P. 164 of Theorem List - Intuitionistic Logic Explorer

Type

Label

Description

Statement

Theorem

clwwlkn2 16301

A closed walk of length 2 represented as word is a word consisting of 2 symbols representing (not necessarily different) vertices connected by (at least) one edge. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 25-Apr-2021.)

|- ( W e. ( 2 ClWWalksN G ) <-> ( (♯ ` W ) = 2 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) )

Theorem

clwwlkext2edg 16302

If a word concatenated with a vertex represents a closed walk (in a graph), there is an edge between this vertex and the last vertex of the word, and between this vertex and the first vertex of the word. (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Revised by AV, 27-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.)

|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( ( W e. Word V /\ Z e. V /\ N e. ( ZZ>= ` 2 ) ) /\ ( W ++ <" Z "> ) e. ( N ClWWalksN G ) ) -> ( { (lastS ` W ) , Z } e. E /\ { Z , ( W ` 0 ) } e. E ) )

Theorem

clwwlknccat 16303

The concatenation of two words representing closed walks anchored at the same vertex represents a closed walk with a length which is the sum of the lengths of the two walks. The resulting walk is a "double loop", starting at the common vertex, coming back to the common vertex by the first walk, following the second walk and finally coming back to the common vertex again. (Contributed by AV, 24-Apr-2022.)

|- ( ( A e. ( M ClWWalksN G ) /\ B e. ( N ClWWalksN G ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( A ++ B ) e. ( ( M + N ) ClWWalksN G ) )

Theorem

umgr2cwwk2dif 16304

If a word represents a closed walk of length at least 2 in a multigraph, the first two symbols of the word must be different. (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 30-Apr-2021.)

|- ( ( G e. UMGraph /\ N e. ( ZZ>= ` 2 ) /\ W e. ( N ClWWalksN G ) ) -> ( W ` 1 ) =/= ( W ` 0 ) )

Theorem

umgr2cwwkdifex 16305*

If a word represents a closed walk of length at least 2 in a undirected simple graph, there must be a symbol different from the first symbol of the word. (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 30-Apr-2021.)

|- ( ( G e. UMGraph /\ N e. ( ZZ>= ` 2 ) /\ W e. ( N ClWWalksN G ) ) -> E. i e. ( 0..^ N ) ( W ` i ) =/= ( W ` 0 ) )

12.3.3.3  Closed walks on a vertex of a fixed length as words

Syntax

cclwwlknon 16306

Extend class notation with closed walks (in an undirected graph) anchored at a fixed vertex and of a fixed length as word over the set of vertices.

class ClWWalksNOn

Definition

df-clwwlknon 16307*

Define the set of all closed walks a graph g, anchored at a fixed vertex v (i.e., a walk starting and ending at the fixed vertex v, also called "a closed walk on vertex v") and having a fixed length n as words over the set of vertices. Such a word corresponds to the sequence v=p(0) p(1) ... p(n-1) of the vertices in a closed walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0)=v . The set ( ( v (ClWWalksNOn ` g ) n ) corresponds to the set of "walks from v to v of length n" in a statement of [Huneke] p. 2. (Contributed by AV, 24-Feb-2022.)

|- ClWWalksNOn = ( g e. _V |-> ( v e. (Vtx ` g ) , n e. NN0 |-> { w e. ( n ClWWalksN g ) | ( w ` 0 ) = v } ) )

Theorem

clwwlknonmpo 16308*

(ClWWalksNOn ` G ) is an operator mapping a vertex v and a nonnegative integer n to the set of closed walks on v of length n as words over the set of vertices in a graph G. (Contributed by AV, 25-Feb-2022.) (Proof shortened by AV, 2-Mar-2024.)

|- (ClWWalksNOn ` G ) = ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )

Theorem

clwwlknon 16309*

The set of closed walks on vertex X of length N in a graph G as words over the set of vertices. (Contributed by Alexander van der Vekens, 14-Sep-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 24-Mar-2022.)

|- ( X (ClWWalksNOn ` G ) N ) = { w e. ( N ClWWalksN G ) | ( w ` 0 ) = X }

Theorem

isclwwlknon 16310

A word over the set of vertices representing a closed walk on vertex X of length N in a graph G. (Contributed by AV, 25-Feb-2022.) (Revised by AV, 24-Mar-2022.)

|- ( W e. ( X (ClWWalksNOn ` G ) N ) <-> ( W e. ( N ClWWalksN G ) /\ ( W ` 0 ) = X ) )

Theorem

clwwlk0on0 16311

There is no word over the set of vertices representing a closed walk on vertex X of length 0 in a graph G. (Contributed by AV, 17-Feb-2022.) (Revised by AV, 25-Feb-2022.)

|- ( X (ClWWalksNOn ` G ) 0 ) = (/)

Theorem

clwwlknonel 16312*

Characterization of a word over the set of vertices representing a closed walk on vertex X of (nonzero) length N in a graph G. This theorem would not hold for N = 0 if W = X = (/). (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 24-Mar-2022.)

|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( N =/= 0 -> ( W e. ( X (ClWWalksNOn ` G ) N ) <-> ( ( W e. Word V /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ (♯ ` W ) = N /\ ( W ` 0 ) = X ) ) )

Theorem

clwwlknonccat 16313

The concatenation of two words representing closed walks on a vertex X represents a closed walk on vertex X. The resulting walk is a "double loop", starting at vertex X, coming back to X by the first walk, following the second walk and finally coming back to X again. (Contributed by AV, 24-Apr-2022.)

|- ( ( A e. ( X (ClWWalksNOn ` G ) M ) /\ B e. ( X (ClWWalksNOn ` G ) N ) ) -> ( A ++ B ) e. ( X (ClWWalksNOn ` G ) ( M + N ) ) )

Theorem

clwwlknon2 16314*

The set of closed walks on vertex X of length 2 in a graph G as words over the set of vertices. (Contributed by AV, 5-Mar-2022.) (Revised by AV, 25-Mar-2022.)

|- C = (ClWWalksNOn ` G )   =>    |- ( X C 2 ) = { w e. ( 2 ClWWalksN G ) | ( w ` 0 ) = X }

Theorem

clwwlknon2x 16315*

The set of closed walks on vertex X of length 2 in a graph G as words over the set of vertices, definition of ClWWalksN expanded. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 25-Mar-2022.)

|- C = (ClWWalksNOn ` G )   &    |- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( X C 2 ) = { w e. Word V | ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E /\ ( w ` 0 ) = X ) }

Theorem

s2elclwwlknon2 16316

Sufficient conditions of a doubleton word to represent a closed walk on vertex X of length 2. (Contributed by AV, 11-May-2022.)

|- C = (ClWWalksNOn ` G )   &    |- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( X e. V /\ Y e. V /\ { X , Y } e. E ) -> <" X Y "> e. ( X C 2 ) )

Theorem

clwwlknonex2lem1 16317

Lemma 1 for clwwlknonex2 16319: Transformation of a special half-open integer range into a union of a smaller half-open integer range and an unordered pair. This Lemma would not hold for N = 2, i.e., (♯ ` W ) = 0, because ( 0..^ ( ( (♯ ` W ) + 2 ) - 1 ) ) = ( 0..^ ( ( 0 + 2 ) - 1 ) ) = ( 0..^ 1 ) = { 0 } =/= { -u 1 , 0 } = ( (/) u. { -u 1 , 0 } ) = ( ( 0..^ ( 0 - 1 ) ) u. { ( 0 - 1 ) , 0 } ) = ( ( 0..^ ( (♯ ` W ) - 1 ) ) u. { ( (♯ ` W ) - 1 ) , (♯ ` W ) } ). (Contributed by AV, 22-Sep-2018.) (Revised by AV, 26-Jan-2022.)

|- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( 0..^ ( ( (♯ ` W ) + 2 ) - 1 ) ) = ( ( 0..^ ( (♯ ` W ) - 1 ) ) u. { ( (♯ ` W ) - 1 ) , (♯ ` W ) } ) )

Theorem

clwwlknonex2lem2 16318*

Lemma 2 for clwwlknonex2 16319: Transformation of a walk and two edges into a walk extended by two vertices/edges. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 27-Jan-2022.)

|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( ( W e. Word V /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ (♯ ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ { X , Y } e. E ) -> A. i e. ( ( 0..^ ( (♯ ` W ) - 1 ) ) u. { ( (♯ ` W ) - 1 ) , (♯ ` W ) } ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E )

Theorem

clwwlknonex2 16319

Extending a closed walk W on vertex X by an additional edge (forth and back) results in a closed walk. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 28-Mar-2022.)

|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) /\ { X , Y } e. E /\ W e. ( X (ClWWalksNOn ` G ) ( N - 2 ) ) ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) )

Theorem

clwwlknonex2e 16320

Extending a closed walk W on vertex X by an additional edge (forth and back) results in a closed walk on vertex X. (Contributed by AV, 17-Apr-2022.)

|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) /\ { X , Y } e. E /\ W e. ( X (ClWWalksNOn ` G ) ( N - 2 ) ) ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( X (ClWWalksNOn ` G ) N ) )

Theorem

clwwlknun 16321*

The set of closed walks of fixed length N in a simple graph G is the union of the closed walks of the fixed length N on each of the vertices of graph G. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 3-Mar-2022.) (Proof shortened by AV, 28-Mar-2022.)

|- V = (Vtx ` G )   =>    |- ( G e. USGraph -> ( N ClWWalksN G ) = U_ x e. V ( x (ClWWalksNOn ` G ) N ) )

12.4  Eulerian paths and the Konigsberg Bridge problem

12.4.1  Eulerian paths

According to Wikipedia ("Eulerian path", 9-Mar-2021, https://en.wikipedia.org/wiki/Eulerian_path): "In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends on the same vertex. ... The term Eulerian graph has two common meanings in graph theory. One meaning is a graph with an Eulerian circuit, and the other is a graph with every vertex of even degree. These definitions coincide for connected graphs. ... A graph that has an Eulerian trail but not an Eulerian circuit is called semi-Eulerian."

Syntax

ceupth 16322

Extend class notation with Eulerian paths.

class EulerPaths

Definition

df-eupth 16323*

Define the set of all Eulerian paths on an arbitrary graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)

|- EulerPaths = ( g e. _V |-> { <. f , p >. | ( f (Trails ` g ) p /\ f : ( 0..^ (♯ ` f ) ) -onto-> dom (iEdg ` g ) ) } )

Theorem

releupth 16324

The set (EulerPaths ` G ) of all Eulerian paths on G is a set of pairs by our definition of an Eulerian path, and so is a relation. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)

|- Rel (EulerPaths ` G )

Theorem

eupthsg 16325*

The Eulerian paths on the graph G. (Contributed by AV, 18-Feb-2021.) (Revised by AV, 29-Oct-2021.)

|- I = (iEdg ` G )   =>    |- ( G e. V -> (EulerPaths ` G ) = { <. f , p >. | ( f (Trails ` G ) p /\ f : ( 0..^ (♯ ` f ) ) -onto-> dom I ) } )

Theorem

eupthv 16326

The classes involved in a Eulerian path are sets. (Contributed by Jim Kingdon, 13-Mar-2026.)

|- ( F (EulerPaths ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )

Theorem

iseupth 16327

The property " <. F , P >. is an Eulerian path on the graph G". An Eulerian path is defined as bijection F from the edges to a set 0 ... ( N - 1 ) and a function P : ( 0 ... N ) --> V into the vertices such that for each 0 <_ k < N, F ( k ) is an edge from P ( k ) to P ( k + 1 ). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.) (Revised by AV, 30-Oct-2021.)

|- I = (iEdg ` G )   =>    |- ( F (EulerPaths ` G ) P <-> ( F (Trails ` G ) P /\ F : ( 0..^ (♯ ` F ) ) -onto-> dom I ) )

Theorem

iseupthf1o 16328

The property " <. F , P >. is an Eulerian path on the graph G". An Eulerian path is defined as bijection F from the edges to a set 0 ... ( N - 1 ) and a function P : ( 0 ... N ) --> V into the vertices such that for each 0 <_ k < N, F ( k ) is an edge from P ( k ) to P ( k + 1 ). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.) (Revised by AV, 30-Oct-2021.)

|- I = (iEdg ` G )   =>    |- ( F (EulerPaths ` G ) P <-> ( F (Walks ` G ) P /\ F : ( 0..^ (♯ ` F ) ) -1-1-onto-> dom I ) )

Theorem

eupthi 16329

Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)

|- I = (iEdg ` G )   =>    |- ( F (EulerPaths ` G ) P -> ( F (Walks ` G ) P /\ F : ( 0..^ (♯ ` F ) ) -1-1-onto-> dom I ) )

Theorem

eupthf1o 16330

The F function in an Eulerian path is a bijection from a half-open range of nonnegative integers to the set of edges. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)

|- I = (iEdg ` G )   =>    |- ( F (EulerPaths ` G ) P -> F : ( 0..^ (♯ ` F ) ) -1-1-onto-> dom I )

Theorem

eupthfi 16331

Any graph with an Eulerian path is of finite size, i.e. with a finite number of edges. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 18-Feb-2021.)

|- I = (iEdg ` G )   =>    |- ( F (EulerPaths ` G ) P -> dom I e. Fin )

Theorem

eupthseg 16332

The N-th edge in an eulerian path is the edge having P ( N ) and P ( N + 1 ) as endpoints . (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)

|- I = (iEdg ` G )   =>    |- ( ( F (EulerPaths ` G ) P /\ N e. ( 0..^ (♯ ` F ) ) ) -> { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) )

Theorem

eupthcl 16333

An Eulerian path has length ♯ ( F ), which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)

|- ( F (EulerPaths ` G ) P -> (♯ ` F ) e. NN0 )

Theorem

eupthistrl 16334

An Eulerian path is a trail. (Contributed by Alexander van der Vekens, 24-Nov-2017.) (Revised by AV, 18-Feb-2021.)

|- ( F (EulerPaths ` G ) P -> F (Trails ` G ) P )

Theorem

eupthiswlk 16335

An Eulerian path is a walk. (Contributed by AV, 6-Apr-2021.)

|- ( F (EulerPaths ` G ) P -> F (Walks ` G ) P )

Theorem

eupthpf 16336

The P function in an Eulerian path is a function from a finite sequence of nonnegative integers to the vertices. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)

|- ( F (EulerPaths ` G ) P -> P : ( 0 ... (♯ ` F ) ) --> (Vtx ` G ) )

Theorem

eupthres 16337

The restriction <. H , Q >. of an Eulerian path <. F , P >. to an initial segment of the path (of length N) forms an Eulerian path on the subgraph S consisting of the edges in the initial segment. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 6-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> F (EulerPaths ` G ) P )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> (iEdg ` S ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- H = ( F prefix N )   &    |- Q = ( P |` ( 0 ... N ) )   &    |- (Vtx ` S ) = V   =>    |- ( ph -> H (EulerPaths ` S ) Q )

Theorem

eupth2lem1 16338

Lemma for eupth2 . (Contributed by Mario Carneiro, 8-Apr-2015.)

|- ( U e. V -> ( U e. if ( A = B , (/) , { A , B } ) <-> ( A =/= B /\ ( U = A \/ U = B ) ) ) )

Theorem

eupth2lem2dc 16339

Lemma for eupth2 . (Contributed by Mario Carneiro, 8-Apr-2015.)

|- ( ph -> B e. X )   &    |- ( ph -> DECID A = B )   &    |- ( ph -> B =/= C )   &    |- ( ph -> B = U )   =>    |- ( ph -> ( -. U e. if ( A = B , (/) , { A , B } ) <-> U e. if ( A = C , (/) , { A , C } ) ) )

Theorem

trlsegvdeglem1 16340

Lemma for trlsegvdeg . (Contributed by AV, 20-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   =>    |- ( ph -> ( ( P ` N ) e. V /\ ( P ` ( N + 1 ) ) e. V ) )

Theorem

trlsegvdeglem2 16341

Lemma for trlsegvdeg . (Contributed by AV, 20-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   =>    |- ( ph -> Fun (iEdg ` X ) )

Theorem

trlsegvdeglem3 16342

Lemma for trlsegvdeg . (Contributed by AV, 20-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   =>    |- ( ph -> Fun (iEdg ` Y ) )

Theorem

trlsegvdeglem4 16343

Lemma for trlsegvdeg . (Contributed by AV, 21-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   =>    |- ( ph -> dom (iEdg ` X ) = ( ( F " ( 0..^ N ) ) i^i dom I ) )

Theorem

trlsegvdeglem5 16344

Lemma for trlsegvdeg . (Contributed by AV, 21-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   =>    |- ( ph -> dom (iEdg ` Y ) = { ( F ` N ) } )

Theorem

trlsegvdeglem6 16345

Lemma for trlsegvdeg . (Contributed by AV, 21-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   =>    |- ( ph -> dom (iEdg ` X ) e. Fin )

Theorem

trlsegvdeglem7 16346

Lemma for trlsegvdeg . (Contributed by AV, 21-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   =>    |- ( ph -> dom (iEdg ` Y ) e. Fin )

Theorem

trlsegvdegfi 16347

The effect on vertex degree of adding one edge to a trail. In the following, a subgraph induced by a segment of a trail is called a "subtrail": For any subtrail Z of a trail <. F , P >. in a pseudograph G which is composed of subtrails X and Y, where Y consists of a single edge, the vertex degree of any vertex U within Z is the sum of the vertex degree of U within X and the vertex degree of U within Y. Note that this theorem would not hold for arbitrary walks (if the last edge was identical with a previous edge, the degree of the vertices incident with this edge would not be increased because of this edge). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   &    |- ( ph -> G e. UPGraph )   &    |- ( ph -> V e. Fin )   =>    |- ( ph -> ( (VtxDeg ` Z ) ` U ) = ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) )

Theorem

eupth2lem3lem1fi 16348

Lemma for eupth2lem3fi 16356. (Contributed by AV, 21-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   &    |- ( ph -> G e. UPGraph )   &    |- ( ph -> V e. Fin )   =>    |- ( ph -> ( (VtxDeg ` X ) ` U ) e. NN0 )

Theorem

eupth2lem3lem2fi 16349

Lemma for eupth2lem3fi 16356. (Contributed by AV, 21-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   &    |- ( ph -> G e. UPGraph )   &    |- ( ph -> V e. Fin )   =>    |- ( ph -> ( (VtxDeg ` Y ) ` U ) e. NN0 )

Theorem

eupth2lem3lem3fi 16350*

Lemma for eupth2lem3fi 16356. If a loop { ( P ` N ) , ( P ` ( N + 1 ) ) } is added to a trail, the degree of the vertices with odd degree remains odd (regarding the subgraphs induced by the involved trails). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 21-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   &    |- ( ph -> G e. UPGraph )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )   &    |- ( ph -> if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )   =>    |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )

Theorem

eupth2lem3lem6fi 16351*

If an edge (not a loop) is added to a trail, the degree of vertices not being end vertices of this edge remains odd if it was odd before (regarding the subgraphs induced by the involved trails). Remark: This seems to be not valid for hyperedges joining more vertices than ( P ` 0 ) and ( P ` N ): if there is a third vertex in the edge, and this vertex is already contained in the trail, then the degree of this vertex could be affected by this edge! (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 25-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   &    |- ( ph -> G e. UPGraph )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )   &    |- ( ph -> ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } )   =>    |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )

Theorem

eupth2lem3lem5 16352

Lemma for eupth2fi 16359. (Contributed by AV, 25-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   &    |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )   &    |- ( ph -> ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } )   =>    |- ( ph -> ( I ` ( F ` N ) ) e. ~P V )

Theorem

eupth2lem3lem4fi 16353*

Lemma for eupth2lem3fi 16356. If an edge (not a loop) is added to a trail, the degree of the end vertices of this edge remains odd if it was odd before (regarding the subgraphs induced by the involved trails). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 25-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   &    |- ( ph -> G e. UMGraph )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )   &    |- ( ph -> if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )   &    |- ( ph -> ( I ` ( F ` N ) ) e. ~P V )   =>    |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U = ( P ` N ) \/ U = ( P ` ( N + 1 ) ) ) ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )

Theorem

eupth2lem3lem7fi 16354*

Lemma for eupth2lem3fi 16356: Combining trlsegvdegfi 16347, eupth2lem3lem3fi 16350, eupth2lem3lem4fi 16353 and eupth2lem3lem6fi 16351. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 27-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> (Vtx ` X ) = V )   &    |- ( ph -> (Vtx ` Y ) = V )   &    |- ( ph -> (Vtx ` Z ) = V )   &    |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )   &    |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )   &    |- ( ph -> G e. UMGraph )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )   &    |- ( ph -> ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } )   =>    |- ( ph -> ( -. 2 || ( (VtxDeg ` Z ) ` U ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )

Theorem

eupthvdres 16355

The vertex degree remains the same for all vertices if the edges are restricted to the edges of an Eulerian path. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> G e. W )   &    |- ( ph -> Fun I )   &    |- ( ph -> F (EulerPaths ` G ) P )   &    |- H = <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >.   =>    |- ( ph -> (VtxDeg ` H ) = (VtxDeg ` G ) )

Theorem

eupth2lem3fi 16356*

Lemma for eupth2fi 16359. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> G e. UMGraph )   &    |- ( ph -> Fun I )   &    |- ( ph -> F (EulerPaths ` G ) P )   &    |- ( ph -> V e. Fin )   &    |- H = <. V , ( I |` ( F " ( 0..^ N ) ) ) >.   &    |- X = <. V , ( I |` ( F " ( 0..^ ( N + 1 ) ) ) ) >.   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> ( N + 1 ) <_ (♯ ` F ) )   &    |- ( ph -> U e. V )   &    |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` H ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )   =>    |- ( ph -> ( -. 2 || ( (VtxDeg ` X ) ` U ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )

Theorem

eupth2lembfi 16357*

Lemma for eupth2fi 16359 (induction basis): There are no vertices of odd degree in an Eulerian path of length 0, having no edge and identical endpoints (the single vertex of the Eulerian path). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> G e. UMGraph )   &    |- ( ph -> Fun I )   &    |- ( ph -> F (EulerPaths ` G ) P )   &    |- ( ph -> V e. Fin )   =>    |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) } = (/) )

Theorem

eupth2lemsfi 16358*

Lemma for eupth2fi 16359 (induction step): The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct, if the Eulerian path shortened by one edge has this property. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> G e. UMGraph )   &    |- ( ph -> Fun I )   &    |- ( ph -> F (EulerPaths ` G ) P )   &    |- ( ph -> V e. Fin )   =>    |- ( ( ph /\ n e. NN0 ) -> ( ( n <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) -> ( ( n + 1 ) <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) ) ) )

Theorem

eupth2fi 16359*

The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> G e. UMGraph )   &    |- ( ph -> Fun I )   &    |- ( ph -> F (EulerPaths ` G ) P )   &    |- ( ph -> V e. Fin )   =>    |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) } = if ( ( P ` 0 ) = ( P ` (♯ ` F ) ) , (/) , { ( P ` 0 ) , ( P ` (♯ ` F ) ) } ) )

Theorem

eulerpathprum 16360*

A graph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 26-Feb-2021.)

|- V = (Vtx ` G )   =>    |- ( ( G e. UMGraph /\ F (EulerPaths ` G ) P /\ V e. Fin ) -> (♯ ` { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) } ) e. { 0 , 2 } )

Theorem

eulerpathum 16361*

A multigraph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 26-Feb-2021.)

|- V = (Vtx ` G )   =>    |- ( ( G e. UMGraph /\ E. j j e. (EulerPaths ` G ) /\ V e. Fin ) -> (♯ ` { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) } ) e. { 0 , 2 } )

12.4.2  The Königsberg Bridge problem

According to Wikipedia ("Seven Bridges of Königsberg", 9-Mar-2021, https://en.wikipedia.org/wiki/Seven_Bridges_of_Koenigsberg): "The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1736 laid the foundations of graph theory and prefigured the idea of topology. The city of Königsberg in [East] Prussia (now Kaliningrad, Russia) was set on both sides of the Pregel River, and included two large islands - Kneiphof and Lomse - which were connected to each other, or to the two mainland portions of the city, by seven bridges. The problem was to devise a walk through the city that would cross each of those bridges once and only once.". Euler proved that the problem has no solution by applying Euler's theorem to the Königsberg graph, which is obtained by replacing each land mass with an abstract "vertex" or node, and each bridge with an abstract connection, an "edge", which connects two land masses/vertices. The Königsberg graph G is a multigraph consisting of 4 vertices and 7 edges, represented by the following ordered pair: G = <. ( 0 ... 3 ) , <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } "> >., see konigsbergumgr 16367. konigsberg 16373 shows that the Königsberg graph has no Eulerian path, thus the Königsberg Bridge problem has no solution.

Theorem

konigsbergvtx 16362

The set of vertices of the Königsberg graph G. (Contributed by AV, 28-Feb-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- (Vtx ` G ) = ( 0 ... 3 )

Theorem

konigsbergiedg 16363

The indexed edges of the Königsberg graph G. (Contributed by AV, 28-Feb-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- (iEdg ` G ) = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">

Theorem

konigsbergiedgwen 16364*

The indexed edges of the Königsberg graph G is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- E e. Word { x e. ~P V | x ~~ 2o }

Theorem

konigsbergssiedgwpren 16365*

Each subset of the indexed edges of the Königsberg graph G is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- ( ( A e. Word _V /\ B e. Word _V /\ E = ( A ++ B ) ) -> A e. Word { x e. ~P V | x ~~ 2o } )

Theorem

konigsbergssiedgwen 16366*

Each subset of the indexed edges of the Königsberg graph G is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- ( ( A e. Word _V /\ B e. Word _V /\ E = ( A ++ B ) ) -> A e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )

Theorem

konigsbergumgr 16367

The Königsberg graph G is a multigraph. (Contributed by AV, 28-Feb-2021.) (Revised by AV, 9-Mar-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- G e. UMGraph

Theorem

konigsberglem1 16368

Lemma 1 for konigsberg 16373: Vertex 0 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- ( (VtxDeg ` G ) ` 0 ) = 3

Theorem

konigsberglem2 16369

Lemma 2 for konigsberg 16373: Vertex 1 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- ( (VtxDeg ` G ) ` 1 ) = 3

Theorem

konigsberglem3 16370

Lemma 3 for konigsberg 16373: Vertex 3 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- ( (VtxDeg ` G ) ` 3 ) = 3

Theorem

konigsberglem4 16371*

Lemma 4 for konigsberg 16373: Vertices 0 , 1 , 3 are vertices of odd degree. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 28-Feb-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- { 0 , 1 , 3 } C_ { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) }

Theorem

konigsberglem5 16372*

Lemma 5 for konigsberg 16373: The set of vertices of odd degree is greater than 2. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 28-Feb-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- 2 < (♯ ` { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) } )

Theorem

konigsberg 16373

The Königsberg Bridge problem. If G is the Königsberg graph, i.e. a graph on four vertices 0 , 1 , 2 , 3, with edges { 0 , 1 } , { 0 , 2 } , { 0 , 3 } , { 1 , 2 } , { 1 , 2 } , { 2 , 3 } , { 2 , 3 }, then vertices 0 , 1 , 3 each have degree three, and 2 has degree five, so there are four vertices of odd degree and thus by eulerpathum 16361 the graph cannot have an Eulerian path. It is sufficient to show that there are 3 vertices of odd degree, since a graph having an Eulerian path can only have 0 or 2 vertices of odd degree. This is Metamath 100 proof #54. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 9-Mar-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- (EulerPaths ` G ) = (/)

PART 13  GUIDES AND MISCELLANEA

13.1  Guides (conventions, explanations, and examples)

13.1.1  Conventions

This section describes the conventions we use. These conventions often refer to existing mathematical practices, which are discussed in more detail in other references. The following sources lay out how mathematics is developed without the law of the excluded middle. Of course, there are a greater number of sources which assume excluded middle and most of what is in them applies here too (especially in a treatment such as ours which is built on first-order logic and set theory, rather than, say, type theory). Studying how a topic is treated in the Metamath Proof Explorer and the references therein is often a good place to start (and is easy to compare with the Intuitionistic Logic Explorer). The textbooks provide a motivation for what we are doing, whereas Metamath lets you see in detail all hidden and implicit steps. Most standard theorems are accompanied by citations. Some closely followed texts include the following:

• Axioms of propositional calculus - Stanford Encyclopedia of Philosophy or [Heyting].
• Axioms of predicate calculus - our axioms are adapted from the ones in the Metamath Proof Explorer.
• Theorems of propositional calculus - [Heyting].
• Theorems of pure predicate calculus - Metamath Proof Explorer.
• Theorems of equality and substitution - Metamath Proof Explorer.
• Axioms of set theory - [Crosilla].
• Development of set theory - Chapter 10 of [HoTT].
• Construction of real and complex numbers - Chapter 11 of [HoTT]; [BauerTaylor].
• Theorems about real numbers - [Geuvers].

Theorem

conventions 16374

Unless there is a reason to diverge, we follow the conventions of the Metamath Proof Explorer (MPE, set.mm). This list of conventions is intended to be read in conjunction with the corresponding conventions in the Metamath Proof Explorer, and only the differences are described below.

• Minimizing axioms and the axiom of choice. We prefer proofs that depend on fewer and/or weaker axioms, even if the proofs are longer. In particular, our choice of IZF (Intuitionistic Zermelo-Fraenkel) over CZF (Constructive Zermelo-Fraenkel, a weaker system) was just an expedient choice because IZF is easier to formalize in Metamath. You can find some development using CZF in BJ's mathbox starting at wbd 16467 (and the section header just above it). As for the axiom of choice, the full axiom of choice implies excluded middle as seen at acexmid 6022, although some authors will use countable choice or dependent choice. For example, countable choice or excluded middle is needed to show that the Cauchy reals coincide with the Dedekind reals - Corollary 11.4.3 of [HoTT], p. (varies).
• Junk/undefined results. Much of the discussion of this topic in the Metamath Proof Explorer applies except that certain techniques are not available to us. For example, the Metamath Proof Explorer will often say "if a function is evaluated within its domain, a certain result follows; if the function is evaluated outside its domain, the same result follows. Since the function must be evaluated within its domain or outside it, the result follows unconditionally" (the use of excluded middle in this argument is perhaps obvious when stated this way). Often, the easiest fix will be to prove we are evaluating functions within their domains, other times it will be possible to use a theorem like relelfvdm 5674 which says that if a function value produces an inhabited set, then the function is being evaluated within its domain.
• Bibliography references. The bibliography for the Intuitionistic Logic Explorer is separate from the one for the Metamath Proof Explorer but feel free to copy-paste a citation in either direction in order to cite it.

Label naming conventions

Here are a few of the label naming conventions:

• Suffixes. We follow the conventions of the Metamath Proof Explorer with a few additions. A biconditional in set.mm which is an implication in iset.mm should have a "r" (for the reverse direction), or "i"/"im" (for the forward direction) appended. A theorem in set.mm which has a decidability condition added should add "dc" to the theorem name. A theorem in set.mm where "nonempty class" is changed to "inhabited class" should add "m" (for member) to the theorem name.
• iset.mm versus set.mm names

  Theorems which are the same as in set.mm should be named the same (that is, where the statement of the theorem is the same; the proof can differ without a new name being called for). Theorems which are different should be named differently (we do have a small number of intentional exceptions to this rule but on the whole it serves us well).

  As for how to choose names so they are different between iset.mm and set.mm, when possible choose a name which reflect the difference in the theorems. For example, if a theorem in set.mm is an equality and the iset.mm analogue is a subset, add "ss" to the iset.mm name. If need be, add "i" to the iset.mm name (usually as a prefix to some portion of the name).

  As with set.mm, we welcome suggestions for better names (such as names which are more consistent with naming conventions).

  We do try to keep set.mm and iset.mm similar where we can. For example, if a theorem exists in both places but the name in set.mm isn't great, we tend to keep that name for iset.mm, or change it in both files together. This is mainly to make it easier to copy theorems, but also to generally help people browse proofs, find theorems, write proofs, etc.

The following table shows some commonly-used abbreviations in labels which are not found in the Metamath Proof Explorer, in alphabetical order. For each abbreviation we provide a mnenomic to help you remember it, the source theorem/assumption defining it, an expression showing what it looks like, whether or not it is a "syntax fragment" (an abbreviation that indicates a particular kind of syntax), and hyperlinks to label examples that use the abbreviation. The abbreviation is bolded if there is a df-NAME definition but the label fragment is not NAME.

For the "g" abbreviation, this is related to the set.mm usage, in which "is a set" conditions are converted from hypotheses to antecedents, but is also used where "is a set" conditions are added relative to similar set.mm theorems.

Abbreviation	Mnenomic/Meaning	Source	Expression	Syntax?	Example(s)
ap	apart	df-pap 7472, df-ap 8767		Yes	apadd1 8793, apne 8808
g	with "is a set" condition			No	1stvalg 6310, brtposg 6425, setsmsbasg 15232
m	inhabited (from "member")		E. x x e. A	No	r19.2m 3580, negm 9854, ctm 7313, basmex 13165
seq3, sum3	recursive sequence	df-seqfrec 10716		Yes	seq3-1 10730, fsum3 11971
tap	tight apartness	df-tap 7474		Yes	df-tap 7474

• Community. The Metamath mailing list also covers the Intuitionistic Logic Explorer and is at: https://groups.google.com/g/metamath 7474.

(Contributed by Jim Kingdon, 24-Feb-2020.) (New usage is discouraged.)

|- ph   =>    |- ph

13.1.2  Definitional examples

Theorem

ex-or 16375

Example for ax-io 716. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

|- ( 2 = 3 \/ 4 = 4 )

Theorem

ex-an 16376

Example for ax-ia1 106. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

|- ( 2 = 2 /\ 3 = 3 )

Theorem

1kp2ke3k 16377

Example for df-dec 9617, 1000 + 2000 = 3000.

This proof disproves (by counterexample) the assertion of Hao Wang, who stated, "There is a theorem in the primitive notation of set theory that corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula would be forbiddingly long... even if (one) knows the definitions and is asked to simplify the long formula according to them, chances are he will make errors and arrive at some incorrect result." (Hao Wang, "Theory and practice in mathematics" , In Thomas Tymoczko, editor, New Directions in the Philosophy of Mathematics, pp 129-152, Birkauser Boston, Inc., Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)

This is noted in Metamath: A Computer Language for Pure Mathematics by Norman Megill (2007) section 1.1.3. Megill then states, "A number of writers have conveyed the impression that the kind of absolute rigor provided by Metamath is an impossible dream, suggesting that a complete, formal verification of a typical theorem would take millions of steps in untold volumes of books... These writers assume, however, that in order to achieve the kind of complete formal verification they desire one must break down a proof into individual primitive steps that make direct reference to the axioms. This is not necessary. There is no reason not to make use of previously proved theorems rather than proving them over and over... A hierarchy of theorems and definitions permits an exponential growth in the formula sizes and primitive proof steps to be described with only a linear growth in the number of symbols used. Of course, this is how ordinary informal mathematics is normally done anyway, but with Metamath it can be done with absolute rigor and precision."

The proof here starts with ( 2 + 1 ) = 3, commutes it, and repeatedly multiplies both sides by ten. This is certainly longer than traditional mathematical proofs, e.g., there are a number of steps explicitly shown here to show that we're allowed to do operations such as multiplication. However, while longer, the proof is clearly a manageable size - even though every step is rigorously derived all the way back to the primitive notions of set theory and logic. And while there's a risk of making errors, the many independent verifiers make it much less likely that an incorrect result will be accepted.

This proof heavily relies on the decimal constructor df-dec 9617 developed by Mario Carneiro in 2015. The underlying Metamath language has an intentionally very small set of primitives; it doesn't even have a built-in construct for numbers. Instead, the digits are defined using these primitives, and the decimal constructor is used to make it easy to express larger numbers as combinations of digits.

(Contributed by David A. Wheeler, 29-Jun-2016.) (Shortened by Mario Carneiro using the arithmetic algorithm in mmj2, 30-Jun-2016.)

|- (;;; 1 0 0 0 + ;;; 2 0 0 0 ) = ;;; 3 0 0 0

Theorem

ex-fl 16378

Example for df-fl 10536. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

|- ( ( |_ ` ( 3 / 2 ) ) = 1 /\ ( |_ ` -u ( 3 / 2 ) ) = -u 2 )

Theorem

ex-ceil 16379

Example for df-ceil 10537. (Contributed by AV, 4-Sep-2021.)

|- ( (⌈ ` ( 3 / 2 ) ) = 2 /\ (⌈ ` -u ( 3 / 2 ) ) = -u 1 )

Theorem

ex-exp 16380

Example for df-exp 10807. (Contributed by AV, 4-Sep-2021.)

|- ( ( 5 ^ 2 ) = ; 2 5 /\ ( -u 3 ^ -u 2 ) = ( 1 / 9 ) )

Theorem

ex-fac 16381

Example for df-fac 10994. (Contributed by AV, 4-Sep-2021.)

|- ( ! ` 5 ) = ;; 1 2 0

Theorem

ex-bc 16382

Example for df-bc 11016. (Contributed by AV, 4-Sep-2021.)

|- ( 5 _C 3 ) = ; 1 0

Theorem

ex-dvds 16383

Example for df-dvds 12372: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)

|- 3 || 6

Theorem

ex-gcd 16384

Example for df-gcd 12548. (Contributed by AV, 5-Sep-2021.)

|- ( -u 6 gcd 9 ) = 3

PART 14  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)

14.1  Mathboxes for user contributions

14.1.1  Mathbox guidelines

Theorem

mathbox 16385

(This theorem is a dummy placeholder for these guidelines. The label of this theorem, "mathbox", is hard-coded into the Metamath program to identify the start of the mathbox section for web page generation.)

A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of iset.mm.

For contributors:

By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of iset.mm.

Guidelines:

Mathboxes in iset.mm follow the same practices as in set.mm, so refer to the mathbox guidelines there for more details.

(Contributed by NM, 20-Feb-2007.) (Revised by the Metamath team, 9-Sep-2023.) (New usage is discouraged.)

|- ph   =>    |- ph

14.2  Mathbox for Matthew House

Theorem

depindlem1 16386*

Lemma for depind 16389. (Contributed by Matthew House, 14-Apr-2026.)

|- ( ph -> P : NN0 --> _V )   &    |- ( ph -> A e. ( P ` 0 ) )   &    |- ( ph -> A. n e. NN0 ( H ` n ) : ( P ` n ) --> ( P ` ( n + 1 ) ) )   &    |- F = seq 0 ( ( x e. _V , h e. _V |-> ( h ` x ) ) , ( m e. NN0 |-> if ( m = 0 , A , ( H ` ( m - 1 ) ) ) ) )   =>    |- ( ph -> ( F : NN0 --> _V /\ ( F ` 0 ) = A /\ A. n e. NN0 ( F ` ( n + 1 ) ) = ( ( H ` n ) ` ( F ` n ) ) ) )

Theorem

depindlem2 16387*

Lemma for depind 16389. (Contributed by Matthew House, 14-Apr-2026.)

|- ( ph -> P : NN0 --> _V )   &    |- ( ph -> A e. ( P ` 0 ) )   &    |- ( ph -> A. n e. NN0 ( H ` n ) : ( P ` n ) --> ( P ` ( n + 1 ) ) )   &    |- F = seq 0 ( ( x e. _V , h e. _V |-> ( h ` x ) ) , ( m e. NN0 |-> if ( m = 0 , A , ( H ` ( m - 1 ) ) ) ) )   =>    |- ( ph -> F e. X_ n e. NN0 ( P ` n ) )

Theorem

depindlem3 16388*

Lemma for depind 16389. (Contributed by Matthew House, 14-Apr-2026.)

|- ( ph -> P : NN0 --> _V )   &    |- ( ph -> A e. ( P ` 0 ) )   &    |- ( ph -> A. n e. NN0 ( H ` n ) : ( P ` n ) --> ( P ` ( n + 1 ) ) )   &    |- F = seq 0 ( ( x e. _V , h e. _V |-> ( h ` x ) ) , ( m e. NN0 |-> if ( m = 0 , A , ( H ` ( m - 1 ) ) ) ) )   =>    |- ( ph -> A. f e. X_ n e. NN0 ( P ` n ) ( ( ( f ` 0 ) = A /\ A. n e. NN0 ( f ` ( n + 1 ) ) = ( ( H ` n ) ` ( f ` n ) ) ) -> f = F ) )

Theorem

depind 16389*

Theorem related to a dependently typed induction principle in type theory. (Contributed by Matthew House, 14-Apr-2026.)

|- ( ph -> P : NN0 --> _V )   &    |- ( ph -> A e. ( P ` 0 ) )   &    |- ( ph -> A. n e. NN0 ( H ` n ) : ( P ` n ) --> ( P ` ( n + 1 ) ) )   =>    |- ( ph -> E! f e. X_ n e. NN0 ( P ` n ) ( ( f ` 0 ) = A /\ A. n e. NN0 ( f ` ( n + 1 ) ) = ( ( H ` n ) ` ( f ` n ) ) ) )

14.3  Mathbox for BJ

14.3.1  Propositional calculus

Theorem

bj-nnsn 16390

As far as implying a negated formula is concerned, a formula is equivalent to its double negation. (Contributed by BJ, 24-Nov-2023.)

|- ( ( ph -> -. ps ) <-> ( -. -. ph -> -. ps ) )

Theorem

bj-nnor 16391

Double negation of a disjunction in terms of implication. (Contributed by BJ, 9-Oct-2019.)

|- ( -. -. ( ph \/ ps ) <-> ( -. ph -> -. -. ps ) )

Theorem

bj-nnim 16392

The double negation of an implication implies the implication with the consequent doubly negated. (Contributed by BJ, 24-Nov-2023.)

|- ( -. -. ( ph -> ps ) -> ( ph -> -. -. ps ) )

Theorem

bj-nnan 16393

The double negation of a conjunction implies the conjunction of the double negations. (Contributed by BJ, 24-Nov-2023.)

|- ( -. -. ( ph /\ ps ) -> ( -. -. ph /\ -. -. ps ) )

Theorem

bj-nnclavius 16394

Clavius law with doubly negated consequent. (Contributed by BJ, 4-Dec-2023.)

|- ( ( -. ph -> ph ) -> -. -. ph )

Theorem

bj-imnimnn 16395

If a formula is implied by both a formula and its negation, then it is not refutable. There is another proof using the inference associated with bj-nnclavius 16394 as its last step. (Contributed by BJ, 27-Oct-2024.)

|- ( ph -> ps )   &    |- ( -. ph -> ps )   =>    |- -. -. ps

14.3.1.1  Stable formulas

Some of the following theorems, like bj-sttru 16397 or bj-stfal 16399 could be deduced from their analogues for decidability, but stability is not provable from decidability in minimal calculus, so direct proofs have their interest.

Theorem

bj-trst 16396

A provable formula is stable. (Contributed by BJ, 24-Nov-2023.)

|- ( ph -> STAB ph )

Theorem

bj-sttru 16397

The true truth value is stable. (Contributed by BJ, 5-Aug-2024.)

|- STAB T.

Theorem

bj-fast 16398

A refutable formula is stable. (Contributed by BJ, 24-Nov-2023.)

|- ( -. ph -> STAB ph )

Theorem

bj-stfal 16399

The false truth value is stable. (Contributed by BJ, 5-Aug-2024.)

|- STAB F.

Theorem

bj-nnst 16400

Double negation of stability of a formula. Intuitionistic logic refutes unstability (but does not prove stability) of any formula. This theorem can also be proved in classical refutability calculus (see https://us.metamath.org/mpeuni/bj-peircestab.html) but not in minimal calculus (see https://us.metamath.org/mpeuni/bj-stabpeirce.html). See nnnotnotr 16645 for the version not using the definition of stability. (Contributed by BJ, 9-Oct-2019.) Prove it in ( -> , -. ) -intuitionistic calculus with definitions (uses of ax-ia1 106, ax-ia2 107, ax-ia3 108 are via sylibr 134, necessary for definition unpackaging), and in ( -> , <-> , -. )-intuitionistic calculus, following a discussion with Jim Kingdon. (Revised by BJ, 27-Oct-2024.)

|- -. -. STAB ph