2 & the Target of G1 tolerates the Target of G2 & bool (G1 \/ G2) c= (bool G1) \/ (bool G2) & not G1 c= G2 holds
G2 c= G1

proof end;

theorem :: GRAPH_1:36

for G1, G2 being Graph st the Source of G1 tolerates the Source of G2 & the Target of G1 tolerates the Target of G2 holds
(bool G1) \/ (bool G2) c= bool (G1 \/ G2)

proof end;

theorem :: GRAPH_1:37

for G1, G, G2 being Graph st G1 in bool G & G2 in bool G holds
G1 \/ G2 in bool G

proof end;