HomeHome Metamath Proof Explorer
Theorem List (p. 400 of 411)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-26652)
  Hilbert Space Explorer  Hilbert Space Explorer
(26653-28175)
  Users' Mathboxes  Users' Mathboxes
(28176-41046)
 

Theorem List for Metamath Proof Explorer - 39901-40000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtrlsonfval 39901* The set of trails between two vertices. (Contributed by Alexander van der Vekens, 4-Nov-2017.) (Revised by AV, 7-Jan-2021.) (Proof shortened by AV, 15-Jan-2021.) (Revised by AV, 21-Mar-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( A  e.  V  /\  B  e.  V ) 
 ->  ( A (TrailsOn `  G ) B )  =  { <. f ,  p >.  |  ( f ( A (WalksOn `  G ) B ) p  /\  f (TrailS `  G ) p ) } )
 
Theoremistrlson 39902 Properties of a pair of functions to be a trail between two given vertices. (Contributed by Alexander van der Vekens, 3-Nov-2017.) (Revised by AV, 7-Jan-2021.) (Revised by AV, 21-Mar-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( ( A  e.  V  /\  B  e.  V )  /\  ( F  e.  U  /\  P  e.  Z ) )  ->  ( F ( A (TrailsOn `  G ) B ) P  <->  ( F ( A (WalksOn `  G ) B ) P  /\  F (TrailS `  G ) P ) ) )
 
Theoremtrlsonprop 39903 Properties of a trail between two vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Revised by AV, 7-Jan-2021.) (Proof shortened by AV, 16-Jan-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  ( F ( A (TrailsOn `  G ) B ) P  ->  ( ( G  e.  _V  /\  A  e.  V  /\  B  e.  V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( F ( A (WalksOn `  G ) B ) P  /\  F (TrailS `  G ) P ) ) )
 
Theoremtrlsonistrl 39904 A trail between two vertices is a trail. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 7-Jan-2021.)
 |-  ( F ( A (TrailsOn `  G ) B ) P  ->  F (TrailS `  G ) P )
 
Theoremtrlsonwlkon 39905 A trail between two vertices is a walk between these vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Revised by AV, 7-Jan-2021.)
 |-  ( F ( A (TrailsOn `  G ) B ) P  ->  F ( A (WalksOn `  G ) B ) P )
 
TheoremtrlOntrl 39906 A trail is a trail between its endpoints. (Contributed by AV, 31-Jan-2021.)
 |-  ( F (TrailS `  G ) P  ->  F ( ( P `  0 ) (TrailsOn `  G )
 ( P `  ( # `
  F ) ) ) P )
 
21.33.8.17  Paths
 
Syntaxcpths 39907 Extend class notation with paths (of a graph).
 class PathS
 
Syntaxcspths 39908 Extend class notation with simple paths (of a graph).
 class SPathS
 
Syntaxcpthson 39909 Extend class notation with paths between two vertices (within a graph).
 class PathsOn
 
Syntaxcspthson 39910 Extend class notation with simple paths between two vertices (within a graph).
 class SPathsOn
 
Definitiondf-pths 39911* Define the set of all paths (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A path is a trail in which all vertices (except possibly the first and last) are distinct. ... use the term simple path to refer to a path which contains no repeated vertices."

According to Bollobas: "... a path is a walk with distinct vertices.", see Notation of [Bollobas] p. 5. (A walk with distinct vertices is actually a simple path, see wlkdvspth 25417).

Therefore, a path can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, which is injective restricted to the set { 1 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the path is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 9-Jan-2021.)

 |- PathS  =  ( g  e.  _V  |->  {
 <. f ,  p >.  |  ( f (TrailS `  g
 ) p  /\  Fun  `' ( p  |`  ( 1..^ ( # `  f
 ) ) )  /\  ( ( p " { 0 ,  ( # `
  f ) }
 )  i^i  ( p " ( 1..^ ( # `  f ) ) ) )  =  (/) ) }
 )
 
Definitiondf-spths 39912* Define the set of all simple paths (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A path is a trail in which all vertices (except possibly the first and last) are distinct. ... use the term simple path to refer to a path which contains no repeated vertices."

Therefore, a simple path can be represented by an injective mapping f from { 1 , ... , n } and an injective mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the simple path is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 9-Jan-2021.)

 |- SPathS  =  ( g  e.  _V  |->  {
 <. f ,  p >.  |  ( f (TrailS `  g
 ) p  /\  Fun  `' p ) } )
 
Definitiondf-pthson 39913* Define the collection of paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 9-Jan-2021.)
 |- PathsOn  =  ( g  e.  _V  |->  ( a  e.  (Vtx `  g ) ,  b  e.  (Vtx `  g )  |->  { <. f ,  p >.  |  ( f ( a (TrailsOn `  g
 ) b ) p 
 /\  f (PathS `  g ) p ) } ) )
 
Definitiondf-spthson 39914* Define the collection of simple paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 9-Jan-2021.)
 |- SPathsOn  =  ( g  e.  _V  |->  ( a  e.  (Vtx `  g ) ,  b  e.  (Vtx `  g )  |->  { <. f ,  p >.  |  ( f ( a (TrailsOn `  g
 ) b ) p 
 /\  f (SPathS `  g
 ) p ) }
 ) )
 
Theorempthsfval 39915* The set of paths (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 31-Jan-2021.)
 |-  ( G  e.  W  ->  (PathS `  G )  =  { <. f ,  p >.  |  ( f (TrailS `  G ) p  /\  Fun  `' ( p  |`  ( 1..^ ( # `  f
 ) ) )  /\  ( ( p " { 0 ,  ( # `
  f ) }
 )  i^i  ( p " ( 1..^ ( # `  f ) ) ) )  =  (/) ) }
 )
 
Theoremspthsfval 39916* The set of simple paths (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 31-Jan-2021.)
 |-  ( G  e.  W  ->  (SPathS `  G )  =  { <. f ,  p >.  |  ( f (TrailS `  G ) p  /\  Fun  `' p ) } )
 
TheoremisPth 39917 Conditions for a pair of functions to be a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 31-Jan-2021.)
 |-  (
 ( G  e.  W  /\  F  e.  U  /\  P  e.  Z )  ->  ( F (PathS `  G ) P  <->  ( F (TrailS `  G ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F ) ) )  /\  ( ( P " { 0 ,  ( # `
  F ) }
 )  i^i  ( P " ( 1..^ ( # `  F ) ) ) )  =  (/) ) ) )
 
TheoremissPth 39918 Conditions for a pair of functions to be a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 31-Jan-2021.)
 |-  (
 ( G  e.  W  /\  F  e.  U  /\  P  e.  Z )  ->  ( F (SPathS `  G ) P  <->  ( F (TrailS `  G ) P  /\  Fun  `' P ) ) )
 
TheoremPthisTrl 39919 A path is a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.)
 |-  ( F (PathS `  G ) P  ->  F (TrailS `  G ) P )
 
TheoremsPthisPth 39920 A simple path is a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.)
 |-  ( F (SPathS `  G ) P  ->  F (PathS `  G ) P )
 
Theorempthis1wlk 39921 A path is a 1-walk (in an undirected graph). (Contributed by AV, 6-Feb-2021.)
 |-  ( F (PathS `  G ) P  ->  F (1Walks `  G ) P )
 
Theorempthdivtx 39922 The inner vertices of a path are distinct from all other vertices. (Contributed by AV, 5-Feb-2021.)
 |-  (
 ( F (PathS `  G ) P  /\  ( I  e.  (
 1..^ ( # `  F ) )  /\  J  e.  ( 0 ... ( # `
  F ) ) 
 /\  I  =/=  J ) )  ->  ( P `
  I )  =/=  ( P `  J ) )
 
Theorempthdadjvtx 39923 The adjacent vertices of a path of length at least 2 are distinct. (Contributed by AV, 5-Feb-2021.)
 |-  (
 ( F (PathS `  G ) P  /\  1  <  ( # `  F )  /\  I  e.  (
 0..^ ( # `  F ) ) )  ->  ( P `  I )  =/=  ( P `  ( I  +  1
 ) ) )
 
Theorem2pthnloop 39924* A path of length at least 2 does not contain a loop. In contrast, a path of length 1 can contain/be a loop, see lppthon 40039. (Contributed by AV, 6-Feb-2021.)
 |-  I  =  (iEdg `  G )   =>    |-  (
 ( F (PathS `  G ) P  /\  1  <  ( # `  F ) )  ->  A. i  e.  ( 0..^ ( # `  F ) ) 2 
 <_  ( # `  ( I `  ( F `  i ) ) ) )
 
Theoremupgr2pthnlp 39925* A path of length at least 2 in a pseudograph does not contain a loop. (Contributed by AV, 6-Feb-2021.)
 |-  I  =  (iEdg `  G )   =>    |-  (
 ( G  e. UPGraph  /\  F (PathS `  G ) P 
 /\  1  <  ( # `
  F ) ) 
 ->  A. i  e.  (
 0..^ ( # `  F ) ) ( # `  ( I `  ( F `  i ) ) )  =  2 )
 
Theoremspthdep 39926 A simple path (at least of length 1) has different start and end points (in an undirected graph). (Contributed by AV, 31-Jan-2021.)
 |-  (
 ( F (SPathS `  G ) P  /\  ( # `  F )  =/=  0
 )  ->  ( P `  0 )  =/=  ( P `  ( # `  F ) ) )
 
TheorempthdepissPth 39927 A path with different start and end points is a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 12-Jan-2021.)
 |-  (
 ( F (PathS `  G ) P  /\  ( P `  0 )  =/=  ( P `  ( # `  F ) ) )  ->  F (SPathS `  G ) P )
 
Theoremupgrwlkdvdelem 39928* Lemma for upgrwlkdvde 39929. Formerly wlkdvspthlem 25416. (Contributed by Alexander van der Vekens, 27-Oct-2017.) (Proof shortened by AV, 17-Jan-2021.)
 |-  (
 ( P : ( 0 ... ( # `  F ) ) -1-1-> V  /\  F  e. Word  dom  I ) 
 ->  ( A. k  e.  ( 0..^ ( # `  F ) ) ( I `  ( F `
  k ) )  =  { ( P `
  k ) ,  ( P `  (
 k  +  1 ) ) }  ->  Fun  `' F ) )
 
Theoremupgrwlkdvde 39929 In a pseudograph, all edges of a walk consisting of different vertices are different. Notice that this theorem would not hold for arbitrary hypergraphs, see the counterexample given in the comment of upgrspths1wlk 39930. (Contributed by AV, 17-Jan-2021.)
 |-  (
 ( G  e. UPGraph  /\  F (1Walks `  G ) P 
 /\  Fun  `' P )  ->  Fun  `' F )
 
Theoremupgrspths1wlk 39930* The set of simple paths in a pseudograph, expressed as walk. Notice that this theorem would not hold for arbitrary hypergraphs, since a walk with distinct vertices does not need to be a trail: let E = { p0, p1, p2 } be a hyperedge, then ( p0, e, p1, e, p2 ) is walk with distinct vertices, but not with distinct edges. Therefore, E is not a trail and, by definition, also no path. (Contributed by AV, 11-Jan-2021.) (Proof shortened by AV, 17-Jan-2021.)
 |-  ( G  e. UPGraph  ->  (SPathS `  G )  =  { <. f ,  p >.  |  (
 f (1Walks `  G ) p  /\  Fun  `' p ) } )
 
Theoremupgrwlkdvspth 39931 A walk consisting of different vertices is a simple path. Formerly wlkdvspth 25417. Notice that this theorem would not hold for arbitrary hypergraphs, see the counterexample given in the comment of upgrspths1wlk 39930. (Contributed by Alexander van der Vekens, 27-Oct-2017.) (Revised by AV, 17-Jan-2021.)
 |-  (
 ( G  e. UPGraph  /\  F (1Walks `  G ) P 
 /\  Fun  `' P )  ->  F (SPathS `  G ) P )
 
Theorempthsonfval 39932* The set of paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( A  e.  V  /\  B  e.  V ) 
 ->  ( A (PathsOn `  G ) B )  =  { <. f ,  p >.  |  ( f ( A (TrailsOn `  G ) B ) p  /\  f (PathS `  G ) p ) } )
 
Theoremspthson 39933* The set of simple paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( A  e.  V  /\  B  e.  V ) 
 ->  ( A (SPathsOn `  G ) B )  =  { <. f ,  p >.  |  ( f ( A (TrailsOn `  G ) B ) p  /\  f (SPathS `  G ) p ) } )
 
Theoremispthson 39934 Properties of a pair of functions to be a path between two given vertices. (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( ( A  e.  V  /\  B  e.  V )  /\  ( F  e.  U  /\  P  e.  Z ) )  ->  ( F ( A (PathsOn `  G ) B ) P  <->  ( F ( A (TrailsOn `  G ) B ) P  /\  F (PathS `  G ) P ) ) )
 
Theoremisspthson 39935 Properties of a pair of functions to be a simple path between two given vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( ( A  e.  V  /\  B  e.  V )  /\  ( F  e.  U  /\  P  e.  Z ) )  ->  ( F ( A (SPathsOn `  G ) B ) P  <->  ( F ( A (TrailsOn `  G ) B ) P  /\  F (SPathS `  G ) P ) ) )
 
Theorempthsonprop 39936 Properties of a path between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 16-Jan-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  ( F ( A (PathsOn `  G ) B ) P  ->  ( ( G  e.  _V  /\  A  e.  V  /\  B  e.  V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( F ( A (TrailsOn `  G ) B ) P  /\  F (PathS `  G ) P ) ) )
 
Theoremspthonprop 39937 Properties of a simple path between two vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-Jan-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  ( F ( A (SPathsOn `  G ) B ) P  ->  ( ( G  e.  _V  /\  A  e.  V  /\  B  e.  V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( F ( A (TrailsOn `  G ) B ) P  /\  F (SPathS `  G ) P ) ) )
 
Theorempthonispth-av 39938 A path between two vertices is a path. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 17-Jan-2021.)
 |-  ( F ( A (PathsOn `  G ) B ) P  ->  F (PathS `  G ) P )
 
Theorempthontrlon 39939 A path between two vertices is a trail between these vertices. (Contributed by AV, 24-Jan-2021.)
 |-  ( F ( A (PathsOn `  G ) B ) P  ->  F ( A (TrailsOn `  G ) B ) P )
 
TheorempthOnpth 39940 A path is a path between its endpoints. (Contributed by AV, 31-Jan-2021.)
 |-  ( F (PathS `  G ) P  ->  F ( ( P `  0 ) (PathsOn `  G )
 ( P `  ( # `
  F ) ) ) P )
 
Theoremisspthonpth-av 39941 A pair of functions is a simple path between two given vertices iff it is a simple path starting and ending at the two vertices. (Contributed by Alexander van der Vekens, 9-Mar-2018.) (Revised by AV, 17-Jan-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( ( A  e.  V  /\  B  e.  V )  /\  ( F  e.  W  /\  P  e.  Z ) )  ->  ( F ( A (SPathsOn `  G ) B ) P  <->  ( F (SPathS `  G ) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) ) )
 
Theoremspthonisspth-av 39942 A simple path between to vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 18-Jan-2021.)
 |-  ( F ( A (SPathsOn `  G ) B ) P  ->  F (SPathS `  G ) P )
 
Theoremspthonpthon 39943 A simple path between two vertices is a path between these vertices. (Contributed by AV, 24-Jan-2021.)
 |-  ( F ( A (SPathsOn `  G ) B ) P  ->  F ( A (PathsOn `  G ) B ) P )
 
Theoremspthonepeq-av 39944 The endpoints of a simple path between two vertices are equal iff the path is of length 0. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 18-Jan-2021.)
 |-  ( F ( A (SPathsOn `  G ) B ) P  ->  ( A  =  B  <->  ( # `  F )  =  0 )
 )
 
Theoremuhgr1wlkspthlem1 39945 Lemma 1 for uhgr1wlkspth 39947. (Contributed by AV, 25-Jan-2021.)
 |-  (
 ( F (1Walks `  G ) P  /\  ( # `  F )  =  1 )  ->  Fun  `' F )
 
Theoremuhgr1wlkspthlem2 39946 Lemma 2 for uhgr1wlkspth 39947. (Contributed by AV, 25-Jan-2021.)
 |-  (
 ( F (1Walks `  G ) P  /\  ( ( # `  F )  =  1  /\  A  =/=  B )  /\  ( ( P `  0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )  ->  Fun  `' P )
 
Theoremuhgr1wlkspth 39947 Any walk of length 1 between two different vertices is a simple path. (Contributed by AV, 25-Jan-2021.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e.  W  /\  ( # `  F )  =  1  /\  A  =/=  B )  ->  ( F ( A (WalksOn `  G ) B ) P  <->  F ( A (SPathsOn `  G ) B ) P ) )
 
Theoremusgr2wlkneq 39948 The vertices and edges are pairwise different in a walk of length 2 in a simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 26-Jan-2021.)
 |-  (
 ( ( G  e. USGraph  /\  F (1Walks `  G ) P )  /\  (
 ( # `  F )  =  2  /\  ( P `  0 )  =/=  ( P `  ( # `
  F ) ) ) )  ->  (
 ( ( P `  0 )  =/=  ( P `  1 )  /\  ( P `  0 )  =/=  ( P `  2 )  /\  ( P `
  1 )  =/=  ( P `  2
 ) )  /\  ( F `  0 )  =/=  ( F `  1
 ) ) )
 
Theoremusgr2wlkspthlem1 39949 Lemma 1 for usgr2wlkspth 39951. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 26-Jan-2021.)
 |-  (
 ( F (1Walks `  G ) P  /\  ( G  e. USGraph  /\  ( # `  F )  =  2  /\  ( P `  0 )  =/=  ( P `  ( # `  F ) ) ) )  ->  Fun  `' F )
 
Theoremusgr2wlkspthlem2 39950 Lemma 2 for usgr2wlkspth 39951. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 27-Jan-2021.)
 |-  (
 ( F (1Walks `  G ) P  /\  ( G  e. USGraph  /\  ( # `  F )  =  2  /\  ( P `  0 )  =/=  ( P `  ( # `  F ) ) ) )  ->  Fun  `' P )
 
Theoremusgr2wlkspth 39951 In a simple graph, any walk of length 2 between two different vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 27-Jan-2021.)
 |-  (
 ( G  e. USGraph  /\  ( # `
  F )  =  2  /\  A  =/=  B )  ->  ( F ( A (WalksOn `  G ) B ) P  <->  F ( A (SPathsOn `  G ) B ) P ) )
 
Theorempthdlem1 39952* Lemma 1 for pthd 39955. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 9-Feb-2021.)
 |-  ( ph  ->  P  e. Word  _V )   &    |-  R  =  ( ( # `  P )  -  1 )   &    |-  ( ph  ->  A. i  e.  ( 0..^ ( # `  P ) ) A. j  e.  ( 1..^ R ) ( i  =/=  j  ->  ( P `  i
 )  =/=  ( P `  j ) ) )   =>    |-  ( ph  ->  Fun  `' ( P  |`  ( 1..^ R ) ) )
 
Theorempthdlem2lem 39953* Lemma for pthdlem2 39954. (Contributed by AV, 10-Feb-2021.)
 |-  ( ph  ->  P  e. Word  _V )   &    |-  R  =  ( ( # `  P )  -  1 )   &    |-  ( ph  ->  A. i  e.  ( 0..^ ( # `  P ) ) A. j  e.  ( 1..^ R ) ( i  =/=  j  ->  ( P `  i
 )  =/=  ( P `  j ) ) )   =>    |-  ( ( ph  /\  ( # `
  P )  e. 
 NN  /\  ( I  =  0  \/  I  =  R ) )  ->  ( P `  I ) 
 e/  ( P "
 ( 1..^ R ) ) )
 
Theorempthdlem2 39954* Lemma 2 for pthd 39955. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 10-Feb-2021.)
 |-  ( ph  ->  P  e. Word  _V )   &    |-  R  =  ( ( # `  P )  -  1 )   &    |-  ( ph  ->  A. i  e.  ( 0..^ ( # `  P ) ) A. j  e.  ( 1..^ R ) ( i  =/=  j  ->  ( P `  i
 )  =/=  ( P `  j ) ) )   =>    |-  ( ph  ->  ( ( P " { 0 ,  R } )  i^i  ( P " (
 1..^ R ) ) )  =  (/) )
 
Theorempthd 39955* Two words representing a trail which also represent a path in a graph. (Contributed by AV, 10-Feb-2021.)
 |-  ( ph  ->  P  e. Word  _V )   &    |-  R  =  ( ( # `  P )  -  1 )   &    |-  ( ph  ->  A. i  e.  ( 0..^ ( # `  P ) ) A. j  e.  ( 1..^ R ) ( i  =/=  j  ->  ( P `  i
 )  =/=  ( P `  j ) ) )   &    |-  ( # `  F )  =  R   &    |-  ( ph  ->  F (TrailS `  G ) P )   =>    |-  ( ph  ->  F (PathS `  G ) P )
 
21.33.8.18  Closed walks
 
Syntaxcclwlks 39956 Extend class notation with closed walks (of a graph).
 class ClWalkS
 
Definitiondf-clwlks 39957* Define the set of all closed walks (in an undirected graph).

According to definition 4 in [Huneke] p. 2: "A walk of length n on (a graph) G is an ordered sequence v0 , v1 , ... v(n) of vertices such that v(i) and v(i+1) are neighbors (i.e are connected by an edge). We say the walk is closed if v(n) = v0".

According to the definition of a walk as two mappings f from { 0 , ... , ( n - 1 ) } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices, a closed walk is represented by the following sequence: p(0) e(f(0)) p(1) e(f(1)) ... p(n-1) e(f(n-1)) p(n)=p(0).

Notice that by this definition, a single vertex is a closed walk of length 0, see also 0clwlk 25572! (Contributed by Alexander van der Vekens, 12-Mar-2018.) (Revised by AV, 16-Feb-2021.)

 |- ClWalkS  =  ( g  e.  _V  |->  {
 <. f ,  p >.  |  ( f (1Walks `  g
 ) p  /\  ( p `  0 )  =  ( p `  ( # `
  f ) ) ) } )
 
TheoremclwlkS 39958* The set of closed walks (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 16-Feb-2021.)
 |-  ( G  e.  X  ->  (ClWalkS `  G )  =  { <. f ,  p >.  |  ( f (1Walks `  G ) p  /\  ( p `
  0 )  =  ( p `  ( # `
  f ) ) ) } )
 
TheoremisclWlk 39959 Properties of a pair of functions to be a closed walk (in an undirected graph) in terms of walks. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 16-Feb-2021.)
 |-  (
 ( G  e.  X  /\  F  e.  Y  /\  P  e.  Z )  ->  ( F (ClWalkS `  G ) P  <->  ( F (1Walks `  G ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) )
 
TheoremisclWlkb 39960 Generalisation of isclwlk 25563: A pair of functions represents a closed walk iff it represents a walk in which the first vertex is equal to the last vertex. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 16-Feb-2021.)
 |-  ( F (ClWalkS `  G ) P 
 <->  ( F (1Walks `  G ) P  /\  ( P `
  0 )  =  ( P `  ( # `
  F ) ) ) )
 
Theoremclwlkis1wlk 39961 A closed walk is a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 16-Feb-2021.)
 |-  ( F (ClWalkS `  G ) P  ->  F (1Walks `  G ) P )
 
Theoremclwlk1wlk 39962 Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 16-Feb-2021.)
 |-  ( W  e.  (ClWalkS `  G )  ->  W  e.  (1Walks `  G ) )
 
Theoremclwlks1wlks 39963 Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 16-Feb-2021.)
 |-  (ClWalkS `  G )  C_  (1Walks `  G )
 
TheoremisclWlke 39964* Properties of a pair of functions to be a closed walk (in an undirected graph). (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 16-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   =>    |-  ( G  e.  X  ->  ( F (ClWalkS `  G ) P  <->  ( ( F  e. Word  dom  I  /\  P : ( 0 ... ( # `  F ) ) --> V ) 
 /\  ( A. k  e.  ( 0..^ ( # `  F ) )if- ( ( P `  k
 )  =  ( P `
  ( k  +  1 ) ) ,  ( I `  ( F `  k ) )  =  { ( P `
  k ) } ,  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  C_  ( I `  ( F `  k ) ) )  /\  ( P `
  0 )  =  ( P `  ( # `
  F ) ) ) ) ) )
 
TheoremclWlkcomp 39965* A closed walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 17-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  F  =  ( 1st `  W )   &    |-  P  =  ( 2nd `  W )   =>    |-  ( ( G  e.  X  /\  W  e.  ( S  X.  T ) ) 
 ->  ( W  e.  (ClWalkS `  G )  <->  ( ( F  e. Word  dom  I  /\  P : ( 0 ... ( # `  F ) ) --> V ) 
 /\  ( A. k  e.  ( 0..^ ( # `  F ) )if- ( ( P `  k
 )  =  ( P `
  ( k  +  1 ) ) ,  ( I `  ( F `  k ) )  =  { ( P `
  k ) } ,  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  C_  ( I `  ( F `  k ) ) )  /\  ( P `
  0 )  =  ( P `  ( # `
  F ) ) ) ) ) )
 
TheoremclWlkcompim 39966* Implications for the properties of the components of a closed walk. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 17-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  F  =  ( 1st `  W )   &    |-  P  =  ( 2nd `  W )   =>    |-  ( W  e.  (ClWalkS `  G )  ->  (
 ( F  e. Word  dom  I 
 /\  P : ( 0 ... ( # `  F ) ) --> V ) 
 /\  ( A. k  e.  ( 0..^ ( # `  F ) )if- ( ( P `  k
 )  =  ( P `
  ( k  +  1 ) ) ,  ( I `  ( F `  k ) )  =  { ( P `
  k ) } ,  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  C_  ( I `  ( F `  k ) ) )  /\  ( P `
  0 )  =  ( P `  ( # `
  F ) ) ) ) )
 
21.33.8.19  Circuits and cycles
 
Syntaxccrcts 39967 Extend class notation with circuits (in a graph).
 class CircuitS
 
Syntaxccycls 39968 Extend class notation with cycles (in a graph).
 class CycleS
 
Definitiondf-crcts 39969* Define the set of all circuits (in an undirected graph).

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A circuit can be a closed walk allowing repetitions of vertices but not edges;"; according to Wikipedia ("Glossary of graph theory terms", https://en.wikipedia.org/wiki/Glossary_of_graph_theory_terms, 3-Oct-2017): "A circuit may refer to ... a trail (a closed tour without repeated edges), ...".

Following Bollobas ("A trail whose endvertices coincide (a closed trail) is called a circuit.", see Definition of [Bollobas] p. 5.), a circuit is a closed trail without repeated edges. So the circuit is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0). (Contributed by Alexander van der Vekens, 3-Oct-2017.) (Revised by AV, 31-Jan-2021.)

 |- CircuitS  =  ( g  e.  _V  |->  {
 <. f ,  p >.  |  ( f (TrailS `  g
 ) p  /\  ( p `  0 )  =  ( p `  ( # `
  f ) ) ) } )
 
Definitiondf-cycls 39970* Define the set of all (simple) cycles (in an undirected graph).

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A simple cycle may be defined either as a closed walk with no repetitions of vertices and edges allowed, other than the repetition of the starting and ending vertex,"

According to Bollobas: "If a walk W = x0 x1 ... x(l) is such that l >= 3, x0=x(l), and the vertices x(i), 0 < i < l, are distinct from each other and x0, then W is said to be a cycle.", see Definition of [Bollobas] p. 5.

However, since a walk consisting of distinct vertices (except the first and the last vertex) is a path, a cycle can be defined as path whose first and last vertices coincide. So a cycle is represented by the following sequence: p(0) e(f(1)) p(1) ... p(n-1) e(f(n)) p(n)=p(0). (Contributed by Alexander van der Vekens, 3-Oct-2017.) (Revised by AV, 31-Jan-2021.)

 |- CycleS  =  ( g  e.  _V  |->  {
 <. f ,  p >.  |  ( f (PathS `  g ) p  /\  ( p `  0 )  =  ( p `  ( # `  f ) ) ) } )
 
TheoremcrctS 39971* The set of circuits (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
 |-  ( G  e.  W  ->  (CircuitS `  G )  =  { <. f ,  p >.  |  ( f (TrailS `  G ) p  /\  ( p `
  0 )  =  ( p `  ( # `
  f ) ) ) } )
 
TheoremcyclS 39972* The set of cycles (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
 |-  ( G  e.  W  ->  (CycleS `  G )  =  { <. f ,  p >.  |  ( f (PathS `  G ) p  /\  ( p `  0 )  =  ( p `  ( # `  f ) ) ) } )
 
TheoremisCrct 39973 Sufficient and necessary conditions for a pair of functions to be a circuit (in an undirected graph): A pair of function "is" (represents) a circuit iff it is a closed trail. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
 |-  (
 ( G  e.  W  /\  F  e.  U  /\  P  e.  Z )  ->  ( F (CircuitS `  G ) P  <->  ( F (TrailS `  G ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) )
 
TheoremisCycl 39974 Sufficient and necessary conditions for a pair of functions to be a cycle (in an undirected graph): A pair of function "is" (represents) a cycle iff it is a closed path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
 |-  (
 ( G  e.  W  /\  F  e.  U  /\  P  e.  Z )  ->  ( F (CycleS `  G ) P  <->  ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) )
 
Theoremcrctprop 39975 The properties of a circuit: A circuit is a closed trail. (Contributed by AV, 31-Jan-2021.)
 |-  ( F (CircuitS `  G ) P  ->  ( F (TrailS `  G ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) )
 
Theoremcyclprop 39976 The properties of a cycle: A cycle is a closed path. (Contributed by AV, 31-Jan-2021.)
 |-  ( F (CycleS `  G ) P  ->  ( F (PathS `  G ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) )
 
Theoremcrctisclwlk 39977 A circuit is a closed walk. (Contributed by AV, 17-Feb-2021.)
 |-  ( F (CircuitS `  G ) P  ->  F (ClWalkS `  G ) P )
 
TheoremcrctisTrl 39978 A circuit is a trail. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
 |-  ( F (CircuitS `  G ) P  ->  F (TrailS `  G ) P )
 
TheoremcyclisPth 39979 A cycle is a path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
 |-  ( F (CycleS `  G ) P  ->  F (PathS `  G ) P )
 
TheoremcyclisWlk 39980 A cycle is a walk. (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 31-Jan-2021.)
 |-  ( F (CycleS `  G ) P  ->  F (1Walks `  G ) P )
 
TheoremcyclisCrct 39981 A cycle is a circuit. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
 |-  ( F (CycleS `  G ) P  ->  F (CircuitS `  G ) P )
 
TheoremcyclnsPth 39982 A (non trivial) cycle is not a simple path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
 |-  ( F  =/=  (/)  ->  ( F (CycleS `  G ) P 
 ->  -.  F (SPathS `  G ) P ) )
 
TheoremcyclisPthon 39983 A cycle is a path starting and ending at its first vertex. (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 31-Jan-2021.)
 |-  ( F (CycleS `  G ) P  ->  F ( ( P `  0 ) (PathsOn `  G )
 ( P `  0
 ) ) P )
 
Theoremlfgrn1cycl 39984* In a loop-free graph there are no cycles with length 1 (consisting of one edge). (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 2-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   =>    |-  ( I : dom  I --> { x  e.  ~P V  |  2 
 <_  ( # `  x ) }  ->  ( F (CycleS `  G ) P  ->  ( # `  F )  =/=  1 ) )
 
Theoremumgrn1cycl 39985 In a multigraph graph (with no loops!) there are no cycles with length 1 (consisting of one edge). (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 2-Feb-2021.)
 |-  (
 ( G  e. UMGraph  /\  F (CycleS `  G ) P )  ->  ( # `  F )  =/=  1 )
 
Theoremuspgrn2crct 39986 In a simple pseudograph there are no circuits with length 2 (consisting of two edges). (Contributed by Alexander van der Vekens, 9-Nov-2017.) (Revised by AV, 3-Feb-2021.)
 |-  (
 ( G  e. USPGraph  /\  F (CircuitS `  G ) P )  ->  ( # `  F )  =/=  2 )
 
Theoremusgrn2cycl 39987 In a simple graph there are no cycles with length 2 (consisting of two edges). (Contributed by Alexander van der Vekens, 9-Nov-2017.) (Revised by AV, 4-Feb-2021.)
 |-  (
 ( G  e. USGraph  /\  F (CycleS `  G ) P )  ->  ( # `  F )  =/=  2 )
 
Theoremcrctcsh1wlkn0lem1 39988 Lemma for crctcsh1wlkn0 39999. (Contributed by AV, 13-Mar-2021.)
 |-  (
 ( A  e.  RR  /\  B  e.  NN )  ->  ( ( A  -  B )  +  1
 )  <_  A )
 
Theoremcrctcsh1wlkn0lem2 39989* Lemma for crctcsh1wlkn0 39999. (Contributed by AV, 12-Mar-2021.)
 |-  ( ph  ->  S  e.  (
 1..^ N ) )   &    |-  Q  =  ( x  e.  ( 0 ... N )  |->  if ( x  <_  ( N  -  S ) ,  ( P `  ( x  +  S ) ) ,  ( P `  ( ( x  +  S )  -  N ) ) ) )   =>    |-  ( ( ph  /\  J  e.  ( 0 ... ( N  -  S ) ) )  ->  ( Q `  J )  =  ( P `  ( J  +  S ) ) )
 
Theoremcrctcsh1wlkn0lem3 39990* Lemma for crctcsh1wlkn0 39999. (Contributed by AV, 12-Mar-2021.)
 |-  ( ph  ->  S  e.  (
 1..^ N ) )   &    |-  Q  =  ( x  e.  ( 0 ... N )  |->  if ( x  <_  ( N  -  S ) ,  ( P `  ( x  +  S ) ) ,  ( P `  ( ( x  +  S )  -  N ) ) ) )   =>    |-  ( ( ph  /\  J  e.  ( ( ( N  -  S )  +  1 ) ... N ) )  ->  ( Q `
  J )  =  ( P `  (
 ( J  +  S )  -  N ) ) )
 
Theoremcrctcsh1wlkn0lem4 39991* Lemma for crctcsh1wlkn0 39999. (Contributed by AV, 12-Mar-2021.)
 |-  ( ph  ->  S  e.  (
 1..^ N ) )   &    |-  Q  =  ( x  e.  ( 0 ... N )  |->  if ( x  <_  ( N  -  S ) ,  ( P `  ( x  +  S ) ) ,  ( P `  ( ( x  +  S )  -  N ) ) ) )   &    |-  H  =  ( F cyclShift  S )   &    |-  N  =  ( # `  F )   &    |-  ( ph  ->  F  e. Word  A )   &    |-  ( ph  ->  A. i  e.  ( 0..^ N )if- ( ( P `  i )  =  ( P `  ( i  +  1 ) ) ,  ( I `  ( F `  i ) )  =  { ( P `
  i ) } ,  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  C_  ( I `  ( F `  i ) ) ) )   =>    |-  ( ph  ->  A. j  e.  ( 0..^ ( N  -  S ) )if- ( ( Q `  j )  =  ( Q `  ( j  +  1 ) ) ,  ( I `  ( H `  j ) )  =  { ( Q `
  j ) } ,  { ( Q `  j ) ,  ( Q `  ( j  +  1 ) ) }  C_  ( I `  ( H `  j ) ) ) )
 
Theoremcrctcsh1wlkn0lem5 39992* Lemma for crctcsh1wlkn0 39999. (Contributed by AV, 12-Mar-2021.)
 |-  ( ph  ->  S  e.  (
 1..^ N ) )   &    |-  Q  =  ( x  e.  ( 0 ... N )  |->  if ( x  <_  ( N  -  S ) ,  ( P `  ( x  +  S ) ) ,  ( P `  ( ( x  +  S )  -  N ) ) ) )   &    |-  H  =  ( F cyclShift  S )   &    |-  N  =  ( # `  F )   &    |-  ( ph  ->  F  e. Word  A )   &    |-  ( ph  ->  A. i  e.  ( 0..^ N )if- ( ( P `  i )  =  ( P `  ( i  +  1 ) ) ,  ( I `  ( F `  i ) )  =  { ( P `
  i ) } ,  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  C_  ( I `  ( F `  i ) ) ) )   =>    |-  ( ph  ->  A. j  e.  ( ( ( N  -  S )  +  1 )..^ N )if- ( ( Q `  j
 )  =  ( Q `
  ( j  +  1 ) ) ,  ( I `  ( H `  j ) )  =  { ( Q `
  j ) } ,  { ( Q `  j ) ,  ( Q `  ( j  +  1 ) ) }  C_  ( I `  ( H `  j ) ) ) )
 
Theoremcrctcsh1wlkn0lem6 39993* Lemma for crctcsh1wlkn0 39999. (Contributed by AV, 12-Mar-2021.)
 |-  ( ph  ->  S  e.  (
 1..^ N ) )   &    |-  Q  =  ( x  e.  ( 0 ... N )  |->  if ( x  <_  ( N  -  S ) ,  ( P `  ( x  +  S ) ) ,  ( P `  ( ( x  +  S )  -  N ) ) ) )   &    |-  H  =  ( F cyclShift  S )   &    |-  N  =  ( # `  F )   &    |-  ( ph  ->  F  e. Word  A )   &    |-  ( ph  ->  A. i  e.  ( 0..^ N )if- ( ( P `  i )  =  ( P `  ( i  +  1 ) ) ,  ( I `  ( F `  i ) )  =  { ( P `
  i ) } ,  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  C_  ( I `  ( F `  i ) ) ) )   &    |-  ( ph  ->  ( P `  N )  =  ( P `  0 ) )   =>    |-  ( ( ph  /\  J  =  ( N  -  S ) ) 
 -> if- ( ( Q `  J )  =  ( Q `  ( J  +  1 ) ) ,  ( I `  ( H `  J ) )  =  { ( Q `
  J ) } ,  { ( Q `  J ) ,  ( Q `  ( J  +  1 ) ) }  C_  ( I `  ( H `  J ) ) ) )
 
Theoremcrctcsh1wlkn0lem7 39994* Lemma for crctcsh1wlkn0 39999. (Contributed by AV, 12-Mar-2021.)
 |-  ( ph  ->  S  e.  (
 1..^ N ) )   &    |-  Q  =  ( x  e.  ( 0 ... N )  |->  if ( x  <_  ( N  -  S ) ,  ( P `  ( x  +  S ) ) ,  ( P `  ( ( x  +  S )  -  N ) ) ) )   &    |-  H  =  ( F cyclShift  S )   &    |-  N  =  ( # `  F )   &    |-  ( ph  ->  F  e. Word  A )   &    |-  ( ph  ->  A. i  e.  ( 0..^ N )if- ( ( P `  i )  =  ( P `  ( i  +  1 ) ) ,  ( I `  ( F `  i ) )  =  { ( P `
  i ) } ,  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  C_  ( I `  ( F `  i ) ) ) )   &    |-  ( ph  ->  ( P `  N )  =  ( P `  0 ) )   =>    |-  ( ph  ->  A. j  e.  ( 0..^ N )if- ( ( Q `  j )  =  ( Q `  ( j  +  1
 ) ) ,  ( I `  ( H `  j ) )  =  { ( Q `  j ) } ,  { ( Q `  j ) ,  ( Q `  ( j  +  1 ) ) }  C_  ( I `  ( H `  j ) ) ) )
 
Theoremcrctcshlem1 39995 Lemma for crctcsh 40002. (Contributed by AV, 10-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  F (CircuitS `  G ) P )   &    |-  N  =  ( # `  F )   =>    |-  ( ph  ->  N  e.  NN0 )
 
Theoremcrctcshlem2 39996 Lemma for crctcsh 40002. (Contributed by AV, 10-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  F (CircuitS `  G ) P )   &    |-  N  =  ( # `  F )   &    |-  ( ph  ->  S  e.  (
 0..^ N ) )   &    |-  H  =  ( F cyclShift  S )   =>    |-  ( ph  ->  ( # `
  H )  =  N )
 
Theoremcrctcshlem3 39997* Lemma for crctcsh 40002. (Contributed by AV, 10-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  F (CircuitS `  G ) P )   &    |-  N  =  ( # `  F )   &    |-  ( ph  ->  S  e.  (
 0..^ N ) )   &    |-  H  =  ( F cyclShift  S )   &    |-  Q  =  ( x  e.  ( 0
 ... N )  |->  if ( x  <_  ( N  -  S ) ,  ( P `  ( x  +  S )
 ) ,  ( P `
  ( ( x  +  S )  -  N ) ) ) )   =>    |-  ( ph  ->  ( G  e.  _V  /\  H  e.  _V  /\  Q  e.  _V ) )
 
Theoremcrctcshlem4 39998* Lemma for crctcsh 40002. (Contributed by AV, 10-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  F (CircuitS `  G ) P )   &    |-  N  =  ( # `  F )   &    |-  ( ph  ->  S  e.  (
 0..^ N ) )   &    |-  H  =  ( F cyclShift  S )   &    |-  Q  =  ( x  e.  ( 0
 ... N )  |->  if ( x  <_  ( N  -  S ) ,  ( P `  ( x  +  S )
 ) ,  ( P `
  ( ( x  +  S )  -  N ) ) ) )   =>    |-  ( ( ph  /\  S  =  0 )  ->  ( H  =  F  /\  Q  =  P ) )
 
Theoremcrctcsh1wlkn0 39999* Cyclically shifting the indices of a circuit  <. F ,  P >. results in a 1-walk  <. H ,  Q >.. (Contributed by AV, 10-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  F (CircuitS `  G ) P )   &    |-  N  =  ( # `  F )   &    |-  ( ph  ->  S  e.  (
 0..^ N ) )   &    |-  H  =  ( F cyclShift  S )   &    |-  Q  =  ( x  e.  ( 0
 ... N )  |->  if ( x  <_  ( N  -  S ) ,  ( P `  ( x  +  S )
 ) ,  ( P `
  ( ( x  +  S )  -  N ) ) ) )   =>    |-  ( ( ph  /\  S  =/=  0 )  ->  H (1Walks `  G ) Q )
 
Theoremcrctcsh1wlk 40000* Cyclically shifting the indices of a circuit  <. F ,  P >. results in a 1-walk  <. H ,  Q >.. (Contributed by AV, 10-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  F (CircuitS `  G ) P )   &    |-  N  =  ( # `  F )   &    |-  ( ph  ->  S  e.  (
 0..^ N ) )   &    |-  H  =  ( F cyclShift  S )   &    |-  Q  =  ( x  e.  ( 0
 ... N )  |->  if ( x  <_  ( N  -  S ) ,  ( P `  ( x  +  S )
 ) ,  ( P `
  ( ( x  +  S )  -  N ) ) ) )   =>    |-  ( ph  ->  H (1Walks `  G ) Q )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41046
  Copyright terms: Public domain < Previous  Next >