Working snapshot of the Phase I graph-theory development toward a formal proof of Euler’s formula in set.mm.
It contains the complete mathbox implementation of finite tree and spanning-tree theory, together with an integration diff showing the required prerequisite promotions. These files are provided as reference material for the planned upstream PR series, not as a single patch intended for review or merge.
| Integration patch for the Mingli Yuan mathbox | |
| Base repository: https://github.com/metamath/set.mm | |
| Base branch: develop | |
| Base commit: 7ddd528c948ae375618ad1ca476b28b9a0eec7d6 | |
| Contents: | |
| - the 255-assertion Mingli Yuan mathbox and split/typesetting integration; | |
| - 40 promoted BTernaryTau assertions; | |
| - the four requested Thierry Arnoux promotions, plus swrdf1, which is a | |
| prerequisite already used by the Phase I development; and | |
| - the Phase I helper wrdf1d. | |
| Apply with: git apply mingli-yuan-mathbox.diff | |
| diff --git a/set.mm b/set.mm | |
| index 836a9723722206b959d3c15700017f650e501455..79b8730cb242e0e2160d85c229a72dbef6c62adf 100644 | |
| --- a/set.mm | |
| +++ b/set.mm | |
| @@ -42073,6 +42073,34 @@ $) | |
| undif2 $p |- ( A u. ( B \ A ) ) = ( A u. B ) $= | |
| ( cdif cun uncom undif1 3eqtri ) ABACZDHADBADABDAHEBAFBAEG $. | |
| + ${ | |
| + srcmpltd.1 $e |- ( ph -> ( C e. A -> ps ) ) $. | |
| + srcmpltd.2 $e |- ( ph -> ( C e. ( B \ A ) -> ps ) ) $. | |
| + $( If a statement is true for every element of a class and for every | |
| + element of its complement relative to a second class, then it is true | |
| + for every element in the second class. (Contributed by BTernaryTau, | |
| + 27-Sep-2023.) $) | |
| + srcmpltd $p |- ( ph -> ( C e. B -> ps ) ) $= | |
| + ( wcel cdif cun elun2 undif2 eleqtrrdi wi elunant sylanbrc syl5 ) EDHZECD | |
| + CIZJZHZABRECDJTEDCKCDLMAECHBNESHBNUABNFGBCSEOPQ $. | |
| + $} | |
| + | |
| + ${ | |
| + prsrcmpltd.1 $e |- ( ph -> ( ( C e. A /\ D e. A ) -> ps ) ) $. | |
| + prsrcmpltd.2 $e |- ( ph -> ( ( C e. A /\ D e. ( B \ A ) ) -> ps ) ) $. | |
| + prsrcmpltd.3 $e |- ( ph -> ( ( C e. ( B \ A ) /\ D e. A ) -> ps ) ) $. | |
| + prsrcmpltd.4 $e |- ( ph -> | |
| + ( ( C e. ( B \ A ) /\ D e. ( B \ A ) ) -> ps ) ) $. | |
| + $( If a statement is true for all pairs of elements of a class, all pairs | |
| + of elements of its complement relative to a second class, and all pairs | |
| + with one element in each, then it is true for all pairs of elements of | |
| + the second class. (Contributed by BTernaryTau, 27-Sep-2023.) $) | |
| + prsrcmpltd $p |- ( ph -> ( ( C e. B /\ D e. B ) -> ps ) ) $= | |
| + ( wcel wi wa expdimp cdif srcmpltd impancom ex impcomd ) AFDKZEDKZBATUABL | |
| + ATMBCDEAECKZTBAUBMBCDFAUBFCKZBGNAUBFDCOZKZBHNPQAEUDKZTBAUFMBCDFAUFUCBINAU | |
| + FUEBJNPQPRS $. | |
| + $} | |
| + | |
| $( Absorption of difference by union. (Contributed by NM, 18-Aug-2013.) $) | |
| undifabs $p |- ( A u. ( A \ B ) ) = A $= | |
| ( cdif cun undif3 unidm difeq1i difdif 3eqtri ) AABCDAADZBACZCAKCAAABEJAKAF | |
| @@ -70297,6 +70325,67 @@ $) | |
| MABCDEPUJULUCUIUKACUIUEUDIZUGKZBCLUKUHUNBCUFUMUGUDUEQRSUGBCUDUATSUBT $. | |
| $} | |
| + ${ | |
| + $d x y A $. $d x y F $. | |
| + $( A one-to-one function in terms of different arguments never having the | |
| + same function value. (Contributed by BTernaryTau, 24-Oct-2023.) $) | |
| + dff15 $p |- ( F : A -1-1-> B <-> | |
| + ( F : A --> B /\ | |
| + -. E. x e. A E. y e. A ( ( F ` x ) = ( F ` y ) /\ x =/= y ) ) ) $= | |
| + ( wf1 wf cv cfv wceq wi wral wa wne wrex wn dff13 iman anbi2i bitri df-ne | |
| + xchbinxr 2ralbii ralnex2 ) CDEFCDEGZAHZEIBHZEIJZUFUGJZKZBCLACLZMUEUHUFUGN | |
| + ZMZBCOACOPZMABCDEQUKUNUEUKUMPZBCLACLUNUJUOABCCUJUHUIPZMUMUHUIRULUPUHUFUGU | |
| + ASUBUCUMABCCUDTST $. | |
| + $} | |
| + | |
| + $( If a function restricted to a class is one-to-one, then for any two | |
| + elements of the class, the values of the function at those elements are | |
| + equal only if the two elements are the same element. (Contributed by | |
| + BTernaryTau, 27-Sep-2023.) $) | |
| + f1resveqaeq $p |- ( ( ( F |` A ) : A -1-1-> B /\ ( C e. A /\ D e. A ) ) -> | |
| + ( ( F ` C ) = ( F ` D ) -> C = D ) ) $= | |
| + ( cres wf1 wcel cfv wceq fvres ad2antrl ad2antll eqeq12d f1veqaeq sylbird | |
| + wa ) ABEAFZGZCAHZDAHZQQZCEIZDEIZJCRIZDRIZJCDJUBUEUCUFUDTUEUCJSUACAEKLUAUFUD | |
| + JSTDAEKMNABCDROP $. | |
| + | |
| + ${ | |
| + f1resrcmplf1dlem.1 $e |- ( ph -> C C_ A ) $. | |
| + f1resrcmplf1dlem.2 $e |- ( ph -> D C_ A ) $. | |
| + f1resrcmplf1dlem.3 $e |- ( ph -> F : A --> B ) $. | |
| + f1resrcmplf1dlem.4 $e |- ( ph -> ( ( F " C ) i^i ( F " D ) ) = (/) ) $. | |
| + $( Lemma for ~ f1resrcmplf1d . (Contributed by BTernaryTau, | |
| + 27-Sep-2023.) $) | |
| + f1resrcmplf1dlem $p |- ( ph -> ( ( X e. C /\ Y e. D ) -> | |
| + ( ( F ` X ) = ( F ` Y ) -> X = Y ) ) ) $= | |
| + ( wcel cfv cima wceq wss fnfvima syl3an1 syl3an2 wi wfn ffnd 3anidm12 wne | |
| + ex wa cin c0 disjne 3expib neneq pm2.21d syl6 syl2and ) AGDMZGFNZFDOZMZHE | |
| + MZHFNZFEOZMZUQVAPZGHPZUAZAUPUSAUPUSAADBQZUPUSIAFBUBZVGUPUSABCFKUCZBDFGRST | |
| + UDUFAUTVCAUTVCAAEBQZUTVCJAVHVJUTVCVIBEFHRSTUDUFAUSVCUGUQVAUEZVFAUSVCVKAUR | |
| + VBUHUIPUSVCVKLURVBUQVAUJSUKVKVDVEUQVAULUMUNUO $. | |
| + $} | |
| + | |
| + ${ | |
| + $d ph x y $. $d x y A $. $d x y F $. | |
| + f1resrcmplf1d.1 $e |- ( ph -> C C_ A ) $. | |
| + f1resrcmplf1d.2 $e |- ( ph -> F : A --> B ) $. | |
| + f1resrcmplf1d.3 $e |- ( ph -> ( F |` C ) : C -1-1-> B ) $. | |
| + f1resrcmplf1d.4 $e |- ( ph -> ( F |` ( A \ C ) ) : ( A \ C ) -1-1-> B ) $. | |
| + f1resrcmplf1d.5 $e |- ( ph -> | |
| + ( ( F " C ) i^i ( F " ( A \ C ) ) ) = (/) ) $. | |
| + $( If a function's restriction to a subclass of its domain and its | |
| + restriction to the relative complement of that subclass are both | |
| + one-to-one, and if the ranges of those two restrictions are disjoint, | |
| + then the function is itself one-to-one. (Contributed by BTernaryTau, | |
| + 28-Sep-2023.) $) | |
| + f1resrcmplf1d $p |- ( ph -> F : A -1-1-> B ) $= | |
| + ( vx vy cv cfv wceq wral wf1 wcel wa cres wf wi f1resveqaeq sylan ex cdif | |
| + difssd f1resrcmplf1dlem cima cin incom eqtr3id prsrcmpltd ralrimivv dff13 | |
| + c0 sylanbrc ) ABCEUAKMZENLMZENOURUSOUBZLBPKBPBCEQGAUTKLBBAUTDBURUSAURDRUS | |
| + DRSZUTADCEDTQVAUTHDCURUSEUCUDUEABCDBDUFZEURUSFABDUGZGJUHABCVBDEURUSVCFGAE | |
| + VBUIZEDUIZUJVEVDUJUPVEVDUKJULUHAURVBRUSVBRSZUTAVBCEVBTQVFUTIVBCURUSEUCUDU | |
| + EUMUNKLBCEUOUQ $. | |
| + $} | |
| + | |
| ${ | |
| $d A a b x y $. $d B x y $. $d G a b x y $. $d V x y $. $d W x y $. | |
| $d X x y $. $d Y x y $. | |
| @@ -93881,6 +93970,21 @@ $) | |
| GHINAJOBGHABKLM $. | |
| $} | |
| + ${ | |
| + $d F p $. $d x y p $. | |
| + $( If a function is equinumerous to ordinal 1, then its converse is also a | |
| + function. (Contributed by BTernaryTau, 8-Oct-2023.) $) | |
| + funen1cnv $p |- ( ( Fun F /\ F ~~ 1o ) -> Fun `' F ) $= | |
| + ( vp vx vy c1o cen wfun ccnv cv csn wceq wex wi wal sylancl syl wa funeqd | |
| + cnveq biimpar wbr en1 cop wrel wcel funrel vsnid elrel sneq funcnvsn gen2 | |
| + 2eximi 19.29r2 exlimivv ax-gen 19.29r funeq imbi12d exlimiv mpan2 impcom | |
| + sylbi ) AEFUAZAGZAHZGZVCABIZJZKZBLZVDVFMZBAUBVJVHGZVHHZGZMZBNZVKVOBVLVHCI | |
| + ZDIZUCZJZKZDLCLZVTHZGZDNCNZVNVLVGVSKZDLCLZWBVLVHUDVGVHUEWGVHUFBUGCDVGVHUH | |
| + OWFWACDVGVSUIULPWDCDVQVRUJUKWBWEQWAWDQZDLCLVNWAWDCDUMWHVNCDWAVNWDWAVMWCVH | |
| + VTSRTUNPOUOVJVPQVIVOQZBLVKVIVOBUPWIVKBVIVKVOVIVDVLVFVNAVHUQVIVEVMAVHSRURT | |
| + USPUTVBVA $. | |
| + $} | |
| + | |
| $( A singleton contains its sole element. (Contributed by Stefan O'Rear, | |
| 16-Aug-2015.) Avoid ~ ax-un . (Revised by BTernaryTau, 24-Sep-2024.) $) | |
| en1uniel $p |- ( S ~~ 1o -> U. S e. S ) $= | |
| @@ -139514,6 +139618,20 @@ $( TODO: The following 14 theorems do not contain ` ZZ ` - these theorems are | |
| these theorems could be moved into a new separate subsection "Positive and | |
| nonnegative integers (cont.)". $) | |
| + $( A number is zero if and only if it's a nonnegative integer that becomes | |
| + negative after subtracting 1. (Contributed by BTernaryTau, | |
| + 30-Sep-2023.) $) | |
| + 0nn0m1nnn0 $p |- ( N = 0 <-> ( N e. NN0 /\ -. ( N - 1 ) e. NN0 ) ) $= | |
| + ( cc0 wceq cn0 wcel c1 cmin co wn wa mpbiri wnel syl cle wbr biimprd adantr | |
| + wb cz wi 0nn0 eleq1 cn 1nn 0mnnnnn0 ax-mp oveq1 neleq1 df-nel sylib jca clt | |
| + nn0z peano2zm elnn0z notbii biimpi annotanannot simprbi syl2an cr 3syl 0red | |
| + zre ltnled mpd 0z zlem1lt sylancl nn0ge0 nn0re letri3d mp2and impbii ) ABCZ | |
| + ADEZAFGHZDEZIZJZVOVPVSVOVPBDEUAABDUBKVOVQDLZVSVOWABFGHZDLZFUCEWCUDFUEUFVOVQ | |
| + WBCWAWCRABFGUGVQWBDUHMKVQDUIUJUKVTABNOZBANOZVOVTVQBULOZWDVTBVQNOZIZWFVPVQSE | |
| + ZWIWGJZIZWHVSVPASEZWIAUMZAUNZMVSWKVRWJVQUOUPUQWIWKJWIWHWIWGURUSUTVPWHWFTVSV | |
| + PWFWHVPVQBVPWLWIVQVAEWMWNVQVDVBVPVCZVEPQVFVPWFWDTVSVPWDWFVPWLBSEWDWFRWMVGAB | |
| + VHVIPQVFVPWEVSAVJQVPWDWEJZVOTVSVPVOWPVPABAVKWOVLPQVMVN $. | |
| + | |
| $( Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.) | |
| (Proof shortened by Mario Carneiro, 16-May-2014.) $) | |
| nnleltp1 $p |- ( ( A e. NN /\ B e. NN ) -> | |
| @@ -160195,6 +160313,34 @@ $) | |
| wf frn hashss 3adant1 eqbrtrd ) CDFZBEFZABCGZHAIJZCKZIJZBIJZLUEUGUHUJMUFABC | |
| DNOUFUGUJUKLPZUEUGUFABCTZULABCQUMUFUIBRULABCUABUIEUBSSUCUD $. | |
| + ${ | |
| + $d x y F $. | |
| + f1resfz0f1d.1 $e |- ( ph -> K e. NN0 ) $. | |
| + f1resfz0f1d.2 $e |- ( ph -> F : ( 0 ... K ) --> V ) $. | |
| + f1resfz0f1d.3 $e |- ( ph -> | |
| + ( F |` ( 1 ... K ) ) : ( 1 ... K ) -1-1-> V ) $. | |
| + f1resfz0f1d.4 $e |- ( ph -> | |
| + ( ( F " { 0 } ) i^i ( F " ( 1 ... K ) ) ) = (/) ) $. | |
| + $( If a function with a sequence of nonnegative integers (starting at 0) as | |
| + its domain is one-to-one when 0 is removed, and if the range of that | |
| + restriction does not contain the function's value at the removed | |
| + integer, then the function is itself one-to-one. (Contributed by | |
| + BTernaryTau, 4-Oct-2023.) $) | |
| + f1resfz0f1d $p |- ( ph -> F : ( 0 ... K ) -1-1-> V ) $= | |
| + ( vx vy cc0 cfz cfv wceq wi cn0 wcel cin c0 cima co wss fz1ssfz0 a1i cres | |
| + c1 csn wf1 cdif wf cv wral 0elfz snssi 3syl fssresd eqidd wb 0nn0 fveqeq2 | |
| + eqeq1 imbi12d fveq2 eqeq2 mp2an mpbir dff13 sylanblrc cun uncom fz0sn0fz1 | |
| + eqeq2d 2ralsng syl eqtr4id wn 0nelfz1 neli disjsn uneqdifeq sylib reseq2d | |
| + eqcomd f1eq123d mpbid imaeq2d ineq2d incom eqtr3id eqtr3d f1resrcmplf1d ) | |
| + AKCLUAZDUFCLUAZBWMWLUBZACUCZUDFGAKUGZDBWPUEZUHZWLWMUIZDBWSUEZUHAWPDWQUJIU | |
| + KZWQMJUKZWQMZNZXAXBNZOZJWPULIWPULZWRAWLDWPBFACPQZKWLQWPWLUBECUMKWLUNUOUPX | |
| + GKWQMZXINZKKNZOZXJKUQKPQZXMXGXLURUSUSXFXIXCNZKXBNZOXLIJKKPPXAKNXDXNXEXOXA | |
| + KXCWQUTXAKXBVAVBXBKNZXNXJXOXKXPXCXIXIXBKWQVCVLXBKKVDVBVMVEVFIJWPDWQVGVHAW | |
| + PWSDDWQWTAWPWSBAWSWPAWMWPVIZWLNZWSWPNZAXQWPWMVIZWLWMWPVJAXHWLXTNECVKVNVOW | |
| + NWMWPRSNZXRXSURWOYAKWMQVPKWMCVQVRWMKVSVFWMWPWLVTVEWAWCZWBYBADUQWDWEABWMTZ | |
| + BWPTZRZYCBWSTZRSAYDYFYCAWPWSBYBWFWGAYEYDYCRSYDYCWHHWIWJWK $. | |
| + $} | |
| + | |
| ${ | |
| resunimafz0.i $e |- ( ph -> Fun I ) $. | |
| resunimafz0.f $e |- ( ph -> F : ( 0 ..^ ( # ` F ) ) --> dom I ) $. | |
| @@ -162646,6 +162792,58 @@ $) | |
| XL $. | |
| $} | |
| + ${ | |
| + $d A i j $. $d B i j $. $d i j ph $. | |
| + ccatf1.s $e |- ( ph -> S e. V ) $. | |
| + ccatf1.a $e |- ( ph -> A e. Word S ) $. | |
| + ccatf1.b $e |- ( ph -> B e. Word S ) $. | |
| + ccatf1.1 $e |- ( ph -> A : dom A -1-1-> S ) $. | |
| + ccatf1.2 $e |- ( ph -> B : dom B -1-1-> S ) $. | |
| + ccatf1.3 $e |- ( ph -> ( ran A i^i ran B ) = (/) ) $. | |
| + $( Conditions for a concatenation to be injective. (Contributed by Thierry | |
| + Arnoux, 11-Dec-2023.) $) | |
| + ccatf1 $p |- ( ph -> ( A ++ B ) : dom ( A ++ B ) -1-1-> S ) $= | |
| + ( vi vj co cfv wceq wcel syl wa adantr cconcat cdm wf cv weq wral wf1 cc0 | |
| + wi chash cfzo cword ccatcl syl2anc wrdf ffdmd ccatval1 syl2an3an ad4ant13 | |
| + simpllr id ad4ant14 3eqtr3d wrddm f1eq2 biimpa dff13 simprbi r19.21bi mpd | |
| + adantllr ex c0 crn wfun f1fun simpr eleqtrrd fvelrn syl2an2r cmin ccatlen | |
| + cin caddc oveq2d eleqtrd ccatval2 syl3anc cn0 lencl nn0zd fzosubel3 elind | |
| + cz eqeltrd ad3antrrr wn noel pm2.21dd wo eleq2d fzospliti ad2antrr mpjaod | |
| + a1i adantlrl 3eqtr3rd cc elfzoelz ad2antlr adantl nn0cnd jca f1veqaeq imp | |
| + zcnd syl21anc subcan2d adantrr ralrimivva sylanbrc ) ABCUANZUBZDYBUCLUDZY | |
| + BOZMUDZYBOZPZLMUEZUIZMYCUFLYCUFYCDYBUGAUHYBUJOZUKNZDYBAYBDULZQZYLDYBUCABY | |
| + MQZCYMQZYNGHDBCUMUNZDYBUORUPAYJLMYCYCAYDYCQZYFYCQZSSZYHYIYTYHSYDUHBUJOZUK | |
| + NZQZYIYDUUAYKUKNZQZAYSYHUUCYIUIYRAYSSZYHSZUUCYIUUGUUCSYFUUBQZYIYFUUDQZAYH | |
| + UUCUUHYIUIZYSAYHSZUUCSZUUHYIUULUUHSZYDBOZYFBOZPZYIUUMYEYGUUNUUOAYHUUCUUHU | |
| + TAUUCYEUUNPZYHUUHAYOYPUUCUUCUUQGHUUCVADDBCYDUQURZUSAUUHYGUUOPZYHUUCAYOYPU | |
| + UHUUHUUSGHUUHVADDBCYFUQURZVBVCAUUCUUHUUPYIUIZYHAUUCSZUVAMUUBAUVAMUUBUFZLU | |
| + UBAUUBDBUGZUVCLUUBUFZABUBZUUBPZUVFDBUGZUVDAYOUVGGDBVDRZIUVGUVHUVDUVFUUBDB | |
| + VEVFUNUVDUUBDBUCUVELMUUBDBVGVHRVIVIVKVJVLVKAYHUUCUUIYIUIZYSUULUUIYIUULUUI | |
| + SZUUNVMQZYIUVKUUNBVNZCVNZWCZVMUVKUVMUVNUUNAUUCUUNUVMQZYHUUIABVOZUUCYDUVFQ | |
| + UVPAUVHUVQIUVFDBVPRZUVBYDUUBUVFAUUCVQAUVGUUCUVITVRYDBVSVTUSUVKUUNYFUUAWAN | |
| + ZCOZUVNUVKYEYGUUNUVTAYHUUCUUIUTAUUCUUQYHUUIUURUSAUUIYGUVTPZYHUUCAUUISZYOY | |
| + PYFUUAUUACUJOZWDNZUKNZQZUWAAYOUUIGTAYPUUIHTUWBYFUUDUWEAUUIVQAUUDUWEPZUUIA | |
| + YKUWDUUAUKAYOYPYKUWDPGHDDBCWBUNWEZTWFZDBCYFWGWHZVBVCAUUIUVTUVNQZYHUUCACVO | |
| + ZUUIUVSCUBZQZUWKAUWMDCUGZUWLJUWMDCVPRZUWBUVSUHUWCUKNZUWMUWBUWFUWCWNQZUVSU | |
| + WQQUWIAUWRUUIAUWCAYPUWCWIQHDCWJRWKZTYFUUAUWCWLUNAUWMUWQPZUUIAYPUWTHDCVDRZ | |
| + TVRZUVSCVSVTVBWOWMAUVOVMPZYHUUCUUIKWPWFUVLWQUVKUUNWRXEWSVLVKUUFUUHUUIWTZY | |
| + HUUCUUFYFYLQZUUAWNQZUXDAYSUXEAYCYLYFAYNYCYLPYQDYBVDRZXAVFAUXFYSAUUAAYOUUA | |
| + WIQGDBWJRZWKZTYFUHYKUUAXBUNZXCXDVLXFAYSYHUUEYIUIYRUUGUUEYIUUGUUESUUHYIUUI | |
| + AYHUUEUUJYSUUKUUESZUUHYIUXKUUHSZUUOVMQZYIUXLUUOUVOVMUXLUVMUVNUUOAUUHUUOUV | |
| + MQZYHUUEAUVQUUHYFUVFQUXNUVRAUUHSYFUUBUVFAUUHVQAUVGUUHUVITVRYFBVSVTVBUXLUU | |
| + OYDUUAWANZCOZUVNUXLYEYGUXPUUOAYHUUEUUHUTAUUEYEUXPPZYHUUHAUUESZYOYPYDUWEQZ | |
| + UXQAYOUUEGTAYPUUEHTUXRYDUUDUWEAUUEVQAUWGUUEUWHTWFZDBCYDWGWHZUSAUUHUUSYHUU | |
| + EUUTVBXGAUUEUXPUVNQZYHUUHAUWLUUEUXOUWMQZUYBUWPUXRUXOUWQUWMUXRUXSUWRUXOUWQ | |
| + QUXTAUWRUUEUWSTYDUUAUWCWLUNAUWTUUEUXATVRZUXOCVSVTUSWOWMAUXCYHUUEUUHKWPWFU | |
| + XMWQUXLUUOWRXEWSVLVKAYHUUEUVJYSUXKUUIYIUXKUUISZYDYFUUAUUEYDXHQUUKUUIUUEYD | |
| + YDUUAYKXIXPXJUUIYFXHQUXKUUIYFYFUUAYKXIXPXKAUUAXHQYHUUEUUIAUUAUXHXLWPUYEUW | |
| + OUYCUWNSZUXPUVTPZUXOUVSPZAUWOYHUUEUUIJWPUYEUYCUWNAUUEUYCYHUUIUYDUSAUUIUWN | |
| + YHUUEUXBVBXMUYEYEYGUXPUVTAYHUUEUUIUTAUUEUXQYHUUIUYAUSAUUIUWAYHUUEUWJVBVCU | |
| + WOUYFSUYGUYHUWMDUXOUVSCXNXOXQXRVLVKUUFUXDYHUUEUXJXCXDVLXFYTUUCUUEWTZYHAYR | |
| + UYIYSAYRSYDYLQZUXFUYIAYRUYJAYCYLYDUXGXAVFAUXFYRUXITYDUHYKUUAXBUNXSTXDVLXT | |
| + LMYCDYBVGYA $. | |
| + $} | |
| + | |
| $( Concatenation of the empty word by the empty word. (Contributed by AV, | |
| 26-Mar-2022.) $) | |
| ccatidid $p |- ( (/) ++ (/) ) = (/) $= | |
| @@ -162830,6 +163028,17 @@ $) | |
| wceq oveq2 fzo01 eqtrdi eqcomi feq2i mpbir fdmi ) BCZDAEZUFDUGFBUGGHZIJZDUG | |
| FZUGDKLUJAMDUGNOUFUIDUGUIUFUHPRZUIUFRAQUKUIBPIJUFUHPBISTUAOUBUCUDUE $. | |
| + ${ | |
| + s1f1.1 $e |- ( ph -> I e. D ) $. | |
| + $( Conditions for a length 1 string to be a one-to-one function. | |
| + (Contributed by Thierry Arnoux, 11-Dec-2023.) $) | |
| + s1f1 $p |- ( ph -> <" I "> : dom <" I "> -1-1-> D ) $= | |
| + ( cs1 cdm wf1 cc0 csn cop wss wf1o cn0 wcel 0nn0 a1i f1osng syl2anc wceq | |
| + syl f1of1 snssd f1ss s1val s1dm eqidd f1eq123d mpbird ) ACEZFZBUIGHIZBHCJ | |
| + IZGZAUKCIZULGZUNBKUMAUKUNULLZUOAHMNZCBNZUPUQAOPDHCMBQRUKUNULUATACBDUBUKUN | |
| + BULUCRAUJUKBBUIULAURUIULSDCBUDTUJUKSACUEPABUFUGUH $. | |
| + $} | |
| + | |
| $( Alternate version of ~ s1dm , having a shorter proof, but requiring that | |
| ` A ` is a set. (Contributed by AV, 9-Jan-2020.) | |
| (Proof modification is discouraged.) (New usage is discouraged.) $) | |
| @@ -163274,6 +163483,45 @@ $) | |
| ZOIZCUKTZGBAPIZOIZCUKTUGUHUNUIUGUKUFFUNCDABQCUKRSUAUJUMUPCUKUJULUOGOCDABUBU | |
| CUDUE $. | |
| + ${ | |
| + $d M i j $. $d N i j $. $d W i j $. $d i j ph $. | |
| + swrdf1.w $e |- ( ph -> W e. Word D ) $. | |
| + swrdf1.m $e |- ( ph -> M e. ( 0 ... N ) ) $. | |
| + swrdf1.n $e |- ( ph -> N e. ( 0 ... ( # ` W ) ) ) $. | |
| + swrdf1.1 $e |- ( ph -> W : dom W -1-1-> D ) $. | |
| + $( Condition for a subword to be injective. (Contributed by Thierry | |
| + Arnoux, 12-Dec-2023.) $) | |
| + swrdf1 $p |- ( ph | |
| + -> ( W substr <. M , N >. ) : dom ( W substr <. M , N >. ) -1-1-> D ) $= | |
| + ( vi vj co cfv wceq cc0 cfzo wcel wa cz ad3antrrr cop csubstr cdm wf wral | |
| + cv wi wf1 cmin cword cfz chash swrdf syl3anc ffdmd fzossz simpllr eleqtrd | |
| + fdmd sselid zcnd simplr elfzelzd caddc wss cuz elfzuz fzoss1 3syl elfzuz3 | |
| + fzoss2 sstrd fzoaddel2 sseldd wrddm syl eleqtrrd swrdfv syl31anc f1veqaeq | |
| + simpr 3eqtr3d anassrs imp syl1111anc addcan2ad ex anasss ralrimivva dff13 | |
| + sylanbrc ) AECDUAUBLZUCZBWLUDJUFZWLMZKUFZWLMZNZWNWPNZUGZKWMUEJWMUEWMBWLUH | |
| + AODCUILZPLZBWLAEBUJQZCODUKLQZDOEULMZUKLQZXBBWLUDFGHCDBEUMUNZUOAWTJKWMWMAW | |
| + NWMQZWPWMQZWTAXHRZXIRZWRWSXKWRRZWNWPCXLWNXLXBSWNOXAUPZXLWNWMXBAXHXIWRUQAW | |
| + MXBNXHXIWRAXBBWLXGUSTZURZUTVAXLWPXLXBSWPXMXLWPWMXBXJXIWRVBXNURZUTVAXLCACS | |
| + QZXHXIWRACODGVCTZVAXLEUCZBEUHZWNCVDLZXSQZWPCVDLZXSQZYAEMZYCEMZNZYAYCNZAXT | |
| + XHXIWRITXLYAOXEPLZXSXLCDPLZYIYAAYJYIVEXHXIWRAYJODPLZYIAXDCOVFMQYJYKVEGCOD | |
| + VGCODVHVIAXFXEDVFMQYKYIVEHDOXEVJDOXEVKVIVLTZXLWNXBQZDSQZXQYAYJQXOAYNXHXIW | |
| + RADOXEHVCTZXRWNDCVMUNVNAXSYINZXHXIWRAXCYPFBEVOVPTZVQXLYCYIXSXLYJYIYCYLXLW | |
| + PXBQZYNXQYCYJQXPYOXRWPDCVMUNVNYQVQXLWOWQYEYFXKWRWAXLXCXDXFYMWOYENAXCXHXIW | |
| + RFTZAXDXHXIWRGTZAXFXHXIWRHTZXOBECDWNVRVSXLXCXDXFYRWQYFNYSYTUUAXPBECDWPVRV | |
| + SWBXTYBRYDRYGYHXTYBYDYGYHUGXSBYAYCEVTWCWDWEWFWGWHWIJKWMBWLWJWK $. | |
| + $} | |
| + | |
| + ${ | |
| + wrdf1d.w $e |- ( ph -> W e. Word D ) $. | |
| + wrdf1d.f $e |- ( ph -> Fun `' W ) $. | |
| + $( A one-to-one word maps its domain into its alphabet. (Contributed by | |
| + Mingli Yuan, 11-Aug-2026.) $) | |
| + wrdf1d $p |- ( ph -> W : dom W -1-1-> D ) $= | |
| + ( cdm wf1 wf ccnv wfun cc0 chash cfv cfzo co cword wcel wceq wrddm syl wa | |
| + eqcomd wrdf feq2dd wb df-f1 a1i mpbir2and ) ACFZBCGZUIBCHZCIJZAKCLMNOZUIB | |
| + CAUIUMACBPQZUIUMRDBCSTUBAUNUMBCHDBCUCTUDEUJUKULUAUEAUIBCUFUGUH $. | |
| + $} | |
| + | |
| $( Value of the subword extractor as function with domain. (Contributed by | |
| Alexander van der Vekens, 28-Mar-2018.) (Proof shortened by AV, | |
| 2-May-2020.) $) | |
| @@ -163291,6 +163539,49 @@ $) | |
| ( cword wcel cc0 cfz co chash cfv w3a cmin cfzo cop csubstr swrdf frnd ) DC | |
| EFAGBHIFBGDJKHIFLGBAMINICDABOPIABCDQR $. | |
| + ${ | |
| + $d M x $. $d N x $. $d V x $. $d W x $. | |
| + $( The range of a subword is a subset of the range of that word. Stronger | |
| + version of ~ swrdrn . (Contributed by Thierry Arnoux, 12-Dec-2023.) $) | |
| + swrdrn2 $p |- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) | |
| + ) ) -> ran ( W substr <. M , N >. ) C_ ran W ) $= | |
| + ( vx cword wcel cc0 cfz co cfv crn cfzo wss cuz adantr cz elfzelzd sseldd | |
| + syl chash w3a cop csubstr cmin cv caddc cmpt swrdval2 rneqd wral wfun cdm | |
| + wa eqidd simpl1 wrdfd ffund elfzuz3 3ad2ant3 fzoss2 elfzuz 3ad2ant2 simpr | |
| + fzoss1 simpl3 simpl2 fzoaddel2 syl3anc wceq wrddm 3ad2ant1 fvelrn syl2anc | |
| + eleqtrrd ralrimiva eqid rnmptss eqsstrd ) DCFGZAHBIJGZBHDUAKZIJGZUBZDABUC | |
| + UDJZLEHBAUEJMJZEUFZAUGJZDKZUHZLZDLZWDWEWJECDABUIUJWDWIWLGZEWFUKWKWLNWDWME | |
| + WFWDWGWFGZUNZDULWHDUMZGWMWOHWBMJZCDWOCWBDWOWBUOVTWAWCWNUPUQURWOWHWQWPWOHB | |
| + MJZWQWHWOWBBOKGZWRWQNWDWSWNWCVTWSWABHWBUSUTPBHWBVATWOABMJZWRWHWOAHOKGZWTW | |
| + RNWDXAWNWAVTXAWCAHBVBVCPAHBVETWOWNBQGAQGWHWTGWDWNVDWOBHWBVTWAWCWNVFRWOAHB | |
| + VTWAWCWNVGRWGBAVHVISSWDWPWQVJZWNVTWAXBWCCDVKVLPVOWHDVMVNVPEWFWIWLWJWJVQVR | |
| + TVS $. | |
| + $} | |
| + | |
| + ${ | |
| + $d M i j y $. $d N i j y $. $d V i j y $. $d W i j y $. | |
| + $( Express the range of a subword. Stronger version of ~ swrdrn2 . | |
| + (Contributed by Thierry Arnoux, 13-Dec-2023.) $) | |
| + swrdrn3 $p |- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) | |
| + ) ) -> ran ( W substr <. M , N >. ) = ( W " ( M ..^ N ) ) ) $= | |
| + ( vy vj vi wcel cc0 co cfv cfzo cv wceq caddc wa cz simpr elfzelzd zcnd | |
| + cword cfz w3a cop csubstr crn cima wrex cmin cmpt simpl3 simpl2 fzoaddel2 | |
| + chash syl3anc pncan3d oveq2d eleqtrrd zsubcld oveq1d eqeq2d fzossz sselid | |
| + fzosubel3 syl2anc npcand eqcomd rspcedvd fveq2d bitr3id rexxfrd eqid fvex | |
| + elrnmpti bitr4di wf wrdf 3ad2ant1 ffnd cuz wss elfzuz 3ad2ant2 fzoss1 syl | |
| + eqcom elfzuz3 3ad2ant3 fzoss2 sstrd fvelimabd swrdval2 rneqd eleq2d eqrdv | |
| + 3bitr4rd ) DCUAHZAIBUBJHZBIDUNKZUBJHZUCZEDABUDUEJZUFZDABLJZUGZXAFMZDKZEMZ | |
| + NZFXDUHZXHGIBAUIJZLJZGMZAOJZDKZUJZUFZHZXHXEHXHXCHXAXJXHXONZGXLUHXRXAXIXSF | |
| + GXNXDXLXAXMXLHZPZXTBQHAQHXNXDHXAXTRYABIWSWQWRWTXTUKSYAAIBWQWRWTXTULSXMBAU | |
| + MUOXAXFXDHZPZXFXNNZXFXFAUIJZAOJZNGYEXLYCXFAAXKOJZLJZHXKQHYEXLHYCXFXDYHXAY | |
| + BRZYCYGBALYCABYCAYCAIBWQWRWTYBULSZTZYCBYCBIWSWQWRWTYBUKSZTUPUQURYCBAYLYJU | |
| + SXFAXKVDVEYCXMYENZPZXNYFXFYNXMYEAOYCYMRUTVAYCYFXFYCXFAYCXFYCXDQXFABVBYIVC | |
| + TYKVFVGVHXIXHXGNXAYDPZXSXHXGWFYOXGXOXHYOXFXNDXAYDRVIVAVJVKGXLXOXHXPXPVLXN | |
| + DVMVNVOXAFIWSLJZXDXHDXAYPCDWQWRYPCDVPWTCDVQVRVSXAXDIBLJZYPXAAIVTKHZXDYQWA | |
| + WRWQYRWTAIBWBWCAIBWDWEXAWSBVTKHZYQYPWAWTWQYSWRBIWSWGWHBIWSWIWEWJWKXAXCXQX | |
| + HXAXBXPGCDABWLWMWNWPWO $. | |
| + $} | |
| + | |
| ${ | |
| $d F i $. $d L i $. $d V i $. $d W i $. | |
| $( The value of the subword extractor is the empty set (undefined) if the | |
| @@ -165470,6 +165761,64 @@ $) | |
| $} | |
| + ${ | |
| + $d L x $. $d V x $. $d W x $. | |
| + $( The reverse of a prefix of a word is equal to the same-length suffix of | |
| + the reverse of that word. (Contributed by BTernaryTau, 2-Dec-2023.) $) | |
| + revpfxsfxrev $p |- ( ( W e. Word V /\ L e. ( 0 ... ( # ` W ) ) ) -> | |
| + ( reverse ` ( W prefix L ) ) = | |
| + ( ( reverse ` W ) substr <. ( ( # ` W ) - L ) , ( # ` W ) >. ) ) $= | |
| + ( wcel cc0 chash cfv cfz co cfzo cmin wfn adantr oveq2d c1 3adant3 oveq1d | |
| + wceq cuz cz vx cword wa cpfx creverse cop csubstr pfxcl revcl 3syl revlen | |
| + wrdfn syl pfxlen eqtrd fneq2d mpbid swrdcl fznn0sub2 lencl sylib eleqtrrd | |
| + cn0 nn0fz0 swrdlen syl3an 3anidm13 cc nn0cnd elfzelz adantl nncand cv w3a | |
| + zcnd simp1 simp3 revfv sylan syl2anc fveq2d 3ad2ant2 elfzoelz 1cnd sub32d | |
| + 3ad2ant3 ubmelm1fzo eqeltrrd pfxfv syld3an3 3eqtrd eleq2d biimp3ar swrdfv | |
| + caddc id syl3anl1 syl3anl3 stoic3 syl3an2 elfzuz3 addlidd eluzsub mp3an2i | |
| + wss fzoss1 nn0zd 3ad2ant1 zsubcld fzo0addel pncan3d eleqtrd sseldd subcld | |
| + 0z subsub4d 3eqtr3d eqtr4d 3expa eqfnfvd ) CBUBZDZAECFGZHIZDZUCZUAEAJIZCA | |
| + UDIZUEGZCUEGZYCAKIZYCUFUGIZYFYIEYIFGZJIZLZYIYGLYBYOYEYBYHYADZYIYADYOBCAUH | |
| + ZBYHUIBYIULUJMYFYNYGYIYFYMAEJYFYMYHFGZAYBYMYRRZYEYBYPYSYQBYHUKUMMBCAUNZUO | |
| + NUPUQYFYLEYLFGZJIZLZYLYGLYBUUCYEYBYJYADZYLYADUUCBCUIZBYJYKYCURBYLULUJMYFU | |
| + UBYGYLYFUUAAEJYFUUAYCYKKIZAYBYEUUAUUFRZYBUUDYEYKYDDZYBYCEYJFGZHIZDZUUGUUE | |
| + AYCUSZYBYCYDUUJYBYCVCDYCYDDBCUTZYCVDVAYBUUIYCEHBCUKNVBZBYJYKYCVEVFVGYFYCA | |
| + YBYCVHDZYEYBYCUUMVIMZYEAVHDZYBYEAAEYCVJZVOZVKVLZUONUPUQYBYEUAVMZYGDZUVAYI | |
| + GZUVAYLGZRYBYEUVBVNZUVCAOKIZUVAKIZCGZUVDUVEUVCYROKIZUVAKIZYHGZUVGYHGZUVHU | |
| + VEYBUVAEYRJIZDZUVCUVKRZYBYEUVBVPZUVEUVAYGUVMYBYEUVBVQZYBYEUVMYGRUVBYFYRAE | |
| + JYTNPVBYBYPUVNUVOYQBYHUVAVRVSVTYBYEUVKUVLRUVBYFUVJUVGYHYFUVIUVFUVAKYFYRAO | |
| + KYTQQWAPYBYEUVBUVGYGDUVLUVHRUVEAUVAKIOKIZUVGYGUVEAUVAOYEYBUUQUVBUUSWBZUVB | |
| + YBUVAVHDYEUVBUVAUVAEAWCVOWFZUVEWDZWEUVBYBUVRYGDYEUVAAWGWFWHUVGABCWIWJWKUV | |
| + EUVDUVAYKWOIZYJGZYCOKIZUWBKIZCGZUVHYBYEUVBUVAEUUFJIZDZUVDUWCRZYBYEUWHUVBY | |
| + FUWGYGUVAYFUUFAEJUUTNWLWMYEYBUUHUWHUWIUULYBUUHYBUUHYBVNZUWHUWIYBUUHUWJUWJ | |
| + WPVGYBYBUUHUUKUWHUWIUUNYBUUDUUHUUKUWHUWIUUEBYJYKYCUVAWNWQWRWSWTWJUVEYBUWB | |
| + EYCJIZDUWCUWFRUVPUVEYKYCJIZUWKUWBYEYBUWLUWKXEZUVBYEYKESGDZUWMETDYEATDZYCE | |
| + AWOIZSGZDUWNXOUURYEYCASGUWQAEYCXAYEUWPASYEAUUSXBWAVBAEYCXCXDYKEYCXFUMWBUV | |
| + EUWBYKAYKWOIZJIZUWLUVEUVBYKTDUWBUWSDUVQUVEYCAYBYEYCTDUVBYBYCUUMXGXHYEYBUW | |
| + OUVBUURWBXIZUVAAYKXJVTUVEUWRYCYKJUVEAYCUVSYBYEUUOUVBUUPPZXKNXLXMBCUWBVRVT | |
| + UVEUWEUVGCUVEUWEUUFOKIZUVAKIZUVGUVEUWDUVAKIYKKIUWDYKKIZUVAKIUWEUXCUVEUWDU | |
| + VAYKUVEYCOUXAUWAXNZUVTUVEYKUWTVOZWEUVEUWDUVAYKUXEUVTUXFXPUVEUXDUXBUVAKUVE | |
| + YCOYKUXAUWAUXFWEQXQUVEUXBUVFUVAKUVEUUFAOKYBYEUUFARUVBUUTPQQUOWAWKXRXSXT | |
| + $. | |
| + $} | |
| + | |
| + $( A subword expressed in terms of reverses and prefixes. (Contributed by | |
| + BTernaryTau, 3-Dec-2023.) $) | |
| + swrdrevpfx $p |- ( ( W e. Word V /\ F e. ( 0 ... L ) /\ | |
| + L e. ( 0 ... ( # ` W ) ) ) -> ( W substr <. F , L >. ) = | |
| + ( reverse ` ( ( reverse ` ( W prefix L ) ) prefix ( L - F ) ) ) ) $= | |
| + ( wcel cc0 cfz co chash cfv w3a cpfx creverse cmin cop csubstr wceq 3adant2 | |
| + wa syl cword fznn0sub2 pfxcl revcl simp3 revlen adantr pfxlen eqtrd 3adant3 | |
| + 3ad2ant1 oveq2d eleqtrrd jca syl3an3 3com23 revpfxsfxrev revrev oveq1d zcnd | |
| + elfzel2 elfzelz nncand 3ad2ant2 opeq12d 3eqtrd wi cuz elfzuz3 eluzfz2 ancli | |
| + swrdpfx syl5 pm2.43i eqtr2d ) DCUAZEZAFBGHZEZBFDIJGHEZKZDBLHZMJZBANHZLHMJZW | |
| + BABOZPHZDWFPHZWAWEWCMJZWCIJZWDNHZWJOZPHZWBWLPHZWGWAWCVPEZWDFWJGHZEZSZWEWMQV | |
| + QVTVSWRVSVQVTWDVREZWRABUBVQVTWSKZWOWQVQVTWOWSVQWBVPEZWOCDBUCZCWBUDTUKWTWDVR | |
| + WPVQVTWSUEWTWJBFGVQVTWJBQZWSVQVTSZWJWBIJZBVQWJXEQZVTVQXAXFXBCWBUFTUGCDBUHUI | |
| + ZUJULUMUNUOUPWDCWCUQTVQVSWMWNQVTVQWIWBWLPVQXAWIWBQXBCWBURTUSUKWAWLWFWBPWAWK | |
| + AWJBWAWKBWDNHZAVQVTWKXHQVSXDWJBWDNXGUSRVSVQXHAQVTVSBAVSBAFBVAUTVSAAFBVBUTVC | |
| + VDUIVQVTXCVSXGRVEULVFWAWGWHQZVQVTWAXIVGVSWAVSBABGHEZSZXDXIVSVQXKVTVSXJVSBAV | |
| + HJEXJAFBVIABVJTVKVDABBCDVLVMRVNVO $. | |
| + | |
| $( | |
| =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= | |
| Repeated symbol words | |
| @@ -450705,6 +451054,60 @@ $) | |
| $} | |
| + ${ | |
| + $d x G $. $d x V $. | |
| + lfuhgr.1 $e |- V = ( Vtx ` G ) $. | |
| + lfuhgr.2 $e |- I = ( iEdg ` G ) $. | |
| + $( A hypergraph is loop-free if and only if every edge connects at least | |
| + two vertices. (Contributed by BTernaryTau, 15-Oct-2023.) $) | |
| + lfuhgr $p |- ( G e. UHGraph -> | |
| + ( I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } <-> | |
| + A. x e. ( Edg ` G ) 2 <_ ( # ` x ) ) ) $= | |
| + ( cuhgr wcel cedg cfv c2 cv chash cle wbr cpw wss wf crn syl ciedg edgval | |
| + crab cdm wral rneqi eqtr4i sseq1i wfun wi uhgrfun fdmrn fss sylbi impbid1 | |
| + ex frn bitrid wb cvtx c0 csn uhgredgss difss2d pweqi sseqtrrdi ssrab baib | |
| + bitr3d ) BGHZBIJZKALMJNOZADPZUCZQZCUDZVNCRZVLAVKUEZVOCSZVNQZVJVQVKVSVNVKB | |
| + UAJZSVSBUBCWAFUFUGUHVJVTVQVJCUIZVTVQUJZCBFUKWBVPVSCRZWCCULWDVTVQVPVSVNCUM | |
| + UPUNTVPVNCUQUOURVJVKVMQZVOVRUSVJVKBUTJZPZVMVJVKWGVAVBBVCVDDWFEVEVFVOWEVRV | |
| + LAVMVKVGVHTVI $. | |
| + | |
| + $( A hypergraph is loop-free if and only if every edge is not a loop. | |
| + (Contributed by BTernaryTau, 15-Oct-2023.) $) | |
| + lfuhgr2 $p |- ( G e. UHGraph -> | |
| + ( I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } <-> | |
| + A. x e. ( Edg ` G ) ( # ` x ) =/= 1 ) ) $= | |
| + ( wcel c2 cfv wbr cpw wral c1 wne wa c0 clt cvv elv cxr cuhgr cdm cv crab | |
| + chash cle wf cedg lfuhgr cc0 cvtx csn cdif uhgredgn0 eldifsni wb hashneq0 | |
| + syl sylibr gt0ne0d cxnn0 hashxnn0 xnn0n0n1ge2b ax-mp biimpi 3exp ralimdv2 | |
| + stoic3 a2d 1xr hashxrcl 1lt2 wi rexri xrltletr mp3an mpan xrltne mp3an12i | |
| + 2re ralimi impbid1 bitr4d ) BUAGZCUBHAUCZUEIZUFJZADKUDCUGWGABUHIZLZWFMNZA | |
| + WHLZABCDEFUIWDWKWIWDWJWGAWHWHWDWEWHGZWJWGWDWLWJWGWDWLWFUJNZWJWGWDWLOZWFWN | |
| + WEPNZUJWFQJZWNWEBUKIKZPULUMGWOWEBUNWEWQPUOURWPWOUPAWERUQSUSUTWMWJOZWGWFVA | |
| + GZWRWGUPWSAWERVBSWFVCVDVEVHVFVIVGWGWJAWHMTGZWFTGZWGMWFQJZWJVJXAAWERVKSZMH | |
| + QJZWGXBVLWTHTGXAXDWGOXBVMVJHVTVNXCMHWFVOVPVQMWFVRVSWAWBWC $. | |
| + $} | |
| + | |
| + ${ | |
| + $d x V $. $d x G a $. | |
| + lfuhgr3.1 $e |- V = ( Vtx ` G ) $. | |
| + lfuhgr3.2 $e |- I = ( iEdg ` G ) $. | |
| + $( A hypergraph is loop-free if and only if none of its edges connect to | |
| + only one vertex. (Contributed by BTernaryTau, 15-Oct-2023.) $) | |
| + lfuhgr3 $p |- ( G e. UHGraph -> | |
| + ( I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } <-> | |
| + -. E. a { a } e. ( Edg ` G ) ) ) $= | |
| + ( wcel cv cfv wbr c1 wral wex wn wceq wa notbii 3bitri eximi cuhgr cdm c2 | |
| + chash cle cpw crab wf wne cedg csn lfuhgr2 df-ne ralbii ralnex df-rex c1o | |
| + wrex cen cvv hashen1 elv en1 bitri anbi2i exbii 19.3v 19.29 sylanbr eleq1 | |
| + wb wal biimpac exlimiv dfclel pm3.22 sylbi excomim 19.40 ax5e anim1i 3syl | |
| + syl impbii bitrdi ) BUAHCUBUCAIZUDJZUEKADUFUGCUHWGLUIZABUJJZMZEIUKZWIHZEN | |
| + ZOZABCDFGULWJWFWIHZWGLPZQZANZOZWOWFWKPZENZQZANZOWNWJWPOZAWIMWPAWIURZOWSWH | |
| + XDAWIWGLUMUNWPAWIUOXEWRWPAWIUPRSWRXCWQXBAWPXAWOWPWFUQUSKZXAWPXFVKAWFUTVAV | |
| + BEWFVCVDVEVFRXCWMXCWMXBWMAXBWOWTQZENZWMWOWOEVLXAXHWOEVGWOWTEVHVIXGWLEWTWO | |
| + WLWFWKWIVJVMTWCVNWMXGANZENXHANXCWLXIEWLWTWOQZANXIAWKWIVOXJXGAWTWOVPTVQTXG | |
| + EAVRXHXBAXHWOENZXAQXBWOWTEVSXKWOXAWOEVTWAWCTWBWDRSWE $. | |
| + $} | |
| + | |
| $( | |
| =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= | |
| Undirected simple graphs | |
| @@ -458519,6 +458922,109 @@ $) | |
| $} | |
| + ${ | |
| + $d L k $. $d P k x $. $d F k x $. $d G k x $. | |
| + $( A prefix of a walk is a walk. (Contributed by BTernaryTau, | |
| + 2-Dec-2023.) $) | |
| + pfxwlk $p |- ( ( F ( Walks ` G ) P /\ L e. ( 0 ... ( # ` F ) ) ) -> | |
| + ( F prefix L ) ( Walks ` G ) ( P prefix ( L + 1 ) ) ) $= | |
| + ( vk vx cfv cc0 cfz co wcel wa c1 caddc wceq wss cfzo adantr syl adantl | |
| + cwlks wbr chash cpfx ciedg cdm cword cvtx wf csn cpr wral eqid wlkf pfxcl | |
| + wif cres wlkp cuz elfzuz3 fzss2 fssresd pfxlen sylan oveq2d feq2d wlkpwrd | |
| + mpbird fzp1elp1 wlklenvp1 eleqtrrd pfxres syl2an2r elfzelz fzval3 reseq2d | |
| + cv cz eqtr4d feq1d wlkprop simp3d eleq2d fveq1d fzossfz a1i sselda fvresd | |
| + eqtr2d fzofzp1 jca ex sylbid imp ancli simpr fveq2d eqcomd simplr wkslem1 | |
| + wi fzoss2 rspcv wb eqeq12 eqeq12d preq12 sseq12d ifpbi123d biimpd sylsyld | |
| + sneq mpd ralrimiva cvv w3a wlkv simp1d iswlkg mpbir3and ) BACUAGZUBZDHBUC | |
| + GZIJZKZLZBDUDJZADMNJZUDJZYAUBZYGCUEGZUFZUGZKZHYGUCGZIJZCUHGZYIUIZEVQZYIGZ | |
| + YSMNJZYIGZOZYSYGGZYKGZYTUJZOZYTUUBUKZUUEPZUPZEHYOQJZULZYFBYMKZYNYBUUMYEAB | |
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| + IZYEABCYQYQUMZURRYEUUQYDPZYBYEYCDUSGZKZUVBDHYCUTZDHYCVASTVBYFYPUUQYQUURYF | |
| + YODHIYBUUMYEYODOUUOYLBDVCVDZVEVFVHYFYPYQYIUURYFYIAHYHQJZUQZUURYBAYQUGKYEY | |
| + HHAUCGZIJZKYIUVHOABCYQUVAVGYFYHHYCMNJZIJZUVJYEYHUVLKYBDHYCVITYBUVJUVLOYEY | |
| + BUVIUVKHIABCVJVERVKYQAYHVLVMYFUUQUVGAYEUUQUVGOZYBYEDVRKUVMDHYCVNHDVOSTVPV | |
| + SZVTVHYFUUJEUUKYFYSUUKKZLZFVQZAGZUVQMNJAGZOUVQBGYKGZUVRUJOUVRUVSUKUVTPUPZ | |
| + FHYCQJZULZUUJYFUWCUVOYBUWCYEYBUUMUUTUWCAFBCYKYQUVAUUNWAWBRRUVPYSAGZYTOZUU | |
| + AAGZUUBOZLZYSBGZYKGZUUEOZLZUWCUWDUWFOZUWJUWDUJZOZUWDUWFUKZUWJPZUPZUUJUVPU | |
| + WHUWKYFUVOUWHYFUVOYSHDQJZKZUWHYFUUKUWSYSYFYODHQUVFVEWCZYFUWTUWHYFUWTLZUWE | |
| + UWGUXBYTYSUURGZUWDYFYTUXCOUWTYFYSYIUURUVNWDRUXBYSUUQAYFUWSUUQYSUWSUUQPYFH | |
| + DWEWFWGWHWIUXBUUBUUAUURGZUWFYFUUBUXDOUWTYFUUAYIUURUVNWDRUXBUUAUUQAUWTUUAU | |
| + UQKYFHDYSWJTWHWIWKWLWMWNUVPUWJYSBUWSUQZGZYKGZUUEYFUVOUWJUXGOZYFUVOUWTUXHU | |
| + XAYFUWTUXHUXBUXGUWJYFYFUUMLZUWTUXGUWJOYFUUMUUPWOUXIUWTLZUXFUWIYKUXJYSUWSB | |
| + UXIUWTWPWHWQVDWRWLWMWNUVPUUDUXFYKUVPYSYGUXEYFUUMUVOYEYGUXEOUUPYBYEUVOWSYL | |
| + BDVLVMWDWQVSWKUVPYSUWBKUWCUWRXAYFUUKUWBYSYFYCYOUSGZKUUKUWBPYFYCUVCUXKYEUV | |
| + DYBUVETYFYODUSUVFWQVKYOHYCXBSWGUWAUWRFYSUWBUVQYSABYKWTXCSUWLUWRUUJUWLUWMU | |
| + WOUWQUUCUUGUUIUWHUWMUUCXDUWKUWDYTUWFUUBXERUWLUWJUUEUWNUUFUWHUWKWPZUWHUWNU | |
| + UFOZUWKUWEUXMUWGUWDYTXLRRXFUWLUWPUUHUWJUUEUWHUWPUUHOUWKUWDUWFYTUUBXGRUXLX | |
| + HXIXJXKXMXNYFCXOKZYJYNYRUULXPXDYBUXNYEYBUXNBXOKAXOKABCXQXRRYIEYGCYKYQXOUV | |
| + AUUNXSSXT $. | |
| + $} | |
| + | |
| + ${ | |
| + $d P k x $. $d F k x $. $d G k x $. | |
| + $( The reverse of a walk is a walk. (Contributed by BTernaryTau, | |
| + 30-Nov-2023.) $) | |
| + revwlk $p |- ( F ( Walks ` G ) P -> | |
| + ( reverse ` F ) ( Walks ` G ) ( reverse ` P ) ) $= | |
| + ( vk vx cfv wcel cc0 chash co c1 caddc wceq cfzo oveq2d eqtrd cmin oveq1d | |
| + syl adantr cwlks wbr creverse ciedg cdm cword cfz cvtx wf csn cpr wss wif | |
| + cv wral eqid wlkf revcl wlkpwrd wrdf 3syl revlen wlklenvp1 cz wlkcl nn0zd | |
| + fzval3 3eqtr4rd feq2d mpbird eleq2d biimpa wa c2 revfv sylan wlklenvm1 cc | |
| + lencl nn0cnd sub1m1 fvoveq1d fveq2d fzonn0p1p1 eleqtrrd syl2an2r elfzoelz | |
| + adantl zcnd 1cnd addcomd subsub4d 3eqtr2d sneqd sneq eqcom cn0 fzossfzop1 | |
| + subcld sseqtrrd sselda sub32d npcand 3eqtr3d eqeq12d bitrid wi wn wkslem1 | |
| + wlkprop simp3d ubmelm1fzo eqeltrrd rspcdva dfifp2 sylib simpld sylbid imp | |
| + notbid simprd prcom preq12d eqtr3id 3sstr4d ifpimpda syldan ralrimiva cvv | |
| + w3a wb wlkv simp1d iswlkg mpbir3and ) BACUAFZUBZBUCFZAUCFZYPUBZYRCUDFZUEZ | |
| + UFZGZHYRIFZUGJZCUHFZYSUIZDUNZYSFZUUIKLJZYSFZMZUUIYRFZUUAFZUUJUJZMZUUJUULU | |
| + KZUUOULZUMZDHUUENJZUOZYQBUUCGZUUDABCUUAUUAUPZUQZUUBBURSYQUUHHYSIFZNJZUUGY | |
| + SUIZYQAUUGUFZGZYSUVIGUVHABCUUGUUGUPZUSZUUGAURUUGYSUTVAYQUUFUVGUUGYSYQUUFH | |
| + BIFZUGJZUVGYQUUEUVMHUGYQUVCUUEUVMMUVEUUBBVBSZOYQHAIFZNJZHUVMKLJZNJZUVGUVN | |
| + YQUVPUVRHNABCVCOZYQUVFUVPHNYQUVJUVFUVPMUVLUUGAVBSOYQUVMVDGUVNUVSMYQUVMABC | |
| + VEZVFHUVMVGSVHPVIVJYQUUTDUVAYQUUIUVAGZUUIHUVMNJZGZUUTYQUWBUWDYQUVAUWCUUIY | |
| + QUUEUVMHNUVOOVKVLYQUWDVMZUUMUUQUUSUWEUUMVMZUUOUVPVNQJZUUIQJZBFZUUAFZUUPUW | |
| + EUUOUWJMZUUMUWEUUNUWIUUAUWEUUNUVMKQJZUUIQJZBFZUWIYQUVCUWDUUNUWNMUVEUUBBUU | |
| + IVOVPYQUWNUWIMUWDYQUWLUWGUUIBQYQUWLUVPKQJZKQJZUWGYQUVMUWOKQABCVQRYQUVJUVP | |
| + VRGZUWPUWGMUVLUVJUVPUUGAVSVTZUVPWAVAZPZWBTPWCZTUWFUULUJZUWHAFZUJZUUPUWJUW | |
| + EUXBUXDMUUMUWEUULUXCUWEUULUWOUUKQJZAFZUXCYQUVJUWDUUKUVQGUULUXFMUVLUWEUUKU | |
| + VSUVQUWDUUKUVSGYQUUIUVMWDWHYQUVQUVSMUWDUVTTWEUUGAUUKVOWFUWEUXEUWHAUWEUXEU | |
| + WOKUUILJZQJUWPUUIQJZUWHUWEUUKUXGUWOQUWEUUIKUWDUUIVRGYQUWDUUIUUIHUVMWGWIWH | |
| + ZUWEWJZWKOUWEUWOKUUIUWEUVPKYQUWQUWDYQUVJUWQUVLUWRSTUXJWSZUXJUXIWLYQUXHUWH | |
| + MUWDYQUWPUWGUUIQUWSRZTWMWCPZWNTUUMUUPUXBMUWEUUJUULWOWHUWEUUMUWJUXDMZUWEUU | |
| + MUXCUWHKLJZAFZMZUXNUUMUULUUJMUWEUXQUUJUULWPUWEUULUXCUUJUXPUXMUWEUUJUWOUUI | |
| + QJZAFZUXPYQUVJUWDUUIUVQGUUJUXSMUVLYQUWCUVQUUIYQUWCUVSUVQYQUVMWQGUWCUVSULU | |
| + WAUVMWRSUVTWTXAUUGAUUIVOWFUWEUXRUXOAUWEUXRKQJZKLJUXHKLJZUXRUXOUWEUXTUXHKL | |
| + UWEUWOUUIKUXKUXIUXJXBRUWEUXRKUWEUWOUUIUXKUXIWSUXJXCYQUYAUXOMUWDYQUXHUWHKL | |
| + UXLRTXDWCPZXEXFZUWEUXQUXNXGZUXQXHZUXCUXPUKZUWJULZXGZUWEUXQUXNUYGUMZUYDUYH | |
| + VMUWEEUNZAFZUYJKLJAFZMUYJBFUUAFZUYKUJMUYKUYLUKUYMULUMZUYIEUWCUWHUYJUWHABU | |
| + UAXIYQUYNEUWCUOZUWDYQUVCUVNUUGAUIUYOAEBCUUAUUGUVKUVDXJXKTUWEUVMUUIQJKQJZU | |
| + WHUWCUWEUYPUWMUWHUWEUVMUUIKYQUVMVRGUWDYQUVMUWAVTTUXIUXJXBUWEUWLUWGUUIQYQU | |
| + WLUWGMUWDUWTTRPUWDUYPUWCGYQUUIUVMXLWHXMXNUXQUXNUYGXOXPZXQXRXSVHPUWEUUMXHZ | |
| + VMUYFUWJUURUUOUWEUYRUYGUWEUYRUYEUYGUWEUUMUXQUYCXTUWEUYDUYHUYQYAXRXSUWEUUR | |
| + UYFMUYRUWEUURUULUUJUKUYFUULUUJYBUWEUULUXCUUJUXPUXMUYBYCYDTUWEUWKUYRUXATYE | |
| + YFYGYHYQCYIGZYTUUDUUHUVBYJYKYQUYSBYIGAYIGABCYLYMYSDYRCUUAUUGYIUVKUVDYNSYO | |
| + $. | |
| + $} | |
| + | |
| + $( Two matching subwords of a walk also represent a walk. (Contributed by | |
| + BTernaryTau, 7-Dec-2023.) $) | |
| + swrdwlk $p |- ( ( F ( Walks ` G ) P /\ B e. ( 0 ... L ) /\ | |
| + L e. ( 0 ... ( # ` F ) ) ) -> ( F substr <. B , L >. ) ( Walks ` G ) | |
| + ( P substr <. B , ( L + 1 ) >. ) ) $= | |
| + ( cfv wbr cc0 cfz co wcel chash cpfx creverse c1 caddc 3ad2ant2 wceq oveq2d | |
| + 3ad2ant1 cwlks w3a cmin cop csubstr pfxwlk 3adant2 revwlk syl fznn0sub2 cdm | |
| + ciedg cword eqid wlkf pfxcl revlen 3syl pfxlen sylan eqtrd eleqtrrd syl2anc | |
| + cz elfzel2 zcnd 1cnd elfzelz addsubd swrdrevpfx syl3an1 cvtx wlkpwrd fzelp1 | |
| + breqtrrd fzp1elp1 3ad2ant3 wlklenvp1 syl3anc 3brtr4d ) CBDUAFZGZAHEIJZKZEHC | |
| + LFZIJKZUBZCEMJZNFZEAUCJZMJZNFZBEOPJZMJZNFZWMAUCJZMJZNFZCAEUDUEJZBAWMUDUEJZW | |
| + AWGWKWQWAGWLWRWAGWGWKWOWJOPJZMJZWQWAWGWIWOWAGZWJHWILFZIJZKWKXBWAGWGWHWNWAGZ | |
| + XCWBWFXFWDBCDEUFUGWNWHDUHUIWGWJWCXEWDWBWJWCKWFAEUJQWGXDEHIWGXDWHLFZEWGCDULF | |
| + ZUKZUMZKZWHXJKXDXGRWBWDXKWFBCDXHXHUNUOZTXICEUPXIWHUQURWBWFXGERZWDWBXKWFXMXL | |
| + XICEUSUTUGVASVBWOWIDWJUFVCWGWPXAWOMWGEOAWGEWDWBEVDKWFAHEVEQVFWGVGWGAWDWBAVD | |
| + KWFAHEVHQVFVISVOWQWKDUHUIWBXKWDWFWSWLRXLAEXICVJVKWGBDVLFZUMKZAHWMIJKZWMHBLF | |
| + ZIJZKWTWRRWBWDXOWFBCDXNXNUNVMTWDWBXPWFAHEVNQWGWMHWEOPJZIJZXRWFWBWMXTKWDEHWE | |
| + VPVQWGXQXSHIWBWDXQXSRWFBCDVRTSVBAWMXNBVJVSVT $. | |
| + | |
| $( | |
| =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= | |
| Walks for loop-free graphs | |
| @@ -467726,6 +468232,358 @@ $) | |
| $} | |
| + ${ | |
| + pthhashvtx.1 $e |- V = ( Vtx ` G ) $. | |
| + $( A graph containing a path has at least as many vertices as there are | |
| + edges in the path. (Contributed by BTernaryTau, 5-Oct-2023.) $) | |
| + pthhashvtx $p |- ( F ( Paths ` G ) P -> ( # ` F ) <_ ( # ` V ) ) $= | |
| + ( cfv wbr c1 co wcel cle cc0 cfz wceq syl cres wss cima c0 ax-mp cmin cn0 | |
| + cpths chash wa caddc hashfz0 cc cwlks pthiswlk wlkcl nn0cn 3syl sylan9eqr | |
| + npcan1 cfn wf1 wfn wlkp ffnd fzfi resfnfinfin sylancl simpr fzssp1 oveq2d | |
| + sseqtrid fssresd adantr ccnv wfun fz1ssfz0 a1i cfzo ctrls cpr cin simp2bi | |
| + wf ispth cz nn0z fzoval reseq2d resabs1 eqtr4di cnveqd funeqd mpbid df-f1 | |
| + sylanbrc csn snsspr1 imass2 wb 0elfz snssd resima2 mpbiri imaeq2d eqtr4id | |
| + sseq1 ineq2d simp3bi eqtrd syl2anr f1resfz0f1d cvv cvtx fvexi hashf1dmcdm | |
| + ssdisj mp3an2 syl2an2r eqbrtrrd 0nn0m1nnn0 biimpri sylan hashge0 eqbrtrdi | |
| + wn pm2.61dan ) BACUCFGZBUDFZHUAIZUBJZYDDUDFZKGYCYFUEZLYEMIZUDFZYDYGKYFYCY | |
| + JYEHUFIZYDYEUGYCYDUBJZYDUHJYKYDNYCBACUIFGZYLABCUJZABCUKOZYDULYDUOUMZUNYCA | |
| + YIPZUPJZYFYIDYQUQZYJYGKGZYCALYDMIZURYIUPJYRYCUUADAYCYMUUADAVSYNABCDEUSOZU | |
| + TLYEVAUUAYIAVBVCYHYQYEDYCYFVDYCYIDYQVSYFYCUUADYIAUUBYCLYKMIYIUUALYEVEYCYK | |
| + YDLMYPVFVGVHZVIYCHYEMIZDYQUUDPZUQZYFYCUUDDUUEVSUUEVJZVKZUUFYCYIDUUDYQUUCU | |
| + UDYIQZYCYEVLZVMVHYCAHYDVNIZPZVJZVKZUUHYCBACVOFGZUUNALYDVPZRZAUUKRZVQZSNZA | |
| + BCVTZVRYCUUMUUGYCUULUUEYCUULAUUDPZUUEYCUUKUUDAYCYLUUKUUDNZYOYLYDWAJUVCYDW | |
| + BHYDWCOOZWDUUIUUEUVBNUUJAUUDYIWETWFWGWHWIUUDDUUEWJWKVIYFYQLWLZRZUUQQZUUQY | |
| + QUUDRZVQZSNUVFUVHVQSNYCYFUVGAUVERZUUQQZUVEUUPQUVKLYDWMUVEUUPAWNTYFUVEYIQU | |
| + VFUVJNUVGUVKWOYFLYIYEWPWQAUVEYIWRUVFUVJUUQXBUMWSYCUVIUUSSYCUVHUURUUQYCUVH | |
| + AUUDRZUURUUIUVHUVLNUUJAUUDYIWRTYCUUKUUDAUVDWTXAXCYCUUOUUNUUTUVAXDXEUVFUUQ | |
| + UVHXLXFXGYRDXHJZYSYTDCXIEXJZYIDYQUPXHXKXMXNXOYCYFYAZUEYDLYGKYCYLUVOYDLNZY | |
| + OUVPYLUVOUEYDXPXQXRUVMLYGKGUVNDXHXSTXTYB $. | |
| + $} | |
| + | |
| + $( A walk is a trivial path if and only if it is both a simple path and a | |
| + cycle. (Contributed by BTernaryTau, 8-Oct-2023.) $) | |
| + spthcycl $p |- ( ( F ( Paths ` G ) P /\ F = (/) ) <-> | |
| + ( F ( SPaths ` G ) P /\ F ( Cycles ` G ) P ) ) $= | |
| + ( cfv wbr c0 wceq wa wfun cc0 chash co c1 caddc adantr cvv wcel sylan syl | |
| + wn cpths cspths ccycls ctrls ccnv pthistrl cwlks pthiswlk c1o cen cvtx eqid | |
| + cfz wlkp ffund wlklenvp1 simp2d hasheq0 biimpar oveq1 0p1e1 eqtrdi eqtrd wb | |
| + simp3d hashen1 biimpa syldan funen1cnv syl2an2r isspth biimpri fveq2 eqcoms | |
| + wlkv iscycl jca spthispth notnot wne cyclnspth com12 con3dimp sylan2 ancoms | |
| + nne sylib impbii ) BACUADEZBFGZHZBACUBDEZBACUCDEZHZWKWLWMWIBACUDDEZWJAUEIZW | |
| + LABCUFWIBACUGDEZWJWPABCUHZWQAIWJAUIUJEZWPWQJBKDZUMLCUKDZAABCXAXAULUNUOWQWJA | |
| + KDZMGZWSWQWJHZXBWTMNLZMWQXBXEGWJABCUPOXDWTJGZXEMGWQBPQZWJXFWQCPQZXGAPQZABCV | |
| + OZUQXGXFWJBPURUSRZXFXEJMNLMWTJMNUTVAVBSVCWQXCWSWQXIXCWSVDWQXHXGXIXJVEAPVFSV | |
| + GVHAVIVJRWLWOWPHABCVKVLVJWIWJJADWTADGZWMWIWQWJXLWRXDXFXLXKXLJWTJWTAVMVNSRWM | |
| + WIXLHABCVPVLVHVQWNWIWJWLWIWMABCVROWMWLWJWLWMWLTZTZWJWLVSWMXNHBFVTZTWJWMXOXM | |
| + XOWMXMABCWAWBWCBFWFWGWDWEVQWH $. | |
| + | |
| + ${ | |
| + $d P k $. $d S k $. $d k F $. $d k G $. | |
| + $( If a walk exists in a subgraph of a graph ` G ` , then that walk also | |
| + exists in ` G ` . (Contributed by BTernaryTau, 22-Oct-2023.) $) | |
| + subgrwlk $p |- ( S SubGraph G -> | |
| + ( F ( Walks ` S ) P -> F ( Walks ` G ) P ) ) $= | |
| + ( vk wbr cwlks cfv ciedg wcel co wceq wss wral w3a wa cvv eqid syl wi cdm | |
| + csubgr cword cc0 chash cfz cvtx wf cv c1 caddc csn cpr wif cfzo wb subgrv | |
| + simpld iswlkg 3simpa cedg cpw subgrprop2 simp2d dmss sswrd 3syl sseld fss | |
| + simp1d expcom syl5 3simpb cres subgrprop fveq1d 3ad2ant1 wrdsymbcl fvresd | |
| + anim12d 3adant1 eqtrd eqeq1d sseq2d ifpbi23d biimpd 3expia ralrimiv ralim | |
| + expimpd jcad sylbid df-3an imbitrrdi simpl2im sylibrd ) BDUBFZCABGHFZCDIH | |
| + ZUAZUCZJZUDCUEHZUFKZDUGHZAUHZEUIZAHZXGUJUKKAHZLZXGCHZWSHZXHULZLZXHXIUMZXL | |
| + MZUNZEUDXCUOKZNZOZCADGHFZWQWRXBXFPZXSPZXTWQWRCBIHZUAZUCZJZXDBUGHZAUHZXJXK | |
| + YDHZXMLZXOYJMZUNZEXRNZOZYCWQBQJZWRYOUPWQYPDQJZBDUQZURAECBYDYHQYHRZYDRZUSS | |
| + WQYOYBXSYOYGYIPWQYBYGYIYNUTWQYGXBYIXFWQYFXACWQYDWSMZYEWTMYFXAMWQYHXEMZUUA | |
| + BVAHZYHVBMZXEWSBUUCDYDYHYSXERZYTWSRZUUCRZVCZVDYDWSVEYEWTVFVGVHWQUUBYIXFTW | |
| + QUUBUUAUUDUUHVJYIUUBXFXDYHXEAVIVKSVTVLYOYGYNPWQXSYGYIYNVMWQYGYNXSWQYGPZYM | |
| + XQTZEXRNYNXSTUUIUUJEXRWQYGXGXRJZUUJWQYGUUKOZYMXQUULXJYKYLXNXPUULYJXLXMUUL | |
| + YJXKWSYEVNZHZXLWQYGYJUUNLUUKWQXKYDUUMWQUUBYDUUMLUUDXEWSBUUCDYDYHYSUUEYTUU | |
| + FUUGVOVDVPVQYGUUKUUNXLLWQYGUUKPXKYEWSXGYECVRVSWAWBZWCUULYJXLXOUUOWDWEWFWG | |
| + WHYMXQEXRWISWJVLWKWLXBXFXSWMWNWQYPYQYAXTUPYRAECDWSXEQUUEUUFUSWOWP $. | |
| + $} | |
| + | |
| + $( If a trail exists in a subgraph of a graph ` G ` , then that trail also | |
| + exists in ` G ` . (Contributed by BTernaryTau, 22-Oct-2023.) $) | |
| + subgrtrl $p |- ( S SubGraph G -> | |
| + ( F ( Trails ` S ) P -> F ( Trails ` G ) P ) ) $= | |
| + ( csubgr wbr cwlks cfv ccnv wfun wa ctrls subgrwlk anim1d istrl 3imtr4g ) B | |
| + DEFZCABGHFZCIJZKCADGHFZSKCABLHFCADLHFQRTSABCDMNACBOACDOP $. | |
| + | |
| + $( If a path exists in a subgraph of a graph ` G ` , then that path also | |
| + exists in ` G ` . (Contributed by BTernaryTau, 22-Oct-2023.) $) | |
| + subgrpth $p |- ( S SubGraph G -> | |
| + ( F ( Paths ` S ) P -> F ( Paths ` G ) P ) ) $= | |
| + ( csubgr wbr ctrls cfv c1 chash cfzo co cres ccnv wfun cima w3a cpths ispth | |
| + idd cc0 cpr cin c0 wceq subgrtrl 3anim123d 3imtr4g ) BDEFZCABGHFZAICJHZKLZM | |
| + NOZAUAUKUBPAULPUCUDUEZQCADGHFZUMUNQCABRHFCADRHFUIUJUOUMUMUNUNABCDUFUIUMTUIU | |
| + NTUGACBSACDSUH $. | |
| + | |
| + $( If a cycle exists in a subgraph of a graph ` G ` , then that cycle also | |
| + exists in ` G ` . (Contributed by BTernaryTau, 23-Oct-2023.) $) | |
| + subgrcycl $p |- ( S SubGraph G -> | |
| + ( F ( Cycles ` S ) P -> F ( Cycles ` G ) P ) ) $= | |
| + ( csubgr wbr cpths cfv cc0 chash wceq ccycls subgrpth anim1d iscycl 3imtr4g | |
| + wa ) BDEFZCABGHFZIAHCJHAHKZQCADGHFZTQCABLHFCADLHFRSUATABCDMNACBOACDOP $. | |
| + | |
| + ${ | |
| + $d A j $. $d G p $. $d A f p $. $d f j G $. | |
| + $( A hypergraph has a cycle of length one if and only if it has a loop. | |
| + (Contributed by BTernaryTau, 13-Oct-2023.) $) | |
| + loop1cycl $p |- ( G e. UHGraph -> | |
| + ( E. f E. p ( f ( Cycles ` G ) p /\ ( # ` f ) = 1 /\ ( p ` 0 ) = A ) <-> | |
| + { A } e. ( Edg ` G ) ) ) $= | |
| + ( vj wcel cv cfv wbr chash c1 wceq cc0 w3a wex csn wi wa syl wrex anabss3 | |
| + cuhgr ccycls cedg cwlks cpths cyclprop fveq2 eqeq2d anbi2d mpan9 pthiswlk | |
| + biimpd anim1i df-3an 3ancomb sylib ciedg wfun wlkl1loop expl eqid uhgrfun | |
| + sylibr syl11 3impb 3adant3 sneq eleq1d 3ad2ant3 sylibd exlimivv com12 cs2 | |
| + wb cs1 wal cdm crn edgval eleq2i elrnrexdm eqcom rexbii imbitrdi biimtrid | |
| + df-rex lp1cycl 3expib eximdv syld s1len ax-gen 19.29r syl6 imp cvv c0 wne | |
| + mpan2 cvtx cpw uhgredgn0 eldifsni snnzb s2fv0 alrimiv syl2anc exbii cword | |
| + cdif s1cli breq1 fveqeq2 3anbi12d rspcev rexex exlimiv s2cli breq2 eqeq1d | |
| + mpan fveq1 3anbi13d eximi ex impbid ) CUBFZBGZDGZCUCHZIZYIJHZKLZMYJHZALZN | |
| + ZDOZBOZAPZCUDHZFZYSYHUUBYQYHUUBQBDYQYHYOPZUUAFZUUBYLYNYHUUDQZYPYLYNRZYIYJ | |
| + CUEHIZYNYOKYJHZLZNZUUEUUFUUGUUIYNNZUUJUUFUUGUUIRZYNRZUUKYLYNUUMUUFUULYNUU | |
| + FYIYJCUFHIZUUIRZUULYLUUNYOYMYJHZLZRZYNUUOYJYICUGYNUURUUOYNUUQUUIUUNYNUUPU | |
| + UHYOYMKYJUHUIUJUMUKUUNUUGUUIYJYICULUNSUNUAUUGUUIYNUOVDUUGUUIYNUPUQUUGYNUU | |
| + IUUECURHZUSZUUGYNUUIRZRUUDYHUUTUUGUVAUUDYJYICUTVAUUSCUUSVBZVCZVEVFSVGYPYL | |
| + UUDUUBVOYNYPUUCYTUUAYOAVHVIVJVKVLVMYHUUBYSYHUUBRZYIAAVNZYKIZYNMUVEHZALZNZ | |
| + BOZYSUVDEGZVPZUVEYKIZUVLJHKLZUVHNZEOZUVJUVDUVMUVNRZUVHRZEOZUVPUVDUVQEOZUV | |
| + HEVQUVSYHUUBUVTYHUUBUVMEOZUVTYHUUBUVKUUSVRZFZUVKUUSHZYTLZRZEOZUWAYHUUBUWE | |
| + EUWBTZUWGYHUUTUUBUWHQUVCUUBYTUUSVSZFZUUTUWHUUAUWIYTCVTWAUUTUWJYTUWDLZEUWB | |
| + TUWHEUUSYTWBUWKUWEEUWBYTUWDWCWDWEWFSUWEEUWBWGWEYHUWFUVMEYHUWCUWEUVMACUUSU | |
| + VKUVBWHWIWJWKUWAUVNEVQUVTUVNEUVKWLWMUVMUVNEWNWTWOWPUVDUVHEUVDAWQFZUVHUVDY | |
| + TWRWSZUWLUVDYTCXAHXBZWRPXKFUWMYTCXCYTUWNWRXDSAXEVDAAWQXFSXGUVQUVHEWNXHUVO | |
| + UVREUVMUVNUVHUOXIVDUVOUVJEUVOUVIBWQXJZTZUVJUVLUWOFUVOUWPUVKXLUVIUVOBUVLUW | |
| + OYIUVLLUVFUVMYNUVNUVHYIUVLUVEYKXMYIUVLKJXNXOXPYBUVIBUWOXQSXRSUVIYRBUVIYQD | |
| + UWOTZYRUVEUWOFUVIUWQAAXSYQUVIDUVEUWOYJUVELZYLUVFYPUVHYNYJUVEYIYKXTUWRYOUV | |
| + GAMYJUVEYCYAYDXPYBYQDUWOXQSYESYFYG $. | |
| + $} | |
| + | |
| + ${ | |
| + 2cycld.1 $e |- P = <" A B C "> $. | |
| + 2cycld.2 $e |- F = <" J K "> $. | |
| + 2cycld.3 $e |- ( ph -> ( A e. V /\ B e. V /\ C e. V ) ) $. | |
| + 2cycld.4 $e |- ( ph -> ( A =/= B /\ B =/= C ) ) $. | |
| + 2cycld.5 $e |- ( ph -> | |
| + ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) ) $. | |
| + 2cycld.6 $e |- V = ( Vtx ` G ) $. | |
| + 2cycld.7 $e |- I = ( iEdg ` G ) $. | |
| + 2cycld.8 $e |- ( ph -> J =/= K ) $. | |
| + 2cycld.9 $e |- ( ph -> A = C ) $. | |
| + $( Construction of a 2-cycle from two given edges in a graph. (Contributed | |
| + by BTernaryTau, 16-Oct-2023.) $) | |
| + 2cycld $p |- ( ph -> F ( Cycles ` G ) P ) $= | |
| + ( cpths cfv wbr cc0 chash wceq ccycls 2pthd wcel w3a wa cs3 fveq1i eqtrid | |
| + s3fv0 3ad2ant1 adantr simpr cs2 fveq2i s2len eqtri fveq12i s3fv2 3ad2ant3 | |
| + c2 eqtr2id 3eqtrd syl2anc iscycl sylanbrc ) AFEGUAUBUCUDEUBZFUEUBZEUBZUFZ | |
| + FEGUGUBUCABCDEFGHIJKLMNOPQRSUHABKUIZCKUIZDKUIZUJZBDUFZVONTVSVTUKVLBDVNVSV | |
| + LBUFZVTVPVQWAVRVPVLUDBCDULZUBBUDEWBLUMBCDKUOUNUPUQVSVTURVSDVNUFZVTVRVPWCV | |
| + QVRVNVFWBUBDVMVFEWBLVMIJUSZUEUBVFFWDUEMUTIJVAVBVCBCDKVDVGVEUQVHVIEFGVJVK | |
| + $. | |
| + $} | |
| + | |
| + ${ | |
| + 2cycl2d.1 $e |- P = <" A B A "> $. | |
| + 2cycl2d.2 $e |- F = <" J K "> $. | |
| + 2cycl2d.3 $e |- ( ph -> ( A e. V /\ B e. V ) ) $. | |
| + 2cycl2d.4 $e |- ( ph -> A =/= B ) $. | |
| + 2cycl2d.5 $e |- ( ph -> | |
| + ( { A , B } C_ ( I ` J ) /\ { A , B } C_ ( I ` K ) ) ) $. | |
| + 2cycl2d.6 $e |- V = ( Vtx ` G ) $. | |
| + 2cycl2d.7 $e |- I = ( iEdg ` G ) $. | |
| + 2cycl2d.8 $e |- ( ph -> J =/= K ) $. | |
| + $( Construction of a 2-cycle from two given edges in a graph. (Contributed | |
| + by BTernaryTau, 16-Oct-2023.) $) | |
| + 2cycl2d $p |- ( ph -> F ( Cycles ` G ) P ) $= | |
| + ( wa wss wcel w3a simpl jccir df-3an sylibr wne necomd jca cpr cfv sseq1i | |
| + prcom anbi2i sylib eqidd 2cycld ) ABCBDEFGHIJKLABJUAZCJUAZSZURSURUSURUBAU | |
| + TURMURUSUCUDURUSURUEUFABCUGCBUGNABCNUHUIABCUJZHGUKTZVAIGUKZTZSVBCBUJZVCTZ | |
| + SOVDVFVBVAVEVCBCUMULUNUOPQRABUPUQ $. | |
| + $} | |
| + | |
| + ${ | |
| + $d ph a b $. $d I a b $. $d J a b $. $d F p a b $. $d G p a b $. | |
| + umgr2cycllem.1 $e |- F = <" J K "> $. | |
| + umgr2cycllem.2 $e |- I = ( iEdg ` G ) $. | |
| + umgr2cycllem.3 $e |- ( ph -> G e. UMGraph ) $. | |
| + umgr2cycllem.4 $e |- ( ph -> J e. dom I ) $. | |
| + umgr2cycllem.5 $e |- ( ph -> J =/= K ) $. | |
| + umgr2cycllem.6 $e |- ( ph -> ( I ` J ) = ( I ` K ) ) $. | |
| + $( Lemma for ~ umgr2cycl . (Contributed by BTernaryTau, 17-Oct-2023.) $) | |
| + umgr2cycllem $p |- ( ph -> E. p F ( Cycles ` G ) p ) $= | |
| + ( va vb cv cfv wa wrex wcel wne cpr wceq cvtx wbr wex cumgr cedg wfun cdm | |
| + ccycls cuhgr umgruhgr uhgrfun 3syl iedgedg syl2anc eqid umgredg wral ax-5 | |
| + wal alral syl r19.29 sylan cs3 w3a simp2 simp3l wss eqimss2 adantl sseq2d | |
| + wi 3ad2ant3 imbitrid adantld 3impib jca 3ad2ant1 3expib exp4c com23 imp4a | |
| + 2cycl2d cvv cword s3cli breq2 rspcev mpan rexex syl8 rexlimdv syl5 expd | |
| + mpd ) ANPZOPZUAZEDQZWSWTUBZUCZRZOCUDQZSZNXFSZBGPZCUKQZUEZGUFZACUGTZXBCUHQ | |
| + ZTZXHJADUIZEDUJTXOAXMCULTXPJCUMDCIUNUOKDCEIUPUQXBXNCXFNOXFURZXNURUSUQAXGX | |
| + LNXFAWSXFTZXGXLXRXGRXRXERZOXFSZAXLXRXROXFUTZXGXTXRXROVBYAXROVAXROXFVCVDXR | |
| + XEOXFVEVFAXSXLOXFAWTXFTZXSBWSWTWSVGZXJUEZXLAYBXRXEYDAXRYBXEYDVOAXRYBXEYDA | |
| + XRYBRZXEYDAYEXEVHZWSWTYCBCDEFXFYCURHAYEXEVIAYEXAXDVJYFXCXBVKZXCFDQZVKZXEA | |
| + YGYEXDYGXAXCXBVLZVMVPAYEXEYIAXEYIYEAXDYIXAXDYGAYIYJAXBYHXCMVNVQVRVRVSVTXQ | |
| + IAYEEFUAXELWAWFWBWCWDWEYDXKGWGWHZSZXLYCYKTYDYLWSWTWSWIXKYDGYCYKXIYCBXJWJW | |
| + KWLXKGYKWMVDWNWOWPWQWOWR $. | |
| + $} | |
| + | |
| + ${ | |
| + $d k I $. $d f j k G p $. | |
| + umgr2cycl.1 $e |- I = ( iEdg ` G ) $. | |
| + $( A multigraph with two distinct edges that connect the same vertices has | |
| + a 2-cycle. (Contributed by BTernaryTau, 17-Oct-2023.) $) | |
| + umgr2cycl $p |- ( ( G e. UMGraph /\ | |
| + E. j e. dom I E. k e. dom I ( ( I ` j ) = ( I ` k ) /\ j =/= k ) ) -> | |
| + E. f E. p ( f ( Cycles ` G ) p /\ ( # ` f ) = 2 ) ) $= | |
| + ( cumgr wcel cv cfv wceq wa wrex wbr chash c2 wex wal syl wne ccycls wral | |
| + cdm ax-5 alral r19.29 sylan w3a cs2 eqid simp1 simp3r simp3l umgr2cycllem | |
| + simp2 s2len ax-gen 19.29r cvv cword s2cli breq1 fveqeq2 rspcev mpan rexex | |
| + anbi12d eximi excomim 3syl sylancl 3expib rexlimdvw syl5 expd rexlimdv | |
| + imp ) DHIZBJZEKCJZEKLZVTWAUAZMZCEUDZNZBWENAJZFJZDUBKZOZWGPKQLZMZFRARZVSWF | |
| + WMBWEVSVTWEIZWFWMWNWFMWNWDMZCWENZVSWMWNWNCWEUCZWFWPWNWNCSWQWNCUEWNCWEUFTW | |
| + NWDCWEUGUHVSWOWMCWEVSWNWDWMVSWNWDUIZVTWAUJZWHWIOZFRZWSPKQLZFSZWMWRWSDEVTW | |
| + AFWSUKGVSWNWDULVSWNWDUPVSWNWBWCUMVSWNWBWCUNUOXBFVTWAUQURXAXCMWTXBMZFRWLAR | |
| + ZFRWMWTXBFUSXDXEFXDWLAUTVAZNZXEWSXFIXDXGVTWAVBWLXDAWSXFWGWSLWJWTWKXBWGWSW | |
| + HWIVCWGWSQPVDVHVEVFWLAXFVGTVIWLFAVJVKVLVMVNVOVPVQVR $. | |
| + $} | |
| + | |
| +$( | |
| +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= | |
| + Acyclic graphs | |
| +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= | |
| +$) | |
| + | |
| + $c AcyclicGraph $. | |
| + | |
| + $( Extend class notation with acyclic graphs. $) | |
| + cacycgr $a class AcyclicGraph $. | |
| + | |
| + ${ | |
| + $d f g p $. | |
| + $( Define the class of all acyclic graphs. A graph is called _acyclic_ if | |
| + it has no (non-trivial) cycles. (Contributed by BTernaryTau, | |
| + 11-Oct-2023.) $) | |
| + df-acycgr $a |- AcyclicGraph = | |
| + { g | -. E. f E. p ( f ( Cycles ` g ) p /\ f =/= (/) ) } $. | |
| + | |
| + $( An alternate definition of the class of all acyclic graphs that requires | |
| + all cycles to be trivial. (Contributed by BTernaryTau, 11-Oct-2023.) $) | |
| + dfacycgr1 $p |- AcyclicGraph = | |
| + { g | A. f A. p ( f ( Cycles ` g ) p -> f = (/) ) } $= | |
| + ( cacycgr cv ccycls cfv wbr c0 wne wa wex cab wceq wal df-acycgr 2exanali | |
| + wn wi df-ne anbi2i 2exbii xchnxbir abbii eqtri ) DAEZCEBEFGHZUFIJZKZCLALZ | |
| + RZBMUGUFINZSCOAOZBMABCPUKUMBUGULRZKZCLALUMUJUGULACQUIUOACUHUNUGUFITUAUBUC | |
| + UDUE $. | |
| + $} | |
| + | |
| + ${ | |
| + $d f g G p $. | |
| + $( The property of being an acyclic graph. (Contributed by BTernaryTau, | |
| + 11-Oct-2023.) $) | |
| + isacycgr $p |- ( G e. W -> ( G e. AcyclicGraph <-> | |
| + -. E. f E. p ( f ( Cycles ` G ) p /\ f =/= (/) ) ) ) $= | |
| + ( vg cv ccycls cfv wbr c0 wne wa wex wn cacycgr wceq fveq2 anbi1d 2exbidv | |
| + breqd notbid df-acycgr elab2g ) AFZDFZEFZGHZIZUDJKZLZDMAMZNUDUEBGHZIZUILZ | |
| + DMAMZNEBOCUFBPZUKUOUPUJUNADUPUHUMUIUPUGULUDUEUFBGQTRSUAAEDUBUC $. | |
| + | |
| + $( The property of being an acyclic graph. (Contributed by BTernaryTau, | |
| + 11-Oct-2023.) $) | |
| + isacycgr1 $p |- ( G e. W -> ( G e. AcyclicGraph <-> | |
| + A. f A. p ( f ( Cycles ` G ) p -> f = (/) ) ) ) $= | |
| + ( vg cv ccycls cfv wbr c0 wceq wal cacycgr fveq2 imbi1d 2albidv dfacycgr1 | |
| + wi breqd elab2g ) AFZDFZEFZGHZIZUAJKZRZDLALUAUBBGHZIZUFRZDLALEBMCUCBKZUGU | |
| + JADUKUEUIUFUKUDUHUAUBUCBGNSOPAEDQT $. | |
| + $} | |
| + | |
| + ${ | |
| + $d P f p $. $d f F p $. $d f G p $. | |
| + $( Any cycle in an acyclic graph is trivial (i.e. has one vertex and no | |
| + edges). (Contributed by BTernaryTau, 12-Oct-2023.) $) | |
| + acycgrcycl $p |- ( ( G e. AcyclicGraph /\ F ( Cycles ` G ) P ) -> | |
| + F = (/) ) $= | |
| + ( vf vp cacycgr wcel ccycls cfv wbr c0 wceq wi wa cv cvv cwlks w3a adantl | |
| + wal cycliswlk syl simp2d simp3d breq1 eqeq1 imbi12d breq2 imbi1d sylan9bb | |
| + wlkv isacycgr1 ibi 19.21bbi adantr vtocl2d ex pm2.43d imp ) CFGZBACHIZJZB | |
| + KLZUTVBVCUTVBVBVCMZUTVBNDOZEOZVAJZVEKLZMZVDDEBAPPVBBPGZUTVBCPGZVJAPGZVBBA | |
| + CQIJVKVJVLRABCUAABCUKUBZUCSVBVLUTVBVKVJVLVMUDSVEBLZVIBVFVAJZVCMVFALZVDVNV | |
| + GVOVHVCVEBVFVAUEVEBKUFUGVPVOVBVCVFABVAUHUIUJUTVIVBUTVIDEUTVIETDTDCFEULUMU | |
| + NUOUPUQURUS $. | |
| + $} | |
| + | |
| + ${ | |
| + $d x V $. $d x G a $. $d f G p a $. | |
| + acycgrislfgr.1 $e |- V = ( Vtx ` G ) $. | |
| + acycgrislfgr.2 $e |- I = ( iEdg ` G ) $. | |
| + $( An acyclic hypergraph is a loop-free hypergraph. (Contributed by | |
| + BTernaryTau, 15-Oct-2023.) $) | |
| + acycgrislfgr $p |- ( ( G e. AcyclicGraph /\ G e. UHGraph ) -> | |
| + I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) $= | |
| + ( va vf vp wcel cuhgr wa cv chash cfv wbr wex wn wceq 2eximi cacycgr crab | |
| + cdm c2 cle cpw wf csn cedg ccycls c0 wne isacycgr biimpac wi c1 loop1cycl | |
| + cc0 w3a 3simpa biimtrrdi exlimdv cvv vex hash1n0 mpan anim2i con3d adantl | |
| + syl6 mpd wb lfuhgr3 mpbird ) BUAJZBKJZLZCUCUDAMNOUEPADUFUBCUGZGMZUHBUIOJZ | |
| + GQZRZVQHMZIMZBUJOPZWCUKULZLZIQHQZRZWBVPVOWIHBKIUMUNVPWIWBUOVOVPWAWHVPWAWE | |
| + WCNOUPSZLZIQHQZWHVPVTWLGVPVTWEWJURWDOVSSZUSZIQHQWLVSHBIUQWNWKHIWEWJWMUTTV | |
| + AVBWKWGHIWJWFWEWCVCJWJWFHVDWCVCVEVFVGTVJVHVIVKVPVRWBVLVOABCDGEFVMVIVN $. | |
| + $} | |
| + | |
| + ${ | |
| + $d x G $. | |
| + $( An acyclic pseudograph is a multigraph. (Contributed by BTernaryTau, | |
| + 15-Oct-2023.) $) | |
| + upgracycumgr $p |- ( ( G e. UPGraph /\ G e. AcyclicGraph ) -> | |
| + G e. UMGraph ) $= | |
| + ( vx cupgr wcel cacycgr ciedg cfv cdm c2 cv chash cle wbr cvtx crab cumgr | |
| + cpw wf wa eqid cuhgr anim1ci acycgrislfgr syl umgrislfupgr biimpri syldan | |
| + upgruhgr ) ACDZAEDZAFGZHIBJKGLMBANGZQOUKRZAPDZUIUJSUJAUADZSUMUIUOUJAUHUBB | |
| + AUKULULTZUKTZUCUDUNUIUMSBAUKULUPUQUEUFUG $. | |
| + $} | |
| + | |
| + ${ | |
| + $d x G $. $d f j k G p $. | |
| + $( An acyclic multigraph is a simple graph. (Contributed by BTernaryTau, | |
| + 17-Oct-2023.) $) | |
| + umgracycusgr $p |- ( ( G e. UMGraph /\ G e. AcyclicGraph ) -> | |
| + G e. USGraph ) $= | |
| + ( vx vj vk vf vp cumgr wcel wa cfv cv chash c2 wceq wne wrex wn wex cc0 | |
| + c0 cacycgr ciedg cdm cvtx cpw crab wf1 cusgr wf umgrf ccycls wbr isacycgr | |
| + eqid biimpa wi umgr2cycl 2ne0 neeq1 mpbiri wb cvv hasheq0 necon3bii sylib | |
| + elv anim2i 2eximi syl con3d adantr dff15 biimpri syl2an2r isusgrs biimprd | |
| + ex mpd ) AGHZAUAHZIZAUBJZUCZBKLJMNBAUDJZUEUFZWBUGZAUHHZVSWCWEWBUIZVTCKZWB | |
| + JDKZWBJNWIWJOIDWCPCWCPZQZWFBWBAWDWDUNZWBUNZUJWAEKZFKAUKJULZWOTOZIZFRERZQZ | |
| + WLVSVTWTEAGFUMUOVSWTWLUPVTVSWKWSVSWKWSVSWKIWPWOLJZMNZIZFRERWSECDAWBFWNUQX | |
| + CWREFXBWQWPXBXASOZWQXBXDMSOURXAMSUSUTXASWOTXASNWOTNVAEWOVBVCVFVDVEVGVHVIV | |
| + QVJVKVRWFWHWLICDWCWEWBVLVMVNVSWFWGUPVTVSWGWFBGWBAWDWMWNVOVPVKVR $. | |
| + $} | |
| + | |
| + $( An acyclic pseudograph is a simple graph. (Contributed by BTernaryTau, | |
| + 17-Oct-2023.) $) | |
| + upgracycusgr $p |- ( ( G e. UPGraph /\ G e. AcyclicGraph ) -> | |
| + G e. USGraph ) $= | |
| + ( cupgr wcel cacycgr cumgr cusgr upgracycumgr umgracycusgr sylancom ) ABCAD | |
| + CAECAFCAGAHI $. | |
| + | |
| + $( A path in an acyclic graph is a simple path. (Contributed by BTernaryTau, | |
| + 21-Oct-2023.) $) | |
| + pthacycspth $p |- ( ( G e. AcyclicGraph /\ F ( Paths ` G ) P ) -> | |
| + F ( SPaths ` G ) P ) $= | |
| + ( cacycgr wcel cpths cfv wbr wa cspths wo ccycls wi c0 cyclispth acycgrcycl | |
| + wceq a1i ex adantr jcad spthcycl simplbi pthisspthorcycl adantl orim2 pm1.2 | |
| + syl6 sylc syl ) CDEZBACFGHZIZBACJGHZUNKZUNUMBACLGHZUNMUNUPKZUOUMUPULBNQZIZU | |
| + NUMUPULURUPULMUMABCORUKUPURMULUKUPURABCPSTUAUSUNUPABCUBUCUHULUQUKABCUDUEUNU | |
| + PUNUFUIUNUGUJ $. | |
| + | |
| + ${ | |
| + $d S f p $. $d f G p $. | |
| + $( The subgraph of an acyclic graph is also acyclic. (Contributed by | |
| + BTernaryTau, 23-Oct-2023.) $) | |
| + acycgrsubgr $p |- ( ( G e. AcyclicGraph /\ S SubGraph G ) -> | |
| + S e. AcyclicGraph ) $= | |
| + ( vf vp csubgr wbr cacycgr wcel cv ccycls cfv c0 wne wa wex subgrcycl cvv | |
| + wn wb isacycgr anim1d 2eximdv con3d subgrv simpl2im simpld 3imtr4d impcom | |
| + syl ) ABEFZBGHZAGHZUJCIZDIZBJKFZUMLMZNZDOCOZRZUMUNAJKFZUPNZDOCOZRZUKULUJV | |
| + BURUJVAUQCDUJUTUOUPUNAUMBPUAUBUCUJAQHZBQHZUKUSSABUDZCBQDTUEUJVDULVCSUJVDV | |
| + EVFUFCAQDTUIUGUH $. | |
| + $} | |
| + | |
| $( | |
| #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# | |
| Eulerian paths and the Konigsberg Bridge problem | |
| @@ -486107,6 +486965,9 @@ htmldef "Circuits" as "Circuits"; | |
| htmldef "Cycles" as "Cycles"; | |
| althtmldef "Cycles" as "Cycles"; | |
| latexdef "Cycles" as "\mathrm{Cycles}"; | |
| +htmldef "AcyclicGraph" as "AcyclicGraph"; | |
| + althtmldef "AcyclicGraph" as "AcyclicGraph"; | |
| + latexdef "AcyclicGraph" as "\mathrm{AcyclicGraph}"; | |
| htmldef "WWalks" as 'WWalks'; | |
| althtmldef "WWalks" as 'WWalks'; | |
| latexdef "WWalks" as "\mathrm{WWalks}"; | |
| @@ -486874,9 +487735,6 @@ htmldef "toTG" as "toTG"; | |
| htmldef "kard" as "kard"; | |
| althtmldef "kard" as "kard"; | |
| latexdef "kard" as "\mathrm{kard}"; | |
| -htmldef "AcyclicGraph" as "AcyclicGraph"; | |
| - althtmldef "AcyclicGraph" as "AcyclicGraph"; | |
| - latexdef "AcyclicGraph" as "\mathrm{AcyclicGraph}"; | |
| /* End of BTernaryTau's mathbox */ | |
| /* Mathbox of Mario Carneiro */ | |
| @@ -490358,6 +491216,15 @@ htmldef "~~>*" as "~~>*"; | |
| latexdef "~~>*" as "\rightsquigarrow *"; | |
| /* End of Glauco Siliprandi's mathbox */ | |
| +/* Mathbox of Mingli Yuan */ | |
| +htmldef "Tree" as "Tree"; | |
| + althtmldef "Tree" as "Tree"; | |
| + latexdef "Tree" as "\mathrm{Tree}"; | |
| +htmldef "SpanningTree" as ' SpanningTree '; | |
| + althtmldef "SpanningTree" as ' SpanningTree '; | |
| + latexdef "SpanningTree" as "\mathbin{\mathrm{SpanningTree}}"; | |
| +/* End of Mingli Yuan's mathbox */ | |
| + | |
| /* Mathbox of Jiamin Zhao */ | |
| htmldef "crossp" as "⊠"; | |
| althtmldef "crossp" as "⊠"; | |
| @@ -516238,16 +517105,6 @@ $) | |
| VBVEREGBDCUNTAVDVBAVLVJVBVDUPEGDCBUQTURABUSUTVA $. | |
| $} | |
| - ${ | |
| - s1f1.1 $e |- ( ph -> I e. D ) $. | |
| - $( Conditions for a length 1 string to be a one-to-one function. | |
| - (Contributed by Thierry Arnoux, 11-Dec-2023.) $) | |
| - s1f1 $p |- ( ph -> <" I "> : dom <" I "> -1-1-> D ) $= | |
| - ( cs1 cdm wf1 cc0 csn cop wss wf1o cn0 wcel 0nn0 a1i f1osng syl2anc wceq | |
| - syl f1of1 snssd f1ss s1val s1dm eqidd f1eq123d mpbird ) ACEZFZBUIGHIZBHCJ | |
| - IZGZAUKCIZULGZUNBKUMAUKUNULLZUOAHMNZCBNZUPUQAOPDHCMBQRUKUNULUATACBDUBUKUN | |
| - BULUCRAUJUKBBUIULAURUIULSDCBUDTUJUKSACUEPABUFUGUH $. | |
| - $} | |
| ${ | |
| s2f1.i $e |- ( ph -> I e. D ) $. | |
| @@ -516309,57 +517166,6 @@ $) | |
| JZUJELGMZUJKNABCOPABCSKQRHTZEUAEKUBUKULUMUCUDUEKUNEUFKGUJEUGUHUI $. | |
| $} | |
| - ${ | |
| - $d A i j $. $d B i j $. $d i j ph $. | |
| - ccatf1.s $e |- ( ph -> S e. V ) $. | |
| - ccatf1.a $e |- ( ph -> A e. Word S ) $. | |
| - ccatf1.b $e |- ( ph -> B e. Word S ) $. | |
| - ccatf1.1 $e |- ( ph -> A : dom A -1-1-> S ) $. | |
| - ccatf1.2 $e |- ( ph -> B : dom B -1-1-> S ) $. | |
| - ccatf1.3 $e |- ( ph -> ( ran A i^i ran B ) = (/) ) $. | |
| - $( Conditions for a concatenation to be injective. (Contributed by Thierry | |
| - Arnoux, 11-Dec-2023.) $) | |
| - ccatf1 $p |- ( ph -> ( A ++ B ) : dom ( A ++ B ) -1-1-> S ) $= | |
| - ( vi vj co cfv wceq wcel syl wa adantr cconcat cdm wf cv weq wral wf1 cc0 | |
| - wi chash cfzo cword ccatcl syl2anc wrdf ffdmd ccatval1 syl2an3an ad4ant13 | |
| - simpllr id ad4ant14 3eqtr3d wrddm f1eq2 biimpa dff13 simprbi r19.21bi mpd | |
| - adantllr ex c0 crn wfun f1fun simpr eleqtrrd fvelrn syl2an2r cmin ccatlen | |
| - cin caddc oveq2d eleqtrd ccatval2 syl3anc cn0 lencl nn0zd fzosubel3 elind | |
| - cz eqeltrd ad3antrrr wn noel pm2.21dd wo eleq2d fzospliti ad2antrr mpjaod | |
| - a1i adantlrl 3eqtr3rd cc elfzoelz ad2antlr adantl nn0cnd jca f1veqaeq imp | |
| - zcnd syl21anc subcan2d adantrr ralrimivva sylanbrc ) ABCUANZUBZDYBUCLUDZY | |
| - BOZMUDZYBOZPZLMUEZUIZMYCUFLYCUFYCDYBUGAUHYBUJOZUKNZDYBAYBDULZQZYLDYBUCABY | |
| - MQZCYMQZYNGHDBCUMUNZDYBUORUPAYJLMYCYCAYDYCQZYFYCQZSSZYHYIYTYHSYDUHBUJOZUK | |
| - NZQZYIYDUUAYKUKNZQZAYSYHUUCYIUIYRAYSSZYHSZUUCYIUUGUUCSYFUUBQZYIYFUUDQZAYH | |
| - UUCUUHYIUIZYSAYHSZUUCSZUUHYIUULUUHSZYDBOZYFBOZPZYIUUMYEYGUUNUUOAYHUUCUUHU | |
| - TAUUCYEUUNPZYHUUHAYOYPUUCUUCUUQGHUUCVADDBCYDUQURZUSAUUHYGUUOPZYHUUCAYOYPU | |
| - UHUUHUUSGHUUHVADDBCYFUQURZVBVCAUUCUUHUUPYIUIZYHAUUCSZUVAMUUBAUVAMUUBUFZLU | |
| - UBAUUBDBUGZUVCLUUBUFZABUBZUUBPZUVFDBUGZUVDAYOUVGGDBVDRZIUVGUVHUVDUVFUUBDB | |
| - VEVFUNUVDUUBDBUCUVELMUUBDBVGVHRVIVIVKVJVLVKAYHUUCUUIYIUIZYSUULUUIYIUULUUI | |
| - SZUUNVMQZYIUVKUUNBVNZCVNZWCZVMUVKUVMUVNUUNAUUCUUNUVMQZYHUUIABVOZUUCYDUVFQ | |
| - UVPAUVHUVQIUVFDBVPRZUVBYDUUBUVFAUUCVQAUVGUUCUVITVRYDBVSVTUSUVKUUNYFUUAWAN | |
| - ZCOZUVNUVKYEYGUUNUVTAYHUUCUUIUTAUUCUUQYHUUIUURUSAUUIYGUVTPZYHUUCAUUISZYOY | |
| - PYFUUAUUACUJOZWDNZUKNZQZUWAAYOUUIGTAYPUUIHTUWBYFUUDUWEAUUIVQAUUDUWEPZUUIA | |
| - YKUWDUUAUKAYOYPYKUWDPGHDDBCWBUNWEZTWFZDBCYFWGWHZVBVCAUUIUVTUVNQZYHUUCACVO | |
| - ZUUIUVSCUBZQZUWKAUWMDCUGZUWLJUWMDCVPRZUWBUVSUHUWCUKNZUWMUWBUWFUWCWNQZUVSU | |
| - WQQUWIAUWRUUIAUWCAYPUWCWIQHDCWJRWKZTYFUUAUWCWLUNAUWMUWQPZUUIAYPUWTHDCVDRZ | |
| - TVRZUVSCVSVTVBWOWMAUVOVMPZYHUUCUUIKWPWFUVLWQUVKUUNWRXEWSVLVKUUFUUHUUIWTZY | |
| - HUUCUUFYFYLQZUUAWNQZUXDAYSUXEAYCYLYFAYNYCYLPYQDYBVDRZXAVFAUXFYSAUUAAYOUUA | |
| - WIQGDBWJRZWKZTYFUHYKUUAXBUNZXCXDVLXFAYSYHUUEYIUIYRUUGUUEYIUUGUUESUUHYIUUI | |
| - AYHUUEUUJYSUUKUUESZUUHYIUXKUUHSZUUOVMQZYIUXLUUOUVOVMUXLUVMUVNUUOAUUHUUOUV | |
| - MQZYHUUEAUVQUUHYFUVFQUXNUVRAUUHSYFUUBUVFAUUHVQAUVGUUHUVITVRYFBVSVTVBUXLUU | |
| - OYDUUAWANZCOZUVNUXLYEYGUXPUUOAYHUUEUUHUTAUUEYEUXPPZYHUUHAUUESZYOYPYDUWEQZ | |
| - UXQAYOUUEGTAYPUUEHTUXRYDUUDUWEAUUEVQAUWGUUEUWHTWFZDBCYDWGWHZUSAUUHUUSYHUU | |
| - EUUTVBXGAUUEUXPUVNQZYHUUHAUWLUUEUXOUWMQZUYBUWPUXRUXOUWQUWMUXRUXSUWRUXOUWQ | |
| - QUXTAUWRUUEUWSTYDUUAUWCWLUNAUWTUUEUXATVRZUXOCVSVTUSWOWMAUXCYHUUEUUHKWPWFU | |
| - XMWQUXLUUOWRXEWSVLVKAYHUUEUVJYSUXKUUIYIUXKUUISZYDYFUUAUUEYDXHQUUKUUIUUEYD | |
| - YDUUAYKXIXPXJUUIYFXHQUXKUUIYFYFUUAYKXIXPXKAUUAXHQYHUUEUUIAUUAUXHXLWPUYEUW | |
| - OUYCUWNSZUXPUVTPZUXOUVSPZAUWOYHUUEUUIJWPUYEUYCUWNAUUEUYCYHUUIUYDUSAUUIUWN | |
| - YHUUEUXBVBXMUYEYEYGUXPUVTAYHUUEUUIUTAUUEUXQYHUUIUYAUSAUUIUWAYHUUEUWJVBVCU | |
| - WOUYFSUYGUYHUWMDUXOUVSCXNXOXQXRVLVKUUFUXDYHUUEUXJXCXDVLXFYTUUCUUEWTZYHAYR | |
| - UYIYSAYRSYDYLQZUXFUYIAYRUYJAYCYLYDUXGXAVFAUXFYRUXITYDUHYKUUAXBUNXSTXDVLXT | |
| - LMYCDYBVGYA $. | |
| - $} | |
| ${ | |
| pfxlsw2ccat.n $e |- N = ( # ` W ) $. | |
| @@ -516538,75 +517344,14 @@ $) | |
| $. | |
| $} | |
| - ${ | |
| - $d M x $. $d N x $. $d V x $. $d W x $. | |
| - $( The range of a subword is a subset of the range of that word. Stronger | |
| - version of ~ swrdrn . (Contributed by Thierry Arnoux, 12-Dec-2023.) $) | |
| - swrdrn2 $p |- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) | |
| - ) ) -> ran ( W substr <. M , N >. ) C_ ran W ) $= | |
| - ( vx cword wcel cc0 cfz co cfv crn cfzo wss cuz adantr cz elfzelzd sseldd | |
| - syl chash w3a cop csubstr cmin cv caddc cmpt swrdval2 rneqd wral wfun cdm | |
| - wa eqidd simpl1 wrdfd ffund elfzuz3 3ad2ant3 fzoss2 elfzuz 3ad2ant2 simpr | |
| - fzoss1 simpl3 simpl2 fzoaddel2 syl3anc wceq wrddm 3ad2ant1 fvelrn syl2anc | |
| - eleqtrrd ralrimiva eqid rnmptss eqsstrd ) DCFGZAHBIJGZBHDUAKZIJGZUBZDABUC | |
| - UDJZLEHBAUEJMJZEUFZAUGJZDKZUHZLZDLZWDWEWJECDABUIUJWDWIWLGZEWFUKWKWLNWDWME | |
| - WFWDWGWFGZUNZDULWHDUMZGWMWOHWBMJZCDWOCWBDWOWBUOVTWAWCWNUPUQURWOWHWQWPWOHB | |
| - MJZWQWHWOWBBOKGZWRWQNWDWSWNWCVTWSWABHWBUSUTPBHWBVATWOABMJZWRWHWOAHOKGZWTW | |
| - RNWDXAWNWAVTXAWCAHBVBVCPAHBVETWOWNBQGAQGWHWTGWDWNVDWOBHWBVTWAWCWNVFRWOAHB | |
| - VTWAWCWNVGRWGBAVHVISSWDWPWQVJZWNVTWAXBWCCDVKVLPVOWHDVMVNVPEWFWIWLWJWJVQVR | |
| - TVS $. | |
| - $} | |
| - ${ | |
| - $d M i j y $. $d N i j y $. $d V i j y $. $d W i j y $. | |
| - $( Express the range of a subword. Stronger version of ~ swrdrn2 . | |
| - (Contributed by Thierry Arnoux, 13-Dec-2023.) $) | |
| - swrdrn3 $p |- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) | |
| - ) ) -> ran ( W substr <. M , N >. ) = ( W " ( M ..^ N ) ) ) $= | |
| - ( vy vj vi wcel cc0 co cfv cfzo cv wceq caddc wa cz simpr elfzelzd zcnd | |
| - cword cfz w3a cop csubstr crn cima wrex cmin cmpt simpl3 simpl2 fzoaddel2 | |
| - chash syl3anc pncan3d oveq2d eleqtrrd zsubcld oveq1d eqeq2d fzossz sselid | |
| - fzosubel3 syl2anc npcand eqcomd rspcedvd fveq2d bitr3id rexxfrd eqid fvex | |
| - elrnmpti bitr4di wf wrdf 3ad2ant1 ffnd cuz wss elfzuz 3ad2ant2 fzoss1 syl | |
| - eqcom elfzuz3 3ad2ant3 fzoss2 sstrd fvelimabd swrdval2 rneqd eleq2d eqrdv | |
| - 3bitr4rd ) DCUAHZAIBUBJHZBIDUNKZUBJHZUCZEDABUDUEJZUFZDABLJZUGZXAFMZDKZEMZ | |
| - NZFXDUHZXHGIBAUIJZLJZGMZAOJZDKZUJZUFZHZXHXEHXHXCHXAXJXHXONZGXLUHXRXAXIXSF | |
| - GXNXDXLXAXMXLHZPZXTBQHAQHXNXDHXAXTRYABIWSWQWRWTXTUKSYAAIBWQWRWTXTULSXMBAU | |
| - MUOXAXFXDHZPZXFXNNZXFXFAUIJZAOJZNGYEXLYCXFAAXKOJZLJZHXKQHYEXLHYCXFXDYHXAY | |
| - BRZYCYGBALYCABYCAYCAIBWQWRWTYBULSZTZYCBYCBIWSWQWRWTYBUKSZTUPUQURYCBAYLYJU | |
| - SXFAXKVDVEYCXMYENZPZXNYFXFYNXMYEAOYCYMRUTVAYCYFXFYCXFAYCXFYCXDQXFABVBYIVC | |
| - TYKVFVGVHXIXHXGNXAYDPZXSXHXGWFYOXGXOXHYOXFXNDXAYDRVIVAVJVKGXLXOXHXPXPVLXN | |
| - DVMVNVOXAFIWSLJZXDXHDXAYPCDWQWRYPCDVPWTCDVQVRVSXAXDIBLJZYPXAAIVTKHZXDYQWA | |
| - WRWQYRWTAIBWBWCAIBWDWEXAWSBVTKHZYQYPWAWTWQYSWRBIWSWGWHBIWSWIWEWJWKXAXCXQX | |
| - HXAXBXPGCDABWLWMWNWPWO $. | |
| - $} | |
| ${ | |
| $d M i j $. $d N i j $. $d W i j $. $d i j ph $. | |
| - swrdf1.w $e |- ( ph -> W e. Word D ) $. | |
| - swrdf1.m $e |- ( ph -> M e. ( 0 ... N ) ) $. | |
| - swrdf1.n $e |- ( ph -> N e. ( 0 ... ( # ` W ) ) ) $. | |
| - swrdf1.1 $e |- ( ph -> W : dom W -1-1-> D ) $. | |
| - $( Condition for a subword to be injective. (Contributed by Thierry | |
| - Arnoux, 12-Dec-2023.) $) | |
| - swrdf1 $p |- ( ph | |
| - -> ( W substr <. M , N >. ) : dom ( W substr <. M , N >. ) -1-1-> D ) $= | |
| - ( vi vj co cfv wceq cc0 cfzo wcel wa cz ad3antrrr cop csubstr cdm wf wral | |
| - cv wi wf1 cmin cword cfz chash swrdf syl3anc ffdmd fzossz simpllr eleqtrd | |
| - fdmd sselid zcnd simplr elfzelzd caddc wss cuz elfzuz fzoss1 3syl elfzuz3 | |
| - fzoss2 sstrd fzoaddel2 sseldd wrddm syl eleqtrrd swrdfv syl31anc f1veqaeq | |
| - simpr 3eqtr3d anassrs imp syl1111anc addcan2ad ex anasss ralrimivva dff13 | |
| - sylanbrc ) AECDUAUBLZUCZBWLUDJUFZWLMZKUFZWLMZNZWNWPNZUGZKWMUEJWMUEWMBWLUH | |
| - AODCUILZPLZBWLAEBUJQZCODUKLQZDOEULMZUKLQZXBBWLUDFGHCDBEUMUNZUOAWTJKWMWMAW | |
| - NWMQZWPWMQZWTAXHRZXIRZWRWSXKWRRZWNWPCXLWNXLXBSWNOXAUPZXLWNWMXBAXHXIWRUQAW | |
| - MXBNXHXIWRAXBBWLXGUSTZURZUTVAXLWPXLXBSWPXMXLWPWMXBXJXIWRVBXNURZUTVAXLCACS | |
| - QZXHXIWRACODGVCTZVAXLEUCZBEUHZWNCVDLZXSQZWPCVDLZXSQZYAEMZYCEMZNZYAYCNZAXT | |
| - XHXIWRITXLYAOXEPLZXSXLCDPLZYIYAAYJYIVEXHXIWRAYJODPLZYIAXDCOVFMQYJYKVEGCOD | |
| - VGCODVHVIAXFXEDVFMQYKYIVEHDOXEVJDOXEVKVIVLTZXLWNXBQZDSQZXQYAYJQXOAYNXHXIW | |
| - RADOXEHVCTZXRWNDCVMUNVNAXSYINZXHXIWRAXCYPFBEVOVPTZVQXLYCYIXSXLYJYIYCYLXLW | |
| - PXBQZYNXQYCYJQXPYOXRWPDCVMUNVNYQVQXLWOWQYEYFXKWRWAXLXCXDXFYMWOYENAXCXHXIW | |
| - RFTZAXDXHXIWRGTZAXFXHXIWRHTZXOBECDWNVRVSXLXCXDXFYRWQYFNYSYTUUAXPBECDWPVRV | |
| - SWBXTYBRYDRYGYHXTYBYDYGYHUGXSBYAYCEVTWCWDWEWFWGWHWIJKWMBWLWJWK $. | |
| + swrdrndisj.w $e |- ( ph -> W e. Word D ) $. | |
| + swrdrndisj.m $e |- ( ph -> M e. ( 0 ... N ) ) $. | |
| + swrdrndisj.n $e |- ( ph -> N e. ( 0 ... ( # ` W ) ) ) $. | |
| + swrdrndisj.f1 $e |- ( ph -> W : dom W -1-1-> D ) $. | |
| swrdrndisj.1 $e |- ( ph -> O e. ( N ... P ) ) $. | |
| swrdrndisj.2 $e |- ( ph -> P e. ( N ... ( # ` W ) ) ) $. | |
| @@ -571519,33 +572264,7 @@ $) | |
| ( cv csn cdif wcel wex wne wa wrex eldifsn exbii df-rex bitr4i ) ADZBCEFGZA | |
| HPBGPCIZJZAHRABKQSAPBCLMRABNO $. | |
| - ${ | |
| - srcmpltd.1 $e |- ( ph -> ( C e. A -> ps ) ) $. | |
| - srcmpltd.2 $e |- ( ph -> ( C e. ( B \ A ) -> ps ) ) $. | |
| - $( If a statement is true for every element of a class and for every | |
| - element of its complement relative to a second class, then it is true | |
| - for every element in the second class. (Contributed by BTernaryTau, | |
| - 27-Sep-2023.) $) | |
| - srcmpltd $p |- ( ph -> ( C e. B -> ps ) ) $= | |
| - ( wcel cdif cun elun2 undif2 eleqtrrdi wi elunant sylanbrc syl5 ) EDHZECD | |
| - CIZJZHZABRECDJTEDCKCDLMAECHBNESHBNUABNFGBCSEOPQ $. | |
| - $} | |
| - ${ | |
| - prsrcmpltd.1 $e |- ( ph -> ( ( C e. A /\ D e. A ) -> ps ) ) $. | |
| - prsrcmpltd.2 $e |- ( ph -> ( ( C e. A /\ D e. ( B \ A ) ) -> ps ) ) $. | |
| - prsrcmpltd.3 $e |- ( ph -> ( ( C e. ( B \ A ) /\ D e. A ) -> ps ) ) $. | |
| - prsrcmpltd.4 $e |- ( ph -> | |
| - ( ( C e. ( B \ A ) /\ D e. ( B \ A ) ) -> ps ) ) $. | |
| - $( If a statement is true for all pairs of elements of a class, all pairs | |
| - of elements of its complement relative to a second class, and all pairs | |
| - with one element in each, then it is true for all pairs of elements of | |
| - the second class. (Contributed by BTernaryTau, 27-Sep-2023.) $) | |
| - prsrcmpltd $p |- ( ph -> ( ( C e. B /\ D e. B ) -> ps ) ) $= | |
| - ( wcel wi wa expdimp cdif srcmpltd impancom ex impcomd ) AFDKZEDKZBATUABL | |
| - ATMBCDEAECKZTBAUBMBCDFAUBFCKZBGNAUBFDCOZKZBHNPQAEUDKZTBAUFMBCDFAUFUCBINAU | |
| - FUEBJNPQPRS $. | |
| - $} | |
| ${ | |
| $d w x y z $. | |
| @@ -571562,66 +572281,9 @@ $) | |
| 'ax-12' 'ax-13' 'ax-ext' 'ax-sep' 'ax-nul'; $) | |
| $} | |
| - ${ | |
| - $d x y A $. $d x y F $. | |
| - $( A one-to-one function in terms of different arguments never having the | |
| - same function value. (Contributed by BTernaryTau, 24-Oct-2023.) $) | |
| - dff15 $p |- ( F : A -1-1-> B <-> | |
| - ( F : A --> B /\ | |
| - -. E. x e. A E. y e. A ( ( F ` x ) = ( F ` y ) /\ x =/= y ) ) ) $= | |
| - ( wf1 wf cv cfv wceq wi wral wa wne wrex wn dff13 iman anbi2i bitri df-ne | |
| - xchbinxr 2ralbii ralnex2 ) CDEFCDEGZAHZEIBHZEIJZUFUGJZKZBCLACLZMUEUHUFUGN | |
| - ZMZBCOACOPZMABCDEQUKUNUEUKUMPZBCLACLUNUJUOABCCUJUHUIPZMUMUHUIRULUPUHUFUGU | |
| - ASUBUCUMABCCUDTST $. | |
| - $} | |
| - $( If a function restricted to a class is one-to-one, then for any two | |
| - elements of the class, the values of the function at those elements are | |
| - equal only if the two elements are the same element. (Contributed by | |
| - BTernaryTau, 27-Sep-2023.) $) | |
| - f1resveqaeq $p |- ( ( ( F |` A ) : A -1-1-> B /\ ( C e. A /\ D e. A ) ) -> | |
| - ( ( F ` C ) = ( F ` D ) -> C = D ) ) $= | |
| - ( cres wf1 wcel cfv wceq fvres ad2antrl ad2antll eqeq12d f1veqaeq sylbird | |
| - wa ) ABEAFZGZCAHZDAHZQQZCEIZDEIZJCRIZDRIZJCDJUBUEUCUFUDTUEUCJSUACAEKLUAUFUD | |
| - JSTDAEKMNABCDROP $. | |
| - ${ | |
| - f1resrcmplf1dlem.1 $e |- ( ph -> C C_ A ) $. | |
| - f1resrcmplf1dlem.2 $e |- ( ph -> D C_ A ) $. | |
| - f1resrcmplf1dlem.3 $e |- ( ph -> F : A --> B ) $. | |
| - f1resrcmplf1dlem.4 $e |- ( ph -> ( ( F " C ) i^i ( F " D ) ) = (/) ) $. | |
| - $( Lemma for ~ f1resrcmplf1d . (Contributed by BTernaryTau, | |
| - 27-Sep-2023.) $) | |
| - f1resrcmplf1dlem $p |- ( ph -> ( ( X e. C /\ Y e. D ) -> | |
| - ( ( F ` X ) = ( F ` Y ) -> X = Y ) ) ) $= | |
| - ( wcel cfv cima wceq wss fnfvima syl3an1 syl3an2 wi wfn ffnd 3anidm12 wne | |
| - ex wa cin c0 disjne 3expib neneq pm2.21d syl6 syl2and ) AGDMZGFNZFDOZMZHE | |
| - MZHFNZFEOZMZUQVAPZGHPZUAZAUPUSAUPUSAADBQZUPUSIAFBUBZVGUPUSABCFKUCZBDFGRST | |
| - UDUFAUTVCAUTVCAAEBQZUTVCJAVHVJUTVCVIBEFHRSTUDUFAUSVCUGUQVAUEZVFAUSVCVKAUR | |
| - VBUHUIPUSVCVKLURVBUQVAUJSUKVKVDVEUQVAULUMUNUO $. | |
| - $} | |
| - ${ | |
| - $d ph x y $. $d x y A $. $d x y F $. | |
| - f1resrcmplf1d.1 $e |- ( ph -> C C_ A ) $. | |
| - f1resrcmplf1d.2 $e |- ( ph -> F : A --> B ) $. | |
| - f1resrcmplf1d.3 $e |- ( ph -> ( F |` C ) : C -1-1-> B ) $. | |
| - f1resrcmplf1d.4 $e |- ( ph -> ( F |` ( A \ C ) ) : ( A \ C ) -1-1-> B ) $. | |
| - f1resrcmplf1d.5 $e |- ( ph -> | |
| - ( ( F " C ) i^i ( F " ( A \ C ) ) ) = (/) ) $. | |
| - $( If a function's restriction to a subclass of its domain and its | |
| - restriction to the relative complement of that subclass are both | |
| - one-to-one, and if the ranges of those two restrictions are disjoint, | |
| - then the function is itself one-to-one. (Contributed by BTernaryTau, | |
| - 28-Sep-2023.) $) | |
| - f1resrcmplf1d $p |- ( ph -> F : A -1-1-> B ) $= | |
| - ( vx vy cv cfv wceq wral wf1 wcel wa cres wf wi f1resveqaeq sylan ex cdif | |
| - difssd f1resrcmplf1dlem cima cin incom eqtr3id prsrcmpltd ralrimivv dff13 | |
| - c0 sylanbrc ) ABCEUAKMZENLMZENOURUSOUBZLBPKBPBCEQGAUTKLBBAUTDBURUSAURDRUS | |
| - DRSZUTADCEDTQVAUTHDCURUSEUCUDUEABCDBDUFZEURUSFABDUGZGJUHABCVBDEURUSVCFGAE | |
| - VBUIZEDUIZUJVEVDUJUPVEVDUKJULUHAURVBRUSVBRSZUTAVBCEVBTQVFUTIVBCURUSEUCUDU | |
| - EUMUNKLBCEUOUQ $. | |
| - $} | |
| ${ | |
| $d B x $. $d C x $. $d F x $. | |
| @@ -571635,20 +572297,6 @@ $) | |
| UMURUSUPUNGZUAUQUMURUSVAUMUSURUTUBUMUNUOHVABCEDUCUNUOUDUEUFUNUPUGUHUI $. | |
| $} | |
| - ${ | |
| - $d F p $. $d x y p $. | |
| - $( If a function is equinumerous to ordinal 1, then its converse is also a | |
| - function. (Contributed by BTernaryTau, 8-Oct-2023.) $) | |
| - funen1cnv $p |- ( ( Fun F /\ F ~~ 1o ) -> Fun `' F ) $= | |
| - ( vp vx vy c1o cen wfun ccnv cv csn wceq wex wi wal sylancl syl wa funeqd | |
| - cnveq biimpar wbr en1 cop wrel wcel funrel vsnid elrel sneq funcnvsn gen2 | |
| - 2eximi 19.29r2 exlimivv ax-gen 19.29r funeq imbi12d exlimiv mpan2 impcom | |
| - sylbi ) AEFUAZAGZAHZGZVCABIZJZKZBLZVDVFMZBAUBVJVHGZVHHZGZMZBNZVKVOBVLVHCI | |
| - ZDIZUCZJZKZDLCLZVTHZGZDNCNZVNVLVGVSKZDLCLZWBVLVHUDVGVHUEWGVHUFBUGCDVGVHUH | |
| - OWFWACDVGVSUIULPWDCDVQVRUJUKWBWEQWAWDQZDLCLVNWAWDCDUMWHVNCDWAVNWDWAVMWCVH | |
| - VTSRTUNPOUOVJVPQVIVOQZBLVKVIVOBUPWIVKBVIVKVOVIVDVLVFVNAVHUQVIVEVMAVHSRURT | |
| - USPUTVBVA $. | |
| - $} | |
| $( If an ordinal class is not a set, then it must be the proper class of all | |
| ordinals. (Contributed by BTernaryTau, 9-Jun-2026.) $) | |
| @@ -573740,47 +574388,7 @@ $) | |
| ( cn0 wcel cz c1 caddc co clt wbr wne wa wb nn0z zltp1ne syl2an ) ACDAEDBED | |
| AFGHZBIJABIJBQKLMBCDANBNABOP $. | |
| - $( A number is zero if and only if it's a nonnegative integer that becomes | |
| - negative after subtracting 1. (Contributed by BTernaryTau, | |
| - 30-Sep-2023.) $) | |
| - 0nn0m1nnn0 $p |- ( N = 0 <-> ( N e. NN0 /\ -. ( N - 1 ) e. NN0 ) ) $= | |
| - ( cc0 wceq cn0 wcel c1 cmin co wn wa mpbiri wnel syl cle wbr biimprd adantr | |
| - wb cz wi 0nn0 eleq1 cn 1nn 0mnnnnn0 ax-mp oveq1 neleq1 df-nel sylib jca clt | |
| - nn0z peano2zm elnn0z notbii biimpi annotanannot simprbi syl2an cr 3syl 0red | |
| - zre ltnled mpd 0z zlem1lt sylancl nn0ge0 nn0re letri3d mp2and impbii ) ABCZ | |
| - ADEZAFGHZDEZIZJZVOVPVSVOVPBDEUAABDUBKVOVQDLZVSVOWABFGHZDLZFUCEWCUDFUEUFVOVQ | |
| - WBCWAWCRABFGUGVQWBDUHMKVQDUIUJUKVTABNOZBANOZVOVTVQBULOZWDVTBVQNOZIZWFVPVQSE | |
| - ZWIWGJZIZWHVSVPASEZWIAUMZAUNZMVSWKVRWJVQUOUPUQWIWKJWIWHWIWGURUSUTVPWHWFTVSV | |
| - PWFWHVPVQBVPWLWIVQVAEWMWNVQVDVBVPVCZVEPQVFVPWFWDTVSVPWDWFVPWLBSEWDWFRWMVGAB | |
| - VHVIPQVFVPWEVSAVJQVPWDWEJZVOTVSVPVOWPVPABAVKWOVLPQVMVN $. | |
| - ${ | |
| - $d x y F $. | |
| - f1resfz0f1d.1 $e |- ( ph -> K e. NN0 ) $. | |
| - f1resfz0f1d.2 $e |- ( ph -> F : ( 0 ... K ) --> V ) $. | |
| - f1resfz0f1d.3 $e |- ( ph -> | |
| - ( F |` ( 1 ... K ) ) : ( 1 ... K ) -1-1-> V ) $. | |
| - f1resfz0f1d.4 $e |- ( ph -> | |
| - ( ( F " { 0 } ) i^i ( F " ( 1 ... K ) ) ) = (/) ) $. | |
| - $( If a function with a sequence of nonnegative integers (starting at 0) as | |
| - its domain is one-to-one when 0 is removed, and if the range of that | |
| - restriction does not contain the function's value at the removed | |
| - integer, then the function is itself one-to-one. (Contributed by | |
| - BTernaryTau, 4-Oct-2023.) $) | |
| - f1resfz0f1d $p |- ( ph -> F : ( 0 ... K ) -1-1-> V ) $= | |
| - ( vx vy cc0 cfz cfv wceq wi cn0 wcel cin c0 cima co wss fz1ssfz0 a1i cres | |
| - c1 csn wf1 cdif wf cv wral 0elfz snssi 3syl fssresd eqidd wb 0nn0 fveqeq2 | |
| - eqeq1 imbi12d fveq2 eqeq2 mp2an mpbir dff13 sylanblrc cun uncom fz0sn0fz1 | |
| - eqeq2d 2ralsng syl eqtr4id wn 0nelfz1 neli disjsn uneqdifeq sylib reseq2d | |
| - eqcomd f1eq123d mpbid imaeq2d ineq2d incom eqtr3id eqtr3d f1resrcmplf1d ) | |
| - AKCLUAZDUFCLUAZBWMWLUBZACUCZUDFGAKUGZDBWPUEZUHZWLWMUIZDBWSUEZUHAWPDWQUJIU | |
| - KZWQMJUKZWQMZNZXAXBNZOZJWPULIWPULZWRAWLDWPBFACPQZKWLQWPWLUBECUMKWLUNUOUPX | |
| - GKWQMZXINZKKNZOZXJKUQKPQZXMXGXLURUSUSXFXIXCNZKXBNZOXLIJKKPPXAKNXDXNXEXOXA | |
| - KXCWQUTXAKXBVAVBXBKNZXNXJXOXKXPXCXIXIXBKWQVCVLXBKKVDVBVMVEVFIJWPDWQVGVHAW | |
| - PWSDDWQWTAWPWSBAWSWPAWMWPVIZWLNZWSWPNZAXQWPWMVIZWLWMWPVJAXHWLXTNECVKVNVOW | |
| - NWMWPRSNZXRXSURWOYAKWMQVPKWMCVQVRWMKVSVFWMWPWLVTVEWAWCZWBYBADUQWDWEABWMTZ | |
| - BWPTZRZYCBWSTZRSAYDYFYCAWPWSBYBWFWGAYEYDYCRSYDYCWHHWIWJWK $. | |
| - $} | |
| $( A finite set is equal to its subset if they are the same size. | |
| (Contributed by BTernaryTau, 3-Oct-2023.) $) | |
| @@ -573792,63 +574400,7 @@ $) | |
| UJUMLZULUIUKUPULUIUKIABNOZUJUIUJMZUMULUIUKUQABPQUIULUJURLUKUIUJRSURUQUMUIUJ | |
| UQUMABUAUBTUCUDTUEUFUGUH $. | |
| - ${ | |
| - $d L x $. $d V x $. $d W x $. | |
| - $( The reverse of a prefix of a word is equal to the same-length suffix of | |
| - the reverse of that word. (Contributed by BTernaryTau, 2-Dec-2023.) $) | |
| - revpfxsfxrev $p |- ( ( W e. Word V /\ L e. ( 0 ... ( # ` W ) ) ) -> | |
| - ( reverse ` ( W prefix L ) ) = | |
| - ( ( reverse ` W ) substr <. ( ( # ` W ) - L ) , ( # ` W ) >. ) ) $= | |
| - ( wcel cc0 chash cfv cfz co cfzo cmin wfn adantr oveq2d c1 3adant3 oveq1d | |
| - wceq cuz cz vx cword wa cpfx creverse cop csubstr pfxcl revcl 3syl revlen | |
| - wrdfn syl pfxlen eqtrd fneq2d mpbid swrdcl fznn0sub2 lencl sylib eleqtrrd | |
| - cn0 nn0fz0 swrdlen syl3an 3anidm13 cc nn0cnd elfzelz adantl nncand cv w3a | |
| - zcnd simp1 simp3 revfv sylan syl2anc fveq2d 3ad2ant2 elfzoelz 1cnd sub32d | |
| - 3ad2ant3 ubmelm1fzo eqeltrrd pfxfv syld3an3 3eqtrd eleq2d biimp3ar swrdfv | |
| - caddc id syl3anl1 syl3anl3 stoic3 syl3an2 elfzuz3 addlidd eluzsub mp3an2i | |
| - wss fzoss1 nn0zd 3ad2ant1 zsubcld fzo0addel pncan3d eleqtrd sseldd subcld | |
| - 0z subsub4d 3eqtr3d eqtr4d 3expa eqfnfvd ) CBUBZDZAECFGZHIZDZUCZUAEAJIZCA | |
| - UDIZUEGZCUEGZYCAKIZYCUFUGIZYFYIEYIFGZJIZLZYIYGLYBYOYEYBYHYADZYIYADYOBCAUH | |
| - ZBYHUIBYIULUJMYFYNYGYIYFYMAEJYFYMYHFGZAYBYMYRRZYEYBYPYSYQBYHUKUMMBCAUNZUO | |
| - NUPUQYFYLEYLFGZJIZLZYLYGLYBUUCYEYBYJYADZYLYADUUCBCUIZBYJYKYCURBYLULUJMYFU | |
| - UBYGYLYFUUAAEJYFUUAYCYKKIZAYBYEUUAUUFRZYBUUDYEYKYDDZYBYCEYJFGZHIZDZUUGUUE | |
| - AYCUSZYBYCYDUUJYBYCVCDYCYDDBCUTZYCVDVAYBUUIYCEHBCUKNVBZBYJYKYCVEVFVGYFYCA | |
| - YBYCVHDZYEYBYCUUMVIMZYEAVHDZYBYEAAEYCVJZVOZVKVLZUONUPUQYBYEUAVMZYGDZUVAYI | |
| - GZUVAYLGZRYBYEUVBVNZUVCAOKIZUVAKIZCGZUVDUVEUVCYROKIZUVAKIZYHGZUVGYHGZUVHU | |
| - VEYBUVAEYRJIZDZUVCUVKRZYBYEUVBVPZUVEUVAYGUVMYBYEUVBVQZYBYEUVMYGRUVBYFYRAE | |
| - JYTNPVBYBYPUVNUVOYQBYHUVAVRVSVTYBYEUVKUVLRUVBYFUVJUVGYHYFUVIUVFUVAKYFYRAO | |
| - KYTQQWAPYBYEUVBUVGYGDUVLUVHRUVEAUVAKIOKIZUVGYGUVEAUVAOYEYBUUQUVBUUSWBZUVB | |
| - YBUVAVHDYEUVBUVAUVAEAWCVOWFZUVEWDZWEUVBYBUVRYGDYEUVAAWGWFWHUVGABCWIWJWKUV | |
| - EUVDUVAYKWOIZYJGZYCOKIZUWBKIZCGZUVHYBYEUVBUVAEUUFJIZDZUVDUWCRZYBYEUWHUVBY | |
| - FUWGYGUVAYFUUFAEJUUTNWLWMYEYBUUHUWHUWIUULYBUUHYBUUHYBVNZUWHUWIYBUUHUWJUWJ | |
| - WPVGYBYBUUHUUKUWHUWIUUNYBUUDUUHUUKUWHUWIUUEBYJYKYCUVAWNWQWRWSWTWJUVEYBUWB | |
| - EYCJIZDUWCUWFRUVPUVEYKYCJIZUWKUWBYEYBUWLUWKXEZUVBYEYKESGDZUWMETDYEATDZYCE | |
| - AWOIZSGZDUWNXOUURYEYCASGUWQAEYCXAYEUWPASYEAUUSXBWAVBAEYCXCXDYKEYCXFUMWBUV | |
| - EUWBYKAYKWOIZJIZUWLUVEUVBYKTDUWBUWSDUVQUVEYCAYBYEYCTDUVBYBYCUUMXGXHYEYBUW | |
| - OUVBUURWBXIZUVAAYKXJVTUVEUWRYCYKJUVEAYCUVSYBYEUUOUVBUUPPZXKNXLXMBCUWBVRVT | |
| - UVEUWEUVGCUVEUWEUUFOKIZUVAKIZUVGUVEUWDUVAKIYKKIUWDYKKIZUVAKIUWEUXCUVEUWDU | |
| - VAYKUVEYCOUXAUWAXNZUVTUVEYKUWTVOZWEUVEUWDUVAYKUXEUVTUXFXPUVEUXDUXBUVAKUVE | |
| - YCOYKUXAUWAUXFWEQXQUVEUXBUVFUVAKUVEUUFAOKYBYEUUFARUVBUUTPQQUOWAWKXRXSXT | |
| - $. | |
| - $} | |
| - $( A subword expressed in terms of reverses and prefixes. (Contributed by | |
| - BTernaryTau, 3-Dec-2023.) $) | |
| - swrdrevpfx $p |- ( ( W e. Word V /\ F e. ( 0 ... L ) /\ | |
| - L e. ( 0 ... ( # ` W ) ) ) -> ( W substr <. F , L >. ) = | |
| - ( reverse ` ( ( reverse ` ( W prefix L ) ) prefix ( L - F ) ) ) ) $= | |
| - ( wcel cc0 cfz co chash cfv w3a cpfx creverse cmin cop csubstr wceq 3adant2 | |
| - wa syl cword fznn0sub2 pfxcl revcl simp3 revlen adantr pfxlen eqtrd 3adant3 | |
| - 3ad2ant1 oveq2d eleqtrrd jca syl3an3 3com23 revpfxsfxrev revrev oveq1d zcnd | |
| - elfzel2 elfzelz nncand 3ad2ant2 opeq12d 3eqtrd wi cuz elfzuz3 eluzfz2 ancli | |
| - swrdpfx syl5 pm2.43i eqtr2d ) DCUAZEZAFBGHZEZBFDIJGHEZKZDBLHZMJZBANHZLHMJZW | |
| - BABOZPHZDWFPHZWAWEWCMJZWCIJZWDNHZWJOZPHZWBWLPHZWGWAWCVPEZWDFWJGHZEZSZWEWMQV | |
| - QVTVSWRVSVQVTWDVREZWRABUBVQVTWSKZWOWQVQVTWOWSVQWBVPEZWOCDBUCZCWBUDTUKWTWDVR | |
| - WPVQVTWSUEWTWJBFGVQVTWJBQZWSVQVTSZWJWBIJZBVQWJXEQZVTVQXAXFXBCWBUFTUGCDBUHUI | |
| - ZUJULUMUNUOUPWDCWCUQTVQVSWMWNQVTVQWIWBWLPVQXAWIWBQXBCWBURTUSUKWAWLWFWBPWAWK | |
| - AWJBWAWKBWDNHZAVQVTWKXHQVSXDWJBWDNXGUSRVSVQXHAQVTVSBAVSBAFBVAUTVSAAFBVBUTVC | |
| - VDUIVQVTXCVSXGRVEULVFWAWGWHQZVQVTWAXIVGVSWAVSBABGHEZSZXDXIVSVQXKVTVSXJVSBAV | |
| - HJEXJAFBVIABVJTVKVDABBCDVLVMRVNVO $. | |
| ${ | |
| $d A a $. | |
| @@ -573880,59 +574432,7 @@ $( | |
| =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= | |
| $) | |
| - ${ | |
| - $d x G $. $d x V $. | |
| - lfuhgr.1 $e |- V = ( Vtx ` G ) $. | |
| - lfuhgr.2 $e |- I = ( iEdg ` G ) $. | |
| - $( A hypergraph is loop-free if and only if every edge connects at least | |
| - two vertices. (Contributed by BTernaryTau, 15-Oct-2023.) $) | |
| - lfuhgr $p |- ( G e. UHGraph -> | |
| - ( I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } <-> | |
| - A. x e. ( Edg ` G ) 2 <_ ( # ` x ) ) ) $= | |
| - ( cuhgr wcel cedg cfv c2 cv chash cle wbr cpw wss wf crn syl ciedg edgval | |
| - crab cdm wral rneqi eqtr4i sseq1i wfun wi uhgrfun fdmrn fss sylbi impbid1 | |
| - ex frn bitrid wb cvtx c0 csn uhgredgss difss2d pweqi sseqtrrdi ssrab baib | |
| - bitr3d ) BGHZBIJZKALMJNOZADPZUCZQZCUDZVNCRZVLAVKUEZVOCSZVNQZVJVQVKVSVNVKB | |
| - UAJZSVSBUBCWAFUFUGUHVJVTVQVJCUIZVTVQUJZCBFUKWBVPVSCRZWCCULWDVTVQVPVSVNCUM | |
| - UPUNTVPVNCUQUOURVJVKVMQZVOVRUSVJVKBUTJZPZVMVJVKWGVAVBBVCVDDWFEVEVFVOWEVRV | |
| - LAVMVKVGVHTVI $. | |
| - $( A hypergraph is loop-free if and only if every edge is not a loop. | |
| - (Contributed by BTernaryTau, 15-Oct-2023.) $) | |
| - lfuhgr2 $p |- ( G e. UHGraph -> | |
| - ( I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } <-> | |
| - A. x e. ( Edg ` G ) ( # ` x ) =/= 1 ) ) $= | |
| - ( wcel c2 cfv wbr cpw wral c1 wne wa c0 clt cvv elv cxr cuhgr cdm cv crab | |
| - chash cle wf cedg lfuhgr cc0 cvtx csn cdif uhgredgn0 eldifsni wb hashneq0 | |
| - syl sylibr gt0ne0d cxnn0 hashxnn0 xnn0n0n1ge2b ax-mp biimpi 3exp ralimdv2 | |
| - stoic3 a2d 1xr hashxrcl 1lt2 wi rexri xrltletr mp3an mpan xrltne mp3an12i | |
| - 2re ralimi impbid1 bitr4d ) BUAGZCUBHAUCZUEIZUFJZADKUDCUGWGABUHIZLZWFMNZA | |
| - WHLZABCDEFUIWDWKWIWDWJWGAWHWHWDWEWHGZWJWGWDWLWJWGWDWLWFUJNZWJWGWDWLOZWFWN | |
| - WEPNZUJWFQJZWNWEBUKIKZPULUMGWOWEBUNWEWQPUOURWPWOUPAWERUQSUSUTWMWJOZWGWFVA | |
| - GZWRWGUPWSAWERVBSWFVCVDVEVHVFVIVGWGWJAWHMTGZWFTGZWGMWFQJZWJVJXAAWERVKSZMH | |
| - QJZWGXBVLWTHTGXAXDWGOXBVMVJHVTVNXCMHWFVOVPVQMWFVRVSWAWBWC $. | |
| - $} | |
| - | |
| - ${ | |
| - $d x V $. $d x G a $. | |
| - lfuhgr3.1 $e |- V = ( Vtx ` G ) $. | |
| - lfuhgr3.2 $e |- I = ( iEdg ` G ) $. | |
| - $( A hypergraph is loop-free if and only if none of its edges connect to | |
| - only one vertex. (Contributed by BTernaryTau, 15-Oct-2023.) $) | |
| - lfuhgr3 $p |- ( G e. UHGraph -> | |
| - ( I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } <-> | |
| - -. E. a { a } e. ( Edg ` G ) ) ) $= | |
| - ( wcel cv cfv wbr c1 wral wex wn wceq wa notbii 3bitri eximi cuhgr cdm c2 | |
| - chash cle cpw crab wf wne cedg csn lfuhgr2 df-ne ralbii ralnex df-rex c1o | |
| - wrex cen cvv hashen1 elv en1 bitri anbi2i exbii 19.3v 19.29 sylanbr eleq1 | |
| - wb wal biimpac exlimiv dfclel pm3.22 sylbi excomim 19.40 ax5e anim1i 3syl | |
| - syl impbii bitrdi ) BUAHCUBUCAIZUDJZUEKADUFUGCUHWGLUIZABUJJZMZEIUKZWIHZEN | |
| - ZOZABCDFGULWJWFWIHZWGLPZQZANZOZWOWFWKPZENZQZANZOWNWJWPOZAWIMWPAWIURZOWSWH | |
| - XDAWIWGLUMUNWPAWIUOXEWRWPAWIUPRSWRXCWQXBAWPXAWOWPWFUQUSKZXAWPXFVKAWFUTVAV | |
| - BEWFVCVDVEVFRXCWMXCWMXBWMAXBWOWTQZENZWMWOWOEVLXAXHWOEVGWOWTEVHVIXGWLEWTWO | |
| - WLWFWKWIVJVMTWCVNWMXGANZENXHANXCWLXIEWLWTWOQZANXIAWKWIVOXJXGAWTWOVPTVQTXG | |
| - EAVRXHXBAXHWOENZXAQXBWOWTEVSXKWOXAWOEVTWAWCTWBWDRSWE $. | |
| - $} | |
| ${ | |
| $d A e a b $. $d B e a b $. $d e E a b $. $d e G a b $. $d e V a b $. | |
| @@ -573987,89 +574487,7 @@ $) | |
| DUGUEBAIZMUIBEAPUJUFUEBAQRSTUAABCDEFGUBUC $. | |
| $} | |
| - ${ | |
| - $d L k $. $d P k x $. $d F k x $. $d G k x $. | |
| - $( A prefix of a walk is a walk. (Contributed by BTernaryTau, | |
| - 2-Dec-2023.) $) | |
| - pfxwlk $p |- ( ( F ( Walks ` G ) P /\ L e. ( 0 ... ( # ` F ) ) ) -> | |
| - ( F prefix L ) ( Walks ` G ) ( P prefix ( L + 1 ) ) ) $= | |
| - ( vk vx cfv cc0 cfz co wcel wa c1 caddc wceq wss cfzo adantr syl adantl | |
| - cwlks wbr chash cpfx ciedg cdm cword cvtx wf csn cpr wral eqid wlkf pfxcl | |
| - wif cres wlkp cuz elfzuz3 fzss2 fssresd pfxlen sylan oveq2d feq2d wlkpwrd | |
| - mpbird fzp1elp1 wlklenvp1 eleqtrrd pfxres syl2an2r elfzelz fzval3 reseq2d | |
| - cv cz eqtr4d feq1d wlkprop simp3d eleq2d fveq1d fzossfz a1i sselda fvresd | |
| - eqtr2d fzofzp1 jca ex sylbid imp ancli simpr fveq2d eqcomd simplr wkslem1 | |
| - wi fzoss2 rspcv wb eqeq12 eqeq12d preq12 sseq12d ifpbi123d biimpd sylsyld | |
| - sneq mpd ralrimiva cvv w3a wlkv simp1d iswlkg mpbir3and ) BACUAGZUBZDHBUC | |
| - GZIJZKZLZBDUDJZADMNJZUDJZYAUBZYGCUEGZUFZUGZKZHYGUCGZIJZCUHGZYIUIZEVQZYIGZ | |
| - YSMNJZYIGZOZYSYGGZYKGZYTUJZOZYTUUBUKZUUEPZUPZEHYOQJZULZYFBYMKZYNYBUUMYEAB | |
| - CYKYKUMZUNZRZYLBDUOSYFYRYPYQAHDIJZUQZUIZYFUUSUUQYQUURUIYFYDYQUUQAYBYDYQAU | |
| - IZYEABCYQYQUMZURRYEUUQYDPZYBYEYCDUSGZKZUVBDHYCUTZDHYCVASTVBYFYPUUQYQUURYF | |
| - YODHIYBUUMYEYODOUUOYLBDVCVDZVEVFVHYFYPYQYIUURYFYIAHYHQJZUQZUURYBAYQUGKYEY | |
| - HHAUCGZIJZKYIUVHOABCYQUVAVGYFYHHYCMNJZIJZUVJYEYHUVLKYBDHYCVITYBUVJUVLOYEY | |
| - BUVIUVKHIABCVJVERVKYQAYHVLVMYFUUQUVGAYEUUQUVGOZYBYEDVRKUVMDHYCVNHDVOSTVPV | |
| - SZVTVHYFUUJEUUKYFYSUUKKZLZFVQZAGZUVQMNJAGZOUVQBGYKGZUVRUJOUVRUVSUKUVTPUPZ | |
| - FHYCQJZULZUUJYFUWCUVOYBUWCYEYBUUMUUTUWCAFBCYKYQUVAUUNWAWBRRUVPYSAGZYTOZUU | |
| - AAGZUUBOZLZYSBGZYKGZUUEOZLZUWCUWDUWFOZUWJUWDUJZOZUWDUWFUKZUWJPZUPZUUJUVPU | |
| - WHUWKYFUVOUWHYFUVOYSHDQJZKZUWHYFUUKUWSYSYFYODHQUVFVEWCZYFUWTUWHYFUWTLZUWE | |
| - UWGUXBYTYSUURGZUWDYFYTUXCOUWTYFYSYIUURUVNWDRUXBYSUUQAYFUWSUUQYSUWSUUQPYFH | |
| - DWEWFWGWHWIUXBUUBUUAUURGZUWFYFUUBUXDOUWTYFUUAYIUURUVNWDRUXBUUAUUQAUWTUUAU | |
| - UQKYFHDYSWJTWHWIWKWLWMWNUVPUWJYSBUWSUQZGZYKGZUUEYFUVOUWJUXGOZYFUVOUWTUXHU | |
| - XAYFUWTUXHUXBUXGUWJYFYFUUMLZUWTUXGUWJOYFUUMUUPWOUXIUWTLZUXFUWIYKUXJYSUWSB | |
| - UXIUWTWPWHWQVDWRWLWMWNUVPUUDUXFYKUVPYSYGUXEYFUUMUVOYEYGUXEOUUPYBYEUVOWSYL | |
| - BDVLVMWDWQVSWKUVPYSUWBKUWCUWRXAYFUUKUWBYSYFYCYOUSGZKUUKUWBPYFYCUVCUXKYEUV | |
| - DYBUVETYFYODUSUVFWQVKYOHYCXBSWGUWAUWRFYSUWBUVQYSABYKWTXCSUWLUWRUUJUWLUWMU | |
| - WOUWQUUCUUGUUIUWHUWMUUCXDUWKUWDYTUWFUUBXERUWLUWJUUEUWNUUFUWHUWKWPZUWHUWNU | |
| - UFOZUWKUWEUXMUWGUWDYTXLRRXFUWLUWPUUHUWJUUEUWHUWPUUHOUWKUWDUWFYTUUBXGRUXLX | |
| - HXIXJXKXMXNYFCXOKZYJYNYRUULXPXDYBUXNYEYBUXNBXOKAXOKABCXQXRRYIEYGCYKYQXOUV | |
| - AUUNXSSXT $. | |
| - $} | |
| - ${ | |
| - $d P k x $. $d F k x $. $d G k x $. | |
| - $( The reverse of a walk is a walk. (Contributed by BTernaryTau, | |
| - 30-Nov-2023.) $) | |
| - revwlk $p |- ( F ( Walks ` G ) P -> | |
| - ( reverse ` F ) ( Walks ` G ) ( reverse ` P ) ) $= | |
| - ( vk vx cfv wcel cc0 chash co c1 caddc wceq cfzo oveq2d eqtrd cmin oveq1d | |
| - syl adantr cwlks wbr creverse ciedg cdm cword cfz cvtx wf csn cpr wss wif | |
| - cv wral eqid wlkf revcl wlkpwrd wrdf 3syl revlen wlklenvp1 cz wlkcl nn0zd | |
| - fzval3 3eqtr4rd feq2d mpbird eleq2d biimpa wa c2 revfv sylan wlklenvm1 cc | |
| - lencl nn0cnd sub1m1 fvoveq1d fveq2d fzonn0p1p1 eleqtrrd syl2an2r elfzoelz | |
| - adantl zcnd 1cnd addcomd subsub4d 3eqtr2d sneqd sneq eqcom cn0 fzossfzop1 | |
| - subcld sseqtrrd sselda sub32d npcand 3eqtr3d eqeq12d bitrid wi wn wkslem1 | |
| - wlkprop simp3d ubmelm1fzo eqeltrrd rspcdva dfifp2 sylib simpld sylbid imp | |
| - notbid simprd prcom preq12d eqtr3id 3sstr4d ifpimpda syldan ralrimiva cvv | |
| - w3a wb wlkv simp1d iswlkg mpbir3and ) BACUAFZUBZBUCFZAUCFZYPUBZYRCUDFZUEZ | |
| - UFZGZHYRIFZUGJZCUHFZYSUIZDUNZYSFZUUIKLJZYSFZMZUUIYRFZUUAFZUUJUJZMZUUJUULU | |
| - KZUUOULZUMZDHUUENJZUOZYQBUUCGZUUDABCUUAUUAUPZUQZUUBBURSYQUUHHYSIFZNJZUUGY | |
| - SUIZYQAUUGUFZGZYSUVIGUVHABCUUGUUGUPZUSZUUGAURUUGYSUTVAYQUUFUVGUUGYSYQUUFH | |
| - BIFZUGJZUVGYQUUEUVMHUGYQUVCUUEUVMMUVEUUBBVBSZOYQHAIFZNJZHUVMKLJZNJZUVGUVN | |
| - YQUVPUVRHNABCVCOZYQUVFUVPHNYQUVJUVFUVPMUVLUUGAVBSOYQUVMVDGUVNUVSMYQUVMABC | |
| - VEZVFHUVMVGSVHPVIVJYQUUTDUVAYQUUIUVAGZUUIHUVMNJZGZUUTYQUWBUWDYQUVAUWCUUIY | |
| - QUUEUVMHNUVOOVKVLYQUWDVMZUUMUUQUUSUWEUUMVMZUUOUVPVNQJZUUIQJZBFZUUAFZUUPUW | |
| - EUUOUWJMZUUMUWEUUNUWIUUAUWEUUNUVMKQJZUUIQJZBFZUWIYQUVCUWDUUNUWNMUVEUUBBUU | |
| - IVOVPYQUWNUWIMUWDYQUWLUWGUUIBQYQUWLUVPKQJZKQJZUWGYQUVMUWOKQABCVQRYQUVJUVP | |
| - VRGZUWPUWGMUVLUVJUVPUUGAVSVTZUVPWAVAZPZWBTPWCZTUWFUULUJZUWHAFZUJZUUPUWJUW | |
| - EUXBUXDMUUMUWEUULUXCUWEUULUWOUUKQJZAFZUXCYQUVJUWDUUKUVQGUULUXFMUVLUWEUUKU | |
| - VSUVQUWDUUKUVSGYQUUIUVMWDWHYQUVQUVSMUWDUVTTWEUUGAUUKVOWFUWEUXEUWHAUWEUXEU | |
| - WOKUUILJZQJUWPUUIQJZUWHUWEUUKUXGUWOQUWEUUIKUWDUUIVRGYQUWDUUIUUIHUVMWGWIWH | |
| - ZUWEWJZWKOUWEUWOKUUIUWEUVPKYQUWQUWDYQUVJUWQUVLUWRSTUXJWSZUXJUXIWLYQUXHUWH | |
| - MUWDYQUWPUWGUUIQUWSRZTWMWCPZWNTUUMUUPUXBMUWEUUJUULWOWHUWEUUMUWJUXDMZUWEUU | |
| - MUXCUWHKLJZAFZMZUXNUUMUULUUJMUWEUXQUUJUULWPUWEUULUXCUUJUXPUXMUWEUUJUWOUUI | |
| - QJZAFZUXPYQUVJUWDUUIUVQGUUJUXSMUVLYQUWCUVQUUIYQUWCUVSUVQYQUVMWQGUWCUVSULU | |
| - WAUVMWRSUVTWTXAUUGAUUIVOWFUWEUXRUXOAUWEUXRKQJZKLJUXHKLJZUXRUXOUWEUXTUXHKL | |
| - UWEUWOUUIKUXKUXIUXJXBRUWEUXRKUWEUWOUUIUXKUXIWSUXJXCYQUYAUXOMUWDYQUXHUWHKL | |
| - UXLRTXDWCPZXEXFZUWEUXQUXNXGZUXQXHZUXCUXPUKZUWJULZXGZUWEUXQUXNUYGUMZUYDUYH | |
| - VMUWEEUNZAFZUYJKLJAFZMUYJBFUUAFZUYKUJMUYKUYLUKUYMULUMZUYIEUWCUWHUYJUWHABU | |
| - UAXIYQUYNEUWCUOZUWDYQUVCUVNUUGAUIUYOAEBCUUAUUGUVKUVDXJXKTUWEUVMUUIQJKQJZU | |
| - WHUWCUWEUYPUWMUWHUWEUVMUUIKYQUVMVRGUWDYQUVMUWAVTTUXIUXJXBUWEUWLUWGUUIQYQU | |
| - WLUWGMUWDUWTTRPUWDUYPUWCGYQUUIUVMXLWHXMXNUXQUXNUYGXOXPZXQXRXSVHPUWEUUMXHZ | |
| - VMUYFUWJUURUUOUWEUYRUYGUWEUYRUYEUYGUWEUUMUXQUYCXTUWEUYDUYHUYQYAXRXSUWEUUR | |
| - UYFMUYRUWEUURUULUUJUKUYFUULUUJYBUWEUULUXCUUJUXPUXMUYBYCYDTUWEUWKUYRUXATYE | |
| - YFYGYHYQCYIGZYTUUDUUHUVBYJYKYQUYSBYIGAYIGABCYLYMYSDYRCUUAUUGYIUVKUVDYNSYO | |
| - $. | |
| - $} | |
| $( Two words represent a walk if and only if their reverses also represent a | |
| walk. (Contributed by BTernaryTau, 4-Dec-2023.) $) | |
| @@ -574079,69 +574497,8 @@ $) | |
| wbr ) CEFGZABFGZHZCADIJZQZCKJZAKJZUAQZACDLUEUCKJZUDKJZUAQTUBUDUCDLRSUFCUGAU | |
| AECMBAMNOP $. | |
| - $( Two matching subwords of a walk also represent a walk. (Contributed by | |
| - BTernaryTau, 7-Dec-2023.) $) | |
| - swrdwlk $p |- ( ( F ( Walks ` G ) P /\ B e. ( 0 ... L ) /\ | |
| - L e. ( 0 ... ( # ` F ) ) ) -> ( F substr <. B , L >. ) ( Walks ` G ) | |
| - ( P substr <. B , ( L + 1 ) >. ) ) $= | |
| - ( cfv wbr cc0 cfz co wcel chash cpfx creverse c1 caddc 3ad2ant2 wceq oveq2d | |
| - 3ad2ant1 cwlks w3a cmin cop csubstr pfxwlk 3adant2 revwlk syl fznn0sub2 cdm | |
| - ciedg cword eqid wlkf pfxcl revlen 3syl pfxlen sylan eqtrd eleqtrrd syl2anc | |
| - cz elfzel2 zcnd 1cnd elfzelz addsubd swrdrevpfx syl3an1 cvtx wlkpwrd fzelp1 | |
| - breqtrrd fzp1elp1 3ad2ant3 wlklenvp1 syl3anc 3brtr4d ) CBDUAFZGZAHEIJZKZEHC | |
| - LFZIJKZUBZCEMJZNFZEAUCJZMJZNFZBEOPJZMJZNFZWMAUCJZMJZNFZCAEUDUEJZBAWMUDUEJZW | |
| - AWGWKWQWAGWLWRWAGWGWKWOWJOPJZMJZWQWAWGWIWOWAGZWJHWILFZIJZKWKXBWAGWGWHWNWAGZ | |
| - XCWBWFXFWDBCDEUFUGWNWHDUHUIWGWJWCXEWDWBWJWCKWFAEUJQWGXDEHIWGXDWHLFZEWGCDULF | |
| - ZUKZUMZKZWHXJKXDXGRWBWDXKWFBCDXHXHUNUOZTXICEUPXIWHUQURWBWFXGERZWDWBXKWFXMXL | |
| - XICEUSUTUGVASVBWOWIDWJUFVCWGWPXAWOMWGEOAWGEWDWBEVDKWFAHEVEQVFWGVGWGAWDWBAVD | |
| - KWFAHEVHQVFVISVOWQWKDUHUIWBXKWDWFWSWLRXLAEXICVJVKWGBDVLFZUMKZAHWMIJKZWMHBLF | |
| - ZIJZKWTWRRWBWDXOWFBCDXNXNUNVMTWDWBXPWFAHEVNQWGWMHWEOPJZIJZXRWFWBWMXTKWDEHWE | |
| - VPVQWGXQXSHIWBWDXQXSRWFBCDVRTSVBAWMXNBVJVSVT $. | |
| - ${ | |
| - pthhashvtx.1 $e |- V = ( Vtx ` G ) $. | |
| - $( A graph containing a path has at least as many vertices as there are | |
| - edges in the path. (Contributed by BTernaryTau, 5-Oct-2023.) $) | |
| - pthhashvtx $p |- ( F ( Paths ` G ) P -> ( # ` F ) <_ ( # ` V ) ) $= | |
| - ( cfv wbr c1 co wcel cle cc0 cfz wceq syl cres wss cima c0 ax-mp cmin cn0 | |
| - cpths chash wa caddc hashfz0 cc cwlks pthiswlk wlkcl nn0cn 3syl sylan9eqr | |
| - npcan1 cfn wf1 wfn wlkp ffnd fzfi resfnfinfin sylancl simpr fzssp1 oveq2d | |
| - sseqtrid fssresd adantr ccnv wfun fz1ssfz0 a1i cfzo ctrls cpr cin simp2bi | |
| - wf ispth cz nn0z fzoval reseq2d resabs1 eqtr4di cnveqd funeqd mpbid df-f1 | |
| - sylanbrc csn snsspr1 imass2 wb 0elfz snssd resima2 mpbiri imaeq2d eqtr4id | |
| - sseq1 ineq2d simp3bi eqtrd syl2anr f1resfz0f1d cvv cvtx fvexi hashf1dmcdm | |
| - ssdisj mp3an2 syl2an2r eqbrtrrd 0nn0m1nnn0 biimpri sylan hashge0 eqbrtrdi | |
| - wn pm2.61dan ) BACUCFGZBUDFZHUAIZUBJZYDDUDFZKGYCYFUEZLYEMIZUDFZYDYGKYFYCY | |
| - JYEHUFIZYDYEUGYCYDUBJZYDUHJYKYDNYCBACUIFGZYLABCUJZABCUKOZYDULYDUOUMZUNYCA | |
| - YIPZUPJZYFYIDYQUQZYJYGKGZYCALYDMIZURYIUPJYRYCUUADAYCYMUUADAVSYNABCDEUSOZU | |
| - TLYEVAUUAYIAVBVCYHYQYEDYCYFVDYCYIDYQVSYFYCUUADYIAUUBYCLYKMIYIUUALYEVEYCYK | |
| - YDLMYPVFVGVHZVIYCHYEMIZDYQUUDPZUQZYFYCUUDDUUEVSUUEVJZVKZUUFYCYIDUUDYQUUCU | |
| - UDYIQZYCYEVLZVMVHYCAHYDVNIZPZVJZVKZUUHYCBACVOFGZUUNALYDVPZRZAUUKRZVQZSNZA | |
| - BCVTZVRYCUUMUUGYCUULUUEYCUULAUUDPZUUEYCUUKUUDAYCYLUUKUUDNZYOYLYDWAJUVCYDW | |
| - BHYDWCOOZWDUUIUUEUVBNUUJAUUDYIWETWFWGWHWIUUDDUUEWJWKVIYFYQLWLZRZUUQQZUUQY | |
| - QUUDRZVQZSNUVFUVHVQSNYCYFUVGAUVERZUUQQZUVEUUPQUVKLYDWMUVEUUPAWNTYFUVEYIQU | |
| - VFUVJNUVGUVKWOYFLYIYEWPWQAUVEYIWRUVFUVJUUQXBUMWSYCUVIUUSSYCUVHUURUUQYCUVH | |
| - AUUDRZUURUUIUVHUVLNUUJAUUDYIWRTYCUUKUUDAUVDWTXAXCYCUUOUUNUUTUVAXDXEUVFUUQ | |
| - UVHXLXFXGYRDXHJZYSYTDCXIEXJZYIDYQUPXHXKXMXNXOYCYFYAZUEYDLYGKYCYLUVOYDLNZY | |
| - OUVPYLUVOUEYDXPXQXRUVMLYGKGUVNDXHXSTXTYB $. | |
| - $} | |
| - $( A walk is a trivial path if and only if it is both a simple path and a | |
| - cycle. (Contributed by BTernaryTau, 8-Oct-2023.) $) | |
| - spthcycl $p |- ( ( F ( Paths ` G ) P /\ F = (/) ) <-> | |
| - ( F ( SPaths ` G ) P /\ F ( Cycles ` G ) P ) ) $= | |
| - ( cfv wbr c0 wceq wa wfun cc0 chash co c1 caddc adantr cvv wcel sylan syl | |
| - wn cpths cspths ccycls ctrls ccnv pthistrl cwlks pthiswlk c1o cen cvtx eqid | |
| - cfz wlkp ffund wlklenvp1 simp2d hasheq0 biimpar oveq1 0p1e1 eqtrdi eqtrd wb | |
| - simp3d hashen1 biimpa syldan funen1cnv syl2an2r isspth biimpri fveq2 eqcoms | |
| - wlkv iscycl jca spthispth notnot wne cyclnspth com12 con3dimp sylan2 ancoms | |
| - nne sylib impbii ) BACUADEZBFGZHZBACUBDEZBACUCDEZHZWKWLWMWIBACUDDEZWJAUEIZW | |
| - LABCUFWIBACUGDEZWJWPABCUHZWQAIWJAUIUJEZWPWQJBKDZUMLCUKDZAABCXAXAULUNUOWQWJA | |
| - KDZMGZWSWQWJHZXBWTMNLZMWQXBXEGWJABCUPOXDWTJGZXEMGWQBPQZWJXFWQCPQZXGAPQZABCV | |
| - OZUQXGXFWJBPURUSRZXFXEJMNLMWTJMNUTVAVBSVCWQXCWSWQXIXCWSVDWQXHXGXIXJVEAPVFSV | |
| - GVHAVIVJRWLWOWPHABCVKVLVJWIWJJADWTADGZWMWIWQWJXLWRXDXFXLXKXLJWTJWTAVMVNSRWM | |
| - WIXLHABCVPVLVHVQWNWIWJWLWIWMABCVROWMWLWJWLWMWLTZTZWJWLVSWMXNHBFVTZTWJWMXOXM | |
| - XOWMXMABCWAWBWCBFWFWGWDWEVQWH $. | |
| $( A non-trivial cycle in a simple graph has a length greater than 2. | |
| (Contributed by BTernaryTau, 24-Sep-2023.) $) | |
| @@ -574173,52 +574530,9 @@ $) | |
| ABCUQABCDEURUSSUT $. | |
| $} | |
| - ${ | |
| - $d P k $. $d S k $. $d k F $. $d k G $. | |
| - $( If a walk exists in a subgraph of a graph ` G ` , then that walk also | |
| - exists in ` G ` . (Contributed by BTernaryTau, 22-Oct-2023.) $) | |
| - subgrwlk $p |- ( S SubGraph G -> | |
| - ( F ( Walks ` S ) P -> F ( Walks ` G ) P ) ) $= | |
| - ( vk wbr cwlks cfv ciedg wcel co wceq wss wral w3a wa cvv eqid syl wi cdm | |
| - csubgr cword cc0 chash cfz cvtx wf cv c1 caddc csn cpr wif cfzo wb subgrv | |
| - simpld iswlkg 3simpa cedg cpw subgrprop2 simp2d dmss sswrd 3syl sseld fss | |
| - simp1d expcom syl5 3simpb cres subgrprop fveq1d 3ad2ant1 wrdsymbcl fvresd | |
| - anim12d 3adant1 eqtrd eqeq1d sseq2d ifpbi23d biimpd 3expia ralrimiv ralim | |
| - expimpd jcad sylbid df-3an imbitrrdi simpl2im sylibrd ) BDUBFZCABGHFZCDIH | |
| - ZUAZUCZJZUDCUEHZUFKZDUGHZAUHZEUIZAHZXGUJUKKAHZLZXGCHZWSHZXHULZLZXHXIUMZXL | |
| - MZUNZEUDXCUOKZNZOZCADGHFZWQWRXBXFPZXSPZXTWQWRCBIHZUAZUCZJZXDBUGHZAUHZXJXK | |
| - YDHZXMLZXOYJMZUNZEXRNZOZYCWQBQJZWRYOUPWQYPDQJZBDUQZURAECBYDYHQYHRZYDRZUSS | |
| - WQYOYBXSYOYGYIPWQYBYGYIYNUTWQYGXBYIXFWQYFXACWQYDWSMZYEWTMYFXAMWQYHXEMZUUA | |
| - BVAHZYHVBMZXEWSBUUCDYDYHYSXERZYTWSRZUUCRZVCZVDYDWSVEYEWTVFVGVHWQUUBYIXFTW | |
| - QUUBUUAUUDUUHVJYIUUBXFXDYHXEAVIVKSVTVLYOYGYNPWQXSYGYIYNVMWQYGYNXSWQYGPZYM | |
| - XQTZEXRNYNXSTUUIUUJEXRWQYGXGXRJZUUJWQYGUUKOZYMXQUULXJYKYLXNXPUULYJXLXMUUL | |
| - YJXKWSYEVNZHZXLWQYGYJUUNLUUKWQXKYDUUMWQUUBYDUUMLUUDXEWSBUUCDYDYHYSUUEYTUU | |
| - FUUGVOVDVPVQYGUUKUUNXLLWQYGUUKPXKYEWSXGYECVRVSWAWBZWCUULYJXLXOUUOWDWEWFWG | |
| - WHYMXQEXRWISWJVLWKWLXBXFXSWMWNWQYPYQYAXTUPYRAECDWSXEQUUEUUFUSWOWP $. | |
| - $} | |
| - $( If a trail exists in a subgraph of a graph ` G ` , then that trail also | |
| - exists in ` G ` . (Contributed by BTernaryTau, 22-Oct-2023.) $) | |
| - subgrtrl $p |- ( S SubGraph G -> | |
| - ( F ( Trails ` S ) P -> F ( Trails ` G ) P ) ) $= | |
| - ( csubgr wbr cwlks cfv ccnv wfun wa ctrls subgrwlk anim1d istrl 3imtr4g ) B | |
| - DEFZCABGHFZCIJZKCADGHFZSKCABLHFCADLHFQRTSABCDMNACBOACDOP $. | |
| - $( If a path exists in a subgraph of a graph ` G ` , then that path also | |
| - exists in ` G ` . (Contributed by BTernaryTau, 22-Oct-2023.) $) | |
| - subgrpth $p |- ( S SubGraph G -> | |
| - ( F ( Paths ` S ) P -> F ( Paths ` G ) P ) ) $= | |
| - ( csubgr wbr ctrls cfv c1 chash cfzo co cres ccnv wfun cima w3a cpths ispth | |
| - idd cc0 cpr cin c0 wceq subgrtrl 3anim123d 3imtr4g ) BDEFZCABGHFZAICJHZKLZM | |
| - NOZAUAUKUBPAULPUCUDUEZQCADGHFZUMUNQCABRHFCADRHFUIUJUOUMUMUNUNABCDUFUIUMTUIU | |
| - NTUGACBSACDSUH $. | |
| - $( If a cycle exists in a subgraph of a graph ` G ` , then that cycle also | |
| - exists in ` G ` . (Contributed by BTernaryTau, 23-Oct-2023.) $) | |
| - subgrcycl $p |- ( S SubGraph G -> | |
| - ( F ( Cycles ` S ) P -> F ( Cycles ` G ) P ) ) $= | |
| - ( csubgr wbr cpths cfv cc0 chash wceq ccycls subgrpth anim1d iscycl 3imtr4g | |
| - wa ) BDEFZCABGHFZIAHCJHAHKZQCADGHFZTQCABLHFCADLHFRSUATABCDMNACBOACDOP $. | |
| ${ | |
| $d V a b c $. $d f G p a b c $. | |
| @@ -574247,126 +574561,10 @@ $) | |
| XNWQWJWJWJWKCWLIXOXHCBWMEWNCWLFGHWOWRWP $. | |
| $} | |
| - ${ | |
| - $d A j $. $d G p $. $d A f p $. $d f j G $. | |
| - $( A hypergraph has a cycle of length one if and only if it has a loop. | |
| - (Contributed by BTernaryTau, 13-Oct-2023.) $) | |
| - loop1cycl $p |- ( G e. UHGraph -> | |
| - ( E. f E. p ( f ( Cycles ` G ) p /\ ( # ` f ) = 1 /\ ( p ` 0 ) = A ) <-> | |
| - { A } e. ( Edg ` G ) ) ) $= | |
| - ( vj wcel cv cfv wbr chash c1 wceq cc0 w3a wex csn wi wa syl wrex anabss3 | |
| - cuhgr ccycls cedg cwlks cpths cyclprop fveq2 eqeq2d anbi2d mpan9 pthiswlk | |
| - biimpd anim1i df-3an 3ancomb sylib ciedg wfun wlkl1loop expl eqid uhgrfun | |
| - sylibr syl11 3impb 3adant3 sneq eleq1d 3ad2ant3 sylibd exlimivv com12 cs2 | |
| - wb cs1 wal cdm crn edgval eleq2i elrnrexdm eqcom rexbii imbitrdi biimtrid | |
| - df-rex lp1cycl 3expib eximdv syld s1len ax-gen 19.29r syl6 imp cvv c0 wne | |
| - mpan2 cvtx cpw uhgredgn0 eldifsni snnzb s2fv0 alrimiv syl2anc exbii cword | |
| - cdif s1cli breq1 fveqeq2 3anbi12d rspcev rexex exlimiv s2cli breq2 eqeq1d | |
| - mpan fveq1 3anbi13d eximi ex impbid ) CUBFZBGZDGZCUCHZIZYIJHZKLZMYJHZALZN | |
| - ZDOZBOZAPZCUDHZFZYSYHUUBYQYHUUBQBDYQYHYOPZUUAFZUUBYLYNYHUUDQZYPYLYNRZYIYJ | |
| - CUEHIZYNYOKYJHZLZNZUUEUUFUUGUUIYNNZUUJUUFUUGUUIRZYNRZUUKYLYNUUMUUFUULYNUU | |
| - FYIYJCUFHIZUUIRZUULYLUUNYOYMYJHZLZRZYNUUOYJYICUGYNUURUUOYNUUQUUIUUNYNUUPU | |
| - UHYOYMKYJUHUIUJUMUKUUNUUGUUIYJYICULUNSUNUAUUGUUIYNUOVDUUGUUIYNUPUQUUGYNUU | |
| - IUUECURHZUSZUUGYNUUIRZRUUDYHUUTUUGUVAUUDYJYICUTVAUUSCUUSVBZVCZVEVFSVGYPYL | |
| - UUDUUBVOYNYPUUCYTUUAYOAVHVIVJVKVLVMYHUUBYSYHUUBRZYIAAVNZYKIZYNMUVEHZALZNZ | |
| - BOZYSUVDEGZVPZUVEYKIZUVLJHKLZUVHNZEOZUVJUVDUVMUVNRZUVHRZEOZUVPUVDUVQEOZUV | |
| - HEVQUVSYHUUBUVTYHUUBUVMEOZUVTYHUUBUVKUUSVRZFZUVKUUSHZYTLZRZEOZUWAYHUUBUWE | |
| - EUWBTZUWGYHUUTUUBUWHQUVCUUBYTUUSVSZFZUUTUWHUUAUWIYTCVTWAUUTUWJYTUWDLZEUWB | |
| - TUWHEUUSYTWBUWKUWEEUWBYTUWDWCWDWEWFSUWEEUWBWGWEYHUWFUVMEYHUWCUWEUVMACUUSU | |
| - VKUVBWHWIWJWKUWAUVNEVQUVTUVNEUVKWLWMUVMUVNEWNWTWOWPUVDUVHEUVDAWQFZUVHUVDY | |
| - TWRWSZUWLUVDYTCXAHXBZWRPXKFUWMYTCXCYTUWNWRXDSAXEVDAAWQXFSXGUVQUVHEWNXHUVO | |
| - UVREUVMUVNUVHUOXIVDUVOUVJEUVOUVIBWQXJZTZUVJUVLUWOFUVOUWPUVKXLUVIUVOBUVLUW | |
| - OYIUVLLUVFUVMYNUVNUVHYIUVLUVEYKXMYIUVLKJXNXOXPYBUVIBUWOXQSXRSUVIYRBUVIYQD | |
| - UWOTZYRUVEUWOFUVIUWQAAXSYQUVIDUVEUWOYJUVELZYLUVFYPUVHYNYJUVEYIYKXTUWRYOUV | |
| - GAMYJUVEYCYAYDXPYBYQDUWOXQSYESYFYG $. | |
| - $} | |
| - ${ | |
| - 2cycld.1 $e |- P = <" A B C "> $. | |
| - 2cycld.2 $e |- F = <" J K "> $. | |
| - 2cycld.3 $e |- ( ph -> ( A e. V /\ B e. V /\ C e. V ) ) $. | |
| - 2cycld.4 $e |- ( ph -> ( A =/= B /\ B =/= C ) ) $. | |
| - 2cycld.5 $e |- ( ph -> | |
| - ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) ) $. | |
| - 2cycld.6 $e |- V = ( Vtx ` G ) $. | |
| - 2cycld.7 $e |- I = ( iEdg ` G ) $. | |
| - 2cycld.8 $e |- ( ph -> J =/= K ) $. | |
| - 2cycld.9 $e |- ( ph -> A = C ) $. | |
| - $( Construction of a 2-cycle from two given edges in a graph. (Contributed | |
| - by BTernaryTau, 16-Oct-2023.) $) | |
| - 2cycld $p |- ( ph -> F ( Cycles ` G ) P ) $= | |
| - ( cpths cfv wbr cc0 chash wceq ccycls 2pthd wcel w3a wa cs3 fveq1i eqtrid | |
| - s3fv0 3ad2ant1 adantr simpr cs2 fveq2i s2len eqtri fveq12i s3fv2 3ad2ant3 | |
| - c2 eqtr2id 3eqtrd syl2anc iscycl sylanbrc ) AFEGUAUBUCUDEUBZFUEUBZEUBZUFZ | |
| - FEGUGUBUCABCDEFGHIJKLMNOPQRSUHABKUIZCKUIZDKUIZUJZBDUFZVONTVSVTUKVLBDVNVSV | |
| - LBUFZVTVPVQWAVRVPVLUDBCDULZUBBUDEWBLUMBCDKUOUNUPUQVSVTURVSDVNUFZVTVRVPWCV | |
| - QVRVNVFWBUBDVMVFEWBLVMIJUSZUEUBVFFWDUEMUTIJVAVBVCBCDKVDVGVEUQVHVIEFGVJVK | |
| - $. | |
| - $} | |
| - ${ | |
| - 2cycl2d.1 $e |- P = <" A B A "> $. | |
| - 2cycl2d.2 $e |- F = <" J K "> $. | |
| - 2cycl2d.3 $e |- ( ph -> ( A e. V /\ B e. V ) ) $. | |
| - 2cycl2d.4 $e |- ( ph -> A =/= B ) $. | |
| - 2cycl2d.5 $e |- ( ph -> | |
| - ( { A , B } C_ ( I ` J ) /\ { A , B } C_ ( I ` K ) ) ) $. | |
| - 2cycl2d.6 $e |- V = ( Vtx ` G ) $. | |
| - 2cycl2d.7 $e |- I = ( iEdg ` G ) $. | |
| - 2cycl2d.8 $e |- ( ph -> J =/= K ) $. | |
| - $( Construction of a 2-cycle from two given edges in a graph. (Contributed | |
| - by BTernaryTau, 16-Oct-2023.) $) | |
| - 2cycl2d $p |- ( ph -> F ( Cycles ` G ) P ) $= | |
| - ( wa wss wcel w3a simpl jccir df-3an sylibr wne necomd jca cpr cfv sseq1i | |
| - prcom anbi2i sylib eqidd 2cycld ) ABCBDEFGHIJKLABJUAZCJUAZSZURSURUSURUBAU | |
| - TURMURUSUCUDURUSURUEUFABCUGCBUGNABCNUHUIABCUJZHGUKTZVAIGUKZTZSVBCBUJZVCTZ | |
| - SOVDVFVBVAVEVCBCUMULUNUOPQRABUPUQ $. | |
| - $} | |
| - ${ | |
| - $d ph a b $. $d I a b $. $d J a b $. $d F p a b $. $d G p a b $. | |
| - umgr2cycllem.1 $e |- F = <" J K "> $. | |
| - umgr2cycllem.2 $e |- I = ( iEdg ` G ) $. | |
| - umgr2cycllem.3 $e |- ( ph -> G e. UMGraph ) $. | |
| - umgr2cycllem.4 $e |- ( ph -> J e. dom I ) $. | |
| - umgr2cycllem.5 $e |- ( ph -> J =/= K ) $. | |
| - umgr2cycllem.6 $e |- ( ph -> ( I ` J ) = ( I ` K ) ) $. | |
| - $( Lemma for ~ umgr2cycl . (Contributed by BTernaryTau, 17-Oct-2023.) $) | |
| - umgr2cycllem $p |- ( ph -> E. p F ( Cycles ` G ) p ) $= | |
| - ( va vb cv cfv wa wrex wcel wne cpr wceq cvtx wbr wex cumgr cedg wfun cdm | |
| - ccycls cuhgr umgruhgr uhgrfun 3syl iedgedg syl2anc eqid umgredg wral ax-5 | |
| - wal alral syl r19.29 sylan cs3 w3a simp2 simp3l wss eqimss2 adantl sseq2d | |
| - wi 3ad2ant3 imbitrid adantld 3impib jca 3ad2ant1 3expib exp4c com23 imp4a | |
| - 2cycl2d cvv cword s3cli breq2 rspcev mpan rexex syl8 rexlimdv syl5 expd | |
| - mpd ) ANPZOPZUAZEDQZWSWTUBZUCZRZOCUDQZSZNXFSZBGPZCUKQZUEZGUFZACUGTZXBCUHQ | |
| - ZTZXHJADUIZEDUJTXOAXMCULTXPJCUMDCIUNUOKDCEIUPUQXBXNCXFNOXFURZXNURUSUQAXGX | |
| - LNXFAWSXFTZXGXLXRXGRXRXERZOXFSZAXLXRXROXFUTZXGXTXRXROVBYAXROVAXROXFVCVDXR | |
| - XEOXFVEVFAXSXLOXFAWTXFTZXSBWSWTWSVGZXJUEZXLAYBXRXEYDAXRYBXEYDVOAXRYBXEYDA | |
| - XRYBRZXEYDAYEXEVHZWSWTYCBCDEFXFYCURHAYEXEVIAYEXAXDVJYFXCXBVKZXCFDQZVKZXEA | |
| - YGYEXDYGXAXCXBVLZVMVPAYEXEYIAXEYIYEAXDYIXAXDYGAYIYJAXBYHXCMVNVQVRVRVSVTXQ | |
| - IAYEEFUAXELWAWFWBWCWDWEYDXKGWGWHZSZXLYCYKTYDYLWSWTWSWIXKYDGYCYKXIYCBXJWJW | |
| - KWLXKGYKWMVDWNWOWPWQWOWR $. | |
| - $} | |
| - ${ | |
| - $d k I $. $d f j k G p $. | |
| - umgr2cycl.1 $e |- I = ( iEdg ` G ) $. | |
| - $( A multigraph with two distinct edges that connect the same vertices has | |
| - a 2-cycle. (Contributed by BTernaryTau, 17-Oct-2023.) $) | |
| - umgr2cycl $p |- ( ( G e. UMGraph /\ | |
| - E. j e. dom I E. k e. dom I ( ( I ` j ) = ( I ` k ) /\ j =/= k ) ) -> | |
| - E. f E. p ( f ( Cycles ` G ) p /\ ( # ` f ) = 2 ) ) $= | |
| - ( cumgr wcel cv cfv wceq wa wrex wbr chash c2 wex wal syl wne ccycls wral | |
| - cdm ax-5 alral r19.29 sylan w3a cs2 eqid simp1 simp3r simp3l umgr2cycllem | |
| - simp2 s2len ax-gen 19.29r cvv cword s2cli breq1 fveqeq2 rspcev mpan rexex | |
| - anbi12d eximi excomim 3syl sylancl 3expib rexlimdvw syl5 expd rexlimdv | |
| - imp ) DHIZBJZEKCJZEKLZVTWAUAZMZCEUDZNZBWENAJZFJZDUBKZOZWGPKQLZMZFRARZVSWF | |
| - WMBWEVSVTWEIZWFWMWNWFMWNWDMZCWENZVSWMWNWNCWEUCZWFWPWNWNCSWQWNCUEWNCWEUFTW | |
| - NWDCWEUGUHVSWOWMCWEVSWNWDWMVSWNWDUIZVTWAUJZWHWIOZFRZWSPKQLZFSZWMWRWSDEVTW | |
| - AFWSUKGVSWNWDULVSWNWDUPVSWNWBWCUMVSWNWBWCUNUOXBFVTWAUQURXAXCMWTXBMZFRWLAR | |
| - ZFRWMWTXBFUSXDXEFXDWLAUTVAZNZXEWSXFIXDXGVTWAVBWLXDAWSXFWGWSLWJWTWKXBWGWSW | |
| - HWIVCWGWSQPVDVHVEVFWLAXFVGTVIWLFAVJVKVLVMVNVOVPVQVR $. | |
| - $} | |
| $( | |
| @@ -574375,62 +574573,10 @@ $( | |
| -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- | |
| $) | |
| - $c AcyclicGraph $. | |
| - $( Extend class notation with acyclic graphs. $) | |
| - cacycgr $a class AcyclicGraph $. | |
| - ${ | |
| - $d f g p $. | |
| - $( Define the class of all acyclic graphs. A graph is called _acyclic_ if | |
| - it has no (non-trivial) cycles. (Contributed by BTernaryTau, | |
| - 11-Oct-2023.) $) | |
| - df-acycgr $a |- AcyclicGraph = | |
| - { g | -. E. f E. p ( f ( Cycles ` g ) p /\ f =/= (/) ) } $. | |
| - $( An alternate definition of the class of all acyclic graphs that requires | |
| - all cycles to be trivial. (Contributed by BTernaryTau, 11-Oct-2023.) $) | |
| - dfacycgr1 $p |- AcyclicGraph = | |
| - { g | A. f A. p ( f ( Cycles ` g ) p -> f = (/) ) } $= | |
| - ( cacycgr cv ccycls cfv wbr c0 wne wa wex cab wceq wal df-acycgr 2exanali | |
| - wn wi df-ne anbi2i 2exbii xchnxbir abbii eqtri ) DAEZCEBEFGHZUFIJZKZCLALZ | |
| - RZBMUGUFINZSCOAOZBMABCPUKUMBUGULRZKZCLALUMUJUGULACQUIUOACUHUNUGUFITUAUBUC | |
| - UDUE $. | |
| - $} | |
| - ${ | |
| - $d f g G p $. | |
| - $( The property of being an acyclic graph. (Contributed by BTernaryTau, | |
| - 11-Oct-2023.) $) | |
| - isacycgr $p |- ( G e. W -> ( G e. AcyclicGraph <-> | |
| - -. E. f E. p ( f ( Cycles ` G ) p /\ f =/= (/) ) ) ) $= | |
| - ( vg cv ccycls cfv wbr c0 wne wa wex wn cacycgr wceq fveq2 anbi1d 2exbidv | |
| - breqd notbid df-acycgr elab2g ) AFZDFZEFZGHZIZUDJKZLZDMAMZNUDUEBGHZIZUILZ | |
| - DMAMZNEBOCUFBPZUKUOUPUJUNADUPUHUMUIUPUGULUDUEUFBGQTRSUAAEDUBUC $. | |
| - | |
| - $( The property of being an acyclic graph. (Contributed by BTernaryTau, | |
| - 11-Oct-2023.) $) | |
| - isacycgr1 $p |- ( G e. W -> ( G e. AcyclicGraph <-> | |
| - A. f A. p ( f ( Cycles ` G ) p -> f = (/) ) ) ) $= | |
| - ( vg cv ccycls cfv wbr c0 wceq wal cacycgr fveq2 imbi1d 2albidv dfacycgr1 | |
| - wi breqd elab2g ) AFZDFZEFZGHZIZUAJKZRZDLALUAUBBGHZIZUFRZDLALEBMCUCBKZUGU | |
| - JADUKUEUIUFUKUDUHUAUBUCBGNSOPAEDQT $. | |
| - $} | |
| - | |
| - ${ | |
| - $d P f p $. $d f F p $. $d f G p $. | |
| - $( Any cycle in an acyclic graph is trivial (i.e. has one vertex and no | |
| - edges). (Contributed by BTernaryTau, 12-Oct-2023.) $) | |
| - acycgrcycl $p |- ( ( G e. AcyclicGraph /\ F ( Cycles ` G ) P ) -> | |
| - F = (/) ) $= | |
| - ( vf vp cacycgr wcel ccycls cfv wbr c0 wceq wi wa cv cvv cwlks w3a adantl | |
| - wal cycliswlk syl simp2d simp3d breq1 eqeq1 imbi12d breq2 imbi1d sylan9bb | |
| - wlkv isacycgr1 ibi 19.21bbi adantr vtocl2d ex pm2.43d imp ) CFGZBACHIZJZB | |
| - KLZUTVBVCUTVBVBVCMZUTVBNDOZEOZVAJZVEKLZMZVDDEBAPPVBBPGZUTVBCPGZVJAPGZVBBA | |
| - CQIJVKVJVLRABCUAABCUKUBZUCSVBVLUTVBVKVJVLVMUDSVEBLZVIBVFVAJZVCMVFALZVDVNV | |
| - GVOVHVCVEBVFVAUEVEBKUFUGVPVOVBVCVFABVAUHUIUJUTVIVBUTVIDEUTVIETDTDCFEULUMU | |
| - NUOUPUQURUS $. | |
| - $} | |
| ${ | |
| $d f g p $. $d f G p $. $d f V p $. | |
| @@ -574495,58 +574641,9 @@ $) | |
| SUNUOVIVQJKWBJKCVQJEUPBUQURUSVLUTWAVMVSVJVKVLJVAVBVCVFVDTTVG $. | |
| $} | |
| - ${ | |
| - $d x V $. $d x G a $. $d f G p a $. | |
| - acycgrislfgr.1 $e |- V = ( Vtx ` G ) $. | |
| - acycgrislfgr.2 $e |- I = ( iEdg ` G ) $. | |
| - $( An acyclic hypergraph is a loop-free hypergraph. (Contributed by | |
| - BTernaryTau, 15-Oct-2023.) $) | |
| - acycgrislfgr $p |- ( ( G e. AcyclicGraph /\ G e. UHGraph ) -> | |
| - I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) $= | |
| - ( va vf vp wcel cuhgr wa cv chash cfv wbr wex wn wceq 2eximi cacycgr crab | |
| - cdm c2 cle cpw wf csn cedg ccycls c0 wne isacycgr biimpac wi c1 loop1cycl | |
| - cc0 w3a 3simpa biimtrrdi exlimdv cvv vex hash1n0 mpan anim2i con3d adantl | |
| - syl6 mpd wb lfuhgr3 mpbird ) BUAJZBKJZLZCUCUDAMNOUEPADUFUBCUGZGMZUHBUIOJZ | |
| - GQZRZVQHMZIMZBUJOPZWCUKULZLZIQHQZRZWBVPVOWIHBKIUMUNVPWIWBUOVOVPWAWHVPWAWE | |
| - WCNOUPSZLZIQHQZWHVPVTWLGVPVTWEWJURWDOVSSZUSZIQHQWLVSHBIUQWNWKHIWEWJWMUTTV | |
| - AVBWKWGHIWJWFWEWCVCJWJWFHVDWCVCVEVFVGTVJVHVIVKVPVRWBVLVOABCDGEFVMVIVN $. | |
| - $} | |
| - ${ | |
| - $d x G $. | |
| - $( An acyclic pseudograph is a multigraph. (Contributed by BTernaryTau, | |
| - 15-Oct-2023.) $) | |
| - upgracycumgr $p |- ( ( G e. UPGraph /\ G e. AcyclicGraph ) -> | |
| - G e. UMGraph ) $= | |
| - ( vx cupgr wcel cacycgr ciedg cfv cdm c2 cv chash cle wbr cvtx crab cumgr | |
| - cpw wf wa eqid cuhgr anim1ci acycgrislfgr syl umgrislfupgr biimpri syldan | |
| - upgruhgr ) ACDZAEDZAFGZHIBJKGLMBANGZQOUKRZAPDZUIUJSUJAUADZSUMUIUOUJAUHUBB | |
| - AUKULULTZUKTZUCUDUNUIUMSBAUKULUPUQUEUFUG $. | |
| - $} | |
| - ${ | |
| - $d x G $. $d f j k G p $. | |
| - $( An acyclic multigraph is a simple graph. (Contributed by BTernaryTau, | |
| - 17-Oct-2023.) $) | |
| - umgracycusgr $p |- ( ( G e. UMGraph /\ G e. AcyclicGraph ) -> | |
| - G e. USGraph ) $= | |
| - ( vx vj vk vf vp cumgr wcel wa cfv cv chash c2 wceq wne wrex wn wex cc0 | |
| - c0 cacycgr ciedg cdm cvtx cpw crab wf1 cusgr wf umgrf ccycls wbr isacycgr | |
| - eqid biimpa wi umgr2cycl 2ne0 neeq1 mpbiri wb cvv hasheq0 necon3bii sylib | |
| - elv anim2i 2eximi syl con3d adantr dff15 biimpri syl2an2r isusgrs biimprd | |
| - ex mpd ) AGHZAUAHZIZAUBJZUCZBKLJMNBAUDJZUEUFZWBUGZAUHHZVSWCWEWBUIZVTCKZWB | |
| - JDKZWBJNWIWJOIDWCPCWCPZQZWFBWBAWDWDUNZWBUNZUJWAEKZFKAUKJULZWOTOZIZFRERZQZ | |
| - WLVSVTWTEAGFUMUOVSWTWLUPVTVSWKWSVSWKWSVSWKIWPWOLJZMNZIZFRERWSECDAWBFWNUQX | |
| - CWREFXBWQWPXBXASOZWQXBXDMSOURXAMSUSUTXASWOTXASNWOTNVAEWOVBVCVFVDVEVGVHVIV | |
| - QVJVKVRWFWHWLICDWCWEWBVLVMVNVSWFWGUPVTVSWGWFBGWBAWDWMWNVOVPVKVR $. | |
| - $} | |
| - $( An acyclic pseudograph is a simple graph. (Contributed by BTernaryTau, | |
| - 17-Oct-2023.) $) | |
| - upgracycusgr $p |- ( ( G e. UPGraph /\ G e. AcyclicGraph ) -> | |
| - G e. USGraph ) $= | |
| - ( cupgr wcel cacycgr cumgr cusgr upgracycumgr umgracycusgr sylancom ) ABCAD | |
| - CAECAFCAGAHI $. | |
| ${ | |
| $d f G p $. $d f V p $. | |
| @@ -574569,28 +574666,7 @@ $) | |
| $. | |
| $} | |
| - $( A path in an acyclic graph is a simple path. (Contributed by BTernaryTau, | |
| - 21-Oct-2023.) $) | |
| - pthacycspth $p |- ( ( G e. AcyclicGraph /\ F ( Paths ` G ) P ) -> | |
| - F ( SPaths ` G ) P ) $= | |
| - ( cacycgr wcel cpths cfv wbr wa cspths wo ccycls wi c0 cyclispth acycgrcycl | |
| - wceq a1i ex adantr jcad spthcycl simplbi pthisspthorcycl adantl orim2 pm1.2 | |
| - syl6 sylc syl ) CDEZBACFGHZIZBACJGHZUNKZUNUMBACLGHZUNMUNUPKZUOUMUPULBNQZIZU | |
| - NUMUPULURUPULMUMABCORUKUPURMULUKUPURABCPSTUAUSUNUPABCUBUCUHULUQUKABCUDUEUNU | |
| - PUNUFUIUNUGUJ $. | |
| - ${ | |
| - $d S f p $. $d f G p $. | |
| - $( The subgraph of an acyclic graph is also acyclic. (Contributed by | |
| - BTernaryTau, 23-Oct-2023.) $) | |
| - acycgrsubgr $p |- ( ( G e. AcyclicGraph /\ S SubGraph G ) -> | |
| - S e. AcyclicGraph ) $= | |
| - ( vf vp csubgr wbr cacycgr wcel cv ccycls cfv c0 wne wa wex subgrcycl cvv | |
| - wn wb isacycgr anim1d 2eximdv con3d subgrv simpl2im simpld 3imtr4d impcom | |
| - syl ) ABEFZBGHZAGHZUJCIZDIZBJKFZUMLMZNZDOCOZRZUMUNAJKFZUPNZDOCOZRZUKULUJV | |
| - BURUJVAUQCDUJUTUOUPUNAUMBPUAUBUCUJAQHZBQHZUKUSSABUDZCBQDTUEUJVDULVCSUJVDV | |
| - EVFUFCAQDTUIUGUH $. | |
| - $} | |
| $( (End of BTernaryTau's mathbox.) $) | |
| $( End $[ set-mbox-bt.mm $] $) | |
| @@ -874674,6 +874750,4391 @@ $( (End of David A. Wheeler's mathbox.) $) | |
| $( End $[ set-mbox-daw.mm $] $) | |
| +$( Begin $[ set-mbox-my.mm $] $) | |
| +$( Skip $[ set-main.mm $] $) | |
| +$( | |
| +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# | |
| + Mathbox for Mingli Yuan | |
| +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# | |
| +$) | |
| + | |
| + | |
| +$( | |
| +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- | |
| + Trees and spanning trees | |
| +-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- | |
| +$) | |
| + | |
| + $( The standard graph-theoretic background for this section can be found in | |
| + section 1.5 of [Diestel]. $) | |
| + | |
| + $c Tree SpanningTree $. | |
| + | |
| + $( Extend class notation with the class of trees. $) | |
| + ctree $a class Tree $. | |
| + | |
| + $( Extend class notation with the spanning-tree relation. $) | |
| + csptr $a class SpanningTree $. | |
| + | |
| + ${ | |
| + $( Define a tree as a nonempty connected acyclic undirected pseudograph. | |
| + The nonemptiness condition is necessary because the empty graph is a | |
| + connected graph by ~ 0conngr . Since an acyclic undirected pseudograph | |
| + is simple by ~ upgracycusgr , this is equivalent to the standard | |
| + simple-graph formulation, Definition in [Diestel] p. 14. (Contributed | |
| + by Mingli Yuan, 11-Aug-2026.) $) | |
| + df-tree $a |- Tree = { g | ( ( g e. UPGraph /\ g e. ConnGraph /\ | |
| + g e. AcyclicGraph ) /\ ( Vtx ` g ) =/= (/) ) } $. | |
| + $} | |
| + | |
| + ${ | |
| + $d g t $. | |
| + $( Define the relation which holds when the first argument is a spanning | |
| + tree of the second argument. The use of ` SubGraph ` preserves indexed | |
| + edge identities, and equality of the vertex sets makes the subgraph | |
| + spanning. See ~ issubgr for the main characterization of ` SubGraph ` . | |
| + For the standard formulation, see Definition in [Diestel] p. 14. | |
| + (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + df-sptr $a |- SpanningTree = { <. t , g >. | | |
| + ( ( t e. Tree /\ t SubGraph g ) /\ | |
| + ( Vtx ` t ) = ( Vtx ` g ) ) } $. | |
| + $} | |
| + | |
| + ${ | |
| + $d g G $. | |
| + $( The property of being a tree. (Contributed by Mingli Yuan, | |
| + 11-Aug-2026.) $) | |
| + istree $p |- ( G e. W -> ( G e. Tree <-> | |
| + ( ( G e. UPGraph /\ G e. ConnGraph /\ G e. AcyclicGraph ) /\ | |
| + ( Vtx ` G ) =/= (/) ) ) ) $= | |
| + ( vg cv cupgr wcel cconngr cacycgr w3a cvtx cfv c0 wne wa ctree 3anbi123d | |
| + wceq eleq1 fveq2 neeq1d anbi12d df-tree elab2g ) CDEFCDGFCDHFICDJKLMNAEFA | |
| + GFAHFIAJKLMNCAOBCDAQCDEFCDGFCDHFIAEFAGFAHFICDJKLMAJKLMCDAQCDEFAEFCDGFAGFC | |
| + DHFAHFCDAERCDAGRCDAHRPCDAQCDJKAJKLCDAJSTUACUBUC $. | |
| + $} | |
| + | |
| + ${ | |
| + $d g t $. $d g G $. $d g T $. $d t G $. $d t T $. | |
| + $( The property of being a spanning tree. (Contributed by Mingli Yuan, | |
| + 11-Aug-2026.) $) | |
| + issptr $p |- ( ( T e. W /\ G e. U ) -> ( T SpanningTree G <-> | |
| + ( ( T e. Tree /\ T SubGraph G ) /\ | |
| + ( Vtx ` T ) = ( Vtx ` G ) ) ) ) $= | |
| + ( vt vg cv ctree wcel csubgr wbr cvtx cfv wceq csptr simpl anbi12d fveq2d | |
| + wa simpr eleq1d breq12d eqeq12d df-sptr brabga ) EGHIEGFGJKSEGLMFGLMNSAHI | |
| + ACJKSALMCLMNSEFACODBEGANFGCNSEGHIEGFGJKSAHIACJKSEGLMFGLMNALMCLMNEGANFGCNS | |
| + EGHIAHIEGFGJKACJKEGANFGCNSEGAHEGANFGCNPUAEGANFGCNSEGAFGCJEGANFGCNPEGANFGC | |
| + NTUBQEGANFGCNSEGLMALMFGLMCLMEGANFGCNSEGALEGANFGCNPREGANFGCNSFGCLEGANFGCNT | |
| + RUCQEFUDUE $. | |
| + $} | |
| + | |
| + $( The defining properties of a tree. (Contributed by Mingli Yuan, | |
| + 11-Aug-2026.) $) | |
| + treeprop $p |- ( G e. Tree -> | |
| + ( ( G e. UPGraph /\ G e. ConnGraph /\ G e. AcyclicGraph ) /\ | |
| + ( Vtx ` G ) =/= (/) ) ) $= | |
| + ( ctree wcel cupgr cconngr cacycgr w3a cvtx cfv c0 wne wa id wb elex istree | |
| + cvv syl mpbid ) ABCZTADCAECAFCGAHIJKLZTMTAQCTUANABOAQPRS $. | |
| + | |
| + ${ | |
| + $d g t $. | |
| + $( The spanning-tree relation is a relation. (Contributed by Mingli Yuan, | |
| + 11-Aug-2026.) $) | |
| + relsptr $p |- Rel SpanningTree $= | |
| + ( vt vg cv ctree wcel csubgr wbr wa cvtx cfv wceq csptr df-sptr relopabiv | |
| + ) ACZDEOBCZFGHOIJPIJKHABLABMN $. | |
| + $} | |
| + | |
| + $( The defining properties of a spanning tree. (Contributed by Mingli Yuan, | |
| + 11-Aug-2026.) $) | |
| + sptrprop $p |- ( T SpanningTree G -> | |
| + ( ( T e. Tree /\ T SubGraph G ) /\ | |
| + ( Vtx ` T ) = ( Vtx ` G ) ) ) $= | |
| + ( csptr wbr ctree wcel csubgr wa cvtx cfv wceq id cvv wb relsptr brrelex12i | |
| + issptr syl mpbid ) ABCDZTAEFABGDHAIJBIJKHZTLTAMFBMFHTUANABCOPAMBMQRS $. | |
| + | |
| + $( A tree is an undirected pseudograph. (Contributed by Mingli Yuan, | |
| + 11-Aug-2026.) $) | |
| + treeupgr $p |- ( G e. Tree -> G e. UPGraph ) $= | |
| + ( ctree wcel cupgr cconngr cacycgr w3a cvtx cfv c0 wne treeprop simpl1 syl | |
| + wa ) ABCADCZAECZAFCZGAHIJKZOPALPQRSMN $. | |
| + | |
| + $( A tree is connected. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + treeconngr $p |- ( G e. Tree -> G e. ConnGraph ) $= | |
| + ( ctree wcel cupgr cconngr cacycgr w3a cvtx cfv c0 wne treeprop simpl2 syl | |
| + wa ) ABCADCZAECZAFCZGAHIJKZOQALPQRSMN $. | |
| + | |
| + $( A tree is acyclic. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + treeacycgr $p |- ( G e. Tree -> G e. AcyclicGraph ) $= | |
| + ( ctree wcel cupgr cconngr cacycgr w3a cvtx cfv c0 wne treeprop simpl3 syl | |
| + wa ) ABCADCZAECZAFCZGAHIJKZORALPQRSMN $. | |
| + | |
| + $( A tree has a nonempty vertex set. (Contributed by Mingli Yuan, | |
| + 11-Aug-2026.) $) | |
| + treevtxne0 $p |- ( G e. Tree -> ( Vtx ` G ) =/= (/) ) $= | |
| + ( ctree wcel cupgr cconngr cacycgr w3a cvtx cfv c0 wne treeprop simpr syl | |
| + wa ) ABCADCAECAFCGZAHIJKZOQALPQMN $. | |
| + | |
| + $( A tree is a simple graph. (Contributed by Mingli Yuan, | |
| + 11-Aug-2026.) $) | |
| + treeusgr $p |- ( G e. Tree -> G e. USGraph ) $= | |
| + ( ctree wcel cupgr cacycgr cusgr treeupgr treeacycgr jca upgracycusgr syl | |
| + wa ) ABCZADCZAECZLAFCMNOAGAHIAJK $. | |
| + | |
| + $( The edgeless graph on one vertex is a tree. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + sngrtree $p |- ( U e. _V -> <. { U } , (/) >. e. Tree ) $= | |
| + ( cvv wcel csn c0 cop ctree cupgr cconngr cacycgr w3a cvtx cfv wne wa snexg | |
| + syl wceq a1i jca upgr0eop opex 0ex opvtxfvi 1conngr cumgr chash c1 usgr0eop | |
| + snex cusgr usgrumgr hashsng eqcomi acycgr1v snnzg eqnetrrd wb istree mpbird | |
| + 3jca ) ABCZADZEFZGCZVDHCZVDICZVDJCZKZVDLMZENZOZVBVIVKVBVFVGVHVBVCBCZVFABPZV | |
| + CBUAQVBVDBCZVJVCRZOVGVBVOVPVOVBVCEUBSZVPVBEVCAUJUCUDZSTVDABUEQVBVDUFCZVCUGM | |
| + UHRZOVHVBVSVTVBVDUKCZVSVBVMWAVNVCBUIQVDULQABUMTVDVCVJVCVRUNZUOQVAVBVCVJEVCV | |
| + JRVBWBSABUPUQTVBVOVEVLURVQVDBUSQUT $. | |
| + | |
| + $( The edgeless graph on a vertex of a graph is a subgraph of that graph. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + sngrsubgr $p |- ( ( G e. _V /\ U e. ( Vtx ` G ) ) -> | |
| + <. { U } , (/) >. SubGraph G ) $= | |
| + ( cvv wcel cvtx cfv wa csn cop wss ciedg wfun cedg wceq w3a a1i jca 0ex crn | |
| + c0 csubgr wbr simpl opex snex opvtxfvi simpr snssd eqsstrd opiedgfvi funeqd | |
| + fun0 mpbird edgval rneqi rn0 3eqtri 3jca egrsubgr syl ) BCDZABEFZDZGZVAAHZT | |
| + IZCDZGZVFEFZVBJZVFKFZLZVFMFZTNZGZOVFBUAUBVDVHVJVOVDVAVGVAVCUCVGVDVETUDPQVDV | |
| + IVEVBVIVENVDTVEAUEZRUFPVDAVBVAVCUGUHUIVDVLVNVDVLTLZVQVDULPVDVKTVKTNVDTVEVPR | |
| + UJZPUKUMVNVDVMVKSTSTVFUNVKTVRUOUPUQPQURVFCBCUSUT $. | |
| + | |
| + $( A spanning tree is a tree. (Contributed by Mingli Yuan, | |
| + 11-Aug-2026.) $) | |
| + sptrtree $p |- ( T SpanningTree G -> T e. Tree ) $= | |
| + ( csptr wbr ctree wcel csubgr wa cvtx cfv wceq sptrprop simpll syl ) ABCDAE | |
| + FZABGDZHAIJBIJKZHOABLOPQMN $. | |
| + | |
| + $( A spanning tree is a subgraph of the graph it spans. (Contributed by | |
| + Mingli Yuan, 11-Aug-2026.) $) | |
| + sptrsubgr $p |- ( T SpanningTree G -> T SubGraph G ) $= | |
| + ( csptr wbr ctree wcel csubgr wa cvtx cfv wceq sptrprop simplr syl ) ABCDAE | |
| + FZABGDZHAIJBIJKZHPABLOPQMN $. | |
| + | |
| + $( A spanning tree has the same vertices as the graph it spans. | |
| + (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + sptrvtx $p |- ( T SpanningTree G -> | |
| + ( Vtx ` T ) = ( Vtx ` G ) ) $= | |
| + ( csptr wbr ctree wcel csubgr wa cvtx cfv wceq sptrprop simpr syl ) ABCDAEF | |
| + ABGDHZAIJBIJKZHPABLOPMN $. | |
| + | |
| + $( A simple path in a subgraph is also a simple path in the containing | |
| + graph. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + subgrspth $p |- ( S SubGraph G -> | |
| + ( F ( SPaths ` S ) P -> F ( SPaths ` G ) P ) ) $= | |
| + ( csubgr wbr ctrls cfv ccnv wfun wa cspths subgrtrl anim1d isspth 3imtr4g ) | |
| + BDEFZCABGHFZAIJZKCADGHFZSKCABLHFCADLHFQRTSABCDMNACBOACDOP $. | |
| + | |
| + $( A simple path between two vertices in a subgraph is also a simple path | |
| + between those vertices in the containing graph. (Contributed by | |
| + Mingli Yuan, 13-Aug-2026.) $) | |
| + subgrspthon $p |- ( S SubGraph G -> | |
| + ( F ( K ( SPathsOn ` S ) M ) P -> | |
| + F ( K ( SPathsOn ` G ) M ) P ) ) $= | |
| + ( wbr cspthson cfv co wa cspths wceq w3a syl cvv wcel eqid jca wss cc0 sylc | |
| + csubgr chash simpl simpr spthonisspth subgrspth cpthson ctrlson spthonpthon | |
| + cwlkson pthontrlon trlsonwlkon cvtx cwlks wlkonprop simp32 simp33 4syl 3jca | |
| + simpld simprd wb ciedg cpw subgrprop2 simp1 spthonprop simp12 sseldd simp13 | |
| + cedg simp2 isspthonpth mpbird ex ) BDUCGZCAEFBHIJGZCAEFDHIJGZVRVSKZVTCADLIG | |
| + ZUAAIEMZCUDIAIFMZNZWAWBWCWDWAVRCABLIGZWBVRVSUEZWAVSWFVRVSUFZEFACBUGOABCDUHU | |
| + BWAWCWDWAVSWCWDKZWHVSCAEFBUIIJGCAEFBUJIJGZCAEFBULIJGZWIEFACBUKEFACBUMEFACBU | |
| + NWKBPQZEBUOIZQZFWMQZNZCPQAPQKZCABUPIGZWCWDNNZWIEFACBWMWMRZUQWSWCWDWPWQWRWCW | |
| + DURWPWQWRWCWDUSSOUTOZVBWAWCWDXAVCVAWAEDUOIZQZFXBQZKZWQKVTWEVDWAXEWQWAXCXDWA | |
| + WMXBEWAVRWMXBTZWGVRXFBVEIZDVEIZTZBVMIZWMVFTZNXFXBXHBXJDXGWMWTXBRZXGRXHRXJRV | |
| + GXFXIXKVHOOZWAVSWNWHVSWPWQWJWFKZNZWNEFACBWMWTVIZWLWNWOWQXNVJOOVKWAWMXBFXMWA | |
| + VSWOWHVSXOWOXPWLWNWOWQXNVLOOVKSWAVSWQWHVSXOWQXPWPWQXNVNOOSEFACDXBPPXLVOOVPV | |
| + Q $. | |
| + | |
| + $( A simple path has no repeated vertices. (Contributed by Mingli Yuan, | |
| + 11-Aug-2026.) $) | |
| + spthf1 $p |- ( F ( SPaths ` G ) P -> Fun `' P ) $= | |
| + ( cspths cfv wbr ctrls ccnv wfun isspth simprbi ) BACDEFBACGEFAHIABCJK $. | |
| + | |
| + ${ | |
| + $d x y $. $d x A $. $d x W $. $d y A $. $d y W $. | |
| + $( Reversing a word with no repeated symbols preserves the absence of | |
| + repeated symbols. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + revinj $p |- ( ( W e. Word A /\ Fun `' W ) -> | |
| + Fun `' ( reverse ` W ) ) $= | |
| + ( vx vy wcel wa cc0 cfv cvv wf1 cmin wral wceq ralrimiva syl cn0 eleqtrrd | |
| + co 3syl jca cword ccnv wfun chash cfzo creverse c1 cv cmpt weq wi a1i cdm | |
| + fvex simpll simpl simpr wrdf1d cfz simplr cz simplll lencl fzoval eleqtrd | |
| + nn0z fznn0sub2 wrddm f1veqaeq cc nn0cn 1cnd subcld elfzonn0 subcanad syld | |
| + biimpd eqid id oveq2d fveq2d f1mpt sylibr revval eqidd f1eq123d mpbird wf | |
| + df-f1 simprbi ) BAUAZEZBUBUCZFZGBUDHZUERZIBUFHZJZWQUBUCZWNWRWPICWPWOUGKRZ | |
| + CUHZKRZBHZUIZJZWNXCIEZCWPLZXCWTDUHZKRZBHZMZCDUJZUKZDWPLZCWPLZFXEWNXGXOWNX | |
| + FCWPXFWNXAWPEZFZXBBUNULNWNXNCWPXQXMDWPXQXHWPEZFZXKXBXIMZXLXSBUMZABJZXBYAE | |
| + ZXIYAEZFZFXKXTUKXSYBYEXSWNYBWNXPXRUOWNABWLWMUPZWLWMUQUROXSYCYDXSXBWPYAXSX | |
| + BGWTUSRZWPXSXAYGEXBYGEXSXAWPYGWNXPXRUTZXSWOVAEZWPYGMXSWLWOPEZYIWLWMXPXRVB | |
| + ZABVCZWOVFSGWOVDOZVEXAWTVGOYMQXSWLYAWPMYKABVHOZQXSXIWPYAXSXIYGWPXSXHYGEXI | |
| + YGEXSXHWPYGXQXRUQZYMVEXHWTVGOYMQYNQTTYAAXBXIBVIOXSXTXLXSWTXAXHXSWOUGXSWLY | |
| + JWOVJEYKYLWOVKSXSVLVMXSXPXAPEXAVJEYHXAWOVNXAVKSXSXRXHPEXHVJEYOXHWOVNXHVKS | |
| + VOVQVPNNTCDWPIXCXJXDXDVRXLXBXIBXLXAXHWTKXLVSVTWAWBWCWNWPWPIIWQXDWNWLWQXDM | |
| + YFCWKBWDOWNWPWEWNIWEWFWGWRWPIWQWHWSWPIWQWIWJO $. | |
| + $} | |
| + | |
| + ${ | |
| + $d B k $. $d E i k $. $d F i k $. $d G i k $. $d L i k $. | |
| + $d P i k $. | |
| + $d V i k $. $d i k $. | |
| + upgrwlkres.v $e |- V = ( Vtx ` G ) $. | |
| + upgrwlkres.e $e |- E = ( iEdg ` G ) $. | |
| + $( Restricting the edge function of a walk to a set containing all of its | |
| + indexed edges preserves the walk. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + upgrwlkres $p |- ( ( ( G e. UPGraph /\ F ( Walks ` G ) P ) /\ | |
| + ran F C_ B ) -> F ( Walks ` <. V , ( E |` B ) >. ) P ) $= | |
| + ( vk cupgr wcel cfv wa wss cdm co wf wceq 3syl syl cwlks wbr crn cres cop | |
| + cword cc0 chash cfz cv c1 caddc cpr cfzo wral w3a simplr wlkf wrdfn simpr | |
| + wfn cin wrdf frn ssind dmres sseqtrrdi wb df-f mpbir2and iswrdi wlkp wfun | |
| + a1i simpllr fnfun wrddm eleqtrrd jca fvelrn sseldd fvres simpl upgrwlkedg | |
| + r19.21bi eqtrd ralrimiva 3jca simpll upgrspanop cvtx fvexi ciedg opvtxfvi | |
| + resex eqcomi opiedgfvi upgriswlk mpbird ) EJKZDBEUALUBZMZDUCZANZMZDBFCAUD | |
| + ZUEZUALUBZDXFOZUFKZUGDUHLZUIPFBQZIUJZDLZXFLZXMBLXMUKULPBLUMZRZIUGXKUNPZUO | |
| + ZUPZXEXJXLXSXEXRXIDQZXJXEYADXRVAZXCXINZXEXADCOZUFKZYBWTXAXDUQZBDECHURZYDD | |
| + USZSXEXCAYDVBXIXEXCAYDXBXDUTXEXAYEXCYDNZYFYGYEXRYDDQYIYDDVCXRYDDVDTSVECAV | |
| + FVGYAYBYCMVHXEXRXIDVIVNVJXIXKDVKTXEXAXLYFBDEFGVLTXEXQIXRXEXMXRKZMZXOXNCLZ | |
| + XPYKXNAKXOYLRYKXCAXNXBXDYJUQYKDVMZXMDOZKZMXNXCKYKYMYOYKYEYBYMYKXAYEWTXAXD | |
| + YJVOZYGTYHXRDVPSYKXMXRYNXEYJUTYKXAYEYNXRRYPYGYDDVQSVRVSXMDVTTWAXNACWBTXEY | |
| + LXPRZIXRXEXBYQIXRUOXBXDWCBIDECHWDTWEWFWGWHXEXGJKZXHXTVHXEWTYRWTXAXDWIACEF | |
| + GHWJTBIDXGXFFXGWKLFXFFFEWKGWLZCACEWMHWLWOZWNWPXGWMLXFXFFYSYTWQWPWRTWS $. | |
| + $( Restricting the edge function of a path to a set containing all of its | |
| + indexed edges preserves the path. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + upgrpthres $p |- ( ( ( G e. UPGraph /\ F ( Paths ` G ) P ) /\ | |
| + ran F C_ B ) -> F ( Paths ` <. V , ( E |` B ) >. ) P ) $= | |
| + ( cpths cfv wbr wa cres ctrls ccnv wfun cima cwlks syl jca cupgr wcel crn | |
| + wss cop c1 chash cfzo co cc0 cpr cin c0 wceq simpll simplr pthiswlk simpr | |
| + w3a upgrwlkres pthistrl istrl simprbi 3syl wb a1i mpbir2and ispth simp2bi | |
| + simp3bi 3jca mpbird ) EUAUBZDBEIJKZLZDUCAUDZLZDBFCAMUEZIJKZDBVRNJKZBUFDUG | |
| + JZUHUIZMOPZBUJWAUKQBWBQULUMUNZUSZVQVTWCWDVQVTDBVRRJKZDOPZVQVMDBERJKZLZVPL | |
| + WFVQWIVPVQVMWHVMVNVPUOVQVNWHVMVNVPUPZBDEUQSTVOVPURTABCDEFGHUTSVQVNDBENJKZ | |
| + WGWJBDEVAWKWHWGBDEVBVCVDVTWFWGLVEVQBDVRVBVFVGVQVNWCWJVNWKWCWDBDEVHZVISVQV | |
| + NWDWJVNWKWCWDWLVJSVKVSWEVEVQBDVRVHVFVL $. | |
| + | |
| + $( Restricting the edge function of a cycle to a set containing all of its | |
| + indexed edges preserves the cycle. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + upgrcyclres $p |- ( ( ( G e. UPGraph /\ F ( Cycles ` G ) P ) /\ | |
| + ran F C_ B ) -> F ( Cycles ` <. V , ( E |` B ) >. ) P ) $= | |
| + ( cupgr wcel ccycls cfv wbr wa crn wss cpths syl jca iscycl cop cc0 chash | |
| + cres wceq simpll simplr cyclispth simpr upgrpthres simprbi a1i mpbir2and | |
| + wb ) EIJZDBEKLMZNZDOAPZNZDBFCAUDUAZKLMZDBUTQLMZUBBLDUCLBLUEZUSUODBEQLMZNZ | |
| + URNVBUSVEURUSUOVDUOUPURUFUSUPVDUOUPURUGZBDEUHRSUQURUISABCDEFGHUJRUSUPVCVF | |
| + UPVDVCBDETUKRVAVBVCNUNUSBDUTTULUM $. | |
| + $} | |
| + | |
| + ${ | |
| + $d E k $. $d F k $. $d G k $. $d H k $. $d I k $. $d P k $. $d W k $. | |
| + $d k x $. | |
| + upgrvtxreswlk.w $e |- W = ( Vtx ` H ) $. | |
| + upgrvtxreswlk.e $e |- E = ( iEdg ` G ) $. | |
| + upgrvtxreswlk.i $e |- I = ( iEdg ` H ) $. | |
| + $( A walk can be transferred between pseudographs with the same indexed | |
| + edges when all its vertices belong to the target graph. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + upgrvtxreswlk $p |- ( ( ( ( G e. UPGraph /\ H e. UPGraph ) /\ E = I ) /\ | |
| + ( F ( Walks ` G ) P /\ ran P C_ W ) ) -> F ( Walks ` H ) P ) $= | |
| + ( vk cupgr wcel wa wceq cwlks cfv co wf syl wbr crn wss cdm cword cc0 cfz | |
| + chash cv c1 caddc cpr cfzo wral w3a simprl wlkf simplr dmeqd wrdeq eqcomd | |
| + id 3syl eleqtrrd cvtx eqid wlkp simprr fss syl2anc simplll jca upgrwlkedg | |
| + ffrn fveq1d eqeq1d ralbidv mpbird 3jca wb simpllr upgriswlk ) DLMZELMZNZB | |
| + FOZNZCADPQUAZAUBZGUCZNZNZCAEPQUAZCFUDZUEZMZUFCUHQZUGRZGASZKUIZCQZFQZWTAQW | |
| + TUJUKRAQULZOZKUFWQUMRZUNZUOZWLWPWSXFWLCBUDZUEZWOWLWHWHCXIMWGWHWJUPZWHVBAC | |
| + DBIUQVCWLXIWOWLXHWNOXIWOOWLBFWEWFWKURZUSXHWNUTTVAVDWLWRWIASZWJWSWLWHWRDVE | |
| + QZASXLXJACDXMXMVFVGWRXMAVNVCWGWHWJVHWRWIGAVIVJWLXFXABQZXCOZKXEUNZWLWCWHNX | |
| + PWLWCWHWCWDWFWKVKXJVLAKCDBIVMTWLXDXOKXEWLXBXNXCWLXNXBWLXABFXKVOVAVPVQVRVS | |
| + WLWDWMXGVTWCWDWFWKWAAKCEFGHJWBTVR $. | |
| + | |
| + $( A path can be transferred between pseudographs with the same indexed | |
| + edges when all its vertices belong to the target graph. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + upgrvtxrespth $p |- ( ( ( ( G e. UPGraph /\ H e. UPGraph ) /\ E = I ) /\ | |
| + ( F ( Paths ` G ) P /\ ran P C_ W ) ) -> F ( Paths ` H ) P ) $= | |
| + ( cupgr wcel wa wceq cpths cfv wbr ctrls jca 3syl crn wss chash cfzo cres | |
| + c1 co ccnv wfun cc0 cpr cin c0 w3a cwlks simpl simprl pthiswlk syl simprr | |
| + cima upgrvtxreswlk pthistrl istrl simprbi wb mpbird ispth simp2bi simp3bi | |
| + a1i id 3jca ) DKLEKLMBFNMZCADOPQZAUAGUBZMZMZCAEOPQZCAERPQZAUFCUCPZUDUGZUE | |
| + UHUIZAUJWAUKVAAWBVAULUMNZUNZVRVTWCWDVRVTCAEUOPQZCUHUIZMZVRWFWGVRVNCADUOPQ | |
| + ZVPMZMWFVRVNWJVNVQUPVRWIVPVRVOWIVNVOVPUQZACDURUSVNVOVPUTSSABCDEFGHIJVBUSV | |
| + RVOCADRPQZWGWKACDVCWLWIWGACDVDVETSVTWHVFVRACEVDVKVGVRVOVOWCWKVOVLZVOWLWCW | |
| + DACDVHZVITVRVOVOWDWKWMVOWLWCWDWNVJTVMVSWEVFVRACEVHVKVG $. | |
| + | |
| + $( A cycle can be transferred between pseudographs with the same indexed | |
| + edges when all its vertices belong to the target graph. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + upgrvtxrescycl $p |- ( ( ( ( G e. UPGraph /\ H e. UPGraph ) /\ E = I ) /\ | |
| + ( F ( Cycles ` G ) P /\ ran P C_ W ) ) -> F ( Cycles ` H ) P ) $= | |
| + ( cupgr wcel wa wceq ccycls cfv wbr cpths syl jca crn wss cc0 chash simpl | |
| + simprl cyclispth simprr upgrvtxrespth id iscycl simprbi 3syl a1i mpbird | |
| + wb ) DKLEKLMBFNMZCADOPQZAUAGUBZMZMZCAEOPQZCAERPQZUCAPCUDPAPNZMZVAVCVDVAUQ | |
| + CADRPQZUSMZMVCVAUQVGUQUTUEVAVFUSVAURVFUQURUSUFZACDUGSUQURUSUHTTABCDEFGHIJ | |
| + UISVAURURVDVHURUJURVFVDACDUKULUMTVBVEUPVAACEUKUNUO $. | |
| + $} | |
| + | |
| + ${ | |
| + $d E i $. $d G i $. $d J i $. $d K i $. $d U i $. $d V i $. | |
| + $d ph i $. | |
| + usgr2iedgvtxdge2d.v $e |- V = ( Vtx ` G ) $. | |
| + usgr2iedgvtxdge2d.e $e |- E = ( iEdg ` G ) $. | |
| + usgr2iedgvtxdge2d.g $e |- ( ph -> G e. USGraph ) $. | |
| + usgr2iedgvtxdge2d.u $e |- ( ph -> U e. V ) $. | |
| + usgr2iedgvtxdge2d.j $e |- ( ph -> J e. dom E ) $. | |
| + usgr2iedgvtxdge2d.k $e |- ( ph -> K e. dom E ) $. | |
| + usgr2iedgvtxdge2d.n $e |- ( ph -> J =/= K ) $. | |
| + usgr2iedgvtxdge2d.a $e |- ( ph -> U e. ( E ` J ) ) $. | |
| + usgr2iedgvtxdge2d.b $e |- ( ph -> U e. ( E ` K ) ) $. | |
| + $( Two different indexed edges incident with a vertex of a simple graph | |
| + force its degree to be at least two. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + usgr2iedgvtxdge2d $p |- ( ph -> | |
| + 2 <_ ( ( VtxDeg ` G ) ` U ) ) $= | |
| + ( vi cfv wcel cvv c2 cv cdm crab chash cvtxdg cle ciedg eqid eqeltri dmex | |
| + fvexi rabex a1i wceq id fveq2d eleq2d elrabd nehash2 cusgr wa vtxdusgrval | |
| + jca syl breqtrrd ) AUABQUBZCRZSZQCUCZUDZUERZBDUFRZRZUGAEFVKTVKTSAVIQVJCCD | |
| + UHRZTIVODUHVOUIULUJUKUMUNAVIBECRZSQEVJVGEUOZVHVPBVQVGECVQUPUQURLOUSAVIBFC | |
| + RZSQFVJVGFUOZVHVRBVSVGFCVSUPUQURMPUSNUTADVASZBGSZVBVNVLUOAVTWAJKVDQVJVMBD | |
| + CGHIVJUIVMUIVCVEVF $. | |
| + $} | |
| + | |
| + ${ | |
| + $d E k $. $d F k $. $d G k $. $d P k $. $d ph k $. | |
| + wlkiedgdomd.e $e |- E = ( iEdg ` G ) $. | |
| + wlkiedgdomd.w $e |- ( ph -> F ( Walks ` G ) P ) $. | |
| + wlkiedgdomd.k $e |- ( ph -> K e. ( 0 ..^ ( # ` F ) ) ) $. | |
| + $( An indexed edge used at a position of a walk belongs to the indexed-edge | |
| + domain of the graph. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + wlkiedgdomd $p |- ( ph -> ( F ` K ) e. dom E ) $= | |
| + ( cc0 chash cfv cfzo co cdm cwlks wbr cword wcel wf wlkf wrdf ffvelcdmd | |
| + 3syl ) AJDKLMNZCOZFDADBEPLQDUFRSUEUFDTHBDECGUAUFDUBUDIUC $. | |
| + $} | |
| + | |
| + ${ | |
| + $d E k $. $d F k $. $d G k $. $d K k $. $d P k $. $d ph k $. | |
| + wlkvtxinedgd.e $e |- E = ( iEdg ` G ) $. | |
| + wlkvtxinedgd.w $e |- ( ph -> F ( Walks ` G ) P ) $. | |
| + wlkvtxinedgd.k $e |- ( ph -> K e. ( 0 ..^ ( # ` F ) ) ) $. | |
| + $( Both vertices adjacent at a position of a walk are incident with the | |
| + indexed edge used there. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + wlkvtxinedgd $p |- ( ph -> | |
| + ( ( P ` K ) e. ( E ` ( F ` K ) ) /\ | |
| + ( P ` ( K + 1 ) ) e. ( E ` ( F ` K ) ) ) ) $= | |
| + ( vk cfv wcel c1 caddc co cpr wss fveq2 fvex a1i cv cc0 cfzo wceq fvoveq1 | |
| + chash preq12d eqidd fveq12d sseq12d cwlks wbr wral wlkvtxeledg syl sseldd | |
| + rspcdva prid1 prid2 jca ) AFBKZFDKZCKZLFMNOZBKZVCLAVAVEPZVCVAAJUAZBKZVGMN | |
| + OBKZPZVGDKZCKZQZVFVCQJUBDUFKUCOZFVGFUDZVJVFVLVCVOVHVAVIVEVGFBRVGFMBNUEUGV | |
| + OVKVBCCVOCUHVGFDRUIUJADBEUKKULVMJVNUMHBJDECGUNUOIUQZVAVFLAVAVEFBSURTUPAVF | |
| + VCVEVPVEVFLAVAVEVDBSUSTUPUT $. | |
| + $} | |
| + | |
| + ${ | |
| + crctlenngt2d.c $e |- ( ph -> F ( Circuits ` G ) P ) $. | |
| + crctlenngt2d.l $e |- ( ph -> 2 < ( # ` F ) ) $. | |
| + $( A circuit whose length is greater than two has positive integral | |
| + length. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + crctlenngt2d $p |- ( ph -> ( # ` F ) e. NN ) $= | |
| + ( chash cfv cn wcel cn0 cc0 clt wbr wa ccrcts cwlks c2 cr a1i wlkcl nn0re | |
| + crctiswlk 3syl 0re 2re syl 2pos lttrd jca wb elnnnn0b mpbird ) ACGHZIJZUN | |
| + KJZLUNMNZOZAUPUQACBDPHNCBDQHNUPEBCDUCBCDUAUDZALRUNLSJAUETRSJAUFTAUPUNSJUS | |
| + UNUBUGLRMNAUHTFUIUJUOURUKAUNULTUM $. | |
| + $} | |
| + | |
| + ${ | |
| + crctendidxd.c $e |- ( ph -> F ( Circuits ` G ) P ) $. | |
| + crctendidxd.l $e |- ( ph -> 2 < ( # ` F ) ) $. | |
| + $( The first and last edge positions of a circuit of length greater than | |
| + two are distinct members of its edge-index range. (Contributed by | |
| + Mingli Yuan, 13-Aug-2026.) $) | |
| + crctendidxd $p |- ( ph -> | |
| + ( 0 e. ( 0 ..^ ( # ` F ) ) /\ | |
| + ( ( # ` F ) - 1 ) e. ( 0 ..^ ( # ` F ) ) /\ | |
| + 0 =/= ( ( # ` F ) - 1 ) ) ) $= | |
| + ( cc0 chash cfv cfzo co wcel c1 cmin wne cn crctlenngt2d syl c2 cr lbfzo0 | |
| + sylibr fzo0end cn0 cle wbr nnnn0 2re a1i nnre ltled jca nn0ge2m1nn necomd | |
| + wa nnne0 3jca ) AGGCHIZJKZLZURMNKZUSLZGVAOAURPLZUTABCDEFQZURUAUBAVCVBVDUR | |
| + UCRAVAGAVAPLZVAGOAURUDLZSURUEUFZUOVEAVFVGAVCVFVDURUGRASURSTLAUHUIAVCURTLV | |
| + DURUJRFUKULURUMRVAUPRUNUQ $. | |
| + $} | |
| + | |
| + ${ | |
| + $d E i k $. $d F i k $. $d G i k $. $d P i k $. $d V i k $. | |
| + $d ph i k $. | |
| + crctfstvdge2d.v $e |- V = ( Vtx ` G ) $. | |
| + crctfstvdge2d.e $e |- E = ( iEdg ` G ) $. | |
| + crctfstvdge2d.g $e |- ( ph -> G e. USGraph ) $. | |
| + crctfstvdge2d.c $e |- ( ph -> F ( Circuits ` G ) P ) $. | |
| + crctfstvdge2d.l $e |- ( ph -> 2 < ( # ` F ) ) $. | |
| + $( In a simple graph, the initial vertex of a circuit of length greater | |
| + than two has degree at least two. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + crctfstvdge2d $p |- ( ph -> | |
| + 2 <_ ( ( VtxDeg ` G ) ` ( P ` 0 ) ) ) $= | |
| + ( cc0 cfv c1 co wbr 3syl wcel syl wa chash cmin ccrcts cwlks wf crctiswlk | |
| + cfz wlkp cn cn0 crctlenngt2d nnnn0 0elfz ffvelcdmd cfzo crctendidxd simp1 | |
| + wne w3a wlkiedgdomd simp2 cdm wf1 ctrls crctistrl jca dff14i wlkvtxinedgd | |
| + trlf1 caddc simpl wceq npcan1 fveq2d crctprop simpr eqcomd eqtrd eqeltrrd | |
| + cc nncn usgr2iedgvtxdge2d ) ALBMZCELDMZDUAMZNUBOZDMZFGHIALWEUGOZFLBADBEUC | |
| + MPZDBEUDMPZWHFBUEJBDEUFZBDEFGUHQAWEUIRZWEUJRLWHRABDEJKUKZWEULWEUMQUNABCDE | |
| + LHAWIWJJWKSZALLWEUOOZRZWFWORZLWFURZUSZWPABDEJKUPZWPWQWRUQSZUTABCDEWFHWNAW | |
| + SWQWTWPWQWRVASZUTAWOCVBZDVCZWSTWDWGURAXDWSAWIDBEVDMPZXDJBDEVEBDECHVIQWTVF | |
| + WOXCDLWFVGSAWCWDCMZRZLNVJOBMXFRZTXGABCDELHWNXAVHXGXHVKSAWFNVJOZBMZWCWGCMZ | |
| + AXJWEBMZWCAXIWEBAWLWEVTRXIWEVLWMWEWAWEVMQVNAWCXLAWIXEWCXLVLZTXMJBDEVOXEXM | |
| + VPQVQVRAWFBMXKRZXJXKRZTXOABCDEWFHWNXBVHXNXOVPSVSWB $. | |
| + $} | |
| + | |
| + ${ | |
| + $d E i k $. $d F i k $. $d G i k $. $d L i k $. $d P i k $. | |
| + $d V i k $. $d ph i k $. | |
| + crctintvdge2d.v $e |- V = ( Vtx ` G ) $. | |
| + crctintvdge2d.e $e |- E = ( iEdg ` G ) $. | |
| + crctintvdge2d.g $e |- ( ph -> G e. USGraph ) $. | |
| + crctintvdge2d.c $e |- ( ph -> F ( Circuits ` G ) P ) $. | |
| + crctintvdge2d.b $e |- ( ph -> L e. ( 1 ..^ ( # ` F ) ) ) $. | |
| + $( In a simple graph, an internal vertex of a circuit has degree at least | |
| + two. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + crctintvdge2d $p |- ( ph -> | |
| + 2 <_ ( ( VtxDeg ` G ) ` ( P ` L ) ) ) $= | |
| + ( cfv c1 cmin co cc0 wbr syl wcel chash cfz wf ccrcts crctiswlk wlkp cfzo | |
| + cwlks fzo0ss1 sseli ffvelcdmd cz elfzoel2 elfzo1elm1fzo0 jca elfzom1elfzo | |
| + elfzofz wa wlkiedgdomd cdm wf1 wne w3a ctrls crctistrl trlf1 elfzoelz zcn | |
| + cc 3syl 1zzd zcnd 0zd ax-1ne0 a1i subneintrd subid1d neeqtrd dff14i caddc | |
| + npcan1 eqcomd fveq2d wlkvtxinedgd simprd eqeltrd simpld usgr2iedgvtxdge2d | |
| + 3jca wceq ) AFBMZCEFNOPZDMZFDMZGHIJAQDUAMZUBPZGFBADBEUHMRZWPGBUCADBEUDMRZ | |
| + WQKBDEUESZBDEGHUFSAFQWOUGPZTZFWPTAFNWOUGPZTZXALXBWTFWOUIUJSZFQWOUQSUKABCD | |
| + EWLIWSAWOULTZWLQWONOPUGPTZURWLWTTZAXEXFAXCXELFNWOUMSAXCXFLFWOUNSUOWLWOUPS | |
| + ZUSABCDEFIWSXDUSAWTCUTZDVAZXGXAWLFVBZVCZURWMWNVBAXJXLADBEVDMRZXJAWRXMKBDE | |
| + VESBDECIVFSAXGXAXKXHXDAWLFQOPFAFNQAXCFULTFVITZLFNWOVGFVHVJZANAVKVLAQAVMVL | |
| + NQVBAVNVOVPAFXOVQVRWIUOWTXIDWLFVSSAWKWLNVTPZBMZWMCMZAFXPBAXPFAXNXPFWJXOFW | |
| + ASWBWCAWLBMXRTXQXRTABCDEWLIWSXHWDWEWFAWKWNCMZTFNVTPBMXSTABCDEFIWSXDWDWGWH | |
| + $. | |
| + $} | |
| + | |
| + ${ | |
| + $d E i k $. $d F i k $. $d G i k $. $d L i k $. $d P i k $. | |
| + $d V i k $. $d ph i k $. | |
| + cyclposvdge2d.v $e |- V = ( Vtx ` G ) $. | |
| + cyclposvdge2d.e $e |- E = ( iEdg ` G ) $. | |
| + cyclposvdge2d.g $e |- ( ph -> G e. USGraph ) $. | |
| + cyclposvdge2d.c $e |- ( ph -> F ( Cycles ` G ) P ) $. | |
| + cyclposvdge2d.n $e |- ( ph -> F =/= (/) ) $. | |
| + cyclposvdge2d.k $e |- ( ph -> L e. ( 0 ... ( # ` F ) ) ) $. | |
| + $( Every vertex position of a nontrivial cycle in a simple graph has degree | |
| + at least two. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + cyclposvdge2d $p |- ( ph -> | |
| + 2 <_ ( ( VtxDeg ` G ) ` ( P ` L ) ) ) $= | |
| + ( cfv c2 wbr wa simpr ex syl c1 chash cfzo cvtxdg cle cusgr adantr ccycls | |
| + co wcel ccrcts simp1 3anidm23 cycliscrct 3syl crctintvdge2d wn cc0 simpll | |
| + wceq wne w3a clt usgrgt2cycl crctfstvdge2d fveq2d breqtrrd ad2antrr cpths | |
| + c0 3jca cyclprop simprd eqcomd eqtrd cfz wo jca elfznelfzo mpjaod pm2.61d | |
| + simpl ) AFUADUBNZUCUIUJZOFBNZEUDNZNZUEPZAWDWHAWDQZBCDEFGHIAEUFUJZWDJUGWIA | |
| + DBEUHNPZDBEUKNPZAWDAAWDWDULUMKBDEUNZUOAWDRUPSAWDUQZWHAWNQZFURUTZWHFWCUTZW | |
| + OWPWHWOWPQZOURBNZWFNZWGUEWRBCDEGHIWRAWJAWNWPUSZJTZWRAWKWLXAKWMUOWRWJWKDVJ | |
| + VAZVBZOWCVCPZWRWJWKXCXBWRAWKXAKTWRAXCXALTVKBDEVDZTVEWRWEWSWFWRFURBWOWPRVF | |
| + VFVGSWOWQWHWOWQQZOWTWGUEXGBCDEGHIXGAWJAWNWQUSZJTZXGAWKWLXHKWMUOXGXDXEXGWJ | |
| + WKXCXIXGAWKXHKTZAXCWNWQLVHVKXFTVEXGWEWSWFXGWEWCBNZWSXGFWCBWOWQRVFXGWSXKXG | |
| + DBEVINPZWSXKUTZXGWKXLXMQXJBDEVLTVMVNVOVFVGSWOFURWCVPUIUJZWNQWPWQVQWOXNWNW | |
| + OAXNAWNWBMTAWNRVRWCFVSTVTSWA $. | |
| + $} | |
| + | |
| + ${ | |
| + $d F f $. $d G f $. $d P f $. | |
| + $( A simple path in an undirected pseudograph, viewed through its vertex | |
| + sequence, is a fixed-length simple-path word. (Contributed by Mingli | |
| + Yuan, 12-Aug-2026.) $) | |
| + spthwspthn $p |- ( ( G e. UPGraph /\ F ( SPaths ` G ) P ) -> | |
| + P e. ( ( # ` F ) WSPathsN G ) ) $= | |
| + ( vf cupgr wcel cspths cfv wbr wa chash cwwspthsn co cwwlksn cv wex cwlks | |
| + wceq syl cvv simpr spthiswlk eqidd jca simpl wlklnwwlkln1 cpths spthispth | |
| + mpd ctrls pthistrl reltrls brrelex1i breq1 spcedv iswspthn a1i mpbir2and | |
| + wi wb ) CEFZBACGHZIZJZABKHZCLMFZAVECNMFZDOZAVBIZDPZVDBACQHIZVEVERZJZVGVDV | |
| + KVLVDVCVKVAVCUAZABCUBSVDVEUCUDVDVAVMVGUSVAVCUEABCVEUFSUIVDVIVCDTBVDVCBTFZ | |
| + VNVCBACUGHIZVOABCUHVPBACUJHZIVOABCUKBAVQCULUMSSSVNVHBAVBUNUOVFVGVJJUTVDDC | |
| + VEAUPUQUR $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G f $. $d P f $. | |
| + $( A word-walk with no repeated vertices in an undirected pseudograph is | |
| + represented by a simple path. (Contributed by Mingli Yuan, | |
| + 11-Aug-2026.) $) | |
| + upgrwwlkdvspth $p |- ( ( G e. UPGraph /\ P e. ( WWalks ` G ) /\ | |
| + Fun `' P ) -> | |
| + E. f f ( SPaths ` G ) P ) $= | |
| + ( cupgr wcel cwwlks cfv ccnv wfun w3a cv cwlks wbr cspths wlkiswwlksupgr2 | |
| + wex imp 3adant3 wa syl simpl simp1 simpr 3jca upgrwlkdvspth ex eximdv mpd | |
| + simp3 ) CDEZACFGEZAHIZJZBKZACLGMZBPZUNACNGMZBPUJUKUPULUJUKUPABCOQRUMUOUQB | |
| + UMUOUQUMUOSZUJUOULJUQURUJUOULURUMUJUMUOUAZUJUKULUBTUMUOUCURUMULUSUJUKULUI | |
| + TUDAUNCUETUFUGUH $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G f $. $d N f $. $d W f $. | |
| + $( A word representing a simple path has no repeated vertices. | |
| + (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wspthnf1 $p |- ( W e. ( N WSPathsN G ) -> Fun `' W ) $= | |
| + ( vf cwwspthsn co wcel cv cspths cfv wbr wex ccnv wfun cn0 cvv wa cwwlksn | |
| + wspthnp simp3d spthf1 exlimiv syl ) CBAEFGDHCAIJKDLCMNCBAEFGBOGAPGQCBARFG | |
| + DHCAIJKDLDABCSTDHCAIJKCMNDCDHAUAUBUC $. | |
| + $} | |
| + | |
| + $( A fixed-length simple-path word is a word over the graph's vertex set. | |
| + (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wspthnwrd $p |- ( W e. ( N WSPathsN G ) -> | |
| + W e. Word ( Vtx ` G ) ) $= | |
| + ( cwwspthsn co wcel cwwlksn cfv cword wspthsswwlkn sseli cn0 chash c1 caddc | |
| + cvtx wceq wwlknbp1 simp2d syl ) CBADEZFCBAGEZFZCAPHIFZUAUBCABJKUCBLFUDCMHBN | |
| + OEQABCRST $. | |
| + | |
| + $( A fixed-length simple-path word is one-to-one on its domain. | |
| + (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wspthnf1m $p |- ( W e. ( N WSPathsN G ) -> | |
| + W : dom W -1-1-> ( Vtx ` G ) ) $= | |
| + ( cwwspthsn co wcel cdm cvtx cfv wf1 wf ccnv wfun cc0 chash cfzo cword wceq | |
| + wspthnwrd syl wrddm eqcomd wrdf feq2dd wspthnf1 wa wb df-f1 a1i mpbir2and ) | |
| + CBADEFZCGZAHIZCJZULUMCKZCLMZUKNCOIPEZULUMCUKCUMQFZUQULRABCSZURULUQUMCUAUBTU | |
| + KURUQUMCKUSUMCUCTUDABCUEUNUOUPUFUGUKULUMCUHUIUJ $. | |
| + | |
| + $( Appending a vertex not occurring in a simple-path word preserves | |
| + injectivity. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wspthnextf1 $p |- ( ( W e. ( N WSPathsN G ) /\ | |
| + S e. ( Vtx ` G ) /\ -. S e. ran W ) -> | |
| + ( W ++ <" S "> ) : dom ( W ++ <" S "> ) -1-1-> ( Vtx ` G ) ) $= | |
| + ( cwwspthsn co wcel cvtx cfv crn w3a cvv 3ad2ant1 3ad2ant2 cdm wf1 cin wceq | |
| + wn c0 cs1 fvex a1i cword wspthnwrd s1cl wspthnf1m id s1f1 s1rn ineq2d simp3 | |
| + csn disjsn sylibr eqtrd ccatf1 ) DCBEFGZABHIZGZADJZGSZKZDAUAZUSLUSLGVCBHUBU | |
| + CURUTDUSUDZGVBBCDUEMUTURVDVEGVBAUSUFNURUTDOUSDPVBBCDUGMUTURVDOUSVDPVBUTUSAU | |
| + TUHUINVCVAVDJZQVAAUMZQZTVCVFVGVAUTURVFVGRVBAUSUJNUKVCVBVHTRURUTVBULVAAUNUOU | |
| + PUQ $. | |
| + | |
| + $( Appending a vertex not occurring in a simple-path word preserves the | |
| + absence of repeated vertices. (Contributed by Mingli Yuan, | |
| + 11-Aug-2026.) $) | |
| + wspthnextfun $p |- ( ( W e. ( N WSPathsN G ) /\ | |
| + S e. ( Vtx ` G ) /\ -. S e. ran W ) -> | |
| + Fun `' ( W ++ <" S "> ) ) $= | |
| + ( cwwspthsn wcel cvtx cfv crn w3a cs1 cconcat cdm wf1 ccnv wfun wspthnextf1 | |
| + co wn wf df-f1 simprbi syl ) DCBERFABGHZFADIFSJDAKLRZMZUDUENZUEOPZABCDQUGUF | |
| + UDUETUHUFUDUEUAUBUC $. | |
| + | |
| + $( A simple-path word can be extended as a walk by an adjacent vertex. | |
| + (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wspthnextwwlk $p |- ( ( W e. ( N WSPathsN G ) /\ | |
| + S e. ( Vtx ` G ) /\ | |
| + { ( lastS ` W ) , S } e. ( Edg ` G ) ) -> | |
| + ( W ++ <" S "> ) e. ( ( N + 1 ) WWalksN G ) ) $= | |
| + ( cwwspthsn co wcel cvtx cfv clsw cpr cedg w3a cwwlksn cs1 cconcat c1 caddc | |
| + wspthsswwlkn eqid sseli 3ad2ant1 simp2 simp3 3jca wwlksnext syl ) DCBEFZGZA | |
| + BHIZGZDJIAKBLIZGZMZDCBNFZGZUKUMMDAOPFCQRFBNFGUNUPUKUMUIUKUPUMUHUODBCSUAUBUI | |
| + UKUMUCUIUKUMUDUEADULBCUJUJTULTUFUG $. | |
| + | |
| + $( An adjacent vertex not occurring in a simple-path word extends it to a | |
| + walk with no repeated vertices. (Contributed by Mingli Yuan, | |
| + 11-Aug-2026.) $) | |
| + wspthnextb $p |- ( ( W e. ( N WSPathsN G ) /\ | |
| + S e. ( Vtx ` G ) /\ | |
| + ( { ( lastS ` W ) , S } e. ( Edg ` G ) /\ | |
| + -. S e. ran W ) ) -> | |
| + ( ( W ++ <" S "> ) e. ( ( N + 1 ) WWalksN G ) /\ | |
| + Fun `' ( W ++ <" S "> ) ) ) $= | |
| + ( cwwspthsn co wcel cvtx cfv clsw cpr cedg crn wn w3a cs1 cconcat 3jca syl | |
| + wa c1 caddc cwwlksn ccnv wfun simp1 simp2 simp3 simpld wspthnextwwlk simprd | |
| + wspthnextfun jca ) DCBEFGZABHIGZDJIAKBLIGZADMGNZTZOZDAPQFZCUAUBFBUCFGZUTUDU | |
| + EZUSUNUOUPOVAUSUNUOUPUNUOURUFZUNUOURUGZUSUPUQUNUOURUHZUIRABCDUJSUSUNUOUQOVB | |
| + USUNUOUQVCVDUSUPUQVEUKRABCDULSUM $. | |
| + | |
| + ${ | |
| + $d G f $. $d S f $. $d W f $. | |
| + $( An adjacent vertex not occurring in a simple path extends that path by | |
| + one edge in an undirected pseudograph. (Contributed by Mingli Yuan, | |
| + 11-Aug-2026.) $) | |
| + wspthnext $p |- ( ( G e. UPGraph /\ | |
| + ( W e. ( N WSPathsN G ) /\ S e. ( Vtx ` G ) /\ | |
| + ( { ( lastS ` W ) , S } e. ( Edg ` G ) /\ -. S e. ran W ) ) ) -> | |
| + ( W ++ <" S "> ) e. ( ( N + 1 ) WSPathsN G ) ) $= | |
| + ( vf cupgr wcel cwwspthsn co cvtx cfv clsw cpr cedg crn wn wa w3a cs1 syl | |
| + cconcat c1 caddc cwwlksn cspths wbr wex ccnv wfun simpr wspthnextb simpld | |
| + cwwlks simpl wwlkswwlksn simprd 3jca upgrwwlkdvspth jca iswspthn sylibr | |
| + cv ) BFGZDCBHIGABJKGDLKAMBNKGADOGPQRZQZDASUAIZCUBUCIZBUDIGZEVBVFBUEKUFEUG | |
| + ZQVFVGBHIGVEVHVIVEVHVFUHUIZVEVDVHVJQVCVDUJABCDUKTZULZVEVCVFBUMKGZVJRVIVEV | |
| + CVMVJVCVDUNVEVHVMVLBVGVFUOTVEVHVJVKUPUQVFEBURTUSEBVGVFUTVA $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G m w $. $d N m w $. $d S m w $. $d W m w $. $d m w $. | |
| + $( A universal upper bound on simple-path lengths applies to a path | |
| + obtained by adjoining one symbol. (Contributed by Mingli Yuan, | |
| + 11-Aug-2026.) $) | |
| + wspthmaxspc $p |- ( A. m A. w | |
| + ( w e. ( m WSPathsN G ) -> m <_ N ) -> | |
| + ( ( W ++ <" S "> ) e. ( ( N + 1 ) WSPathsN G ) -> | |
| + ( N + 1 ) <_ N ) ) $= | |
| + ( c1 caddc co cvv wcel cconcat cv cwwspthsn cle wbr wi wal ovex wceq cs1 | |
| + wa simpr simpl oveq1d eleq12d breq1d imbi12d spc2gv mp2an ) EGHIZJKFBUAZL | |
| + IZJKAMZCMZDNIZKZUOEOPZQZARCRUMUKDNIZKZUKEOPZQZQEGHSFULLSUSVCCAUKUMJJUOUKT | |
| + ZUNUMTZUBZUQVAURVBVFUNUMUPUTVDVEUCVFUOUKDNVDVEUDZUEUFVFUOUKEOVGUGUHUIUJ | |
| + $. | |
| + $} | |
| + | |
| + $( The length parameter of a fixed-length word walk is a nonnegative | |
| + integer. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wwlknn0 $p |- ( W e. ( N WWalksN G ) -> N e. NN0 ) $= | |
| + ( cwwlksn co wcel cn0 cvtx cfv cword chash c1 caddc wceq wwlknbp1 simp1d ) | |
| + CBADEFBGFCAHIJFCKIBLMENABCOP $. | |
| + | |
| + $( The word representing a fixed-length walk has one more symbol than the | |
| + stated length. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wwlknlen $p |- ( W e. ( N WWalksN G ) -> | |
| + ( # ` W ) = ( N + 1 ) ) $= | |
| + ( cwwlksn co wcel cn0 cvtx cfv cword chash c1 caddc wceq wwlknbp1 simp3d ) | |
| + CBADEFBGFCAHIJFCKIBLMENABCOP $. | |
| + | |
| + $( The domain of a word representing a fixed-length walk is the closed | |
| + integer interval from zero through the stated length. (Contributed by | |
| + Mingli Yuan, 12-Aug-2026.) $) | |
| + wwlkndm $p |- ( W e. ( N WWalksN G ) -> | |
| + dom W = ( 0 ... N ) ) $= | |
| + ( cwwlksn co wcel cdm cc0 chash cfv cfzo cfz cvtx cword wceq c1 caddc eqtrd | |
| + cn0 syl wwlknbp1 simp2d wrddm wwlknlen oveq2d cz wwlknn0 nn0z fzval3 eqcomd | |
| + ) CBADEFZCGZHCIJZKEZHBLEZUKCAMJZNFZULUNOUKBSFZUQUMBPQEZOABCUAUBUPCUCTUKUNHU | |
| + SKEZUOUKUMUSHKABCUDUEUKUOUTUKBUFFZUOUTOUKURVAABCUGBUHTHBUITUJRR $. | |
| + | |
| + $( A fixed-length simple-path word is a function on the corresponding | |
| + closed integer interval. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + wspthnfn $p |- ( W e. ( N WSPathsN G ) -> | |
| + W Fn ( 0 ... N ) ) $= | |
| + ( cwwspthsn co wcel cdm wfn cc0 cfz cvtx cfv wf1 wspthnf1m f1fn syl cwwlksn | |
| + wceq wspthsswwlkn sseli wwlkndm fneq2d mpbid ) CBADEZFZCCGZHZCIBJEZHUEUFAKL | |
| + ZCMUGABCNUFUICOPUEUFUHCUECBAQEZFUFUHRUDUJCABSTABCUAPUBUC $. | |
| + | |
| + $( The edge sequence of a fixed-length simple path has the stated length. | |
| + (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wspthnlen $p |- ( ( W e. ( N WWalksN G ) /\ | |
| + F ( SPaths ` G ) W ) -> ( # ` F ) = N ) $= | |
| + ( cwwlksn co wcel cspths cfv wbr wa chash c1 cmin caddc cwlks wceq simpl cc | |
| + syl simpr spthiswlk wlklenvm1 wwlknlen oveq1d cn0 wwlknn0 nn0cn 3syl pncan1 | |
| + 3eqtrd ) DCBEFGADBHIJKALIDLIMNFCMOFMNFCDCBEFGADBHIJKADBPIJALIDLIMNFQDCBEFGA | |
| + DBHIJKADBHIJADBPIJDCBEFGADBHIJUADABUBTDABUCTDCBEFGADBHIJKDLICMOFMNDCBEFGADB | |
| + HIJKDCBEFGDLICMOFQDCBEFGADBHIJRBCDUDTUEDCBEFGADBHIJKCSGCMOFMNFCQDCBEFGADBHI | |
| + JKDCBEFGCUFGCSGDCBEFGADBHIJRBCDUGCUHUICUJTUK $. | |
| + | |
| + ${ | |
| + wspthnleb.v $e |- V = ( Vtx ` G ) $. | |
| + $( A fixed-length simple path has at most as many edges as its graph has | |
| + vertices. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wspthnleb $p |- ( ( W e. ( N WWalksN G ) /\ | |
| + F ( SPaths ` G ) W ) -> | |
| + N <_ ( # ` V ) ) $= | |
| + ( cwwlksn co wcel cspths cfv wbr wa chash wspthnlen cpths simpr spthispth | |
| + cle syl pthhashvtx eqbrtrrd ) ECBGHIZAEBJKLZMZANKZCDNKZSABCEOUEAEBPKLZUFU | |
| + GSLUEUDUHUCUDQEABRTEABDFUATUB $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G f $. $d N f $. $d V f $. $d W f $. | |
| + wspthnle.v $e |- V = ( Vtx ` G ) $. | |
| + $( The length of a simple path represented as a word is bounded by the | |
| + number of graph vertices. (Contributed by Mingli Yuan, | |
| + 11-Aug-2026.) $) | |
| + wspthnle $p |- ( W e. ( N WSPathsN G ) -> | |
| + N <_ ( # ` V ) ) $= | |
| + ( vf cwwspthsn co wcel cv cspths cfv wbr wex chash cle cn0 cvv wa syl w3a | |
| + cwwlksn wspthnp simp3 simp2 wspthnleb sylan ex exlimdv mpd ) DBAGHIZFJZDA | |
| + KLMZFNZBCOLPMZUKBQIARISZDBAUBHIZUNUAZUNFABDUCZUPUQUNUDTUKUMUOFUKUMUOUKUQU | |
| + MUOUKURUQUSUPUQUNUETULABCDEUFUGUHUIUJ $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A f $. $d A w $. $d G f $. $d G w $. | |
| + vtx0wspth.v $e |- V = ( Vtx ` G ) $. | |
| + $( A graph vertex, represented by its singleton word, is a simple path of | |
| + length zero. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + vtx0wspth $p |- ( ( G e. _V /\ A e. V ) -> | |
| + <" A "> e. ( 0 WSPathsN G ) ) $= | |
| + ( vf vw cvv wcel wa cc0 co cv cfv chash c1 wceq a1i syl c0 cfzo cwwspthsn | |
| + cs1 cwwlksn cspths wbr wex cvtx cword crab id fveq2d eqeq1d simpr eleqtrd | |
| + s1cl s1len elrabd wwlksn0s eleqtrrd cfz wrdf csn fz0sn oveq2i fzo01 eqtri | |
| + wf eqtr4i feq2d mpbird wb simpl 0spth 0ex breq1 spcev jca iswspthn sylibr | |
| + ) BGHZACHZIZAUBZJBUCKZHZELZWCBUDMZUEZEUFZIWCJBUAKHWBWEWIWBWCFLZNMZOPZFBUG | |
| + MZUHZUIZWDWBWLWCNMZOPZFWCWNWJWCPZWKWPOWRWJWCNWRUJUKULWBAWMHWCWNHWBACWMVTW | |
| + AUMZCWMPWBDQUNAWMUORWQWBAUPZQUQWDWOPWBFBURQUSWBSWCWGUEZWIWBXAJJUTKZCWCVGZ | |
| + WBXCJWPTKZCWCVGZWBWCCUHHZXEWBWAXFWSACUORCWCVARWBXBXDCWCXBXDPWBXBJVBZXDVCX | |
| + DJOTKXGWPOJTWTVDVEVFVHQVIVJWBVTXAXCVKVTWAVLWCBCGDVMRVJWHXAESVNWFSWCWGVOVP | |
| + RVQEBJWCVRVS $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G f $. $d N f $. $d W f $. | |
| + $( The length parameter of a fixed-length simple path is a nonnegative | |
| + integer. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wspthnn0 $p |- ( W e. ( N WSPathsN G ) -> N e. NN0 ) $= | |
| + ( vf cwwspthsn co cn0 cvv wa cwwlksn cv cspths cfv wbr wex wspthnp simp1d | |
| + wcel simpld ) CBAEFRZBGRZAHRZTUAUBICBAJFRDKCALMNDODABCPQS $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G x $. $d N x $. $d W x $. | |
| + wspthnevtx.v $e |- V = ( Vtx ` G ) $. | |
| + $( The first and last symbols of a fixed-length simple-path word are | |
| + vertices of the graph. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wspthnevtx $p |- ( W e. ( N WSPathsN G ) -> | |
| + ( ( W ` 0 ) e. V /\ ( W ` N ) e. V ) ) $= | |
| + ( cwwspthsn co wcel cwwlksn cc0 cfv wa wspthsswwlkn sseli wwlknllvtx syl | |
| + ) DBAFGZHDBAIGZHJDKCHBDKCHLQRDABMNABCDEOP $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A f $. $d A w $. $d G f $. $d G w $. | |
| + wspth0ne0.v $e |- V = ( Vtx ` G ) $. | |
| + $( If a graph has a vertex, then its set of zero-length simple paths is | |
| + nonempty. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wspth0ne0 $p |- ( ( G e. _V /\ A e. V ) -> | |
| + ( 0 WSPathsN G ) =/= (/) ) $= | |
| + ( cvv wcel wa cc0 cwwspthsn co cs1 vtx0wspth ne0d ) BEFACFGHBIJAKABCDLM | |
| + $. | |
| + $} | |
| + | |
| + ${ | |
| + $d E l $. $d F l $. $d G l $. $d P l $. $d V l $. $d X l $. | |
| + $d ph l $. | |
| + cyclvtxdge2d.v $e |- V = ( Vtx ` G ) $. | |
| + cyclvtxdge2d.e $e |- E = ( iEdg ` G ) $. | |
| + cyclvtxdge2d.g $e |- ( ph -> G e. USGraph ) $. | |
| + cyclvtxdge2d.c $e |- ( ph -> F ( Cycles ` G ) P ) $. | |
| + cyclvtxdge2d.n $e |- ( ph -> F =/= (/) ) $. | |
| + cyclvtxdge2d.x $e |- ( ph -> X e. ran P ) $. | |
| + $( Every vertex occurring in a nontrivial cycle of a simple graph has | |
| + degree at least two. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + cyclvtxdge2d $p |- ( ph -> | |
| + 2 <_ ( ( VtxDeg ` G ) ` X ) ) $= | |
| + ( vl cfv wceq c2 wbr wcel syl cv cvtxdg cle cdm crn wrex wi cc0 chash cfz | |
| + wfun co wf ccycls cwlks cycliswlk wlkp 3syl ffun elrnrexdm wa cusgr simpl | |
| + mpd c0 wne simprl fdm eleqtrd cyclposvdge2d simprr eqcomd fveq2d breqtrd | |
| + rexlimddv ) AGNUAZBOZPZQGEUBOZOZUCRNBUDZAGBUESZVRNWAUFZMABUKZWBWCUGAUHDUI | |
| + OUJULZFBUMZWDADBEUNORZDBEUOORZWFKBDEUPZBDEFHUQZURWEFBUSTNBGUTTVDAVPWASZVR | |
| + VAZVAZQVQVSOVTUCWMBCDEVPFHIWMAEVBSAWLVCZJTWMAWGWNKTWMADVEVFWNLTWMVPWAWEAW | |
| + KVRVGWMWFWAWEPWMAWGWFWNKWGWHWFWIWJTURWEFBVHTVIVJWMVQGVSWMGVQAWKVRVKVLVMVN | |
| + VO $. | |
| + $} | |
| + | |
| + ${ | |
| + revfstd.w $e |- ( ph -> W e. Word A ) $. | |
| + revfstd.n $e |- ( ph -> N e. NN0 ) $. | |
| + revfstd.z $e |- ( ph -> 0 e. ( 0 ..^ ( # ` W ) ) ) $. | |
| + revfstd.l $e |- ( ph -> ( ( # ` W ) - 1 ) = N ) $. | |
| + $( The first symbol of a reversed nonempty word is the last symbol of the | |
| + original word, in deduction form. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + revfstd $p |- ( ph -> ( ( reverse ` W ) ` 0 ) = ( W ` N ) ) $= | |
| + ( cc0 creverse cfv chash c1 cmin co cword wcel syl cc fveq2d cfzo wa wceq | |
| + jca revfv eqcomd cn0 nn0cn eqeltrrd subid1d 3eqtrd ) AIDJKKZDLKZMNOZINOZD | |
| + KZUNDKCDKADBPQZIIUMUAOQZUBULUPUCAUQUREGUDBDIUERAUOUNDAUNACUNSAUNCHUFACUGQ | |
| + CSQFCUHRUIUJTAUNCDHTUK $. | |
| + $} | |
| + | |
| + ${ | |
| + revlstd.w $e |- ( ph -> W e. Word A ) $. | |
| + revlstd.n $e |- ( ph -> N e. NN0 ) $. | |
| + revlstd.b $e |- ( ph -> N e. ( 0 ..^ ( # ` W ) ) ) $. | |
| + revlstd.l $e |- ( ph -> ( ( # ` W ) - 1 ) = N ) $. | |
| + $( The last symbol of a reversed nonempty word is the first symbol of the | |
| + original word, in deduction form. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + revlstd $p |- ( ph -> ( ( reverse ` W ) ` N ) = ( W ` 0 ) ) $= | |
| + ( creverse cfv chash c1 cmin co cc0 cword wcel cfzo syl fveq2d wceq revfv | |
| + wa jca oveq1d cn0 cc nn0cn subidd 3eqtrd ) ACDIJJZDKJZLMNZCMNZDJZCCMNZDJO | |
| + DJADBPQZCOULRNQZUCUKUOUAAUQUREGUDBDCUBSAUNUPDAUMCCMHUETAUPODACACUFQCUGQFC | |
| + UHSUITUJ $. | |
| + $} | |
| + | |
| + ${ | |
| + revspthond.v $e |- V = ( Vtx ` G ) $. | |
| + revspthond.g $e |- ( ph -> G e. UPGraph ) $. | |
| + revspthond.p $e |- ( ph -> F ( K ( SPathsOn ` G ) M ) P ) $. | |
| + $( Reversing a simple path in a pseudograph exchanges its two endpoints. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + revspthond $p |- ( ph -> ( reverse ` F ) | |
| + ( M ( SPathsOn ` G ) K ) ( reverse ` P ) ) $= | |
| + ( cfv co wbr wceq w3a wcel 3syl syl wa cvv creverse cspthson cspths chash | |
| + cc0 cupgr cwlks ccnv wfun spthonisspth spthiswlk revwlk cword wlkpwrd jca | |
| + spthf1 revinj 3jca upgrwlkdvspth cn0 wlkcl cn cfzo caddc wlklenvp1 eqcomd | |
| + c1 nn0p1nn eqeltrrd lbfzo0 cmin wlklenvm1 revfstd cwlkson cpthson ctrlson | |
| + sylibr spthonpthon pthontrlon trlsonwlkon 4syl wlkonprop simp33 eqtrd cdm | |
| + ciedg eqid wlkf revlen fveq2d fzo0end revlstd simp32 3eqtrd wb spthonprop | |
| + simp13 simp12 fvexd isspthonpth mpbird ) ACUAKZBUAKZFEDUBKZLMZXBXCDUCKZMZ | |
| + UEXCKZFNZXBUDKZXCKZENZOZAXGXIXLADUFPZXBXCDUGKZMZXCUHUIZOXGAXNXPXQIACBXOMZ | |
| + XPACBEFXDLMZCBXFMZXRJEFBCDUJZBCDUKQZBCDULRABGUMPZBUHUIZSXQAYCYDAXRYCYBBCD | |
| + GHUNRZAXSXTYDJYABCDUPQUOGBUQRURXCXBDUSRAXHCUDKZBKZFAGYFBYEAXRYFUTPZYBBCDV | |
| + ARZABUDKZVBPZUEUEYJVCLZPAYFVGVDLZYJVBAYJYMAXRYJYMNYBBCDVERVFAYHYMVBPYIYFV | |
| + HRVIZYJVJVQAYFYJVGVKLZAXRYFYONYBBCDVLRVFZVMACBEFDVNKLMZDTPZEGPZFGPZOZCTPB | |
| + TPSZXRUEBKZENZYGFNZOOZUUEAXSCBEFDVOKLMCBEFDVPKLMZYQJEFBCDVREFBCDVSEFBCDVT | |
| + WAZEFBCDGHWBZUUAUUBXRUUDUUEWCQWDAXKYFXCKUUCEAXJYFXCACDWFKZWEZUMPZXJYFNAXR | |
| + UULYBBCDUUJUUJWGWHRUUKCWIRWJAGYFBYEYIAYOYFYLYPAYKYOYLPYNYJWKRVIYPWLAYQUUF | |
| + UUDUUHUUIUUAUUBXRUUDUUEWMQWNURAYTYSSZXBTPZXCTPZSZSXEXMWOAUUMUUPAYTYSAXSUU | |
| + AUUBUUGXTSZOZYTJEFBCDGHWPZYRYSYTUUBUUQWQQAXSUURYSJUUSYRYSYTUUBUUQWRQUOAUU | |
| + NUUOACUAWSABUAWSUOUOFEXCXBDGTTHWTRXA $. | |
| + $} | |
| + | |
| + ${ | |
| + $d E x $. $d G x $. $d V x $. $d W x $. $d ph x $. | |
| + upgrvtxsupd.v $e |- V = ( Vtx ` G ) $. | |
| + upgrvtxsupd.e $e |- E = ( iEdg ` G ) $. | |
| + upgrvtxsupd.g $e |- ( ph -> G e. UPGraph ) $. | |
| + upgrvtxsupd.w $e |- W e. _V $. | |
| + upgrvtxsupd.s $e |- ( ph -> V C_ W ) $. | |
| + $( Enlarging the vertex set of a pseudograph while keeping its indexed | |
| + edges produces a pseudograph. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + upgrvtxsupd $p |- ( ph -> <. W , E >. e. UPGraph ) $= | |
| + ( vx cupgr wcel cfv cpw cdif crab syl cvv ciedg cop cdm chash cle wbr csn | |
| + cv c2 c0 wf upgrf wss sspwd ssdifd rabss2 fssd opex a1i cvtx eqid eqeltri | |
| + wb fvexi opvtxfvi eqcomi opiedgfvi isupgr mpbird ) AEBUAZLMZBUBZKUGUCNUHU | |
| + DUEZKEOZUIUFZPZQZBUJZAVKVLKDOZVNPZQZVPBACLMVKVTBUJHKBCDFGUKRAVSVOULVTVPUL | |
| + AVRVMVNADEJUMUNVLKVSVOUORUPAVISMZVJVQVBWAAEBUQURKSBVIEVIUSNEBEIBCTNZSGWBC | |
| + TWBUTVCVAZVDVEVITNBBEIWCVFVEVGRVH $. | |
| + $} | |
| + | |
| + ${ | |
| + $d E G T V W x $. | |
| + grvtxsup.v $e |- V = ( Vtx ` G ) $. | |
| + grvtxsup.e $e |- E = ( iEdg ` G ) $. | |
| + grvtxsup.w $e |- W e. _V $. | |
| + grvtxsup.t $e |- T = <. W , E >. $. | |
| + $( The vertex set after enlarging only the vertex set of a graph. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grvtxsupvtx $p |- ( Vtx ` T ) = W $= | |
| + ( cvtx cfv cop fveq2i ciedg cvv eqid fvexi eqeltri opvtxfvi eqtri ) AJKEB | |
| + LZJKEAUAJIMBEHBCNKZOGUBCNUBPQRST $. | |
| + | |
| + $( The indexed-edge function is unchanged when only the vertex set is | |
| + enlarged. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grvtxsupiedg $p |- ( iEdg ` T ) = E $= | |
| + ( ciedg cfv cop fveq2i cvv eqid fvexi eqeltri opiedgfvi eqtri ) AJKEBLZJK | |
| + BATJIMBEHBCJKZNGUACJUAOPQRS $. | |
| + | |
| + $( A pseudograph remains a pseudograph after enlarging only its vertex | |
| + set. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grvtxsup $p |- ( ( G e. UPGraph /\ V C_ W ) -> T e. UPGraph ) $= | |
| + ( cupgr wcel wss wa cop simpl simpr upgrvtxsupd wceq a1i eleq1d mpbird ) | |
| + CJKZDELZMZAJKEBNZJKUDBCDEFGUBUCOHUBUCPQUDAUEJAUERUDISTUA $. | |
| + | |
| + $( The old graph is a subgraph of the graph obtained by enlarging only its | |
| + vertex set. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grvtxsupsubgr $p |- ( ( G e. UPGraph /\ V C_ W ) -> | |
| + G SubGraph T ) $= | |
| + ( cupgr wcel wss wa csubgr wbr simpr cvv syl cfv eqcomi ssidd jca wfun wb | |
| + cuhgr w3a grvtxsup elex simpl upgruhgr 3jca cvtx grvtxsupvtx grvtxsupiedg | |
| + uhgrfun ciedg uhgrissubgr mpbird ) CJKZDELZMZCANOZUTBBLZMZVAUTVCUSUTPVABU | |
| + AUBVAAQKZBUCZCUEKZUFVBVDUDVAVEVFVGVAAJKVEABCDEFGHIUGAJUHRVAVGVFVAUSVGUSUT | |
| + UICUJRZBCGUORVHUKEBCABDQFAULSEABCDEFGHIUMTGAUPSBABCDEFGHIUNTUQRUR $. | |
| + | |
| + $( A simple graph remains simple after enlarging only its vertex set. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grvtxsupusgr $p |- ( ( G e. USGraph /\ V C_ W ) -> | |
| + T e. USGraph ) $= | |
| + ( vx cusgr wcel wss wa cfv cpw crab wf1 syl cupgr cdm cv chash wceq simpl | |
| + c2 usgrfs simpr sspw rabss2 3syl f1ss syl2anc wb usgrupgr anim1i grvtxsup | |
| + cvv elex cvtx grvtxsupvtx eqcomi ciedg grvtxsupiedg isusgrs mpbird ) CKLZ | |
| + DEMZNZAKLZBUAZJUBUCOUFUDZJEPZQZBRZVIVKVLJDPZQZBRZVQVNMZVOVIVGVRVGVHUEJBCD | |
| + FGUGSVIVHVPVMMVSVGVHUHDEUIVLJVPVMUJUKVKVQVNBULUMVIAURLZVJVOUNVICTLZVHNATL | |
| + VTVGWAVHCUOUPABCDEFGHIUQATUSUKJURBAEAUTOEABCDEFGHIVAVBAVCOBABCDEFGHIVDVBV | |
| + ESVF $. | |
| + $} | |
| + | |
| + ${ | |
| + $d E x $. $d G x $. $d N x $. $d V x $. | |
| + usgrnewvtxedgnrn.v $e |- V = ( Vtx ` G ) $. | |
| + usgrnewvtxedgnrn.e $e |- E = ( iEdg ` G ) $. | |
| + $( An old edge of a simple graph cannot contain a vertex outside the | |
| + graph. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + usgrnewvtxedgnrn $p |- ( ( G e. USGraph /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) -> { U , N } e/ ran E ) $= | |
| + ( wcel cvv wnel wa w3a crn wn cfv simpl syl simpr eleq2i sylibr cusgr cpr | |
| + cvtx cuhgr cedg simp1 usgruhgr ciedg edgval rneqi eqtr4i 3syl prid2g 3jca | |
| + simp3 uhgredgrnv df-nel bicomi a1d mt2d ex pm2.01d ) CUAHZAEHZDIHZDEJZKZL | |
| + ZADUBZBMZHZNZVIVJJVHVKVHVKVLVHVKKZVKDEHZVMDCUCOZHZVNVMCUDHZVICUEOZHZDVIHZ | |
| + LVPVMVQVSVTVMVCVQVMVHVCVHVKPZVCVDVGUFQCUGQVMVKVSVHVKRVRVJVIVRCUHOZMVJCUIB | |
| + WBGUJUKSTVMVEVTVMVHVGVEWAVCVDVGUOZVEVFPULADIUMQUNVICDUPQEVODFSTVMVNNZVKVM | |
| + VFWDVMVHVFWAVHVGVFWCVEVFRQQVFWDDEUQURTUSUTVAVBVIVJUQT $. | |
| + $} | |
| + | |
| + ${ | |
| + $d E i $. $d G i $. $d N i $. $d V i $. | |
| + uhgrnewvtxiedgnel.v $e |- V = ( Vtx ` G ) $. | |
| + uhgrnewvtxiedgnel.e $e |- E = ( iEdg ` G ) $. | |
| + $( A vertex outside a hypergraph is incident with none of its indexed | |
| + edges. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + uhgrnewvtxiedgnel $p |- ( ( G e. UHGraph /\ -. N e. V ) -> | |
| + -. E. i e. dom E N e. ( E ` i ) ) $= | |
| + ( cuhgr wcel wn wa cv cfv cdm wrex simpr wi simpl w3a syl wss simp1 simp2 | |
| + jca uhgrss simp3 sseldd 3exp rexlimdv con3d mpd ) CHIZDEIZJZKZUNDALZBMZIZ | |
| + ABNZOZJZULUNPUOULUNVAQULUNRULUTUMULURUMAUSULUPUSIZURUMULVBURSZUQEDVCULVBK | |
| + UQEUAVCULVBULVBURUBULVBURUCUDBUPCEFGUETULVBURUFUGUHUIUJTUK $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A x $. $d E x $. $d G x $. $d J x $. $d N x $. | |
| + $d P x $. $d S x $. $d U x $. $d V x $. | |
| + grleaf.v $e |- V = ( Vtx ` G ) $. | |
| + grleaf.e $e |- E = ( iEdg ` G ) $. | |
| + grleaf.a $e |- A = ( V u. { N } ) $. | |
| + grleaf.p $e |- P = ( E u. { <. J , { U , N } >. } ) $. | |
| + grleaf.s $e |- S = <. A , P >. $. | |
| + $( The vertex set of a graph obtained by adjoining a new leaf. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafvtx $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) -> ( Vtx ` S ) = A ) $= | |
| + ( wcel cvv wa cvtx cfv eqeltri cusgr wnel w3a cdm cop wceq fveq2i a1i csn | |
| + cun fvex snex unex cpr ciedg jca opvtxfv syl eqtrd ) FUAODIOHPOHIUBQUCGPO | |
| + GEUDUBQQZCRSZABUEZRSZAVAVCUFUTCVBRNUGUHUTAPOZBPOZQVCAUFUTVDVEVDUTAIHUIZUJ | |
| + PLIVFIFRSPJFRUKTHULUMTUHVEUTBEGDHUNUEZUIZUJPMEVHEFUOSPKFUOUKTVGULUMTUHUPB | |
| + APPUQURUS $. | |
| + | |
| + $( The indexed-edge function of a graph obtained by adjoining a new leaf. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafiedg $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) -> ( iEdg ` S ) = P ) $= | |
| + ( wcel cvv wnel wa ciedg cfv cusgr w3a cdm cop wceq fveq2i cvtx grleafvtx | |
| + a1i fvexd eqeltrrd cpr csn cun fvex eqeltri snex unex jca opiedgfv eqtrd | |
| + syl ) FUAODIOHPOHIQRUBGPOGEUCQRRZCSTZABUDZSTZBVDVFUEVCCVESNUFUIVCAPOZBPOZ | |
| + RVFBUEVCVGVHVCCUGTAPABCDEFGHIJKLMNUHVCCUGUJUKVHVCBEGDHULUDZUMZUNPMEVJEFST | |
| + PKFSUOUPVIUQURUPUIUSBAPPUTVBVA $. | |
| + | |
| + $( The old vertex set is contained in the vertex set obtained by adjoining | |
| + a new leaf. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafvss $p |- V C_ A $= | |
| + ( csn cun ssun1 sseqtrri ) IIHOZPAISQLR $. | |
| + | |
| + $( Old indexed edges remain proper pairs in the enlarged vertex set. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafrnss $p |- ( G e. USGraph -> | |
| + ran E C_ { x e. ~P A | ( # ` x ) = 2 } ) $= | |
| + ( cusgr wcel cpw crab wss crn cv chash cfv c2 wceq cdm wf1 usgrfs f1f syl | |
| + wf frnd grleafvss sspwi rabss2 ax-mp a1i sstrd ) GPQZFUAAUBUCUDUEUFZAJRZS | |
| + ZVAABRZSZUTFUGZVCFUTVFVCFUHVFVCFULAFGJKLUIVFVCFUJUKUMVCVETZUTVBVDTVGJBBCD | |
| + EFGHIJKLMNOUNUOVAAVBVDUPUQURUS $. | |
| + | |
| + $( The edge joining an old vertex to the new vertex is a proper pair in | |
| + the enlarged vertex set. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + grleafnewedg $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) -> | |
| + { U , N } e. { x e. ~P A | ( # ` x ) = 2 } ) $= | |
| + ( wcel cvv wa chash syl cusgr wnel w3a cdm cv cfv c2 wceq cpr cpw fveqeq2 | |
| + a1i wss csn cun ssun1 sseqtrri simpl2 sseldd simpl3l snidg elun2 eleqtrrd | |
| + prex prssd elpwd wne elex simpl3r jca elnelne2 3jca hashprb bicomi sylibr | |
| + elrabd ) GUAPZEJPZIQPZIJUBZRZUCHQPHFUDUBRZRZAUEZSUFUGUHEIUIZSUFUGUHZAWEBU | |
| + JWDWEUGSUKWCWEBQWEQPWCEIVDULWCEIBWCJBEJBUMWCJJIUNZUOZBJWGUPMUQULVQVRWAWBU | |
| + RZUSWCIWHBWCIWGPZIWHPWCVSWJVSVTVQVRWBUTZIQVATIWGJVBTBWHUHWCMULVCVEVFWCEQP | |
| + ZVSEIVGZUCZWFWCWLVSWMWCVRWLWIEJVHTWKWCVRVTRWMWCVRVTWIVSVTVQVRWBVIVJEIJVKT | |
| + VLWNWFEIVMVNVOVP $. | |
| + | |
| + $( The indexed-edge function obtained by adjoining a fresh leaf edge is a | |
| + bijection onto the old range together with that edge. (Contributed by | |
| + Mingli Yuan, 13-Aug-2026.) $) | |
| + grleaff1o $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) -> | |
| + P : ( dom E u. { J } ) -1-1-onto-> | |
| + ( ran E u. { { U , N } } ) ) $= | |
| + ( wcel cvv wnel wa w3a syl cdm crn wf1o cpr csn cun simpl1 usgrf1o simprl | |
| + cusgr prex a1i jca simprr simpl usgrnewvtxedgnrn 3jca f1ounsn ) FUJOZDIOZ | |
| + HPOHIQRZSZGPOZGEUAZQZRZRZVDEUBZEUCZVCDHUDZPOZRZVEVJVHQZRZSVDGUEUFVHVJUEUF | |
| + BUCVGVIVLVNVGUSVIUSUTVAVFUGEFKUHTVGVCVKVBVCVEUIVKVGDHUKULUMVGVEVMVBVCVEUN | |
| + VGVBVMVBVFUODEFHIJKUPTUMUQVDVHBEPPGVJMURT $. | |
| + | |
| + $( The enlarged indexed-edge function is one-to-one into the proper pairs | |
| + of the enlarged vertex set. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + grleaff1 $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) -> | |
| + P : dom P -1-1-> { x e. ~P A | ( # ` x ) = 2 } ) $= | |
| + ( wcel cvv wa wf1 syl cusgr wnel w3a cdm cv chash cfv c2 wceq cpw csn cun | |
| + crab crn cpr wss wf1o grleaff1o f1of1 grleafrnss grleafnewedg snssd unssd | |
| + simpl1 jca f1ss eqidd f1odm eqcomd f1eq123d bicomd mpbird ) GUAPZEJPZIQPI | |
| + JUBRZUCHQPHFUDZUBRZRZCUDZAUEUFUGUHUIABUJUMZCSZVPHUKULZVTCSZVRWBFUNZEIUOZU | |
| + KZULZCSZWGVTUPZRWCVRWHWIVRWBWGCUQZWHBCDEFGHIJKLMNOURZWBWGCUSTVRWDWFVTVRVM | |
| + WDVTUPVMVNVOVQVDABCDEFGHIJKLMNOUTTVRWEVTABCDEFGHIJKLMNOVAVBVCVEWBWGVTCVFT | |
| + VRWCWAVRWBVSVTVTCCVRCVGVRVSWBVRWJVSWBUIWKWBWGCVHTVIVRVTVGVJVKVL $. | |
| + | |
| + $( Adjoining one freshly indexed edge from an old vertex to a new vertex | |
| + preserves simplicity. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + grleafusgr $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) -> S e. USGraph ) $= | |
| + ( vx cusgr wcel cvv wa eqeltri wnel w3a cdm cop wceq a1i cv chash cfv cpw | |
| + c2 crab wf1 grleaff1 wb csn cun cvtx fvex snex unex ciedg pm3.2i isusgrop | |
| + cpr ax-mp sylibr eqeltrd ) FPQDIQHRQHIUASUBGRQGEUCUASSZCABUDZPCVJUEVINUFV | |
| + IBUCOUGUHUIUKUEOAUJULBUMZVJPQZOABCDEFGHIJKLMNUNARQZBRQZSVLVKUOVMVNAIHUPZU | |
| + QRLIVOIFURUIRJFURUSTHUTVATBEGDHVEUDZUPZUQRMEVQEFVBUIRKFVBUSTVPUTVATVCBARR | |
| + OVDVFVGVH $. | |
| + | |
| + $( The original graph is a subgraph of the graph obtained by adjoining a | |
| + new leaf. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafsubgr $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) -> G SubGraph S ) $= | |
| + ( cusgr wcel cvv wa wss syl wnel w3a cdm wbr cvtx cfv ciedg grleafvss a1i | |
| + csubgr grleafvtx sseqtrrd cpr cop csn cun ssun1 sseqtrri grleafiedg cuhgr | |
| + wfun grleafusgr usgruhgr eqid uhgrfun simpl1 3jca uhgrissubgr mpbir2and | |
| + wb ) FOPZDIPZHQPHIUARZUBGQPGEUCUARZRZFCUJUDZICUEUFZSZECUGUFZSZVOIAVQIASVO | |
| + ABCDEFGHIJKLMNUHUIABCDEFGHIJKLMNUKULVOEBVSEBSVOEEGDHUMUNUOZUPBEWAUQMURUIA | |
| + BCDEFGHIJKLMNUSULVOCOPZVSVAZFUTPZUBVPVRVTRVJVOWBWCWDABCDEFGHIJKLMNVBZVOCU | |
| + TPZWCVOWBWFWECVCTVSCVSVDZVETVOVKWDVKVLVMVNVFFVCTVGVQVSFCEIOJVQVDKWGVHTVI | |
| + $. | |
| + | |
| + $( The new vertex has degree one after a fresh leaf is adjoined. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafvtxd1 $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) -> | |
| + ( ( VtxDeg ` S ) ` N ) = 1 ) $= | |
| + ( vx wcel cvv cfv syl wceq cusgr wnel wa w3a cdm cvtxdg cop c1 co cc0 cpr | |
| + cxad ciedg cvtx eqid wfun cuhgr simpl1 usgruhgr uhgrfun csn cun fvex snex | |
| + eqeltri a1i jca opiedgfv funeqd mpbird grleafvtx eqtr4d grleafiedg eqcomd | |
| + unex opvtxfv eqidd uneq12d 3eqtrd simprl simprr dmeqd simpl3l snidg elun2 | |
| + neleq12d eqcomi eleqtrd eleqtrrd cv chash crab grleafnewedg fveqeq2 elrab | |
| + cpw c2 simplbi pweqd prid2g cle cr 2re leidd wne simpl3r elnelne2 hashprg | |
| + simpl2 wb bicomd breqtrrd p1evtxdp1 df-nel sylib uhgrnewvtxiedgnel fveq1d | |
| + wrex wn eleq2d rexeqbidv notbid vtxd0nedgb oveq1d cxr 1xr xaddlid ) FUAPZ | |
| + DIPZHQPZHIUBZUCZUDZGQPZGEUEZUBZUCZUCZHCUFRRHAEUGZUFRZRZUHULUIUJUHULUIZUHY | |
| + RHDHUKZCYSYSUMRZGYSUNRZQUUEUOZUUDUOZYRUUDUPEUPZYRFUQPZUUHYRYHUUIYHYIYLYQU | |
| + RFUSSZEFKUTSYRUUDEYRAQPZEQPZUCZUUDETYRUUKUULUUKYRAIHVAZVBZQLIUUNIFUNRQJFU | |
| + NVCVEHVDVOVEVFUULYREFUMRQKFUMVCVEVFVGZEAQQVHSZVIVJYRCUNRAUUEABCDEFGHIJKLM | |
| + NVKYRUUMUUEATUUPEAQQVPSZVLYRCUMRBEGUUCUGVAZVBZUUDUUSVBABCDEFGHIJKLMNVMBUU | |
| + TTYRMVFYREUUDUUSUUSYRUUDEUUQVNYRUUSVQVRVSYMYNYPVTYRGUUDUEZUBYPYMYNYPWAYRG | |
| + GUVAYOYRGVQYRUUDEUUQWBZWFVJYRHAUUEYRHUUOAYRHUUNPZHUUOPYRYJUVCYJYKYHYIYQWC | |
| + ZHQWDSHUUNIWESUUOATYRAUUOLWGVFWHUURWIZYRUUCAWPZUUEWPYRUUCOWJZWKRWQTZOUVFW | |
| + LPZUUCUVFPZOABCDEFGHIJKLMNWMUVIUVJUUCWKRZWQTZUVHUVLOUUCUVFUVGUUCWQWKWNWOW | |
| + RSYRUUEAUURWSWIYRYJHUUCPUVDDHQWTSYRWQWQUVKXAYRWQWQXBPYRXCVFXDYRUVLDHXEZYR | |
| + YIYKUCUVMYRYIYKYHYIYLYQXIZYJYKYHYIYQXFZVGDHIXGSYRUVMUVLYRYIYJUCUVMUVLXJYR | |
| + YIYJUVNUVDVGDHIQXHSXKVJXLXMYRUUAUJUHULYRUUAUJTZHUVGUUDRZPZOUVAXRZXSZYRUVT | |
| + HUVGERZPZOYOXRZXSZYRUUIHIPXSZUCUWDYRUUIUWEUUJYRYKUWEUVOHIXNXOVGOEFHIJKXPS | |
| + YRUVSUWCYRUVRUWBOUVAYOUVBYRUVQUWAHYRUVGUUDEUUQXQXTYAYBVJYRHUUEPUVPUVTXJUV | |
| + EYTHOYSUUDUUEUUFUUGYTUOYCSVJYDYRUHYEPZUUBUHTUWFYRYFVFUHYGSVS $. | |
| + | |
| + $} | |
| + | |
| + ${ | |
| + $d G n $. $d L n $. $d V n $. | |
| + wspthlen.v $e |- V = ( Vtx ` G ) $. | |
| + wspthlen.l $e |- L = { n e. ( 0 ... ( # ` V ) ) | | |
| + ( n WSPathsN G ) =/= (/) } $. | |
| + $( The set of attainable simple-path lengths bounded by the number of | |
| + graph vertices is finite. (Contributed by Mingli Yuan, | |
| + 11-Aug-2026.) $) | |
| + wspthlenfi $p |- L e. Fin $= | |
| + ( cv cwwspthsn co c0 wne cc0 chash cfv cfz crab cfn wcel fzfi rabfi ax-mp | |
| + eqeltri ) CAGBHIJKZALDMNZOIZPZQFUEQRUFQRLUDSUCAUETUAUB $. | |
| + | |
| + $( Attainable simple-path lengths are real numbers. (Contributed by | |
| + Mingli Yuan, 11-Aug-2026.) $) | |
| + wspthlenssre $p |- L C_ RR $= | |
| + ( cn0 cr cc0 chash cfv cfz co cv cwwspthsn c0 wne ssrab3 fz0ssnn0 sstri | |
| + nn0ssre ) CGHCIDJKZLMZGANBOMPQAUCCFRUBSTUAT $. | |
| + | |
| + $( If the finite graph has a vertex, then its set of attainable | |
| + simple-path lengths is nonempty. (Contributed by Mingli Yuan, | |
| + 11-Aug-2026.) $) | |
| + wspthlenne0 $p |- ( ( G e. _V /\ V e. Fin /\ A e. V ) -> | |
| + L =/= (/) ) $= | |
| + ( cvv wcel cfn w3a cc0 cv cwwspthsn co c0 wne chash wceq syl cfv cfz crab | |
| + oveq1 neeq1d cn0 simp2 hashcl 0elfz simp1 jca wspth0ne0 elrabd eqcomi a1i | |
| + wa simp3 eleqtrd ne0d ) CHIZEJIZAEIZKZDLVCLBMZCNOZPQZBLERUAZUBOZUCZDVCVFL | |
| + CNOZPQZBLVHVDLSVEVJPVDLCNUDUEVCVGUFIZLVHIVCVAVLUTVAVBUGEUHTVGUITVCUTVBUPV | |
| + KVCUTVBUTVAVBUJUTVAVBUQUKACEFULTUMVIDSVCDVIGUNUOURUS $. | |
| + | |
| + ${ | |
| + $d L m $. $d m n $. | |
| + $( A finite graph with a vertex has a greatest attainable simple-path | |
| + length. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wspthlenmax $p |- ( ( G e. _V /\ V e. Fin /\ A e. V ) -> | |
| + E. n e. L A. m e. L m <_ n ) $= | |
| + ( cvv wcel cfn w3a cr wss c0 wne cv cle wbr a1i wspthlenssre wspthlenfi | |
| + wral wrex wspthlenne0 3jca fimaxre syl ) DIJFKJAFJLZEMNZEKJZEOPZLBQCQRS | |
| + BEUCCEUDUIUJUKULUJUICDEFGHUATUKUICDEFGHUBTACDEFGHUEUFCBEUGUH $. | |
| + $} | |
| + | |
| + ${ | |
| + $d N n $. | |
| + $( The length of every simple path in a finite graph belongs to the set | |
| + of attainable bounded lengths. (Contributed by Mingli Yuan, | |
| + 11-Aug-2026.) $) | |
| + wspthlenel $p |- ( ( V e. Fin /\ W e. ( N WSPathsN G ) ) -> | |
| + N e. L ) $= | |
| + ( cfn wcel cwwspthsn co wa cv c0 wne cc0 wceq cn0 syl chash crab neeq1d | |
| + cfv cfz oveq1 cle wbr simpr wspthnn0 hashcl wspthnle elfz2nn0 syl3anbrc | |
| + simpl ne0d elrabd eqcomi a1i eleqtrd ) EIJZFDBKLZJZMZDANZBKLZOPZAQEUAUD | |
| + ZUELZUBZCVDVGVBOPADVIVEDRVFVBOVEDBKUFUCVDDSJZVHSJZDVHUGUHZDVIJVDVCVKVAV | |
| + CUIZBDFUJTVDVAVLVAVCUOEUKTVDVCVMVNBDEFGULTDVHUMUNVDVBFVNUPUQVJCRVDCVJHU | |
| + RUSUT $. | |
| + $} | |
| + | |
| + ${ | |
| + $d N n $. | |
| + $( An attainable length has at least one simple path. (Contributed by | |
| + Mingli Yuan, 11-Aug-2026.) $) | |
| + wspthlenwne0 $p |- ( N e. L -> | |
| + ( N WSPathsN G ) =/= (/) ) $= | |
| + ( wcel cc0 chash cfv cfz co cwwspthsn c0 wne cv wceq oveq1 neeq1d | |
| + elrab2 simprbi ) DCHDIEJKLMZHDBNMZOPZAQZBNMZOPUEADUCCUFDRUGUDOUFDBNSTGU | |
| + AUB $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G w $. $d N n $. $d N w $. | |
| + $( An attainable length is represented by a simple-path word. | |
| + (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wspthlenwex $p |- ( N e. L -> | |
| + E. w w e. ( N WSPathsN G ) ) $= | |
| + ( wcel cwwspthsn co c0 wne cv wex wspthlenwne0 n0 sylib ) EDIECJKZLMANS | |
| + IAOBCDEFGHPASQR $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G k $. $d k n $. $d k V $. | |
| + $( The bound variable in the definition of attainable path lengths can | |
| + be changed. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wspthlenlcv $p |- L = { k e. ( 0 ... ( # ` V ) ) | | |
| + ( k WSPathsN G ) =/= (/) } $= | |
| + ( cv cwwspthsn co c0 wne cc0 chash cfv cfz crab weq oveq1 neeq1d eqtri | |
| + cbvrabv ) DBHZCIJZKLZBMENOPJZQAHZCIJZKLZAUFQGUEUIBAUFBARUDUHKUCUGCISTUB | |
| + UA $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G k $. $d G p $. $d k L $. $d k n $. $d k V $. $d n p $. | |
| + $( An attainable length represented by the defining bound variable has | |
| + a simple-path word. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wspthlenwexn $p |- ( n e. L -> | |
| + E. p p e. ( n WSPathsN G ) ) $= | |
| + ( vk cv wspthlenlcv wspthlenwex ) EHBCAIDFHABCDFGJK $. | |
| + $} | |
| + | |
| + ${ | |
| + $d k L $. $d k M $. $d k N $. $d M n $. | |
| + $( A greatest attainable length bounds the length of every simple path. | |
| + (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wspthlenle $p |- ( ( V e. Fin /\ A. k e. L k <_ N /\ | |
| + W e. ( M WSPathsN G ) ) -> M <_ N ) $= | |
| + ( cfn wcel cv cle wbr wral cwwspthsn wa jca syl co w3a simp1 wspthlenel | |
| + simp3 simp2 breq1 rspcva ) GKLZAMZFNOZADPZHECQUALZUBZEDLZULREFNOZUNUOUL | |
| + UNUIUMRUOUNUIUMUIULUMUCUIULUMUESBCDEGHIJUDTUIULUMUFSUKUPAEDUJEFNUGUHT | |
| + $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G m w $. $d k L $. $d k m w $. $d k N $. $d L m w $. $d m n w $. | |
| + $d m N w $. $d m V w $. | |
| + $( A greatest attainable length bounds all simple paths. | |
| + (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wspthlenallle $p |- ( ( V e. Fin /\ A. k e. L k <_ N ) -> | |
| + A. m A. w ( w e. ( m WSPathsN G ) -> m <_ N ) ) $= | |
| + ( cfn wcel cv cle wbr wral wa cwwspthsn co wi wspthlenle alrimivv | |
| + 3expia ) HKLZBMGNOBFPZQAMZCMZERSLZUGGNOZTCAUDUEUHUIBDEFUGGHUFIJUAUCUB | |
| + $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A n $. $d G m w $. $d k L $. $d k m n w $. $d L m w $. | |
| + $d m n w $. $d m V w $. | |
| + $( A finite graph with a vertex has a maximum simple-path length. | |
| + (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wspthmax $p |- ( ( G e. _V /\ V e. Fin /\ A e. V ) -> | |
| + E. n e. L A. m A. w | |
| + ( w e. ( m WSPathsN G ) -> m <_ n ) ) $= | |
| + ( vk cvv wcel cfn w3a cv cle wbr wal wa syl cwwspthsn co wi simpl simp2 | |
| + wral simprr jca wspthlenallle wspthlenmax reximddv ) EKLZGMLZBGLZNZJODO | |
| + ZPQJFUFZAOCOZEUAUBLURUPPQUCARCRZDFUOUPFLZUQSZSZUMUQSUSVBUMUQVBUOUMUOVAU | |
| + DULUMUNUETUOUTUQUGUHAJCDEFUPGHIUITBJDEFGHIUJUK $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A n p $. $d G m p w $. $d k L m n p w $. $d L m p w $. | |
| + $d m n p w $. $d m V p w $. $d n p $. $d p V w $. | |
| + $( A finite graph with a vertex has a longest simple path represented by | |
| + a word. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wspthmaxw $p |- ( ( G e. _V /\ V e. Fin /\ A e. V ) -> | |
| + E. n e. L E. p ( p e. ( n WSPathsN G ) /\ | |
| + A. m A. w | |
| + ( w e. ( m WSPathsN G ) -> m <_ n ) ) ) $= | |
| + ( cvv wcel cfn cv cwwspthsn co wal wa wex syl w3a cle wbr simprl simprr | |
| + wi wspthlenwexn alrimiv jca 19.29r wspthmax reximddv ) EKLGMLBGLUAZANCN | |
| + ZEOPLUNDNZUBUCUFAQCQZHNUOEOPLZUPRHSZDFUMUOFLZUPRRZUQHSZUPHQZRURUTVAVBUT | |
| + USVAUMUSUPUDDEFGHIJUGTUTUPHUMUSUPUEUHUIUQUPHUJTABCDEFGIJUKUL $. | |
| + $} | |
| + $} | |
| + | |
| + ${ | |
| + $d G m w $. $d N m w $. $d S m w $. $d W m w $. $d m w $. | |
| + $( Every vertex adjacent to the final vertex of a maximum simple path is | |
| + already on the path. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wspthmaxend $p |- ( ( ( G e. UPGraph /\ | |
| + W e. ( N WSPathsN G ) /\ | |
| + A. m A. w ( w e. ( m WSPathsN G ) -> m <_ N ) ) /\ | |
| + ( S e. ( Vtx ` G ) /\ | |
| + { ( lastS ` W ) , S } e. ( Edg ` G ) ) ) -> | |
| + S e. ran W ) $= | |
| + ( wcel cwwspthsn co cv cle wbr wal w3a cfv wa wn jca syl cr cupgr cvtx clsw | |
| + wi cpr cedg crn caddc cconcat simpll3 simpll1 simpll2 simplrl simplrr simpr | |
| + c1 cs1 3jca wspthnext wspthmaxspc sylc ex clt cn0 wspthnn0 nn0re 3syl ltp1d | |
| + simpl2 wb peano2re ltnle mpbid a1d pm2.65d notnotrd ) DUAGZFEDHIGZAJCJZDHIG | |
| + VSEKLUDAMCMZNZBDUBOGZFUCOBUEDUFOGZPZPZBFUGGZWEWFQZEUPUHIZEKLZWEWGWIWEWGPZVT | |
| + FBUQUIIWHDHIGZWIVQVRVTWDWGUJWJVQVRWBWCWGPZNZPWKWJVQWMVQVRVTWDWGUKWJVRWBWLVQ | |
| + VRVTWDWGULWAWBWCWGUMWJWCWGWAWBWCWGUNWEWGUORURRBDEFUSSABCDEFUTVAVBWEWIQZWGWE | |
| + EWHVCLZWNWEEWEVREVDGETGZVQVRVTWDVIDEFVEEVFVGZVHWEWPWHTGZPWOWNVJWEWPWRWQWEWP | |
| + WRWQEVKSREWHVLSVMVNVOVP $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G i $. $d N i $. $d W i $. | |
| + $( The final vertex of a positive-length simple path is adjacent to its | |
| + predecessor. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wspthendpred $p |- ( ( W e. ( N WSPathsN G ) /\ N e. NN ) -> | |
| + { ( lastS ` W ) , ( W ` ( N - 1 ) ) } e. ( Edg ` G ) ) $= | |
| + ( vi cwwspthsn co wcel wa cfv c1 cpr wceq syl eqcomd preq12d caddc fveq2d | |
| + simpr eqid eqeltrd cn clsw cmin cedg cwwlksn simpl cn0 cvv cspths wbr wex | |
| + cv wspthnp simp2d wwlknlsw eqidd prcom a1i eqtrd cc nncn npcan1 cfzo wral | |
| + cc0 cvtx cword chash wwlknp simp3d fzo0end oveq1d eleq1d rspcdv mpd ) CBA | |
| + EFGZBUAGZHZCUBIZBJUCFZCIZKZWABCIZKZAUDIZVRWBWCWAKZWDVRVSWCWAWAVRWCVSVRCBA | |
| + UEFGZWCVSLVRVPWGVPVQUFVPBUGGAUHGHWGDULZCAUIIUJDUKDABCUMUNMZABCUOMNVRWAUPZ | |
| + OWFWDLVRWCWAUQURUSVRWDWAVTJPFZCIZKZWEVRWAWAWCWLWJVRBWKCVRWKBVRBUTGZWKBLVR | |
| + VQWNVPVQRZBVAMBVBMNQOVRWHCIZWHJPFZCIZKZWEGZDVEBVCFZVDZWMWEGZVRWGXBWIWGCAV | |
| + FIZVGGCVHIBJPFLXBDWEABXDCXDSWESVIVJMVRWTXCDVTXAVRVQVTXAGWOBVKMVRWHVTLZHZW | |
| + SWMWEXFWPWAWRWLXFWHVTCVRXERZQXFWQWKCXFWHVTJPXGVLQOVMVNVOTT $. | |
| + $} | |
| + | |
| + ${ | |
| + $d F k $. $d G k $. $d H k $. $d I k $. $d J k $. $d P k $. $d Q k $. | |
| + $d S k $. $d V k $. $d ph k $. | |
| + wlkccat1d.v $e |- V = ( Vtx ` G ) $. | |
| + wlkccat1d.i $e |- I = ( iEdg ` G ) $. | |
| + wlkccat1d.h $e |- H = ( F ++ <" J "> ) $. | |
| + wlkccat1d.q $e |- Q = ( P ++ <" S "> ) $. | |
| + wlkccat1d.g $e |- ( ph -> G e. UPGraph ) $. | |
| + wlkccat1d.w $e |- ( ph -> F ( Walks ` G ) P ) $. | |
| + wlkccat1d.j $e |- ( ph -> J e. dom I ) $. | |
| + wlkccat1d.s $e |- ( ph -> S e. V ) $. | |
| + wlkccat1d.e $e |- ( ph -> ( I ` J ) = | |
| + { ( lastS ` P ) , S } ) $. | |
| + $( Append an indexed edge and its terminal vertex to the two words | |
| + representing a walk. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + wlkccat1d $p |- ( ph -> H ( Walks ` G ) Q ) $= | |
| + ( cfv vk cwlks wbr cdm cword wcel cc0 chash cfz co wf cv c1 cpr wceq cfzo | |
| + caddc wral w3a cs1 cconcat a1i eqcomd wlkf syl ccatws1cl syl2anc eqeltrrd | |
| + wa wlkpwrd jca wrdffz fveq2d wrdlenccats1lenm1 eqtrd wlklenvp1 ccatws1len | |
| + oveq1d 3eqtrd oveq2d feq2d mpbid csn cupgr upgrwlkedg fveq1d adantr simpr | |
| + cmin cun ccats1val1 cn0 wlkcl fzossfzop1 sseldd eleq2d fzonn0p1p1 preq12d | |
| + wss eqeq12d ralbidva clsw eqidd 3jca ccats1val2 wlklenvm1 wne ccatval1lsw | |
| + c0 s1cl wlkn0 ccatws1ls wb id ralunsn mpbird cuz elnn0uz sylib fzosplitsn | |
| + raleqtrrdv upgriswlk ) AGCFUBTZUCZGHUDZUEZUFZUGGUHTZUIUJZJCUKZUAULZGTZHTZ | |
| + YKCTZYKUMUQUJZCTZUNZUOZUAUGYHUPUJZURZUSZAYGYJYTAEIUTVAUJZGYFAGUUBGUUBUOZA | |
| + MVBZVCAEYFUFZIYEUFZUUBYFUFAEBYCUCZUUEPBEFHLVDVEZQYEEIVFVGVHAUGCUHTZUMWIUJ | |
| + ZUIUJZJCUKZYJACJUEZUFUULABDUTZVAUJZCUUMACUUOCUUOUOZANVBZVCABUUMUFZDJUFZVI | |
| + ZUUOUUMUFAUURUUSAUUGUURPBEFJKVJVEZRVKZJBDVFVEVHJCVLVEAUUKYIJCAUUJYHUGUIAU | |
| + UJBUHTZEUHTZUMUQUJZYHAUUJUUOUHTZUMWIUJZUVCAUUIUVFUMWIACUUOUHUUQVMVRAUURUV | |
| + GUVCUOUVADJBVNVEVOAUUGUVCUVEUOZPBEFVPVEZAYHUVEAYHUUBUHTZUVEAGUUBUHUUDVMAU | |
| + UEUVJUVEUOUUHYEEIVQVEVOZVCVSVTWAWBAYRUAUGUVDUPUJZUVDWCWJZYSAYRUAUVMURZYRU | |
| + AUVLURZUVDGTZHTZUVDCTZUVECTZUNZUOZVIZAUVOUWAAYKETZHTZYKBTZYOBTZUNZUOZUAUV | |
| + LURZUVOAFWDUFZUUGVIUWIAUWJUUGOPVKBUAEFHLWEVEAUWHYRUAUVLAYKUVLUFZVIZUWDYMU | |
| + WGYQUWLUWCYLHUWLYLUWCUWLYLYKUUBTZUWCUWLYKGUUBUUCUWLMVBWFUWLUUEUWKVIUWMUWC | |
| + UOUWLUUEUWKAUUEUWKUUHWGAUWKWHZVKIYKYEEWKVEVOVCVMUWLUWEYNUWFYPUWLYNUWEUWLY | |
| + NYKUUOTZUWEUWLYKCUUOUUPUWLNVBZWFUWLUURYKUGUVCUPUJZUFZVIUWOUWEUOUWLUURUWRA | |
| + UURUWKUVAWGZUWLYKUGUVEUPUJZUFUWRUWLUVLUWTYKUWLUVDWLUFZUVLUWTWSAUXAUWKAUUG | |
| + UXAPBEFWMVEZWGUVDWNVEUWNWOUWLUWTUWQYKUWLUVEUVCUGUPUWLUVCUVEAUVHUWKUVIWGVC | |
| + VTZWPWBVKDYKJBWKVEVOVCUWLYPUWFUWLYPYOUUOTZUWFUWLYOCUUOUWPWFUWLUURYOUWQUFZ | |
| + VIUXDUWFUOUWLUURUXEUWSUWLYOUWTUFZUXEUWLUWKUXFUWNYKUVDWQVEUWLUWTUWQYOUXCWP | |
| + WBVKDYOJBWKVEVOVCWRWTXAWBAUVQIHTBXBTZDUNUVTAUVPIHAUVPUVDUUBTZIAUVDGUUBUUD | |
| + WFAUUEUUFUVDUVDUOZUSUXHIUOAUUEUUFUXIUUHQAUVDXCXDIUVDYEEXEVEVOVMSAUXGUVRDU | |
| + VSAUVRUXGAUVRUVDUUOTUVCUMWIUJZUUOTZUXGAUVDCUUOUUQWFAUVDUXJUUOAUUGUVDUXJUO | |
| + PBEFXFVEVMAUURUUNUUMUFZBXIXGZUSUXKUXGUOAUURUXLUXMUVAAUUSUXLRDJXJVEAUUGUXM | |
| + PBEFXKVEXDBUUNJXHVEVSVCAUVSDAUVSUVEUUOTUVCUUOTZDAUVECUUOUUQWFAUVEUVCUUOAU | |
| + VCUVEUVIVCVMAUUTUXNDUOUVBJBDXLVEVSVCWRVSVKAUXAUVNUWBXMUXBYRUWAUAUVLUVDWLY | |
| + KUVDUOZYMUVQYQUVTUXOYLUVPHUXOYKUVDGUXOXNZVMVMUXOYNUVRYPUVSUXOYKUVDCUXPVMU | |
| + XOYOUVECUXOYKUVDUMUQUXPVRVMWRWTXOVEXPAYSUWTUVMAYHUVEUGUPUVKVTAUVDUGXQTUFZ | |
| + UWTUVMUOAUXAUXQUXBUVDXRXSUGUVDXTVEVOYAXDAUWJYDUUAXMOCUAGFHJKLYBVEXP $. | |
| + $} | |
| + | |
| + ${ | |
| + ccatfstlen.q $e |- Q = ( P ++ <" ( P ` 0 ) "> ) $. | |
| + ccatfstlen.r $e |- R = ( ( # ` Q ) - 1 ) $. | |
| + ccatfstlen.p $e |- ( ph -> P e. Word V ) $. | |
| + $( The predecessor of the length of a word closed by its first symbol is | |
| + the original word length. (Contributed by Mingli Yuan, | |
| + 11-Aug-2026.) $) | |
| + ccatfstlen $p |- ( ph -> R = ( # ` P ) ) $= | |
| + ( chash cfv c1 cmin co caddc wceq a1i cc0 cs1 wcel syl cconcat ccatws1len | |
| + fveq2d cword eqtrd oveq1d cc cn0 lencl nn0cn 3syl pncan1 3eqtrd ) ADCIJZK | |
| + LMZBIJZKNMZKLMZUPDUOOAGPAUNUQKLAUNBQBJZRUAMZIJZUQACUTICUTOAFPUCABEUDSZVAU | |
| + QOHEBUSUBTUEUFAUPUGSZURUPOAVBUPUHSVCHEBUIUPUJUKUPULTUM $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A x $. $d E x $. $d G x $. $d J x $. $d N x $. | |
| + $d P x $. $d S x $. $d U x $. $d V x $. | |
| + grleafiedgfv.v $e |- V = ( Vtx ` G ) $. | |
| + grleafiedgfv.e $e |- E = ( iEdg ` G ) $. | |
| + grleafiedgfv.a $e |- A = ( V u. { N } ) $. | |
| + grleafiedgfv.p $e |- P = | |
| + ( E u. { <. J , { U , N } >. } ) $. | |
| + grleafiedgfv.s $e |- S = <. A , P >. $. | |
| + $( The freshly indexed leaf edge has the expected endpoint pair. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafiedgfv $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) -> | |
| + ( ( iEdg ` S ) ` J ) = { U , N } ) $= | |
| + ( wcel cvv wnel wa w3a cfv cusgr cdm ciedg cpr cop grleafiedg fveq1d wceq | |
| + csn cun a1i wn simprl prex simprr df-nel sylib 3jca fsnunfv syl 3eqtrd ) | |
| + FUAODIOHPOHIQRSZGPOZGEUBZQZRRZGCUCTZTGBTGEGDHUDZUEUIUJZTZVHVFGVGBABCDEFGH | |
| + IJKLMNUFUGVFGBVIBVIUHVFMUKUGVFVCVHPOZGVDOULZSVJVHUHVFVCVKVLVBVCVEUMVKVFDH | |
| + UNUKVFVEVLVBVCVEUOGVDUPUQUREPPGVHUSUTVA $. | |
| + | |
| + $( Restricting the enlarged indexed-edge function to the old domain | |
| + recovers the old indexed-edge function. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + grleafiedgres $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) -> | |
| + ( ( iEdg ` S ) |` dom E ) = E ) $= | |
| + ( vx wcel wa cres wceq syl cusgr cvv w3a cdm ciedg cfv cpr cop grleafiedg | |
| + wnel csn cun reseq1d a1i wfn wn simpl1 cv chash c2 cpw crab usgrfs simprr | |
| + wf1 f1fn df-nel sylib jca fsnunres 3eqtrd ) FUAPZDIPZHUBPHIUJQZUCZGUBPZGE | |
| + UDZUJZQZQZCUEUFZVQRBVQREGDHUGZUHUKULZVQRZEVTWABVQABCDEFGHIJKLMNUIUMVTBWCV | |
| + QBWCSVTMUNUMVTEVQUOZGVQPUPZQWDESVTWEWFVTVLWEVLVMVNVSUQVLVQOURUSUFUTSOIVAV | |
| + BZEVEWEOEFIJKVCVQWGEVFTTVTVRWFVOVPVRVDGVQVGVHVIVQEGWBVJTVK $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A l $. $d E l $. $d F l $. $d G l $. $d J l $. | |
| + $d N l $. $d P l $. $d Q l $. $d S l $. $d U l $. $d V l $. | |
| + grleafcyclnran.v $e |- V = ( Vtx ` G ) $. | |
| + grleafcyclnran.e $e |- E = ( iEdg ` G ) $. | |
| + grleafcyclnran.a $e |- A = ( V u. { N } ) $. | |
| + grleafcyclnran.p $e |- P = | |
| + ( E u. { <. J , { U , N } >. } ) $. | |
| + grleafcyclnran.s $e |- S = <. A , P >. $. | |
| + $( A nontrivial cycle in a graph with an adjoined leaf does not contain | |
| + the new leaf vertex. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafcyclnran $p |- ( ( ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) /\ | |
| + F ( Cycles ` S ) Q ) /\ F =/= (/) ) -> | |
| + N e/ ran Q ) $= | |
| + ( wcel wa cfv syl ctree cvv wnel w3a cdm ccycls wbr c0 wne crn wn cle clt | |
| + c2 c1 1lt2 1re 2re ltnlei mpbi a1i cvtxdg ciedg cvtx cusgr simplll simpl1 | |
| + eqid treeusgr simpl2 simpl3 3jca simpr jca grleafusgr simplr cyclvtxdge2d | |
| + simpllr wceq grleafvtxd1 breq2d mpbid ex mtod df-nel sylibr ) HUAQZEKQZJU | |
| + BQJKUCRZUDZIUBQIFUEUCRZRZGCDUFSUGZRZGUHUIZRZJCUJZQZUKJWQUCWPWRUNUOULUGZWS | |
| + UKZWPUOUNUMUGWTUPUOUNUQURUSUTVAWPWRWSWPWRRZUNJDVBSSZULUGWSXACDVCSZGDDVDSZ | |
| + JXDVHXCVHXAHVEQZWHWIUDZWKRZDVEQXAXFWKXAXEWHWIXAWGXEXAWLWGWLWMWOWRVFZWGWHW | |
| + IWKVGTHVITXAWLWHXHWGWHWIWKVJTXAWLWIXHWGWHWIWKVKTVLXAWLWKXHWJWKVMTVNZABDEF | |
| + HIJKLMNOPVOTWLWMWOWRVRWNWOWRVPWPWRVMVQXAXBUOUNULXAXGXBUOVSXIABDEFHIJKLMNO | |
| + PVTTWAWBWCWDJWQWEWF $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A k x $. $d E k x $. $d F k x $. $d G k x $. $d J k x $. | |
| + $d N k x $. $d P k x $. $d Q k x $. $d S k x $. | |
| + $d U k x $. $d V k x $. $d k x $. | |
| + grleafcycloldvtxrn.v $e |- V = ( Vtx ` G ) $. | |
| + grleafcycloldvtxrn.e $e |- E = ( iEdg ` G ) $. | |
| + grleafcycloldvtxrn.a $e |- A = ( V u. { N } ) $. | |
| + grleafcycloldvtxrn.p $e |- P = | |
| + ( E u. { <. J , { U , N } >. } ) $. | |
| + grleafcycloldvtxrn.s $e |- S = <. A , P >. $. | |
| + $( Every vertex of a nontrivial cycle remains in the old vertex set after | |
| + a leaf is adjoined. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + grleafcycloldvtxrn $p |- ( ( ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) /\ | |
| + F ( Cycles ` S ) Q ) /\ F =/= (/) ) -> | |
| + ran Q C_ V ) $= | |
| + ( wcel wa cfv syl cvv wnel w3a cdm ccycls wbr c0 wne crn csn cun cdif wss | |
| + ctree wn cvtx cc0 chash cfz co wf cwlks simplr cycliswlk eqid wlkp eqcomi | |
| + frn wceq a1i cusgr simplll simp1 treeusgr simp2 simp3 3jca simpllr eqcomd | |
| + grleafvtx eqtrd sseqtrrd grleafcyclnran df-nel sylib sylibr difun2 simp3r | |
| + jca ssdifsn simpl simpl3 difsn ) HUNQZEKQZJUAQZJKUBZRZUCZIUAQIFUDUBRZRZGC | |
| + DUESUFZRGUGUHZRZCUIZKJUJZUKZXFULZKXDXEXGUMZJXEQUOZRXEXHUMXDXIXJXDXEDUPSZX | |
| + GXDUQGURSUSUTZXKCVAZXEXKUMXDGCDVBSUFZXMXDXBXNXAXBXCVCZCGDVDTCGDXKXKVEVFTX | |
| + LXKCVHTXDXGAXKXGAVIXDAXGNVGVJXDXKAXDHVKQZWOWRUCZWTRXKAVIXDXQWTXDXPWOWRXDW | |
| + NXPXDWSWNWSWTXBXCVLZWNWOWRVMTZHVNTXDWSWOXRWNWOWRVOTZXDWSWRXRWNWOWRVPTVQWS | |
| + WTXBXCVRZWIABDEFHIJKLMNOPVTTVSWAWBXDJXEUBXJABCDEFGHIJKLMNOPWCJXEWDWEWIXEX | |
| + GJWJWFXDXHKXDXHKXFULZKXHYBVIXDKXFWGVJXDJKQUOZYBKVIXDWQYCXDWNWOWQUCZWTRZWQ | |
| + XDYEXBRYEXDYEXBXDYDWTXDWNWOWQXSXTXDWSWQXRWNWOWPWQWHTVQYAWIXOWIYEXBWKTWNWO | |
| + WQWTWLTJKWDWEJKWMTWAVSWB $. | |
| + $} | |
| + | |
| + ${ | |
| + $d P i j $. $d Q i j $. $d V i j $. $d i j $. $d ph i j $. | |
| + ccatfstpthcond.q $e |- Q = ( P ++ <" ( P ` 0 ) "> ) $. | |
| + ccatfstpthcond.p $e |- ( ph -> P e. Word V ) $. | |
| + ccatfstpthcond.f $e |- ( ph -> Fun `' P ) $. | |
| + $( Closing an injective word by repeating its first symbol preserves the | |
| + vertex-distinctness condition needed for a path. (Contributed by | |
| + Mingli Yuan, 11-Aug-2026.) $) | |
| + ccatfstpthcond $p |- ( ph -> | |
| + A. i e. ( 0 ..^ ( ( # ` P ) + 1 ) ) A. j e. ( 1 ..^ ( # ` P ) ) | |
| + ( i =/= j -> ( Q ` i ) =/= ( Q ` j ) ) ) $= | |
| + ( cfv wi cc0 co wcel wa wceq a1i syl jca 3syl cv wne chash caddc cfzo weq | |
| + c1 cs1 cconcat fveq1d cword simplll simplr ccats1val1 eqtrd simpr fzo0ss1 | |
| + eqcomd simpl simplrr sselid 3eqtrd wf1 wf ccnv wfun wb mpbir2and f1veqaeq | |
| + wrdf df-f1 mpd cvv w3a wrdv fvex 3jca ccats1val2 cn clt wbr elfzo1 biimpi | |
| + exp31 simp2 lbfzo0 sylibr simp1 nnne0 pm2.21ddne simprl cuz lencl elnn0uz | |
| + wo cn0 fzosplitsni mpbid mpjaod necon3d ralrimivva ) ADUAZEUAZUBXBCJZXCCJ | |
| + ZUBKDELBUCJZUGUDMUEMZUGXFUEMZAXBXGNZXCXHNZOZOZXDXEXBXCXLXBLXFUEMZNZXDXEPZ | |
| + DEUFZKXBXFPZXLXNXOXPXLXNOZXOOZXBBJZXCBJZPZXPXSXTXDXEYAXSXDXTXSXDXBBLBJZUH | |
| + UIMZJZXTXSXBCYDCYDPZXSGQZUJXSBFUKNZXNOYEXTPXSYHXNXSAYHAXKXNXOULZHRZXLXNXO | |
| + UMZSYCXBFBUNRUOURXRXOUPXSXEXCYDJZYAXSXCCYDYGUJXSYHXCXMNZOZYLYAPZXSYHYMYJX | |
| + SXHXMXCXFUQZXSXRXJXRXOUSAXIXJXNUTRVAZSYCXCFBUNZRUOVBXSXMFBVCZXNYMOZOYBXPK | |
| + XSYSYTXSYSXMFBVDZBVEVFZXSAUUAYIAYHUUAHFBVJZRRXSAUUBYIIRYSUUAUUBOVGZXSXMFB | |
| + VKZQVHXSXNYMYKYQSSXMFXBXCBVIRVLWDXLXQXOXPXLXQOZXOOZXPXCLUUGLXCUUGYCYAPZLX | |
| + CPZUUGYCXDXEYAUUGXDYCUUGXDYEYCUUGXBCYDYFUUGGQZUJUUGBVMUKNZYCVMNZXQVNYEYCP | |
| + UUGUUKUULXQUUGAYHUUKAXKXQXOULZHFBVOTUULUUGLBVPQXLXQXOUMVQYCXBVMBVRRUOURUU | |
| + FXOUPUUGXEYLYAUUGXCCYDUUJUJUUGYNYOUUGYHYMUUGAYHUUMHRUUGXHXMXCYPUUGUUFXJUU | |
| + FXOUSAXIXJXQUTRZVAZSYRRUOVBUUGYSLXMNZYMOZOUUHUUIKUUGYSUUQUUGYSUUAUUBUUGAY | |
| + HUUAUUMHUUCTUUGAUUBUUMIRUUDUUGUUEQVHUUGUUPYMUUGXFVSNZUUPUUGXJXCVSNZUURXCX | |
| + FVTWAZVNZUURUUNXJUVAXFXCWBWCZUUSUURUUTWETXFWFWGUUOSSXMFLXCBVIRVLURUUGUUSX | |
| + CLUBUUGXJUVAUUSUUNUVBUUSUURUUTWHTXCWIRWJWDXLXIXNXQWOZAXIXJWKXLXFLWLJNZXIU | |
| + VCVGXLYHXFWPNZUVDXLAYHAXKUSHRFBWMUVEUVDXFWNWCTLXFXBWQRWRWSWTXA $. | |
| + $} | |
| + | |
| + ${ | |
| + spthiedgf1d.i $e |- I = ( iEdg ` G ) $. | |
| + spthiedgf1d.p $e |- ( ph -> F ( SPaths ` G ) P ) $. | |
| + $( The edge enumeration of a simple path is one-to-one on its domain. | |
| + (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + spthiedgf1d $p |- ( ph -> | |
| + F : dom F -1-1-> dom I ) $= | |
| + ( cc0 chash cfv cfzo co cdm wf1 ctrls wbr cspths 3syl syl eqidd spthispth | |
| + cpths pthistrl trlf1 cwlks wceq spthiswlk cword wcel wlkf eqcomd f1eq123d | |
| + wrddm mpbid ) AHCIJKLZEMZCNZCMZUPCNACBDOJPZUQACBDQJPZCBDUBJPUSGBCDUABCDUC | |
| + RBCDEFUDSAUOURUPUPCCACTAURUOAUTCBDUEJPZURUOUFZGBCDUGVACUPUHUIVBBCDEFUJUPC | |
| + UMSRUKAUPTULUN $. | |
| + $} | |
| + | |
| + ${ | |
| + $d F i j $. $d G i j $. $d I i j $. $d J i j $. $d P i j $. $d i j $. | |
| + $d ph i j $. | |
| + spthccats1f1d.i $e |- I = ( iEdg ` G ) $. | |
| + spthccats1f1d.p $e |- ( ph -> F ( SPaths ` G ) P ) $. | |
| + spthccats1f1d.j $e |- ( ph -> J e. dom I ) $. | |
| + spthccats1f1d.n $e |- ( ph -> -. J e. ran F ) $. | |
| + $( Appending a fresh edge index to a simple path preserves injectivity of | |
| + the edge enumeration. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + spthccats1f1d $p |- ( ph -> | |
| + ( F ++ <" J "> ) : dom ( F ++ <" J "> ) -1-1-> dom I ) $= | |
| + ( cvv wcel ciedg cfv wbr syl crn cin c0 wceq cs1 fvex eqeltri dmex cspths | |
| + cdm a1i cwlks cword spthiswlk wlkf 3syl s1cl spthiedgf1d s1f1 s1rn ineq2d | |
| + csn wn disjsn sylibr eqtrd ccatf1 ) ACFUAZEUFZKVEKLAEEDMNKGDMUBUCUDUGACBD | |
| + UENOCBDUHNOCVEUIZLHBCDUJBCDEGUKULAFVELZVDVFLIFVEUMPABCDEGHUNAVEFIUOACQZVD | |
| + QZRVHFURZRZSAVIVJVHAVGVIVJTIFVEUPPUQAFVHLUSVKSTJVHFUTVAVBVC $. | |
| + $} | |
| + | |
| + ${ | |
| + wlklswd.w $e |- ( ph -> F ( Walks ` G ) P ) $. | |
| + $( The last symbol of the vertex word of a walk is its final indexed | |
| + vertex. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + wlklswd $p |- ( ph -> | |
| + ( lastS ` P ) = ( P ` ( # ` F ) ) ) $= | |
| + ( clsw cfv chash c1 cmin co caddc cvv wcel wceq w3a cwlks wbr syl fveq2d | |
| + wlkv simp3 lsw wlklenvp1 oveq1d cc cn0 wlkcl nn0cn 3syl pncan1 3eqtrd ) A | |
| + BFGZBHGZIJKZBGZCHGZILKZIJKZBGUQBGABMNZUMUPOADMNZCMNZUTPZUTACBDQGRZVCEBCDU | |
| + ASVAVBUTUBSBMUCSAUOUSBAUNURIJAVDUNUROEBCDUDSUETAUSUQBAUQUFNZUSUQOAVDUQUGN | |
| + VEEBCDUHUQUIUJUQUKSTUL $. | |
| + $} | |
| + | |
| + $( A walk between two vertices has the indicated first and last vertices. | |
| + (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + wlkonendpts $p |- ( F ( K ( WalksOn ` G ) M ) P -> | |
| + ( ( P ` 0 ) = K /\ ( P ` ( # ` F ) ) = M ) ) $= | |
| + ( cwlkson cfv co wbr cvv wcel cvtx w3a wa cwlks cc0 wceq chash eqid syl | |
| + wlkonprop simp3 3simpc ) BADECFGHICJKDCLGZKEUDKMZBJKAJKNZBACOGIZPAGDQZBRGAG | |
| + EQZMZMZUHUINZDEABCUDUDSUAUKUJULUEUFUJUBUGUHUIUCTT $. | |
| + | |
| + ${ | |
| + $d F i j $. $d G i j $. $d H i j $. $d I i j $. $d J i j $. | |
| + $d K i j $. $d M i j $. $d N i j $. $d P i j $. $d Q i j $. | |
| + $d V i j $. $d i j $. $d ph i j $. | |
| + spthccatspthond.v $e |- V = ( Vtx ` G ) $. | |
| + spthccatspthond.i $e |- I = ( iEdg ` G ) $. | |
| + spthccatspthond.h $e |- H = ( F ++ <" J "> ) $. | |
| + spthccatspthond.q $e |- Q = ( P ++ <" N "> ) $. | |
| + spthccatspthond.g $e |- ( ph -> G e. UPGraph ) $. | |
| + spthccatspthond.p $e |- ( ph -> | |
| + F ( K ( SPathsOn ` G ) M ) P ) $. | |
| + spthccatspthond.j $e |- ( ph -> J e. dom I ) $. | |
| + spthccatspthond.e $e |- ( ph -> ( I ` J ) = { M , N } ) $. | |
| + spthccatspthond.f $e |- ( ph -> -. J e. ran F ) $. | |
| + spthccatspthond.n $e |- ( ph -> N e. V ) $. | |
| + spthccatspthond.r $e |- ( ph -> -. N e. ran P ) $. | |
| + $( Extending a simple path by a fresh indexed edge to a new terminal | |
| + vertex produces a simple path with the expected endpoints. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + spthccatspthond $p |- ( ph -> | |
| + H ( K ( SPathsOn ` G ) N ) Q ) $= | |
| + ( cspthson cfv co wbr cspths cc0 wceq chash cupgr wcel cwlks spthonisspth | |
| + w3a ccnv wfun spthiswlk 3syl cpr clsw wlklswd cwlkson cpthson spthonpthon | |
| + ctrlson pthontrlon trlsonwlkon wlkonendpts simpr syl eqtrd eqcomd preq12d | |
| + wa eqidd wlkccat1d cs1 cconcat cwwspthsn crn wn spthwspthn syl2anc eqcomi | |
| + cvtx a1i eleqtrrd wspthnextfun syl3anc cnveqd funeqd mpbird upgrwlkdvspth | |
| + fveq1d cword cfzo wlkpwrd cn c1 caddc wlklenvp1 cn0 wlkcl nn0p1nn eqeltrd | |
| + lbfzo0 biimpri ccats1val1 simpl fveq2d cdm wlkf ccatws1len ccats1val2 cvv | |
| + 3jca wb spthonprop simp12 jca ccatws1clv isspthonpth ) AFCIKEUDUEZUFUGZFC | |
| + EUHUEZUGZUICUEZIUJZFUKUEZCUEZKUJZUPZAYHYJYMAEULUMZFCEUNUEZUGCUQZURZYHQABC | |
| + KDEFGHLMNOPQADBIJYEUFUGZDBYGUGZDBYPUGZRIJBDEUOZBDEUSZUTZSUBAHGUEJKVABVBUE | |
| + ZKVATAJUUEKKAUUEJAUUEDUKUEZBUEZJABDEUUDVCAYSDBIJEVDUEUFUGZUUGJUJZRYSDBIJE | |
| + VEUEUFUGDBIJEVGUEUFUGZUUHIJBDEVFIJBDEVHIJBDEVIUTZUUHUIBUEZIUJZUUIVPZUUIBD | |
| + EIJVJZUUMUUIVKVLUTVMVNAKVQVOVMVRAYRBKVSVTUFZUQZURZABUUFEWAUFUMZKEWGUEZUMK | |
| + BWBUMWCUURAYOYTUUSQAYSYTRUUBVLBDEWDWEAKLUUTUBUUTLUJALUUTMWFWHWIUCKEUUFBWJ | |
| + WKAYQUUQACUUPCUUPUJAPWHZWLWMWNCFEWOWKAYIUULIAYIUIUUPUEZUULAUICUUPUVAWPABL | |
| + WQUMZUIUIBUKUEZWRUFUMZUVBUULUJAYSYTUVCRUUBYTUUAUVCUUCBDELMWSVLUTZAUVDWTUM | |
| + ZUVEAUVDUUFXAXBUFZWTAYSYTUVDUVHUJZRUUBYTUUAUVIUUCBDEXCVLUTZAUUFXDUMZUVHWT | |
| + UMAYSYTUVKRUUBYTUUAUVKUUCBDEXEVLUTUUFXFVLXGUVEUVGUVDXHXIVLKUILBXJWEVMAYSU | |
| + UHUUMRUUKUUHUUNUUMUUOUUMUUIXKVLUTVMAYLYKUUPUEZKAYKCUUPUVAWPAUVLUVDUUPUEZK | |
| + AYKUVDUUPAYKUVHUVDAYKDHVSVTUFZUKUEZUVHAFUVNUKFUVNUJAOWHZXLADGXMZWQUMZUVOU | |
| + VHUJAYSYTUVRRUUBYTUUAUVRUUCBDEGNXNVLUTZUVQDHXOVLVMAUVDUVHUVJVNVMXLAUVCKLU | |
| + MZUVDUVDUJUVMKUJUVFUBAUVDVQKUVDLBXPWKVMVMXRAILUMZUVTVPZFXQWQZUMZCUWCUMZVP | |
| + ZVPYFYNXSAUWBUWFAUWAUVTAYSEXQUMZUWAJLUMZUPDXQUMBXQUMVPZUUJYTVPZUPUWARIJBD | |
| + ELMXTUWGUWAUWHUWIUWJYAUTUBYBAUWDUWEAFUVNUWCUVPAUVRUVNUWCUMUVSUVQDHYCVLXGA | |
| + CUUPUWCUVAAUVCUUPUWCUMUVFLBKYCVLXGYBYBIKCFELUWCUWCMYDVLWN $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A i $. $d E i $. $d F i $. $d G i $. $d J i $. $d M i $. | |
| + $d N i $. $d P i $. $d Q i $. $d S i $. $d U i $. $d V i $. | |
| + grleafspthond.v $e |- V = ( Vtx ` G ) $. | |
| + grleafspthond.e $e |- E = ( iEdg ` G ) $. | |
| + grleafspthond.a $e |- A = ( V u. { N } ) $. | |
| + grleafspthond.p $e |- P = ( E u. { <. J , { U , N } >. } ) $. | |
| + grleafspthond.s $e |- S = <. A , P >. $. | |
| + $( A simple path in the original graph can be extended across the new | |
| + leaf edge. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafspthond $p |- ( ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) /\ | |
| + F ( M ( SPathsOn ` G ) U ) Q ) -> | |
| + ( F ++ <" J "> ) ( M ( SPathsOn ` S ) N ) | |
| + ( Q ++ <" N "> ) ) $= | |
| + ( wcel cfv syl ctree cvv wnel wa w3a cdm cspthson co wbr cs1 cconcat cvtx | |
| + ciedg eqid cusgr cupgr simpll1 treeusgr simpll2 simpll3 simplr grleafusgr | |
| + 3jca jca usgrupgr simpr csubgr wi grleafsubgr subgrspthon mpd csn simplrl | |
| + cun snidg elun2 crn cpr wf1o wceq grleaff1o f1odm grleafiedg eqcomd dmeqd | |
| + eleqtrrd eleqtrd cop fveq1d a1i wn elex simpl prex simplrr df-nel fsnunfv | |
| + biimpi 3eqtrd cc0 chash cfzo cword wf cwlks cspths spthonisspth spthiswlk | |
| + wlkf wrdf frnd ssneldd grleafvtx cfz wlkp spthccatspthond ) HUARZELRZKUBR | |
| + ZKLUCZUDZUEZIUBRZIFUFZUCZUDZUDZGCJEHUGSUHUIZUDZCCKUJUKUHZGDGIUJUKUHZDUMSZ | |
| + IJEKDULSZYMUNYLUNYKUNYJUNYIDUORZDUPRYIHUORZXRYAUEZYFUDZYNYIYPYFYIYOXRYAYI | |
| + XQYOXQXRYAYFYHUQHURTXQXRYAYFYHUSZXQXRYAYFYHUTZVCYBYFYHVAVDZABDEFHIKLMNOPQ | |
| + VBTDVETYIYHGCJEDUGSUHUIZYGYHVFZYIHDVGUIZYHUUAVHYIYQUUCYTABDEFHIKLMNOPQVIT | |
| + CHGDJEVJTVKYIIBUFZYLUFYIIYDIVLZVNZUUDYIIUUERZIUUFRYIYCUUGYBYCYEYHVMZIUBVO | |
| + TIUUEYDVPTYIUUFFVQEKVRZVLVNZBVSZUUDUUFVTYIYQUUKYTABDEFHIKLMNOPQWATUUFUUJB | |
| + WBTWFYIBYLYIYLBYIYQYLBVTYTABDEFHIKLMNOPQWCTZWDWEWGYIIYLSIBSIFIUUIWHVLVNZS | |
| + ZUUIYIIYLBUULWIYIIBUUMBUUMVTYIPWJWIYIYCUUIUBRZIYDRWKZUEUUNUUIVTYIYCUUOUUP | |
| + UUHYIEUBRZXSUDZUUOYIUUQXSYIXRUUQYRELWLTYIYAXSYSXSXTWMTZVDUUOUUREKWNWJTYIY | |
| + EUUPYBYCYEYHWOYEUUPIYDWPWRTZVCFUBUBIUUIWQTWSYIGVQYDIYIWTGXASZXBUHZYDGYIGY | |
| + DXCRZUVBYDGXDYIGCHXESUIZUVCYIGCHXFSUIZUVDYIYHUVEUUBJECGHXGTCGHXHTZCGHFNXI | |
| + TYDGXJTXKUUTXLYIKAYMYIKLKVLZVNZAYIKUVGRZKUVHRYIXSUVIUUSKUBVOTKUVGLVPTYIAU | |
| + VHAUVHVTYIOWJWDWGYIYMAYIYQYMAVTYTABDEFHIKLMNOPQXMTWDWGYICVQLKYIWTUVAXNUHZ | |
| + LCYIUVDUVJLCXDUVFCGHLMXOTXKYIXTKLRWKZYIYAXTYSXSXTVFTXTUVKKLWPWRTXLXP $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A k x $. $d E k x $. $d F k x $. $d G k x $. | |
| + $d J k x $. $d K k x $. $d N k x $. $d P k x $. | |
| + $d Q k x $. $d S k x $. $d U k x $. $d V k x $. $d k x $. | |
| + grleafnewedgrnd.v $e |- V = ( Vtx ` G ) $. | |
| + grleafnewedgrnd.e $e |- E = ( iEdg ` G ) $. | |
| + grleafnewedgrnd.a $e |- A = ( V u. { N } ) $. | |
| + grleafnewedgrnd.p $e |- P = | |
| + ( E u. { <. J , { U , N } >. } ) $. | |
| + grleafnewedgrnd.s $e |- S = <. A , P >. $. | |
| + $( If the new leaf edge occurs in a walk, then the new leaf vertex occurs | |
| + in its vertex word. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafnewedgrnd $p |- ( ( ( ( ( G e. USGraph /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) /\ | |
| + F ( Walks ` S ) Q ) /\ J e. ran F ) -> N e. ran Q ) $= | |
| + ( wcel wa cfv syl vk vx cusgr cvv wnel w3a cdm cwlks wbr crn cv wceq wfun | |
| + wrex simplr ciedg cword eqid wlkf cc0 chash cfzo co wfn wrdfn fnfun simpr | |
| + jca elrnrexdm imp c1 caddc cpr weq id fveq2 fvoveq1 preq12d eqeq12d cupgr | |
| + fveq2d wral simplll grleafusgr usgrupgr simpllr upgrwlkedg simprl eleqtrd | |
| + wrddm rspcdva cfz cvtx wf wlkp ffn elfzofz fnfvelrnd fzofzp1 prssd simpl3 | |
| + eqsstrd simpl prid2g cop csn cun a1i fveq1d wn simpll simplrl prex df-nel | |
| + simplrr sylib 3jca eqtrd eqcomd grleafiedg simprr 3eqtrd sseldd rexlimddv | |
| + fsnunfv ) HUCQZEKQZJUDQZJKUEZRZUFZIUDQZIFUGZUEZRZRZGCDUHSUIZRZIGUJQZRZIUA | |
| + UKZGSZULZJCUJZQUAGUGZYTGUMZYSRUUCUAUUEUNZYTUUFYSYTYQUUFYPYQYSUOYQGDUPSZUG | |
| + ZUQQZUUFCGDUUHUUHURZUSZUUJGUTGVASZVBVCZVDUUFUUIGVEUUNGVFTTTYRYSVGVHUUFYSU | |
| + UGUAGIVIVJTYTUUAUUEQZUUCRZRZUUBUUHSZUUDJUUQUURUUACSZUUAVKVLVCZCSZVMZUUDUU | |
| + QUBUKZGSZUUHSZUVCCSZUVCVKVLVCCSZVMZULZUURUVBULUBUUNUUAUBUAVNZUVEUURUVHUVB | |
| + UVJUVDUUBUUHUVJUVCUUAGUVJVOWAWAUVJUVFUUSUVGUVAUVCUUACVPUVCUUAVKCVLVQVRVSU | |
| + UQDVTQZYQRUVIUBUUNWBUUQUVKYQUUQDUCQZUVKUUQYPUVLYPYQYSUUPWCZABDEFHIJKLMNOP | |
| + WDTDWETYPYQYSUUPWFZVHCUBGDUUHUUKWGTUUQUUAUUEUUNYTUUOUUCWHUUQUUJUUEUUNULUU | |
| + QYQUUJUVNUULTUUIGWJTWIZWKUUQUUSUVAUUDUUQUTUUMWLVCZUUACUUQUVPDWMSZCWNZCUVP | |
| + VDUUQYQUVRUVNCGDUVQUVQURWOTUVPUVQCWPTZUUQUUAUUNQZUUAUVPQUVOUUAUTUUMWQTWRU | |
| + UQUVPUUTCUVSUUQUVTUUTUVPQUVOUTUUMUUAWSTWRWTXBUUQJEJVMZUURUUQYHJUWAQUUQYJY | |
| + HUUQYPYJUVMYFYGYJYOXATYHYIXCTEJUDXDTUUQUWAIBSZIUUHSZUURUUQUWBUWAUUQUWBIFI | |
| + UWAXEXFXGZSZUWAUUQIBUWDBUWDULUUQOXHXIUUQYLUWAUDQZIYMQXJZUFUWEUWAULUUQYLUW | |
| + FUWGUUQYRYLYRYSUUPXKZYKYLYNYQXLTUWFUUQEJXMXHUUQYNUWGUUQYRYNUWHYKYLYNYQXOT | |
| + IYMXNXPXQFUDUDIUWAYETXRXSUUQUWCUWBUUQIUUHBUUQYPUUHBULUVMABDEFHIJKLMNOPXTT | |
| + XIXSUUQIUUBUUHYTUUOUUCYAWAYBWIYCYD $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A x $. $d E x $. $d G x $. $d J x $. $d N x $. | |
| + $d P x $. $d S x $. $d U x $. $d V x $. | |
| + grleafiedgdm.v $e |- V = ( Vtx ` G ) $. | |
| + grleafiedgdm.e $e |- E = ( iEdg ` G ) $. | |
| + grleafiedgdm.a $e |- A = ( V u. { N } ) $. | |
| + grleafiedgdm.p $e |- P = | |
| + ( E u. { <. J , { U , N } >. } ) $. | |
| + grleafiedgdm.s $e |- S = <. A , P >. $. | |
| + $( The edge indices after adjoining a freshly indexed leaf edge are the | |
| + old edge indices together with the new index. (Contributed by | |
| + Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafiedgdm $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) -> | |
| + dom ( iEdg ` S ) = ( dom E u. { J } ) ) $= | |
| + ( wcel cvv wnel wa cdm csn cusgr w3a ciedg cfv grleafiedg dmeqd wf1o wceq | |
| + cun crn cpr grleaff1o f1odm syl eqtrd ) FUAODIOHPOHIQRUBGPOGESZQRRZCUCUDZ | |
| + SBSZUPGTUIZUQURBABCDEFGHIJKLMNUEUFUQUTEUJDHUKTUIZBUGUSUTUHABCDEFGHIJKLMNU | |
| + LUTVABUMUNUO $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A k x $. $d E k x $. $d F k x $. $d G k x $. $d J k x $. | |
| + $d N k x $. $d P k x $. $d Q k x $. $d S k x $. | |
| + $d U k x $. $d V k x $. $d k x $. | |
| + grleafcyclnewedgnran.v $e |- V = ( Vtx ` G ) $. | |
| + grleafcyclnewedgnran.e $e |- E = ( iEdg ` G ) $. | |
| + grleafcyclnewedgnran.a $e |- A = ( V u. { N } ) $. | |
| + grleafcyclnewedgnran.p $e |- P = | |
| + ( E u. { <. J , { U , N } >. } ) $. | |
| + grleafcyclnewedgnran.s $e |- S = <. A , P >. $. | |
| + $( A nontrivial cycle in a graph with an adjoined leaf does not contain | |
| + the new edge. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafcyclnewedgnran $p |- ( ( ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) /\ | |
| + F ( Cycles ` S ) Q ) /\ F =/= (/) ) -> | |
| + J e/ ran F ) $= | |
| + ( wcel wnel wa syl ctree cvv w3a cdm ccycls cfv wbr c0 wne grleafcyclnran | |
| + crn wn df-nel bicomi cusgr cwlks simplll simpl simp1 treeusgr simp2 simp3 | |
| + sylibr 3jca id simpr jca simpllr cycliswlk grleafnewedgrnd ex mtod ) HUAQ | |
| + ZEKQZJUBQJKRSZUCZIUBQIFUDRSZSZGCDUEUFUGZSGUHUIZSZIGUKZQZULIWBRWAWCJCUKZQZ | |
| + WAJWDRZWEULZABCDEFGHIJKLMNOPUJWFWGJWDUMUNVCWAWCWEWAWCSZHUOQZVNVOUCZVQSZGC | |
| + DUPUFUGZSZWCSWEWHWMWCWHWKWLWHWJVQWHWIVNVOWHVMWIWHVPVMWHVRVPVRVSVTWCUQZVPV | |
| + QURTZVMVNVOUSTHUTTWHVPVNWOVMVNVOVATWHVPVOWOVMVNVOVBTVDWHVRVQWHWHVRWHVEZWN | |
| + TVPVQVFTVGWHVSWLWHWHVSWPVRVSVTWCVHTCGDVITVGWAWCVFVGABCDEFGHIJKLMNOPVJTVKV | |
| + LIWBUMVC $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A k x $. $d E k x $. $d F k x $. $d G k x $. $d J k x $. | |
| + $d N k x $. $d P k x $. $d Q k x $. $d S k x $. | |
| + $d U k x $. $d V k x $. $d k x $. | |
| + grleafcycloldrn.v $e |- V = ( Vtx ` G ) $. | |
| + grleafcycloldrn.e $e |- E = ( iEdg ` G ) $. | |
| + grleafcycloldrn.a $e |- A = ( V u. { N } ) $. | |
| + grleafcycloldrn.p $e |- P = | |
| + ( E u. { <. J , { U , N } >. } ) $. | |
| + grleafcycloldrn.s $e |- S = <. A , P >. $. | |
| + $( Every indexed edge of a nontrivial cycle remains in the old edge-index | |
| + set after a leaf is adjoined. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + grleafcycloldrn $p |- ( ( ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) /\ | |
| + F ( Cycles ` S ) Q ) /\ F =/= (/) ) -> | |
| + ran F C_ dom E ) $= | |
| + ( wcel wa cfv syl cvv wnel w3a cdm ccycls wbr c0 wne crn csn cun cdif wss | |
| + ctree wn ciedg cc0 chash cfzo co wf cword simplr cycliswlk eqid wlkf wrdf | |
| + cwlks frn cusgr wceq simplll simp1 treeusgr simp2 simp3 3jca simpll simpr | |
| + jca grleafiedgdm eqcomd sseqtrrd grleafcyclnewedgnran df-nel sylib sylibr | |
| + ssdifsn difun2 a1i simprr difsn eqtrd ) HUNQZEKQZJUAQJKUBRZUCZIUAQZIFUDZU | |
| + BZRZRZGCDUESUFZRGUGUHZRZGUIZWSIUJZUKZXGULZWSXEXFXHUMZIXFQUOZRXFXIUMXEXJXK | |
| + XEXFDUPSZUDZXHXEUQGURSUSUTZXMGVAZXFXMUMXEGXMVBQZXOXEGCDVHSUFZXPXEXCXQXBXC | |
| + XDVCCGDVDTCGDXLXLVEVFTXMGVGTXNXMGVITXEXMXHXEHVJQZWOWPUCZXARXMXHVKXEXSXAXE | |
| + XRWOWPXEWNXRXEWQWNWQXAXCXDVLZWNWOWPVMTHVNTXEWQWOXTWNWOWPVOTXEWQWPXTWNWOWP | |
| + VPTVQXEXBXAXBXCXDVRZWQXAVSTVTABDEFHIJKLMNOPWATWBWCXEIXFUBXKABCDEFGHIJKLMN | |
| + OPWDIXFWEWFVTXFXHIWHWGXEXIWSXEXIWSXGULZWSXIYBVKXEWSXGWIWJXEIWSQUOZYBWSVKX | |
| + EWTYCXEXBWTYAWQWRWTWKTIWSWEWFIWSWLTWMWBWC $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A k x $. $d E k x $. $d F k x $. $d G k x $. $d J k x $. | |
| + $d N k x $. $d P k x $. $d Q k x $. $d S k x $. | |
| + $d U k x $. $d V k x $. $d k x $. | |
| + grleafcyclresedg.v $e |- V = ( Vtx ` G ) $. | |
| + grleafcyclresedg.e $e |- E = ( iEdg ` G ) $. | |
| + grleafcyclresedg.a $e |- A = ( V u. { N } ) $. | |
| + grleafcyclresedg.p $e |- P = | |
| + ( E u. { <. J , { U , N } >. } ) $. | |
| + grleafcyclresedg.s $e |- S = <. A , P >. $. | |
| + $( Restricting a nontrivial cycle after adjoining a leaf recovers a cycle | |
| + on the old indexed-edge function. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + grleafcyclresedg $p |- ( ( ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) /\ | |
| + F ( Cycles ` S ) Q ) /\ F =/= (/) ) -> | |
| + F ( Cycles ` <. A , E >. ) Q ) $= | |
| + ( wcel wa cfv syl ctree cvv wnel w3a cdm ccycls wbr c0 wne cop cvtx ciedg | |
| + cupgr crn wss cusgr simplll simp1 treeusgr simp2 simp3 simpllr grleafusgr | |
| + cres 3jca usgrupgr simplr grleafcycloldrn eqid upgrcyclres wceq grleafvtx | |
| + jca grleafiedgres opeq12d eqcomd fveq2d breqd mpbird ) HUAQZEKQZJUBQJKUCR | |
| + ZUDZIUBQIFUEZUCRZRZGCDUFSUGZRGUHUIZRZGCAFUJZUFSZUGGCDUKSZDULSZWDVDZUJZUFS | |
| + ZUGZWIDUMQZWGRZGUNWDUOZRWQWIWSWTWIWRWGWIDUPQZWRWIHUPQZWAWBUDZWERZXAWIXCWE | |
| + WIXBWAWBWIVTXBWIWCVTWCWEWGWHUQZVTWAWBURTHUSTWIWCWAXEVTWAWBUTTWIWCWBXEVTWA | |
| + WBVATVEWCWEWGWHVBVMZABDEFHIJKLMNOPVCTDVFTWFWGWHVGVMABCDEFGHIJKLMNOPVHVMWD | |
| + CWMGDWLWLVIWMVIVJTWIWKWPGCWIWJWOUFWIWOWJWIWLAWNFWIXDWLAVKXFABDEFHIJKLMNOP | |
| + VLTWIXDWNFVKXFABDEFHIJKLMNOPVNTVOVPVQVRVS $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A x $. $d E x $. $d G x $. $d N x $. $d V x $. | |
| + grleafbaseupgr.v $e |- V = ( Vtx ` G ) $. | |
| + grleafbaseupgr.e $e |- E = ( iEdg ` G ) $. | |
| + grleafbaseupgr.a $e |- A = ( V u. { N } ) $. | |
| + $( Keeping the old indexed edges while adjoining a vertex produces an | |
| + undirected pseudograph. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + grleafbaseupgr $p |- ( G e. UPGraph -> | |
| + <. A , E >. e. UPGraph ) $= | |
| + ( cupgr wcel id csn cun cvv cvtx fvexi snex unex eqeltri wss sseqtrri a1i | |
| + ssun1 upgrvtxsupd ) CIJZBCEAFGUEKAEDLZMZNHEUFECOFPDQRSEATUEEUGAEUFUCHUAUB | |
| + UD $. | |
| + | |
| + $( Keeping the old indexed edges while adjoining a vertex does not change | |
| + the indexed-edge function. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + grleafbaseiedg $p |- ( iEdg ` <. A , E >. ) = E $= | |
| + ( csn cun cvv cvtx fvexi snex unex eqeltri ciedg cfv eqid opiedgfvi ) BAA | |
| + EDIZJKHEUAECLFMDNOPBCQRZKGUBCQUBSMPT $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A k x $. $d E k x $. $d F k x $. $d G k x $. $d J k x $. | |
| + $d N k x $. $d P k x $. $d Q k x $. $d S k x $. | |
| + $d U k x $. $d V k x $. $d k x $. | |
| + grleafcyclres.v $e |- V = ( Vtx ` G ) $. | |
| + grleafcyclres.e $e |- E = ( iEdg ` G ) $. | |
| + grleafcyclres.a $e |- A = ( V u. { N } ) $. | |
| + grleafcyclres.p $e |- P = | |
| + ( E u. { <. J , { U , N } >. } ) $. | |
| + grleafcyclres.s $e |- S = <. A , P >. $. | |
| + $( Every nontrivial cycle after adjoining a leaf is already a cycle of the | |
| + old tree. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafcyclres $p |- ( ( ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) /\ | |
| + F ( Cycles ` S ) Q ) /\ F =/= (/) ) -> | |
| + F ( Cycles ` G ) Q ) $= | |
| + ( wcel wa cfv syl ctree cvv wnel w3a cdm ccycls wbr c0 wne cop cupgr wceq | |
| + crn wss simplll simp1 treeupgr grleafbaseupgr jca eqid grleafcycloldvtxrn | |
| + a1i grleafcyclresedg ciedg grleafbaseiedg eqcomi upgrvtxrescycl ) HUAQZEK | |
| + QZJUBQJKUCRZUDZIUBQIFUEUCRZRGCDUFSUGZRGUHUIZRZAFUJZUKQZHUKQZRZFFULZRZGCVP | |
| + UFSUGZCUMKUNZRZRGCHUFSUGVOWAWDVOVSVTVOVQVRVOVRVQVOVHVRVOVKVHVKVLVMVNUOVHV | |
| + IVJUPTHUQTZAFHJKLMNURTWEUSVTVOFUTVBUSVOWBWCABCDEFGHIJKLMNOPVCABCDEFGHIJKL | |
| + MNOPVAUSUSCFGVPHFKLVPVDSFAFHJKLMNVEVFMVGT $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A x $. $d E x $. $d F x $. $d G x $. $d J x $. | |
| + $d N x $. $d P x $. $d Q x $. $d S x $. $d U x $. $d V x $. | |
| + grleafcycltriv.v $e |- V = ( Vtx ` G ) $. | |
| + grleafcycltriv.e $e |- E = ( iEdg ` G ) $. | |
| + grleafcycltriv.a $e |- A = ( V u. { N } ) $. | |
| + grleafcycltriv.p $e |- P = | |
| + ( E u. { <. J , { U , N } >. } ) $. | |
| + grleafcycltriv.s $e |- S = <. A , P >. $. | |
| + $( Every cycle after adjoining a leaf to a tree is trivial. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafcycltriv $p |- ( ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) /\ | |
| + F ( Cycles ` S ) Q ) -> F = (/) ) $= | |
| + ( wcel wa c0 syl cvv wnel w3a cdm ccycls cfv wbr wne wceq cacycgr simplll | |
| + ctree simp1 treeacycgr grleafcyclres jca acycgrcycl ex nne biimpi pm2.61d | |
| + wn wi a1i ) HULQZEKQZJUAQJKUBRZUCZIUAQIFUDUBRZRGCDUEUFUGZRZGSUHZGSUIZVKVL | |
| + VMVKVLRZHUJQZGCHUEUFUGZRVMVNVOVPVNVEVOVNVHVEVHVIVJVLUKVEVFVGUMTHUNTABCDEF | |
| + GHIJKLMNOPUOUPCGHUQTURVLVBZVMVCVKVQVMGSUSUTVDVA $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A f p x $. $d E f p x $. $d G f p x $. $d J f p x $. | |
| + $d N f p x $. $d P f p x $. $d S f p x $. $d U f p x $. | |
| + $d V f p x $. $d f p x $. | |
| + grleafacycgr.v $e |- V = ( Vtx ` G ) $. | |
| + grleafacycgr.e $e |- E = ( iEdg ` G ) $. | |
| + grleafacycgr.a $e |- A = ( V u. { N } ) $. | |
| + grleafacycgr.p $e |- P = | |
| + ( E u. { <. J , { U , N } >. } ) $. | |
| + grleafacycgr.s $e |- S = <. A , P >. $. | |
| + $( Adjoining a leaf to a tree preserves acyclicity. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafacycgr $p |- ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) -> | |
| + S e. AcyclicGraph ) $= | |
| + ( vf vp wcel cvv wa cusgr ctree wnel w3a cdm cacycgr cv ccycls cfv wbr c0 | |
| + wceq wi wal grleafcycltriv ex alrimivv wb simpl1 treeusgr syl simpl2 3jca | |
| + simpl3 simpr jca grleafusgr elexd isacycgr1 mpbird ) FUAQZDIQZHRQHIUBSZUC | |
| + ZGRQGEUDUBSZSZCUEQZOUFZPUFZCUGUHUIZVQUJUKZULZPUMOUMZVOWAOPVOVSVTABVRCDEVQ | |
| + FGHIJKLMNUNUOUPVOCRQVPWBUQVOCTVOFTQZVKVLUCZVNSCTQVOWDVNVOWCVKVLVOVJWCVJVK | |
| + VLVNURFUSUTVJVKVLVNVAVJVKVLVNVCVBVMVNVDVEABCDEFGHIJKLMNVFUTVGOCRPVHUTVI | |
| + $. | |
| + $} | |
| + | |
| + ${ | |
| + $d P i j $. $d Q i j $. $d R i j $. $d V i j $. $d i j $. $d ph i j $. | |
| + ccatfstpthcondr.q $e |- Q = ( P ++ <" ( P ` 0 ) "> ) $. | |
| + ccatfstpthcondr.r $e |- R = ( ( # ` Q ) - 1 ) $. | |
| + ccatfstpthcondr.p $e |- ( ph -> P e. Word V ) $. | |
| + ccatfstpthcondr.f $e |- ( ph -> Fun `' P ) $. | |
| + $( The vertex-distinctness condition for closing an injective word, | |
| + expressed with the length bounds of the closed word. (Contributed by | |
| + Mingli Yuan, 11-Aug-2026.) $) | |
| + ccatfstpthcondr $p |- ( ph -> | |
| + A. i e. ( 0 ..^ ( # ` Q ) ) A. j e. ( 1 ..^ R ) | |
| + ( i =/= j -> ( Q ` i ) =/= ( Q ` j ) ) ) $= | |
| + ( cv wne cfv c1 cfzo co wral cc0 chash wi caddc ccatfstpthcond ccatfstlen | |
| + eqcomd oveq2d raleqdv ralbidv mpbid cs1 cconcat wceq a1i cword ccatws1len | |
| + fveq2d wcel syl eqtrd raleqtrdv ) AELZFLZMVACNVBCNMUAZFODPQZRZESBTNZOUBQZ | |
| + PQZSCTNZPQAVCFOVFPQZRZEVHRVEEVHRABCEFGHJKUCAVKVEEVHAVCFVJVDAVFDOPADVFABCD | |
| + GHIJUDUEUFUGUHUIAVGVISPAVIVGAVIBSBNZUJUKQZTNZVGACVMTCVMULAHUMUPABGUNUQVNV | |
| + GULJGBVLUOURUSUEUFUT $. | |
| + $} | |
| + | |
| + ${ | |
| + ccatfstclosed.q $e |- Q = ( P ++ <" ( P ` 0 ) "> ) $. | |
| + ccatfstclosed.r $e |- R = ( ( # ` Q ) - 1 ) $. | |
| + ccatfstclosed.p $e |- ( ph -> P e. Word V ) $. | |
| + $( A word closed by repeating its first symbol has equal endpoints. | |
| + (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + ccatfstclosed $p |- ( ph -> | |
| + ( Q ` 0 ) = ( Q ` R ) ) $= | |
| + ( cc0 cfv cs1 cconcat wceq a1i fveq1d cword wcel syl eqtrd cvv ccat1st1st | |
| + eqidd chash w3a wrdv fvex ccatfstlen 3jca ccats1val2 eqcomd eqcomi 3eqtrd | |
| + co ) AICJZIBJZUODCJZAUNIBUOKLUMZJZUOAICUQCUQMAFNOABEPQZURUOMHEBUARSAUOUBA | |
| + UODUQJZUPAUTUOABTPQZUOTQZDBUCJMZUDUTUOMAVAVBVCAUSVAHEBUERVBAIBUFNABCDEFGH | |
| + UGUHUODTBUIRUJADUQCUQCMACUQFUKNOSUL $. | |
| + $} | |
| + | |
| + ${ | |
| + $d F i j $. $d P i j $. $d R i j $. $d ph i j $. | |
| + pthdfd.p $e |- ( ph -> P e. Word _V ) $. | |
| + pthdfd.r $e |- R = ( ( # ` P ) - 1 ) $. | |
| + pthdfd.s $e |- ( ph -> | |
| + A. i e. ( 0 ..^ ( # ` P ) ) A. j e. ( 1 ..^ R ) | |
| + ( i =/= j -> ( P ` i ) =/= ( P ` j ) ) ) $. | |
| + pthdfd.f $e |- ( ph -> ( # ` F ) = R ) $. | |
| + pthdfd.t $e |- ( ph -> F ( Trails ` G ) P ) $. | |
| + $( Deduction-form version of ~ pthd for the edge-count equality. | |
| + (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + pthdfd $p |- ( ph -> F ( Paths ` G ) P ) $= | |
| + ( cfv wbr c1 cfzo co cc0 cima c0 ctrls chash cres ccnv wfun cpr cin cpths | |
| + wceq w3a pthdlem1 eqidd eqcomd oveq2d reseq12d cnveqd funeqd mpbid preq2d | |
| + pthdlem2 imaeq12d ineq12d eqeq1d 3jca ispth sylibr ) AFBGUAMNZBOFUBMZPQZU | |
| + CZUDZUEZBRVHUFZSZBVISZUGZTUIZUJFBGUHMNAVGVLVQLABOCPQZUCZUDZUEVLABCDEHIJUK | |
| + AVTVKAVSVJABBVRVIABULZACVHOPAVHCKUMZUNZUOUPUQURABRCUFZSZBVRSZUGZTUIVQABCD | |
| + EHIJUTAWGVPTAWEVNWFVOABBWDVMWAACVHRWBUSVAABBVRVIWAWCVAVBVCURVDBFGVEVF $. | |
| + $} | |
| + | |
| + ${ | |
| + $d F i j k $. $d G i j k $. $d H i j k $. $d I i j k $. $d J i j k $. | |
| + $d P i j k $. $d Q i j k $. $d R i j $. $d V i j k $. $d i j $. | |
| + $d ph i j k $. | |
| + spthccatcycld.v $e |- V = ( Vtx ` G ) $. | |
| + spthccatcycld.i $e |- I = ( iEdg ` G ) $. | |
| + spthccatcycld.h $e |- H = ( F ++ <" J "> ) $. | |
| + spthccatcycld.q $e |- Q = ( P ++ <" ( P ` 0 ) "> ) $. | |
| + spthccatcycld.r $e |- R = ( ( # ` Q ) - 1 ) $. | |
| + spthccatcycld.g $e |- ( ph -> G e. UPGraph ) $. | |
| + spthccatcycld.p $e |- ( ph -> F ( SPaths ` G ) P ) $. | |
| + spthccatcycld.j $e |- ( ph -> J e. dom I ) $. | |
| + spthccatcycld.e $e |- ( ph -> ( I ` J ) = | |
| + { ( lastS ` P ) , ( P ` 0 ) } ) $. | |
| + spthccatcycld.n $e |- ( ph -> -. J e. ran F ) $. | |
| + $( Closing a simple path with a fresh indexed edge produces a cycle. | |
| + (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + spthccatcycld $p |- ( ph -> H ( Cycles ` G ) Q ) $= | |
| + ( vi vj cpths cfv wbr cc0 chash wceq wa ccycls cs1 cconcat cvv cword wcel | |
| + co cspths cwlks spthiswlk wlkpwrd 3syl ccatws1clv eleq1i sylibr ccnv wfun | |
| + syl spthf1 ccatfstpthcondr cmin wlkepvtx simpl wlkccat1d wlklenvm1 eqcomi | |
| + c1 a1i eqtrd ctrls cdm spthccats1f1d wf df-f1 simprbi cnveqi funeqi istrl | |
| + wf1 jca pthdfd ccatfstclosed fveq2d eqcomd iscycl ) AGCFUCUDUEZUFCUDZGUGU | |
| + DZCUDZUHZUIGCFUJUDUEAWOWSACDUAUBGFABUFBUDZUKULUPZUMUNZUOZCXBUOABJUNUOZXCA | |
| + EBFUQUDUEZEBFURUDZUEZXDQBEFUSZBEFJKUTVAZJBWTVBVGCXAXBNVCVDOABCDUAUBJNOXIA | |
| + XEBVEVFQBEFVHVGVIAWQCUGUDVPVJUPZDAGCXFUEZWQXJUHABCWTEFGHIJKLMNPAXEXGQXHVG | |
| + ZRAXGWTJUOZXLXGXMEUGUDBUDJUOZUIXMBEFJKVKXMXNVLVGVGSVMZCGFVNVGXJDUHADXJOVO | |
| + VQVRZAXKGVEZVFZUIGCFVSUDUEAXKXRXOAEIUKULUPZVEZVFZXRAXSVTZHVTZXSWHZYAABEFH | |
| + ILQRTWAYDYBYCXSWBYAYBYCXSWCWDVGXQXTGXSMWEWFVDWICGFWGVDWJAWPDCUDZWRABCDJNO | |
| + XIWKAWRYEAWQDCXPWLWMVRWICGFWNVD $. | |
| + $} | |
| + | |
| + ${ | |
| + $d F i j k $. $d G i j k $. $d H i j k $. $d I i j k $. $d J i j k $. | |
| + $d P i j k $. $d Q i j k $. $d R i j $. $d V i j k $. $d i j $. | |
| + $d ph i j k $. | |
| + acycgrspthcld.v $e |- V = ( Vtx ` G ) $. | |
| + acycgrspthcld.i $e |- I = ( iEdg ` G ) $. | |
| + acycgrspthcld.h $e |- H = ( F ++ <" J "> ) $. | |
| + acycgrspthcld.q $e |- Q = ( P ++ <" ( P ` 0 ) "> ) $. | |
| + acycgrspthcld.r $e |- R = ( ( # ` Q ) - 1 ) $. | |
| + acycgrspthcld.g $e |- ( ph -> G e. UPGraph ) $. | |
| + acycgrspthcld.a $e |- ( ph -> G e. AcyclicGraph ) $. | |
| + acycgrspthcld.p $e |- ( ph -> F ( SPaths ` G ) P ) $. | |
| + acycgrspthcld.j $e |- ( ph -> J e. dom I ) $. | |
| + acycgrspthcld.e $e |- ( ph -> ( I ` J ) = | |
| + { ( lastS ` P ) , ( P ` 0 ) } ) $. | |
| + $( In an acyclic graph, an indexed edge closing a simple path is already | |
| + used by that path. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| + acycgrspthcld $p |- ( ph -> J e. ran F ) $= | |
| + ( crn wcel idd wn wa wceq cacycgr ccycls cfv wbr adantr cupgr cspths clsw | |
| + c0 cdm cc0 cpr simpr spthccatcycld jca acycgrcycl syl cconcat cword cwlks | |
| + cs1 spthiswlk wlkf 3syl wne ccatws1n0 df-ne biimpi eqeq1i notbii pm2.21dd | |
| + co sylibr ex pm2.61d ) AIEUAUBZWBAWBUCAWBUDZWBAWCUEZGUOUFZWBWDFUGUBZGCFUH | |
| + UIUJZUEWEWDWFWGAWFWCQUKWDBCDEFGHIJKLMNOAFULUBWCPUKAEBFUMUIUJZWCRUKAIHUPZU | |
| + BWCSUKAIHUIBUNUIUQBUIURUFWCTUKAWCUSUTVACGFVBVCWDEIVGVDVRZUOUFZUDZWEUDWDEW | |
| + IVEUBZWLAWMWCAWHEBFVFUIUJWMRBEFVHBEFHLVIVJUKWMWJUOVKZWLWIEIVLWNWLWJUOVMVN | |
| + VCVCWEWKGWJUOMVOVPVSVQVTWA $. | |
| + $} | |
| + | |
| + ${ | |
| + $d B i j $. $d F i j $. $d G i j $. $d H i j $. $d I i j $. | |
| + $d L i j $. $d P i j $. $d Q i j $. $d V i j $. $d i j $. $d ph i j $. | |
| + swrdspthd.v $e |- V = ( Vtx ` G ) $. | |
| + swrdspthd.i $e |- I = ( iEdg ` G ) $. | |
| + swrdspthd.h $e |- H = ( F substr <. B , L >. ) $. | |
| + swrdspthd.q $e |- Q = ( P substr <. B , ( L + 1 ) >. ) $. | |
| + swrdspthd.p $e |- ( ph -> F ( SPaths ` G ) P ) $. | |
| + swrdspthd.b $e |- ( ph -> B e. ( 0 ... L ) ) $. | |
| + swrdspthd.l $e |- ( ph -> L e. ( 0 ... ( # ` F ) ) ) $. | |
| + $( A continuous segment of a simple path is a simple path. (Contributed | |
| + by Mingli Yuan, 11-Aug-2026.) $) | |
| + swrdspthd $p |- ( ph -> H ( SPaths ` G ) Q ) $= | |
| + ( wbr co cc0 ctrls cfv ccnv wfun wa cspths cwlks cop csubstr caddc cfz wcel | |
| + c1 chash spthiswlk syl swrdwlk syl3anc breq12i sylibr cdm cword spthiedgf1d | |
| + wf1 wlkf 3syl swrdf1 wf df-f1 simprbi cnveqi funeqi jca istrl wlkpwrd sseli | |
| + fzssp1 fzp1elp1 wceq wlklenvp1 oveq2d eleqtrrd spthf1 wrdf1d isspth ) AGDFU | |
| + AUBRZDUCZUDZUEGDFUFUBZRAWFWHAGDFUGUBZRZGUCZUDZUEWFAWKWMAEBIUHUISZCBIUMUJSZU | |
| + HUISZWJRZWKAECWJRZBTIUKSZULZITEUNUBZUKSULZWQAECWIRZWROCEFUOZUPPQBCEFIUQURGW | |
| + NDWPWJMNUSUTAWNUCZUDZWMAWNVAZHVAZWNVDZXFAXHBIEAXCWREXHVBULOXDCEFHLVEVFPQACE | |
| + FHLOVCVGXIXGXHWNVHXFXGXHWNVIVJUPWLXEGWNMVKVLUTVMDGFVNUTAWPUCZUDZWHAWPVAZJWP | |
| + VDZXKAJBWOCAXCWRCJVBULOXDCEFJKVOVFZAWTBTWOUKSZULPWSXOBTIVQVPUPAWOTXAUMUJSZU | |
| + KSZTCUNUBZUKSAXBWOXQULQITXAVRUPAXRXPTUKAXCWRXRXPVSOXDCEFVTVFWAWBAJCXNAXCCUC | |
| + UDOCEFWCUPWDVGXMXLJWPVHXKXLJWPVIVJUPWGXJDWPNVKVLUTVMDGFWEUT $. | |
| + $} | |
| + | |
| + ${ | |
| + swrdfv0d.w $e |- ( ph -> W e. Word D ) $. | |
| + swrdfv0d.m $e |- ( ph -> M e. ( 0 ..^ N ) ) $. | |
| + swrdfv0d.n $e |- ( ph -> N e. ( 0 ... ( # ` W ) ) ) $. | |
| + $( The first symbol of a subword, in deduction form. (Contributed by | |
| + Mingli Yuan, 12-Aug-2026.) $) | |
| + swrdfv0d $p |- ( ph -> | |
| + ( ( W substr <. M , N >. ) ` 0 ) = ( W ` M ) ) $= | |
| + ( cword wcel cc0 cfzo co chash cfv cfz w3a cop csubstr wceq 3jca swrdfv0 | |
| + syl ) AEBIJZCKDLMJZDKENOPMJZQKECDRSMOCEOTAUDUEUFFGHUABECDUBUC $. | |
| + $} | |
| + | |
| + ${ | |
| + swrdlswd.w $e |- ( ph -> W e. Word D ) $. | |
| + swrdlswd.m $e |- ( ph -> M e. ( 0 ..^ N ) ) $. | |
| + swrdlswd.n $e |- ( ph -> N e. ( 0 ... ( # ` W ) ) ) $. | |
| + $( The last symbol of a nonempty subword, in deduction form. | |
| + (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + swrdlswd $p |- ( ph -> | |
| + ( lastS ` ( W substr <. M , N >. ) ) = | |
| + ( W ` ( N - 1 ) ) ) $= | |
| + ( co cfv chash c1 cmin wcel wceq syl cc0 cfz 3syl cc csubstr cword swrdcl | |
| + cop clsw lsw cfzo elfzofz 3jca swrdlen oveq1d fveq2d caddc wa cn fzonnsub | |
| + w3a fzo0end jca swrdfv cz elfzoel2 zcn elfzoelz 1cnd nnpcan eqtrd 3eqtrd | |
| + ) AECDUDUAIZUEJZVIKJZLMIZVIJZDCMIZLMIZVIJZDLMIZEJZAVIBUBZNZVJVMOAEVSNZVTF | |
| + BECDUCPVIVSUFPAVLVOVIAVKVNLMAWACQDRINZDQEKJRINZUQZVKVNOAWAWBWCFACQDUGINZW | |
| + BGCQDUHPHUIZBECDUJPUKULAVPVOCUMIZEJZVRAWDVOQVNUGINZUNVPWHOAWDWIWFAWEVNUON | |
| + WIGCQDUPVNURSUSBECDVOUTPAWGVQEADTNZCTNZLTNZUQWGVQOAWJWKWLAWEDVANWJGCQDVBD | |
| + VCSAWECVANWKGCQDVDCVCSAVEUIDCLVFPULVGVH $. | |
| + $} | |
| + | |
| + ${ | |
| + f1veqaeqd.f $e |- ( ph -> F : A -1-1-> B ) $. | |
| + f1veqaeqd.c $e |- ( ph -> C e. A ) $. | |
| + f1veqaeqd.d $e |- ( ph -> D e. A ) $. | |
| + f1veqaeqd.e $e |- ( ph -> ( F ` C ) = ( F ` D ) ) $. | |
| + $( Equality of arguments of a one-to-one function, in deduction form. | |
| + (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + f1veqaeqd $p |- ( ph -> C = D ) $= | |
| + ( cfv wceq wf1 wcel wa wi jca f1veqaeq syl mpd ) ADFKEFKLZDELZJABCFMZDBNZ | |
| + EBNZOZOUAUBPAUCUFGAUDUEHIQQBCDEFRST $. | |
| + $} | |
| + | |
| + ${ | |
| + f1endprd.f $e |- ( ph -> | |
| + P : ( 0 ... N ) -1-1-> V ) $. | |
| + f1endprd.b $e |- ( ph -> B e. ( 0 ..^ N ) ) $. | |
| + f1endprd.k $e |- ( ph -> K e. ( B ..^ N ) ) $. | |
| + f1endprd.e $e |- ( ph -> | |
| + { ( P ` K ) , ( P ` ( K + 1 ) ) } = | |
| + { ( P ` N ) , ( P ` B ) } ) $. | |
| + $( An edge from the final value of an injective finite sequence back to | |
| + an earlier value is its final consecutive pair. (Contributed by | |
| + Mingli Yuan, 12-Aug-2026.) $) | |
| + f1endprd $p |- ( ph -> B = ( N - 1 ) ) $= | |
| + ( cfv wceq c1 co wa cc0 adantr wcel syl fvex cmin cfz wf1 cuz wss elfzouz | |
| + caddc fzoss1 sseld mpd elfzofz cn0 cn clt wbr elfzo0 simp2bi nnnn0 nn0fz0 | |
| + cfzo sylib simprl f1veqaeqd wn wne cz cr elfzoelz zre 3syl elfzolt2 ltned | |
| + df-ne pm2.21d ex oveq1d fzofzp1 fz0fzelfz0 syl2anc simprr eqtr3d zcn 1cnd | |
| + cc elfzoel2 addlsub mpbid cpr wo wb preq12b a1i mpjaod ) ADCKZECKZLZDMUGN | |
| + ZCKZBCKZLZOZBEMUANLZWNWSLZWRWOLZOZAXAXBAXAOZDELZXBXFPEUBNZFDECAXHFCUCZXAG | |
| + QADXHRZXAADPEUTNZRZXJADBEUTNZRZXLIAXMXKDABPUDKRZXMXKUEABXKRZXOHBPEUFSBPEU | |
| + HSUIUJDPEUKSZQAEXHRZXAAEULRZXRAEUMRZXSAXPXTHXPBULRXTBEUNUOBEUPUQSEURSEUSV | |
| + AZQAWPWTVBVCXFXGXBAXGVDZXAADEVEYBADEAXNDVFRDVGRIDBEVHDVIVJAXNDEUNUOIDBEVK | |
| + SVLDEVMVAQVNUJVOAXEXBAXEOZBMUGNZELXBYCWQYDEYCDBMUGYCXHFDBCAXIXEGQZAXJXEXQ | |
| + QABXHRZXEAXPYFHBPEUKSZQAXCXDVBVCVPYCXHFWQECYEAWQXHRZXEAYFWQBEUBNRZYHYGAXN | |
| + YIIBEDVQSEWQBVRVSQAXRXEYAQAXCXDVTVCWAYCBMEABWDRZXEAXPBVFRYJHBPEVHBWBVJQYC | |
| + WCAEWDRZXEAXPEVFRYKHBPEWEEWBVJQWFWGVOAWNWRWHWOWSWHLZXAXEWIZJYLYMWJAWNWRWO | |
| + WSDCTWQCTECTBCTWKWLWGWM $. | |
| + $} | |
| + | |
| + ${ | |
| + $d B k $. $d F k $. $d G k $. $d I k $. $d K k $. $d P k $. | |
| + $d ph k $. | |
| + spthendprd.i $e |- I = ( iEdg ` G ) $. | |
| + spthendprd.g $e |- ( ph -> G e. UPGraph ) $. | |
| + spthendprd.p $e |- ( ph -> F ( SPaths ` G ) P ) $. | |
| + spthendprd.b $e |- ( ph -> B e. ( 0 ..^ ( # ` F ) ) ) $. | |
| + spthendprd.k $e |- ( ph -> K e. ( B ..^ ( # ` F ) ) ) $. | |
| + spthendprd.e $e |- ( ph -> ( I ` ( F ` K ) ) = | |
| + { ( P ` ( # ` F ) ) , ( P ` B ) } ) $. | |
| + $( An indexed edge from the final vertex of a simple path back to an | |
| + earlier vertex is the final indexed edge of the path. (Contributed by | |
| + Mingli Yuan, 12-Aug-2026.) $) | |
| + spthendprd $p |- ( ph -> B = ( ( # ` F ) - 1 ) ) $= | |
| + ( vk cfv cc0 co syl fveq2d wcel chash cvtx cspths wbr cfz wf1 spthdifv c1 | |
| + caddc cpr cv wceq cfzo wa simpr oveq1d preq12d eqeq12d cuz elfzouz fzoss1 | |
| + wss sseld mpd cupgr cwlks wral upgrwlkedg syl2anc rspcdv2 eqtr3d f1endprd | |
| + spthiswlk ) ABCGDUAOZEUBOZADCEUCOUDZPVNUEQVOCUFJCDEUGRKLAGDOZFOZGCOZGUHUI | |
| + QZCOZUJZVNCOBCOUJANUKZDOZFOZWCCOZWCUHUIQZCOZUJZULZVRWBULNGPVNUMQZAWCGULZU | |
| + NZWEVRWIWBWMWDVQFWMWCGDAWLUOZSSWMWFVSWHWAWMWCGCWNSWMWGVTCWMWCGUHUIWNUPSUQ | |
| + URAGBVNUMQZTGWKTLAWOWKGABPUSOTZWOWKVBABWKTWPKBPVNUTRBPVNVARVCVDAEVETDCEVF | |
| + OUDZWJNWKVGIAVPWQJCDEVMRCNDEFHVHVIVJMVKVL $. | |
| + $} | |
| + | |
| + ${ | |
| + spthswrdendd.q $e |- Q = | |
| + ( P substr <. B , ( ( # ` F ) + 1 ) >. ) $. | |
| + spthswrdendd.p $e |- ( ph -> F ( SPaths ` G ) P ) $. | |
| + spthswrdendd.b $e |- ( ph -> B e. ( 0 ... ( # ` F ) ) ) $. | |
| + $( The endpoints of the final segment of a simple path are the selected | |
| + earlier vertex and the final vertex. (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + spthswrdendd $p |- ( ph -> | |
| + { ( lastS ` Q ) , ( Q ` 0 ) } = | |
| + { ( P ` ( # ` F ) ) , ( P ` B ) } ) $= | |
| + ( clsw cfv chash cc0 c1 co wceq a1i wcel 3syl syl cop csubstr cmin fveq2d | |
| + caddc cvtx cspths wbr cwlks cword spthiswlk eqid wlkpwrd cfz cz cn0 wlkcl | |
| + cfzo nn0z fzval3 eleqtrd wlklenvp1 lencl sylib eqeltrrd swrdlswd cc nn0cn | |
| + nn0fz0 pncan1 3eqtrd fveq1i swrdfv0d eqidd preq12d ) ADJKZELKZCKZMDKZBCKZ | |
| + AVPCBVQNUEOZUAUBOZJKWANUCOZCKVRADWBJDWBPAGQUDAFUFKZBWACAECFUGKUHZECFUIKUH | |
| + ZCWDUJRZHCEFUKZCEFWDWDULUMZSZABMVQUNOZMWAUROZIAVQUORZWKWLPAWFVQUPRZWMAWEW | |
| + FHWHTZCEFUQZVQUSSMVQUTTVAZACLKZWAMWRUNOZAWFWRWAPWOCEFVBTAWRUPRZWRWSRAWFWG | |
| + WTWOWIWDCVCSWRVIVDVEZVFAWCVQCAVQVGRZWCVQPAWNXBAWEWFWNHWHWPSVQVHTVQVJTUDVK | |
| + AVSMWBKZVTVTVSXCPAMDWBGVLQAWDBWACWJWQXAVMAVTVNVKVO $. | |
| + $} | |
| + | |
| + ${ | |
| + $d B i j k $. $d F i j k $. $d G i j k $. $d H i j k $. $d I i j k $. | |
| + $d J i j k $. $d P i j k $. $d Q i j k $. $d i j $. $d ph i j k $. | |
| + acycgrspthendrnd.i $e |- I = ( iEdg ` G ) $. | |
| + acycgrspthendrnd.h $e |- H = | |
| + ( F substr <. B , ( # ` F ) >. ) $. | |
| + acycgrspthendrnd.q $e |- Q = | |
| + ( P substr <. B , ( ( # ` F ) + 1 ) >. ) $. | |
| + acycgrspthendrnd.g $e |- ( ph -> G e. UPGraph ) $. | |
| + acycgrspthendrnd.a $e |- ( ph -> G e. AcyclicGraph ) $. | |
| + acycgrspthendrnd.p $e |- ( ph -> F ( SPaths ` G ) P ) $. | |
| + acycgrspthendrnd.b $e |- ( ph -> B e. ( 0 ..^ ( # ` F ) ) ) $. | |
| + acycgrspthendrnd.j $e |- ( ph -> J e. dom I ) $. | |
| + acycgrspthendrnd.e $e |- ( ph -> ( I ` J ) = | |
| + { ( P ` ( # ` F ) ) , ( P ` B ) } ) $. | |
| + $( In an acyclic graph, an edge from the final vertex of a simple path to | |
| + an earlier vertex already occurs in the corresponding final segment. | |
| + (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + acycgrspthendrnd $p |- ( ph -> J e. ran H ) $= | |
| + ( cfv co cc0 cs1 cconcat chash c1 cmin cvtx eqid cfzo cfz elfzofz syl cn0 | |
| + wcel cspths wbr cwlks spthiswlk wlkcl nn0fz0 sylib swrdspthd spthswrdendd | |
| + 3syl cpr clsw eqtr4d acycgrspthcld ) ADDUADSZUBUCTZVJUDSUEUFTZGFGIUBUCTZH | |
| + IFUGSZVMUHZJVLUHVJUHVKUHMNABCDEFGHEUDSZVMVNJKLOABUAVOUITUNBUAVOUJTZUNPBUA | |
| + VOUKULZAVOUMUNZVOVPUNAECFUOSUPECFUQSUPVROCEFURCEFUSVDVOUTVAVBQAIHSVOCSBCS | |
| + VEDVFSVIVERABCDEFLOVQVCVGVH $. | |
| + $} | |
| + | |
| + ${ | |
| + spthswrdrnd.h $e |- H = | |
| + ( F substr <. B , ( # ` F ) >. ) $. | |
| + spthswrdrnd.p $e |- ( ph -> F ( SPaths ` G ) P ) $. | |
| + spthswrdrnd.b $e |- ( ph -> B e. ( 0 ..^ ( # ` F ) ) ) $. | |
| + $( The range of the final segment of a simple path is the image of the | |
| + corresponding half-open index interval. (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + spthswrdrnd $p |- ( ph -> | |
| + ran H = ( F " ( B ..^ ( # ` F ) ) ) ) $= | |
| + ( crn chash cfv cop co cfzo wceq wcel cc0 wbr 3syl csubstr cima rneqi a1i | |
| + ciedg cdm cword cfz cspths cwlks spthiswlk eqid wlkf elfzofz wlkcl nn0fz0 | |
| + syl cn0 sylib swrdrn3 syl3anc eqtrd ) AFJZDBDKLZMUANZJZDBVDONUBZVCVFPAFVE | |
| + GUCUDADEUELZUFZUGQZBRVDUHNZQZVDVKQZVFVGPADCEUILSZDCEUJLSZVJHCDEUKZCDEVHVH | |
| + ULUMTABRVDONQVLIBRVDUNUQAVDURQZVMAVNVOVQHVPCDEUOTVDUPUSBVDVIDUTVAVB $. | |
| + $} | |
| + | |
| + ${ | |
| + $d B i j k $. $d F i j k $. $d G i j k $. $d H i j k $. $d I i j k $. | |
| + $d J i j k $. $d P i j k $. $d i j $. $d ph i j k $. | |
| + acycgrspthendd.i $e |- I = ( iEdg ` G ) $. | |
| + acycgrspthendd.h $e |- H = | |
| + ( F substr <. B , ( # ` F ) >. ) $. | |
| + acycgrspthendd.g $e |- ( ph -> G e. UPGraph ) $. | |
| + acycgrspthendd.a $e |- ( ph -> G e. AcyclicGraph ) $. | |
| + acycgrspthendd.p $e |- ( ph -> F ( SPaths ` G ) P ) $. | |
| + acycgrspthendd.b $e |- ( ph -> B e. ( 0 ..^ ( # ` F ) ) ) $. | |
| + acycgrspthendd.j $e |- ( ph -> J e. dom I ) $. | |
| + acycgrspthendd.e $e |- ( ph -> ( I ` J ) = | |
| + { ( P ` ( # ` F ) ) , ( P ` B ) } ) $. | |
| + $( In an acyclic graph, every edge from the final vertex of a simple path | |
| + to an earlier vertex joins it to the immediately preceding vertex. | |
| + (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + acycgrspthendd $p |- ( ph -> | |
| + B = ( ( # ` F ) - 1 ) ) $= | |
| + ( vk cfv co wcel cv wceq chash c1 cmin cfzo wfun cima wrex cc0 cdm cspths | |
| + wf cword wbr cwlks spthiswlk wlkf 3syl wrdf syl ffun crn cop csubstr eqid | |
| + caddc acycgrspthendrnd spthswrdrnd eleqtrd fvelima wa cupgr adantr simprl | |
| + syl2anc cpr simprr fveq2d eqtrd spthendprd rexlimddv ) AQUAZDRZHUBZBDUCRZ | |
| + UDUESUBQBWFUFSZADUGZHDWGUHZTWEQWGUIAUJWFUFSZGUKZDUMZWHADWKUNTZWLADCEULRUO | |
| + ZDCEUPRUOWMMCDEUQCDEGIURUSWKDUTVAWJWKDVBVAAHFVCWIABCCBWFUDVGSVDVESZDEFGHI | |
| + JWOVFKLMNOPVHABCDEFJMNVIVJQHWGDVKVPAWCWGTZWEVLZVLZBCDEGWCIAEVMTWQKVNAWNWQ | |
| + MVNABWJTWQNVNAWPWEVOWRWDGRHGRZWFCRBCRVQZWRWDHGAWPWEVRVSAWSWTUBWQPVNVTWAWB | |
| + $. | |
| + $} | |
| + | |
| + ${ | |
| + $d B i j k $. $d F i j k $. $d G i j k $. $d H i j k $. $d I i j k $. | |
| + $d J i j k $. $d W i j k $. $d i j $. $d ph i j k $. | |
| + acycgrwspthendlem.i $e |- I = ( iEdg ` G ) $. | |
| + acycgrwspthendlem.g $e |- ( ph -> G e. UPGraph ) $. | |
| + acycgrwspthendlem.a $e |- ( ph -> G e. AcyclicGraph ) $. | |
| + acycgrwspthendlem.w $e |- ( ph -> | |
| + W e. ( N WSPathsN G ) ) $. | |
| + acycgrwspthendlem.n $e |- ( ph -> N e. NN ) $. | |
| + acycgrwspthendlem.s $e |- ( ph -> | |
| + S e. ( G NeighbVtx ( lastS ` W ) ) ) $. | |
| + acycgrwspthendlem.p $e |- ( ph -> F ( SPaths ` G ) W ) $. | |
| + acycgrwspthendlem.b $e |- ( ph -> B e. ( 0 ... N ) ) $. | |
| + acycgrwspthendlem.v $e |- ( ph -> ( W ` B ) = S ) $. | |
| + acycgrwspthendlem.j $e |- ( ph -> J e. dom I ) $. | |
| + acycgrwspthendlem.e $e |- ( ph -> ( I ` J ) = | |
| + { ( W ` N ) , ( W ` B ) } ) $. | |
| + $( Lemma for ~ acycgrwspthendd . (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + acycgrwspthendlem $p |- ( ph -> | |
| + S = ( W ` ( N - 1 ) ) ) $= | |
| + ( cfv c1 cmin co eqcomd chash cop csubstr eqid cc0 cfzo wne cfz wcel clsw | |
| + wa cnbgr wnel nbgrnself2 df-nel mpbi a1i jca nelne2 syl cwwlksn cwwspthsn | |
| + wn wceq wspthsswwlkn sseli wwlknlsw 3netr4d wi necon3d mpd fzofzim cspths | |
| + fveq2 wbr wspthnlen oveq2d eleqtrrd cpr fveq2d eqidd eqtrd acycgrspthendd | |
| + preq12d oveq1d ) ACBIUAZHUBUCUDZIUAAWKCRUEABWLIABDUFUAZUBUCUDWLABIDEDBWMU | |
| + GUHUDZFGJWNUIKLPABUJHUKUDZUJWMUKUDABHULZBUJHUMUDUNZUPBWOUNAWPWQAWKHIUAZUL | |
| + WPACIUOUAZWKWRACEWSUQUDZUNZWSWTUNVHZUPCWSULAXAXBOXBAWSWTURXBEWSUSWSWTUTVA | |
| + VBVCCWSWTVDVERAIHEVFUDZUNZWRWSVIAIHEVGUDZUNXDMXEXCIEHVJVKVEZEHIVLVEVMABHW | |
| + KWRBHVIWKWRVIVNABHIVSVBVOVPQVCBHVQVEAWMHUJUKAXDDIEVRUAVTZUPWMHVIAXDXGXFPV | |
| + CDEHIWAVEZWBWCSAGFUAWRWKWDWMIUAZWKWDTAWRXIWKWKAHWMIAWMHXHUEWEAWKWFWIWGWHA | |
| + WMHUBUCXHWJWGWEWG $. | |
| + $} | |
| + | |
| + ${ | |
| + acycgrwspthendedgd.g $e |- ( ph -> G e. UPGraph ) $. | |
| + acycgrwspthendedgd.a $e |- ( ph -> G e. AcyclicGraph ) $. | |
| + acycgrwspthendedgd.w $e |- ( ph -> | |
| + W e. ( N WSPathsN G ) ) $. | |
| + acycgrwspthendedgd.s $e |- ( ph -> | |
| + S e. ( G NeighbVtx ( lastS ` W ) ) ) $. | |
| + acycgrwspthendedgd.v $e |- ( ph -> ( W ` B ) = S ) $. | |
| + $( A neighbor of the final vertex represented at a path index determines | |
| + the corresponding graph edge. (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + acycgrwspthendedgd $p |- ( ph -> | |
| + { ( W ` N ) , ( W ` B ) } e. ( Edg ` G ) ) $= | |
| + ( cfv cpr clsw cedg cwwlksn co wcel wceq syl cwwspthsn wspthsswwlkn sseli | |
| + wwlknlsw preq12d prcom eqtrd cnbgr cusgr wb cupgr cacycgr wa upgracycusgr | |
| + a1i jca eqid nbusgreledg mpbid eqeltrd ) AEFLZBFLZMZCFNLZMZDOLZAVCVDCMZVE | |
| + AVAVDVBCAFEDPQZRZVAVDSAFEDUAQZRVIIVJVHFDEUBUCTDEFUDTKUEVGVESAVDCUFUOUGACD | |
| + VDUHQRZVEVFRZJADUIRZVKVLUJADUKRZDULRZUMVMAVNVOGHUPDUNTVFDVDCVFUQURTUSUT | |
| + $. | |
| + $} | |
| + | |
| + ${ | |
| + $d B i $. $d F i $. $d G i $. $d I i $. $d N i $. $d S i $. $d W i $. | |
| + $d ph i $. | |
| + acycgrwspthendlem2.i $e |- I = ( iEdg ` G ) $. | |
| + acycgrwspthendlem2.g $e |- ( ph -> G e. UPGraph ) $. | |
| + acycgrwspthendlem2.a $e |- ( ph -> G e. AcyclicGraph ) $. | |
| + acycgrwspthendlem2.w $e |- ( ph -> | |
| + W e. ( N WSPathsN G ) ) $. | |
| + acycgrwspthendlem2.n $e |- ( ph -> N e. NN ) $. | |
| + acycgrwspthendlem2.s $e |- ( ph -> | |
| + S e. ( G NeighbVtx ( lastS ` W ) ) ) $. | |
| + acycgrwspthendlem2.p $e |- ( ph -> F ( SPaths ` G ) W ) $. | |
| + acycgrwspthendlem2.b $e |- ( ph -> B e. ( 0 ... N ) ) $. | |
| + acycgrwspthendlem2.v $e |- ( ph -> ( W ` B ) = S ) $. | |
| + acycgrwspthendlem2.e $e |- ( ph -> | |
| + { ( W ` N ) , ( W ` B ) } e. ( Edg ` G ) ) $. | |
| + $( Lemma for ~ acycgrwspthendd . (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + acycgrwspthendlem2 $p |- ( ph -> | |
| + S = ( W ` ( N - 1 ) ) ) $= | |
| + ( wcel adantr vi cfv cpr cv wceq c1 cmin co cdm cedg cuhgr cupgr upgruhgr | |
| + wrex wb syl uhgredgiedgb mpbid wa cacycgr cwwspthsn clsw cnbgr cspths wbr | |
| + cn cc0 cfz simprl simprr eqcomd acycgrwspthendlem rexlimddv ) AGHUBBHUBZU | |
| + CZUAUDZFUBZUEZCGUFUGUHHUBUEUAFUIZAVOEUJUBSZVRUAVSUNZRAEUKSZVTWAUOAEULSZWB | |
| + JEUMUPUAVOEFIUQUPURAVPVSSZVRUSZUSZBCDEFVPGHIAWCWEJTAEUTSWEKTAHGEVAUHSWELT | |
| + AGVFSWEMTACEHVBUBVCUHSWENTADHEVDUBVEWEOTABVGGVHUHSWEPTAVNCUEWEQTAWDVRVIWF | |
| + VOVQAWDVRVJVKVLVM $. | |
| + $} | |
| + | |
| + ${ | |
| + acycgrwspthendlem3.i $e |- I = ( iEdg ` G ) $. | |
| + acycgrwspthendlem3.g $e |- ( ph -> G e. UPGraph ) $. | |
| + acycgrwspthendlem3.a $e |- ( ph -> G e. AcyclicGraph ) $. | |
| + acycgrwspthendlem3.w $e |- ( ph -> | |
| + W e. ( N WSPathsN G ) ) $. | |
| + acycgrwspthendlem3.n $e |- ( ph -> N e. NN ) $. | |
| + acycgrwspthendlem3.s $e |- ( ph -> | |
| + S e. ( G NeighbVtx ( lastS ` W ) ) ) $. | |
| + acycgrwspthendlem3.p $e |- ( ph -> F ( SPaths ` G ) W ) $. | |
| + acycgrwspthendlem3.b $e |- ( ph -> B e. ( 0 ... N ) ) $. | |
| + acycgrwspthendlem3.v $e |- ( ph -> ( W ` B ) = S ) $. | |
| + $( Lemma for ~ acycgrwspthendd . (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + acycgrwspthendlem3 $p |- ( ph -> | |
| + S = ( W ` ( N - 1 ) ) ) $= | |
| + ( acycgrwspthendedgd acycgrwspthendlem2 ) ABCDEFGHIJKLMNOPQABCEGHJKLNQRS | |
| + $. | |
| + $} | |
| + | |
| + ${ | |
| + $d F b $. $d G b $. $d I b $. $d N b $. $d S b $. $d W b $. | |
| + $d ph b $. | |
| + acycgrwspthendlem4.i $e |- I = ( iEdg ` G ) $. | |
| + acycgrwspthendlem4.g $e |- ( ph -> G e. UPGraph ) $. | |
| + acycgrwspthendlem4.a $e |- ( ph -> G e. AcyclicGraph ) $. | |
| + acycgrwspthendlem4.w $e |- ( ph -> | |
| + W e. ( N WSPathsN G ) ) $. | |
| + acycgrwspthendlem4.n $e |- ( ph -> N e. NN ) $. | |
| + acycgrwspthendlem4.s $e |- ( ph -> | |
| + S e. ( G NeighbVtx ( lastS ` W ) ) ) $. | |
| + acycgrwspthendlem4.r $e |- ( ph -> S e. ran W ) $. | |
| + acycgrwspthendlem4.p $e |- ( ph -> F ( SPaths ` G ) W ) $. | |
| + $( Lemma for ~ acycgrwspthendd . (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + acycgrwspthendlem4 $p |- ( ph -> | |
| + S = ( W ` ( N - 1 ) ) ) $= | |
| + ( vb cfv co wcel adantr cv wceq c1 cmin cc0 cfz crn wrex wfn wb cwwspthsn | |
| + wspthnfn syl fvelrnb mpbid wa cupgr cacycgr cn cnbgr cspths simprl simprr | |
| + clsw wbr acycgrwspthendlem3 rexlimddv ) APUAZGQBUBZBFUCUDRGQUBPUEFUFRZABG | |
| + UGSZVIPVJUHZNAGVJUIZVKVLUJAGFDUKRSZVMKDFGULUMPVJBGUNUMUOAVHVJSZVIUPZUPVHB | |
| + CDEFGHADUQSVPITADURSVPJTAVNVPKTAFUSSVPLTABDGVDQUTRSVPMTACGDVAQVEVPOTAVOVI | |
| + VBAVOVIVCVFVG $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G b f i $. $d N b f i $. $d S b f i $. $d W b f i $. $d b f i $. | |
| + $( In an acyclic graph, a neighbor of the final vertex which already | |
| + occurs on a positive-length simple path is its predecessor. | |
| + (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + acycgrwspthendd $p |- ( | |
| + ( ( G e. UPGraph /\ G e. AcyclicGraph /\ | |
| + W e. ( N WSPathsN G ) ) /\ | |
| + ( N e. NN /\ S e. ( G NeighbVtx ( lastS ` W ) ) /\ | |
| + S e. ran W ) ) -> | |
| + S = ( W ` ( N - 1 ) ) ) $= | |
| + ( vf cupgr wcel cacycgr cwwspthsn co w3a cn clsw cfv cnbgr crn cspths syl | |
| + wa cv wbr c1 cmin wceq wex simpl simp3 cn0 cvv cwwlksn wspthnp ciedg eqid | |
| + simp3d simpll1 simpll2 simpll3 simplr1 simplr2 simplr3 acycgrwspthendlem4 | |
| + simpr exlimddv ) BFGZBHGZDCBIJGZKZCLGZABDMNOJGZADPGZKZSZETZDBQNUAZACUBUCJ | |
| + DNUDEVLVFVNEUEZVLVGVFVGVKUFVDVEVFUGRVFCUHGBUIGSDCBUJJGVOEBCDUKUNRVLVNSAVM | |
| + BBULNZCDVPUMVDVEVFVKVNUOVDVEVFVKVNUPVDVEVFVKVNUQVHVIVJVGVNURVHVIVJVGVNUSV | |
| + HVIVJVGVNUTVLVNVBVAVC $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G m w $. $d N m w $. $d S m w $. $d W m w $. $d m w $. | |
| + $( Every neighbor of the final vertex of a positive-length maximum simple | |
| + path in an acyclic graph is its predecessor. (Contributed by Mingli | |
| + Yuan, 12-Aug-2026.) $) | |
| + acycgrwspthmaxend $p |- ( | |
| + ( ( G e. UPGraph /\ G e. AcyclicGraph /\ | |
| + W e. ( N WSPathsN G ) ) /\ | |
| + ( N e. NN /\ | |
| + A. m A. w ( w e. ( m WSPathsN G ) -> m <_ N ) /\ | |
| + S e. ( G NeighbVtx ( lastS ` W ) ) ) ) -> | |
| + S = ( W ` ( N - 1 ) ) ) $= | |
| + ( wcel cwwspthsn co w3a cv wal cfv wa wceq cpr 3jca eqid syl jca cupgr cn | |
| + cacycgr cle wbr wi clsw cnbgr crn c1 cmin simpl simpr1 simpr3 cvtx simpl1 | |
| + cedg simpl3 simpr2 nbgrisvtx prcom a1i wb simpl2 upgracycusgr nbusgreledg | |
| + cusgr mpbid eqeltrd wspthmaxend acycgrwspthendd ) DUAGZDUCGZFEDHIGZJZEUBG | |
| + ZAKCKZDHIGVQEUDUEUFALCLZBDFUGMZUHIGZJZNZVOVPVTBFUIGZJZNBEUJUKIFMOWBVOWDVO | |
| + WAULWBVPVTWCVOVPVRVTUMVOVPVRVTUNZWBVLVNVRJZBDUOMZGZVSBPZDUQMZGZNZNWCWBWFW | |
| + LWBVLVNVRVLVMVNWAUPZVLVMVNWAURVOVPVRVTUSQWBWHWKWBVTWHWEDVSBWGWGRUTSWBWIBV | |
| + SPZWJWIWNOWBVSBVAVBWBVTWNWJGZWEWBDVGGZVTWOVCWBVLVMNWPWBVLVMWMVLVMVNWAVDTD | |
| + VESWJDVSBWJRVFSVHVITTABCDEFVJSQTBDEFVKS $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G m s w $. $d N m s w $. $d W m s w $. $d m s w $. | |
| + $( The neighbors of the final vertex of a positive-length maximum simple | |
| + path in an acyclic graph form the singleton containing its predecessor. | |
| + (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + acycgrwspthmaxendnb $p |- ( | |
| + ( ( G e. UPGraph /\ G e. AcyclicGraph /\ | |
| + W e. ( N WSPathsN G ) ) /\ | |
| + ( N e. NN /\ | |
| + A. m A. w ( w e. ( m WSPathsN G ) -> m <_ N ) ) ) -> | |
| + ( G NeighbVtx ( lastS ` W ) ) = | |
| + { ( W ` ( N - 1 ) ) } ) $= | |
| + ( vs cupgr wcel cacycgr cwwspthsn co w3a cv wal wa cfv wceq jca syl cpr | |
| + cn cle wbr wi clsw cnbgr c1 cmin simpll simplrl simplrr acycgrwspthmaxend | |
| + csn simpr 3jca elsn sylibr ex ssrdv cedg prcom simpl3 simprl wspthendpred | |
| + vex a1i eqeltrd cusgr simpl1 simpl2 upgracycusgr nbusgreledg mpbird snssd | |
| + wb eqid eqssd ) CGHZCIHZEDCJKHZLZDUAHZAMBMZCJKHWCDUBUCUDANBNZOZOZCEUEPZUF | |
| + KZDUGUHKEPZUMZWFFWHWJWFFMZWHHZWKWJHZWFWLOZWKWIQZWMWNWAWBWDWLLZOWOWNWAWPWA | |
| + WEWLUIWNWBWDWLWAWBWDWLUJWAWBWDWLUKWFWLUNUORAWKBCDEULSWKWIFVEUPUQURUSWFWIW | |
| + HWFWIWHHZWIWGTZCUTPZHZWFWRWGWITZWSWRXAQWFWIWGVAVFWFVTWBOXAWSHWFVTWBVRVSVT | |
| + WEVBWAWBWDVCRCDEVDSVGWFCVHHZWQWTVOWFVRVSOXBWFVRVSVRVSVTWEVIVRVSVTWEVJRCVK | |
| + SWSCWGWIWSVPVLSVMVNVQ $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G x $. $d N x $. $d W x $. | |
| + $( The final symbol of a fixed-length simple-path word is a vertex of the | |
| + graph. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + wspthnlswvtx $p |- ( W e. ( N WSPathsN G ) -> | |
| + ( lastS ` W ) e. ( Vtx ` G ) ) $= | |
| + ( cwwspthsn co wcel cfv clsw cvtx cwwlksn wspthsswwlkn sseli wwlknlsw syl | |
| + wceq cc0 wa eqid wspthnevtx simpr eqeltrrd ) CBADEZFZBCGZCHGZAIGZUCCBAJEZ | |
| + FUDUEOUBUGCABKLABCMNUCPCGUFFZUDUFFZQUIABUFCUFRSUHUITNUA $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G s $. $d N s $. $d W s $. $d G m w $. $d N m w $. $d W m w $. | |
| + $d m s w $. | |
| + $( The neighbors of the final vertex of a maximum simple path in an | |
| + acyclic graph occur on that path. (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + acycgrwspthmaxendnbss $p |- ( | |
| + ( ( G e. UPGraph /\ G e. AcyclicGraph /\ | |
| + W e. ( N WSPathsN G ) ) /\ | |
| + A. m A. w ( w e. ( m WSPathsN G ) -> m <_ N ) ) -> | |
| + ( G NeighbVtx ( lastS ` W ) ) C_ ran W ) $= | |
| + ( vs cupgr wcel cwwspthsn co w3a cv wi wal wa cfv cpr eqid a1i syl simpl1 | |
| + cacycgr cle wbr clsw cnbgr cvtx cedg simpl3 simpr 3jca nbgrisvtx cusgr wb | |
| + crn simpl2 upgracycusgr nbusgreledg eleq1i biimpi sylbid jcad wspthmaxend | |
| + jca prcom syl6an ssrdv ) CGHZCUBHZEDCIJHZKZALBLZCIJHVLDUCUDMANBNZOZFCEUEP | |
| + ZUFJZEUOZVNVHVJVMKFLZVPHZVRCUGPZHZVOVRQZCUHPZHZOVRVQHVNVHVJVMVHVIVJVMUAZV | |
| + HVIVJVMUIVKVMUJUKVNVSWAWDVSWAMVNCVOVRVTVTRULSVNVSVRVOQZWCHZWDVNCUMHZVSWGU | |
| + NVNVHVIOWHVNVHVIWEVHVIVJVMUPVDCUQTWCCVOVRWCRURTWGWDMVNWGWDWFWBWCVRVOVEUSU | |
| + TSVAVBAVRBCDEVCVFVG $. | |
| + $} | |
| + | |
| + $( A zero-length simple path is the singleton word containing its final | |
| + vertex. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + wspth0s1 $p |- ( W e. ( 0 WSPathsN G ) -> | |
| + W = <" ( lastS ` W ) "> ) $= | |
| + ( cc0 cwwspthsn co wcel cfv cs1 clsw cvtx cword chash c1 wa wspthnwrd caddc | |
| + wceq cwwlksn syl eqtrd wspthsswwlkn sseli wwlknlen 0p1e1 a1i jca eqs1 s1eqd | |
| + wwlknlsw ) BCADEZFZBCBGZHZBIGZHUKBAJGZKFZBLGZMQZNBUMQUKUPURACBOUKUQCMPEZMUK | |
| + BCAREZFZUQUSQUJUTBACUAUBZACBUCSUSMQUKUDUETUFUOBUGSUKULUNUKVAULUNQVBACBUISUH | |
| + T $. | |
| + | |
| + $( The range of a zero-length simple path consists of its final vertex. | |
| + (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + wspth0rn $p |- ( W e. ( 0 WSPathsN G ) -> | |
| + ran W = { ( lastS ` W ) } ) $= | |
| + ( cc0 cwwspthsn wcel crn clsw cfv cs1 wspth0s1 rneqd cvtx wceq wspthnlswvtx | |
| + co csn s1rn syl eqtrd ) BCADOEZBFBGHZIZFZUAPZTBUBABJKTUAALHZEUCUDMACBNUAUEQ | |
| + RS $. | |
| + | |
| + $( A simple path whose length parameter is zero has singleton range. | |
| + (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + wspthlen0rn $p |- ( ( W e. ( N WSPathsN G ) /\ N = 0 ) -> | |
| + ran W = { ( lastS ` W ) } ) $= | |
| + ( cwwspthsn co wcel cc0 wceq wa crn clsw cfv csn simpl simpr oveq1d eleqtrd | |
| + wspth0rn syl ) CBADEZFZBGHZIZCGADEZFCJCKLMHUCCTUDUAUBNUCBGADUAUBOPQACRS $. | |
| + | |
| + ${ | |
| + $d G m s w $. $d W m s w $. $d m s w $. | |
| + $( The final vertex of a zero-length maximum simple path in an acyclic | |
| + graph has no neighbors. (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + acycgrwspthmaxendnb0 $p |- ( | |
| + ( ( G e. UPGraph /\ G e. AcyclicGraph /\ | |
| + W e. ( 0 WSPathsN G ) ) /\ | |
| + A. m A. w ( w e. ( m WSPathsN G ) -> m <_ 0 ) ) -> | |
| + ( G NeighbVtx ( lastS ` W ) ) = (/) ) $= | |
| + ( vs cupgr wcel cacycgr cc0 cwwspthsn co w3a cv cle wbr wal a1i wceq syl6 | |
| + wi wa clsw cfv cnbgr wnel nbgrnself2 csn crn acycgrwspthmaxendnbss simpl3 | |
| + wn wspth0rn syl sseqtrd sseld wb vex elsn sylibd eqcom biimpi eleq1a mpdd | |
| + nnel biimpri mt2d eq0rdv ) CFGZCHGZDICJKGZLAMBMZCJKGVKINOTAPBPZUAZECDUBUC | |
| + ZUDKZVMEMZVOGZVNVOUEZVRVMCVNUFQVMVQVNVOGZVRUKZVMVQVNVPRZVSVMVQVPVNRZWAVMV | |
| + QVPVNUGZGZWBVMVOWCVPVMVODUHZWCABCIDUIVMVJWEWCRVHVIVJVLUJCDULUMUNUOWDWBUPV | |
| + MVPVNEUQURQUSWBWAVPVNUTVASVQWAVSTTVMVPVOVNVBQVCVTVSVNVOVDVESVFVG $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G m w $. $d W m w $. $d m w $. | |
| + $( The final vertex of a zero-length maximum simple path in an acyclic | |
| + graph has degree zero. (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + acycgrwspthmaxendvd0 $p |- ( | |
| + ( ( G e. UPGraph /\ G e. AcyclicGraph /\ | |
| + W e. ( 0 WSPathsN G ) ) /\ | |
| + A. m A. w ( w e. ( m WSPathsN G ) -> m <_ 0 ) ) -> | |
| + ( ( VtxDeg ` G ) ` ( lastS ` W ) ) = 0 ) $= | |
| + ( cupgr wcel cacycgr cc0 cwwspthsn co w3a cv wal wa cfv chash wceq jca c0 | |
| + syl cle wi clsw cnbgr cvtxdg cusgr cvtx simpl1 simpl2 upgracycusgr simpl3 | |
| + wbr wspthnlswvtx hashnbusgrvd acycgrwspthmaxendnb0 fveq2d hash0 a1i eqtrd | |
| + eqid eqtr3d ) CEFZCGFZDHCIJFZKALBLZCIJFVEHUAULUBAMBMZNZCDUCOZUDJZPOZVHCUE | |
| + OOZHVGCUFFZVHCUGOZFZNVJVKQVGVLVNVGVBVCNVLVGVBVCVBVCVDVFUHVBVCVDVFUIRCUJTV | |
| + GVDVNVBVCVDVFUKCHDUMTRVHCVMVMUTUNTVGVJSPOZHVGVISPABCDUOUPVOHQVGUQURUSVA | |
| + $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G m s w $. $d N m s w $. $d W m s w $. $d m s w $. | |
| + $( If a maximum simple path in an acyclic graph has length zero, then its | |
| + final vertex has no neighbors. (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + acycgrwspthmaxendnb0g $p |- ( | |
| + ( ( G e. UPGraph /\ G e. AcyclicGraph /\ | |
| + W e. ( N WSPathsN G ) ) /\ | |
| + ( N = 0 /\ | |
| + A. m A. w ( w e. ( m WSPathsN G ) -> m <_ N ) ) ) -> | |
| + ( G NeighbVtx ( lastS ` W ) ) = (/) ) $= | |
| + ( vs cupgr wcel cacycgr cwwspthsn co wceq cv wi wal wa a1i jca syl syl6 | |
| + w3a cc0 cle wbr clsw cfv wnel nbgrnself2 wn csn crn acycgrwspthmaxendnbss | |
| + cnbgr wss simpl simprr simpl3 simprl wspthlen0rn sseqtrd sseld vex sylibd | |
| + wb elsn eqcom biimpi eleq1a mpdd nnel biimpri mt2d eq0rdv ) CGHZCIHZEDCJK | |
| + HZUAZDUBLZAMBMZCJKHVSDUCUDNAOBOZPZPZFCEUEUFZUMKZWBFMZWDHZWCWDUGZWGWBCWCUH | |
| + QWBWFWCWDHZWGUIZWBWFWCWELZWHWBWFWEWCLZWJWBWFWEWCUJZHZWKWBWDWLWEWBWDEUKZWL | |
| + WBVQVTPWDWNUNWBVQVTVQWAUOVQVRVTUPRABCDEULSWBVPVRPWNWLLWBVPVRVNVOVPWAUQVQV | |
| + RVTURRCDEUSSUTVAWMWKVDWBWEWCFVBVEQVCWKWJWEWCVFVGTWFWJWHNNWBWEWDWCVHQVIWIW | |
| + HWCWDVJVKTVLVM $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G m w $. $d N m w $. $d W m w $. $d m w $. | |
| + $( If a maximum simple path in an acyclic graph has length zero, then its | |
| + final vertex has degree zero. (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + acycgrwspthmaxendvd0g $p |- ( | |
| + ( ( G e. UPGraph /\ G e. AcyclicGraph /\ | |
| + W e. ( N WSPathsN G ) ) /\ | |
| + ( N = 0 /\ | |
| + A. m A. w ( w e. ( m WSPathsN G ) -> m <_ N ) ) ) -> | |
| + ( ( VtxDeg ` G ) ` ( lastS ` W ) ) = 0 ) $= | |
| + ( cupgr wcel cacycgr cwwspthsn co cc0 wceq cv wal wa cfv chash jca syl c0 | |
| + w3a cle wi clsw cnbgr cvtxdg cusgr cvtx simpl1 simpl2 upgracycusgr simpl3 | |
| + wbr wspthnlswvtx eqid hashnbusgrvd acycgrwspthmaxendnb0g fveq2d hash0 a1i | |
| + eqtrd eqtr3d ) CFGZCHGZEDCIJGZUADKLAMBMZCIJGVFDUBUMUCANBNOZOZCEUDPZUEJZQP | |
| + ZVICUFPPZKVHCUGGZVICUHPZGZOVKVLLVHVMVOVHVCVDOVMVHVCVDVCVDVEVGUIVCVDVEVGUJ | |
| + RCUKSVHVEVOVCVDVEVGULCDEUNSRVICVNVNUOUPSVHVKTQPZKVHVJTQABCDEUQURVPKLVHUSU | |
| + TVAVB $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G m w $. $d N m w $. $d V m w $. $d W m w $. $d m w $. | |
| + acycgrwspthmaxnn.v $e |- V = ( Vtx ` G ) $. | |
| + $( A maximum simple path in a connected acyclic graph with at least two | |
| + vertices has positive length. (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + acycgrwspthmaxnn $p |- ( | |
| + ( ( ( G e. UPGraph /\ G e. ConnGraph /\ G e. AcyclicGraph ) /\ | |
| + ( 1 < ( # ` V ) /\ W e. ( N WSPathsN G ) ) ) /\ | |
| + A. m A. w ( w e. ( m WSPathsN G ) -> m <_ N ) ) -> | |
| + N e. NN ) $= | |
| + ( wcel w3a cfv wbr cwwspthsn co wa cv wal cc0 syl jca a1d cconngr cacycgr | |
| + cupgr c1 chash clt cle wi cn0 wne cn simplrr wspthnn0 clsw cvtxdg simpll2 | |
| + simpll1 simpll3 upgracycumgr wspthnlswvtx eleq2i simplrl vdn0conngrumgrv2 | |
| + cumgr cvtx sylibr wceq 3jcad idd simpr jcad acycgrwspthmaxendvd0g necon3d | |
| + syl6 mpd elnnne0 ) CUCHZCUAHZCUBHZIZUDEUEJUFKZFDCLMHZNZNZAOBOZCLMHWEDUGKU | |
| + HAPBPZNZDUIHZDQUJZNDUKHWGWHWIWGWBWHVTWAWBWFULZCDFUMRWGFUNJZCUOJJZQUJZWIWG | |
| + VRCVDHZNZWKEHZWANZNWMWGWOWQWGVRWNVQVRVSWCWFUPWGVQVSNWNWGVQVSVQVRVSWCWFUQZ | |
| + VQVRVSWCWFURZSCUSRSWGWPWAWGWKCVEJZHZWPWGWBXAWJCDFUTREWTWKGVAVFVTWAWBWFVBS | |
| + SCWKEGVCRWGDQWLQWGDQVGZVQVSWBIZXBWFNZNWLQVGWGXBXCXDWGXBVQVSWBWGVQXBWRTWGV | |
| + SXBWSTWGWBXBWJTVHWGXBXBWFWGXBVIWGWFXBWDWFVJTVKVKABCDFVLVNVMVOSDVPVF $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G m w $. $d N m w $. $d W m w $. $d m w $. | |
| + $( The final vertex of a positive-length maximum simple path in an acyclic | |
| + graph has degree one. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + acycgrwspthmaxendvd1 $p |- ( | |
| + ( ( G e. UPGraph /\ G e. AcyclicGraph /\ | |
| + W e. ( N WSPathsN G ) ) /\ | |
| + ( N e. NN /\ | |
| + A. m A. w ( w e. ( m WSPathsN G ) -> m <_ N ) ) ) -> | |
| + ( ( VtxDeg ` G ) ` ( lastS ` W ) ) = 1 ) $= | |
| + ( cupgr wcel cacycgr cwwspthsn co cv wal wa cfv chash c1 wceq jca syl cvv | |
| + w3a cn cle wbr wi clsw cvtxdg cnbgr cmin cusgr simpl1 simpl2 upgracycusgr | |
| + csn cvtx simpl3 wspthnlswvtx eqid hashnbusgrvd eqcomd acycgrwspthmaxendnb | |
| + fveq2d fvexd hashsng 3eqtrd ) CFGZCHGZEDCIJGZUADUBGAKBKZCIJGVIDUCUDUEALBL | |
| + MZMZEUFNZCUGNNZCVLUHJZONZDPUIJZENZUNZONZPVKVOVMVKCUJGZVLCUONZGZMVOVMQVKVT | |
| + WBVKVFVGMVTVKVFVGVFVGVHVJUKVFVGVHVJULRCUMSVKVHWBVFVGVHVJUPCDEUQSRVLCWAWAU | |
| + RUSSUTVKVNVROABCDEVAVBVKVQTGVSPQVKVPEVCVQTVDSVE $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G m w $. $d N m w $. $d V m w $. $d W m w $. $d m w $. | |
| + treewspthmaxendvd1.v $e |- V = ( Vtx ` G ) $. | |
| + $( The final vertex of a maximum simple path in a tree with at least two | |
| + vertices has degree one. (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + treewspthmaxendvd1 $p |- ( | |
| + ( ( G e. Tree /\ 1 < ( # ` V ) /\ | |
| + W e. ( N WSPathsN G ) ) /\ | |
| + A. m A. w ( w e. ( m WSPathsN G ) -> m <_ N ) ) -> | |
| + ( ( VtxDeg ` G ) ` ( lastS ` W ) ) = 1 ) $= | |
| + ( wcel c1 cfv wbr cwwspthsn co w3a cv wal wa syl 3jca jca ctree chash clt | |
| + cle wi cupgr cacycgr cn clsw cvtxdg wceq simpl1 treeacycgr simpl3 cconngr | |
| + treeupgr treeconngr simpl2 simpr acycgrwspthmaxnn acycgrwspthmaxendvd1 ) | |
| + CUAHZIEUBJUCKZFDCLMHZNZAOBOZCLMHVFDUDKUEAPBPZQZCUFHZCUGHZVDNZDUHHZVGQZQFU | |
| + IJCUJJJIUKVHVKVMVHVIVJVDVHVBVIVBVCVDVGULZCUPRZVHVBVJVNCUMRZVBVCVDVGUNZSVH | |
| + VLVGVHVICUOHZVJNZVCVDQZQZVGQVLVHWAVGVHVSVTVHVIVRVJVOVHVBVRVNCUQRVPSVHVCVD | |
| + VBVCVDVGURVQTTVEVGUSZTABCDEFGUTRWBTTABCDFVAR $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G m p v w $. $d n m p v w $. $d V m p v w $. $d m p v w $. | |
| + $d p v w $. | |
| + treewspthmaxleaf.v $e |- V = ( Vtx ` G ) $. | |
| + $( A maximum simple path in a tree with at least two vertices supplies a | |
| + vertex of degree one. (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + treewspthmaxleaf $p |- ( | |
| + ( ( G e. Tree /\ 1 < ( # ` V ) ) /\ | |
| + ( p e. ( n WSPathsN G ) /\ | |
| + A. m A. w ( w e. ( m WSPathsN G ) -> m <_ n ) ) ) -> | |
| + E. v e. V ( ( VtxDeg ` G ) ` v ) = 1 ) $= | |
| + ( ctree wcel c1 cfv wbr wa cv cwwspthsn co wal wceq syl clt cle wi cvtxdg | |
| + chash clsw id fveq2d eqeq1d cvtx simprl wspthnlswvtx eleq2i sylibr simpll | |
| + w3a simplr 3jca simprr jca treewspthmaxendvd1 rspcedvdw ) EIJZKFUELUAMZNZ | |
| + GOZDOZEPQJZAOCOZEPQJVIVGUBMUCARCRZNZNZBOZEUDLZLZKSVFUFLZVNLZKSZBVPFVMVPSZ | |
| + VOVQKVSVMVPVNVSUGUHUIVLVPEUJLZJZVPFJVLVHWAVEVHVJUKZEVGVFULTFVTVPHUMUNVLVC | |
| + VDVHUPZVJNVRVLWCVJVLVCVDVHVCVDVKUOVCVDVKUQWBURVEVHVJUSUTACEVGFVFHVATVB $. | |
| + | |
| + $( If a tree with at least two vertices has a maximum simple path, then it | |
| + has a vertex of degree one. (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + treewspthmaxleafex $p |- ( ( G e. Tree /\ 1 < ( # ` V ) ) -> | |
| + ( E. p ( p e. ( n WSPathsN G ) /\ | |
| + A. m A. w ( w e. ( m WSPathsN G ) -> m <_ n ) ) -> | |
| + E. v e. V ( ( VtxDeg ` G ) ` v ) = 1 ) ) $= | |
| + ( ctree wcel c1 chash cfv clt wbr wa cv cwwspthsn co wal cle wi wceq wrex | |
| + cvtxdg treewspthmaxleaf ex exlimdv ) EIJKFLMNOPZGQDQZERSJAQCQZERSJUKUJUAO | |
| + UBATCTPZBQEUEMMKUCBFUDZGUIULUMABCDEFGHUFUGUH $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G a k m n p v w $. $d L a m p v w $. $d V a m p v w $. | |
| + $d a k n p v w $. $d k L $. $d k n $. $d k V $. $d L n $. | |
| + $d m n p v w $. $d V n $. $d n p v $. $d p v w $. | |
| + treeleaf.v $e |- V = ( Vtx ` G ) $. | |
| + treeleaf.l $e |- L = { n e. ( 0 ... ( # ` V ) ) | | |
| + ( n WSPathsN G ) =/= (/) } $. | |
| + $( A finite tree with at least two vertices has a vertex of degree one. | |
| + This follows Observation in [Diestel] p. 14. | |
| + (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + treeleaf $p |- ( ( G e. Tree /\ V e. Fin /\ 1 < ( # ` V ) ) -> | |
| + E. v e. V ( ( VtxDeg ` G ) ` v ) = 1 ) $= | |
| + ( va vp vw vm ctree wcel c1 cfv wbr cv c0 syl wa cfn chash clt w3a cvtxdg | |
| + wceq wrex cvtx wne simp1 treevtxne0 neeq1i sylibr cwwspthsn co cle wi wal | |
| + wex cvv simpl1 elex simpl2 simpr 3jca wspthmaxw simpll treewspthmaxleafex | |
| + simp3 jca simprr sylc rexlimddv n0limd ) CLMZEUAMZNEUBOUCPZUDZAQCUEOONUFA | |
| + EUGZHEVRCUHOZRUIZERUIVRVOWAVOVPVQUJZCUKSEVTRFULUMVRHQZEMZTZIQBQZCUNUOMJQK | |
| + QZCUNUOMWGWFUPPUQJURKURTIUSZVSBDWECUTMZVPWDUDWHBDUGWEWIVPWDWEVOWIVOVPVQWD | |
| + VACLVBSVOVPVQWDVCVRWDVDVEJWCKBCDEIFGVFSWEWFDMZWHTZTZVOVQTZWHVSWLVRWMVRWDW | |
| + KVGVRVOVQWBVOVPVQVIVJSWEWJWHVKJAKBCEIFVHVLVMVN $. | |
| + $} | |
| + | |
| + ${ | |
| + $d B i $. $d G i $. $d N i $. $d W i $. | |
| + $( An internal vertex of a fixed-length simple path in a simple graph has | |
| + two distinct neighbors on the path. (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + usgrwspthintnb $p |- ( ( G e. USGraph /\ | |
| + W e. ( N WSPathsN G ) /\ B e. ( 1 ... ( N - 1 ) ) ) -> | |
| + ( ( W ` ( B - 1 ) ) e. ( G NeighbVtx ( W ` B ) ) /\ | |
| + ( W ` ( B + 1 ) ) e. ( G NeighbVtx ( W ` B ) ) /\ | |
| + ( W ` ( B - 1 ) ) =/= ( W ` ( B + 1 ) ) ) ) $= | |
| + ( vi wcel co c1 cfz w3a cfv caddc cpr cc0 wceq syl cn0 wa 3syl cr cmin cv | |
| + cusgr cwwspthsn cnbgr wne cedg cfzo wral simp2 cwwlksn wspthsswwlkn sseli | |
| + cvtx cword chash eqid wwlknp simp3 fzelp1 cc wspthnn0 nn0cn npcan1 oveq2d | |
| + eleqtrd fz1fzo0m1 simpr fveq2d oveq1d cz simpl elfzelz zcn preq12d eleq1d | |
| + eqtrd rspcdv mpd wb simp1 nbusgreledg mpbird prcom a1i cuz elfznn elnn0uz | |
| + nnnn0 biimpi nn0z zre elfzel2 nn0re cle wbr elfzle2 ltm1d lelttrd eqeltrd | |
| + cn elfzod cdm wf1 wspthnf1m nnm1nn0 peano2rem lem1d letrd elfz2nn0 sylibr | |
| + 3jca wfn wspthnfn fndm eleqtrrd peano2nn clt zltp1le bicomd zltlem1 bitrd | |
| + jca peano2re ltp1d lttrd ltned dff14i ) BUCFZDCBUDGZFZAHCHUAGZIGFZJZAHUAG | |
| + ZDKZBADKZUEGZFZAHLGZDKZYRFZYPUUAUFZYNYSYPYQMZBUGKZFZYNEUBZDKZUUGHLGZDKZMZ | |
| + UUEFZENCUHGZUIZUUFYNYKUUNYIYKYMUJZYKDCBUKGZFZUUNYJUUPDBCULUMZUUQDBUNKZUOF | |
| + ZDUPKCHLGOZUUNJUUNEUUEBCUUSDUUSUQUUEUQZURUUTUVAUUNUSPZPPYNUULUUFEYOUUMYNA | |
| + HCIGZFYOUUMFYNAHYLHLGZIGZUVDYNYMAUVFFYIYKYMUSZAHYLUTPYNUVECHIYNCVAFZUVECO | |
| + YNCQFZUVHYNYKUVIUUOBCDVBZPZCVCPCVDPVEVFACVGPYNUUGYOOZRZUUKUUDUUEUVMUUHYPU | |
| + UJYQUVMUUGYODYNUVLVHZVIUVMUUIADUVMUUIYOHLGZAUVMUUGYOHLUVNVJUVMAVAFZUVOAOU | |
| + VMAVKFZUVPUVMYMUVQUVMYNYMYNUVLVLUVGPAHYLVMZPAVNPAVDPVQVIVOVPVRVSYNYIYSUUF | |
| + VTYIYKYMWAZUUEBYQYPUVBWBPWCYNUUBUUAYQMZUUEFZYNUVTYQUUAMZUUEUVTUWBOYNUUAYQ | |
| + WDWEYNUUNUWBUUEFZYNUUQUUNYNYKUUQUUOUURPUVCPYNUULUWCEAUUMYNANCYNAQFZANWFKF | |
| + ZYNAXAFZUWDYNYMUWFUVGAYLWGZPAWIPUWDUWEAWHWJPYNYKUVICVKFZUUOUVJCWKSZYNAYLC | |
| + YNYMUVQATFZUVGUVRAWLSZYNYMYLVKFYLTFUVGAHYLWMYLWLSZYNYKUVICTFUUOUVJCWNSZYN | |
| + YMAYLWOWPZUVGAHYLWQPZYNCUWMWRWSXBYNUUGAOZRZUUKUWBUUEUWQUUHYQUUJUUAUWQUUGA | |
| + DYNUWPVHZVIUWQUUIYTDUWQUUGAHLUWRVJVIVOVPVRVSWTYNYIUUBUWAVTUVSUUEBYQUUAUVB | |
| + WBPWCYNDXCZUUSDXDZYOUWSFZYTUWSFZYOYTUFZJZRUUCYNUWTUXDYNYKUWTUUOBCDXEPYNUX | |
| + AUXBUXCYNYONCIGZUWSYNYOQFZUVIYOCWOWPZJYOUXEFYNUXFUVIUXGYNYMUWFUXFUVGUWGAX | |
| + FSUVKYNYOACYNUWJYOTFUWKAXGPZUWKUWMYNAUWKXHYNAYLCUWKUWLUWMUWOYNCUWMXHXIXIX | |
| + LYOCXJXKYNYKDUXEXMUWSUXEOUUOBCDXNUXEDXOSZXPYNYTUXEUWSYNYTQFZUVIYTCWOWPZJY | |
| + TUXEFYNUXJUVIUXKYNYTXAFZUXJYNYMUWFUXLUVGUWGAXQSYTWIPUVKYNUXKUWNUWOYNUXKAC | |
| + XRWPZUWNYNUXMUXKYNUVQUWHRZUXMUXKVTYNUVQUWHYNYMUVQUVGUVRPUWIYCZACXSPXTYNUX | |
| + NUXMUWNVTUXOACYAPYBWCXLYTCXJXKUXIXPYNYOYTUXHYNYOAYTUXHUWKYNUWJYTTFUWKAYDP | |
| + YNAUWKWRYNAUWKYEYFYGXLYCUWSUUSDYOYTYHPXL $. | |
| + $} | |
| + | |
| + ${ | |
| + $d B i $. $d G i $. $d N i $. $d W i $. | |
| + $( An internal vertex of a fixed-length simple path in a simple graph has | |
| + degree at least two. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + usgrwspthintvdge2 $p |- ( ( G e. USGraph /\ | |
| + W e. ( N WSPathsN G ) /\ B e. ( 1 ... ( N - 1 ) ) ) -> | |
| + 2 <_ ( ( VtxDeg ` G ) ` ( W ` B ) ) ) $= | |
| + ( cusgr wcel cwwspthsn co c1 cmin cfz w3a c2 cfv cnbgr cvtxdg cle cvv syl | |
| + chash caddc ovex a1i wne usgrwspthintnb simp1d simp2d simp3d nehash2 cvtx | |
| + wceq simp1 nbgrsym sylibr eqid nbgrisvtx jca hashnbusgrvd eqcomd breqtrrd | |
| + wa ) BEFZDCBGHFZAICIJHKHFZLZMBADNZOHZTNZVFBPNNZQVEAIJHDNZAIUAHDNZVGRVGRFV | |
| + EBVFOUBUCVEVJVGFZVKVGFZVJVKUDZABCDUEZUFZVEVLVMVNVOUGVEVLVMVNVOUHUIVEVHVIV | |
| + EVBVFBUJNZFZVAVHVIUKVEVBVRVBVCVDULVEVFBVJOHFZVRVEVLVSVPBVJVFUMUNBVJVFVQVQ | |
| + UOZUPSUQVFBVQVTURSUSUT $. | |
| + $} | |
| + | |
| + ${ | |
| + $( A degree-one vertex cannot occur internally on a fixed-length simple | |
| + path in a simple graph. (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + usgrwspthvd1nint $p |- ( ( G e. USGraph /\ | |
| + W e. ( N WSPathsN G ) /\ | |
| + ( ( VtxDeg ` G ) ` ( W ` B ) ) = 1 ) -> | |
| + -. B e. ( 1 ... ( N - 1 ) ) ) $= | |
| + ( cusgr wcel cwwspthsn co cfv cvtxdg c1 wceq w3a cmin cfz wa cle wbr clt | |
| + c2 wn 1lt2 1re 2re ltnlei mpbi simpl1 simpl2 simpr 3jca usgrwspthintvdge2 | |
| + syl simpl3 eqcomd breqtrrd mto imnani ) BEFZDCBGHFZADIBJIIZKLZMZAKCKNHOHF | |
| + ZVBVCPZTKQRZKTSRVEUAUBKTUCUDUEUFVDTUTKQVDURUSVCMTUTQRVDURUSVCURUSVAVCUGUR | |
| + USVAVCUHVBVCUIUJABCDUKULVDUTKURUSVAVCUMUNUOUPUQ $. | |
| + $} | |
| + | |
| + $( A positive index in a half-open integer range belongs to the corresponding | |
| + closed range ending at the predecessor. (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + elfzo1elfzm1 $p |- ( K e. ( 1 ..^ N ) -> | |
| + K e. ( 1 ... ( N - 1 ) ) ) $= | |
| + ( c1 cfzo co wcel cmin cz 1z a1i elfzoel2 mpdan elfzoelz elfzole1 elfzolem1 | |
| + zsubcl syl elfzd ) ACBDEFZACBCGEZCHFZSIJSBHFZTHFZACBKUBUAUCUAUBIJBCPLQACBMA | |
| + CBNACBOR $. | |
| + | |
| + $( A degree-one vertex cannot occur internally in the vertex sequence of a | |
| + simple path in a simple graph. (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + usgrspthvd1nint $p |- ( ( G e. USGraph /\ F ( SPaths ` G ) P /\ | |
| + ( ( VtxDeg ` G ) ` ( P ` B ) ) = 1 ) -> | |
| + -. B e. ( 1 ... ( ( # ` F ) - 1 ) ) ) $= | |
| + ( cusgr wcel cspths cfv wbr cvtxdg c1 wceq w3a chash cwwspthsn cmin cfz syl | |
| + co wn simp1 cupgr usgrupgr simp2 jca spthwspthn simp3 3jca usgrwspthvd1nint | |
| + wa ) DEFZCBDGHIZABHDJHHKLZMZUKBCNHZDOSFZUMMAKUOKPSQSFTUNUKUPUMUKULUMUAZUNDU | |
| + BFZULUJUPUNURULUNUKURUQDUCRUKULUMUDUEBCDUFRUKULUMUGUHADUOBUIR $. | |
| + | |
| + $( A degree-one vertex cannot occur at a positive nonfinal index in the | |
| + vertex sequence of a simple path in a simple graph. (Contributed by | |
| + Mingli Yuan, 12-Aug-2026.) $) | |
| + usgrspthvd1nintfzo $p |- ( ( G e. USGraph /\ F ( SPaths ` G ) P /\ | |
| + ( ( VtxDeg ` G ) ` ( P ` B ) ) = 1 ) -> | |
| + -. B e. ( 1 ..^ ( # ` F ) ) ) $= | |
| + ( cusgr wcel cspths cfv wbr cvtxdg c1 wceq w3a chash co cfz usgrspthvd1nint | |
| + cmin cfzo elfzo1elfzm1 nsyl ) DEFCBDGHIABHDJHHKLMAKCNHZKROPOFAKUBSOFABCDQAU | |
| + BTUA $. | |
| + | |
| + $( A degree-one vertex on a simple path in a simple graph occurs only at an | |
| + endpoint of the path. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + usgrspthvd1endidx $p |- ( ( ( G e. USGraph /\ F ( SPaths ` G ) P /\ | |
| + ( ( VtxDeg ` G ) ` ( P ` B ) ) = 1 ) /\ | |
| + B e. ( 0 ... ( # ` F ) ) ) -> | |
| + ( B = 0 \/ B = ( # ` F ) ) ) $= | |
| + ( cusgr wcel cspths cfv wbr cvtxdg c1 wceq w3a cc0 chash cfz co wa wo cfzo | |
| + wn usgrspthvd1nintfzo adantr wb simpr elfznelfzob syl bicomd mpbird ) DEFCB | |
| + DGHIABHDJHHKLMZANCOHZPQFZRZANLAUKLSZAKUKTQFUAZUJUOULABCDUBUCUMUOUNUMULUOUNU | |
| + DUJULUEUKAUFUGUHUI $. | |
| + | |
| + $( A path between two vertices in an acyclic graph is a simple path between | |
| + the same vertices. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + pthonacycspthon $p |- ( ( G e. AcyclicGraph /\ | |
| + F ( K ( PathsOn ` G ) M ) P ) -> | |
| + F ( K ( SPathsOn ` G ) M ) P ) $= | |
| + ( cacycgr wcel cpthson cfv co wbr wa cspthson ctrlson cspths pthontrlon jca | |
| + cvv w3a syl adantl cpths pthonispth pthacycspth sylan2 cvtx eqid pthsonprop | |
| + wb simpr simp1 3simpc simp2 isspthson mpbird ) CFGZBADECHIJKZLZBADECMIJKZBA | |
| + DECNIJKZBACOIKZLZURUTVAUQUTUPDEABCPUAUQUPBACUBIKZVADEABCUCABCUDUEQURDCUFIZG | |
| + ZEVDGZLZBRGARGLZLZUSVBUIURUQVIUPUQUJUQCRGZVEVFSZVHUTVCLZSZVIDEABCVDVDUGZUHV | |
| + MVGVHVMVKVGVKVHVLUKVJVEVFULTVKVHVLUMQTTDEARBCVDRVNUNTUO $. | |
| + | |
| + ${ | |
| + $d G f k m p $. $d K f k m p $. $d M f k m p $. $d V f k m p $. | |
| + $d f k m p $. | |
| + conngrpthon.v $e |- V = ( Vtx ` G ) $. | |
| + $( Any two vertices of a connected graph are joined by a path. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + conngrpthon $p |- ( ( G e. ConnGraph /\ K e. V /\ M e. V ) -> | |
| + E. f E. p f ( K ( PathsOn ` G ) M ) p ) $= | |
| + ( vk vm cconngr wcel wa cv co wbr wex wral cvv syl wceq w3a cpthson simp1 | |
| + cfv 3simpc wb elexd isconngr mpbid id oveq1d breqd 2exbidv oveq2d rspc2va | |
| + jca ) BJKZCEKZDEKZUAZURUSLZAMZFMZHMZIMZBUBUDZNZOZFPAPZIEQHEQZLVBVCCDVFNZO | |
| + ZFPAPZUTVAVJUQURUSUEUTUQVJUQURUSUCZUTBRKUQVJUFUTBJVNUGAHIBERFGUHSUIUPVIVM | |
| + VBVCCVEVFNZOZFPAPHICDEEVDCTZVHVPAFVQVGVOVBVCVQVDCVEVFVQUJUKULUMVEDTZVPVLA | |
| + FVRVOVKVBVCVRVEDCVFVRUJUNULUMUOS $. | |
| + $} | |
| + | |
| + ${ | |
| + $d f p $. $d f F $. $d f G $. $d f K $. $d f M $. $d f P $. $d p F $. | |
| + $d p G $. $d p K $. $d p M $. $d p P $. | |
| + $( A displayed path supplies the existential witnesses used in the | |
| + definition of connectedness. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + pthonex $p |- ( F ( K ( PathsOn ` G ) M ) P -> | |
| + E. f E. p f ( K ( PathsOn ` G ) M ) p ) $= | |
| + ( cpthson cfv co wbr cvv wcel wa cv wex cvtx w3a ctrlson cpths pthsonprop | |
| + eqid simp2 syl id breq12 spc2egv sylc ) CAEFDHIJZKZCLMALMNZUJBOZGOZUIKZGP | |
| + BPUJDLMEDQIZMFUOMRZUKCAEFDSIJKCADTIKNZRUKEFACDUOUOUBUAUPUKUQUCUDUJUEUNUJB | |
| + GCALLULCUMAUIUFUGUH $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A m n $. $d F m n $. $d G m n $. $d N m n $. $d P m n $. | |
| + $d U m n $. $d V m n $. | |
| + pthcutidx.v $e |- V = ( Vtx ` G ) $. | |
| + $( A path from a set to a vertex outside the set has two consecutive | |
| + vertices on opposite sides of the set. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + pthcutidx $p |- ( ( ( A C_ V /\ U e. A /\ N e. ( V \ A ) ) /\ | |
| + F ( U ( PathsOn ` G ) N ) P ) -> | |
| + E. m e. ( 0 ..^ ( # ` F ) ) | |
| + ( ( P ` m ) e. A /\ ( P ` ( m + 1 ) ) e. ( V \ A ) ) ) $= | |
| + ( wcel cfv co wa cc0 wnel cn0 syl wb a1i jca vn wss w3a cpthson wbr chash | |
| + cdif cv cfz wn c1 caddc wrex cfzo wceq eleq1 fveq2 eqidd neleq12d anbi12d | |
| + weq cwlkson ctrlson simpr pthontrlon trlsonwlkon wlkonendpts simpl simpl2 | |
| + 3syl eqeltrd nnel bicomi mpbid intnand wne cwlks wlkoniswlk simpl3 eldifn | |
| + wlkcl df-nel mpbird elnelne2 wlkon2n0 elnnne0 nn0fz0 rspcedvdw nn0min nfv | |
| + cn nfre1 simprrr cuz simprl nn0uz eleqtrd fzofzp1b simprrl simplr simplrl | |
| + simprr elfzofz mtand wi wlkpvtx mpd eldifd 3impb 3exp rspe syl6d rexlimd | |
| + ) AHUBZCAJZGHAUGZJZUCZEBCGFUDKLUEZMZDUHZNEUFKZUILZJZYABKZAOZMZUJZYAUKULLZ | |
| + YCJZYIBKZAOZMZMZDPUMYEAJZYKXPJZMZDNYBUNLZUMZXTUAUHZYCJZYTBKZAOZMZNYCJZNBK | |
| + ZAOZMYGYMDUAYTNUOZUUAUUEUUCUUGYTNYCUPUUHUUBUUFAAYTNBUQUUHAURUSUTUADVAZUUA | |
| + YDUUCYFYTYAYCUPUUIUUBYEAAYTYABUQUUIAURUSUTYTYIUOZUUAYJUUCYLYTYIYCUPUUJUUB | |
| + YKAAYTYIBUQUUJAURUSUTXTUUGUUEXTUUFAJZUUGUJZXTUUFCAXTEBCGFVBKLUEZUUFCUOZXT | |
| + XSEBCGFVCKLUEZUUMXRXSVDCGBEFVEZCGBEFVFZVJZUUMUUNYBBKZGUOZMZUUNBEFCGVGZUUN | |
| + UUTVHQQXNXOXQXSVIZVKUUKUULRXTUULUUKUUFAVLVMSVNVOXTUUDYBYCJZUUSAOZMUAYBWKY | |
| + TYBUOZUUAUVDUUCUVEYTYBYCUPUVFUUBUUSAAYTYBBUQUVFAURUSUTXTYBWKJZYBPJZYBNVPZ | |
| + MZXTUVHUVIXTUUMEBFVQKUEZUVHUURCGBEFVRZBEFWAVJZXTUUMCGVPZMUVIXTUUMUVNUURXT | |
| + XOGAOZMUVNXTXOUVOUVCXTUVOGAJUJZXTXQUVPXNXOXQXSVSGHAVTQUVOUVPRXTGAWBSWCZTC | |
| + GAWDQTCGBEFWEQTUVGUVJRXTYBWFSWCXTUVDUVEXTUVHUVDUVMUVHUVDRXTYBWGSVNXTUVEUV | |
| + OUVQXTUUSGAAXTUUMUUTUURUUMUVAUUTUVBUUNUUTVDQQXTAURUSWCTWHWIXTYNYSDPXTDWJY | |
| + QDYRWLXTYAPJZYNYAYRJZYQMZYSXTUVRYNUVTXTUVRYNUVTXTUVRYNMZMZUVSYQUWBUVSYJUW | |
| + BYMYJXTUVRYHYMWMZYJYLVHZQZUWBYANWNKZJZUVSYJRZUWBYAPUWFXTUVRYNWOPUWFUOZUWB | |
| + WPSWQNYBYAWRZQWCUWBYOYPUWBYFUJZYOUWBYFYGXTUVRYHYMWSUWBYFMZYDYFUWLUVSYDUWL | |
| + UVSYJUWLYMYJUWLUWAYMXTUWAYFWTUVRYHYMXBQUWDQUWLUWGUWHUWLYAPUWFXTUVRYNYFXAU | |
| + WIUWLWPSWQUWJQWCYANYBXCQUWBYFVDTXDUWKYORUWBYEAVLSVNUWBYKHAUWBYJYKHJZUWEUW | |
| + BUVKYJUWMXEUWBXSUUMUVKXRXSUWAWTXSUUOUUMUUPUUQQUVLVJBEFYIHIXFQXGUWBYLYKAJU | |
| + JZUWBYMYLUWCYJYLVDQYLUWNRUWBYKAWBSVNXHTTXIXJUVTYSXEXTYQDYRXKSXLXMXG $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A m $. $d E m $. $d F m $. $d G m $. $d N m $. $d P m $. $d U m $. | |
| + $d V m $. $d E k $. $d F k $. $d G k $. $d P k $. $d k m $. | |
| + pthcutiedg.v $e |- V = ( Vtx ` G ) $. | |
| + pthcutiedg.e $e |- E = ( iEdg ` G ) $. | |
| + $( A path crossing a vertex cut contains an indexed edge whose endpoints | |
| + lie on opposite sides of the cut. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + pthcutiedg $p |- ( ( G e. UPGraph /\ | |
| + ( ( A C_ V /\ U e. A /\ N e. ( V \ A ) ) /\ | |
| + F ( U ( PathsOn ` G ) N ) P ) ) -> | |
| + E. m e. ( 0 ..^ ( # ` F ) ) | |
| + ( ( ( P ` m ) e. A /\ ( P ` ( m + 1 ) ) e. ( V \ A ) ) /\ | |
| + ( ( F ` m ) e. dom E /\ | |
| + ( E ` ( F ` m ) ) = { ( P ` m ) , ( P ` ( m + 1 ) ) } ) ) ) $= | |
| + ( vk wcel cfv co wbr wa c1 syl fveq2d cupgr wss cdif w3a cpthson cv caddc | |
| + cc0 chash cfzo wrex cdm cpr wceq simpr pthcutidx ctrlson cwlks simpl 4syl | |
| + cwlkson pthontrlon trlsonwlkon wlkoniswlk wlkiedgdomd weq preq12d eqeq12d | |
| + simplr id oveq1d wral simplll simpllr 3syl upgrwlkedg rspcdva ex reximdva | |
| + jca mpd ) GUAMZAIUBCAMHIAUCZMUDZFBCHGUENOPZQZQZDUFZBNZAMWHRUGOZBNZWCMQZDU | |
| + HFUINUJOZUKZWLWHFNZEULMZWOENZWIWKUMZUNZQZQZDWMUKWGWFWNWBWFUOZABCDFGHIJUPS | |
| + WGWLXADWMWGWHWMMZQZWLXAXDWLQZWLWTXDWLUOXEWPWSXEBEFGWHKXEWEFBCHGUQNOPZFBCH | |
| + GVANOPZFBGURNPZXEXDWGWFWEXDWLUSWGXCUSXBWDWEUOZUTCHBFGVBZCHBFGVCZCHBFGVDZU | |
| + TWGXCWLVIZVEXELUFZFNZENZXNBNZXNRUGOZBNZUMZUNZWSLWMWHLDVFZXPWQXTWRYBXOWOEY | |
| + BXNWHFYBVJZTTYBXQWIXSWKYBXNWHBYCTYBXRWJBYBXNWHRUGYCVKTVGVHXEWBXHQYALWMVLX | |
| + EWBXHWBWFXCWLVMXEWEXHXEWFWEWBWFXCWLVNXISWEXFXGXHXJXKXLVOSVTBLFGEKVPSXMVQV | |
| + TVTVRVSWA $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A j n u $. $d B j n u $. $d E j n u $. $d J j n u $. $d N j n u $. | |
| + $d U j n u $. $d j n u $. | |
| + $( An indexed edge with specified endpoints supplies the corresponding | |
| + restricted existential witnesses. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + cutiedgex $p |- ( ( ( U e. A /\ N e. B ) /\ | |
| + ( J e. dom E /\ ( E ` J ) = { U , N } ) ) -> | |
| + E. j e. dom E E. u e. A E. n e. B | |
| + ( E ` j ) = { u , n } ) $= | |
| + ( wcel wa cfv cpr wceq cv wrex simpr eqeq2d rspcedv mpd cdm simprr simplr | |
| + preq2d simpll preq1d rexbidv simprl fveq2d eqeq1d 2rexbidv ) DBJZICJZKZHG | |
| + UAZJZHGLZDIMZNZKZKZUQAOZFOZMZNZFCPZABPZEOZGLZVDNZFCPABPZEUOPVAUQDVCMZNZFC | |
| + PZVGVAUSVNUNUPUSUBVAVMUSFICULUMUTUCVAVCINZKZVLURUQVPVCIDVAVOQUDRSTVAVFVNA | |
| + DBULUMUTUEVAVBDNZKZVEVMFCVRVDVLUQVRVBDVCVAVQQUFRUGSTVAVKVGEHUOUNUPUSUHVAV | |
| + HHNZKZVJVEAFBCVTVIUQVDVTVHHGVAVSQUIUJUKST $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A m $. $d E m $. $d F m $. $d G m $. $d N m $. $d P m $. $d U m $. | |
| + $d V m $. $d A j n u $. $d E j n u $. $d F j n u $. $d P j n u $. | |
| + $d V j n u $. $d j m n u $. | |
| + pthcutiedgex.v $e |- V = ( Vtx ` G ) $. | |
| + pthcutiedgex.e $e |- E = ( iEdg ` G ) $. | |
| + $( A path crossing a vertex cut supplies an edge with one endpoint on | |
| + each side of the cut. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + pthcutiedgex $p |- ( ( G e. UPGraph /\ | |
| + ( ( A C_ V /\ U e. A /\ N e. ( V \ A ) ) /\ | |
| + F ( U ( PathsOn ` G ) N ) P ) ) -> | |
| + E. j e. dom E E. u e. A E. n e. ( V \ A ) | |
| + ( E ` j ) = { u , n } ) $= | |
| + ( vm wcel cfv co wa cv wrex cupgr wss cdif w3a cpthson wbr caddc cdm wceq | |
| + c1 cpr cc0 chash cfzo pthcutiedg simp3 cutiedgex syl 3exp rexlimdv mpd ) | |
| + IUAOBKUBDBOJKBUCZOUDHCDJIUEPQUFRRZNSZCPZBOVDUJUGQCPZVBORVDHPZGUHZOVGGPVEV | |
| + FUKUIRRZNULHUMPUNQZTESGPASFSUKUIFVBTABTEVHTZBCDNGHIJKLMUOVCVIVKNVJVCVDVJO | |
| + ZVIVKVCVLVIUDVIVKVCVLVIUPABVBVEEFGVGVFUQURUSUTVA $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A f p $. $d E f p $. $d G f p $. $d N f p $. $d U f p $. | |
| + $d V f p $. $d f j n p u $. $d A j n u $. $d E j n u $. $d V j n u $. | |
| + conngrcutiedgpt.v $e |- V = ( Vtx ` G ) $. | |
| + conngrcutiedgpt.e $e |- E = ( iEdg ` G ) $. | |
| + $( In a connected graph, two specified vertices on opposite sides of a | |
| + cut are joined through an edge crossing the cut. (Contributed by | |
| + Mingli Yuan, 13-Aug-2026.) $) | |
| + conngrcutiedgpt $p |- ( ( ( G e. UPGraph /\ G e. ConnGraph ) /\ | |
| + ( A C_ V /\ U e. A /\ N e. ( V \ A ) ) ) -> | |
| + E. j e. dom E E. u e. A E. n e. ( V \ A ) | |
| + ( E ` j ) = { u , n } ) $= | |
| + ( vf vp wcel wa w3a cv cfv wex wrex cupgr cconngr wss cdif cpthson co wbr | |
| + cpr wceq cdm simplr simpr1 simpr2 sseldd eldifad 3jca conngrpthon simplll | |
| + simpr3 syl simpr jca pthcutiedgex ex exlimdvv mpd ) GUANZGUBNZOZBIUCZCBNZ | |
| + HIBUDZNZPZOZLQZMQZCHGUERUFUGZMSLSZDQFRAQEQUHUIEVLTABTDFUJTZVOVHCINZHINZPV | |
| + SVOVHWAWBVGVHVNUKVOBICVIVJVKVMULVIVJVKVMUMUNVOHIBVIVJVKVMUSUOUPLGCHIMJUQU | |
| + TVOVRVTLMVOVRVTVOVROZVGVNVROZOVTWCVGWDVGVHVNVRURWCVNVRVIVNVRUKVOVRVAVBVBA | |
| + BVQCDEFVPGHIJKVCUTVDVEVF $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A x y $. $d E x y $. $d G x y $. $d V x y $. $d j n u x y $. | |
| + $d x y $. $d A j n u $. $d E j n u $. $d V j n u $. | |
| + conngrcutiedg.v $e |- V = ( Vtx ` G ) $. | |
| + conngrcutiedg.e $e |- E = ( iEdg ` G ) $. | |
| + $( Every nontrivial vertex cut of a connected graph is crossed by an | |
| + indexed edge. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + conngrcutiedg $p |- ( ( ( G e. UPGraph /\ G e. ConnGraph ) /\ | |
| + ( A C_ V /\ A =/= (/) /\ A =/= V ) ) -> | |
| + E. j e. dom E E. u e. A E. n e. ( V \ A ) | |
| + ( E ` j ) = { u , n } ) $= | |
| + ( vx vy wcel wa c0 wne w3a cv wex wrex n0 cupgr cconngr wss cfv wceq cdif | |
| + cpr cdm simpr2 sylib wi simpr1 simpr3 jca pssdifn0 syl simp1l simp3 simp2 | |
| + simp1r1 3jca conngrcutiedgpt 3exp exlimdv mpd ) FUALFUBLMZBGUCZBNOZBGOZPZ | |
| + MZJQZBLZJRZCQEUDAQDQUGUEDGBUFZSABSCEUHSZVKVHVNVFVGVHVIUIJBTUJVKVMVPJVKKQZ | |
| + VOLZKRZVMVPUKZVKVONOZVSVKVGVIMWAVKVGVIVFVGVHVIULVFVGVHVIUMUNBGUOUPKVOTUJV | |
| + KVRVTKVKVRVMVPVKVRVMPZVFVGVMVRPZMVPWBVFWCVFVJVRVMUQWBVGVMVRVGVHVIVFVRVMUT | |
| + VKVRVMURVKVRVMUSVAUNABVLCDEFVQGHIVBUPVCVDVEVDVE $. | |
| + $} | |
| + | |
| + $( The vertex set of a subgraph is a subclass of that of the ambient graph. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + subgrvtxss $p |- ( T SubGraph G -> | |
| + ( Vtx ` T ) C_ ( Vtx ` G ) ) $= | |
| + ( csubgr wbr cvtx cfv wss ciedg cedg cpw w3a eqid subgrprop2 simp1 syl ) AB | |
| + CDAEFZBEFZGZAHFZBHFZGZAIFZPJGZKRQTAUBBSPPLQLSLTLUBLMRUAUCNO $. | |
| + | |
| + $( The indexed-edge function of a subgraph is a subclass of that of the | |
| + ambient graph. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + subgriedgss $p |- ( T SubGraph G -> | |
| + ( iEdg ` T ) C_ ( iEdg ` G ) ) $= | |
| + ( csubgr wbr cvtx cfv wss ciedg cedg cpw w3a eqid subgrprop2 simp2 syl ) AB | |
| + CDAEFZBEFZGZAHFZBHFZGZAIFZPJGZKUAQTAUBBSPPLQLSLTLUBLMRUAUCNO $. | |
| + | |
| + $( An indexed edge retained by a subgraph has the same value in the | |
| + subgraph and the ambient hypergraph. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + subgriedgfv $p |- ( ( G e. UHGraph /\ T SubGraph G /\ | |
| + J e. dom ( iEdg ` T ) ) -> | |
| + ( ( iEdg ` T ) ` J ) = ( ( iEdg ` G ) ` J ) ) $= | |
| + ( cuhgr wcel csubgr wbr ciedg cfv cdm w3a wfun wss wceq simp1 uhgrfun simp2 | |
| + eqid syl subgriedgss simp3 3jca funssfv eqcomd ) BDEZABFGZCAHIZJEZKZCBHIZIZ | |
| + CUGIZUIUJLZUGUJMZUHKUKULNUIUMUNUHUIUEUMUEUFUHOUJBUJRPSUIUFUNUEUFUHQABTSUEUF | |
| + UHUAUBCUJUGUCSUD $. | |
| + | |
| + $( An edge of the ambient graph incident with a vertex outside a tree cannot | |
| + already be indexed by the tree. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + trnewedgnin $p |- ( ( ( ( G e. UHGraph /\ T e. Tree ) /\ | |
| + T SubGraph G ) /\ ( J e. dom ( iEdg ` G ) /\ | |
| + ( ( iEdg ` G ) ` J ) = { U , N } /\ | |
| + ( N e. _V /\ N e/ ( Vtx ` T ) ) ) ) -> | |
| + J e/ dom ( iEdg ` T ) ) $= | |
| + ( cuhgr wcel ctree wa csubgr wbr ciedg cfv cdm wceq cvv wnel w3a syl eqid | |
| + cpr cvtx simpr3r cpw simplr simp3l prid2g simp-4l simpllr simpr subgriedgfv | |
| + 3jca simplr2 eqtrd eleqtrrd c0 csn subgruhgredgd eldifad elelpwi syl2anc ex | |
| + nelcon3d mpd ) CFGZAHGZIZACJKZIZDCLMZNGZDVJMZBEUAZOZEPGZEAUBMZQZIZRZIZVQDAL | |
| + MZNZQVOVQVKVNVIUCVTDWBEVPVTDWBGZEVPGZVTWCIZEDWAMZGWFVPUDZGWDWEEVMWFWEVOEVMG | |
| + WEVSVOVIVSWCUEVKVNVOVQUFSBEPUGSWEWFVLVMWEVEVHWCRWFVLOWEVEVHWCVEVFVHVSWCUHZV | |
| + GVHVSWCUIZVTWCUJZULACDUKSVKVNVRVIWCUMUNUOWEWFWGUPUQWEACWAVPDVPTWATWHWIWJURU | |
| + SEWFVPUTVAVBVCVD $. | |
| + | |
| + ${ | |
| + $d E x $. $d G x $. $d I x $. $d J x $. $d N x $. | |
| + $d T x $. $d U x $. $d V x $. $d W x $. | |
| + trleafext.v $e |- V = ( Vtx ` T ) $. | |
| + trleafext.e $e |- E = ( iEdg ` T ) $. | |
| + trleafext.w $e |- W = ( Vtx ` G ) $. | |
| + trleafext.i $e |- I = ( iEdg ` G ) $. | |
| + trleafext.a $e |- A = ( V u. { N } ) $. | |
| + trleafext.p $e |- P = ( E u. { <. J , { U , N } >. } ) $. | |
| + trleafext.s $e |- S = <. A , P >. $. | |
| + $( The hypotheses supplied by an ambient crossing edge satisfy the fresh | |
| + leaf-construction interface. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + trleafhyp $p |- ( ( ( ( G e. USGraph /\ T e. Tree /\ | |
| + T SubGraph G ) /\ ( U e. V /\ N e. ( W \ V ) ) ) /\ | |
| + ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> | |
| + ( ( T e. Tree /\ U e. V /\ ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) ) $= | |
| + ( wcel cusgr ctree csubgr wbr w3a cdif wa cdm cfv cpr cvv simpll2 simplrl | |
| + wceq wnel simplrr elex syl wn eldifn df-nel sylibr jca simprl cuhgr ciedg | |
| + 3jca cvtx simpll1 usgruhgr simpll3 dmeq ax-mp eqcomi eleq2 fveq1i eqtr3id | |
| + wb simprr neleq2 bicomi trnewedgnin ) GUATZDUBTZDGUCUDZUEZEKTZJLKUFZTZUGZ | |
| + UGZIHUHZTZIHUIZEJUJZUNZUGZUGZWDWGJUKTZJKUOZUGZUEIUKTZIFUHZUOZUGWRWDWGXAWC | |
| + WDWEWJWQULZWFWGWIWQUMWRWSWTWRWIWSWFWGWIWQUPZJWHUQURZWRJKTUSZWTWRWIXHXFJLK | |
| + UTURJKVAVBZVCVGWRXBXDWRWMXBWKWMWPVDZIWLUQURWRGVETZWDUGZWEUGZIGVFUIZUHZTZI | |
| + XNUIZWOUNZWSJDVHUIZUOZUGZUEZUGZXDWRXMYBWRXLWEWRXKWDWRWCXKWCWDWEWJWQVIGVJU | |
| + RXEVCWCWDWEWJWQVKVCWRXPXRYAWRWMXPXJXOWLUNXPWMVRWLXOHXNUNWLXOUNPHXNVLVMVNX | |
| + OWLIVOVMVBWRXQWNWOIHXNPVPWKWMWPVSVQWRWSXTXGWRWTXTXIWTXTKXSUNWTXTVRMKXSJVT | |
| + VMWAVBVCVGVCYCIDVFUIZUHZUOZXDDEGIJWBXCYEUNZXDYFVRFYDUNYGNFYDVLVMXCYEIVTVM | |
| + VBURVCVC $. | |
| + | |
| + $} | |
| + | |
| + ${ | |
| + $d G a b f p $. $d K a b f p $. $d M a b f p $. | |
| + $d a b f p $. | |
| + $( In an acyclic undirected pseudograph, existence of a path is symmetric | |
| + in its endpoints. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + acycgrpthonsym $p |- ( ( ( G e. UPGraph /\ G e. AcyclicGraph ) /\ | |
| + E. a E. b a ( K ( PathsOn ` G ) M ) b ) -> | |
| + E. f E. p f ( M ( PathsOn ` G ) K ) p ) $= | |
| + ( cupgr wcel cacycgr wa cv cpthson cfv co wbr wex creverse syl syl6 simpr | |
| + cspthson cvtx simpll id simplr jca pthonacycspthon revspthond spthonpthon | |
| + eqid ex pthonex exlimdvv imp ) BHIZBJIZKZFLZGLZCDBMNZOPZGQFQALELDCVAOZPEQ | |
| + AQZURVBVDFGURVBUSRNZUTRNZVCPZVDURVBVEVFDCBUBNZOPZVGURVBVIURVBKZUTUSBCDBUC | |
| + NZVKUKVJUPUPUPUQVBUDUPUESVJUQVBKUSUTCDVHOPVJUQVBUPUQVBUFURVBUAUGUTUSBCDUH | |
| + SUIULDCVFVEBUJTVFAVEBDCEUMTUNUO $. | |
| + $} | |
| + | |
| + ${ | |
| + $d f p $. $d f G $. $d f K $. $d f M $. $d f N $. $d p G $. $d p K $. | |
| + $d p M $. $d p N $. | |
| + $( Substitution of the first endpoint in a path-existence assertion. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + pthonexeq1 $p |- ( K = N -> | |
| + ( E. f E. p f ( K ( PathsOn ` G ) M ) p <-> | |
| + E. f E. p f ( N ( PathsOn ` G ) M ) p ) ) $= | |
| + ( wceq cv cpthson cfv co wbr id oveq1d breqd 2exbidv ) CEGZAHZFHZCDBIJZKZ | |
| + LRSEDTKZLAFQUAUBRSQCEDTQMNOP $. | |
| + | |
| + $( Substitution of the second endpoint in a path-existence assertion. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + pthonexeq2 $p |- ( M = N -> | |
| + ( E. f E. p f ( K ( PathsOn ` G ) M ) p <-> | |
| + E. f E. p f ( K ( PathsOn ` G ) N ) p ) ) $= | |
| + ( wceq cv cpthson cfv co wbr id oveq2d breqd 2exbidv ) DEGZAHZFHZCDBIJZKZ | |
| + LRSCETKZLAFQUAUBRSQDECTQMNOP $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A a b f p $. $d E a b f p $. $d F a b f p $. | |
| + $d G a b f p $. $d J a b f p $. $d K a b f p $. | |
| + $d M a b f p $. $d N a b f p $. $d P a b f p $. | |
| + $d Q a b f p $. $d S a b f p $. $d U a b f p $. | |
| + $d V a b f p $. $d a b f p $. | |
| + grleafpth.v $e |- V = ( Vtx ` G ) $. | |
| + grleafpth.e $e |- E = ( iEdg ` G ) $. | |
| + grleafpth.a $e |- A = ( V u. { N } ) $. | |
| + grleafpth.p $e |- P = ( E u. { <. J , { U , N } >. } ) $. | |
| + grleafpth.s $e |- S = <. A , P >. $. | |
| + $( A path whose endpoints are old vertices remains a path after a leaf is | |
| + adjoined. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafoldpthond $p |- ( ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) /\ | |
| + F ( K ( PathsOn ` G ) M ) Q ) -> | |
| + F ( K ( PathsOn ` S ) M ) Q ) $= | |
| + ( wcel wa ctree cvv wnel w3a cdm cpthson cfv co wbr cspthson csubgr cusgr | |
| + simpll1 treeusgr syl simpll2 simpll3 simplr jca grleafsubgr cacycgr simpr | |
| + 3jca treeacycgr pthonacycspthon subgrspthon syl5com mpd spthonpthon ) HUA | |
| + SZEMSZLUBSLMUCTZUDZIUBSIFUEUCTZTZGCJKHUFUGUHUIZTZGCJKDUJUGUHUIZGCJKDUFUGU | |
| + HUIVQHDUKUIZVRVQHULSZVKVLUDZVNTVSVQWAVNVQVTVKVLVQVJVTVJVKVLVNVPUMZHUNUOVJ | |
| + VKVLVNVPUPVJVKVLVNVPUQVCVMVNVPURUSABDEFHILMNOPQRUTUOVQGCJKHUJUGUHUIZVSVRV | |
| + QHVASZVPTWCVQWDVPVQVJWDWBHVDUOVOVPVBUSCGHJKVEUOCHGDJKVFVGVHJKCGDVIUO $. | |
| + | |
| + $( Any two old vertices remain joined by a path after a leaf is adjoined. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafoldoldpthon $p |- ( ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) /\ ( K e. V /\ M e. V ) ) -> | |
| + E. f E. p f ( K ( PathsOn ` S ) M ) p ) $= | |
| + ( wcel wa ctree cvv wnel w3a cdm cv cpthson cfv co wbr cconngr treeconngr | |
| + wex simpll1 syl simprl simprr 3jca conngrpthon simpll jca grleafoldpthond | |
| + simpr ex 2eximdv mpd ) GUASZDLSZKUBSKLUCTZUDHUBSHFUEUCTZTZILSZJLSZTZTZEUF | |
| + ZMUFZIJGUGUHUIUJZMUMEUMZVPVQIJCUGUHUIUJZMUMEUMVOGUKSZVLVMUDVSVOWAVLVMVOVG | |
| + WAVGVHVIVJVNUNGULUOVKVLVMUPVKVLVMUQUREGIJLMNUSUOVOVRVTEMVOVRVTVOVRTZVKVRT | |
| + VTWBVKVRVKVNVRUTVOVRVCVAABVQCDFVPGHIJKLNOPQRVBUOVDVEVF $. | |
| + | |
| + $( A path from an old vertex to the attachment vertex extends to the new | |
| + leaf. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafoldnewpthond $p |- ( ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) /\ | |
| + F ( K ( PathsOn ` G ) U ) Q ) -> | |
| + ( F ++ <" J "> ) ( K ( PathsOn ` S ) N ) | |
| + ( Q ++ <" N "> ) ) $= | |
| + ( wcel wa co ctree cvv w3a cdm cpthson cfv wbr cs1 cconcat cspthson simpl | |
| + wnel cacycgr simpll1 treeacycgr syl simpr jca pthonacycspthon spthonpthon | |
| + grleafspthond ) HUARZELRZKUBRKLULSZUCIUBRIFUDULSZSZGCJEHUEUFTUGZSZGIUHUIT | |
| + ZCKUHUITZJKDUJUFTUGZVIVJJKDUEUFTUGVHVFGCJEHUJUFTUGZSVKVHVFVLVFVGUKVHHUMRZ | |
| + VGSVLVHVMVGVHVBVMVBVCVDVEVGUNHUOUPVFVGUQURCGHJEUSUPURABCDEFGHIJKLMNOPQVAU | |
| + PJKVJVIDUTUP $. | |
| + | |
| + $( Every old vertex is joined to the new leaf after adjoining the leaf. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafoldnewpthon $p |- ( ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) /\ K e. V ) -> | |
| + E. f E. p f ( K ( PathsOn ` S ) N ) p ) $= | |
| + ( va wcel wa vb ctree cvv wnel w3a cdm cv cpthson cfv wbr cconngr simpll1 | |
| + co wex treeconngr syl simpll2 3jca conngrpthon cconcat grleafoldnewpthond | |
| + simpr cs1 simpll jca ex pthonex syl6 exlimdvv mpd ) GUBSZDKSZJUCSJKUDTZUE | |
| + HUCSHFUFUDTZTZIKSZTZRUGZUAUGZIDGUHUIUMUJZUAUNRUNZEUGLUGIJCUHUIUMZUJLUNEUN | |
| + ZVQGUKSZVPVLUEWAVQWDVPVLVQVKWDVKVLVMVNVPULGUOUPVOVPVBVKVLVMVNVPUQURRGIDKU | |
| + AMUSUPVQVTWCRUAVQVTVRHVCUTUMZVSJVCUTUMZWBUJZWCVQVTWGVQVTTZVOVTTWGWHVOVTVO | |
| + VPVTVDVQVTVBVEABVSCDFVRGHIJKMNOPQVAUPVFWFEWECIJLVGVHVIVJ $. | |
| + | |
| + $( The new leaf is joined to every old vertex after adjoining the leaf. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafnewoldpthon $p |- ( ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) /\ K e. V ) -> | |
| + E. f E. p f ( N ( PathsOn ` S ) K ) p ) $= | |
| + ( wcel wa syl va vb ctree cvv w3a cdm cupgr cacycgr cv cpthson cfv co wbr | |
| + wnel wex simpll1 treeusgr simpll2 simpll3 3jca simplr grleafusgr usgrupgr | |
| + cusgr jca simpl grleafacycgr grleafoldnewpthon acycgrpthonsym ) GUCRZDKRZ | |
| + JUDRJKUNSZUEZHUDRHFUFUNSZSZIKRZSZCUGRZCUHRZSZUAUIUBUIIJCUJUKZULUMUBUOUAUO | |
| + ZSEUILUIJIWAULUMLUOEUOVQVTWBVQVRVSVQCVDRZVRVQGVDRZVKVLUEZVNSWCVQWEVNVQWDV | |
| + KVLVQVJWDVJVKVLVNVPUPGUQTVJVKVLVNVPURVJVKVLVNVPUSUTVMVNVPVAVEABCDFGHJKMNO | |
| + PQVBTCVCTVQVOVSVOVPVFABCDFGHJKMNOPQVGTVEABCDUAFGHIJKUBMNOPQVHVEECIJLUAUBV | |
| + IT $. | |
| + | |
| + $( Membership in the enlarged vertex set is membership in the old vertex | |
| + set or equality with the new leaf. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + grleafela $p |- ( K e. A -> ( K e. V \/ K = N ) ) $= | |
| + ( wcel csn wo biimpi syl cun wceq eleq2i elun id elsni orim12i ) HAPZHJIQ | |
| + ZUAZPZHJPZHIUBZRZUHUKAUJHMUCSUKULHUIPZRZUNUKUPHJUIUDSULULUOUMULUEHIUFUGTT | |
| + $. | |
| + | |
| + $( The adjoined leaf belongs to the vertex set of the enlarged graph. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafnewvtx $p |- ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) -> N e. ( Vtx ` S ) ) $= | |
| + ( wcel cvv wnel wa w3a syl ctree cdm cfv csn cun simpl3 simpl snidg elun2 | |
| + cvtx wceq a1i eleqtrrd cusgr simpl1 treeusgr simpl2 3jca simpr grleafvtx | |
| + jca ) FUAOZDIOZHPOZHIQZRZSZGPOGEUBQRZRZHACUJUCZVIHIHUDZUEZAVIVDHVLOZVIVFV | |
| + DVBVCVFVHUFZVDVEUGTVDHVKOVMHPUHHVKIUITTAVLUKVILULUMVIFUNOZVCVFSZVHRVJAUKV | |
| + IVPVHVIVOVCVFVIVBVOVBVCVFVHUOFUPTVBVCVFVHUQVNURVGVHUSVAABCDEFGHIJKLMNUTTU | |
| + M $. | |
| + | |
| + $( The old-to-new endpoint case for paths in the enlarged graph. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafpthcaseon $p |- ( ( ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) /\ K e. V ) /\ M = N ) -> | |
| + E. f E. p f ( K ( PathsOn ` S ) M ) p ) $= | |
| + ( wcel wa ctree cvv wnel w3a cdm wceq cv cpthson cfv co grleafoldnewpthon | |
| + wbr wex simpl syl wb simpr pthonexeq2 mpbird ) GUASDLSKUBSKLUCTUDHUBSHFUE | |
| + UCTTILSTZJKUFZTZEUGZMUGZIJCUHUIZUJULMUMEUMZVCVDIKVEUJULMUMEUMZVBUTVGUTVAU | |
| + NABCDEFGHIKLMNOPQRUKUOVBVAVFVGUPUTVAUQECIJKMURUOUS $. | |
| + | |
| + $( The new-to-old endpoint case for paths in the enlarged graph. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafpthcaseno $p |- ( ( ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) /\ M e. V ) /\ K = N ) -> | |
| + E. f E. p f ( K ( PathsOn ` S ) M ) p ) $= | |
| + ( wcel wa ctree cvv wnel w3a cdm wceq cv cpthson cfv co grleafnewoldpthon | |
| + wbr wex simpl syl wb simpr pthonexeq1 mpbird ) GUASDLSKUBSKLUCTUDHUBSHFUE | |
| + UCTTJLSTZIKUFZTZEUGZMUGZIJCUHUIZUJULMUMEUMZVCVDKJVEUJULMUMEUMZVBUTVGUTVAU | |
| + NABCDEFGHJKLMNOPQRUKUOVBVAVFVGUPUTVAUQECIJKMURUOUS $. | |
| + | |
| + $( The new-to-new endpoint case for paths in the enlarged graph. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafpthcasenn $p |- ( ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) /\ ( K = N /\ M = N ) ) -> | |
| + E. f E. p f ( K ( PathsOn ` S ) M ) p ) $= | |
| + ( wcel wex ctree cvv wnel wa w3a cdm wceq cv cpthson cfv wbr grleafnewvtx | |
| + co simpl cvtx eqid 0pthonv syl simprr pthonexeq2 mpbird simprl pthonexeq1 | |
| + wb ) GUASDLSKUBSKLUCUDUEHUBSHFUFUCUDUDZIKUGZJKUGZUDZUDZEUHZMUHZIJCUIUJZUM | |
| + UKMTETZVJVKKJVLUMUKMTETZVIVNVJVKKKVLUMUKMTETZVIVEVOVEVHUNVEKCUOUJZSVOABCD | |
| + FGHKLNOPQRULECKVPMVPUPUQURURVIVGVNVOVDVEVFVGUSECKJKMUTURVAVIVFVMVNVDVEVFV | |
| + GVBECIJKMVCURVA $. | |
| + | |
| + $( Every pair of vertices in the enlarged graph is joined by a path. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafallpthon $p |- ( ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) /\ ( K e. A /\ M e. A ) ) -> | |
| + E. f E. p f ( K ( PathsOn ` S ) M ) p ) $= | |
| + ( wcel wa ctree cvv wnel w3a cdm wceq wo cv cpthson cfv co wbr wex simprl | |
| + grleafela syl simprr jca wi simpl grleafoldoldpthon grleafpthcaseno exp31 | |
| + ex com23 impd grleafpthcaseon grleafpthcasenn ccased mpd ) GUASDLSKUBSKLU | |
| + CTUDHUBSHFUEUCTTZIASZJASZTZTZILSZIKUFZUGZJLSZJKUFZUGZTZEUHMUHIJCUIUJUKULM | |
| + UMEUMZVOVRWAVOVLVRVKVLVMUNABCDFGHIKLNOPQRUOUPVOVMWAVKVLVMUQABCDFGHJKLNOPQ | |
| + RUOUPURVOVKWBWCUSVKVNUTVKVPVSVQVTWCVKVPVSTWCABCDEFGHIJKLMNOPQRVAVDVKVQVSW | |
| + CVKVSVQWCVKVSVQWCABCDEFGHIJKLMNOPQRVBVCVEVFVKVPVTWCVKVPVTWCABCDEFGHIJKLMN | |
| + OPQRVGVCVFVKVQVTTWCABCDEFGHIJKLMNOPQRVHVDVIUPVJ $. | |
| + | |
| + $( Adjoining a fresh leaf to a tree produces an undirected simple graph. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleaftreeusgr $p |- ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) -> S e. USGraph ) $= | |
| + ( wcel cvv wnel wa w3a cusgr ctree cdm simpl1 treeusgr simpl2 simpl3 3jca | |
| + syl simpr jca grleafusgr ) FUAOZDIOZHPOHIQRZSZGPOGEUBQRZRZFTOZUMUNSZUPRCT | |
| + OUQUSUPUQURUMUNUQULURULUMUNUPUCFUDUHULUMUNUPUEULUMUNUPUFUGUOUPUIUJABCDEFG | |
| + HIJKLMNUKUH $. | |
| + | |
| + $( The vertex set after adjoining a fresh leaf to a tree is the prescribed | |
| + enlarged vertex set. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleaftreevtx $p |- ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) -> ( Vtx ` S ) = A ) $= | |
| + ( wcel cvv wnel wa w3a syl ctree cdm cvtx cfv wceq simpl1 treeusgr simpl2 | |
| + cusgr simpl3 3jca simpr jca grleafvtx ) FUAOZDIOZHPOHIQRZSZGPOGEUBQRZRZFU | |
| + IOZUPUQSZUSRCUCUDAUEUTVBUSUTVAUPUQUTUOVAUOUPUQUSUFFUGTUOUPUQUSUHUOUPUQUSU | |
| + JUKURUSULUMABCDEFGHIJKLMNUNT $. | |
| + | |
| + ${ | |
| + $d A f k m p $. $d E f k m p $. $d G f k m p $. | |
| + $d J f k m p $. $d N f k m p $. $d U f k m p $. | |
| + $d P f k m p $. $d V f k m p $. $d S f k m p $. | |
| + $d f k m p $. | |
| + $( Every pair of vertices in a tree enlarged by a fresh leaf is joined | |
| + by a path. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafallpthonvtx $p |- ( ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) /\ | |
| + ( k e. ( Vtx ` S ) /\ m e. ( Vtx ` S ) ) ) -> | |
| + E. f E. p f ( k ( PathsOn ` S ) m ) p ) $= | |
| + ( wcel wa ctree cvv wnel w3a cdm cv cfv cpthson co wbr wex simpl simprl | |
| + cvtx wceq grleaftreevtx syl eleqtrd simprr jca grleafallpthon ) IUASDLS | |
| + KUBSKLUCTUDJUBSJHUEUCTTZFUFZCUNUGZSZGUFZVDSZTZTZVBVCASZVFASZTZTEUFMUFVC | |
| + VFCUHUGUIUJMUKEUKVIVBVLVBVHULZVIVJVKVIVCVDAVBVEVGUMVIVBVDAUOVMABCDHIJKL | |
| + NOPQRUPUQZURVIVFVDAVBVEVGUSVNURUTUTABCDEHIJVCVFKLMNOPQRVAUQ $. | |
| + | |
| + $( Adjoining a fresh leaf to a tree preserves connectedness. | |
| + (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + grleafconngr $p |- ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) -> S e. ConnGraph ) $= | |
| + ( vf vp vk vm wcel cvv ctree wnel wa w3a cdm cconngr cv cpthson cfv wbr | |
| + co cvtx wral grleafallpthonvtx ralrimivva wb cusgr grleaftreeusgr elexd | |
| + wex eqid isconngr syl mpbird ) FUASDISHTSHIUBUCUDGTSGEUEUBUCUCZCUFSZOUG | |
| + PUGQUGRUGCUHUIUKUJPUTOUTZRCULUIZUMQVHUMZVEVGQRVHVHABCDOQREFGHIPJKLMNUNU | |
| + OVECTSVFVIUPVECUQABCDEFGHIJKLMNURUSOQRCVHTPVHVAVBVCVD $. | |
| + $} | |
| + | |
| + $( Adjoining a fresh leaf to a tree produces a tree. This is the standard | |
| + leaf-extension step; compare [Diestel] p. 14. (Contributed by Mingli | |
| + Yuan, 13-Aug-2026.) $) | |
| + grleaftree $p |- ( ( ( G e. Tree /\ U e. V /\ | |
| + ( N e. _V /\ N e/ V ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) -> S e. Tree ) $= | |
| + ( ctree wcel cvv wnel wa syl w3a cdm cupgr cconngr cacycgr cvtx cfv cusgr | |
| + c0 wne grleaftreeusgr usgrupgr grleafconngr grleafacycgr grleafnewvtx jca | |
| + 3jca ne0i wb elexd istree mpbird ) FOPDIPHQPHIRSUAGQPGEUBRSSZCOPZCUCPZCUD | |
| + PZCUEPZUAZCUFUGZUIUJZSZVCVHVJVCVEVFVGVCCUHPVEABCDEFGHIJKLMNUKZCULTABCDEFG | |
| + HIJKLMNUMABCDEFGHIJKLMNUNUQVCHVIPVJABCDEFGHIJKLMNUOVIHURTUPVCCQPVDVKUSVCC | |
| + UHVLUTCQVATVB $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A x $. $d E x $. $d G x $. $d I x $. $d J x $. | |
| + $d N x $. $d P x $. $d S x $. $d T x $. $d U x $. | |
| + $d V x $. $d W x $. | |
| + trleafsubgr.v $e |- V = ( Vtx ` T ) $. | |
| + trleafsubgr.e $e |- E = ( iEdg ` T ) $. | |
| + trleafsubgr.w $e |- W = ( Vtx ` G ) $. | |
| + trleafsubgr.i $e |- I = ( iEdg ` G ) $. | |
| + trleafsubgr.a $e |- A = ( V u. { N } ) $. | |
| + trleafsubgr.p $e |- P = ( E u. { <. J , { U , N } >. } ) $. | |
| + trleafsubgr.s $e |- S = <. A , P >. $. | |
| + $( The vertex set of a fresh-leaf extension is a subclass of that of the | |
| + ambient graph. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| + trleafvtxss $p |- ( ( ( ( G e. USGraph /\ T e. Tree /\ | |
| + T SubGraph G ) /\ ( U e. V /\ N e. ( W \ V ) ) ) /\ | |
| + ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> | |
| + ( Vtx ` S ) C_ W ) $= | |
| + ( wcel cusgr csubgr wbr w3a cdif wa cdm cfv cpr wceq cvtx wss csn | |
| + simpll3 ctree cun subgrvtxss syl wb pm3.2i sseq12 ax-mp sylibr eldifi | |
| + snssd unssd simplrr a1i sseq1 mpbird cvv wnel trleafhyp grleaftreevtx | |
| + sseq1d ) | |
| + GUATZDUOTZDGUBUCZUDZEKTZJLKUETZUFZUFIHUGTIHUHEJUIUJUFZUFZCUKUHZLULAL | |
| + ULZWDWFKJUMZUPZLULZWDKWGLWDDUKUHZGUKUHZULZKLULZWDVRWLVPVQVRWBWCUNDGUQ | |
| + URKWJUJZLWKUJZUFWMWLUSWNWOMOUTKWJLWKVAVBVCWDJLWDWAJLTVSVTWAWCVGJLKVDU | |
| + RVEVFWDAWHUJZWFWIUSWPWDQVHAWHLVIURVJWDWEALWDVQVTJVKTJKVLUFUDIVKTIFUGV | |
| + LUFUFWEAUJABCDEFGHIJKLMNOPQRSVMABCEFDIJKMNQRSVNURVOVJ $. | |
| + | |
| + $( The indexed-edge function of a fresh-leaf extension is a subclass of | |
| + that of the ambient graph. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + trleafiedgss $p |- ( ( ( ( G e. USGraph /\ T e. Tree /\ | |
| + T SubGraph G ) /\ ( U e. V /\ N e. ( W \ V ) ) ) /\ | |
| + ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> | |
| + ( iEdg ` S ) C_ I ) $= | |
| + ( wa cusgr wcel ctree wbr w3a cdif cdm cfv cpr wceq ciedg wss cop csn cun | |
| + csubgr simpll3 subgriedgss syl wb pm3.2i sseq12 ax-mp sylibr simprr opeq2 | |
| + wfun simpll1 usgrfun funeq simprl jca funfvop eqeltrrd snssd unssd mpbird | |
| + a1i sseq1 cvv wnel trleafhyp simpl1 treeusgr simpl2 3jca simpr grleafiedg | |
| + simpl3 sseq1d ) GUAUBZDUCUBZDGUPUDZUEEKUBZJLKUFUBTZTZIHUGUBZIHUHZEJUIZUJZ | |
| + TZTZCUKUHZHULBHULZXBXDFIWSUMZUNZUOZHULZXBFXFHXBDUKUHZGUKUHZULZFHULZXBWMXK | |
| + WKWLWMWOXAUQDGURUSFXIUJZHXJUJZTXLXKUTXMXNNPVAFXIHXJVBVCVDXBXEHXBIWRUMZXEH | |
| + XBWTXOXEUJWPWQWTVEWRWSIVFUSXBHVGZWQTXOHUBXBXPWQXBXJVGZXPXBWKXQWKWLWMWOXAV | |
| + HGVIUSXNXPXQUTPHXJVJVCVDWPWQWTVKVLIHVMUSVNVOVPXBBXGUJZXDXHUTXRXBRVRBXGHVS | |
| + USVQXBXCBHXBDUAUBZWNJVTUBJKWATZUEZIVTUBIFUGWATZTZXCBUJXBWLWNXTUEZYBTZYCAB | |
| + CDEFGHIJKLMNOPQRSWBYEYAYBYEXSWNXTYEWLXSWLWNXTYBWCDWDUSWLWNXTYBWEWLWNXTYBW | |
| + IWFYDYBWGVLUSABCEFDIJKMNQRSWHUSWJVQ $. | |
| + | |
| + $( A fresh leaf of a tree, chosen from an ambient simple graph, remains a | |
| + subgraph of the ambient graph. (Contributed by Mingli Yuan, | |
| + 13-Aug-2026.) $) | |
| + trleafsubgr $p |- ( ( ( ( G e. USGraph /\ T e. Tree /\ | |
| + T SubGraph G ) /\ ( U e. V /\ N e. ( W \ V ) ) ) /\ | |
| + ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> | |
| + S SubGraph G ) $= | |
| + ( wcel cusgr ctree csubgr wbr w3a cdif wa cdm cfv cpr wceq cvtx wss ciedg | |
| + trleafvtxss trleafiedgss jca cvv wfun cuhgr wb simpll1 elex id usgrfun wi | |
| + syl funeq ax-mp a1i wnel trleafhyp grleaftree treeusgr usgruhgr 3jca eqid | |
| + sylibr uhgrissubgr mpbird ) GUATZDUBTZDGUCUDZUEEKTZJLKUFTUGZUGIHUHTIHUIEJ | |
| + UJUKUGZUGZCGUCUDZCULUIZLUMZCUNUIZHUMZUGZWGWJWLABCDEFGHIJKLMNOPQRSUOABCDEF | |
| + GHIJKLMNOPQRSUPUQWGGURTZHUSZCUTTZUEWHWMVAWGWNWOWPWGWAWNWAWBWCWEWFVBZGUAVC | |
| + VGWGGUNUIZUSZWOWGWAWSWGWAWAWQWAVDVGGVEVGHWRUKZWTVFZWOWSVAZWTVDXBXAWTXBPHW | |
| + RVHVIVJVIVRWGCUBTZWPWGWBWDJURTJKVKUGUEIURTIFUHVKUGUGXCABCDEFGHIJKLMNOPQRS | |
| + VLABCEFDIJKMNQRSVMVGXCCUATWPCVNCVOVGVGVPLHCGWKWIURWIVQOWKVQPVSVGVT $. | |
| + $} | |
| + | |
| + $( The hypotheses used to extend a tree select exactly the ambient | |
| + fresh-leaf data needed by ~ trleafhyp . (Contributed by Mingli Yuan, | |
| + 14-Aug-2026.) $) | |
| + trextleafdata $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + W e. Fin ) /\ ( T e. Tree /\ T SubGraph G /\ A =/= W ) ) /\ | |
| + ( U e. A /\ N e. ( W \ A ) ) ) /\ | |
| + ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> | |
| + ( ( ( G e. USGraph /\ T e. Tree /\ T SubGraph G ) /\ | |
| + ( U e. A /\ N e. ( W \ A ) ) ) /\ | |
| + ( J e. dom I /\ ( I ` J ) = { U , N } ) ) ) $= | |
| + ( cusgr wcel cconngr cfn w3a ctree csubgr wbr wne wa syl jca cdif cdm simpl | |
| + cfv cpr wceq simpll1 simplr1 simplr2 3jca simplr simpr ) DIJZDKJZHLJZMZBNJZ | |
| + BDOPZAHQZMZRZCAJGHAUAJRZRZFEUBJFEUDCGUEUFRZRZUMUQURMZVBRVDVEVFVBVEUMUQURVEV | |
| + CUMVCVDUCZUMUNUOUTVBUGSVEVCUQVGUQURUSUPVBUHSVEVCURVGUQURUSUPVBUISUJVAVBVDUK | |
| + TVCVDULT $. | |
| + | |
| + ${ | |
| + $d A J N U $. $d E J N U $. $d G J N U $. | |
| + $d I J N U $. $d T J N U $. $d W J N U $. | |
| + $d J N U $. | |
| + trextfresh.a $e |- A = ( Vtx ` T ) $. | |
| + trextfresh.e $e |- E = ( iEdg ` T ) $. | |
| + trextfresh.w $e |- W = ( Vtx ` G ) $. | |
| + trextfresh.i $e |- I = ( iEdg ` G ) $. | |
| + trextfresh.x $e |- X = | |
| + <. ( A u. { N } ) , ( E u. { <. J , { U , N } >. } ) >. $. | |
| + $( Extension data give a genuinely fresh leaf of the old tree. | |
| + (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| + trextfresh $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + W e. Fin ) /\ ( T e. Tree /\ T SubGraph G /\ A =/= W ) ) /\ | |
| + ( U e. A /\ N e. ( W \ A ) ) ) /\ | |
| + ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> | |
| + ( ( T e. Tree /\ U e. A /\ ( N e. _V /\ N e/ A ) ) /\ | |
| + ( J e. _V /\ J e/ dom E ) ) ) $= | |
| + ( wcel w3a wa cdm cvv cusgr cconngr cfn ctree csubgr wbr wne cdif cfv cpr | |
| + wceq wnel trextleafdata csn cun cop eqid trleafhyp syl ) EUAPZEUBPIUCPQBU | |
| + DPZBEUEUFZAIUGQRCAPZHIAUHPRZRGFSPGFUICHUJZUKRZRUTVAVBQVDRVFRVAVCHTPHAULRQ | |
| + GTPGDSULRRABCEFGHIUMAHUNUOZDGVEUPUNUOZJBCDEFGHAIKLMNVGUQVHUQOURUS $. | |
| + | |
| + $( The graph obtained from the extension data is a tree. | |
| + (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| + trexttree $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + W e. Fin ) /\ ( T e. Tree /\ T SubGraph G /\ A =/= W ) ) /\ | |
| + ( U e. A /\ N e. ( W \ A ) ) ) /\ | |
| + ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> X e. Tree ) $= | |
| + ( wcel w3a ctree wa cdm cusgr cconngr cfn csubgr wbr wne cdif cfv cpr cvv | |
| + wceq wnel trextfresh csn cun cop eqid grleaftree syl ) EUAPEUBPIUCPQBRPZB | |
| + EUDUEAIUFQSCAPZHIAUGPSSGFTPGFUHCHUIZUKSSUTVAHUJPHAULSQGUJPGDTULSSJRPABCDE | |
| + FGHIJKLMNOUMAHUNUOZDGVBUPUNUOZJCDBGHAKLVCUQVDUQOURUS $. | |
| + | |
| + $( The graph obtained from the extension data remains a subgraph of the | |
| + ambient graph. (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| + trextxsubgr $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + W e. Fin ) /\ ( T e. Tree /\ T SubGraph G /\ A =/= W ) ) /\ | |
| + ( U e. A /\ N e. ( W \ A ) ) ) /\ | |
| + ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> X SubGraph G ) $= | |
| + ( wcel w3a csubgr wbr wa cusgr cconngr cfn wne cdif cdm cfv trextleafdata | |
| + ctree cpr wceq csn cun cop eqid trleafsubgr syl ) EUAPZEUBPIUCPQBUIPZBERS | |
| + ZAIUDQTCAPHIAUEPTZTGFUFPGFUGCHUJZUKTZTURUSUTQVATVCTJERSABCEFGHIUHAHULUMZD | |
| + GVBUNULUMZJBCDEFGHAIKLMNVDUOVEUOOUPUQ $. | |
| + | |
| + $( The extension adjoins precisely the selected new vertex. | |
| + (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| + trextvtx $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + W e. Fin ) /\ ( T e. Tree /\ T SubGraph G /\ A =/= W ) ) /\ | |
| + ( U e. A /\ N e. ( W \ A ) ) ) /\ | |
| + ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> | |
| + ( Vtx ` X ) = ( A u. { N } ) ) $= | |
| + ( wcel w3a wa cdm cfv cusgr cconngr cfn ctree csubgr wbr wne cdif cpr cvv | |
| + wceq wnel cvtx csn cun trextfresh cop eqid grleaftreevtx syl ) EUAPEUBPIU | |
| + CPQBUDPZBEUEUFAIUGQRCAPZHIAUHPRRGFSPGFTCHUIZUKRRVAVBHUJPHAULRQGUJPGDSULRR | |
| + JUMTAHUNUOZUKABCDEFGHIJKLMNOUPVDDGVCUQUNUOZJCDBGHAKLVDURVEUROUSUT $. | |
| + $} | |
| + | |
| + ${ | |
| + trextvtxss.a $e |- A = ( Vtx ` T ) $. | |
| + trextvtxss.w $e |- W = ( Vtx ` G ) $. | |
| + $( The old tree's vertices form a subclass of the ambient vertex set. | |
| + (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| + trextvtxss $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + W e. Fin ) /\ ( T e. Tree /\ T SubGraph G /\ A =/= W ) ) /\ | |
| + ( U e. A /\ N e. ( W \ A ) ) ) /\ | |
| + ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> A C_ W ) $= | |
| + ( cusgr wcel cconngr cfn w3a wa cfv cvtx wss syl ctree csubgr wbr wne cdm | |
| + cdif cpr wceq simpllr simp2 subgrvtxss sseq12i bicomi biimpi ) DKLDMLHNLO | |
| + ZBUALZBDUBUCZAHUDZOZPCALGHAUFLPZPFEUELFEQCGUGUHPZPZBRQZDRQZSZAHSZVBUQVEVB | |
| + USUQUOUSUTVAUIUPUQURUJTBDUKTVEVFVFVEAVCHVDIJULUMUNT $. | |
| + | |
| + $( The old tree has a finite vertex set. | |
| + (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| + trextafin $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + W e. Fin ) /\ ( T e. Tree /\ T SubGraph G /\ A =/= W ) ) /\ | |
| + ( U e. A /\ N e. ( W \ A ) ) ) /\ | |
| + ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> A e. Fin ) $= | |
| + ( cusgr wcel cconngr cfn w3a ctree csubgr wbr wa syl wne cdif cdm cfv cpr | |
| + wceq wss simplll simp3 trextvtxss jca ssfi ) DKLZDMLZHNLZOZBPLBDQRAHUAOZS | |
| + CALGHAUBLSZSFEUCLFEUDCGUEUFSZSZUOAHUGZSANLUTUOVAUTUPUOUPUQURUSUHUMUNUOUIT | |
| + ABCDEFGHIJUJUKHAULT $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A J N U s $. $d E J N U s $. $d G J N U s $. | |
| + $d I J N U s $. $d T J N U s $. $d W J N U s $. | |
| + $d X s $. $d J N U s $. | |
| + trextsubgrexlem.a $e |- A = ( Vtx ` T ) $. | |
| + trextsubgrexlem.e $e |- E = ( iEdg ` T ) $. | |
| + trextsubgrexlem.w $e |- W = ( Vtx ` G ) $. | |
| + trextsubgrexlem.i $e |- I = ( iEdg ` G ) $. | |
| + trextsubgrexlem.x $e |- X = | |
| + <. ( A u. { N } ) , ( E u. { <. J , { U , N } >. } ) >. $. | |
| + $( A crossing edge extends a finite tree subgraph by exactly one vertex. | |
| + (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| + trextsubgrexlem $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + W e. Fin ) /\ ( T e. Tree /\ T SubGraph G /\ A =/= W ) ) /\ | |
| + ( U e. A /\ N e. ( W \ A ) ) ) /\ | |
| + ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> | |
| + E. s ( s e. Tree /\ s SubGraph G /\ | |
| + ( # ` ( Vtx ` s ) ) = ( ( # ` A ) + 1 ) ) ) $= | |
| + ( wcel wa cfv chash cusgr cconngr cfn w3a ctree wbr wne cdif cdm cpr wceq | |
| + csubgr cv cvtx c1 caddc co cvv trexttree syl trextxsubgr csn cun trextvtx | |
| + elex fveq2d simplrr trextafin eldifn jca hashunsng eqtrd 3jca eleq1 breq1 | |
| + wn imp 2fveq3 eqeq1d 3anbi123d spcedv ) EUAQEUBQIUCQUDBUEQBEULUFAIUGUDRZC | |
| + AQZHIAUHZQZRRGFUIQGFSCHUJUKRZRZKUMZUEQZWHEULUFZWHUNSTSZATSUOUPUQZUKZUDJUE | |
| + QZJEULUFZJUNSZTSZWLUKZUDKURJWGWNJURQABCDEFGHIJLMNOPUSZJUEVEUTWGWNWOWRWSAB | |
| + CDEFGHIJLMNOPVAWGWQAHVBVCZTSZWLWGWPWTTABCDEFGHIJLMNOPVDVFWGHURQZAUCQZHAQV | |
| + PZRZRXAWLUKZWGXBXEWGWEXBWBWCWEWFVGZHWDVEUTWGXCXDABCEFGHILNVHWGWEXDXGHIAVI | |
| + UTVJVJXBXEXFAHURVKVQUTVLVMWHJUKZWIWNWJWOWMWRWHJUEVNWHJEULVOXHWKWQWLWHJTUN | |
| + VRVSVTWA $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A j n u $. $d G j n u $. $d I j n u $. | |
| + $d T j n u $. $d W j n u $. $d j n u $. | |
| + trextcut.a $e |- A = ( Vtx ` T ) $. | |
| + trextcut.w $e |- W = ( Vtx ` G ) $. | |
| + trextcut.i $e |- I = ( iEdg ` G ) $. | |
| + $( A proper tree subgraph of a connected simple graph has an ambient | |
| + indexed edge crossing its vertex cut. (Contributed by Mingli Yuan, | |
| + 14-Aug-2026.) $) | |
| + trextcut $p |- ( ( ( G e. USGraph /\ G e. ConnGraph /\ W e. Fin ) /\ | |
| + ( T e. Tree /\ T SubGraph G /\ A =/= W ) ) -> | |
| + E. j e. dom I E. u e. A E. n e. ( W \ A ) | |
| + ( I ` j ) = { u , n } ) $= | |
| + ( wcel w3a wne wa c0 cv cfv wrex syl cusgr cconngr cfn ctree csubgr cupgr | |
| + wbr wss cpr wceq cdm simpl1 usgrupgr simpl2 jca simpr2 subgrvtxss sseq12i | |
| + cdif cvtx bicomi biimpi simpr1 treevtxne0 neeq1i simpr3 conngrcutiedg | |
| + 3jca ) FUALZFUBLZHUCLZMZCUDLZCFUEUGZBHNZMZOZFUFLZVJOZBHUHZBPNZVOMZODQGRAQ | |
| + EQUIUJEHBUSSABSDGUKSVQVSWBVQVRVJVQVIVRVIVJVKVPULFUMTVIVJVKVPUNUOVQVTWAVOV | |
| + QCUTRZFUTRZUHZVTVQVNWEVLVMVNVOUPCFUQTWEVTVTWEBWCHWDIJURVAVBTVQWCPNZWAVQVM | |
| + WFVLVMVNVOVCCVDTWFWAWAWFBWCPIVEVAVBTVLVMVNVOVFVHUOABDEGFHJKVGT $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A j n s u $. $d E j n s u $. $d G j n s u $. | |
| + $d I j n s u $. $d T j n s u $. $d W j n s u $. | |
| + $d j n s u $. | |
| + trextsubgrex.a $e |- A = ( Vtx ` T ) $. | |
| + trextsubgrex.e $e |- E = ( iEdg ` T ) $. | |
| + trextsubgrex.w $e |- W = ( Vtx ` G ) $. | |
| + trextsubgrex.i $e |- I = ( iEdg ` G ) $. | |
| + $( A proper finite tree subgraph of a connected simple graph can be | |
| + extended to a tree subgraph with one more vertex. (Contributed by | |
| + Mingli Yuan, 14-Aug-2026.) $) | |
| + trextsubgrex $p |- ( ( ( G e. USGraph /\ G e. ConnGraph /\ W e. Fin ) /\ | |
| + ( T e. Tree /\ T SubGraph G /\ A =/= W ) ) -> | |
| + E. s ( s e. Tree /\ s SubGraph G /\ | |
| + ( # ` ( Vtx ` s ) ) = ( ( # ` A ) + 1 ) ) ) $= | |
| + ( vj vu vn wcel w3a wa cv cfv jca cusgr cconngr cfn ctree csubgr wbr wceq | |
| + wne cpr cdif wrex cdm cvtx chash c1 caddc co wex trextcut wi simp1 simp22 | |
| + simp23 simp21 simp3 csn cun cop eqid trextsubgrexlem syl 3exp rexlimdvvva | |
| + imp mpd ) DUAODUBOFUCOPBUDOBDUEUFAFUHPQZLRZESMRZNRZUIZUGZNFAUJZUKMAUKLEUL | |
| + ZUKGRZUDOWDDUEUFWDUMSUNSAUNSUOUPUQUGPGURZMABLNDEFHJKUSVPWAWELMNWCAWBVPVQW | |
| + COZVRAOZVSWBOZPZWAWEUTVPWIWAWEVPWIWAPZVPWGWHQZQZWFWAQZQWEWJWLWMWJVPWKVPWI | |
| + WAVAWJWGWHVPWFWGWHWAVBVPWFWGWHWAVCTTWJWFWAVPWFWGWHWAVDVPWIWAVETTABVRCDEVQ | |
| + VSFAVSVFVGCVQVTVHVFVGVHZGHIJKWNVIVJVKVLVNVMVO $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A Y s $. $d E Y s $. $d G Y s $. | |
| + $d I Y s $. $d T Y s $. $d W Y s $. | |
| + trextsubgrhash.a $e |- A = ( Vtx ` T ) $. | |
| + trextsubgrhash.e $e |- E = ( iEdg ` T ) $. | |
| + trextsubgrhash.w $e |- W = ( Vtx ` G ) $. | |
| + trextsubgrhash.i $e |- I = ( iEdg ` G ) $. | |
| + $( A finite tree subgraph whose vertex count is still below the ambient | |
| + count can be extended by one vertex. (Contributed by Mingli Yuan, | |
| + 14-Aug-2026.) $) | |
| + trextsubgrhash $p |- ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + W e. Fin ) /\ | |
| + ( T e. Tree /\ T SubGraph G /\ ( # ` A ) = Y ) ) /\ | |
| + ( Y e. NN /\ ( Y + 1 ) <_ ( # ` W ) ) ) -> | |
| + E. s ( s e. Tree /\ s SubGraph G /\ | |
| + ( # ` ( Vtx ` s ) ) = ( Y + 1 ) ) ) $= | |
| + ( wcel w3a wbr chash cfv wceq wa syl cusgr cconngr cfn ctree csubgr cn c1 | |
| + caddc co cle cv cvtx wex simpll simplr1 simplr2 cr simplr3 simprl eqeltrd | |
| + wne nnre clt simprr cz nnz simpll3 hashcl nn0z 3syl zltp1led mpbird ltned | |
| + eqbrtrd fveq2 necon3i 3jca trextsubgrex simpr1 simpr2 simpr3 simpl oveq1d | |
| + cn0 jca eqtrd ex eximdv mpd ) DUAMZDUBMZFUCMZNZBUDMZBDUEOZAPQZGRZNZSZGUFM | |
| + ZGUGUHUIZFPQZUJOZSZSZHUKZUDMZXFDUEOZXFULQPQZWPUGUHUIZRZNZHUMZXGXHXIXARZNZ | |
| + HUMXEWMWNWOAFVAZNZSXMXEWMXQWMWRXDUNXEWNWOXPWNWOWQWMXDUOWNWOWQWMXDUPXEWPXB | |
| + VAXPXEWPXBXEWPGUQWNWOWQWMXDURZXEWTGUQMWSWTXCUSZGVBTUTXEWPGXBVCXRXEGXBVCOX | |
| + CWSWTXCVDXEGXBXEWTGVEMXSGVFTXEWLXBWDMXBVEMWJWKWLWRXDVGFVHXBVIVJVKVLVNVMAF | |
| + WPXBAFPVOVPTVQWEABCDEFHIJKLVRTXEXLXOHXEXLXOXEXLSZXGXHXNXEXGXHXKVSXEXGXHXK | |
| + VTXTXIXJXAXEXGXHXKWAXTWPGUGUHXTXEWQXEXLWBXRTWCWFVQWGWHWI $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G T Y s $. | |
| + $( A finite tree subgraph whose vertex count is still below that of the | |
| + ambient graph can be extended by one vertex. This version expands the | |
| + vertex- and indexed-edge-set abbreviations for use in induction. | |
| + (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| + trextsubgrhashfv $p |- ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ | |
| + ( T e. Tree /\ T SubGraph G /\ | |
| + ( # ` ( Vtx ` T ) ) = Y ) ) /\ | |
| + ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) -> | |
| + E. s ( s e. Tree /\ s SubGraph G /\ | |
| + ( # ` ( Vtx ` s ) ) = ( Y + 1 ) ) ) $= | |
| + ( cvtx cfv ciedg eqid trextsubgrhash ) AEFZAAGFZBBGFZBEFZCDJHKHMHLHI $. | |
| + $} | |
| + | |
| + ${ | |
| + $d F b $. $d G b $. $d K b $. $d M b $. $d N b $. $d P b $. | |
| + $( A degree-one vertex on a simple path between two vertices is one of its | |
| + endpoints. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + usgrspthonvd1inend $p |- ( ( G e. USGraph /\ | |
| + F ( K ( SPathsOn ` G ) M ) P /\ | |
| + ( ( ( VtxDeg ` G ) ` N ) = 1 /\ N e. ran P ) ) -> | |
| + ( N = K \/ N = M ) ) $= | |
| + ( vb wcel cfv co wbr c1 wceq wa w3a wo cc0 syl fveq2d simpl cvtxdg crn cv | |
| + cusgr cspthson chash cfz wrex simp3r wfn wb wf1 cspths simp2 spthonisspth | |
| + cvtx spthdifv fvelrnb mpbid simplrr eqcomd cwlkson simpl2 cpthson ctrlson | |
| + f1fn simpr spthonpthon pthontrlon trlsonwlkon 3syl wlkonendpts 3eqtrd orc | |
| + ex syl6 simprd olc simpl1 simprr simp3l 3jca simprl jca usgrspthvd1endidx | |
| + eqtrd mpjaod rexlimddv ) CUDHZBADECUEIJKZFCUAIZIZLMZFAUBHZNZOZGUCZAIZFMZF | |
| + DMZFEMZPZGQBUFIZUGJZWPWNWSGXDUHZWIWJWMWNUIWPAXDUJZWNXEUKWPXDCUPIZAULZXFWP | |
| + BACUMIKZXHWPWJXIWIWJWOUNDEABCUOZRABCUQRXDXGAVFRGXDFAURRUSWPWQXDHZWSNZNZWQ | |
| + QMZXBWQXCMZXMXNWTXBXMXNWTXMXNNZFWRQAIZDXPWRFWPXKWSXNUTVAXPWQQAXMXNVGSXPXQ | |
| + DMZXCAIZEMZNZXRXPBADECVBIJKZYAXPXMYBXMXNTXMWJYBWIWJWOXLVCZWJBADECVDIJKBAD | |
| + ECVEIJKYBDEABCVHDEABCVIDEABCVJVKRZRABCDEVLZRXRXTTRVMVOWTXAVNVPXMXOXAXBXMX | |
| + OXAXMXONZFWRXSEYFWRFWPXKWSXOUTVAYFWQXCAXMXOVGSYFYBXTYFXMYBXMXOTYDRYBXRXTY | |
| + EVQRVMVOXAWTVRVPXMWIXIWRWKIZLMZOZXKNXNXOPXMYIXKXMWIXIYHWIWJWOXLVSXMWJXIYC | |
| + XJRXMYGWLLXMWRFWKWPXKWSVTSXMWPWMWPXLTWIWJWMWNWARWFWBWPXKWSWCWDWQABCWERWGW | |
| + H $. | |
| + | |
| + $( A degree-one vertex different from both endpoints of a simple path is | |
| + not on the path. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + usgrspthonvd1nran $p |- ( ( G e. USGraph /\ | |
| + F ( K ( SPathsOn ` G ) M ) P /\ | |
| + ( ( ( VtxDeg ` G ) ` N ) = 1 /\ N =/= K /\ N =/= M ) ) -> | |
| + N e/ ran P ) $= | |
| + ( cusgr wcel cspthson cfv wceq wne w3a wn wa df-ne sylibr sylib jca syl | |
| + co wbr cvtxdg c1 crn wnel simp32 bicomi simp33 pm4.56 simpl1 simpl2 simpl | |
| + wo simp31 simpr 3jca usgrspthonvd1inend ex mtod df-nel ) CGHZBADECIJUAUBZ | |
| + FCUCJJUDKZFDLZFELZMZMZFAUEZHZNFVIUFVHVJFDKZFEKZUNZVHVKNZVLNZOVMNVHVNVOVHV | |
| + EVNVBVCVDVEVFUGVEVNFDPUHQVHVFVOVBVCVDVEVFUIFEPRSVKVLUJRVHVJVMVHVJOZVBVCVD | |
| + VJOZMVMVPVBVCVQVBVCVGVJUKVBVCVGVJULVPVDVJVPVHVDVHVJUMVBVCVDVEVFUOTVHVJUPS | |
| + UQABCDEFURTUSUTFVIVAQ $. | |
| + $} | |
| + | |
| + ${ | |
| + $d E a b f i k m p $. $d F a b f i k m p $. $d G a b f i k m p $. | |
| + $d I a b f i k m p $. $d K a b f i k m p $. $d M a b f i k m p $. | |
| + $d N a b f i k m p $. $d P a b f i k m p $. $d S a b f i k m p $. | |
| + $d V a b f i k m p $. $d a b f i k m p $. | |
| + usgrspthres.v $e |- V = ( Vtx ` G ) $. | |
| + usgrspthres.e $e |- E = ( iEdg ` G ) $. | |
| + usgrspthres.i $e |- I = { i e. dom E | N e/ ( E ` i ) } $. | |
| + usgrspthres.s $e |- S = <. ( V \ { N } ) , ( E |` I ) >. $. | |
| + $( A simple path avoiding a removed vertex remains a simple path in the | |
| + corresponding induced subgraph. (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + usgrspthres $p |- ( ( ( G e. USGraph /\ N e. V ) /\ | |
| + ( F ( SPaths ` G ) P /\ N e/ ran P ) ) -> | |
| + F ( SPaths ` S ) P ) $= | |
| + ( vk wcel wa cfv wceq syl jca cusgr cspths wbr wnel cupgr cwlks ccnv wfun | |
| + crn w3a simpl usgrres usgrupgr 3syl ciedg cdm cword cc0 chash cfz co cvtx | |
| + wf cv c1 caddc cpr cfzo wral cvv simprl spthiswlk wlkf simpr wrdsymbcl wn | |
| + wrdv wne simplrl eqid wlkp ffun elfzofz fdmd eleqtrrd simplrr 3jca necomd | |
| + nelrnfvne fzofzp1 nelprd df-nel sylibr eqidd weq 2fveq3 id fveq2d preq12d | |
| + fvoveq1d eqeq12d simpll upgrwlkedg rspcdva neleq12d mpbird wb crab eleq2i | |
| + fveq2 neleq2 elrab2 bitri a1i ralrimiva iswrdsymb cres vtxdginducedm1lem2 | |
| + csn cdif eqcomi wrdeqi eleqtrd wfn wss ffn frn simprr biimpi ssdifsn df-f | |
| + uhgrspan1lem2 feq3 ax-mp sylib simplll vtxdginducedm1lem3 eqtrd upgriswlk | |
| + spthf1 upgrwlkdvspth ) FUAOZHIOZPZEAFUBQUCZHAUIZUDZPZPZBUEOZEABUFQUCZAUGU | |
| + HZUJEABUBQUCUUIUUJUUKUULUUIUUDBUAOUUJUUDUUHUKBCDGFHIJKLMULBUMUNZUUIUUKEBU | |
| + OQZUPZUQZOZUREUSQZUTVAZBVBQZAVCZNVDZEQZUUNQZUVBAQZUVBVEVFVAZAQZVGZRZNURUU | |
| + RVHVAZVIZUJZUUIUUQUVAUVKUUIEGUQZUUPUUIEVJUQOZUVCGOZNUVJVIZPEUVMOUUIUVNUVP | |
| + UUIEDUPZUQOZUVNUUIUUEEAFUFQUCZUVRUUDUUEUUGVKZAEFVLZAEFDKVMZUNZUVQEVQSUUIU | |
| + VONUVJUUIUVBUVJOZPZUVOUVCUVQOZHUVCDQZUDZPZUWEUWFUWHUWEUVRUWDPZUWFUWEUVRUW | |
| + DUWEUUIUVRUUIUWDUKZUWCSUUIUWDVNZTUVBUVQEVOZSUWEUWHHUVHUDZUWEHUVHOVPUWNUWE | |
| + HUVEUVGUWEUVEHUWEAUHZUVBAUPZOZUUGUJUVEHVRUWEUWOUWQUUGUWEUUSFVBQZAVCZUWOUW | |
| + EUVSUWSUWEUUEUVSUUDUUEUUGUWDVSUWASZAEFUWRUWRVTWASZUUSUWRAWBSZUWEUVBUUSUWP | |
| + UWEUWDUVBUUSOUWLUVBURUURWCSUWEUUSUWRAUXAWDZWEUUDUUEUUGUWDWFZWGAUVBHWISWHU | |
| + WEUVGHUWEUWOUVFUWPOZUUGUJUVGHVRUWEUWOUXEUUGUXBUWEUVFUUSUWPUWEUWDUVFUUSOUW | |
| + LURUURUVBWJSUXCWEUXDWGAUVFHWISWHWKHUVHWLWMZUWEHHUWGUVHUWEHWNUWECVDZEQDQZU | |
| + XGAQZUXGVEVFVAAQZVGZRZUWGUVHRZCUVJUVBCNWOZUXHUWGUXKUVHUXGUVBDEWPUXNUXIUVE | |
| + UXJUVGUXNUXGUVBAUXNWQZWRUXNUXGUVBVEAVFUXOWTWSXAZUWEUUIUXLCUVJVIZUWKUUIFUE | |
| + OZUVSPZUXQUUIUXRUVSUUIUUBUXRUUBUUCUUHXBFUMZSUUIUUEUVSUVTUWASTACEFDKXCZSSU | |
| + WLXDXEXFTUVOUWIXGUWEUVOUVCHUXGDQZUDZCUVQXHZOZUWIGUYDUVCLXIUYCUWHCUVCUVQUY | |
| + DUXGUVCRZUYBUWGRUYCUWHXGUXGUVCDXJUYBUWGHXKSUYDVTZXLXMXNXFXOTNGEXPSUVMUUPR | |
| + UUIGUUOUUOGDGXQZBCDFGIHXSXTZHIJKUYIVTZLUYHVTZMXRYAYBXNYCUUIUUSUYIAVCZUVAU | |
| + UIUYLAUUSYDZUUFUYIYEZPZUUIUYMUYNUUIUUSIAVCZUYMUUIUUEUVSUYPUVTUWAAEFIJWAUN | |
| + ZUUSIAYFSUUIUYNUUFIYEZHUUFOVPZPZUUIUYRUYSUUIUYPUYRUYQUUSIAYGSUUIUUGUYSUUD | |
| + UUEUUGYHUUGUYSHUUFWLYISTUYNUYTXGUUIUUFIHYJXNXFTUYLUYOXGUUIUUSUYIAYKXNXFUY | |
| + IUUTRUYLUVAXGUUTUYIBCGFDHIJKLMYLYAUYIUUTUUSAYMYNYOUUIUVINUVJUWEUVDUWGUVHU | |
| + WEUVOUVDUWGRUWEUVCUYDGUWEUWIUYEUWEUWFUWHUWEUWJUWFUWEUVRUWDUWEUVSUVRUWTUWB | |
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| + UVTAEFYTSWGAEBUUAS $. | |
| + | |
| + $( A simple path between two vertices which avoids a removed vertex remains | |
| + a path between the same vertices in the corresponding induced subgraph. | |
| + (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + usgrspthonres $p |- ( ( ( G e. USGraph /\ N e. V ) /\ | |
| + ( F ( K ( SPathsOn ` G ) M ) P /\ N e/ ran P ) ) -> | |
| + F ( K ( PathsOn ` S ) M ) P ) $= | |
| + ( wa cfv co wbr 3syl cusgr wcel cspthson crn wnel cc0 cpthson wceq simprl | |
| + cwlkson ctrlson spthonpthon trlsonwlkon wlkonendpts simpld simprd oveq12d | |
| + chash pthontrlon cspths cpths simpl spthonisspth syl simprr jca spthispth | |
| + usgrspthres pthonpth breqdi ) FUAUBJKUBPZEAHIFUCQRSZJAUDUEZPZPZUFAQZEURQA | |
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| + MVEVFVFABCDEFGJKLMNOVHVDAEBVGAEBVITVJ $. | |
| + | |
| + $( Removing a degree-one vertex from a tree leaves a connected graph. | |
| + (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + treeleafresconngr $p |- ( ( G e. Tree /\ N e. V /\ | |
| + ( ( VtxDeg ` G ) ` N ) = 1 ) -> S e. ConnGraph ) $= | |
| + ( vf vp wcel cfv cv wex wa syl jca vk vm va ctree cvtxdg wceq w3a cconngr | |
| + vb c1 cpthson co wbr csn cdif wral simprl eldifi simprr simpl1 treeconngr | |
| + cvv wb isconngr mpbid weq id oveq1d breqd 2exbidv oveq2d rspc2va cspthson | |
| + elex cusgr crn wnel treeusgr simpl2 adantr cacycgr simpl treeacycgr simpr | |
| + pthonacycspthon 3syl simpl3 eldifsni 3jca usgrspthonvd1nran usgrspthonres | |
| + wne necomd ex 2eximdv ralrimivva simp1 simp2 usgrres uhgrspan1lem2 eqcomi | |
| + mpd cvtx mpbird ) DUDNZFGNZFDUEOOUJUFZUGZAUHNZLPZMPZUAPZUBPZAUKOULUMZMQLQ | |
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| + XHXEUUFXEXFXGWQUUKSXEXFXGWRTABCEDFGHIJKWSSAVOVNSLUAUBAXQVBMAXCOXQABEDCFGH | |
| + IJKWTXAVDSXD $. | |
| + | |
| + $( Removing a vertex from a tree leaves an acyclic graph. | |
| + (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + treevtxresacycgr $p |- ( ( G e. Tree /\ N e. V ) -> | |
| + S e. AcyclicGraph ) $= | |
| + ( ctree wcel wa cacycgr csubgr wbr simpl syl jca treeacycgr treeupgr 3syl | |
| + cuhgr cupgr upgruhgr simpr uhgrspan1 acycgrsubgr ) DLMZFGMZNZDOMZADPQZNAO | |
| + MULUMUNULUJUMUJUKRZDUASULDUDMZUKNUNULUPUKULUJDUEMUPUODUBDUFUCUJUKUGTABEDC | |
| + FGHIJKUHSTADUIS $. | |
| + | |
| + $( Removing a leaf from a finite tree with at least two vertices leaves a | |
| + tree. See Observation in [Diestel] p. 14. (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + treeleafres $p |- ( ( ( G e. Tree /\ V e. Fin /\ | |
| + 1 < ( # ` V ) ) /\ | |
| + ( N e. V /\ ( ( VtxDeg ` G ) ` N ) = 1 ) ) -> | |
| + S e. Tree ) $= | |
| + ( vb wcel cfv w3a wa c0 wne syl jca ctree cfn c1 chash clt wbr wceq cupgr | |
| + cvtxdg cconngr cacycgr cvtx cusgr simpl1 treeusgr simprl usgrres usgrupgr | |
| + 3syl simprr 3jca treeleafresconngr treevtxresacycgr csn uhgrspan1lem2 a1i | |
| + cdif cv wrex simpl2 simpl3 hashgt12el2 simpl simpr necomd eldifsn biimpri | |
| + ne0i ex rexlimiv eqnetrd wb istree mpbird ) DUAMZGUBMZUCGUDNUEUFZOZFGMZFD | |
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| + XMXJXGVMXKFXFXJXGVNVOTXLXNXFGFVPVQSXEXFVRSVSVTSWATWLWNWMWTWBXDAUHWCSWD $. | |
| + $} | |
| + | |
| + ${ | |
| + $d E i $. $d G i $. $d N i $. $d V i $. | |
| + treeleafiedghash.v $e |- V = ( Vtx ` G ) $. | |
| + treeleafiedghash.e $e |- E = ( iEdg ` G ) $. | |
| + $( A leaf of a finite tree is incident with exactly one indexed edge. | |
| + (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + treeleafiedghash $p |- ( ( ( G e. Tree /\ V e. Fin ) /\ | |
| + ( N e. V /\ ( ( VtxDeg ` G ) ` N ) = 1 ) ) -> | |
| + ( # ` { i e. dom E | N e. ( E ` i ) } ) = 1 ) $= | |
| + ( wcel cfn wa cfv c1 wceq crab chash caddc co cc0 syl c0 ctree cvtxdg cdm | |
| + cv csn cfusgr cupgr w3a cusgr simpll treeusgr simplr isfusgr fusgrfupgrfs | |
| + jca sylibr simp3 3syl dmfi simprl eqid vtxdgfival eqidd usgrnloop0 fveq2d | |
| + hash0 a1i eqtrd oveq12d cn0 cc rabfi hashcl addridd 3eqtrd eqcomd simprr | |
| + nn0cn ) CUAHZEIHZJZDEHZDCUBKKZLMZJZJZDAUDBKZHZABUCZNZOKZWCLWFWCWKWFWCWKWG | |
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| + VOVPWAWBWDVQVH $. | |
| + $} | |
| + | |
| + ${ | |
| + $d E i $. $d G i $. $d I i $. $d J i $. $d N i $. $d V i $. | |
| + treeleafedgstep.v $e |- V = ( Vtx ` G ) $. | |
| + treeleafedgstep.e $e |- E = ( iEdg ` G ) $. | |
| + treeleafedgstep.k $e |- K = ( V \ { N } ) $. | |
| + treeleafedgstep.i $e |- I = { i e. dom E | N e/ ( E ` i ) } $. | |
| + treeleafedgstep.p $e |- P = ( E |` I ) $. | |
| + treeleafedgstep.s $e |- S = <. K , P >. $. | |
| + treeleafedgstep.j $e |- J = { i e. dom E | N e. ( E ` i ) } $. | |
| + $( Removing a leaf from a finite tree removes exactly one indexed edge. | |
| + (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + treeleafedgstep $p |- ( ( ( G e. Tree /\ V e. Fin ) /\ | |
| + ( N e. V /\ ( ( VtxDeg ` G ) ` N ) = 1 ) ) -> | |
| + ( # ` dom E ) = ( ( # ` dom P ) + 1 ) ) $= | |
| + ( wcel wa cfv ctree cfn cvtxdg c1 wceq cdm chash caddc co wfun ciedg eqid | |
| + cvv fvexi eqeltri a1i cuhgr simpll treeupgr upgruhgr 3syl uhgrfun syl jca | |
| + cupgr hashfundm eqcomd simprl simplr cfusgr cusgr treeusgr isfusgr sylibr | |
| + w3a fusgrfupgrfs simp3 finsumvtxdg2ssteplem1 cres funres funeqd mpbird cv | |
| + resex crab fveq2i treeleafiedghash eqtrd oveq12d 3eqtrd ) EUARZJUBRZSZIJR | |
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| + WJ $. | |
| + $} | |
| + | |
| + ${ | |
| + tree1v0iedghash.v $e |- V = ( Vtx ` G ) $. | |
| + tree1v0iedghash.i $e |- I = ( iEdg ` G ) $. | |
| + $( A tree with one vertex has no indexed edges. | |
| + (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + tree1v0iedghash $p |- ( ( G e. Tree /\ ( # ` V ) = 1 ) -> | |
| + ( # ` dom I ) = 0 ) $= | |
| + ( ctree wcel chash cfv c1 wceq wa cdm cedg cc0 cvv ciedg eqid fvexi syl | |
| + eqeltri dmex cusgr wf1o simpl treeusgr usgrf1oedg hasheqf1od eqidd anim1i | |
| + a1i usgr1v0e 3eqtrd ) AFGZCHIJKZLZBMZHIANIZHIZUSOUPUQURPBUQPGUPBBAQIZPEUT | |
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| + KUNVAUOVBUJURACDVCULTUM $. | |
| + $} | |
| + | |
| + ${ | |
| + $d E g i $. $d G g i $. $d I g i $. $d J g i $. $d K g i $. | |
| + $d N g i $. $d P g i $. $d S g i $. $d V g i $. $d Y g i $. | |
| + treeiedghashleaf.v $e |- V = ( Vtx ` G ) $. | |
| + treeiedghashleaf.e $e |- E = ( iEdg ` G ) $. | |
| + treeiedghashleaf.k $e |- K = ( V \ { N } ) $. | |
| + treeiedghashleaf.i $e |- I = { i e. dom E | N e/ ( E ` i ) } $. | |
| + treeiedghashleaf.p $e |- P = ( E |` I ) $. | |
| + treeiedghashleaf.s $e |- S = <. K , P >. $. | |
| + treeiedghashleaf.j $e |- J = { i e. dom E | N e. ( E ` i ) } $. | |
| + treeiedghashleaf.h $e |- A. g ( ( g e. Tree /\ | |
| + ( Vtx ` g ) e. Fin /\ ( # ` ( Vtx ` g ) ) = Y ) -> | |
| + ( ( # ` dom ( iEdg ` g ) ) + 1 ) = ( # ` ( Vtx ` g ) ) ) $. | |
| + $( The induction hypothesis for a smaller tree proves the indexed-edge | |
| + count after removing a leaf. (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + treeiedghashleaf $p |- ( ( ( Y e. NN /\ ( G e. Tree /\ V e. Fin /\ | |
| + ( # ` V ) = ( Y + 1 ) ) ) /\ | |
| + ( N e. V /\ ( ( VtxDeg ` G ) ` N ) = 1 ) ) -> | |
| + ( ( # ` dom E ) + 1 ) = ( # ` V ) ) $= | |
| + ( cn wcel ctree cfn chash cfv c1 caddc co wceq w3a cvtxdg cdm simpl simpr | |
| + wa 3simpa 3syl jca treeleafedgstep syl ciedg cvtx cop a1i fveq2d cvv cdif | |
| + csn eqid fvexi eqeltri difexi cres resex opiedgfv eqtrd eqcomd oveq1d clt | |
| + dmeqd wbr simp1 simp2 simpll cc0 nngt0 cr wb 1re ltaddpos2 mpbid breqtrrd | |
| + simp3 opeq12i eqtri treeleafres uhgrspan1lem2 diffi eqeltrd eqtr4d fveq2i | |
| + nnre 3jca cn0 wi elex simprl nnnn0 hashdifsnp1 mpd wal opex eleq1d eqeq1d | |
| + cv id 3anbi123d eqeq12d imbi12d spcv ax-mp 3eqtrd ) LUAUBZFUCUBZKUDUBZKUE | |
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| + VEXKUVPYCUWAYCVSYOYGYHUVMVRVQ $. | |
| + $} | |
| + | |
| + ${ | |
| + $d E g i $. $d G g i $. $d I g i $. $d J g i $. $d K g i $. | |
| + $d N g i $. $d P g i $. $d S g i $. $d V g i $. $d Y g i $. | |
| + treeiedghashleafd.v $e |- V = ( Vtx ` G ) $. | |
| + treeiedghashleafd.e $e |- E = ( iEdg ` G ) $. | |
| + treeiedghashleafd.k $e |- K = ( V \ { N } ) $. | |
| + treeiedghashleafd.i $e |- I = { i e. dom E | N e/ ( E ` i ) } $. | |
| + treeiedghashleafd.p $e |- P = ( E |` I ) $. | |
| + treeiedghashleafd.s $e |- S = <. K , P >. $. | |
| + treeiedghashleafd.j $e |- J = { i e. dom E | N e. ( E ` i ) } $. | |
| + treeiedghashleafd.h $e |- ( ph -> A. g ( ( g e. Tree /\ | |
| + ( Vtx ` g ) e. Fin /\ ( # ` ( Vtx ` g ) ) = Y ) -> | |
| + ( ( # ` dom ( iEdg ` g ) ) + 1 ) = ( # ` ( Vtx ` g ) ) ) ) $. | |
| + $( Deduction form of the leaf-removal induction step for the indexed-edge | |
| + count of a finite tree. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + treeiedghashleafd $p |- ( ( ph /\ | |
| + ( ( Y e. NN /\ ( G e. Tree /\ V e. Fin /\ | |
| + ( # ` V ) = ( Y + 1 ) ) ) /\ | |
| + ( N e. V /\ ( ( VtxDeg ` G ) ` N ) = 1 ) ) ) -> | |
| + ( ( # ` dom E ) + 1 ) = ( # ` V ) ) $= | |
| + ( cn wcel ctree cfn chash cfv c1 caddc co wceq w3a cvtxdg cdm simpr simpl | |
| + 3simpa 3syl jca ancom biimpi treeleafedgstep syl ciedg cvtx cres cop cdif | |
| + wa csn opeq12i eqtri uhgrspan1lem3 eqtr4i dmeqi fveq2i a1i oveq1d clt wbr | |
| + simp1 simp2 simpll cc0 nngt0 cr wb nnre 1re ltaddpos2 mpbid breqtrrd 3jca | |
| + simp3 treeleafres uhgrspan1lem2 diffi eqeltrd fveq2d cvv cn0 simprl nnnn0 | |
| + wi elex hashdifsnp1 mpd eqtrd cv wal opex eqeltri eleq1d eqeq1d 3anbi123d | |
| + id dmeqd eqeq12d imbi12d spcv 3eqtrd eqcomd ) AMUBUCZGUDUCZLUEUCZLUFUGZMU | |
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| + QVCWRYNUUNUURUFUGZMYNUUMUURUFUVOWSYNYHUVPMUKZUVMYNLWTUCZYKMXAUCZULYHUVQXD | |
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| + DXIZUDUCZUWAVEUGZUEUCZUWCUFUGZMUKZULZUWAVDUGZUNZUFUGZUHUIUJZUWEUKZXDZDXJU | |
| + VCUVDXDZAYNUPUAUWMUWNDCCUUPWTSJBXKXLUWACUKZUWGUVCUWLUVDUWOUWBUUTUWDUVAUWF | |
| + UVBUWOUWACUDUWOXPZXMUWOUWCUUMUEUWOUWACVEUWPWSZXMUWOUWEUUNMUWOUWCUUMUFUWQW | |
| + SZXNXOUWOUWKUULUWEUUNUWOUWJUUKUHUIUWOUWIUUJUFUWOUWHUUIUWOUWACVDUWPWSXQWSV | |
| + RUWRXRXSXTURXGUUNYTUKYOUUMJUFUUMUURJUVNPVNVPVQYAYOYNYTMUKUUBYNYTUVPMYTUVP | |
| + UKYNJUURUFPVPVQUVTXHVCYAVRYOYFYGYOYNYHUUBUVMVCYBXH $. | |
| + $} | |
| + | |
| + ${ | |
| + $d Y g h $. | |
| + $( Change the bound graph variable in the induction predicate for the | |
| + indexed-edge count of a finite tree. (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + treeiedghashcbv $p |- ( A. g ( ( g e. Tree /\ ( Vtx ` g ) e. Fin /\ | |
| + ( # ` ( Vtx ` g ) ) = Y ) -> | |
| + ( ( # ` dom ( iEdg ` g ) ) + 1 ) = ( # ` ( Vtx ` g ) ) ) -> | |
| + A. h ( ( h e. Tree /\ ( Vtx ` h ) e. Fin /\ | |
| + ( # ` ( Vtx ` h ) ) = Y ) -> | |
| + ( ( # ` dom ( iEdg ` h ) ) + 1 ) = ( # ` ( Vtx ` h ) ) ) ) $= | |
| + ( cv ctree wcel cvtx cfv cfn chash wceq w3a ciedg cdm caddc eleq1d fveq2d | |
| + c1 co wi id eqeq1d 3anbi123d dmeqd oveq1d eqeq12d imbi12d biimpd cbvalivw | |
| + weq ) ADZEFZUKGHZIFZUMJHZCKZLZUKMHZNZJHZROSZUOKZTZBDZEFZVDGHZIFZVFJHZCKZL | |
| + ZVDMHZNZJHZROSZVHKZTZABABUJZVCVPVQUQVJVBVOVQULVEUNVGUPVIVQUKVDEVQUAZPVQUM | |
| + VFIVQUKVDGVRQZPVQUOVHCVQUMVFJVSQZUBUCVQVAVNUOVHVQUTVMROVQUSVLJVQURVKVQUKV | |
| + DMVRQUDQUEVTUFUGUHUI $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G g h i j k n $. $d Y g h i j k n $. | |
| + $( The induction hypothesis for trees with ` Y ` vertices proves the | |
| + indexed-edge count for a fixed tree with ` Y + 1 ` vertices. | |
| + (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + treeiedghashstep $p |- ( Y e. NN -> | |
| + ( A. g ( ( g e. Tree /\ ( Vtx ` g ) e. Fin /\ | |
| + ( # ` ( Vtx ` g ) ) = Y ) -> | |
| + ( ( # ` dom ( iEdg ` g ) ) + 1 ) = ( # ` ( Vtx ` g ) ) ) -> | |
| + ( ( G e. Tree /\ ( Vtx ` G ) e. Fin /\ | |
| + ( # ` ( Vtx ` G ) ) = ( Y + 1 ) ) -> | |
| + ( ( # ` dom ( iEdg ` G ) ) + 1 ) = | |
| + ( # ` ( Vtx ` G ) ) ) ) ) $= | |
| + ( vn vj vi vh vk wcel cv cfv chash wceq w3a c1 co syl wa jca eqid cn cvtx | |
| + ctree cfn ciedg cdm caddc wal cvtxdg wrex clt wbr simp3 simp1 simp2 nngt0 | |
| + wi cc0 cr wb nnre 1re a1i ltaddpos2 mpbid breqtrrd 3jca cwwspthsn wne cfz | |
| + c0 crab weq id oveq1d neeq1d cbvrabv treeleaf simpll simplr wnel cres csn | |
| + simpr cop eqidd fveq2d neleq12d eleq12d treeiedghashcbv treeiedghashleafd | |
| + cdif 3ad2ant2 ex rexlimdva mpd 3exp ) CUAIZAJZUCIWSUBKZUDIWTLKZCMNWSUEKUF | |
| + LKOUGPXAMUQAUHZBUCIZBUBKZUDIZXDLKZCOUGPZMZNZBUEKZUFZLKOUGPXFMZWRXBXINZDJZ | |
| + BUIKKOMZDXDUJZXLXMXCXEOXFUKULZNXPXMXCXEXQXMXIXCWRXBXIUMZXCXEXHUNQXMXIXEXR | |
| + XCXEXHUOQXMOXGXFUKXMWROXGUKULZWRXBXIUNZWRURCUKULZXSCUPWRCUSIZOUSIZRYAXSUT | |
| + WRYBYCCVAYCWRVBVCSCOVDQVEQXMXIXHXRXCXEXHUMQVFVGDEBFJZBVHPZVKVIZFURXFVJPZV | |
| + LXDXDTZYFEJZBVHPZVKVIFEYGFEVMZYEYJVKYKYDYIBVHYKVNVOVPVQVRQXMXOXLDXDXMXNXD | |
| + IZRZXOXLYMXORZXMWRXIRZYLXORZRZRXLYNXMYQXMYLXOVSZYNYOYPYNWRXIYNXMWRYRXTQYN | |
| + XMXIYRXRQSYNYLXOXMYLXOVTYMXOWDSSSXMXJXNYDXJKZWAZFXKVLZWBZXDXNWCWLZUUBWEZG | |
| + HXJBUUAXNYSIZFXKVLUUCXNXDCYHXJTUUCTYTXNHJZXJKZWAFHXKFHVMZXNXNYSUUGUUHXNWF | |
| + ZUUHYDUUFXJUUHVNWGZWHVQUUBTUUDTUUEXNUUGIFHXKUUHXNXNYSUUGUUIUUJWIVQXBWRGJZ | |
| + UCIUUKUBKZUDIUULLKZCMNUUKUEKUFLKOUGPUUMMUQGUHXIAGCWJWMWKQWNWOWPWQ $. | |
| + $} | |
| + | |
| + ${ | |
| + $d Y g h $. | |
| + $( The indexed-edge count induction predicate for finite trees is closed | |
| + under successor. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + treeiedghashsucc $p |- ( Y e. NN -> | |
| + ( A. h ( ( h e. Tree /\ ( Vtx ` h ) e. Fin /\ | |
| + ( # ` ( Vtx ` h ) ) = Y ) -> | |
| + ( ( # ` dom ( iEdg ` h ) ) + 1 ) = ( # ` ( Vtx ` h ) ) ) -> | |
| + A. h ( ( h e. Tree /\ ( Vtx ` h ) e. Fin /\ | |
| + ( # ` ( Vtx ` h ) ) = ( Y + 1 ) ) -> | |
| + ( ( # ` dom ( iEdg ` h ) ) + 1 ) = | |
| + ( # ` ( Vtx ` h ) ) ) ) ) $= | |
| + ( vg cn wcel cv ctree cvtx cfv cfn chash w3a ciedg cdm c1 caddc co wi wal | |
| + wceq treeiedghashstep alrimdv treeiedghashcbv syl6 ) BDEZAFZGEZUFHIZJEZUH | |
| + KIZBTLUFMINKIOPQUJTZRASZCFZGEUMHIZJEUNKIZBOPQZTLUMMINKIOPQUOTRZCSUGUIUJUP | |
| + TLUKRASUEULUQCAUMBUAUBCAUPUCUD $. | |
| + $} | |
| + | |
| + $( The indexed-edge count identity holds for every tree with one vertex. | |
| + (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + treeiedghashbase $p |- A. h ( ( h e. Tree /\ | |
| + ( Vtx ` h ) e. Fin /\ ( # ` ( Vtx ` h ) ) = 1 ) -> | |
| + ( ( # ` dom ( iEdg ` h ) ) + 1 ) = ( # ` ( Vtx ` h ) ) ) $= | |
| + ( cv ctree wcel cvtx cfv cfn chash c1 wceq w3a ciedg cdm caddc co wi cc0 wa | |
| + simp1 eqid simp3 jca tree1v0iedghash syl oveq1d 0p1e1 eqcomd 3eqtrd ax-gen | |
| + a1i ) ABZCDZUKEFZGDZUMHFZIJZKZUKLFZMHFZINOZUOJPAUQUTQINOZIUOUQUSQINUQULUPRU | |
| + SQJUQULUPULUNUPSULUNUPUAZUBUKURUMUMTURTUCUDUEVAIJUQUFUJUQUOIVBUGUHUI $. | |
| + | |
| + ${ | |
| + $d A h $. $d B h $. | |
| + $( Congruence of the indexed-edge count induction predicate in its vertex | |
| + count parameter. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + treeiedghashbid $p |- ( A = B -> | |
| + ( A. h ( ( h e. Tree /\ ( Vtx ` h ) e. Fin /\ | |
| + ( # ` ( Vtx ` h ) ) = A ) -> | |
| + ( ( # ` dom ( iEdg ` h ) ) + 1 ) = ( # ` ( Vtx ` h ) ) ) <-> | |
| + A. h ( ( h e. Tree /\ ( Vtx ` h ) e. Fin /\ | |
| + ( # ` ( Vtx ` h ) ) = B ) -> | |
| + ( ( # ` dom ( iEdg ` h ) ) + 1 ) = | |
| + ( # ` ( Vtx ` h ) ) ) ) ) $= | |
| + ( wceq cv ctree wcel cfv cfn chash w3a ciedg cdm c1 caddc co wi id eqeq2d | |
| + cvtx 3anbi3d imbi1d albidv ) ABDZCEZFGZUETHZIGZUGJHZADZKZUELHMJHNOPUIDZQU | |
| + FUHUIBDZKZULQCUDUKUNULUDUJUMUFUHUDABUIUDRSUAUBUC $. | |
| + $} | |
| + | |
| + ${ | |
| + $d h x y $. $d Y h x $. | |
| + $( The indexed-edge count identity holds for every positive finite-tree | |
| + vertex count. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| + treeiedghashn $p |- ( Y e. NN -> | |
| + A. h ( ( h e. Tree /\ ( Vtx ` h ) e. Fin /\ | |
| + ( # ` ( Vtx ` h ) ) = Y ) -> | |
| + ( ( # ` dom ( iEdg ` h ) ) + 1 ) = | |
| + ( # ` ( Vtx ` h ) ) ) ) $= | |
| + ( vx vy cv ctree wcel cvtx cfv cfn chash wceq ciedg caddc treeiedghashbid | |
| + w3a c1 co wi wal cdm treeiedghashbase treeiedghashsucc nnind ) AEZFGZUEHI | |
| + ZJGZUGKIZCEZLPUEMIUAKIQNRUILZSATUFUHUIQLPUKSATUFUHUIDEZLPUKSATUFUHUIULQNR | |
| + ZLPUKSATUFUHUIBLPUKSATCDBUJQAOUJULAOUJUMAOUJBAOAUBAULUCUD $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G t u $. $d W t u $. $d t u $. | |
| + sptrbase.w $e |- W = ( Vtx ` G ) $. | |
| + $( A nonempty simple graph has a one-vertex tree subgraph. | |
| + (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| + sptrbase $p |- ( ( G e. USGraph /\ W =/= (/) ) -> | |
| + E. t ( t e. Tree /\ t SubGraph G /\ | |
| + ( # ` ( Vtx ` t ) ) = 1 ) ) $= | |
| + ( vu cusgr wcel c0 wa cv ctree csubgr cvtx cfv chash c1 wceq cvv elex syl | |
| + wne wex wbr w3a simpr bicomi sylibr csn cop simpl sngrtree simprl eleqtrd | |
| + n0 id a1i jca sngrsubgr snexg 0ex opvtxfv fveq2d hashsng eqtrd 3jca eleq1 | |
| + breq1 2fveq3 eqeq1d 3anbi123d spcedv expcom exlimdv mpd ) BFGZCHUAZIZEJZC | |
| + GZEUBZAJZKGZWABLUCZWAMNONZPQZUDZAUBZVQVPVTVOVPUEVPVTECUNUFUGVQVSWGEVSVQWG | |
| + VSVQIZWFVRUHZHUIZKGZWJBLUCZWJMNZONZPQZUDARWJWHWKWJRGWHVRRGZWKWHVSWPVSVQUJ | |
| + ZVRCSTZVRUKTZWJKSTWHWKWLWOWSWHBRGZVRBMNZGZIWLWHWTXBWHVOWTVSVOVPULBFSTWHVS | |
| + XBWQVSVRCXAVSUOCXAQVSDUPUMTUQVRBURTWHWPWOWRWPWNWIONPWPWMWIOWPWIRGZHRGZIWM | |
| + WIQWPXCXDVRRUSXDWPUTUPUQHWIRRVATVBVRRVCVDTVEWAWJQZWBWKWCWLWEWOWAWJKVFWAWJ | |
| + BLVGXEWDWNPWAWJOMVHVIVJVKVLVMVN $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G t $. | |
| + $( A nonempty simple graph has a one-vertex tree subgraph, with the vertex | |
| + set abbreviation expanded. (Contributed by Mingli Yuan, | |
| + 14-Aug-2026.) $) | |
| + sptrbasefv $p |- ( ( G e. USGraph /\ ( Vtx ` G ) =/= (/) ) -> | |
| + E. t ( t e. Tree /\ t SubGraph G /\ | |
| + ( # ` ( Vtx ` t ) ) = 1 ) ) $= | |
| + ( cvtx cfv eqid sptrbase ) ABBCDZGEF $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G Y h t $. | |
| + $( Substitution in the tree-subgraph existence predicate. | |
| + (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| + sptrcbvlem $p |- ( h = t -> | |
| + ( ( h e. Tree /\ h SubGraph G /\ | |
| + ( # ` ( Vtx ` h ) ) = Y ) <-> | |
| + ( t e. Tree /\ t SubGraph G /\ | |
| + ( # ` ( Vtx ` t ) ) = Y ) ) ) $= | |
| + ( weq ctree wcel csubgr wbr cvtx cfv chash wceq eleq1 breq1 fveq2d eqeq1d | |
| + cv id 3anbi123d ) BAEZBRZFGARZFGUBCHIUCCHIUBJKZLKZDMUCJKZLKZDMUBUCFNUBUCC | |
| + HOUAUEUGDUAUDUFLUAUBUCJUASPPQT $. | |
| + | |
| + $( Change the bound variable in the tree-subgraph existence predicate. | |
| + (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| + sptrcbv $p |- ( E. h ( h e. Tree /\ h SubGraph G /\ | |
| + ( # ` ( Vtx ` h ) ) = Y ) <-> | |
| + E. t ( t e. Tree /\ t SubGraph G /\ | |
| + ( # ` ( Vtx ` t ) ) = Y ) ) $= | |
| + ( cv ctree wcel csubgr wbr cvtx cfv chash wceq w3a sptrcbvlem cbvexvw ) B | |
| + EZFGQCHIQJKLKDMNAEZFGRCHIRJKLKDMNBAABCDOP $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G Y s t $. | |
| + $( An existentially given tree satisfying the extension hypotheses can be | |
| + enlarged by one vertex. (Contributed by Mingli Yuan, | |
| + 14-Aug-2026.) $) | |
| + sptrgrow $p |- ( E. t ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ | |
| + ( t e. Tree /\ t SubGraph G /\ | |
| + ( # ` ( Vtx ` t ) ) = Y ) ) /\ | |
| + ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) -> | |
| + E. s ( s e. Tree /\ s SubGraph G /\ | |
| + ( # ` ( Vtx ` s ) ) = ( Y + 1 ) ) ) $= | |
| + ( cusgr wcel cconngr cvtx cfv cfn w3a cv ctree csubgr chash wceq wa cn c1 | |
| + wbr caddc co cle wex trextsubgrhashfv exlimiv ) BEFBGFBHIZJFKALZMFUHBNTUH | |
| + HIOICPKQCRFCSUAUBZUGOIUCTQQDLZMFUJBNTUJHIOIUIPKDUDAUHBCDUEUF $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G Y $. | |
| + $( A successor cardinal bound implies the corresponding predecessor | |
| + bound for a nonempty finite graph. (Contributed by Mingli Yuan, | |
| + 14-Aug-2026.) $) | |
| + sptryle $p |- ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| + ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) -> | |
| + Y <_ ( # ` ( Vtx ` G ) ) ) $= | |
| + ( cusgr wcel cconngr cvtx cfv cfn w3a c0 wne wa cn caddc cle wbr nnre syl | |
| + c1 cr co chash simprl simplr simpll3 hashnncl mpbird simprr p1le syl3anc | |
| + wb ) ACDZAEDZAFGZHDZIZUNJKZLZBMDZBSNUAUNUBGZOPZLZLZBTDZUTTDZVABUTOPVCUSVD | |
| + URUSVAUCBQRVCUTMDZVEVCVFUQUPUQVBUDVCUOVFUQUKULUMUOUQVBUEUNUFRUGUTQRURUSVA | |
| + UHBUTUIUJ $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G Y h $. | |
| + $( The bounded induction hypothesis supplies a tree at the predecessor | |
| + cardinal. (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| + sptrih $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| + ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) /\ | |
| + ( Y <_ ( # ` ( Vtx ` G ) ) -> | |
| + E. h ( h e. Tree /\ h SubGraph G /\ | |
| + ( # ` ( Vtx ` h ) ) = Y ) ) ) -> | |
| + E. h ( h e. Tree /\ h SubGraph G /\ | |
| + ( # ` ( Vtx ` h ) ) = Y ) ) $= | |
| + ( cusgr wcel cconngr cvtx cfv cfn w3a c0 wne wa cn c1 caddc chash cle wbr | |
| + co cv ctree csubgr wceq wex wi sptryle id mpan9 ) BDEBFEBGHZIEJUJKLMCNECO | |
| + PTUJQHZRSMMCUKRSZULAUAZUBEUMBUCSUMGHQHCUDJAUEZUFZUNBCUGUOUHUI $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G Y t $. | |
| + $( The induction data and a predecessor tree imply the hypotheses of one | |
| + extension step. (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| + sptrdatain $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| + ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) /\ | |
| + ( t e. Tree /\ t SubGraph G /\ | |
| + ( # ` ( Vtx ` t ) ) = Y ) ) -> | |
| + ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ | |
| + ( t e. Tree /\ t SubGraph G /\ | |
| + ( # ` ( Vtx ` t ) ) = Y ) ) /\ | |
| + ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) ) $= | |
| + ( cusgr wcel cconngr cvtx cfv cfn w3a c0 wne wa cn c1 caddc chash wbr jca | |
| + co cle cv ctree csubgr wceq simplll simpr simplr ) BDEBFEBGHZIEJZUIKLZMZC | |
| + NECOPTUIQHUARMZMZAUBZUCEUOBUDRUOGHQHCUEJZMZUJUPMUMUQUJUPUJUKUMUPUFUNUPUGS | |
| + ULUMUPUHS $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G Y h t $. | |
| + $( Rename the bound variable of the predecessor-tree witness under the | |
| + fixed induction data. (Contributed by Mingli Yuan, | |
| + 14-Aug-2026.) $) | |
| + sptrcbvan $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| + ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) /\ | |
| + E. h ( h e. Tree /\ h SubGraph G /\ | |
| + ( # ` ( Vtx ` h ) ) = Y ) ) -> | |
| + ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| + ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) /\ | |
| + E. t ( t e. Tree /\ t SubGraph G /\ | |
| + ( # ` ( Vtx ` t ) ) = Y ) ) ) $= | |
| + ( cusgr wcel cconngr cvtx cfv cfn w3a c0 wa chash wbr cv ctree csubgr wex | |
| + wceq wne cn c1 caddc co cle sptrcbv anbi2i biimpi ) CEFCGFCHIZJFKUJLUAMDU | |
| + BFDUCUDUEUJNIUFOMMZBPZQFULCROULHINIDTKBSZMUKAPZQFUNCROUNHINIDTKASZMUMUOUK | |
| + ABCDUGUHUI $. | |
| + | |
| + $( Move the fixed induction data inside the existential quantifier and | |
| + discard the unused nonemptiness hypothesis. (Contributed by | |
| + Mingli Yuan, 14-Aug-2026.) $) | |
| + sptrdatat $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| + ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) /\ | |
| + E. t ( t e. Tree /\ t SubGraph G /\ | |
| + ( # ` ( Vtx ` t ) ) = Y ) ) -> | |
| + E. t ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ | |
| + ( t e. Tree /\ t SubGraph G /\ | |
| + ( # ` ( Vtx ` t ) ) = Y ) ) /\ | |
| + ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) ) $= | |
| + ( cusgr wcel cconngr cvtx cfv cfn w3a c0 wne wa cn c1 caddc chash wbr wex | |
| + co cle cv ctree csubgr wceq 19.42v sptrdatain eximi sylbir ) BDEBFEBGHZIE | |
| + JZUJKLMCNECOPTUJQHUARMZMZAUBZUCEUNBUDRUNGHQHCUEJZASMUMUOMZASUKUOMULMZASUM | |
| + UOAUFUPUQAABCUGUHUI $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G Y h $. | |
| + $( Package the fixed induction data with the predecessor-tree witness. | |
| + (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| + sptrdatah $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| + ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) /\ | |
| + ( Y <_ ( # ` ( Vtx ` G ) ) -> | |
| + E. h ( h e. Tree /\ h SubGraph G /\ | |
| + ( # ` ( Vtx ` h ) ) = Y ) ) ) -> | |
| + ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| + ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) /\ | |
| + E. h ( h e. Tree /\ h SubGraph G /\ | |
| + ( # ` ( Vtx ` h ) ) = Y ) ) ) $= | |
| + ( cusgr wcel cconngr cvtx cfv cfn w3a c0 wne wa cn c1 caddc chash cle wbr | |
| + co cv ctree csubgr wceq wex wi simpl sptrih jca ) BDEBFEBGHZIEJUJKLMCNECO | |
| + PTUJQHZRSMMZCUKRSAUAZUBEUMBUCSUMGHQHCUDJAUEZUFZMULUNULUOUGABCUHUI $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G h $. | |
| + $( Under the standing graph hypotheses, the bounded induction predicate | |
| + holds at one. (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| + sptrbaseh $p |- ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) -> | |
| + E. h ( h e. Tree /\ h SubGraph G /\ | |
| + ( # ` ( Vtx ` h ) ) = 1 ) ) $= | |
| + ( cusgr wcel cconngr cvtx cfv cfn w3a c0 wne wa cv ctree csubgr wbr chash | |
| + c1 wceq wex simp1l 3anandirs simpr jca sptrbasefv syl ) BCDZBEDZBFGZHDZIZ | |
| + UIJKZLZUGULLAMZNDUNBOPUNFGQGRSIATUMUGULUGUHUJULUGUGULUHULLUJULLUAUBUKULUC | |
| + UDABUEUF $. | |
| + | |
| + $( Base case of the bounded vertex-cardinality induction. | |
| + (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| + sptrindbase $p |- ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| + 1 <_ ( # ` ( Vtx ` G ) ) ) -> | |
| + E. h ( h e. Tree /\ h SubGraph G /\ | |
| + ( # ` ( Vtx ` h ) ) = 1 ) ) $= | |
| + ( cusgr wcel cconngr cvtx cfv cfn w3a c0 wne wa cv ctree csubgr wbr chash | |
| + c1 wceq wex cle sptrbaseh adantr ) BCDBEDBFGZHDIUDJKLAMZNDUEBOPUEFGQGRSIA | |
| + TRUDQGUAPABUBUC $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G Y h t $. $d G Y s t $. | |
| + $( The bounded spanning-tree construction predicate is closed under the | |
| + successor step. (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| + sptrstep $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| + ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) /\ | |
| + ( Y <_ ( # ` ( Vtx ` G ) ) -> | |
| + E. h ( h e. Tree /\ h SubGraph G /\ | |
| + ( # ` ( Vtx ` h ) ) = Y ) ) ) -> | |
| + E. s ( s e. Tree /\ s SubGraph G /\ | |
| + ( # ` ( Vtx ` s ) ) = ( Y + 1 ) ) ) $= | |
| + ( vt cusgr wcel cconngr cvtx cfv w3a wa chash cle wbr cv ctree csubgr wex | |
| + wceq cfn c0 cn c1 caddc co wi sptrdatah sptrcbvan sptrdatat sptrgrow 4syl | |
| + wne ) BFGBHGBIJZUAGKZUNUBUMLCUCGCUDUEUFZUNMJZNOLZLZCUQNOAPZQGUTBROUTIJMJC | |
| + TKASZUGLUSVALUSEPZQGVBBROVBIJMJCTKZESLUOVCLURLESDPZQGVDBROVDIJMJUPTKDSABC | |
| + UHEABCUIEBCUJEBCDUKUL $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G Y h $. | |
| + $( Rearrange the hypotheses of the induction successor step into the form | |
| + required by ~ sptrstep . (Contributed by Mingli Yuan, | |
| + 14-Aug-2026.) $) | |
| + sptrstepant $p |- ( ( Y e. NN /\ | |
| + ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| + Y <_ ( # ` ( Vtx ` G ) ) ) -> | |
| + E. h ( h e. Tree /\ h SubGraph G /\ | |
| + ( # ` ( Vtx ` h ) ) = Y ) ) /\ | |
| + ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| + ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) -> | |
| + ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| + ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) /\ | |
| + ( Y <_ ( # ` ( Vtx ` G ) ) -> | |
| + E. h ( h e. Tree /\ h SubGraph G /\ | |
| + ( # ` ( Vtx ` h ) ) = Y ) ) ) ) $= | |
| + ( cn wcel cusgr cconngr cvtx cfv cfn w3a c0 wne wa chash cle wbr cv jca | |
| + wi ctree csubgr wceq wex c1 caddc co simp3l simp1 simp3r simp2 mpand ) CD | |
| + EZBFEBGEBHIZJEKUNLMNZCUNOIZPQZNARZUAEURBUBQURHIOICUCKAUDZTZUOCUEUFUGUPPQZ | |
| + NZKZUOUMVANZNUQUSTVCUOVDUMUTUOVAUHZVCUMVAUMUTVBUIUMUTUOVAUJSSVCUOUQUSVEUM | |
| + UTVBUKULS $. | |
| + | |
| + $( Successor case of the bounded vertex-cardinality induction. | |
| + (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| + sptrindsucc $p |- ( Y e. NN -> | |
| + ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| + Y <_ ( # ` ( Vtx ` G ) ) ) -> | |
| + E. h ( h e. Tree /\ h SubGraph G /\ | |
| + ( # ` ( Vtx ` h ) ) = Y ) ) -> | |
| + ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| + ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) -> | |
| + E. h ( h e. Tree /\ h SubGraph G /\ | |
| + ( # ` ( Vtx ` h ) ) = ( Y + 1 ) ) ) ) ) $= | |
| + ( cn wcel cusgr cconngr cvtx cfv cfn w3a c0 wne wa chash cle wbr wceq wex | |
| + wi cv ctree csubgr c1 caddc co sptrstepant sptrstep syl 3exp ) CDEZBFEBGE | |
| + BHIZJEKULLMNZCULOIZPQZNAUAZUBEZUPBUCQZUPHIOIZCRKASZTZUMCUDUEUFZUNPQZNZUQU | |
| + RUSVBRKASZUKVAVDKUMUKVCNNUOUTTNVEABCUGABCAUHUIUJ $. | |
| + $} | |
| + | |
| + ${ | |
| + $d A h $. $d B h $. | |
| + $( Congruence of the bounded spanning-tree construction predicate in its | |
| + vertex-count parameter. (Contributed by Mingli Yuan, | |
| + 14-Aug-2026.) $) | |
| + sptrindbid $p |- ( A = B -> | |
| + ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| + A <_ ( # ` ( Vtx ` G ) ) ) -> | |
| + E. h ( h e. Tree /\ h SubGraph G /\ | |
| + ( # ` ( Vtx ` h ) ) = A ) ) <-> | |
| + ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| + B <_ ( # ` ( Vtx ` G ) ) ) -> | |
| + E. h ( h e. Tree /\ h SubGraph G /\ | |
| + ( # ` ( Vtx ` h ) ) = B ) ) ) ) $= | |
| + ( wceq cusgr wcel cconngr cvtx cfv cfn w3a c0 wne wa chash cle wbr cv wex | |
| + ctree csubgr breq1 anbi2d id eqeq2d 3anbi3d exbidv imbi12d ) ABEZDFGDHGDI | |
| + JZKGLUKMNOZAUKPJZQRZOULBUMQRZOCSZUAGZUPDUBRZUPIJPJZAEZLZCTUQURUSBEZLZCTUJ | |
| + UNUOULABUMQUCUDUJVAVCCUJUTVBUQURUJABUSUJUEUFUGUHUI $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G h x y $. $d Y h x $. | |
| + $( Every positive vertex count not exceeding that of the ambient graph is | |
| + attained by a tree subgraph. (Contributed by Mingli Yuan, | |
| + 14-Aug-2026.) $) | |
| + sptrindn $p |- ( Y e. NN -> | |
| + ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| + Y <_ ( # ` ( Vtx ` G ) ) ) -> | |
| + E. h ( h e. Tree /\ h SubGraph G /\ | |
| + ( # ` ( Vtx ` h ) ) = Y ) ) ) $= | |
| + ( vx vy cusgr wcel cvtx cfv w3a wa cv chash cle wbr wceq wi c1 sptrindbid | |
| + wex cconngr cfn c0 wne ctree csubgr caddc sptrindbase sptrindsucc nnind | |
| + co ) BFGBUAGBHIZUBGJULUCUDKZDLZULMIZNOKALZUEGZUPBUFOZUPHIMIZUNPJATQUMRUON | |
| + OKUQURUSRPJATQUMELZUONOKUQURUSUTPJATQUMUTRUGUKZUONOKUQURUSVAPJATQUMCUONOK | |
| + UQURUSCPJATQDECUNRABSUNUTABSUNVAABSUNCABSABUHABUTUIUJ $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G h $. | |
| + $( The bounded construction reaches the full vertex cardinality of the | |
| + ambient finite nonempty connected graph. (Contributed by Mingli Yuan, | |
| + 14-Aug-2026.) $) | |
| + sptrfull $p |- ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) -> | |
| + E. h ( h e. Tree /\ h SubGraph G /\ | |
| + ( # ` ( Vtx ` h ) ) = ( # ` ( Vtx ` G ) ) ) ) $= | |
| + ( cusgr wcel cconngr cvtx cfv cfn w3a c0 wne wa chash cn cle wbr cv ctree | |
| + csubgr syl wceq wex simpr wb simpl simp3 hashnncl mpbird id cr nnre leidd | |
| + jca sptrindn sylc ) BCDZBEDZBFGZHDZIZURJKZLZURMGZNDZVBVCVCOPZLAQZRDVFBSPV | |
| + FFGMGVCUAIAUBVBVDVAUTVAUCVBUSVDVAUDVBUTUSUTVAUEUPUQUSUFTURUGTUHZVBVBVEVBU | |
| + IVBVCVBVDVCUJDVGVCUKTULUMABVCUNUO $. | |
| + $} | |
| + | |
| + $( A finite set is equal to a subset having the same cardinality. This | |
| + lemma keeps ~ sptrvtxeq independent of other mathboxes. (Contributed by | |
| + Mingli Yuan, 14-Aug-2026.) $) | |
| + sptrvtxeqlem $p |- ( ( B e. Fin /\ A C_ B /\ | |
| + ( # ` A ) = ( # ` B ) ) -> A = B ) $= | |
| + ( cfn wcel wss chash cfv wceq w3a ssfi 3adant3 wi wa cen wbr hashen biimp3a | |
| + pm3.2 3ad2ant2 expcom fisseneq 3expa sylsyld 3expb com23 3impia 3com23 mpd | |
| + ) BCDZABEZAFGBFGHZIACDZABHZUIUJULUKBAJKUIUKUJULUMLZUIUKUJUNUIUKMZULUJUMULUO | |
| + UJUMLZULUIUKUPULUIUKIABNOZUJUIUJMZUMULUIUKUQABPQUIULUJURLUKUIUJRSURUQUMUIUJ | |
| + UQUMABUAUBTUCUDTUEUFUGUH $. | |
| + | |
| + $( A finite ambient vertex set and equality of vertex cardinalities force a | |
| + subgraph to contain every ambient vertex. (Contributed by Mingli Yuan, | |
| + 14-Aug-2026.) $) | |
| + sptrvtxeq $p |- ( ( ( Vtx ` G ) e. Fin /\ T SubGraph G /\ | |
| + ( # ` ( Vtx ` T ) ) = ( # ` ( Vtx ` G ) ) ) -> | |
| + ( Vtx ` T ) = ( Vtx ` G ) ) $= | |
| + ( cvtx cfv cfn wcel wbr chash wceq w3a wss simp1 simp2 subgrvtxss syl simp3 | |
| + csubgr 3jca sptrvtxeqlem ) | |
| + BCDZEFZABQGZACDZHDTHDIZJZUAUCTKZUDJUCTIUEUAUFUDUAUBUDLUEUBUFUAUBUDMABNOUAUB | |
| + UDPRUCTSO $. | |
| + | |
| + $( A tree subgraph of a finite simple graph with the full vertex cardinality | |
| + is a spanning tree. (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| + sptrwitness $p |- ( ( ( G e. USGraph /\ ( Vtx ` G ) e. Fin ) /\ | |
| + ( T e. Tree /\ T SubGraph G /\ | |
| + ( # ` ( Vtx ` T ) ) = ( # ` ( Vtx ` G ) ) ) ) -> | |
| + T SpanningTree G ) $= | |
| + ( cusgr wcel cvtx cfv cfn ctree csubgr wbr chash wceq w3a csptr simpr simp1 | |
| + wa syl simp2 jca simplr simp3 3jca sptrvtxeq wb simpll issptr mpbird ) BCDZ | |
| + BEFZGDZQZAHDZABIJZAEFZKFUJKFLZMZQZABNJZUMUNQZUOUJLZQZURUTVAURUMUNURUQUMULUQ | |
| + OZUMUNUPPRZURUQUNVCUMUNUPSRZTURUKUNUPMVAURUKUNUPUIUKUQUAVEURUQUPVCUMUNUPUBR | |
| + UCABUDRTURUMUIQUSVBUEURUMUIVDUIUKUQUFTACBHUGRUH $. | |
| + | |
| + ${ | |
| + $d G h t $. | |
| + $( Every finite nonempty connected simple graph has a spanning tree. See | |
| + Result in [Diestel] p. 14. (Contributed by Mingli Yuan, | |
| + 14-Aug-2026.) $) | |
| + sptriex $p |- ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| + ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) -> | |
| + E. h h SpanningTree G ) $= | |
| + ( vt cusgr wcel cconngr cvtx cfv cfn w3a c0 wne wa ctree csubgr wbr chash | |
| + cv csptr jca wex sptrfull weq simpll1 simpll3 simpr sptrwitness syl breq1 | |
| + wceq biimprd syl5 spimevw exlimddv ) BDEZBFEZBGHZIEZJUQKLZMZCRZNEVABOPVAG | |
| + HQHUQQHUJJZARZBSPZAUACCBUBUTVBMZVDACVEVABSPZACUCZVDVEUOURMZVBMVFVEVHVBVEU | |
| + OURUOUPURUSVBUDUOUPURUSVBUETUTVBUFTVABUGUHVGVDVFVCVABSUIUKULUMUN $. | |
| + $} | |
| + | |
| + ${ | |
| + $d G h $. | |
| + $( A finite tree has one fewer indexed edge than vertices. See Corollary | |
| + 1.5.3 in [Diestel] p. 14. (Contributed by Mingli Yuan, | |
| + 12-Aug-2026.) $) | |
| + treeiedghash $p |- ( ( G e. Tree /\ ( Vtx ` G ) e. Fin ) -> | |
| + ( ( # ` dom ( iEdg ` G ) ) + 1 ) = ( # ` ( Vtx ` G ) ) ) $= | |
| + ( vh ctree wcel cvtx cfv cfn wa chash wceq w3a ciedg c1 caddc co wi simpr | |
| + cdm syl fveq2d cv cn c0 simpl treevtxne0 wb hashnncl mpbird treeiedghashn | |
| + wal id eleq1d eqeq1d 3anbi123d dmeqd oveq1d 2fveq3 eqeq12d imbi12d biimpd | |
| + wne spcimdv mpd eqidd 3jca syl5com ) ACDZAEFZGDZHZVGVIVHIFZVKJZKZALFZRZIF | |
| + ZMNOZVKJZPZVRVJBUAZCDZVTEFZGDZWBIFZVKJZKZVTLFZRZIFZMNOZWDJZPZBUJZVSVJVKUB | |
| + DZWMVJWNVHUCVAZVJVGWOVGVIUDZAUESVJVIWNWOUFVGVIQZVHUGSUHBVKUISVJWLVSBACWPV | |
| + JVTAJZHWRWLVSPVJWRQWRWLVSWRWFVMWKVRWRWAVGWCVIWEVLWRVTACWRUKZULWRWBVHGWRVT | |
| + AEWSTZULWRWDVKVKWRWBVHIWTTUMUNWRWJVQWDVKWRWIVPMNWRWHVOIWRWGVNWRVTALWSTUOT | |
| + UPVTAIEUQURUSUTSVBVCVJVMVSVRVJVGVIVLWPWQVJVKVDVEVSUKVFVC $. | |
| + $} | |
| + | |
| + $( A finite spanning tree has one fewer indexed edge than the vertices of | |
| + the graph it spans. (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| + sptriedghash $p |- ( ( T SpanningTree G /\ ( Vtx ` G ) e. Fin ) -> | |
| + ( ( # ` dom ( iEdg ` T ) ) + 1 ) = ( # ` ( Vtx ` G ) ) ) $= | |
| + ( csptr wbr cvtx cfv cfn wcel wa ciedg cdm chash c1 caddc co ctree sptrtree | |
| + wceq simpl syl sptrvtx simpr eqeltrd jca treeiedghash fveq2d eqtrd ) ABCDZB | |
| + EFZGHZIZAJFKLFMNOZAEFZLFZUILFUKAPHZUMGHZIULUNRUKUOUPUKUHUOUHUJSZABQTUKUMUIG | |
| + UKUHUMUIRUQABUATZUHUJUBUCUDAUETUKUMUILURUFUG $. | |
| + | |
| +$( (End of Mingli Yuan's mathbox.) $) | |
| +$( End $[ set-mbox-my.mm $] $) | |
| + | |
| + | |
| + | |
| $( Begin $[ set-mbox-jz.mm $] $) | |
| $( Skip $[ set-main.mm $] $) | |
| $( |
| $[ set-main.mm $] | |
| $( | |
| #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# | |
| Mathbox for Mingli Yuan | |
| #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# | |
| $) | |
| $( | |
| -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- | |
| Trees and spanning trees | |
| -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- | |
| $) | |
| $( The standard graph-theoretic background for this section can be found in | |
| section 1.5 of [Diestel]. $) | |
| $c Tree SpanningTree $. | |
| $( Extend class notation with the class of trees. $) | |
| ctree $a class Tree $. | |
| $( Extend class notation with the spanning-tree relation. $) | |
| csptr $a class SpanningTree $. | |
| ${ | |
| $( Define a tree as a nonempty connected acyclic undirected pseudograph. | |
| The nonemptiness condition is necessary because the empty graph is a | |
| connected graph by ~ 0conngr . Since an acyclic undirected pseudograph | |
| is simple by ~ upgracycusgr , this is equivalent to the standard | |
| simple-graph formulation, Definition in [Diestel] p. 14. (Contributed | |
| by Mingli Yuan, 11-Aug-2026.) $) | |
| df-tree $a |- Tree = { g | ( ( g e. UPGraph /\ g e. ConnGraph /\ | |
| g e. AcyclicGraph ) /\ ( Vtx ` g ) =/= (/) ) } $. | |
| $} | |
| ${ | |
| $d g t $. | |
| $( Define the relation which holds when the first argument is a spanning | |
| tree of the second argument. The use of ` SubGraph ` preserves indexed | |
| edge identities, and equality of the vertex sets makes the subgraph | |
| spanning. See ~ issubgr for the main characterization of ` SubGraph ` . | |
| For the standard formulation, see Definition in [Diestel] p. 14. | |
| (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| df-sptr $a |- SpanningTree = { <. t , g >. | | |
| ( ( t e. Tree /\ t SubGraph g ) /\ | |
| ( Vtx ` t ) = ( Vtx ` g ) ) } $. | |
| $} | |
| ${ | |
| $d g G $. | |
| $( The property of being a tree. (Contributed by Mingli Yuan, | |
| 11-Aug-2026.) $) | |
| istree $p |- ( G e. W -> ( G e. Tree <-> | |
| ( ( G e. UPGraph /\ G e. ConnGraph /\ G e. AcyclicGraph ) /\ | |
| ( Vtx ` G ) =/= (/) ) ) ) $= | |
| ( vg cv cupgr wcel cconngr cacycgr w3a cvtx cfv c0 wne wa ctree 3anbi123d | |
| wceq eleq1 fveq2 neeq1d anbi12d df-tree elab2g ) CDEFCDGFCDHFICDJKLMNAEFA | |
| GFAHFIAJKLMNCAOBCDAQCDEFCDGFCDHFIAEFAGFAHFICDJKLMAJKLMCDAQCDEFAEFCDGFAGFC | |
| DHFAHFCDAERCDAGRCDAHRPCDAQCDJKAJKLCDAJSTUACUBUC $. | |
| $} | |
| ${ | |
| $d g t $. $d g G $. $d g T $. $d t G $. $d t T $. | |
| $( The property of being a spanning tree. (Contributed by Mingli Yuan, | |
| 11-Aug-2026.) $) | |
| issptr $p |- ( ( T e. W /\ G e. U ) -> ( T SpanningTree G <-> | |
| ( ( T e. Tree /\ T SubGraph G ) /\ | |
| ( Vtx ` T ) = ( Vtx ` G ) ) ) ) $= | |
| ( vt vg cv ctree wcel csubgr wbr cvtx cfv wceq csptr simpl anbi12d fveq2d | |
| wa simpr eleq1d breq12d eqeq12d df-sptr brabga ) EGHIEGFGJKSEGLMFGLMNSAHI | |
| ACJKSALMCLMNSEFACODBEGANFGCNSEGHIEGFGJKSAHIACJKSEGLMFGLMNALMCLMNEGANFGCNS | |
| EGHIAHIEGFGJKACJKEGANFGCNSEGAHEGANFGCNPUAEGANFGCNSEGAFGCJEGANFGCNPEGANFGC | |
| NTUBQEGANFGCNSEGLMALMFGLMCLMEGANFGCNSEGALEGANFGCNPREGANFGCNSFGCLEGANFGCNT | |
| RUCQEFUDUE $. | |
| $} | |
| $( The defining properties of a tree. (Contributed by Mingli Yuan, | |
| 11-Aug-2026.) $) | |
| treeprop $p |- ( G e. Tree -> | |
| ( ( G e. UPGraph /\ G e. ConnGraph /\ G e. AcyclicGraph ) /\ | |
| ( Vtx ` G ) =/= (/) ) ) $= | |
| ( ctree wcel cupgr cconngr cacycgr w3a cvtx cfv c0 wne wa id wb elex istree | |
| cvv syl mpbid ) ABCZTADCAECAFCGAHIJKLZTMTAQCTUANABOAQPRS $. | |
| ${ | |
| $d g t $. | |
| $( The spanning-tree relation is a relation. (Contributed by Mingli Yuan, | |
| 11-Aug-2026.) $) | |
| relsptr $p |- Rel SpanningTree $= | |
| ( vt vg cv ctree wcel csubgr wbr wa cvtx cfv wceq csptr df-sptr relopabiv | |
| ) ACZDEOBCZFGHOIJPIJKHABLABMN $. | |
| $} | |
| $( The defining properties of a spanning tree. (Contributed by Mingli Yuan, | |
| 11-Aug-2026.) $) | |
| sptrprop $p |- ( T SpanningTree G -> | |
| ( ( T e. Tree /\ T SubGraph G ) /\ | |
| ( Vtx ` T ) = ( Vtx ` G ) ) ) $= | |
| ( csptr wbr ctree wcel csubgr wa cvtx cfv wceq id cvv wb relsptr brrelex12i | |
| issptr syl mpbid ) ABCDZTAEFABGDHAIJBIJKHZTLTAMFBMFHTUANABCOPAMBMQRS $. | |
| $( A tree is an undirected pseudograph. (Contributed by Mingli Yuan, | |
| 11-Aug-2026.) $) | |
| treeupgr $p |- ( G e. Tree -> G e. UPGraph ) $= | |
| ( ctree wcel cupgr cconngr cacycgr w3a cvtx cfv c0 wne treeprop simpl1 syl | |
| wa ) ABCADCZAECZAFCZGAHIJKZOPALPQRSMN $. | |
| $( A tree is connected. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| treeconngr $p |- ( G e. Tree -> G e. ConnGraph ) $= | |
| ( ctree wcel cupgr cconngr cacycgr w3a cvtx cfv c0 wne treeprop simpl2 syl | |
| wa ) ABCADCZAECZAFCZGAHIJKZOQALPQRSMN $. | |
| $( A tree is acyclic. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| treeacycgr $p |- ( G e. Tree -> G e. AcyclicGraph ) $= | |
| ( ctree wcel cupgr cconngr cacycgr w3a cvtx cfv c0 wne treeprop simpl3 syl | |
| wa ) ABCADCZAECZAFCZGAHIJKZORALPQRSMN $. | |
| $( A tree has a nonempty vertex set. (Contributed by Mingli Yuan, | |
| 11-Aug-2026.) $) | |
| treevtxne0 $p |- ( G e. Tree -> ( Vtx ` G ) =/= (/) ) $= | |
| ( ctree wcel cupgr cconngr cacycgr w3a cvtx cfv c0 wne treeprop simpr syl | |
| wa ) ABCADCAECAFCGZAHIJKZOQALPQMN $. | |
| $( A tree is a simple graph. (Contributed by Mingli Yuan, | |
| 11-Aug-2026.) $) | |
| treeusgr $p |- ( G e. Tree -> G e. USGraph ) $= | |
| ( ctree wcel cupgr cacycgr cusgr treeupgr treeacycgr jca upgracycusgr syl | |
| wa ) ABCZADCZAECZLAFCMNOAGAHIAJK $. | |
| $( The edgeless graph on one vertex is a tree. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| sngrtree $p |- ( U e. _V -> <. { U } , (/) >. e. Tree ) $= | |
| ( cvv wcel csn c0 cop ctree cupgr cconngr cacycgr w3a cvtx cfv wne wa snexg | |
| syl wceq a1i jca upgr0eop opex 0ex opvtxfvi 1conngr cumgr chash c1 usgr0eop | |
| snex cusgr usgrumgr hashsng eqcomi acycgr1v snnzg eqnetrrd wb istree mpbird | |
| 3jca ) ABCZADZEFZGCZVDHCZVDICZVDJCZKZVDLMZENZOZVBVIVKVBVFVGVHVBVCBCZVFABPZV | |
| CBUAQVBVDBCZVJVCRZOVGVBVOVPVOVBVCEUBSZVPVBEVCAUJUCUDZSTVDABUEQVBVDUFCZVCUGM | |
| UHRZOVHVBVSVTVBVDUKCZVSVBVMWAVNVCBUIQVDULQABUMTVDVCVJVCVRUNZUOQVAVBVCVJEVCV | |
| JRVBWBSABUPUQTVBVOVEVLURVQVDBUSQUT $. | |
| $( The edgeless graph on a vertex of a graph is a subgraph of that graph. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| sngrsubgr $p |- ( ( G e. _V /\ U e. ( Vtx ` G ) ) -> | |
| <. { U } , (/) >. SubGraph G ) $= | |
| ( cvv wcel cvtx cfv wa csn cop wss ciedg wfun cedg wceq w3a a1i jca 0ex crn | |
| c0 csubgr wbr simpl opex snex opvtxfvi simpr snssd eqsstrd opiedgfvi funeqd | |
| fun0 mpbird edgval rneqi rn0 3eqtri 3jca egrsubgr syl ) BCDZABEFZDZGZVAAHZT | |
| IZCDZGZVFEFZVBJZVFKFZLZVFMFZTNZGZOVFBUAUBVDVHVJVOVDVAVGVAVCUCVGVDVETUDPQVDV | |
| IVEVBVIVENVDTVEAUEZRUFPVDAVBVAVCUGUHUIVDVLVNVDVLTLZVQVDULPVDVKTVKTNVDTVEVPR | |
| UJZPUKUMVNVDVMVKSTSTVFUNVKTVRUOUPUQPQURVFCBCUSUT $. | |
| $( A spanning tree is a tree. (Contributed by Mingli Yuan, | |
| 11-Aug-2026.) $) | |
| sptrtree $p |- ( T SpanningTree G -> T e. Tree ) $= | |
| ( csptr wbr ctree wcel csubgr wa cvtx cfv wceq sptrprop simpll syl ) ABCDAE | |
| FZABGDZHAIJBIJKZHOABLOPQMN $. | |
| $( A spanning tree is a subgraph of the graph it spans. (Contributed by | |
| Mingli Yuan, 11-Aug-2026.) $) | |
| sptrsubgr $p |- ( T SpanningTree G -> T SubGraph G ) $= | |
| ( csptr wbr ctree wcel csubgr wa cvtx cfv wceq sptrprop simplr syl ) ABCDAE | |
| FZABGDZHAIJBIJKZHPABLOPQMN $. | |
| $( A spanning tree has the same vertices as the graph it spans. | |
| (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| sptrvtx $p |- ( T SpanningTree G -> | |
| ( Vtx ` T ) = ( Vtx ` G ) ) $= | |
| ( csptr wbr ctree wcel csubgr wa cvtx cfv wceq sptrprop simpr syl ) ABCDAEF | |
| ABGDHZAIJBIJKZHPABLOPMN $. | |
| $( A simple path in a subgraph is also a simple path in the containing | |
| graph. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| subgrspth $p |- ( S SubGraph G -> | |
| ( F ( SPaths ` S ) P -> F ( SPaths ` G ) P ) ) $= | |
| ( csubgr wbr ctrls cfv ccnv wfun wa cspths subgrtrl anim1d isspth 3imtr4g ) | |
| BDEFZCABGHFZAIJZKCADGHFZSKCABLHFCADLHFQRTSABCDMNACBOACDOP $. | |
| $( A simple path between two vertices in a subgraph is also a simple path | |
| between those vertices in the containing graph. (Contributed by | |
| Mingli Yuan, 13-Aug-2026.) $) | |
| subgrspthon $p |- ( S SubGraph G -> | |
| ( F ( K ( SPathsOn ` S ) M ) P -> | |
| F ( K ( SPathsOn ` G ) M ) P ) ) $= | |
| ( wbr cspthson cfv co wa cspths wceq w3a syl cvv wcel eqid jca wss cc0 sylc | |
| csubgr chash simpl simpr spthonisspth subgrspth cpthson ctrlson spthonpthon | |
| cwlkson pthontrlon trlsonwlkon cvtx cwlks wlkonprop simp32 simp33 4syl 3jca | |
| simpld simprd wb ciedg cpw subgrprop2 simp1 spthonprop simp12 sseldd simp13 | |
| cedg simp2 isspthonpth mpbird ex ) BDUCGZCAEFBHIJGZCAEFDHIJGZVRVSKZVTCADLIG | |
| ZUAAIEMZCUDIAIFMZNZWAWBWCWDWAVRCABLIGZWBVRVSUEZWAVSWFVRVSUFZEFACBUGOABCDUHU | |
| BWAWCWDWAVSWCWDKZWHVSCAEFBUIIJGCAEFBUJIJGZCAEFBULIJGZWIEFACBUKEFACBUMEFACBU | |
| NWKBPQZEBUOIZQZFWMQZNZCPQAPQKZCABUPIGZWCWDNNZWIEFACBWMWMRZUQWSWCWDWPWQWRWCW | |
| DURWPWQWRWCWDUSSOUTOZVBWAWCWDXAVCVAWAEDUOIZQZFXBQZKZWQKVTWEVDWAXEWQWAXCXDWA | |
| WMXBEWAVRWMXBTZWGVRXFBVEIZDVEIZTZBVMIZWMVFTZNXFXBXHBXJDXGWMWTXBRZXGRXHRXJRV | |
| GXFXIXKVHOOZWAVSWNWHVSWPWQWJWFKZNZWNEFACBWMWTVIZWLWNWOWQXNVJOOVKWAWMXBFXMWA | |
| VSWOWHVSXOWOXPWLWNWOWQXNVLOOVKSWAVSWQWHVSXOWQXPWPWQXNVNOOSEFACDXBPPXLVOOVPV | |
| Q $. | |
| $( A simple path has no repeated vertices. (Contributed by Mingli Yuan, | |
| 11-Aug-2026.) $) | |
| spthf1 $p |- ( F ( SPaths ` G ) P -> Fun `' P ) $= | |
| ( cspths cfv wbr ctrls ccnv wfun isspth simprbi ) BACDEFBACGEFAHIABCJK $. | |
| ${ | |
| $d x y $. $d x A $. $d x W $. $d y A $. $d y W $. | |
| $( Reversing a word with no repeated symbols preserves the absence of | |
| repeated symbols. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| revinj $p |- ( ( W e. Word A /\ Fun `' W ) -> | |
| Fun `' ( reverse ` W ) ) $= | |
| ( vx vy wcel wa cc0 cfv cvv wf1 cmin wral wceq ralrimiva syl cn0 eleqtrrd | |
| co 3syl jca cword ccnv wfun chash cfzo creverse c1 cv cmpt weq wi a1i cdm | |
| fvex simpll simpl simpr wrdf1d cfz simplr cz simplll lencl fzoval eleqtrd | |
| nn0z fznn0sub2 wrddm f1veqaeq cc nn0cn 1cnd subcld elfzonn0 subcanad syld | |
| biimpd eqid id oveq2d fveq2d f1mpt sylibr revval eqidd f1eq123d mpbird wf | |
| df-f1 simprbi ) BAUAZEZBUBUCZFZGBUDHZUERZIBUFHZJZWQUBUCZWNWRWPICWPWOUGKRZ | |
| CUHZKRZBHZUIZJZWNXCIEZCWPLZXCWTDUHZKRZBHZMZCDUJZUKZDWPLZCWPLZFXEWNXGXOWNX | |
| FCWPXFWNXAWPEZFZXBBUNULNWNXNCWPXQXMDWPXQXHWPEZFZXKXBXIMZXLXSBUMZABJZXBYAE | |
| ZXIYAEZFZFXKXTUKXSYBYEXSWNYBWNXPXRUOWNABWLWMUPZWLWMUQUROXSYCYDXSXBWPYAXSX | |
| BGWTUSRZWPXSXAYGEXBYGEXSXAWPYGWNXPXRUTZXSWOVAEZWPYGMXSWLWOPEZYIWLWMXPXRVB | |
| ZABVCZWOVFSGWOVDOZVEXAWTVGOYMQXSWLYAWPMYKABVHOZQXSXIWPYAXSXIYGWPXSXHYGEXI | |
| YGEXSXHWPYGXQXRUQZYMVEXHWTVGOYMQYNQTTYAAXBXIBVIOXSXTXLXSWTXAXHXSWOUGXSWLY | |
| JWOVJEYKYLWOVKSXSVLVMXSXPXAPEXAVJEYHXAWOVNXAVKSXSXRXHPEXHVJEYOXHWOVNXHVKS | |
| VOVQVPNNTCDWPIXCXJXDXDVRXLXBXIBXLXAXHWTKXLVSVTWAWBWCWNWPWPIIWQXDWNWLWQXDM | |
| YFCWKBWDOWNWPWEWNIWEWFWGWRWPIWQWHWSWPIWQWIWJO $. | |
| $} | |
| ${ | |
| $d B k $. $d E i k $. $d F i k $. $d G i k $. $d L i k $. | |
| $d P i k $. | |
| $d V i k $. $d i k $. | |
| upgrwlkres.v $e |- V = ( Vtx ` G ) $. | |
| upgrwlkres.e $e |- E = ( iEdg ` G ) $. | |
| $( Restricting the edge function of a walk to a set containing all of its | |
| indexed edges preserves the walk. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| upgrwlkres $p |- ( ( ( G e. UPGraph /\ F ( Walks ` G ) P ) /\ | |
| ran F C_ B ) -> F ( Walks ` <. V , ( E |` B ) >. ) P ) $= | |
| ( vk cupgr wcel cfv wa wss cdm co wf wceq 3syl syl cwlks wbr crn cres cop | |
| cword cc0 chash cfz cv c1 caddc cpr cfzo wral w3a simplr wlkf wrdfn simpr | |
| wfn cin wrdf frn ssind dmres sseqtrrdi wb df-f mpbir2and iswrdi wlkp wfun | |
| a1i simpllr fnfun wrddm eleqtrrd jca fvelrn sseldd fvres simpl upgrwlkedg | |
| r19.21bi eqtrd ralrimiva 3jca simpll upgrspanop cvtx fvexi ciedg opvtxfvi | |
| resex eqcomi opiedgfvi upgriswlk mpbird ) EJKZDBEUALUBZMZDUCZANZMZDBFCAUD | |
| ZUEZUALUBZDXFOZUFKZUGDUHLZUIPFBQZIUJZDLZXFLZXMBLXMUKULPBLUMZRZIUGXKUNPZUO | |
| ZUPZXEXJXLXSXEXRXIDQZXJXEYADXRVAZXCXINZXEXADCOZUFKZYBWTXAXDUQZBDECHURZYDD | |
| USZSXEXCAYDVBXIXEXCAYDXBXDUTXEXAYEXCYDNZYFYGYEXRYDDQYIYDDVCXRYDDVDTSVECAV | |
| FVGYAYBYCMVHXEXRXIDVIVNVJXIXKDVKTXEXAXLYFBDEFGVLTXEXQIXRXEXMXRKZMZXOXNCLZ | |
| XPYKXNAKXOYLRYKXCAXNXBXDYJUQYKDVMZXMDOZKZMXNXCKYKYMYOYKYEYBYMYKXAYEWTXAXD | |
| YJVOZYGTYHXRDVPSYKXMXRYNXEYJUTYKXAYEYNXRRYPYGYDDVQSVRVSXMDVTTWAXNACWBTXEY | |
| LXPRZIXRXEXBYQIXRUOXBXDWCBIDECHWDTWEWFWGWHXEXGJKZXHXTVHXEWTYRWTXAXDWIACEF | |
| GHWJTBIDXGXFFXGWKLFXFFFEWKGWLZCACEWMHWLWOZWNWPXGWMLXFXFFYSYTWQWPWRTWS $. | |
| $( Restricting the edge function of a path to a set containing all of its | |
| indexed edges preserves the path. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| upgrpthres $p |- ( ( ( G e. UPGraph /\ F ( Paths ` G ) P ) /\ | |
| ran F C_ B ) -> F ( Paths ` <. V , ( E |` B ) >. ) P ) $= | |
| ( cpths cfv wbr wa cres ctrls ccnv wfun cima cwlks syl jca cupgr wcel crn | |
| wss cop c1 chash cfzo co cc0 cpr cin c0 wceq simpll simplr pthiswlk simpr | |
| w3a upgrwlkres pthistrl istrl simprbi 3syl wb a1i mpbir2and ispth simp2bi | |
| simp3bi 3jca mpbird ) EUAUBZDBEIJKZLZDUCAUDZLZDBFCAMUEZIJKZDBVRNJKZBUFDUG | |
| JZUHUIZMOPZBUJWAUKQBWBQULUMUNZUSZVQVTWCWDVQVTDBVRRJKZDOPZVQVMDBERJKZLZVPL | |
| WFVQWIVPVQVMWHVMVNVPUOVQVNWHVMVNVPUPZBDEUQSTVOVPURTABCDEFGHUTSVQVNDBENJKZ | |
| WGWJBDEVAWKWHWGBDEVBVCVDVTWFWGLVEVQBDVRVBVFVGVQVNWCWJVNWKWCWDBDEVHZVISVQV | |
| NWDWJVNWKWCWDWLVJSVKVSWEVEVQBDVRVHVFVL $. | |
| $( Restricting the edge function of a cycle to a set containing all of its | |
| indexed edges preserves the cycle. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| upgrcyclres $p |- ( ( ( G e. UPGraph /\ F ( Cycles ` G ) P ) /\ | |
| ran F C_ B ) -> F ( Cycles ` <. V , ( E |` B ) >. ) P ) $= | |
| ( cupgr wcel ccycls cfv wbr wa crn wss cpths syl jca iscycl cop cc0 chash | |
| cres wceq simpll simplr cyclispth simpr upgrpthres simprbi a1i mpbir2and | |
| wb ) EIJZDBEKLMZNZDOAPZNZDBFCAUDUAZKLMZDBUTQLMZUBBLDUCLBLUEZUSUODBEQLMZNZ | |
| URNVBUSVEURUSUOVDUOUPURUFUSUPVDUOUPURUGZBDEUHRSUQURUISABCDEFGHUJRUSUPVCVF | |
| UPVDVCBDETUKRVAVBVCNUNUSBDUTTULUM $. | |
| $} | |
| ${ | |
| $d E k $. $d F k $. $d G k $. $d H k $. $d I k $. $d P k $. $d W k $. | |
| $d k x $. | |
| upgrvtxreswlk.w $e |- W = ( Vtx ` H ) $. | |
| upgrvtxreswlk.e $e |- E = ( iEdg ` G ) $. | |
| upgrvtxreswlk.i $e |- I = ( iEdg ` H ) $. | |
| $( A walk can be transferred between pseudographs with the same indexed | |
| edges when all its vertices belong to the target graph. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| upgrvtxreswlk $p |- ( ( ( ( G e. UPGraph /\ H e. UPGraph ) /\ E = I ) /\ | |
| ( F ( Walks ` G ) P /\ ran P C_ W ) ) -> F ( Walks ` H ) P ) $= | |
| ( vk cupgr wcel wa wceq cwlks cfv co wf syl wbr crn wss cdm cword cc0 cfz | |
| chash cv c1 caddc cpr cfzo wral w3a simprl wlkf simplr dmeqd wrdeq eqcomd | |
| id 3syl eleqtrrd cvtx eqid wlkp simprr fss syl2anc simplll jca upgrwlkedg | |
| ffrn fveq1d eqeq1d ralbidv mpbird 3jca wb simpllr upgriswlk ) DLMZELMZNZB | |
| FOZNZCADPQUAZAUBZGUCZNZNZCAEPQUAZCFUDZUEZMZUFCUHQZUGRZGASZKUIZCQZFQZWTAQW | |
| TUJUKRAQULZOZKUFWQUMRZUNZUOZWLWPWSXFWLCBUDZUEZWOWLWHWHCXIMWGWHWJUPZWHVBAC | |
| DBIUQVCWLXIWOWLXHWNOXIWOOWLBFWEWFWKURZUSXHWNUTTVAVDWLWRWIASZWJWSWLWHWRDVE | |
| QZASXLXJACDXMXMVFVGWRXMAVNVCWGWHWJVHWRWIGAVIVJWLXFXABQZXCOZKXEUNZWLWCWHNX | |
| PWLWCWHWCWDWFWKVKXJVLAKCDBIVMTWLXDXOKXEWLXBXNXCWLXNXBWLXABFXKVOVAVPVQVRVS | |
| WLWDWMXGVTWCWDWFWKWAAKCEFGHJWBTVR $. | |
| $( A path can be transferred between pseudographs with the same indexed | |
| edges when all its vertices belong to the target graph. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| upgrvtxrespth $p |- ( ( ( ( G e. UPGraph /\ H e. UPGraph ) /\ E = I ) /\ | |
| ( F ( Paths ` G ) P /\ ran P C_ W ) ) -> F ( Paths ` H ) P ) $= | |
| ( cupgr wcel wa wceq cpths cfv wbr ctrls jca 3syl crn wss chash cfzo cres | |
| c1 co ccnv wfun cc0 cpr cin c0 w3a cwlks simpl simprl pthiswlk syl simprr | |
| cima upgrvtxreswlk pthistrl istrl simprbi wb mpbird ispth simp2bi simp3bi | |
| a1i id 3jca ) DKLEKLMBFNMZCADOPQZAUAGUBZMZMZCAEOPQZCAERPQZAUFCUCPZUDUGZUE | |
| UHUIZAUJWAUKVAAWBVAULUMNZUNZVRVTWCWDVRVTCAEUOPQZCUHUIZMZVRWFWGVRVNCADUOPQ | |
| ZVPMZMWFVRVNWJVNVQUPVRWIVPVRVOWIVNVOVPUQZACDURUSVNVOVPUTSSABCDEFGHIJVBUSV | |
| RVOCADRPQZWGWKACDVCWLWIWGACDVDVETSVTWHVFVRACEVDVKVGVRVOVOWCWKVOVLZVOWLWCW | |
| DACDVHZVITVRVOVOWDWKWMVOWLWCWDWNVJTVMVSWEVFVRACEVHVKVG $. | |
| $( A cycle can be transferred between pseudographs with the same indexed | |
| edges when all its vertices belong to the target graph. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| upgrvtxrescycl $p |- ( ( ( ( G e. UPGraph /\ H e. UPGraph ) /\ E = I ) /\ | |
| ( F ( Cycles ` G ) P /\ ran P C_ W ) ) -> F ( Cycles ` H ) P ) $= | |
| ( cupgr wcel wa wceq ccycls cfv wbr cpths syl jca crn wss cc0 chash simpl | |
| simprl cyclispth simprr upgrvtxrespth id iscycl simprbi 3syl a1i mpbird | |
| wb ) DKLEKLMBFNMZCADOPQZAUAGUBZMZMZCAEOPQZCAERPQZUCAPCUDPAPNZMZVAVCVDVAUQ | |
| CADRPQZUSMZMVCVAUQVGUQUTUEVAVFUSVAURVFUQURUSUFZACDUGSUQURUSUHTTABCDEFGHIJ | |
| UISVAURURVDVHURUJURVFVDACDUKULUMTVBVEUPVAACEUKUNUO $. | |
| $} | |
| ${ | |
| $d E i $. $d G i $. $d J i $. $d K i $. $d U i $. $d V i $. | |
| $d ph i $. | |
| usgr2iedgvtxdge2d.v $e |- V = ( Vtx ` G ) $. | |
| usgr2iedgvtxdge2d.e $e |- E = ( iEdg ` G ) $. | |
| usgr2iedgvtxdge2d.g $e |- ( ph -> G e. USGraph ) $. | |
| usgr2iedgvtxdge2d.u $e |- ( ph -> U e. V ) $. | |
| usgr2iedgvtxdge2d.j $e |- ( ph -> J e. dom E ) $. | |
| usgr2iedgvtxdge2d.k $e |- ( ph -> K e. dom E ) $. | |
| usgr2iedgvtxdge2d.n $e |- ( ph -> J =/= K ) $. | |
| usgr2iedgvtxdge2d.a $e |- ( ph -> U e. ( E ` J ) ) $. | |
| usgr2iedgvtxdge2d.b $e |- ( ph -> U e. ( E ` K ) ) $. | |
| $( Two different indexed edges incident with a vertex of a simple graph | |
| force its degree to be at least two. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| usgr2iedgvtxdge2d $p |- ( ph -> | |
| 2 <_ ( ( VtxDeg ` G ) ` U ) ) $= | |
| ( vi cfv wcel cvv c2 cv cdm crab chash cvtxdg cle ciedg eqid eqeltri dmex | |
| fvexi rabex a1i wceq id fveq2d eleq2d elrabd nehash2 cusgr wa vtxdusgrval | |
| jca syl breqtrrd ) AUABQUBZCRZSZQCUCZUDZUERZBDUFRZRZUGAEFVKTVKTSAVIQVJCCD | |
| UHRZTIVODUHVOUIULUJUKUMUNAVIBECRZSQEVJVGEUOZVHVPBVQVGECVQUPUQURLOUSAVIBFC | |
| RZSQFVJVGFUOZVHVRBVSVGFCVSUPUQURMPUSNUTADVASZBGSZVBVNVLUOAVTWAJKVDQVJVMBD | |
| CGHIVJUIVMUIVCVEVF $. | |
| $} | |
| ${ | |
| $d E k $. $d F k $. $d G k $. $d P k $. $d ph k $. | |
| wlkiedgdomd.e $e |- E = ( iEdg ` G ) $. | |
| wlkiedgdomd.w $e |- ( ph -> F ( Walks ` G ) P ) $. | |
| wlkiedgdomd.k $e |- ( ph -> K e. ( 0 ..^ ( # ` F ) ) ) $. | |
| $( An indexed edge used at a position of a walk belongs to the indexed-edge | |
| domain of the graph. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| wlkiedgdomd $p |- ( ph -> ( F ` K ) e. dom E ) $= | |
| ( cc0 chash cfv cfzo co cdm cwlks wbr cword wcel wf wlkf wrdf ffvelcdmd | |
| 3syl ) AJDKLMNZCOZFDADBEPLQDUFRSUEUFDTHBDECGUAUFDUBUDIUC $. | |
| $} | |
| ${ | |
| $d E k $. $d F k $. $d G k $. $d K k $. $d P k $. $d ph k $. | |
| wlkvtxinedgd.e $e |- E = ( iEdg ` G ) $. | |
| wlkvtxinedgd.w $e |- ( ph -> F ( Walks ` G ) P ) $. | |
| wlkvtxinedgd.k $e |- ( ph -> K e. ( 0 ..^ ( # ` F ) ) ) $. | |
| $( Both vertices adjacent at a position of a walk are incident with the | |
| indexed edge used there. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| wlkvtxinedgd $p |- ( ph -> | |
| ( ( P ` K ) e. ( E ` ( F ` K ) ) /\ | |
| ( P ` ( K + 1 ) ) e. ( E ` ( F ` K ) ) ) ) $= | |
| ( vk cfv wcel c1 caddc co cpr wss fveq2 fvex a1i cv cc0 cfzo wceq fvoveq1 | |
| chash preq12d eqidd fveq12d sseq12d cwlks wbr wral wlkvtxeledg syl sseldd | |
| rspcdva prid1 prid2 jca ) AFBKZFDKZCKZLFMNOZBKZVCLAVAVEPZVCVAAJUAZBKZVGMN | |
| OBKZPZVGDKZCKZQZVFVCQJUBDUFKUCOZFVGFUDZVJVFVLVCVOVHVAVIVEVGFBRVGFMBNUEUGV | |
| OVKVBCCVOCUHVGFDRUIUJADBEUKKULVMJVNUMHBJDECGUNUOIUQZVAVFLAVAVEFBSURTUPAVF | |
| VCVEVPVEVFLAVAVEVDBSUSTUPUT $. | |
| $} | |
| ${ | |
| crctlenngt2d.c $e |- ( ph -> F ( Circuits ` G ) P ) $. | |
| crctlenngt2d.l $e |- ( ph -> 2 < ( # ` F ) ) $. | |
| $( A circuit whose length is greater than two has positive integral | |
| length. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| crctlenngt2d $p |- ( ph -> ( # ` F ) e. NN ) $= | |
| ( chash cfv cn wcel cn0 cc0 clt wbr wa ccrcts cwlks c2 cr a1i wlkcl nn0re | |
| crctiswlk 3syl 0re 2re syl 2pos lttrd jca wb elnnnn0b mpbird ) ACGHZIJZUN | |
| KJZLUNMNZOZAUPUQACBDPHNCBDQHNUPEBCDUCBCDUAUDZALRUNLSJAUETRSJAUFTAUPUNSJUS | |
| UNUBUGLRMNAUHTFUIUJUOURUKAUNULTUM $. | |
| $} | |
| ${ | |
| crctendidxd.c $e |- ( ph -> F ( Circuits ` G ) P ) $. | |
| crctendidxd.l $e |- ( ph -> 2 < ( # ` F ) ) $. | |
| $( The first and last edge positions of a circuit of length greater than | |
| two are distinct members of its edge-index range. (Contributed by | |
| Mingli Yuan, 13-Aug-2026.) $) | |
| crctendidxd $p |- ( ph -> | |
| ( 0 e. ( 0 ..^ ( # ` F ) ) /\ | |
| ( ( # ` F ) - 1 ) e. ( 0 ..^ ( # ` F ) ) /\ | |
| 0 =/= ( ( # ` F ) - 1 ) ) ) $= | |
| ( cc0 chash cfv cfzo co wcel c1 cmin wne cn crctlenngt2d syl c2 cr lbfzo0 | |
| sylibr fzo0end cn0 cle wbr nnnn0 2re a1i nnre ltled jca nn0ge2m1nn necomd | |
| wa nnne0 3jca ) AGGCHIZJKZLZURMNKZUSLZGVAOAURPLZUTABCDEFQZURUAUBAVCVBVDUR | |
| UCRAVAGAVAPLZVAGOAURUDLZSURUEUFZUOVEAVFVGAVCVFVDURUGRASURSTLAUHUIAVCURTLV | |
| DURUJRFUKULURUMRVAUPRUNUQ $. | |
| $} | |
| ${ | |
| $d E i k $. $d F i k $. $d G i k $. $d P i k $. $d V i k $. | |
| $d ph i k $. | |
| crctfstvdge2d.v $e |- V = ( Vtx ` G ) $. | |
| crctfstvdge2d.e $e |- E = ( iEdg ` G ) $. | |
| crctfstvdge2d.g $e |- ( ph -> G e. USGraph ) $. | |
| crctfstvdge2d.c $e |- ( ph -> F ( Circuits ` G ) P ) $. | |
| crctfstvdge2d.l $e |- ( ph -> 2 < ( # ` F ) ) $. | |
| $( In a simple graph, the initial vertex of a circuit of length greater | |
| than two has degree at least two. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| crctfstvdge2d $p |- ( ph -> | |
| 2 <_ ( ( VtxDeg ` G ) ` ( P ` 0 ) ) ) $= | |
| ( cc0 cfv c1 co wbr 3syl wcel syl wa chash cmin ccrcts cwlks wf crctiswlk | |
| cfz wlkp cn cn0 crctlenngt2d nnnn0 0elfz ffvelcdmd cfzo crctendidxd simp1 | |
| wne w3a wlkiedgdomd simp2 cdm wf1 ctrls crctistrl jca dff14i wlkvtxinedgd | |
| trlf1 caddc simpl wceq npcan1 fveq2d crctprop simpr eqcomd eqtrd eqeltrrd | |
| cc nncn usgr2iedgvtxdge2d ) ALBMZCELDMZDUAMZNUBOZDMZFGHIALWEUGOZFLBADBEUC | |
| MPZDBEUDMPZWHFBUEJBDEUFZBDEFGUHQAWEUIRZWEUJRLWHRABDEJKUKZWEULWEUMQUNABCDE | |
| LHAWIWJJWKSZALLWEUOOZRZWFWORZLWFURZUSZWPABDEJKUPZWPWQWRUQSZUTABCDEWFHWNAW | |
| SWQWTWPWQWRVASZUTAWOCVBZDVCZWSTWDWGURAXDWSAWIDBEVDMPZXDJBDEVEBDECHVIQWTVF | |
| WOXCDLWFVGSAWCWDCMZRZLNVJOBMXFRZTXGABCDELHWNXAVHXGXHVKSAWFNVJOZBMZWCWGCMZ | |
| AXJWEBMZWCAXIWEBAWLWEVTRXIWEVLWMWEWAWEVMQVNAWCXLAWIXEWCXLVLZTXMJBDEVOXEXM | |
| VPQVQVRAWFBMXKRZXJXKRZTXOABCDEWFHWNXBVHXNXOVPSVSWB $. | |
| $} | |
| ${ | |
| $d E i k $. $d F i k $. $d G i k $. $d L i k $. $d P i k $. | |
| $d V i k $. $d ph i k $. | |
| crctintvdge2d.v $e |- V = ( Vtx ` G ) $. | |
| crctintvdge2d.e $e |- E = ( iEdg ` G ) $. | |
| crctintvdge2d.g $e |- ( ph -> G e. USGraph ) $. | |
| crctintvdge2d.c $e |- ( ph -> F ( Circuits ` G ) P ) $. | |
| crctintvdge2d.b $e |- ( ph -> L e. ( 1 ..^ ( # ` F ) ) ) $. | |
| $( In a simple graph, an internal vertex of a circuit has degree at least | |
| two. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| crctintvdge2d $p |- ( ph -> | |
| 2 <_ ( ( VtxDeg ` G ) ` ( P ` L ) ) ) $= | |
| ( cfv c1 cmin co cc0 wbr syl wcel chash cfz wf ccrcts crctiswlk wlkp cfzo | |
| cwlks fzo0ss1 sseli ffvelcdmd cz elfzoel2 elfzo1elm1fzo0 jca elfzom1elfzo | |
| elfzofz wa wlkiedgdomd cdm wf1 wne w3a ctrls crctistrl trlf1 elfzoelz zcn | |
| cc 3syl 1zzd zcnd 0zd ax-1ne0 a1i subneintrd subid1d neeqtrd dff14i caddc | |
| npcan1 eqcomd fveq2d wlkvtxinedgd simprd eqeltrd simpld usgr2iedgvtxdge2d | |
| 3jca wceq ) AFBMZCEFNOPZDMZFDMZGHIJAQDUAMZUBPZGFBADBEUHMRZWPGBUCADBEUDMRZ | |
| WQKBDEUESZBDEGHUFSAFQWOUGPZTZFWPTAFNWOUGPZTZXALXBWTFWOUIUJSZFQWOUQSUKABCD | |
| EWLIWSAWOULTZWLQWONOPUGPTZURWLWTTZAXEXFAXCXELFNWOUMSAXCXFLFWOUNSUOWLWOUPS | |
| ZUSABCDEFIWSXDUSAWTCUTZDVAZXGXAWLFVBZVCZURWMWNVBAXJXLADBEVDMRZXJAWRXMKBDE | |
| VESBDECIVFSAXGXAXKXHXDAWLFQOPFAFNQAXCFULTFVITZLFNWOVGFVHVJZANAVKVLAQAVMVL | |
| NQVBAVNVOVPAFXOVQVRWIUOWTXIDWLFVSSAWKWLNVTPZBMZWMCMZAFXPBAXPFAXNXPFWJXOFW | |
| ASWBWCAWLBMXRTXQXRTABCDEWLIWSXHWDWEWFAWKWNCMZTFNVTPBMXSTABCDEFIWSXDWDWGWH | |
| $. | |
| $} | |
| ${ | |
| $d E i k $. $d F i k $. $d G i k $. $d L i k $. $d P i k $. | |
| $d V i k $. $d ph i k $. | |
| cyclposvdge2d.v $e |- V = ( Vtx ` G ) $. | |
| cyclposvdge2d.e $e |- E = ( iEdg ` G ) $. | |
| cyclposvdge2d.g $e |- ( ph -> G e. USGraph ) $. | |
| cyclposvdge2d.c $e |- ( ph -> F ( Cycles ` G ) P ) $. | |
| cyclposvdge2d.n $e |- ( ph -> F =/= (/) ) $. | |
| cyclposvdge2d.k $e |- ( ph -> L e. ( 0 ... ( # ` F ) ) ) $. | |
| $( Every vertex position of a nontrivial cycle in a simple graph has degree | |
| at least two. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| cyclposvdge2d $p |- ( ph -> | |
| 2 <_ ( ( VtxDeg ` G ) ` ( P ` L ) ) ) $= | |
| ( cfv c2 wbr wa simpr ex syl c1 chash cfzo cvtxdg cle cusgr adantr ccycls | |
| co wcel ccrcts simp1 3anidm23 cycliscrct 3syl crctintvdge2d wn cc0 simpll | |
| wceq wne w3a clt usgrgt2cycl crctfstvdge2d fveq2d breqtrrd ad2antrr cpths | |
| c0 3jca cyclprop simprd eqcomd eqtrd cfz wo jca elfznelfzo mpjaod pm2.61d | |
| simpl ) AFUADUBNZUCUIUJZOFBNZEUDNZNZUEPZAWDWHAWDQZBCDEFGHIAEUFUJZWDJUGWIA | |
| DBEUHNPZDBEUKNPZAWDAAWDWDULUMKBDEUNZUOAWDRUPSAWDUQZWHAWNQZFURUTZWHFWCUTZW | |
| OWPWHWOWPQZOURBNZWFNZWGUEWRBCDEGHIWRAWJAWNWPUSZJTZWRAWKWLXAKWMUOWRWJWKDVJ | |
| VAZVBZOWCVCPZWRWJWKXCXBWRAWKXAKTWRAXCXALTVKBDEVDZTVEWRWEWSWFWRFURBWOWPRVF | |
| VFVGSWOWQWHWOWQQZOWTWGUEXGBCDEGHIXGAWJAWNWQUSZJTZXGAWKWLXHKWMUOXGXDXEXGWJ | |
| WKXCXIXGAWKXHKTZAXCWNWQLVHVKXFTVEXGWEWSWFXGWEWCBNZWSXGFWCBWOWQRVFXGWSXKXG | |
| DBEVINPZWSXKUTZXGWKXLXMQXJBDEVLTVMVNVOVFVGSWOFURWCVPUIUJZWNQWPWQVQWOXNWNW | |
| OAXNAWNWBMTAWNRVRWCFVSTVTSWA $. | |
| $} | |
| ${ | |
| $d F f $. $d G f $. $d P f $. | |
| $( A simple path in an undirected pseudograph, viewed through its vertex | |
| sequence, is a fixed-length simple-path word. (Contributed by Mingli | |
| Yuan, 12-Aug-2026.) $) | |
| spthwspthn $p |- ( ( G e. UPGraph /\ F ( SPaths ` G ) P ) -> | |
| P e. ( ( # ` F ) WSPathsN G ) ) $= | |
| ( vf cupgr wcel cspths cfv wbr wa chash cwwspthsn co cwwlksn cv wex cwlks | |
| wceq syl cvv simpr spthiswlk eqidd jca simpl wlklnwwlkln1 cpths spthispth | |
| mpd ctrls pthistrl reltrls brrelex1i breq1 spcedv iswspthn a1i mpbir2and | |
| wi wb ) CEFZBACGHZIZJZABKHZCLMFZAVECNMFZDOZAVBIZDPZVDBACQHIZVEVERZJZVGVDV | |
| KVLVDVCVKVAVCUAZABCUBSVDVEUCUDVDVAVMVGUSVAVCUEABCVEUFSUIVDVIVCDTBVDVCBTFZ | |
| VNVCBACUGHIZVOABCUHVPBACUJHZIVOABCUKBAVQCULUMSSSVNVHBAVBUNUOVFVGVJJUTVDDC | |
| VEAUPUQUR $. | |
| $} | |
| ${ | |
| $d G f $. $d P f $. | |
| $( A word-walk with no repeated vertices in an undirected pseudograph is | |
| represented by a simple path. (Contributed by Mingli Yuan, | |
| 11-Aug-2026.) $) | |
| upgrwwlkdvspth $p |- ( ( G e. UPGraph /\ P e. ( WWalks ` G ) /\ | |
| Fun `' P ) -> | |
| E. f f ( SPaths ` G ) P ) $= | |
| ( cupgr wcel cwwlks cfv ccnv wfun w3a cv cwlks wbr cspths wlkiswwlksupgr2 | |
| wex imp 3adant3 wa syl simpl simp1 simpr 3jca upgrwlkdvspth ex eximdv mpd | |
| simp3 ) CDEZACFGEZAHIZJZBKZACLGMZBPZUNACNGMZBPUJUKUPULUJUKUPABCOQRUMUOUQB | |
| UMUOUQUMUOSZUJUOULJUQURUJUOULURUMUJUMUOUAZUJUKULUBTUMUOUCURUMULUSUJUKULUI | |
| TUDAUNCUETUFUGUH $. | |
| $} | |
| ${ | |
| $d G f $. $d N f $. $d W f $. | |
| $( A word representing a simple path has no repeated vertices. | |
| (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wspthnf1 $p |- ( W e. ( N WSPathsN G ) -> Fun `' W ) $= | |
| ( vf cwwspthsn co wcel cv cspths cfv wbr wex ccnv wfun cn0 cvv wa cwwlksn | |
| wspthnp simp3d spthf1 exlimiv syl ) CBAEFGDHCAIJKDLCMNCBAEFGBOGAPGQCBARFG | |
| DHCAIJKDLDABCSTDHCAIJKCMNDCDHAUAUBUC $. | |
| $} | |
| $( A fixed-length simple-path word is a word over the graph's vertex set. | |
| (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wspthnwrd $p |- ( W e. ( N WSPathsN G ) -> | |
| W e. Word ( Vtx ` G ) ) $= | |
| ( cwwspthsn co wcel cwwlksn cfv cword wspthsswwlkn sseli cn0 chash c1 caddc | |
| cvtx wceq wwlknbp1 simp2d syl ) CBADEZFCBAGEZFZCAPHIFZUAUBCABJKUCBLFUDCMHBN | |
| OEQABCRST $. | |
| $( A fixed-length simple-path word is one-to-one on its domain. | |
| (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wspthnf1m $p |- ( W e. ( N WSPathsN G ) -> | |
| W : dom W -1-1-> ( Vtx ` G ) ) $= | |
| ( cwwspthsn co wcel cdm cvtx cfv wf1 wf ccnv wfun cc0 chash cfzo cword wceq | |
| wspthnwrd syl wrddm eqcomd wrdf feq2dd wspthnf1 wa wb df-f1 a1i mpbir2and ) | |
| CBADEFZCGZAHIZCJZULUMCKZCLMZUKNCOIPEZULUMCUKCUMQFZUQULRABCSZURULUQUMCUAUBTU | |
| KURUQUMCKUSUMCUCTUDABCUEUNUOUPUFUGUKULUMCUHUIUJ $. | |
| $( Appending a vertex not occurring in a simple-path word preserves | |
| injectivity. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wspthnextf1 $p |- ( ( W e. ( N WSPathsN G ) /\ | |
| S e. ( Vtx ` G ) /\ -. S e. ran W ) -> | |
| ( W ++ <" S "> ) : dom ( W ++ <" S "> ) -1-1-> ( Vtx ` G ) ) $= | |
| ( cwwspthsn co wcel cvtx cfv crn w3a cvv 3ad2ant1 3ad2ant2 cdm wf1 cin wceq | |
| wn c0 cs1 fvex a1i cword wspthnwrd s1cl wspthnf1m id s1f1 s1rn ineq2d simp3 | |
| csn disjsn sylibr eqtrd ccatf1 ) DCBEFGZABHIZGZADJZGSZKZDAUAZUSLUSLGVCBHUBU | |
| CURUTDUSUDZGVBBCDUEMUTURVDVEGVBAUSUFNURUTDOUSDPVBBCDUGMUTURVDOUSVDPVBUTUSAU | |
| TUHUINVCVAVDJZQVAAUMZQZTVCVFVGVAUTURVFVGRVBAUSUJNUKVCVBVHTRURUTVBULVAAUNUOU | |
| PUQ $. | |
| $( Appending a vertex not occurring in a simple-path word preserves the | |
| absence of repeated vertices. (Contributed by Mingli Yuan, | |
| 11-Aug-2026.) $) | |
| wspthnextfun $p |- ( ( W e. ( N WSPathsN G ) /\ | |
| S e. ( Vtx ` G ) /\ -. S e. ran W ) -> | |
| Fun `' ( W ++ <" S "> ) ) $= | |
| ( cwwspthsn wcel cvtx cfv crn w3a cs1 cconcat cdm wf1 ccnv wfun wspthnextf1 | |
| co wn wf df-f1 simprbi syl ) DCBERFABGHZFADIFSJDAKLRZMZUDUENZUEOPZABCDQUGUF | |
| UDUETUHUFUDUEUAUBUC $. | |
| $( A simple-path word can be extended as a walk by an adjacent vertex. | |
| (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wspthnextwwlk $p |- ( ( W e. ( N WSPathsN G ) /\ | |
| S e. ( Vtx ` G ) /\ | |
| { ( lastS ` W ) , S } e. ( Edg ` G ) ) -> | |
| ( W ++ <" S "> ) e. ( ( N + 1 ) WWalksN G ) ) $= | |
| ( cwwspthsn co wcel cvtx cfv clsw cpr cedg w3a cwwlksn cs1 cconcat c1 caddc | |
| wspthsswwlkn eqid sseli 3ad2ant1 simp2 simp3 3jca wwlksnext syl ) DCBEFZGZA | |
| BHIZGZDJIAKBLIZGZMZDCBNFZGZUKUMMDAOPFCQRFBNFGUNUPUKUMUIUKUPUMUHUODBCSUAUBUI | |
| UKUMUCUIUKUMUDUEADULBCUJUJTULTUFUG $. | |
| $( An adjacent vertex not occurring in a simple-path word extends it to a | |
| walk with no repeated vertices. (Contributed by Mingli Yuan, | |
| 11-Aug-2026.) $) | |
| wspthnextb $p |- ( ( W e. ( N WSPathsN G ) /\ | |
| S e. ( Vtx ` G ) /\ | |
| ( { ( lastS ` W ) , S } e. ( Edg ` G ) /\ | |
| -. S e. ran W ) ) -> | |
| ( ( W ++ <" S "> ) e. ( ( N + 1 ) WWalksN G ) /\ | |
| Fun `' ( W ++ <" S "> ) ) ) $= | |
| ( cwwspthsn co wcel cvtx cfv clsw cpr cedg crn wn w3a cs1 cconcat 3jca syl | |
| wa c1 caddc cwwlksn ccnv wfun simp1 simp2 simp3 simpld wspthnextwwlk simprd | |
| wspthnextfun jca ) DCBEFGZABHIGZDJIAKBLIGZADMGNZTZOZDAPQFZCUAUBFBUCFGZUTUDU | |
| EZUSUNUOUPOVAUSUNUOUPUNUOURUFZUNUOURUGZUSUPUQUNUOURUHZUIRABCDUJSUSUNUOUQOVB | |
| USUNUOUQVCVDUSUPUQVEUKRABCDULSUM $. | |
| ${ | |
| $d G f $. $d S f $. $d W f $. | |
| $( An adjacent vertex not occurring in a simple path extends that path by | |
| one edge in an undirected pseudograph. (Contributed by Mingli Yuan, | |
| 11-Aug-2026.) $) | |
| wspthnext $p |- ( ( G e. UPGraph /\ | |
| ( W e. ( N WSPathsN G ) /\ S e. ( Vtx ` G ) /\ | |
| ( { ( lastS ` W ) , S } e. ( Edg ` G ) /\ -. S e. ran W ) ) ) -> | |
| ( W ++ <" S "> ) e. ( ( N + 1 ) WSPathsN G ) ) $= | |
| ( vf cupgr wcel cwwspthsn co cvtx cfv clsw cpr cedg crn wn wa w3a cs1 syl | |
| cconcat c1 caddc cwwlksn cspths wbr wex ccnv wfun simpr wspthnextb simpld | |
| cwwlks simpl wwlkswwlksn simprd 3jca upgrwwlkdvspth jca iswspthn sylibr | |
| cv ) BFGZDCBHIGABJKGDLKAMBNKGADOGPQRZQZDASUAIZCUBUCIZBUDIGZEVBVFBUEKUFEUG | |
| ZQVFVGBHIGVEVHVIVEVHVFUHUIZVEVDVHVJQVCVDUJABCDUKTZULZVEVCVFBUMKGZVJRVIVEV | |
| CVMVJVCVDUNVEVHVMVLBVGVFUOTVEVHVJVKUPUQVFEBURTUSEBVGVFUTVA $. | |
| $} | |
| ${ | |
| $d G m w $. $d N m w $. $d S m w $. $d W m w $. $d m w $. | |
| $( A universal upper bound on simple-path lengths applies to a path | |
| obtained by adjoining one symbol. (Contributed by Mingli Yuan, | |
| 11-Aug-2026.) $) | |
| wspthmaxspc $p |- ( A. m A. w | |
| ( w e. ( m WSPathsN G ) -> m <_ N ) -> | |
| ( ( W ++ <" S "> ) e. ( ( N + 1 ) WSPathsN G ) -> | |
| ( N + 1 ) <_ N ) ) $= | |
| ( c1 caddc co cvv wcel cconcat cv cwwspthsn cle wbr wi wal ovex wceq cs1 | |
| wa simpr simpl oveq1d eleq12d breq1d imbi12d spc2gv mp2an ) EGHIZJKFBUAZL | |
| IZJKAMZCMZDNIZKZUOEOPZQZARCRUMUKDNIZKZUKEOPZQZQEGHSFULLSUSVCCAUKUMJJUOUKT | |
| ZUNUMTZUBZUQVAURVBVFUNUMUPUTVDVEUCVFUOUKDNVDVEUDZUEUFVFUOUKEOVGUGUHUIUJ | |
| $. | |
| $} | |
| $( The length parameter of a fixed-length word walk is a nonnegative | |
| integer. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wwlknn0 $p |- ( W e. ( N WWalksN G ) -> N e. NN0 ) $= | |
| ( cwwlksn co wcel cn0 cvtx cfv cword chash c1 caddc wceq wwlknbp1 simp1d ) | |
| CBADEFBGFCAHIJFCKIBLMENABCOP $. | |
| $( The word representing a fixed-length walk has one more symbol than the | |
| stated length. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wwlknlen $p |- ( W e. ( N WWalksN G ) -> | |
| ( # ` W ) = ( N + 1 ) ) $= | |
| ( cwwlksn co wcel cn0 cvtx cfv cword chash c1 caddc wceq wwlknbp1 simp3d ) | |
| CBADEFBGFCAHIJFCKIBLMENABCOP $. | |
| $( The domain of a word representing a fixed-length walk is the closed | |
| integer interval from zero through the stated length. (Contributed by | |
| Mingli Yuan, 12-Aug-2026.) $) | |
| wwlkndm $p |- ( W e. ( N WWalksN G ) -> | |
| dom W = ( 0 ... N ) ) $= | |
| ( cwwlksn co wcel cdm cc0 chash cfv cfzo cfz cvtx cword wceq c1 caddc eqtrd | |
| cn0 syl wwlknbp1 simp2d wrddm wwlknlen oveq2d cz wwlknn0 nn0z fzval3 eqcomd | |
| ) CBADEFZCGZHCIJZKEZHBLEZUKCAMJZNFZULUNOUKBSFZUQUMBPQEZOABCUAUBUPCUCTUKUNHU | |
| SKEZUOUKUMUSHKABCUDUEUKUOUTUKBUFFZUOUTOUKURVAABCUGBUHTHBUITUJRR $. | |
| $( A fixed-length simple-path word is a function on the corresponding | |
| closed integer interval. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| wspthnfn $p |- ( W e. ( N WSPathsN G ) -> | |
| W Fn ( 0 ... N ) ) $= | |
| ( cwwspthsn co wcel cdm wfn cc0 cfz cvtx cfv wf1 wspthnf1m f1fn syl cwwlksn | |
| wceq wspthsswwlkn sseli wwlkndm fneq2d mpbid ) CBADEZFZCCGZHZCIBJEZHUEUFAKL | |
| ZCMUGABCNUFUICOPUEUFUHCUECBAQEZFUFUHRUDUJCABSTABCUAPUBUC $. | |
| $( The edge sequence of a fixed-length simple path has the stated length. | |
| (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wspthnlen $p |- ( ( W e. ( N WWalksN G ) /\ | |
| F ( SPaths ` G ) W ) -> ( # ` F ) = N ) $= | |
| ( cwwlksn co wcel cspths cfv wbr wa chash c1 cmin caddc cwlks wceq simpl cc | |
| syl simpr spthiswlk wlklenvm1 wwlknlen oveq1d cn0 wwlknn0 nn0cn 3syl pncan1 | |
| 3eqtrd ) DCBEFGADBHIJKALIDLIMNFCMOFMNFCDCBEFGADBHIJKADBPIJALIDLIMNFQDCBEFGA | |
| DBHIJKADBHIJADBPIJDCBEFGADBHIJUADABUBTDABUCTDCBEFGADBHIJKDLICMOFMNDCBEFGADB | |
| HIJKDCBEFGDLICMOFQDCBEFGADBHIJRBCDUDTUEDCBEFGADBHIJKCSGCMOFMNFCQDCBEFGADBHI | |
| JKDCBEFGCUFGCSGDCBEFGADBHIJRBCDUGCUHUICUJTUK $. | |
| ${ | |
| wspthnleb.v $e |- V = ( Vtx ` G ) $. | |
| $( A fixed-length simple path has at most as many edges as its graph has | |
| vertices. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wspthnleb $p |- ( ( W e. ( N WWalksN G ) /\ | |
| F ( SPaths ` G ) W ) -> | |
| N <_ ( # ` V ) ) $= | |
| ( cwwlksn co wcel cspths cfv wbr wa chash wspthnlen cpths simpr spthispth | |
| cle syl pthhashvtx eqbrtrrd ) ECBGHIZAEBJKLZMZANKZCDNKZSABCEOUEAEBPKLZUFU | |
| GSLUEUDUHUCUDQEABRTEABDFUATUB $. | |
| $} | |
| ${ | |
| $d G f $. $d N f $. $d V f $. $d W f $. | |
| wspthnle.v $e |- V = ( Vtx ` G ) $. | |
| $( The length of a simple path represented as a word is bounded by the | |
| number of graph vertices. (Contributed by Mingli Yuan, | |
| 11-Aug-2026.) $) | |
| wspthnle $p |- ( W e. ( N WSPathsN G ) -> | |
| N <_ ( # ` V ) ) $= | |
| ( vf cwwspthsn co wcel cv cspths cfv wbr wex chash cle cn0 cvv wa syl w3a | |
| cwwlksn wspthnp simp3 simp2 wspthnleb sylan ex exlimdv mpd ) DBAGHIZFJZDA | |
| KLMZFNZBCOLPMZUKBQIARISZDBAUBHIZUNUAZUNFABDUCZUPUQUNUDTUKUMUOFUKUMUOUKUQU | |
| MUOUKURUQUSUPUQUNUETULABCDEUFUGUHUIUJ $. | |
| $} | |
| ${ | |
| $d A f $. $d A w $. $d G f $. $d G w $. | |
| vtx0wspth.v $e |- V = ( Vtx ` G ) $. | |
| $( A graph vertex, represented by its singleton word, is a simple path of | |
| length zero. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| vtx0wspth $p |- ( ( G e. _V /\ A e. V ) -> | |
| <" A "> e. ( 0 WSPathsN G ) ) $= | |
| ( vf vw cvv wcel wa cc0 co cv cfv chash c1 wceq a1i syl c0 cfzo cwwspthsn | |
| cs1 cwwlksn cspths wbr wex cvtx cword crab id fveq2d eqeq1d simpr eleqtrd | |
| s1cl s1len elrabd wwlksn0s eleqtrrd cfz wrdf csn fz0sn oveq2i fzo01 eqtri | |
| wf eqtr4i feq2d mpbird wb simpl 0spth 0ex breq1 spcev jca iswspthn sylibr | |
| ) BGHZACHZIZAUBZJBUCKZHZELZWCBUDMZUEZEUFZIWCJBUAKHWBWEWIWBWCFLZNMZOPZFBUG | |
| MZUHZUIZWDWBWLWCNMZOPZFWCWNWJWCPZWKWPOWRWJWCNWRUJUKULWBAWMHWCWNHWBACWMVTW | |
| AUMZCWMPWBDQUNAWMUORWQWBAUPZQUQWDWOPWBFBURQUSWBSWCWGUEZWIWBXAJJUTKZCWCVGZ | |
| WBXCJWPTKZCWCVGZWBWCCUHHZXEWBWAXFWSACUORCWCVARWBXBXDCWCXBXDPWBXBJVBZXDVCX | |
| DJOTKXGWPOJTWTVDVEVFVHQVIVJWBVTXAXCVKVTWAVLWCBCGDVMRVJWHXAESVNWFSWCWGVOVP | |
| RVQEBJWCVRVS $. | |
| $} | |
| ${ | |
| $d G f $. $d N f $. $d W f $. | |
| $( The length parameter of a fixed-length simple path is a nonnegative | |
| integer. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wspthnn0 $p |- ( W e. ( N WSPathsN G ) -> N e. NN0 ) $= | |
| ( vf cwwspthsn co cn0 cvv wa cwwlksn cv cspths cfv wbr wex wspthnp simp1d | |
| wcel simpld ) CBAEFRZBGRZAHRZTUAUBICBAJFRDKCALMNDODABCPQS $. | |
| $} | |
| ${ | |
| $d G x $. $d N x $. $d W x $. | |
| wspthnevtx.v $e |- V = ( Vtx ` G ) $. | |
| $( The first and last symbols of a fixed-length simple-path word are | |
| vertices of the graph. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wspthnevtx $p |- ( W e. ( N WSPathsN G ) -> | |
| ( ( W ` 0 ) e. V /\ ( W ` N ) e. V ) ) $= | |
| ( cwwspthsn co wcel cwwlksn cc0 cfv wa wspthsswwlkn sseli wwlknllvtx syl | |
| ) DBAFGZHDBAIGZHJDKCHBDKCHLQRDABMNABCDEOP $. | |
| $} | |
| ${ | |
| $d A f $. $d A w $. $d G f $. $d G w $. | |
| wspth0ne0.v $e |- V = ( Vtx ` G ) $. | |
| $( If a graph has a vertex, then its set of zero-length simple paths is | |
| nonempty. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wspth0ne0 $p |- ( ( G e. _V /\ A e. V ) -> | |
| ( 0 WSPathsN G ) =/= (/) ) $= | |
| ( cvv wcel wa cc0 cwwspthsn co cs1 vtx0wspth ne0d ) BEFACFGHBIJAKABCDLM | |
| $. | |
| $} | |
| ${ | |
| $d E l $. $d F l $. $d G l $. $d P l $. $d V l $. $d X l $. | |
| $d ph l $. | |
| cyclvtxdge2d.v $e |- V = ( Vtx ` G ) $. | |
| cyclvtxdge2d.e $e |- E = ( iEdg ` G ) $. | |
| cyclvtxdge2d.g $e |- ( ph -> G e. USGraph ) $. | |
| cyclvtxdge2d.c $e |- ( ph -> F ( Cycles ` G ) P ) $. | |
| cyclvtxdge2d.n $e |- ( ph -> F =/= (/) ) $. | |
| cyclvtxdge2d.x $e |- ( ph -> X e. ran P ) $. | |
| $( Every vertex occurring in a nontrivial cycle of a simple graph has | |
| degree at least two. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| cyclvtxdge2d $p |- ( ph -> | |
| 2 <_ ( ( VtxDeg ` G ) ` X ) ) $= | |
| ( vl cfv wceq c2 wbr wcel syl cv cvtxdg cle cdm crn wrex wi cc0 chash cfz | |
| wfun co wf ccycls cwlks cycliswlk wlkp 3syl ffun elrnrexdm wa cusgr simpl | |
| mpd c0 wne simprl fdm eleqtrd cyclposvdge2d simprr eqcomd fveq2d breqtrd | |
| rexlimddv ) AGNUAZBOZPZQGEUBOZOZUCRNBUDZAGBUESZVRNWAUFZMABUKZWBWCUGAUHDUI | |
| OUJULZFBUMZWDADBEUNORZDBEUOORZWFKBDEUPZBDEFHUQZURWEFBUSTNBGUTTVDAVPWASZVR | |
| VAZVAZQVQVSOVTUCWMBCDEVPFHIWMAEVBSAWLVCZJTWMAWGWNKTWMADVEVFWNLTWMVPWAWEAW | |
| KVRVGWMWFWAWEPWMAWGWFWNKWGWHWFWIWJTURWEFBVHTVIVJWMVQGVSWMGVQAWKVRVKVLVMVN | |
| VO $. | |
| $} | |
| ${ | |
| revfstd.w $e |- ( ph -> W e. Word A ) $. | |
| revfstd.n $e |- ( ph -> N e. NN0 ) $. | |
| revfstd.z $e |- ( ph -> 0 e. ( 0 ..^ ( # ` W ) ) ) $. | |
| revfstd.l $e |- ( ph -> ( ( # ` W ) - 1 ) = N ) $. | |
| $( The first symbol of a reversed nonempty word is the last symbol of the | |
| original word, in deduction form. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| revfstd $p |- ( ph -> ( ( reverse ` W ) ` 0 ) = ( W ` N ) ) $= | |
| ( cc0 creverse cfv chash c1 cmin co cword wcel syl cc fveq2d cfzo wa wceq | |
| jca revfv eqcomd cn0 nn0cn eqeltrrd subid1d 3eqtrd ) AIDJKKZDLKZMNOZINOZD | |
| KZUNDKCDKADBPQZIIUMUAOQZUBULUPUCAUQUREGUDBDIUERAUOUNDAUNACUNSAUNCHUFACUGQ | |
| CSQFCUHRUIUJTAUNCDHTUK $. | |
| $} | |
| ${ | |
| revlstd.w $e |- ( ph -> W e. Word A ) $. | |
| revlstd.n $e |- ( ph -> N e. NN0 ) $. | |
| revlstd.b $e |- ( ph -> N e. ( 0 ..^ ( # ` W ) ) ) $. | |
| revlstd.l $e |- ( ph -> ( ( # ` W ) - 1 ) = N ) $. | |
| $( The last symbol of a reversed nonempty word is the first symbol of the | |
| original word, in deduction form. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| revlstd $p |- ( ph -> ( ( reverse ` W ) ` N ) = ( W ` 0 ) ) $= | |
| ( creverse cfv chash c1 cmin co cc0 cword wcel cfzo syl fveq2d wceq revfv | |
| wa jca oveq1d cn0 cc nn0cn subidd 3eqtrd ) ACDIJJZDKJZLMNZCMNZDJZCCMNZDJO | |
| DJADBPQZCOULRNQZUCUKUOUAAUQUREGUDBDCUBSAUNUPDAUMCCMHUETAUPODACACUFQCUGQFC | |
| UHSUITUJ $. | |
| $} | |
| ${ | |
| revspthond.v $e |- V = ( Vtx ` G ) $. | |
| revspthond.g $e |- ( ph -> G e. UPGraph ) $. | |
| revspthond.p $e |- ( ph -> F ( K ( SPathsOn ` G ) M ) P ) $. | |
| $( Reversing a simple path in a pseudograph exchanges its two endpoints. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| revspthond $p |- ( ph -> ( reverse ` F ) | |
| ( M ( SPathsOn ` G ) K ) ( reverse ` P ) ) $= | |
| ( cfv co wbr wceq w3a wcel 3syl syl wa cvv creverse cspthson cspths chash | |
| cc0 cupgr cwlks ccnv wfun spthonisspth spthiswlk revwlk cword wlkpwrd jca | |
| spthf1 revinj 3jca upgrwlkdvspth cn0 wlkcl cn cfzo caddc wlklenvp1 eqcomd | |
| c1 nn0p1nn eqeltrrd lbfzo0 cmin wlklenvm1 revfstd cwlkson cpthson ctrlson | |
| sylibr spthonpthon pthontrlon trlsonwlkon 4syl wlkonprop simp33 eqtrd cdm | |
| ciedg eqid wlkf revlen fveq2d fzo0end revlstd simp32 3eqtrd wb spthonprop | |
| simp13 simp12 fvexd isspthonpth mpbird ) ACUAKZBUAKZFEDUBKZLMZXBXCDUCKZMZ | |
| UEXCKZFNZXBUDKZXCKZENZOZAXGXIXLADUFPZXBXCDUGKZMZXCUHUIZOXGAXNXPXQIACBXOMZ | |
| XPACBEFXDLMZCBXFMZXRJEFBCDUJZBCDUKQZBCDULRABGUMPZBUHUIZSXQAYCYDAXRYCYBBCD | |
| GHUNRZAXSXTYDJYABCDUPQUOGBUQRURXCXBDUSRAXHCUDKZBKZFAGYFBYEAXRYFUTPZYBBCDV | |
| ARZABUDKZVBPZUEUEYJVCLZPAYFVGVDLZYJVBAYJYMAXRYJYMNYBBCDVERVFAYHYMVBPYIYFV | |
| HRVIZYJVJVQAYFYJVGVKLZAXRYFYONYBBCDVLRVFZVMACBEFDVNKLMZDTPZEGPZFGPZOZCTPB | |
| TPSZXRUEBKZENZYGFNZOOZUUEAXSCBEFDVOKLMCBEFDVPKLMZYQJEFBCDVREFBCDVSEFBCDVT | |
| WAZEFBCDGHWBZUUAUUBXRUUDUUEWCQWDAXKYFXCKUUCEAXJYFXCACDWFKZWEZUMPZXJYFNAXR | |
| UULYBBCDUUJUUJWGWHRUUKCWIRWJAGYFBYEYIAYOYFYLYPAYKYOYLPYNYJWKRVIYPWLAYQUUF | |
| UUDUUHUUIUUAUUBXRUUDUUEWMQWNURAYTYSSZXBTPZXCTPZSZSXEXMWOAUUMUUPAYTYSAXSUU | |
| AUUBUUGXTSZOZYTJEFBCDGHWPZYRYSYTUUBUUQWQQAXSUURYSJUUSYRYSYTUUBUUQWRQUOAUU | |
| NUUOACUAWSABUAWSUOUOFEXCXBDGTTHWTRXA $. | |
| $} | |
| ${ | |
| $d E x $. $d G x $. $d V x $. $d W x $. $d ph x $. | |
| upgrvtxsupd.v $e |- V = ( Vtx ` G ) $. | |
| upgrvtxsupd.e $e |- E = ( iEdg ` G ) $. | |
| upgrvtxsupd.g $e |- ( ph -> G e. UPGraph ) $. | |
| upgrvtxsupd.w $e |- W e. _V $. | |
| upgrvtxsupd.s $e |- ( ph -> V C_ W ) $. | |
| $( Enlarging the vertex set of a pseudograph while keeping its indexed | |
| edges produces a pseudograph. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| upgrvtxsupd $p |- ( ph -> <. W , E >. e. UPGraph ) $= | |
| ( vx cupgr wcel cfv cpw cdif crab syl cvv ciedg cop cdm chash cle wbr csn | |
| cv c2 c0 wf upgrf wss sspwd ssdifd rabss2 fssd opex a1i cvtx eqid eqeltri | |
| wb fvexi opvtxfvi eqcomi opiedgfvi isupgr mpbird ) AEBUAZLMZBUBZKUGUCNUHU | |
| DUEZKEOZUIUFZPZQZBUJZAVKVLKDOZVNPZQZVPBACLMVKVTBUJHKBCDFGUKRAVSVOULVTVPUL | |
| AVRVMVNADEJUMUNVLKVSVOUORUPAVISMZVJVQVBWAAEBUQURKSBVIEVIUSNEBEIBCTNZSGWBC | |
| TWBUTVCVAZVDVEVITNBBEIWCVFVEVGRVH $. | |
| $} | |
| ${ | |
| $d E G T V W x $. | |
| grvtxsup.v $e |- V = ( Vtx ` G ) $. | |
| grvtxsup.e $e |- E = ( iEdg ` G ) $. | |
| grvtxsup.w $e |- W e. _V $. | |
| grvtxsup.t $e |- T = <. W , E >. $. | |
| $( The vertex set after enlarging only the vertex set of a graph. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grvtxsupvtx $p |- ( Vtx ` T ) = W $= | |
| ( cvtx cfv cop fveq2i ciedg cvv eqid fvexi eqeltri opvtxfvi eqtri ) AJKEB | |
| LZJKEAUAJIMBEHBCNKZOGUBCNUBPQRST $. | |
| $( The indexed-edge function is unchanged when only the vertex set is | |
| enlarged. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grvtxsupiedg $p |- ( iEdg ` T ) = E $= | |
| ( ciedg cfv cop fveq2i cvv eqid fvexi eqeltri opiedgfvi eqtri ) AJKEBLZJK | |
| BATJIMBEHBCJKZNGUACJUAOPQRS $. | |
| $( A pseudograph remains a pseudograph after enlarging only its vertex | |
| set. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grvtxsup $p |- ( ( G e. UPGraph /\ V C_ W ) -> T e. UPGraph ) $= | |
| ( cupgr wcel wss wa cop simpl simpr upgrvtxsupd wceq a1i eleq1d mpbird ) | |
| CJKZDELZMZAJKEBNZJKUDBCDEFGUBUCOHUBUCPQUDAUEJAUERUDISTUA $. | |
| $( The old graph is a subgraph of the graph obtained by enlarging only its | |
| vertex set. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grvtxsupsubgr $p |- ( ( G e. UPGraph /\ V C_ W ) -> | |
| G SubGraph T ) $= | |
| ( cupgr wcel wss wa csubgr wbr simpr cvv syl cfv eqcomi ssidd jca wfun wb | |
| cuhgr w3a grvtxsup elex simpl upgruhgr 3jca cvtx grvtxsupvtx grvtxsupiedg | |
| uhgrfun ciedg uhgrissubgr mpbird ) CJKZDELZMZCANOZUTBBLZMZVAUTVCUSUTPVABU | |
| AUBVAAQKZBUCZCUEKZUFVBVDUDVAVEVFVGVAAJKVEABCDEFGHIUGAJUHRVAVGVFVAUSVGUSUT | |
| UICUJRZBCGUORVHUKEBCABDQFAULSEABCDEFGHIUMTGAUPSBABCDEFGHIUNTUQRUR $. | |
| $( A simple graph remains simple after enlarging only its vertex set. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grvtxsupusgr $p |- ( ( G e. USGraph /\ V C_ W ) -> | |
| T e. USGraph ) $= | |
| ( vx cusgr wcel wss wa cfv cpw crab wf1 syl cupgr cdm cv chash wceq simpl | |
| c2 usgrfs simpr sspw rabss2 3syl f1ss syl2anc wb usgrupgr anim1i grvtxsup | |
| cvv elex cvtx grvtxsupvtx eqcomi ciedg grvtxsupiedg isusgrs mpbird ) CKLZ | |
| DEMZNZAKLZBUAZJUBUCOUFUDZJEPZQZBRZVIVKVLJDPZQZBRZVQVNMZVOVIVGVRVGVHUEJBCD | |
| FGUGSVIVHVPVMMVSVGVHUHDEUIVLJVPVMUJUKVKVQVNBULUMVIAURLZVJVOUNVICTLZVHNATL | |
| VTVGWAVHCUOUPABCDEFGHIUQATUSUKJURBAEAUTOEABCDEFGHIVAVBAVCOBABCDEFGHIVDVBV | |
| ESVF $. | |
| $} | |
| ${ | |
| $d E x $. $d G x $. $d N x $. $d V x $. | |
| usgrnewvtxedgnrn.v $e |- V = ( Vtx ` G ) $. | |
| usgrnewvtxedgnrn.e $e |- E = ( iEdg ` G ) $. | |
| $( An old edge of a simple graph cannot contain a vertex outside the | |
| graph. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| usgrnewvtxedgnrn $p |- ( ( G e. USGraph /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) -> { U , N } e/ ran E ) $= | |
| ( wcel cvv wnel wa w3a crn wn cfv simpl syl simpr eleq2i sylibr cusgr cpr | |
| cvtx cuhgr cedg simp1 usgruhgr ciedg edgval rneqi eqtr4i 3syl prid2g 3jca | |
| simp3 uhgredgrnv df-nel bicomi a1d mt2d ex pm2.01d ) CUAHZAEHZDIHZDEJZKZL | |
| ZADUBZBMZHZNZVIVJJVHVKVHVKVLVHVKKZVKDEHZVMDCUCOZHZVNVMCUDHZVICUEOZHZDVIHZ | |
| LVPVMVQVSVTVMVCVQVMVHVCVHVKPZVCVDVGUFQCUGQVMVKVSVHVKRVRVJVIVRCUHOZMVJCUIB | |
| WBGUJUKSTVMVEVTVMVHVGVEWAVCVDVGUOZVEVFPULADIUMQUNVICDUPQEVODFSTVMVNNZVKVM | |
| VFWDVMVHVFWAVHVGVFWCVEVFRQQVFWDDEUQURTUSUTVAVBVIVJUQT $. | |
| $} | |
| ${ | |
| $d E i $. $d G i $. $d N i $. $d V i $. | |
| uhgrnewvtxiedgnel.v $e |- V = ( Vtx ` G ) $. | |
| uhgrnewvtxiedgnel.e $e |- E = ( iEdg ` G ) $. | |
| $( A vertex outside a hypergraph is incident with none of its indexed | |
| edges. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| uhgrnewvtxiedgnel $p |- ( ( G e. UHGraph /\ -. N e. V ) -> | |
| -. E. i e. dom E N e. ( E ` i ) ) $= | |
| ( cuhgr wcel wn wa cv cfv cdm wrex simpr wi simpl w3a syl wss simp1 simp2 | |
| jca uhgrss simp3 sseldd 3exp rexlimdv con3d mpd ) CHIZDEIZJZKZUNDALZBMZIZ | |
| ABNZOZJZULUNPUOULUNVAQULUNRULUTUMULURUMAUSULUPUSIZURUMULVBURSZUQEDVCULVBK | |
| UQEUAVCULVBULVBURUBULVBURUCUDBUPCEFGUETULVBURUFUGUHUIUJTUK $. | |
| $} | |
| ${ | |
| $d A x $. $d E x $. $d G x $. $d J x $. $d N x $. | |
| $d P x $. $d S x $. $d U x $. $d V x $. | |
| grleaf.v $e |- V = ( Vtx ` G ) $. | |
| grleaf.e $e |- E = ( iEdg ` G ) $. | |
| grleaf.a $e |- A = ( V u. { N } ) $. | |
| grleaf.p $e |- P = ( E u. { <. J , { U , N } >. } ) $. | |
| grleaf.s $e |- S = <. A , P >. $. | |
| $( The vertex set of a graph obtained by adjoining a new leaf. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafvtx $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) -> ( Vtx ` S ) = A ) $= | |
| ( wcel cvv wa cvtx cfv eqeltri cusgr wnel w3a cdm cop wceq fveq2i a1i csn | |
| cun fvex snex unex cpr ciedg jca opvtxfv syl eqtrd ) FUAODIOHPOHIUBQUCGPO | |
| GEUDUBQQZCRSZABUEZRSZAVAVCUFUTCVBRNUGUHUTAPOZBPOZQVCAUFUTVDVEVDUTAIHUIZUJ | |
| PLIVFIFRSPJFRUKTHULUMTUHVEUTBEGDHUNUEZUIZUJPMEVHEFUOSPKFUOUKTVGULUMTUHUPB | |
| APPUQURUS $. | |
| $( The indexed-edge function of a graph obtained by adjoining a new leaf. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafiedg $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) -> ( iEdg ` S ) = P ) $= | |
| ( wcel cvv wnel wa ciedg cfv cusgr w3a cdm cop wceq fveq2i cvtx grleafvtx | |
| a1i fvexd eqeltrrd cpr csn cun fvex eqeltri snex unex jca opiedgfv eqtrd | |
| syl ) FUAODIOHPOHIQRUBGPOGEUCQRRZCSTZABUDZSTZBVDVFUEVCCVESNUFUIVCAPOZBPOZ | |
| RVFBUEVCVGVHVCCUGTAPABCDEFGHIJKLMNUHVCCUGUJUKVHVCBEGDHULUDZUMZUNPMEVJEFST | |
| PKFSUOUPVIUQURUPUIUSBAPPUTVBVA $. | |
| $( The old vertex set is contained in the vertex set obtained by adjoining | |
| a new leaf. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafvss $p |- V C_ A $= | |
| ( csn cun ssun1 sseqtrri ) IIHOZPAISQLR $. | |
| $( Old indexed edges remain proper pairs in the enlarged vertex set. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafrnss $p |- ( G e. USGraph -> | |
| ran E C_ { x e. ~P A | ( # ` x ) = 2 } ) $= | |
| ( cusgr wcel cpw crab wss crn cv chash cfv c2 wceq cdm wf1 usgrfs f1f syl | |
| wf frnd grleafvss sspwi rabss2 ax-mp a1i sstrd ) GPQZFUAAUBUCUDUEUFZAJRZS | |
| ZVAABRZSZUTFUGZVCFUTVFVCFUHVFVCFULAFGJKLUIVFVCFUJUKUMVCVETZUTVBVDTVGJBBCD | |
| EFGHIJKLMNOUNUOVAAVBVDUPUQURUS $. | |
| $( The edge joining an old vertex to the new vertex is a proper pair in | |
| the enlarged vertex set. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| grleafnewedg $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) -> | |
| { U , N } e. { x e. ~P A | ( # ` x ) = 2 } ) $= | |
| ( wcel cvv wa chash syl cusgr wnel w3a cdm cv cfv c2 wceq cpr cpw fveqeq2 | |
| a1i wss csn cun ssun1 sseqtrri simpl2 sseldd simpl3l snidg elun2 eleqtrrd | |
| prex prssd elpwd wne elex simpl3r jca elnelne2 3jca hashprb bicomi sylibr | |
| elrabd ) GUAPZEJPZIQPZIJUBZRZUCHQPHFUDUBRZRZAUEZSUFUGUHEIUIZSUFUGUHZAWEBU | |
| JWDWEUGSUKWCWEBQWEQPWCEIVDULWCEIBWCJBEJBUMWCJJIUNZUOZBJWGUPMUQULVQVRWAWBU | |
| RZUSWCIWHBWCIWGPZIWHPWCVSWJVSVTVQVRWBUTZIQVATIWGJVBTBWHUHWCMULVCVEVFWCEQP | |
| ZVSEIVGZUCZWFWCWLVSWMWCVRWLWIEJVHTWKWCVRVTRWMWCVRVTWIVSVTVQVRWBVIVJEIJVKT | |
| VLWNWFEIVMVNVOVP $. | |
| $( The indexed-edge function obtained by adjoining a fresh leaf edge is a | |
| bijection onto the old range together with that edge. (Contributed by | |
| Mingli Yuan, 13-Aug-2026.) $) | |
| grleaff1o $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) -> | |
| P : ( dom E u. { J } ) -1-1-onto-> | |
| ( ran E u. { { U , N } } ) ) $= | |
| ( wcel cvv wnel wa w3a syl cdm crn wf1o cpr csn cun simpl1 usgrf1o simprl | |
| cusgr prex a1i jca simprr simpl usgrnewvtxedgnrn 3jca f1ounsn ) FUJOZDIOZ | |
| HPOHIQRZSZGPOZGEUAZQZRZRZVDEUBZEUCZVCDHUDZPOZRZVEVJVHQZRZSVDGUEUFVHVJUEUF | |
| BUCVGVIVLVNVGUSVIUSUTVAVFUGEFKUHTVGVCVKVBVCVEUIVKVGDHUKULUMVGVEVMVBVCVEUN | |
| VGVBVMVBVFUODEFHIJKUPTUMUQVDVHBEPPGVJMURT $. | |
| $( The enlarged indexed-edge function is one-to-one into the proper pairs | |
| of the enlarged vertex set. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| grleaff1 $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) -> | |
| P : dom P -1-1-> { x e. ~P A | ( # ` x ) = 2 } ) $= | |
| ( wcel cvv wa wf1 syl cusgr wnel w3a cdm cv chash cfv c2 wceq cpw csn cun | |
| crab crn cpr wss wf1o grleaff1o f1of1 grleafrnss grleafnewedg snssd unssd | |
| simpl1 jca f1ss eqidd f1odm eqcomd f1eq123d bicomd mpbird ) GUAPZEJPZIQPI | |
| JUBRZUCHQPHFUDZUBRZRZCUDZAUEUFUGUHUIABUJUMZCSZVPHUKULZVTCSZVRWBFUNZEIUOZU | |
| KZULZCSZWGVTUPZRWCVRWHWIVRWBWGCUQZWHBCDEFGHIJKLMNOURZWBWGCUSTVRWDWFVTVRVM | |
| WDVTUPVMVNVOVQVDABCDEFGHIJKLMNOUTTVRWEVTABCDEFGHIJKLMNOVAVBVCVEWBWGVTCVFT | |
| VRWCWAVRWBVSVTVTCCVRCVGVRVSWBVRWJVSWBUIWKWBWGCVHTVIVRVTVGVJVKVL $. | |
| $( Adjoining one freshly indexed edge from an old vertex to a new vertex | |
| preserves simplicity. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| grleafusgr $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) -> S e. USGraph ) $= | |
| ( vx cusgr wcel cvv wa eqeltri wnel w3a cdm cop wceq a1i cv chash cfv cpw | |
| c2 crab wf1 grleaff1 wb csn cun cvtx fvex snex unex ciedg pm3.2i isusgrop | |
| cpr ax-mp sylibr eqeltrd ) FPQDIQHRQHIUASUBGRQGEUCUASSZCABUDZPCVJUEVINUFV | |
| IBUCOUGUHUIUKUEOAUJULBUMZVJPQZOABCDEFGHIJKLMNUNARQZBRQZSVLVKUOVMVNAIHUPZU | |
| QRLIVOIFURUIRJFURUSTHUTVATBEGDHVEUDZUPZUQRMEVQEFVBUIRKFVBUSTVPUTVATVCBARR | |
| OVDVFVGVH $. | |
| $( The original graph is a subgraph of the graph obtained by adjoining a | |
| new leaf. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafsubgr $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) -> G SubGraph S ) $= | |
| ( cusgr wcel cvv wa wss syl wnel w3a cdm wbr cvtx cfv ciedg grleafvss a1i | |
| csubgr grleafvtx sseqtrrd cpr cop csn cun ssun1 sseqtrri grleafiedg cuhgr | |
| wfun grleafusgr usgruhgr eqid uhgrfun simpl1 3jca uhgrissubgr mpbir2and | |
| wb ) FOPZDIPZHQPHIUARZUBGQPGEUCUARZRZFCUJUDZICUEUFZSZECUGUFZSZVOIAVQIASVO | |
| ABCDEFGHIJKLMNUHUIABCDEFGHIJKLMNUKULVOEBVSEBSVOEEGDHUMUNUOZUPBEWAUQMURUIA | |
| BCDEFGHIJKLMNUSULVOCOPZVSVAZFUTPZUBVPVRVTRVJVOWBWCWDABCDEFGHIJKLMNVBZVOCU | |
| TPZWCVOWBWFWECVCTVSCVSVDZVETVOVKWDVKVLVMVNVFFVCTVGVQVSFCEIOJVQVDKWGVHTVI | |
| $. | |
| $( The new vertex has degree one after a fresh leaf is adjoined. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafvtxd1 $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) -> | |
| ( ( VtxDeg ` S ) ` N ) = 1 ) $= | |
| ( vx wcel cvv cfv syl wceq cusgr wnel wa w3a cdm cvtxdg cop c1 co cc0 cpr | |
| cxad ciedg cvtx eqid wfun cuhgr simpl1 usgruhgr uhgrfun csn cun fvex snex | |
| eqeltri a1i jca opiedgfv funeqd mpbird grleafvtx eqtr4d grleafiedg eqcomd | |
| unex opvtxfv eqidd uneq12d 3eqtrd simprl simprr dmeqd simpl3l snidg elun2 | |
| neleq12d eqcomi eleqtrd eleqtrrd cv chash crab grleafnewedg fveqeq2 elrab | |
| cpw c2 simplbi pweqd prid2g cle cr 2re leidd wne simpl3r elnelne2 hashprg | |
| simpl2 wb bicomd breqtrrd p1evtxdp1 df-nel sylib uhgrnewvtxiedgnel fveq1d | |
| wrex wn eleq2d rexeqbidv notbid vtxd0nedgb oveq1d cxr 1xr xaddlid ) FUAPZ | |
| DIPZHQPZHIUBZUCZUDZGQPZGEUEZUBZUCZUCZHCUFRRHAEUGZUFRZRZUHULUIUJUHULUIZUHY | |
| RHDHUKZCYSYSUMRZGYSUNRZQUUEUOZUUDUOZYRUUDUPEUPZYRFUQPZUUHYRYHUUIYHYIYLYQU | |
| RFUSSZEFKUTSYRUUDEYRAQPZEQPZUCZUUDETYRUUKUULUUKYRAIHVAZVBZQLIUUNIFUNRQJFU | |
| NVCVEHVDVOVEVFUULYREFUMRQKFUMVCVEVFVGZEAQQVHSZVIVJYRCUNRAUUEABCDEFGHIJKLM | |
| NVKYRUUMUUEATUUPEAQQVPSZVLYRCUMRBEGUUCUGVAZVBZUUDUUSVBABCDEFGHIJKLMNVMBUU | |
| TTYRMVFYREUUDUUSUUSYRUUDEUUQVNYRUUSVQVRVSYMYNYPVTYRGUUDUEZUBYPYMYNYPWAYRG | |
| GUVAYOYRGVQYRUUDEUUQWBZWFVJYRHAUUEYRHUUOAYRHUUNPZHUUOPYRYJUVCYJYKYHYIYQWC | |
| ZHQWDSHUUNIWESUUOATYRAUUOLWGVFWHUURWIZYRUUCAWPZUUEWPYRUUCOWJZWKRWQTZOUVFW | |
| LPZUUCUVFPZOABCDEFGHIJKLMNWMUVIUVJUUCWKRZWQTZUVHUVLOUUCUVFUVGUUCWQWKWNWOW | |
| RSYRUUEAUURWSWIYRYJHUUCPUVDDHQWTSYRWQWQUVKXAYRWQWQXBPYRXCVFXDYRUVLDHXEZYR | |
| YIYKUCUVMYRYIYKYHYIYLYQXIZYJYKYHYIYQXFZVGDHIXGSYRUVMUVLYRYIYJUCUVMUVLXJYR | |
| YIYJUVNUVDVGDHIQXHSXKVJXLXMYRUUAUJUHULYRUUAUJTZHUVGUUDRZPZOUVAXRZXSZYRUVT | |
| HUVGERZPZOYOXRZXSZYRUUIHIPXSZUCUWDYRUUIUWEUUJYRYKUWEUVOHIXNXOVGOEFHIJKXPS | |
| YRUVSUWCYRUVRUWBOUVAYOUVBYRUVQUWAHYRUVGUUDEUUQXQXTYAYBVJYRHUUEPUVPUVTXJUV | |
| EYTHOYSUUDUUEUUFUUGYTUOYCSVJYDYRUHYEPZUUBUHTUWFYRYFVFUHYGSVS $. | |
| $} | |
| ${ | |
| $d G n $. $d L n $. $d V n $. | |
| wspthlen.v $e |- V = ( Vtx ` G ) $. | |
| wspthlen.l $e |- L = { n e. ( 0 ... ( # ` V ) ) | | |
| ( n WSPathsN G ) =/= (/) } $. | |
| $( The set of attainable simple-path lengths bounded by the number of | |
| graph vertices is finite. (Contributed by Mingli Yuan, | |
| 11-Aug-2026.) $) | |
| wspthlenfi $p |- L e. Fin $= | |
| ( cv cwwspthsn co c0 wne cc0 chash cfv cfz crab cfn wcel fzfi rabfi ax-mp | |
| eqeltri ) CAGBHIJKZALDMNZOIZPZQFUEQRUFQRLUDSUCAUETUAUB $. | |
| $( Attainable simple-path lengths are real numbers. (Contributed by | |
| Mingli Yuan, 11-Aug-2026.) $) | |
| wspthlenssre $p |- L C_ RR $= | |
| ( cn0 cr cc0 chash cfv cfz co cv cwwspthsn c0 wne ssrab3 fz0ssnn0 sstri | |
| nn0ssre ) CGHCIDJKZLMZGANBOMPQAUCCFRUBSTUAT $. | |
| $( If the finite graph has a vertex, then its set of attainable | |
| simple-path lengths is nonempty. (Contributed by Mingli Yuan, | |
| 11-Aug-2026.) $) | |
| wspthlenne0 $p |- ( ( G e. _V /\ V e. Fin /\ A e. V ) -> | |
| L =/= (/) ) $= | |
| ( cvv wcel cfn w3a cc0 cv cwwspthsn co c0 wne chash wceq syl cfv cfz crab | |
| oveq1 neeq1d cn0 simp2 hashcl 0elfz simp1 jca wspth0ne0 elrabd eqcomi a1i | |
| wa simp3 eleqtrd ne0d ) CHIZEJIZAEIZKZDLVCLBMZCNOZPQZBLERUAZUBOZUCZDVCVFL | |
| CNOZPQZBLVHVDLSVEVJPVDLCNUDUEVCVGUFIZLVHIVCVAVLUTVAVBUGEUHTVGUITVCUTVBUPV | |
| KVCUTVBUTVAVBUJUTVAVBUQUKACEFULTUMVIDSVCDVIGUNUOURUS $. | |
| ${ | |
| $d L m $. $d m n $. | |
| $( A finite graph with a vertex has a greatest attainable simple-path | |
| length. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wspthlenmax $p |- ( ( G e. _V /\ V e. Fin /\ A e. V ) -> | |
| E. n e. L A. m e. L m <_ n ) $= | |
| ( cvv wcel cfn w3a cr wss c0 wne cv cle wbr a1i wspthlenssre wspthlenfi | |
| wral wrex wspthlenne0 3jca fimaxre syl ) DIJFKJAFJLZEMNZEKJZEOPZLBQCQRS | |
| BEUCCEUDUIUJUKULUJUICDEFGHUATUKUICDEFGHUBTACDEFGHUEUFCBEUGUH $. | |
| $} | |
| ${ | |
| $d N n $. | |
| $( The length of every simple path in a finite graph belongs to the set | |
| of attainable bounded lengths. (Contributed by Mingli Yuan, | |
| 11-Aug-2026.) $) | |
| wspthlenel $p |- ( ( V e. Fin /\ W e. ( N WSPathsN G ) ) -> | |
| N e. L ) $= | |
| ( cfn wcel cwwspthsn co wa cv c0 wne cc0 wceq cn0 syl chash crab neeq1d | |
| cfv cfz oveq1 cle wbr simpr wspthnn0 hashcl wspthnle elfz2nn0 syl3anbrc | |
| simpl ne0d elrabd eqcomi a1i eleqtrd ) EIJZFDBKLZJZMZDANZBKLZOPZAQEUAUD | |
| ZUELZUBZCVDVGVBOPADVIVEDRVFVBOVEDBKUFUCVDDSJZVHSJZDVHUGUHZDVIJVDVCVKVAV | |
| CUIZBDFUJTVDVAVLVAVCUOEUKTVDVCVMVNBDEFGULTDVHUMUNVDVBFVNUPUQVJCRVDCVJHU | |
| RUSUT $. | |
| $} | |
| ${ | |
| $d N n $. | |
| $( An attainable length has at least one simple path. (Contributed by | |
| Mingli Yuan, 11-Aug-2026.) $) | |
| wspthlenwne0 $p |- ( N e. L -> | |
| ( N WSPathsN G ) =/= (/) ) $= | |
| ( wcel cc0 chash cfv cfz co cwwspthsn c0 wne cv wceq oveq1 neeq1d | |
| elrab2 simprbi ) DCHDIEJKLMZHDBNMZOPZAQZBNMZOPUEADUCCUFDRUGUDOUFDBNSTGU | |
| AUB $. | |
| $} | |
| ${ | |
| $d G w $. $d N n $. $d N w $. | |
| $( An attainable length is represented by a simple-path word. | |
| (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wspthlenwex $p |- ( N e. L -> | |
| E. w w e. ( N WSPathsN G ) ) $= | |
| ( wcel cwwspthsn co c0 wne cv wex wspthlenwne0 n0 sylib ) EDIECJKZLMANS | |
| IAOBCDEFGHPASQR $. | |
| $} | |
| ${ | |
| $d G k $. $d k n $. $d k V $. | |
| $( The bound variable in the definition of attainable path lengths can | |
| be changed. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wspthlenlcv $p |- L = { k e. ( 0 ... ( # ` V ) ) | | |
| ( k WSPathsN G ) =/= (/) } $= | |
| ( cv cwwspthsn co c0 wne cc0 chash cfv cfz crab weq oveq1 neeq1d eqtri | |
| cbvrabv ) DBHZCIJZKLZBMENOPJZQAHZCIJZKLZAUFQGUEUIBAUFBARUDUHKUCUGCISTUB | |
| UA $. | |
| $} | |
| ${ | |
| $d G k $. $d G p $. $d k L $. $d k n $. $d k V $. $d n p $. | |
| $( An attainable length represented by the defining bound variable has | |
| a simple-path word. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wspthlenwexn $p |- ( n e. L -> | |
| E. p p e. ( n WSPathsN G ) ) $= | |
| ( vk cv wspthlenlcv wspthlenwex ) EHBCAIDFHABCDFGJK $. | |
| $} | |
| ${ | |
| $d k L $. $d k M $. $d k N $. $d M n $. | |
| $( A greatest attainable length bounds the length of every simple path. | |
| (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wspthlenle $p |- ( ( V e. Fin /\ A. k e. L k <_ N /\ | |
| W e. ( M WSPathsN G ) ) -> M <_ N ) $= | |
| ( cfn wcel cv cle wbr wral cwwspthsn wa jca syl co w3a simp1 wspthlenel | |
| simp3 simp2 breq1 rspcva ) GKLZAMZFNOZADPZHECQUALZUBZEDLZULREFNOZUNUOUL | |
| UNUIUMRUOUNUIUMUIULUMUCUIULUMUESBCDEGHIJUDTUIULUMUFSUKUPAEDUJEFNUGUHT | |
| $. | |
| $} | |
| ${ | |
| $d G m w $. $d k L $. $d k m w $. $d k N $. $d L m w $. $d m n w $. | |
| $d m N w $. $d m V w $. | |
| $( A greatest attainable length bounds all simple paths. | |
| (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wspthlenallle $p |- ( ( V e. Fin /\ A. k e. L k <_ N ) -> | |
| A. m A. w ( w e. ( m WSPathsN G ) -> m <_ N ) ) $= | |
| ( cfn wcel cv cle wbr wral wa cwwspthsn co wi wspthlenle alrimivv | |
| 3expia ) HKLZBMGNOBFPZQAMZCMZERSLZUGGNOZTCAUDUEUHUIBDEFUGGHUFIJUAUCUB | |
| $. | |
| $} | |
| ${ | |
| $d A n $. $d G m w $. $d k L $. $d k m n w $. $d L m w $. | |
| $d m n w $. $d m V w $. | |
| $( A finite graph with a vertex has a maximum simple-path length. | |
| (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wspthmax $p |- ( ( G e. _V /\ V e. Fin /\ A e. V ) -> | |
| E. n e. L A. m A. w | |
| ( w e. ( m WSPathsN G ) -> m <_ n ) ) $= | |
| ( vk cvv wcel cfn w3a cv cle wbr wal wa syl cwwspthsn co wi simpl simp2 | |
| wral simprr jca wspthlenallle wspthlenmax reximddv ) EKLZGMLZBGLZNZJODO | |
| ZPQJFUFZAOCOZEUAUBLURUPPQUCARCRZDFUOUPFLZUQSZSZUMUQSUSVBUMUQVBUOUMUOVAU | |
| DULUMUNUETUOUTUQUGUHAJCDEFUPGHIUITBJDEFGHIUJUK $. | |
| $} | |
| ${ | |
| $d A n p $. $d G m p w $. $d k L m n p w $. $d L m p w $. | |
| $d m n p w $. $d m V p w $. $d n p $. $d p V w $. | |
| $( A finite graph with a vertex has a longest simple path represented by | |
| a word. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wspthmaxw $p |- ( ( G e. _V /\ V e. Fin /\ A e. V ) -> | |
| E. n e. L E. p ( p e. ( n WSPathsN G ) /\ | |
| A. m A. w | |
| ( w e. ( m WSPathsN G ) -> m <_ n ) ) ) $= | |
| ( cvv wcel cfn cv cwwspthsn co wal wa wex syl w3a cle wbr simprl simprr | |
| wi wspthlenwexn alrimiv jca 19.29r wspthmax reximddv ) EKLGMLBGLUAZANCN | |
| ZEOPLUNDNZUBUCUFAQCQZHNUOEOPLZUPRHSZDFUMUOFLZUPRRZUQHSZUPHQZRURUTVAVBUT | |
| USVAUMUSUPUDDEFGHIJUGTUTUPHUMUSUPUEUHUIUQUPHUJTABCDEFGIJUKUL $. | |
| $} | |
| $} | |
| ${ | |
| $d G m w $. $d N m w $. $d S m w $. $d W m w $. $d m w $. | |
| $( Every vertex adjacent to the final vertex of a maximum simple path is | |
| already on the path. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wspthmaxend $p |- ( ( ( G e. UPGraph /\ | |
| W e. ( N WSPathsN G ) /\ | |
| A. m A. w ( w e. ( m WSPathsN G ) -> m <_ N ) ) /\ | |
| ( S e. ( Vtx ` G ) /\ | |
| { ( lastS ` W ) , S } e. ( Edg ` G ) ) ) -> | |
| S e. ran W ) $= | |
| ( wcel cwwspthsn co cv cle wbr wal w3a cfv wa wn jca syl cr cupgr cvtx clsw | |
| wi cpr cedg crn caddc cconcat simpll3 simpll1 simpll2 simplrl simplrr simpr | |
| c1 cs1 3jca wspthnext wspthmaxspc sylc ex clt cn0 wspthnn0 nn0re 3syl ltp1d | |
| simpl2 wb peano2re ltnle mpbid a1d pm2.65d notnotrd ) DUAGZFEDHIGZAJCJZDHIG | |
| VSEKLUDAMCMZNZBDUBOGZFUCOBUEDUFOGZPZPZBFUGGZWEWFQZEUPUHIZEKLZWEWGWIWEWGPZVT | |
| FBUQUIIWHDHIGZWIVQVRVTWDWGUJWJVQVRWBWCWGPZNZPWKWJVQWMVQVRVTWDWGUKWJVRWBWLVQ | |
| VRVTWDWGULWAWBWCWGUMWJWCWGWAWBWCWGUNWEWGUORURRBDEFUSSABCDEFUTVAVBWEWIQZWGWE | |
| EWHVCLZWNWEEWEVREVDGETGZVQVRVTWDVIDEFVEEVFVGZVHWEWPWHTGZPWOWNVJWEWPWRWQWEWP | |
| WRWQEVKSREWHVLSVMVNVOVP $. | |
| $} | |
| ${ | |
| $d G i $. $d N i $. $d W i $. | |
| $( The final vertex of a positive-length simple path is adjacent to its | |
| predecessor. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wspthendpred $p |- ( ( W e. ( N WSPathsN G ) /\ N e. NN ) -> | |
| { ( lastS ` W ) , ( W ` ( N - 1 ) ) } e. ( Edg ` G ) ) $= | |
| ( vi cwwspthsn co wcel wa cfv c1 cpr wceq syl eqcomd preq12d caddc fveq2d | |
| simpr eqid eqeltrd cn clsw cmin cedg cwwlksn simpl cn0 cvv cspths wbr wex | |
| cv wspthnp simp2d wwlknlsw eqidd prcom a1i eqtrd cc nncn npcan1 cfzo wral | |
| cc0 cvtx cword chash wwlknp simp3d fzo0end oveq1d eleq1d rspcdv mpd ) CBA | |
| EFGZBUAGZHZCUBIZBJUCFZCIZKZWABCIZKZAUDIZVRWBWCWAKZWDVRVSWCWAWAVRWCVSVRCBA | |
| UEFGZWCVSLVRVPWGVPVQUFVPBUGGAUHGHWGDULZCAUIIUJDUKDABCUMUNMZABCUOMNVRWAUPZ | |
| OWFWDLVRWCWAUQURUSVRWDWAVTJPFZCIZKZWEVRWAWAWCWLWJVRBWKCVRWKBVRBUTGZWKBLVR | |
| VQWNVPVQRZBVAMBVBMNQOVRWHCIZWHJPFZCIZKZWEGZDVEBVCFZVDZWMWEGZVRWGXBWIWGCAV | |
| FIZVGGCVHIBJPFLXBDWEABXDCXDSWESVIVJMVRWTXCDVTXAVRVQVTXAGWOBVKMVRWHVTLZHZW | |
| SWMWEXFWPWAWRWLXFWHVTCVRXERZQXFWQWKCXFWHVTJPXGVLQOVMVNVOTT $. | |
| $} | |
| ${ | |
| $d F k $. $d G k $. $d H k $. $d I k $. $d J k $. $d P k $. $d Q k $. | |
| $d S k $. $d V k $. $d ph k $. | |
| wlkccat1d.v $e |- V = ( Vtx ` G ) $. | |
| wlkccat1d.i $e |- I = ( iEdg ` G ) $. | |
| wlkccat1d.h $e |- H = ( F ++ <" J "> ) $. | |
| wlkccat1d.q $e |- Q = ( P ++ <" S "> ) $. | |
| wlkccat1d.g $e |- ( ph -> G e. UPGraph ) $. | |
| wlkccat1d.w $e |- ( ph -> F ( Walks ` G ) P ) $. | |
| wlkccat1d.j $e |- ( ph -> J e. dom I ) $. | |
| wlkccat1d.s $e |- ( ph -> S e. V ) $. | |
| wlkccat1d.e $e |- ( ph -> ( I ` J ) = | |
| { ( lastS ` P ) , S } ) $. | |
| $( Append an indexed edge and its terminal vertex to the two words | |
| representing a walk. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| wlkccat1d $p |- ( ph -> H ( Walks ` G ) Q ) $= | |
| ( cfv vk cwlks wbr cdm cword wcel cc0 chash cfz co wf cv c1 cpr wceq cfzo | |
| caddc wral w3a cs1 cconcat a1i eqcomd wlkf syl ccatws1cl syl2anc eqeltrrd | |
| wa wlkpwrd jca wrdffz fveq2d wrdlenccats1lenm1 eqtrd wlklenvp1 ccatws1len | |
| oveq1d 3eqtrd oveq2d feq2d mpbid csn cupgr upgrwlkedg fveq1d adantr simpr | |
| cmin cun ccats1val1 cn0 wlkcl fzossfzop1 sseldd eleq2d fzonn0p1p1 preq12d | |
| wss eqeq12d ralbidva clsw eqidd 3jca ccats1val2 wlklenvm1 wne ccatval1lsw | |
| c0 s1cl wlkn0 ccatws1ls wb id ralunsn mpbird cuz elnn0uz sylib fzosplitsn | |
| raleqtrrdv upgriswlk ) AGCFUBTZUCZGHUDZUEZUFZUGGUHTZUIUJZJCUKZUAULZGTZHTZ | |
| YKCTZYKUMUQUJZCTZUNZUOZUAUGYHUPUJZURZUSZAYGYJYTAEIUTVAUJZGYFAGUUBGUUBUOZA | |
| MVBZVCAEYFUFZIYEUFZUUBYFUFAEBYCUCZUUEPBEFHLVDVEZQYEEIVFVGVHAUGCUHTZUMWIUJ | |
| ZUIUJZJCUKZYJACJUEZUFUULABDUTZVAUJZCUUMACUUOCUUOUOZANVBZVCABUUMUFZDJUFZVI | |
| ZUUOUUMUFAUURUUSAUUGUURPBEFJKVJVEZRVKZJBDVFVEVHJCVLVEAUUKYIJCAUUJYHUGUIAU | |
| UJBUHTZEUHTZUMUQUJZYHAUUJUUOUHTZUMWIUJZUVCAUUIUVFUMWIACUUOUHUUQVMVRAUURUV | |
| GUVCUOUVADJBVNVEVOAUUGUVCUVEUOZPBEFVPVEZAYHUVEAYHUUBUHTZUVEAGUUBUHUUDVMAU | |
| UEUVJUVEUOUUHYEEIVQVEVOZVCVSVTWAWBAYRUAUGUVDUPUJZUVDWCWJZYSAYRUAUVMURZYRU | |
| AUVLURZUVDGTZHTZUVDCTZUVECTZUNZUOZVIZAUVOUWAAYKETZHTZYKBTZYOBTZUNZUOZUAUV | |
| LURZUVOAFWDUFZUUGVIUWIAUWJUUGOPVKBUAEFHLWEVEAUWHYRUAUVLAYKUVLUFZVIZUWDYMU | |
| WGYQUWLUWCYLHUWLYLUWCUWLYLYKUUBTZUWCUWLYKGUUBUUCUWLMVBWFUWLUUEUWKVIUWMUWC | |
| UOUWLUUEUWKAUUEUWKUUHWGAUWKWHZVKIYKYEEWKVEVOVCVMUWLUWEYNUWFYPUWLYNUWEUWLY | |
| NYKUUOTZUWEUWLYKCUUOUUPUWLNVBZWFUWLUURYKUGUVCUPUJZUFZVIUWOUWEUOUWLUURUWRA | |
| UURUWKUVAWGZUWLYKUGUVEUPUJZUFUWRUWLUVLUWTYKUWLUVDWLUFZUVLUWTWSAUXAUWKAUUG | |
| UXAPBEFWMVEZWGUVDWNVEUWNWOUWLUWTUWQYKUWLUVEUVCUGUPUWLUVCUVEAUVHUWKUVIWGVC | |
| VTZWPWBVKDYKJBWKVEVOVCUWLYPUWFUWLYPYOUUOTZUWFUWLYOCUUOUWPWFUWLUURYOUWQUFZ | |
| VIUXDUWFUOUWLUURUXEUWSUWLYOUWTUFZUXEUWLUWKUXFUWNYKUVDWQVEUWLUWTUWQYOUXCWP | |
| WBVKDYOJBWKVEVOVCWRWTXAWBAUVQIHTBXBTZDUNUVTAUVPIHAUVPUVDUUBTZIAUVDGUUBUUD | |
| WFAUUEUUFUVDUVDUOZUSUXHIUOAUUEUUFUXIUUHQAUVDXCXDIUVDYEEXEVEVOVMSAUXGUVRDU | |
| VSAUVRUXGAUVRUVDUUOTUVCUMWIUJZUUOTZUXGAUVDCUUOUUQWFAUVDUXJUUOAUUGUVDUXJUO | |
| PBEFXFVEVMAUURUUNUUMUFZBXIXGZUSUXKUXGUOAUURUXLUXMUVAAUUSUXLRDJXJVEAUUGUXM | |
| PBEFXKVEXDBUUNJXHVEVSVCAUVSDAUVSUVEUUOTUVCUUOTZDAUVECUUOUUQWFAUVEUVCUUOAU | |
| VCUVEUVIVCVMAUUTUXNDUOUVBJBDXLVEVSVCWRVSVKAUXAUVNUWBXMUXBYRUWAUAUVLUVDWLY | |
| KUVDUOZYMUVQYQUVTUXOYLUVPHUXOYKUVDGUXOXNZVMVMUXOYNUVRYPUVSUXOYKUVDCUXPVMU | |
| XOYOUVECUXOYKUVDUMUQUXPVRVMWRWTXOVEXPAYSUWTUVMAYHUVEUGUPUVKVTAUVDUGXQTUFZ | |
| UWTUVMUOAUXAUXQUXBUVDXRXSUGUVDXTVEVOYAXDAUWJYDUUAXMOCUAGFHJKLYBVEXP $. | |
| $} | |
| ${ | |
| ccatfstlen.q $e |- Q = ( P ++ <" ( P ` 0 ) "> ) $. | |
| ccatfstlen.r $e |- R = ( ( # ` Q ) - 1 ) $. | |
| ccatfstlen.p $e |- ( ph -> P e. Word V ) $. | |
| $( The predecessor of the length of a word closed by its first symbol is | |
| the original word length. (Contributed by Mingli Yuan, | |
| 11-Aug-2026.) $) | |
| ccatfstlen $p |- ( ph -> R = ( # ` P ) ) $= | |
| ( chash cfv c1 cmin co caddc wceq a1i cc0 cs1 wcel syl cconcat ccatws1len | |
| fveq2d cword eqtrd oveq1d cc cn0 lencl nn0cn 3syl pncan1 3eqtrd ) ADCIJZK | |
| LMZBIJZKNMZKLMZUPDUOOAGPAUNUQKLAUNBQBJZRUAMZIJZUQACUTICUTOAFPUCABEUDSZVAU | |
| QOHEBUSUBTUEUFAUPUGSZURUPOAVBUPUHSVCHEBUIUPUJUKUPULTUM $. | |
| $} | |
| ${ | |
| $d A x $. $d E x $. $d G x $. $d J x $. $d N x $. | |
| $d P x $. $d S x $. $d U x $. $d V x $. | |
| grleafiedgfv.v $e |- V = ( Vtx ` G ) $. | |
| grleafiedgfv.e $e |- E = ( iEdg ` G ) $. | |
| grleafiedgfv.a $e |- A = ( V u. { N } ) $. | |
| grleafiedgfv.p $e |- P = | |
| ( E u. { <. J , { U , N } >. } ) $. | |
| grleafiedgfv.s $e |- S = <. A , P >. $. | |
| $( The freshly indexed leaf edge has the expected endpoint pair. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafiedgfv $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) -> | |
| ( ( iEdg ` S ) ` J ) = { U , N } ) $= | |
| ( wcel cvv wnel wa w3a cfv cusgr cdm ciedg cpr cop grleafiedg fveq1d wceq | |
| csn cun a1i wn simprl prex simprr df-nel sylib 3jca fsnunfv syl 3eqtrd ) | |
| FUAODIOHPOHIQRSZGPOZGEUBZQZRRZGCUCTZTGBTGEGDHUDZUEUIUJZTZVHVFGVGBABCDEFGH | |
| IJKLMNUFUGVFGBVIBVIUHVFMUKUGVFVCVHPOZGVDOULZSVJVHUHVFVCVKVLVBVCVEUMVKVFDH | |
| UNUKVFVEVLVBVCVEUOGVDUPUQUREPPGVHUSUTVA $. | |
| $( Restricting the enlarged indexed-edge function to the old domain | |
| recovers the old indexed-edge function. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| grleafiedgres $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) -> | |
| ( ( iEdg ` S ) |` dom E ) = E ) $= | |
| ( vx wcel wa cres wceq syl cusgr cvv w3a cdm ciedg cfv cpr cop grleafiedg | |
| wnel csn cun reseq1d a1i wfn wn simpl1 cv chash c2 cpw crab usgrfs simprr | |
| wf1 f1fn df-nel sylib jca fsnunres 3eqtrd ) FUAPZDIPZHUBPHIUJQZUCZGUBPZGE | |
| UDZUJZQZQZCUEUFZVQRBVQREGDHUGZUHUKULZVQRZEVTWABVQABCDEFGHIJKLMNUIUMVTBWCV | |
| QBWCSVTMUNUMVTEVQUOZGVQPUPZQWDESVTWEWFVTVLWEVLVMVNVSUQVLVQOURUSUFUTSOIVAV | |
| BZEVEWEOEFIJKVCVQWGEVFTTVTVRWFVOVPVRVDGVQVGVHVIVQEGWBVJTVK $. | |
| $} | |
| ${ | |
| $d A l $. $d E l $. $d F l $. $d G l $. $d J l $. | |
| $d N l $. $d P l $. $d Q l $. $d S l $. $d U l $. $d V l $. | |
| grleafcyclnran.v $e |- V = ( Vtx ` G ) $. | |
| grleafcyclnran.e $e |- E = ( iEdg ` G ) $. | |
| grleafcyclnran.a $e |- A = ( V u. { N } ) $. | |
| grleafcyclnran.p $e |- P = | |
| ( E u. { <. J , { U , N } >. } ) $. | |
| grleafcyclnran.s $e |- S = <. A , P >. $. | |
| $( A nontrivial cycle in a graph with an adjoined leaf does not contain | |
| the new leaf vertex. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafcyclnran $p |- ( ( ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) /\ | |
| F ( Cycles ` S ) Q ) /\ F =/= (/) ) -> | |
| N e/ ran Q ) $= | |
| ( wcel wa cfv syl ctree cvv wnel w3a cdm ccycls wbr c0 wne crn wn cle clt | |
| c2 c1 1lt2 1re 2re ltnlei mpbi a1i cvtxdg ciedg cvtx cusgr simplll simpl1 | |
| eqid treeusgr simpl2 simpl3 3jca simpr jca grleafusgr simplr cyclvtxdge2d | |
| simpllr wceq grleafvtxd1 breq2d mpbid ex mtod df-nel sylibr ) HUAQZEKQZJU | |
| BQJKUCRZUDZIUBQIFUEUCRZRZGCDUFSUGZRZGUHUIZRZJCUJZQZUKJWQUCWPWRUNUOULUGZWS | |
| UKZWPUOUNUMUGWTUPUOUNUQURUSUTVAWPWRWSWPWRRZUNJDVBSSZULUGWSXACDVCSZGDDVDSZ | |
| JXDVHXCVHXAHVEQZWHWIUDZWKRZDVEQXAXFWKXAXEWHWIXAWGXEXAWLWGWLWMWOWRVFZWGWHW | |
| IWKVGTHVITXAWLWHXHWGWHWIWKVJTXAWLWIXHWGWHWIWKVKTVLXAWLWKXHWJWKVMTVNZABDEF | |
| HIJKLMNOPVOTWLWMWOWRVRWNWOWRVPWPWRVMVQXAXBUOUNULXAXGXBUOVSXIABDEFHIJKLMNO | |
| PVTTWAWBWCWDJWQWEWF $. | |
| $} | |
| ${ | |
| $d A k x $. $d E k x $. $d F k x $. $d G k x $. $d J k x $. | |
| $d N k x $. $d P k x $. $d Q k x $. $d S k x $. | |
| $d U k x $. $d V k x $. $d k x $. | |
| grleafcycloldvtxrn.v $e |- V = ( Vtx ` G ) $. | |
| grleafcycloldvtxrn.e $e |- E = ( iEdg ` G ) $. | |
| grleafcycloldvtxrn.a $e |- A = ( V u. { N } ) $. | |
| grleafcycloldvtxrn.p $e |- P = | |
| ( E u. { <. J , { U , N } >. } ) $. | |
| grleafcycloldvtxrn.s $e |- S = <. A , P >. $. | |
| $( Every vertex of a nontrivial cycle remains in the old vertex set after | |
| a leaf is adjoined. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| grleafcycloldvtxrn $p |- ( ( ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) /\ | |
| F ( Cycles ` S ) Q ) /\ F =/= (/) ) -> | |
| ran Q C_ V ) $= | |
| ( wcel wa cfv syl cvv wnel w3a cdm ccycls wbr c0 wne crn csn cun cdif wss | |
| ctree wn cvtx cc0 chash cfz co wf cwlks simplr cycliswlk eqid wlkp eqcomi | |
| frn wceq a1i cusgr simplll simp1 treeusgr simp2 simp3 3jca simpllr eqcomd | |
| grleafvtx eqtrd sseqtrrd grleafcyclnran df-nel sylib sylibr difun2 simp3r | |
| jca ssdifsn simpl simpl3 difsn ) HUNQZEKQZJUAQZJKUBZRZUCZIUAQIFUDUBRZRZGC | |
| DUESUFZRGUGUHZRZCUIZKJUJZUKZXFULZKXDXEXGUMZJXEQUOZRXEXHUMXDXIXJXDXEDUPSZX | |
| GXDUQGURSUSUTZXKCVAZXEXKUMXDGCDVBSUFZXMXDXBXNXAXBXCVCZCGDVDTCGDXKXKVEVFTX | |
| LXKCVHTXDXGAXKXGAVIXDAXGNVGVJXDXKAXDHVKQZWOWRUCZWTRXKAVIXDXQWTXDXPWOWRXDW | |
| NXPXDWSWNWSWTXBXCVLZWNWOWRVMTZHVNTXDWSWOXRWNWOWRVOTZXDWSWRXRWNWOWRVPTVQWS | |
| WTXBXCVRZWIABDEFHIJKLMNOPVTTVSWAWBXDJXEUBXJABCDEFGHIJKLMNOPWCJXEWDWEWIXEX | |
| GJWJWFXDXHKXDXHKXFULZKXHYBVIXDKXFWGVJXDJKQUOZYBKVIXDWQYCXDWNWOWQUCZWTRZWQ | |
| XDYEXBRYEXDYEXBXDYDWTXDWNWOWQXSXTXDWSWQXRWNWOWPWQWHTVQYAWIXOWIYEXBWKTWNWO | |
| WQWTWLTJKWDWEJKWMTWAVSWB $. | |
| $} | |
| ${ | |
| $d P i j $. $d Q i j $. $d V i j $. $d i j $. $d ph i j $. | |
| ccatfstpthcond.q $e |- Q = ( P ++ <" ( P ` 0 ) "> ) $. | |
| ccatfstpthcond.p $e |- ( ph -> P e. Word V ) $. | |
| ccatfstpthcond.f $e |- ( ph -> Fun `' P ) $. | |
| $( Closing an injective word by repeating its first symbol preserves the | |
| vertex-distinctness condition needed for a path. (Contributed by | |
| Mingli Yuan, 11-Aug-2026.) $) | |
| ccatfstpthcond $p |- ( ph -> | |
| A. i e. ( 0 ..^ ( ( # ` P ) + 1 ) ) A. j e. ( 1 ..^ ( # ` P ) ) | |
| ( i =/= j -> ( Q ` i ) =/= ( Q ` j ) ) ) $= | |
| ( cfv wi cc0 co wcel wa wceq a1i syl jca 3syl cv wne chash caddc cfzo weq | |
| c1 cs1 cconcat fveq1d cword simplll simplr ccats1val1 eqtrd simpr fzo0ss1 | |
| eqcomd simpl simplrr sselid 3eqtrd wf1 wf ccnv wfun wb mpbir2and f1veqaeq | |
| wrdf df-f1 mpd cvv w3a wrdv fvex 3jca ccats1val2 cn clt wbr elfzo1 biimpi | |
| exp31 simp2 lbfzo0 sylibr simp1 nnne0 pm2.21ddne simprl cuz lencl elnn0uz | |
| wo cn0 fzosplitsni mpbid mpjaod necon3d ralrimivva ) ADUAZEUAZUBXBCJZXCCJ | |
| ZUBKDELBUCJZUGUDMUEMZUGXFUEMZAXBXGNZXCXHNZOZOZXDXEXBXCXLXBLXFUEMZNZXDXEPZ | |
| DEUFZKXBXFPZXLXNXOXPXLXNOZXOOZXBBJZXCBJZPZXPXSXTXDXEYAXSXDXTXSXDXBBLBJZUH | |
| UIMZJZXTXSXBCYDCYDPZXSGQZUJXSBFUKNZXNOYEXTPXSYHXNXSAYHAXKXNXOULZHRZXLXNXO | |
| UMZSYCXBFBUNRUOURXRXOUPXSXEXCYDJZYAXSXCCYDYGUJXSYHXCXMNZOZYLYAPZXSYHYMYJX | |
| SXHXMXCXFUQZXSXRXJXRXOUSAXIXJXNUTRVAZSYCXCFBUNZRUOVBXSXMFBVCZXNYMOZOYBXPK | |
| XSYSYTXSYSXMFBVDZBVEVFZXSAUUAYIAYHUUAHFBVJZRRXSAUUBYIIRYSUUAUUBOVGZXSXMFB | |
| VKZQVHXSXNYMYKYQSSXMFXBXCBVIRVLWDXLXQXOXPXLXQOZXOOZXPXCLUUGLXCUUGYCYAPZLX | |
| CPZUUGYCXDXEYAUUGXDYCUUGXDYEYCUUGXBCYDYFUUGGQZUJUUGBVMUKNZYCVMNZXQVNYEYCP | |
| UUGUUKUULXQUUGAYHUUKAXKXQXOULZHFBVOTUULUUGLBVPQXLXQXOUMVQYCXBVMBVRRUOURUU | |
| FXOUPUUGXEYLYAUUGXCCYDUUJUJUUGYNYOUUGYHYMUUGAYHUUMHRUUGXHXMXCYPUUGUUFXJUU | |
| FXOUSAXIXJXQUTRZVAZSYRRUOVBUUGYSLXMNZYMOZOUUHUUIKUUGYSUUQUUGYSUUAUUBUUGAY | |
| HUUAUUMHUUCTUUGAUUBUUMIRUUDUUGUUEQVHUUGUUPYMUUGXFVSNZUUPUUGXJXCVSNZUURXCX | |
| FVTWAZVNZUURUUNXJUVAXFXCWBWCZUUSUURUUTWETXFWFWGUUOSSXMFLXCBVIRVLURUUGUUSX | |
| CLUBUUGXJUVAUUSUUNUVBUUSUURUUTWHTXCWIRWJWDXLXIXNXQWOZAXIXJWKXLXFLWLJNZXIU | |
| VCVGXLYHXFWPNZUVDXLAYHAXKUSHRFBWMUVEUVDXFWNWCTLXFXBWQRWRWSWTXA $. | |
| $} | |
| ${ | |
| spthiedgf1d.i $e |- I = ( iEdg ` G ) $. | |
| spthiedgf1d.p $e |- ( ph -> F ( SPaths ` G ) P ) $. | |
| $( The edge enumeration of a simple path is one-to-one on its domain. | |
| (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| spthiedgf1d $p |- ( ph -> | |
| F : dom F -1-1-> dom I ) $= | |
| ( cc0 chash cfv cfzo co cdm wf1 ctrls wbr cspths 3syl syl eqidd spthispth | |
| cpths pthistrl trlf1 cwlks wceq spthiswlk cword wcel wlkf eqcomd f1eq123d | |
| wrddm mpbid ) AHCIJKLZEMZCNZCMZUPCNACBDOJPZUQACBDQJPZCBDUBJPUSGBCDUABCDUC | |
| RBCDEFUDSAUOURUPUPCCACTAURUOAUTCBDUEJPZURUOUFZGBCDUGVACUPUHUIVBBCDEFUJUPC | |
| UMSRUKAUPTULUN $. | |
| $} | |
| ${ | |
| $d F i j $. $d G i j $. $d I i j $. $d J i j $. $d P i j $. $d i j $. | |
| $d ph i j $. | |
| spthccats1f1d.i $e |- I = ( iEdg ` G ) $. | |
| spthccats1f1d.p $e |- ( ph -> F ( SPaths ` G ) P ) $. | |
| spthccats1f1d.j $e |- ( ph -> J e. dom I ) $. | |
| spthccats1f1d.n $e |- ( ph -> -. J e. ran F ) $. | |
| $( Appending a fresh edge index to a simple path preserves injectivity of | |
| the edge enumeration. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| spthccats1f1d $p |- ( ph -> | |
| ( F ++ <" J "> ) : dom ( F ++ <" J "> ) -1-1-> dom I ) $= | |
| ( cvv wcel ciedg cfv wbr syl crn cin c0 wceq cs1 fvex eqeltri dmex cspths | |
| cdm a1i cwlks cword spthiswlk wlkf 3syl s1cl spthiedgf1d s1f1 s1rn ineq2d | |
| csn wn disjsn sylibr eqtrd ccatf1 ) ACFUAZEUFZKVEKLAEEDMNKGDMUBUCUDUGACBD | |
| UENOCBDUHNOCVEUIZLHBCDUJBCDEGUKULAFVELZVDVFLIFVEUMPABCDEGHUNAVEFIUOACQZVD | |
| QZRVHFURZRZSAVIVJVHAVGVIVJTIFVEUPPUQAFVHLUSVKSTJVHFUTVAVBVC $. | |
| $} | |
| ${ | |
| wlklswd.w $e |- ( ph -> F ( Walks ` G ) P ) $. | |
| $( The last symbol of the vertex word of a walk is its final indexed | |
| vertex. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| wlklswd $p |- ( ph -> | |
| ( lastS ` P ) = ( P ` ( # ` F ) ) ) $= | |
| ( clsw cfv chash c1 cmin co caddc cvv wcel wceq w3a cwlks wbr syl fveq2d | |
| wlkv simp3 lsw wlklenvp1 oveq1d cc cn0 wlkcl nn0cn 3syl pncan1 3eqtrd ) A | |
| BFGZBHGZIJKZBGZCHGZILKZIJKZBGUQBGABMNZUMUPOADMNZCMNZUTPZUTACBDQGRZVCEBCDU | |
| ASVAVBUTUBSBMUCSAUOUSBAUNURIJAVDUNUROEBCDUDSUETAUSUQBAUQUFNZUSUQOAVDUQUGN | |
| VEEBCDUHUQUIUJUQUKSTUL $. | |
| $} | |
| $( A walk between two vertices has the indicated first and last vertices. | |
| (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| wlkonendpts $p |- ( F ( K ( WalksOn ` G ) M ) P -> | |
| ( ( P ` 0 ) = K /\ ( P ` ( # ` F ) ) = M ) ) $= | |
| ( cwlkson cfv co wbr cvv wcel cvtx w3a wa cwlks cc0 wceq chash eqid syl | |
| wlkonprop simp3 3simpc ) BADECFGHICJKDCLGZKEUDKMZBJKAJKNZBACOGIZPAGDQZBRGAG | |
| EQZMZMZUHUINZDEABCUDUDSUAUKUJULUEUFUJUBUGUHUIUCTT $. | |
| ${ | |
| $d F i j $. $d G i j $. $d H i j $. $d I i j $. $d J i j $. | |
| $d K i j $. $d M i j $. $d N i j $. $d P i j $. $d Q i j $. | |
| $d V i j $. $d i j $. $d ph i j $. | |
| spthccatspthond.v $e |- V = ( Vtx ` G ) $. | |
| spthccatspthond.i $e |- I = ( iEdg ` G ) $. | |
| spthccatspthond.h $e |- H = ( F ++ <" J "> ) $. | |
| spthccatspthond.q $e |- Q = ( P ++ <" N "> ) $. | |
| spthccatspthond.g $e |- ( ph -> G e. UPGraph ) $. | |
| spthccatspthond.p $e |- ( ph -> | |
| F ( K ( SPathsOn ` G ) M ) P ) $. | |
| spthccatspthond.j $e |- ( ph -> J e. dom I ) $. | |
| spthccatspthond.e $e |- ( ph -> ( I ` J ) = { M , N } ) $. | |
| spthccatspthond.f $e |- ( ph -> -. J e. ran F ) $. | |
| spthccatspthond.n $e |- ( ph -> N e. V ) $. | |
| spthccatspthond.r $e |- ( ph -> -. N e. ran P ) $. | |
| $( Extending a simple path by a fresh indexed edge to a new terminal | |
| vertex produces a simple path with the expected endpoints. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| spthccatspthond $p |- ( ph -> | |
| H ( K ( SPathsOn ` G ) N ) Q ) $= | |
| ( cspthson cfv co wbr cspths cc0 wceq chash cupgr wcel cwlks spthonisspth | |
| w3a ccnv wfun spthiswlk 3syl cpr clsw wlklswd cwlkson cpthson spthonpthon | |
| ctrlson pthontrlon trlsonwlkon wlkonendpts simpr syl eqtrd eqcomd preq12d | |
| wa eqidd wlkccat1d cs1 cconcat cwwspthsn crn wn spthwspthn syl2anc eqcomi | |
| cvtx a1i eleqtrrd wspthnextfun syl3anc cnveqd funeqd mpbird upgrwlkdvspth | |
| fveq1d cword cfzo wlkpwrd cn c1 caddc wlklenvp1 cn0 wlkcl nn0p1nn eqeltrd | |
| lbfzo0 biimpri ccats1val1 simpl fveq2d cdm wlkf ccatws1len ccats1val2 cvv | |
| 3jca wb spthonprop simp12 jca ccatws1clv isspthonpth ) AFCIKEUDUEZUFUGZFC | |
| EUHUEZUGZUICUEZIUJZFUKUEZCUEZKUJZUPZAYHYJYMAEULUMZFCEUNUEZUGCUQZURZYHQABC | |
| KDEFGHLMNOPQADBIJYEUFUGZDBYGUGZDBYPUGZRIJBDEUOZBDEUSZUTZSUBAHGUEJKVABVBUE | |
| ZKVATAJUUEKKAUUEJAUUEDUKUEZBUEZJABDEUUDVCAYSDBIJEVDUEUFUGZUUGJUJZRYSDBIJE | |
| VEUEUFUGDBIJEVGUEUFUGZUUHIJBDEVFIJBDEVHIJBDEVIUTZUUHUIBUEZIUJZUUIVPZUUIBD | |
| EIJVJZUUMUUIVKVLUTVMVNAKVQVOVMVRAYRBKVSVTUFZUQZURZABUUFEWAUFUMZKEWGUEZUMK | |
| BWBUMWCUURAYOYTUUSQAYSYTRUUBVLBDEWDWEAKLUUTUBUUTLUJALUUTMWFWHWIUCKEUUFBWJ | |
| WKAYQUUQACUUPCUUPUJAPWHZWLWMWNCFEWOWKAYIUULIAYIUIUUPUEZUULAUICUUPUVAWPABL | |
| WQUMZUIUIBUKUEZWRUFUMZUVBUULUJAYSYTUVCRUUBYTUUAUVCUUCBDELMWSVLUTZAUVDWTUM | |
| ZUVEAUVDUUFXAXBUFZWTAYSYTUVDUVHUJZRUUBYTUUAUVIUUCBDEXCVLUTZAUUFXDUMZUVHWT | |
| UMAYSYTUVKRUUBYTUUAUVKUUCBDEXEVLUTUUFXFVLXGUVEUVGUVDXHXIVLKUILBXJWEVMAYSU | |
| UHUUMRUUKUUHUUNUUMUUOUUMUUIXKVLUTVMAYLYKUUPUEZKAYKCUUPUVAWPAUVLUVDUUPUEZK | |
| AYKUVDUUPAYKUVHUVDAYKDHVSVTUFZUKUEZUVHAFUVNUKFUVNUJAOWHZXLADGXMZWQUMZUVOU | |
| VHUJAYSYTUVRRUUBYTUUAUVRUUCBDEGNXNVLUTZUVQDHXOVLVMAUVDUVHUVJVNVMXLAUVCKLU | |
| MZUVDUVDUJUVMKUJUVFUBAUVDVQKUVDLBXPWKVMVMXRAILUMZUVTVPZFXQWQZUMZCUWCUMZVP | |
| ZVPYFYNXSAUWBUWFAUWAUVTAYSEXQUMZUWAJLUMZUPDXQUMBXQUMVPZUUJYTVPZUPUWARIJBD | |
| ELMXTUWGUWAUWHUWIUWJYAUTUBYBAUWDUWEAFUVNUWCUVPAUVRUVNUWCUMUVSUVQDHYCVLXGA | |
| CUUPUWCUVAAUVCUUPUWCUMUVFLBKYCVLXGYBYBIKCFELUWCUWCMYDVLWN $. | |
| $} | |
| ${ | |
| $d A i $. $d E i $. $d F i $. $d G i $. $d J i $. $d M i $. | |
| $d N i $. $d P i $. $d Q i $. $d S i $. $d U i $. $d V i $. | |
| grleafspthond.v $e |- V = ( Vtx ` G ) $. | |
| grleafspthond.e $e |- E = ( iEdg ` G ) $. | |
| grleafspthond.a $e |- A = ( V u. { N } ) $. | |
| grleafspthond.p $e |- P = ( E u. { <. J , { U , N } >. } ) $. | |
| grleafspthond.s $e |- S = <. A , P >. $. | |
| $( A simple path in the original graph can be extended across the new | |
| leaf edge. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafspthond $p |- ( ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) /\ | |
| F ( M ( SPathsOn ` G ) U ) Q ) -> | |
| ( F ++ <" J "> ) ( M ( SPathsOn ` S ) N ) | |
| ( Q ++ <" N "> ) ) $= | |
| ( wcel cfv syl ctree cvv wnel wa w3a cdm cspthson co wbr cs1 cconcat cvtx | |
| ciedg eqid cusgr cupgr simpll1 treeusgr simpll2 simpll3 simplr grleafusgr | |
| 3jca jca usgrupgr simpr csubgr wi grleafsubgr subgrspthon mpd csn simplrl | |
| cun snidg elun2 crn cpr wf1o wceq grleaff1o f1odm grleafiedg eqcomd dmeqd | |
| eleqtrrd eleqtrd cop fveq1d a1i wn elex simpl prex simplrr df-nel fsnunfv | |
| biimpi 3eqtrd cc0 chash cfzo cword wf cwlks cspths spthonisspth spthiswlk | |
| wlkf wrdf frnd ssneldd grleafvtx cfz wlkp spthccatspthond ) HUARZELRZKUBR | |
| ZKLUCZUDZUEZIUBRZIFUFZUCZUDZUDZGCJEHUGSUHUIZUDZCCKUJUKUHZGDGIUJUKUHZDUMSZ | |
| IJEKDULSZYMUNYLUNYKUNYJUNYIDUORZDUPRYIHUORZXRYAUEZYFUDZYNYIYPYFYIYOXRYAYI | |
| XQYOXQXRYAYFYHUQHURTXQXRYAYFYHUSZXQXRYAYFYHUTZVCYBYFYHVAVDZABDEFHIKLMNOPQ | |
| VBTDVETYIYHGCJEDUGSUHUIZYGYHVFZYIHDVGUIZYHUUAVHYIYQUUCYTABDEFHIKLMNOPQVIT | |
| CHGDJEVJTVKYIIBUFZYLUFYIIYDIVLZVNZUUDYIIUUERZIUUFRYIYCUUGYBYCYEYHVMZIUBVO | |
| TIUUEYDVPTYIUUFFVQEKVRZVLVNZBVSZUUDUUFVTYIYQUUKYTABDEFHIKLMNOPQWATUUFUUJB | |
| WBTWFYIBYLYIYLBYIYQYLBVTYTABDEFHIKLMNOPQWCTZWDWEWGYIIYLSIBSIFIUUIWHVLVNZS | |
| ZUUIYIIYLBUULWIYIIBUUMBUUMVTYIPWJWIYIYCUUIUBRZIYDRWKZUEUUNUUIVTYIYCUUOUUP | |
| UUHYIEUBRZXSUDZUUOYIUUQXSYIXRUUQYRELWLTYIYAXSYSXSXTWMTZVDUUOUUREKWNWJTYIY | |
| EUUPYBYCYEYHWOYEUUPIYDWPWRTZVCFUBUBIUUIWQTWSYIGVQYDIYIWTGXASZXBUHZYDGYIGY | |
| DXCRZUVBYDGXDYIGCHXESUIZUVCYIGCHXFSUIZUVDYIYHUVEUUBJECGHXGTCGHXHTZCGHFNXI | |
| TYDGXJTXKUUTXLYIKAYMYIKLKVLZVNZAYIKUVGRZKUVHRYIXSUVIUUSKUBVOTKUVGLVPTYIAU | |
| VHAUVHVTYIOWJWDWGYIYMAYIYQYMAVTYTABDEFHIKLMNOPQXMTWDWGYICVQLKYIWTUVAXNUHZ | |
| LCYIUVDUVJLCXDUVFCGHLMXOTXKYIXTKLRWKZYIYAXTYSXSXTVFTXTUVKKLWPWRTXLXP $. | |
| $} | |
| ${ | |
| $d A k x $. $d E k x $. $d F k x $. $d G k x $. | |
| $d J k x $. $d K k x $. $d N k x $. $d P k x $. | |
| $d Q k x $. $d S k x $. $d U k x $. $d V k x $. $d k x $. | |
| grleafnewedgrnd.v $e |- V = ( Vtx ` G ) $. | |
| grleafnewedgrnd.e $e |- E = ( iEdg ` G ) $. | |
| grleafnewedgrnd.a $e |- A = ( V u. { N } ) $. | |
| grleafnewedgrnd.p $e |- P = | |
| ( E u. { <. J , { U , N } >. } ) $. | |
| grleafnewedgrnd.s $e |- S = <. A , P >. $. | |
| $( If the new leaf edge occurs in a walk, then the new leaf vertex occurs | |
| in its vertex word. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafnewedgrnd $p |- ( ( ( ( ( G e. USGraph /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) /\ | |
| F ( Walks ` S ) Q ) /\ J e. ran F ) -> N e. ran Q ) $= | |
| ( wcel wa cfv syl vk vx cusgr cvv wnel w3a cdm cwlks wbr crn cv wceq wfun | |
| wrex simplr ciedg cword eqid wlkf cc0 chash cfzo co wfn wrdfn fnfun simpr | |
| jca elrnrexdm imp c1 caddc cpr weq id fveq2 fvoveq1 preq12d eqeq12d cupgr | |
| fveq2d wral simplll grleafusgr usgrupgr simpllr upgrwlkedg simprl eleqtrd | |
| wrddm rspcdva cfz cvtx wf wlkp ffn elfzofz fnfvelrnd fzofzp1 prssd simpl3 | |
| eqsstrd simpl prid2g cop csn cun a1i fveq1d wn simpll simplrl prex df-nel | |
| simplrr sylib 3jca eqtrd eqcomd grleafiedg simprr 3eqtrd sseldd rexlimddv | |
| fsnunfv ) HUCQZEKQZJUDQZJKUEZRZUFZIUDQZIFUGZUEZRZRZGCDUHSUIZRZIGUJQZRZIUA | |
| UKZGSZULZJCUJZQUAGUGZYTGUMZYSRUUCUAUUEUNZYTUUFYSYTYQUUFYPYQYSUOYQGDUPSZUG | |
| ZUQQZUUFCGDUUHUUHURZUSZUUJGUTGVASZVBVCZVDUUFUUIGVEUUNGVFTTTYRYSVGVHUUFYSU | |
| UGUAGIVIVJTYTUUAUUEQZUUCRZRZUUBUUHSZUUDJUUQUURUUACSZUUAVKVLVCZCSZVMZUUDUU | |
| QUBUKZGSZUUHSZUVCCSZUVCVKVLVCCSZVMZULZUURUVBULUBUUNUUAUBUAVNZUVEUURUVHUVB | |
| UVJUVDUUBUUHUVJUVCUUAGUVJVOWAWAUVJUVFUUSUVGUVAUVCUUACVPUVCUUAVKCVLVQVRVSU | |
| UQDVTQZYQRUVIUBUUNWBUUQUVKYQUUQDUCQZUVKUUQYPUVLYPYQYSUUPWCZABDEFHIJKLMNOP | |
| WDTDWETYPYQYSUUPWFZVHCUBGDUUHUUKWGTUUQUUAUUEUUNYTUUOUUCWHUUQUUJUUEUUNULUU | |
| QYQUUJUVNUULTUUIGWJTWIZWKUUQUUSUVAUUDUUQUTUUMWLVCZUUACUUQUVPDWMSZCWNZCUVP | |
| VDUUQYQUVRUVNCGDUVQUVQURWOTUVPUVQCWPTZUUQUUAUUNQZUUAUVPQUVOUUAUTUUMWQTWRU | |
| UQUVPUUTCUVSUUQUVTUUTUVPQUVOUTUUMUUAWSTWRWTXBUUQJEJVMZUURUUQYHJUWAQUUQYJY | |
| HUUQYPYJUVMYFYGYJYOXATYHYIXCTEJUDXDTUUQUWAIBSZIUUHSZUURUUQUWBUWAUUQUWBIFI | |
| UWAXEXFXGZSZUWAUUQIBUWDBUWDULUUQOXHXIUUQYLUWAUDQZIYMQXJZUFUWEUWAULUUQYLUW | |
| FUWGUUQYRYLYRYSUUPXKZYKYLYNYQXLTUWFUUQEJXMXHUUQYNUWGUUQYRYNUWHYKYLYNYQXOT | |
| IYMXNXPXQFUDUDIUWAYETXRXSUUQUWCUWBUUQIUUHBUUQYPUUHBULUVMABDEFHIJKLMNOPXTT | |
| XIXSUUQIUUBUUHYTUUOUUCYAWAYBWIYCYD $. | |
| $} | |
| ${ | |
| $d A x $. $d E x $. $d G x $. $d J x $. $d N x $. | |
| $d P x $. $d S x $. $d U x $. $d V x $. | |
| grleafiedgdm.v $e |- V = ( Vtx ` G ) $. | |
| grleafiedgdm.e $e |- E = ( iEdg ` G ) $. | |
| grleafiedgdm.a $e |- A = ( V u. { N } ) $. | |
| grleafiedgdm.p $e |- P = | |
| ( E u. { <. J , { U , N } >. } ) $. | |
| grleafiedgdm.s $e |- S = <. A , P >. $. | |
| $( The edge indices after adjoining a freshly indexed leaf edge are the | |
| old edge indices together with the new index. (Contributed by | |
| Mingli Yuan, 13-Aug-2026.) $) | |
| grleafiedgdm $p |- ( ( ( G e. USGraph /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) -> | |
| dom ( iEdg ` S ) = ( dom E u. { J } ) ) $= | |
| ( wcel cvv wnel wa cdm csn cusgr w3a ciedg cfv grleafiedg dmeqd wf1o wceq | |
| cun crn cpr grleaff1o f1odm syl eqtrd ) FUAODIOHPOHIQRUBGPOGESZQRRZCUCUDZ | |
| SBSZUPGTUIZUQURBABCDEFGHIJKLMNUEUFUQUTEUJDHUKTUIZBUGUSUTUHABCDEFGHIJKLMNU | |
| LUTVABUMUNUO $. | |
| $} | |
| ${ | |
| $d A k x $. $d E k x $. $d F k x $. $d G k x $. $d J k x $. | |
| $d N k x $. $d P k x $. $d Q k x $. $d S k x $. | |
| $d U k x $. $d V k x $. $d k x $. | |
| grleafcyclnewedgnran.v $e |- V = ( Vtx ` G ) $. | |
| grleafcyclnewedgnran.e $e |- E = ( iEdg ` G ) $. | |
| grleafcyclnewedgnran.a $e |- A = ( V u. { N } ) $. | |
| grleafcyclnewedgnran.p $e |- P = | |
| ( E u. { <. J , { U , N } >. } ) $. | |
| grleafcyclnewedgnran.s $e |- S = <. A , P >. $. | |
| $( A nontrivial cycle in a graph with an adjoined leaf does not contain | |
| the new edge. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafcyclnewedgnran $p |- ( ( ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) /\ | |
| F ( Cycles ` S ) Q ) /\ F =/= (/) ) -> | |
| J e/ ran F ) $= | |
| ( wcel wnel wa syl ctree cvv w3a cdm ccycls cfv wbr c0 wne grleafcyclnran | |
| crn wn df-nel bicomi cusgr cwlks simplll simpl simp1 treeusgr simp2 simp3 | |
| sylibr 3jca id simpr jca simpllr cycliswlk grleafnewedgrnd ex mtod ) HUAQ | |
| ZEKQZJUBQJKRSZUCZIUBQIFUDRSZSZGCDUEUFUGZSGUHUIZSZIGUKZQZULIWBRWAWCJCUKZQZ | |
| WAJWDRZWEULZABCDEFGHIJKLMNOPUJWFWGJWDUMUNVCWAWCWEWAWCSZHUOQZVNVOUCZVQSZGC | |
| DUPUFUGZSZWCSWEWHWMWCWHWKWLWHWJVQWHWIVNVOWHVMWIWHVPVMWHVRVPVRVSVTWCUQZVPV | |
| QURTZVMVNVOUSTHUTTWHVPVNWOVMVNVOVATWHVPVOWOVMVNVOVBTVDWHVRVQWHWHVRWHVEZWN | |
| TVPVQVFTVGWHVSWLWHWHVSWPVRVSVTWCVHTCGDVITVGWAWCVFVGABCDEFGHIJKLMNOPVJTVKV | |
| LIWBUMVC $. | |
| $} | |
| ${ | |
| $d A k x $. $d E k x $. $d F k x $. $d G k x $. $d J k x $. | |
| $d N k x $. $d P k x $. $d Q k x $. $d S k x $. | |
| $d U k x $. $d V k x $. $d k x $. | |
| grleafcycloldrn.v $e |- V = ( Vtx ` G ) $. | |
| grleafcycloldrn.e $e |- E = ( iEdg ` G ) $. | |
| grleafcycloldrn.a $e |- A = ( V u. { N } ) $. | |
| grleafcycloldrn.p $e |- P = | |
| ( E u. { <. J , { U , N } >. } ) $. | |
| grleafcycloldrn.s $e |- S = <. A , P >. $. | |
| $( Every indexed edge of a nontrivial cycle remains in the old edge-index | |
| set after a leaf is adjoined. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| grleafcycloldrn $p |- ( ( ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) /\ | |
| F ( Cycles ` S ) Q ) /\ F =/= (/) ) -> | |
| ran F C_ dom E ) $= | |
| ( wcel wa cfv syl cvv wnel w3a cdm ccycls wbr c0 wne crn csn cun cdif wss | |
| ctree wn ciedg cc0 chash cfzo co wf cword simplr cycliswlk eqid wlkf wrdf | |
| cwlks frn cusgr wceq simplll simp1 treeusgr simp2 simp3 3jca simpll simpr | |
| jca grleafiedgdm eqcomd sseqtrrd grleafcyclnewedgnran df-nel sylib sylibr | |
| ssdifsn difun2 a1i simprr difsn eqtrd ) HUNQZEKQZJUAQJKUBRZUCZIUAQZIFUDZU | |
| BZRZRZGCDUESUFZRGUGUHZRZGUIZWSIUJZUKZXGULZWSXEXFXHUMZIXFQUOZRXFXIUMXEXJXK | |
| XEXFDUPSZUDZXHXEUQGURSUSUTZXMGVAZXFXMUMXEGXMVBQZXOXEGCDVHSUFZXPXEXCXQXBXC | |
| XDVCCGDVDTCGDXLXLVEVFTXMGVGTXNXMGVITXEXMXHXEHVJQZWOWPUCZXARXMXHVKXEXSXAXE | |
| XRWOWPXEWNXRXEWQWNWQXAXCXDVLZWNWOWPVMTHVNTXEWQWOXTWNWOWPVOTXEWQWPXTWNWOWP | |
| VPTVQXEXBXAXBXCXDVRZWQXAVSTVTABDEFHIJKLMNOPWATWBWCXEIXFUBXKABCDEFGHIJKLMN | |
| OPWDIXFWEWFVTXFXHIWHWGXEXIWSXEXIWSXGULZWSXIYBVKXEWSXGWIWJXEIWSQUOZYBWSVKX | |
| EWTYCXEXBWTYAWQWRWTWKTIWSWEWFIWSWLTWMWBWC $. | |
| $} | |
| ${ | |
| $d A k x $. $d E k x $. $d F k x $. $d G k x $. $d J k x $. | |
| $d N k x $. $d P k x $. $d Q k x $. $d S k x $. | |
| $d U k x $. $d V k x $. $d k x $. | |
| grleafcyclresedg.v $e |- V = ( Vtx ` G ) $. | |
| grleafcyclresedg.e $e |- E = ( iEdg ` G ) $. | |
| grleafcyclresedg.a $e |- A = ( V u. { N } ) $. | |
| grleafcyclresedg.p $e |- P = | |
| ( E u. { <. J , { U , N } >. } ) $. | |
| grleafcyclresedg.s $e |- S = <. A , P >. $. | |
| $( Restricting a nontrivial cycle after adjoining a leaf recovers a cycle | |
| on the old indexed-edge function. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| grleafcyclresedg $p |- ( ( ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) /\ | |
| F ( Cycles ` S ) Q ) /\ F =/= (/) ) -> | |
| F ( Cycles ` <. A , E >. ) Q ) $= | |
| ( wcel wa cfv syl ctree cvv wnel w3a cdm ccycls wbr c0 wne cop cvtx ciedg | |
| cupgr crn wss cusgr simplll simp1 treeusgr simp2 simp3 simpllr grleafusgr | |
| cres 3jca usgrupgr simplr grleafcycloldrn eqid upgrcyclres wceq grleafvtx | |
| jca grleafiedgres opeq12d eqcomd fveq2d breqd mpbird ) HUAQZEKQZJUBQJKUCR | |
| ZUDZIUBQIFUEZUCRZRZGCDUFSUGZRGUHUIZRZGCAFUJZUFSZUGGCDUKSZDULSZWDVDZUJZUFS | |
| ZUGZWIDUMQZWGRZGUNWDUOZRWQWIWSWTWIWRWGWIDUPQZWRWIHUPQZWAWBUDZWERZXAWIXCWE | |
| WIXBWAWBWIVTXBWIWCVTWCWEWGWHUQZVTWAWBURTHUSTWIWCWAXEVTWAWBUTTWIWCWBXEVTWA | |
| WBVATVEWCWEWGWHVBVMZABDEFHIJKLMNOPVCTDVFTWFWGWHVGVMABCDEFGHIJKLMNOPVHVMWD | |
| CWMGDWLWLVIWMVIVJTWIWKWPGCWIWJWOUFWIWOWJWIWLAWNFWIXDWLAVKXFABDEFHIJKLMNOP | |
| VLTWIXDWNFVKXFABDEFHIJKLMNOPVNTVOVPVQVRVS $. | |
| $} | |
| ${ | |
| $d A x $. $d E x $. $d G x $. $d N x $. $d V x $. | |
| grleafbaseupgr.v $e |- V = ( Vtx ` G ) $. | |
| grleafbaseupgr.e $e |- E = ( iEdg ` G ) $. | |
| grleafbaseupgr.a $e |- A = ( V u. { N } ) $. | |
| $( Keeping the old indexed edges while adjoining a vertex produces an | |
| undirected pseudograph. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| grleafbaseupgr $p |- ( G e. UPGraph -> | |
| <. A , E >. e. UPGraph ) $= | |
| ( cupgr wcel id csn cun cvv cvtx fvexi snex unex eqeltri wss sseqtrri a1i | |
| ssun1 upgrvtxsupd ) CIJZBCEAFGUEKAEDLZMZNHEUFECOFPDQRSEATUEEUGAEUFUCHUAUB | |
| UD $. | |
| $( Keeping the old indexed edges while adjoining a vertex does not change | |
| the indexed-edge function. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| grleafbaseiedg $p |- ( iEdg ` <. A , E >. ) = E $= | |
| ( csn cun cvv cvtx fvexi snex unex eqeltri ciedg cfv eqid opiedgfvi ) BAA | |
| EDIZJKHEUAECLFMDNOPBCQRZKGUBCQUBSMPT $. | |
| $} | |
| ${ | |
| $d A k x $. $d E k x $. $d F k x $. $d G k x $. $d J k x $. | |
| $d N k x $. $d P k x $. $d Q k x $. $d S k x $. | |
| $d U k x $. $d V k x $. $d k x $. | |
| grleafcyclres.v $e |- V = ( Vtx ` G ) $. | |
| grleafcyclres.e $e |- E = ( iEdg ` G ) $. | |
| grleafcyclres.a $e |- A = ( V u. { N } ) $. | |
| grleafcyclres.p $e |- P = | |
| ( E u. { <. J , { U , N } >. } ) $. | |
| grleafcyclres.s $e |- S = <. A , P >. $. | |
| $( Every nontrivial cycle after adjoining a leaf is already a cycle of the | |
| old tree. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafcyclres $p |- ( ( ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) /\ | |
| F ( Cycles ` S ) Q ) /\ F =/= (/) ) -> | |
| F ( Cycles ` G ) Q ) $= | |
| ( wcel wa cfv syl ctree cvv wnel w3a cdm ccycls wbr c0 wne cop cupgr wceq | |
| crn wss simplll simp1 treeupgr grleafbaseupgr jca eqid grleafcycloldvtxrn | |
| a1i grleafcyclresedg ciedg grleafbaseiedg eqcomi upgrvtxrescycl ) HUAQZEK | |
| QZJUBQJKUCRZUDZIUBQIFUEUCRZRGCDUFSUGZRGUHUIZRZAFUJZUKQZHUKQZRZFFULZRZGCVP | |
| UFSUGZCUMKUNZRZRGCHUFSUGVOWAWDVOVSVTVOVQVRVOVRVQVOVHVRVOVKVHVKVLVMVNUOVHV | |
| IVJUPTHUQTZAFHJKLMNURTWEUSVTVOFUTVBUSVOWBWCABCDEFGHIJKLMNOPVCABCDEFGHIJKL | |
| MNOPVAUSUSCFGVPHFKLVPVDSFAFHJKLMNVEVFMVGT $. | |
| $} | |
| ${ | |
| $d A x $. $d E x $. $d F x $. $d G x $. $d J x $. | |
| $d N x $. $d P x $. $d Q x $. $d S x $. $d U x $. $d V x $. | |
| grleafcycltriv.v $e |- V = ( Vtx ` G ) $. | |
| grleafcycltriv.e $e |- E = ( iEdg ` G ) $. | |
| grleafcycltriv.a $e |- A = ( V u. { N } ) $. | |
| grleafcycltriv.p $e |- P = | |
| ( E u. { <. J , { U , N } >. } ) $. | |
| grleafcycltriv.s $e |- S = <. A , P >. $. | |
| $( Every cycle after adjoining a leaf to a tree is trivial. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafcycltriv $p |- ( ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) /\ | |
| F ( Cycles ` S ) Q ) -> F = (/) ) $= | |
| ( wcel wa c0 syl cvv wnel w3a cdm ccycls cfv wbr wne wceq cacycgr simplll | |
| ctree simp1 treeacycgr grleafcyclres jca acycgrcycl ex nne biimpi pm2.61d | |
| wn wi a1i ) HULQZEKQZJUAQJKUBRZUCZIUAQIFUDUBRZRGCDUEUFUGZRZGSUHZGSUIZVKVL | |
| VMVKVLRZHUJQZGCHUEUFUGZRVMVNVOVPVNVEVOVNVHVEVHVIVJVLUKVEVFVGUMTHUNTABCDEF | |
| GHIJKLMNOPUOUPCGHUQTURVLVBZVMVCVKVQVMGSUSUTVDVA $. | |
| $} | |
| ${ | |
| $d A f p x $. $d E f p x $. $d G f p x $. $d J f p x $. | |
| $d N f p x $. $d P f p x $. $d S f p x $. $d U f p x $. | |
| $d V f p x $. $d f p x $. | |
| grleafacycgr.v $e |- V = ( Vtx ` G ) $. | |
| grleafacycgr.e $e |- E = ( iEdg ` G ) $. | |
| grleafacycgr.a $e |- A = ( V u. { N } ) $. | |
| grleafacycgr.p $e |- P = | |
| ( E u. { <. J , { U , N } >. } ) $. | |
| grleafacycgr.s $e |- S = <. A , P >. $. | |
| $( Adjoining a leaf to a tree preserves acyclicity. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafacycgr $p |- ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) -> | |
| S e. AcyclicGraph ) $= | |
| ( vf vp wcel cvv wa cusgr ctree wnel w3a cdm cacycgr cv ccycls cfv wbr c0 | |
| wceq wi wal grleafcycltriv ex alrimivv wb simpl1 treeusgr syl simpl2 3jca | |
| simpl3 simpr jca grleafusgr elexd isacycgr1 mpbird ) FUAQZDIQZHRQHIUBSZUC | |
| ZGRQGEUDUBSZSZCUEQZOUFZPUFZCUGUHUIZVQUJUKZULZPUMOUMZVOWAOPVOVSVTABVRCDEVQ | |
| FGHIJKLMNUNUOUPVOCRQVPWBUQVOCTVOFTQZVKVLUCZVNSCTQVOWDVNVOWCVKVLVOVJWCVJVK | |
| VLVNURFUSUTVJVKVLVNVAVJVKVLVNVCVBVMVNVDVEABCDEFGHIJKLMNVFUTVGOCRPVHUTVI | |
| $. | |
| $} | |
| ${ | |
| $d P i j $. $d Q i j $. $d R i j $. $d V i j $. $d i j $. $d ph i j $. | |
| ccatfstpthcondr.q $e |- Q = ( P ++ <" ( P ` 0 ) "> ) $. | |
| ccatfstpthcondr.r $e |- R = ( ( # ` Q ) - 1 ) $. | |
| ccatfstpthcondr.p $e |- ( ph -> P e. Word V ) $. | |
| ccatfstpthcondr.f $e |- ( ph -> Fun `' P ) $. | |
| $( The vertex-distinctness condition for closing an injective word, | |
| expressed with the length bounds of the closed word. (Contributed by | |
| Mingli Yuan, 11-Aug-2026.) $) | |
| ccatfstpthcondr $p |- ( ph -> | |
| A. i e. ( 0 ..^ ( # ` Q ) ) A. j e. ( 1 ..^ R ) | |
| ( i =/= j -> ( Q ` i ) =/= ( Q ` j ) ) ) $= | |
| ( cv wne cfv c1 cfzo co wral cc0 chash wi caddc ccatfstpthcond ccatfstlen | |
| eqcomd oveq2d raleqdv ralbidv mpbid cs1 cconcat wceq a1i cword ccatws1len | |
| fveq2d wcel syl eqtrd raleqtrdv ) AELZFLZMVACNVBCNMUAZFODPQZRZESBTNZOUBQZ | |
| PQZSCTNZPQAVCFOVFPQZRZEVHRVEEVHRABCEFGHJKUCAVKVEEVHAVCFVJVDAVFDOPADVFABCD | |
| GHIJUDUEUFUGUHUIAVGVISPAVIVGAVIBSBNZUJUKQZTNZVGACVMTCVMULAHUMUPABGUNUQVNV | |
| GULJGBVLUOURUSUEUFUT $. | |
| $} | |
| ${ | |
| ccatfstclosed.q $e |- Q = ( P ++ <" ( P ` 0 ) "> ) $. | |
| ccatfstclosed.r $e |- R = ( ( # ` Q ) - 1 ) $. | |
| ccatfstclosed.p $e |- ( ph -> P e. Word V ) $. | |
| $( A word closed by repeating its first symbol has equal endpoints. | |
| (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| ccatfstclosed $p |- ( ph -> | |
| ( Q ` 0 ) = ( Q ` R ) ) $= | |
| ( cc0 cfv cs1 cconcat wceq a1i fveq1d cword wcel syl eqtrd cvv ccat1st1st | |
| eqidd chash w3a wrdv fvex ccatfstlen 3jca ccats1val2 eqcomd eqcomi 3eqtrd | |
| co ) AICJZIBJZUODCJZAUNIBUOKLUMZJZUOAICUQCUQMAFNOABEPQZURUOMHEBUARSAUOUBA | |
| UODUQJZUPAUTUOABTPQZUOTQZDBUCJMZUDUTUOMAVAVBVCAUSVAHEBUERVBAIBUFNABCDEFGH | |
| UGUHUODTBUIRUJADUQCUQCMACUQFUKNOSUL $. | |
| $} | |
| ${ | |
| $d F i j $. $d P i j $. $d R i j $. $d ph i j $. | |
| pthdfd.p $e |- ( ph -> P e. Word _V ) $. | |
| pthdfd.r $e |- R = ( ( # ` P ) - 1 ) $. | |
| pthdfd.s $e |- ( ph -> | |
| A. i e. ( 0 ..^ ( # ` P ) ) A. j e. ( 1 ..^ R ) | |
| ( i =/= j -> ( P ` i ) =/= ( P ` j ) ) ) $. | |
| pthdfd.f $e |- ( ph -> ( # ` F ) = R ) $. | |
| pthdfd.t $e |- ( ph -> F ( Trails ` G ) P ) $. | |
| $( Deduction-form version of ~ pthd for the edge-count equality. | |
| (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| pthdfd $p |- ( ph -> F ( Paths ` G ) P ) $= | |
| ( cfv wbr c1 cfzo co cc0 cima c0 ctrls chash cres ccnv wfun cpr cin cpths | |
| wceq w3a pthdlem1 eqidd eqcomd oveq2d reseq12d cnveqd funeqd mpbid preq2d | |
| pthdlem2 imaeq12d ineq12d eqeq1d 3jca ispth sylibr ) AFBGUAMNZBOFUBMZPQZU | |
| CZUDZUEZBRVHUFZSZBVISZUGZTUIZUJFBGUHMNAVGVLVQLABOCPQZUCZUDZUEVLABCDEHIJUK | |
| AVTVKAVSVJABBVRVIABULZACVHOPAVHCKUMZUNZUOUPUQURABRCUFZSZBVRSZUGZTUIVQABCD | |
| EHIJUTAWGVPTAWEVNWFVOABBWDVMWAACVHRWBUSVAABBVRVIWAWCVAVBVCURVDBFGVEVF $. | |
| $} | |
| ${ | |
| $d F i j k $. $d G i j k $. $d H i j k $. $d I i j k $. $d J i j k $. | |
| $d P i j k $. $d Q i j k $. $d R i j $. $d V i j k $. $d i j $. | |
| $d ph i j k $. | |
| spthccatcycld.v $e |- V = ( Vtx ` G ) $. | |
| spthccatcycld.i $e |- I = ( iEdg ` G ) $. | |
| spthccatcycld.h $e |- H = ( F ++ <" J "> ) $. | |
| spthccatcycld.q $e |- Q = ( P ++ <" ( P ` 0 ) "> ) $. | |
| spthccatcycld.r $e |- R = ( ( # ` Q ) - 1 ) $. | |
| spthccatcycld.g $e |- ( ph -> G e. UPGraph ) $. | |
| spthccatcycld.p $e |- ( ph -> F ( SPaths ` G ) P ) $. | |
| spthccatcycld.j $e |- ( ph -> J e. dom I ) $. | |
| spthccatcycld.e $e |- ( ph -> ( I ` J ) = | |
| { ( lastS ` P ) , ( P ` 0 ) } ) $. | |
| spthccatcycld.n $e |- ( ph -> -. J e. ran F ) $. | |
| $( Closing a simple path with a fresh indexed edge produces a cycle. | |
| (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| spthccatcycld $p |- ( ph -> H ( Cycles ` G ) Q ) $= | |
| ( vi vj cpths cfv wbr cc0 chash wceq wa ccycls cs1 cconcat cvv cword wcel | |
| co cspths cwlks spthiswlk wlkpwrd 3syl ccatws1clv eleq1i sylibr ccnv wfun | |
| syl spthf1 ccatfstpthcondr cmin wlkepvtx simpl wlkccat1d wlklenvm1 eqcomi | |
| c1 a1i eqtrd ctrls cdm spthccats1f1d wf df-f1 simprbi cnveqi funeqi istrl | |
| wf1 jca pthdfd ccatfstclosed fveq2d eqcomd iscycl ) AGCFUCUDUEZUFCUDZGUGU | |
| DZCUDZUHZUIGCFUJUDUEAWOWSACDUAUBGFABUFBUDZUKULUPZUMUNZUOZCXBUOABJUNUOZXCA | |
| EBFUQUDUEZEBFURUDZUEZXDQBEFUSZBEFJKUTVAZJBWTVBVGCXAXBNVCVDOABCDUAUBJNOXIA | |
| XEBVEVFQBEFVHVGVIAWQCUGUDVPVJUPZDAGCXFUEZWQXJUHABCWTEFGHIJKLMNPAXEXGQXHVG | |
| ZRAXGWTJUOZXLXGXMEUGUDBUDJUOZUIXMBEFJKVKXMXNVLVGVGSVMZCGFVNVGXJDUHADXJOVO | |
| VQVRZAXKGVEZVFZUIGCFVSUDUEAXKXRXOAEIUKULUPZVEZVFZXRAXSVTZHVTZXSWHZYAABEFH | |
| ILQRTWAYDYBYCXSWBYAYBYCXSWCWDVGXQXTGXSMWEWFVDWICGFWGVDWJAWPDCUDZWRABCDJNO | |
| XIWKAWRYEAWQDCXPWLWMVRWICGFWNVD $. | |
| $} | |
| ${ | |
| $d F i j k $. $d G i j k $. $d H i j k $. $d I i j k $. $d J i j k $. | |
| $d P i j k $. $d Q i j k $. $d R i j $. $d V i j k $. $d i j $. | |
| $d ph i j k $. | |
| acycgrspthcld.v $e |- V = ( Vtx ` G ) $. | |
| acycgrspthcld.i $e |- I = ( iEdg ` G ) $. | |
| acycgrspthcld.h $e |- H = ( F ++ <" J "> ) $. | |
| acycgrspthcld.q $e |- Q = ( P ++ <" ( P ` 0 ) "> ) $. | |
| acycgrspthcld.r $e |- R = ( ( # ` Q ) - 1 ) $. | |
| acycgrspthcld.g $e |- ( ph -> G e. UPGraph ) $. | |
| acycgrspthcld.a $e |- ( ph -> G e. AcyclicGraph ) $. | |
| acycgrspthcld.p $e |- ( ph -> F ( SPaths ` G ) P ) $. | |
| acycgrspthcld.j $e |- ( ph -> J e. dom I ) $. | |
| acycgrspthcld.e $e |- ( ph -> ( I ` J ) = | |
| { ( lastS ` P ) , ( P ` 0 ) } ) $. | |
| $( In an acyclic graph, an indexed edge closing a simple path is already | |
| used by that path. (Contributed by Mingli Yuan, 11-Aug-2026.) $) | |
| acycgrspthcld $p |- ( ph -> J e. ran F ) $= | |
| ( crn wcel idd wn wa wceq cacycgr ccycls cfv wbr adantr cupgr cspths clsw | |
| c0 cdm cc0 cpr simpr spthccatcycld jca acycgrcycl syl cconcat cword cwlks | |
| cs1 spthiswlk wlkf 3syl wne ccatws1n0 df-ne biimpi eqeq1i notbii pm2.21dd | |
| co sylibr ex pm2.61d ) AIEUAUBZWBAWBUCAWBUDZWBAWCUEZGUOUFZWBWDFUGUBZGCFUH | |
| UIUJZUEWEWDWFWGAWFWCQUKWDBCDEFGHIJKLMNOAFULUBWCPUKAEBFUMUIUJZWCRUKAIHUPZU | |
| BWCSUKAIHUIBUNUIUQBUIURUFWCTUKAWCUSUTVACGFVBVCWDEIVGVDVRZUOUFZUDZWEUDWDEW | |
| IVEUBZWLAWMWCAWHEBFVFUIUJWMRBEFVHBEFHLVIVJUKWMWJUOVKZWLWIEIVLWNWLWJUOVMVN | |
| VCVCWEWKGWJUOMVOVPVSVQVTWA $. | |
| $} | |
| ${ | |
| $d B i j $. $d F i j $. $d G i j $. $d H i j $. $d I i j $. | |
| $d L i j $. $d P i j $. $d Q i j $. $d V i j $. $d i j $. $d ph i j $. | |
| swrdspthd.v $e |- V = ( Vtx ` G ) $. | |
| swrdspthd.i $e |- I = ( iEdg ` G ) $. | |
| swrdspthd.h $e |- H = ( F substr <. B , L >. ) $. | |
| swrdspthd.q $e |- Q = ( P substr <. B , ( L + 1 ) >. ) $. | |
| swrdspthd.p $e |- ( ph -> F ( SPaths ` G ) P ) $. | |
| swrdspthd.b $e |- ( ph -> B e. ( 0 ... L ) ) $. | |
| swrdspthd.l $e |- ( ph -> L e. ( 0 ... ( # ` F ) ) ) $. | |
| $( A continuous segment of a simple path is a simple path. (Contributed | |
| by Mingli Yuan, 11-Aug-2026.) $) | |
| swrdspthd $p |- ( ph -> H ( SPaths ` G ) Q ) $= | |
| ( wbr co cc0 ctrls cfv ccnv wfun wa cspths cwlks cop csubstr caddc cfz wcel | |
| c1 chash spthiswlk syl swrdwlk syl3anc breq12i sylibr cdm cword spthiedgf1d | |
| wf1 wlkf 3syl swrdf1 wf df-f1 simprbi cnveqi funeqi jca istrl wlkpwrd sseli | |
| fzssp1 fzp1elp1 wceq wlklenvp1 oveq2d eleqtrrd spthf1 wrdf1d isspth ) AGDFU | |
| AUBRZDUCZUDZUEGDFUFUBZRAWFWHAGDFUGUBZRZGUCZUDZUEWFAWKWMAEBIUHUISZCBIUMUJSZU | |
| HUISZWJRZWKAECWJRZBTIUKSZULZITEUNUBZUKSULZWQAECWIRZWROCEFUOZUPPQBCEFIUQURGW | |
| NDWPWJMNUSUTAWNUCZUDZWMAWNVAZHVAZWNVDZXFAXHBIEAXCWREXHVBULOXDCEFHLVEVFPQACE | |
| FHLOVCVGXIXGXHWNVHXFXGXHWNVIVJUPWLXEGWNMVKVLUTVMDGFVNUTAWPUCZUDZWHAWPVAZJWP | |
| VDZXKAJBWOCAXCWRCJVBULOXDCEFJKVOVFZAWTBTWOUKSZULPWSXOBTIVQVPUPAWOTXAUMUJSZU | |
| KSZTCUNUBZUKSAXBWOXQULQITXAVRUPAXRXPTUKAXCWRXRXPVSOXDCEFVTVFWAWBAJCXNAXCCUC | |
| UDOCEFWCUPWDVGXMXLJWPVHXKXLJWPVIVJUPWGXJDWPNVKVLUTVMDGFWEUT $. | |
| $} | |
| ${ | |
| swrdfv0d.w $e |- ( ph -> W e. Word D ) $. | |
| swrdfv0d.m $e |- ( ph -> M e. ( 0 ..^ N ) ) $. | |
| swrdfv0d.n $e |- ( ph -> N e. ( 0 ... ( # ` W ) ) ) $. | |
| $( The first symbol of a subword, in deduction form. (Contributed by | |
| Mingli Yuan, 12-Aug-2026.) $) | |
| swrdfv0d $p |- ( ph -> | |
| ( ( W substr <. M , N >. ) ` 0 ) = ( W ` M ) ) $= | |
| ( cword wcel cc0 cfzo co chash cfv cfz w3a cop csubstr wceq 3jca swrdfv0 | |
| syl ) AEBIJZCKDLMJZDKENOPMJZQKECDRSMOCEOTAUDUEUFFGHUABECDUBUC $. | |
| $} | |
| ${ | |
| swrdlswd.w $e |- ( ph -> W e. Word D ) $. | |
| swrdlswd.m $e |- ( ph -> M e. ( 0 ..^ N ) ) $. | |
| swrdlswd.n $e |- ( ph -> N e. ( 0 ... ( # ` W ) ) ) $. | |
| $( The last symbol of a nonempty subword, in deduction form. | |
| (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| swrdlswd $p |- ( ph -> | |
| ( lastS ` ( W substr <. M , N >. ) ) = | |
| ( W ` ( N - 1 ) ) ) $= | |
| ( co cfv chash c1 cmin wcel wceq syl cc0 cfz 3syl cc csubstr cword swrdcl | |
| cop clsw lsw cfzo elfzofz 3jca swrdlen oveq1d fveq2d caddc wa cn fzonnsub | |
| w3a fzo0end jca swrdfv cz elfzoel2 zcn elfzoelz 1cnd nnpcan eqtrd 3eqtrd | |
| ) AECDUDUAIZUEJZVIKJZLMIZVIJZDCMIZLMIZVIJZDLMIZEJZAVIBUBZNZVJVMOAEVSNZVTF | |
| BECDUCPVIVSUFPAVLVOVIAVKVNLMAWACQDRINZDQEKJRINZUQZVKVNOAWAWBWCFACQDUGINZW | |
| BGCQDUHPHUIZBECDUJPUKULAVPVOCUMIZEJZVRAWDVOQVNUGINZUNVPWHOAWDWIWFAWEVNUON | |
| WIGCQDUPVNURSUSBECDVOUTPAWGVQEADTNZCTNZLTNZUQWGVQOAWJWKWLAWEDVANWJGCQDVBD | |
| VCSAWECVANWKGCQDVDCVCSAVEUIDCLVFPULVGVH $. | |
| $} | |
| ${ | |
| f1veqaeqd.f $e |- ( ph -> F : A -1-1-> B ) $. | |
| f1veqaeqd.c $e |- ( ph -> C e. A ) $. | |
| f1veqaeqd.d $e |- ( ph -> D e. A ) $. | |
| f1veqaeqd.e $e |- ( ph -> ( F ` C ) = ( F ` D ) ) $. | |
| $( Equality of arguments of a one-to-one function, in deduction form. | |
| (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| f1veqaeqd $p |- ( ph -> C = D ) $= | |
| ( cfv wceq wf1 wcel wa wi jca f1veqaeq syl mpd ) ADFKEFKLZDELZJABCFMZDBNZ | |
| EBNZOZOUAUBPAUCUFGAUDUEHIQQBCDEFRST $. | |
| $} | |
| ${ | |
| f1endprd.f $e |- ( ph -> | |
| P : ( 0 ... N ) -1-1-> V ) $. | |
| f1endprd.b $e |- ( ph -> B e. ( 0 ..^ N ) ) $. | |
| f1endprd.k $e |- ( ph -> K e. ( B ..^ N ) ) $. | |
| f1endprd.e $e |- ( ph -> | |
| { ( P ` K ) , ( P ` ( K + 1 ) ) } = | |
| { ( P ` N ) , ( P ` B ) } ) $. | |
| $( An edge from the final value of an injective finite sequence back to | |
| an earlier value is its final consecutive pair. (Contributed by | |
| Mingli Yuan, 12-Aug-2026.) $) | |
| f1endprd $p |- ( ph -> B = ( N - 1 ) ) $= | |
| ( cfv wceq c1 co wa cc0 adantr wcel syl fvex cmin cfz wf1 cuz wss elfzouz | |
| caddc fzoss1 sseld mpd elfzofz cn0 cn clt wbr elfzo0 simp2bi nnnn0 nn0fz0 | |
| cfzo sylib simprl f1veqaeqd wn wne cz cr elfzoelz zre 3syl elfzolt2 ltned | |
| df-ne pm2.21d ex oveq1d fzofzp1 fz0fzelfz0 syl2anc simprr eqtr3d zcn 1cnd | |
| cc elfzoel2 addlsub mpbid cpr wo wb preq12b a1i mpjaod ) ADCKZECKZLZDMUGN | |
| ZCKZBCKZLZOZBEMUANLZWNWSLZWRWOLZOZAXAXBAXAOZDELZXBXFPEUBNZFDECAXHFCUCZXAG | |
| QADXHRZXAADPEUTNZRZXJADBEUTNZRZXLIAXMXKDABPUDKRZXMXKUEABXKRZXOHBPEUFSBPEU | |
| HSUIUJDPEUKSZQAEXHRZXAAEULRZXRAEUMRZXSAXPXTHXPBULRXTBEUNUOBEUPUQSEURSEUSV | |
| AZQAWPWTVBVCXFXGXBAXGVDZXAADEVEYBADEAXNDVFRDVGRIDBEVHDVIVJAXNDEUNUOIDBEVK | |
| SVLDEVMVAQVNUJVOAXEXBAXEOZBMUGNZELXBYCWQYDEYCDBMUGYCXHFDBCAXIXEGQZAXJXEXQ | |
| QABXHRZXEAXPYFHBPEUKSZQAXCXDVBVCVPYCXHFWQECYEAWQXHRZXEAYFWQBEUBNRZYHYGAXN | |
| YIIBEDVQSEWQBVRVSQAXRXEYAQAXCXDVTVCWAYCBMEABWDRZXEAXPBVFRYJHBPEVHBWBVJQYC | |
| WCAEWDRZXEAXPEVFRYKHBPEWEEWBVJQWFWGVOAWNWRWHWOWSWHLZXAXEWIZJYLYMWJAWNWRWO | |
| WSDCTWQCTECTBCTWKWLWGWM $. | |
| $} | |
| ${ | |
| $d B k $. $d F k $. $d G k $. $d I k $. $d K k $. $d P k $. | |
| $d ph k $. | |
| spthendprd.i $e |- I = ( iEdg ` G ) $. | |
| spthendprd.g $e |- ( ph -> G e. UPGraph ) $. | |
| spthendprd.p $e |- ( ph -> F ( SPaths ` G ) P ) $. | |
| spthendprd.b $e |- ( ph -> B e. ( 0 ..^ ( # ` F ) ) ) $. | |
| spthendprd.k $e |- ( ph -> K e. ( B ..^ ( # ` F ) ) ) $. | |
| spthendprd.e $e |- ( ph -> ( I ` ( F ` K ) ) = | |
| { ( P ` ( # ` F ) ) , ( P ` B ) } ) $. | |
| $( An indexed edge from the final vertex of a simple path back to an | |
| earlier vertex is the final indexed edge of the path. (Contributed by | |
| Mingli Yuan, 12-Aug-2026.) $) | |
| spthendprd $p |- ( ph -> B = ( ( # ` F ) - 1 ) ) $= | |
| ( vk cfv cc0 co syl fveq2d wcel chash cvtx cspths wbr cfz wf1 spthdifv c1 | |
| caddc cpr cv wceq cfzo wa simpr oveq1d preq12d eqeq12d cuz elfzouz fzoss1 | |
| wss sseld mpd cupgr cwlks wral upgrwlkedg syl2anc rspcdv2 eqtr3d f1endprd | |
| spthiswlk ) ABCGDUAOZEUBOZADCEUCOUDZPVNUEQVOCUFJCDEUGRKLAGDOZFOZGCOZGUHUI | |
| QZCOZUJZVNCOBCOUJANUKZDOZFOZWCCOZWCUHUIQZCOZUJZULZVRWBULNGPVNUMQZAWCGULZU | |
| NZWEVRWIWBWMWDVQFWMWCGDAWLUOZSSWMWFVSWHWAWMWCGCWNSWMWGVTCWMWCGUHUIWNUPSUQ | |
| URAGBVNUMQZTGWKTLAWOWKGABPUSOTZWOWKVBABWKTWPKBPVNUTRBPVNVARVCVDAEVETDCEVF | |
| OUDZWJNWKVGIAVPWQJCDEVMRCNDEFHVHVIVJMVKVL $. | |
| $} | |
| ${ | |
| spthswrdendd.q $e |- Q = | |
| ( P substr <. B , ( ( # ` F ) + 1 ) >. ) $. | |
| spthswrdendd.p $e |- ( ph -> F ( SPaths ` G ) P ) $. | |
| spthswrdendd.b $e |- ( ph -> B e. ( 0 ... ( # ` F ) ) ) $. | |
| $( The endpoints of the final segment of a simple path are the selected | |
| earlier vertex and the final vertex. (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| spthswrdendd $p |- ( ph -> | |
| { ( lastS ` Q ) , ( Q ` 0 ) } = | |
| { ( P ` ( # ` F ) ) , ( P ` B ) } ) $= | |
| ( clsw cfv chash cc0 c1 co wceq a1i wcel 3syl syl cop csubstr cmin fveq2d | |
| caddc cvtx cspths wbr cwlks cword spthiswlk eqid wlkpwrd cfz cz cn0 wlkcl | |
| cfzo nn0z fzval3 eleqtrd wlklenvp1 lencl sylib eqeltrrd swrdlswd cc nn0cn | |
| nn0fz0 pncan1 3eqtrd fveq1i swrdfv0d eqidd preq12d ) ADJKZELKZCKZMDKZBCKZ | |
| AVPCBVQNUEOZUAUBOZJKWANUCOZCKVRADWBJDWBPAGQUDAFUFKZBWACAECFUGKUHZECFUIKUH | |
| ZCWDUJRZHCEFUKZCEFWDWDULUMZSZABMVQUNOZMWAUROZIAVQUORZWKWLPAWFVQUPRZWMAWEW | |
| FHWHTZCEFUQZVQUSSMVQUTTVAZACLKZWAMWRUNOZAWFWRWAPWOCEFVBTAWRUPRZWRWSRAWFWG | |
| WTWOWIWDCVCSWRVIVDVEZVFAWCVQCAVQVGRZWCVQPAWNXBAWEWFWNHWHWPSVQVHTVQVJTUDVK | |
| AVSMWBKZVTVTVSXCPAMDWBGVLQAWDBWACWJWQXAVMAVTVNVKVO $. | |
| $} | |
| ${ | |
| $d B i j k $. $d F i j k $. $d G i j k $. $d H i j k $. $d I i j k $. | |
| $d J i j k $. $d P i j k $. $d Q i j k $. $d i j $. $d ph i j k $. | |
| acycgrspthendrnd.i $e |- I = ( iEdg ` G ) $. | |
| acycgrspthendrnd.h $e |- H = | |
| ( F substr <. B , ( # ` F ) >. ) $. | |
| acycgrspthendrnd.q $e |- Q = | |
| ( P substr <. B , ( ( # ` F ) + 1 ) >. ) $. | |
| acycgrspthendrnd.g $e |- ( ph -> G e. UPGraph ) $. | |
| acycgrspthendrnd.a $e |- ( ph -> G e. AcyclicGraph ) $. | |
| acycgrspthendrnd.p $e |- ( ph -> F ( SPaths ` G ) P ) $. | |
| acycgrspthendrnd.b $e |- ( ph -> B e. ( 0 ..^ ( # ` F ) ) ) $. | |
| acycgrspthendrnd.j $e |- ( ph -> J e. dom I ) $. | |
| acycgrspthendrnd.e $e |- ( ph -> ( I ` J ) = | |
| { ( P ` ( # ` F ) ) , ( P ` B ) } ) $. | |
| $( In an acyclic graph, an edge from the final vertex of a simple path to | |
| an earlier vertex already occurs in the corresponding final segment. | |
| (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| acycgrspthendrnd $p |- ( ph -> J e. ran H ) $= | |
| ( cfv co cc0 cs1 cconcat chash c1 cmin cvtx eqid cfzo cfz elfzofz syl cn0 | |
| wcel cspths wbr cwlks spthiswlk wlkcl nn0fz0 sylib swrdspthd spthswrdendd | |
| 3syl cpr clsw eqtr4d acycgrspthcld ) ADDUADSZUBUCTZVJUDSUEUFTZGFGIUBUCTZH | |
| IFUGSZVMUHZJVLUHVJUHVKUHMNABCDEFGHEUDSZVMVNJKLOABUAVOUITUNBUAVOUJTZUNPBUA | |
| VOUKULZAVOUMUNZVOVPUNAECFUOSUPECFUQSUPVROCEFURCEFUSVDVOUTVAVBQAIHSVOCSBCS | |
| VEDVFSVIVERABCDEFLOVQVCVGVH $. | |
| $} | |
| ${ | |
| spthswrdrnd.h $e |- H = | |
| ( F substr <. B , ( # ` F ) >. ) $. | |
| spthswrdrnd.p $e |- ( ph -> F ( SPaths ` G ) P ) $. | |
| spthswrdrnd.b $e |- ( ph -> B e. ( 0 ..^ ( # ` F ) ) ) $. | |
| $( The range of the final segment of a simple path is the image of the | |
| corresponding half-open index interval. (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| spthswrdrnd $p |- ( ph -> | |
| ran H = ( F " ( B ..^ ( # ` F ) ) ) ) $= | |
| ( crn chash cfv cop co cfzo wceq wcel cc0 wbr 3syl csubstr cima rneqi a1i | |
| ciedg cdm cword cfz cspths cwlks spthiswlk eqid wlkf elfzofz wlkcl nn0fz0 | |
| syl cn0 sylib swrdrn3 syl3anc eqtrd ) AFJZDBDKLZMUANZJZDBVDONUBZVCVFPAFVE | |
| GUCUDADEUELZUFZUGQZBRVDUHNZQZVDVKQZVFVGPADCEUILSZDCEUJLSZVJHCDEUKZCDEVHVH | |
| ULUMTABRVDONQVLIBRVDUNUQAVDURQZVMAVNVOVQHVPCDEUOTVDUPUSBVDVIDUTVAVB $. | |
| $} | |
| ${ | |
| $d B i j k $. $d F i j k $. $d G i j k $. $d H i j k $. $d I i j k $. | |
| $d J i j k $. $d P i j k $. $d i j $. $d ph i j k $. | |
| acycgrspthendd.i $e |- I = ( iEdg ` G ) $. | |
| acycgrspthendd.h $e |- H = | |
| ( F substr <. B , ( # ` F ) >. ) $. | |
| acycgrspthendd.g $e |- ( ph -> G e. UPGraph ) $. | |
| acycgrspthendd.a $e |- ( ph -> G e. AcyclicGraph ) $. | |
| acycgrspthendd.p $e |- ( ph -> F ( SPaths ` G ) P ) $. | |
| acycgrspthendd.b $e |- ( ph -> B e. ( 0 ..^ ( # ` F ) ) ) $. | |
| acycgrspthendd.j $e |- ( ph -> J e. dom I ) $. | |
| acycgrspthendd.e $e |- ( ph -> ( I ` J ) = | |
| { ( P ` ( # ` F ) ) , ( P ` B ) } ) $. | |
| $( In an acyclic graph, every edge from the final vertex of a simple path | |
| to an earlier vertex joins it to the immediately preceding vertex. | |
| (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| acycgrspthendd $p |- ( ph -> | |
| B = ( ( # ` F ) - 1 ) ) $= | |
| ( vk cfv co wcel cv wceq chash c1 cmin cfzo wfun cima wrex cc0 cdm cspths | |
| wf cword wbr cwlks spthiswlk wlkf 3syl wrdf syl ffun crn cop csubstr eqid | |
| caddc acycgrspthendrnd spthswrdrnd eleqtrd fvelima wa cupgr adantr simprl | |
| syl2anc cpr simprr fveq2d eqtrd spthendprd rexlimddv ) AQUAZDRZHUBZBDUCRZ | |
| UDUESUBQBWFUFSZADUGZHDWGUHZTWEQWGUIAUJWFUFSZGUKZDUMZWHADWKUNTZWLADCEULRUO | |
| ZDCEUPRUOWMMCDEUQCDEGIURUSWKDUTVAWJWKDVBVAAHFVCWIABCCBWFUDVGSVDVESZDEFGHI | |
| JWOVFKLMNOPVHABCDEFJMNVIVJQHWGDVKVPAWCWGTZWEVLZVLZBCDEGWCIAEVMTWQKVNAWNWQ | |
| MVNABWJTWQNVNAWPWEVOWRWDGRHGRZWFCRBCRVQZWRWDHGAWPWEVRVSAWSWTUBWQPVNVTWAWB | |
| $. | |
| $} | |
| ${ | |
| $d B i j k $. $d F i j k $. $d G i j k $. $d H i j k $. $d I i j k $. | |
| $d J i j k $. $d W i j k $. $d i j $. $d ph i j k $. | |
| acycgrwspthendlem.i $e |- I = ( iEdg ` G ) $. | |
| acycgrwspthendlem.g $e |- ( ph -> G e. UPGraph ) $. | |
| acycgrwspthendlem.a $e |- ( ph -> G e. AcyclicGraph ) $. | |
| acycgrwspthendlem.w $e |- ( ph -> | |
| W e. ( N WSPathsN G ) ) $. | |
| acycgrwspthendlem.n $e |- ( ph -> N e. NN ) $. | |
| acycgrwspthendlem.s $e |- ( ph -> | |
| S e. ( G NeighbVtx ( lastS ` W ) ) ) $. | |
| acycgrwspthendlem.p $e |- ( ph -> F ( SPaths ` G ) W ) $. | |
| acycgrwspthendlem.b $e |- ( ph -> B e. ( 0 ... N ) ) $. | |
| acycgrwspthendlem.v $e |- ( ph -> ( W ` B ) = S ) $. | |
| acycgrwspthendlem.j $e |- ( ph -> J e. dom I ) $. | |
| acycgrwspthendlem.e $e |- ( ph -> ( I ` J ) = | |
| { ( W ` N ) , ( W ` B ) } ) $. | |
| $( Lemma for ~ acycgrwspthendd . (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| acycgrwspthendlem $p |- ( ph -> | |
| S = ( W ` ( N - 1 ) ) ) $= | |
| ( cfv c1 cmin co eqcomd chash cop csubstr eqid cc0 cfzo wne cfz wcel clsw | |
| wa cnbgr wnel nbgrnself2 df-nel mpbi a1i jca nelne2 syl cwwlksn cwwspthsn | |
| wn wceq wspthsswwlkn sseli wwlknlsw 3netr4d wi necon3d mpd fzofzim cspths | |
| fveq2 wbr wspthnlen oveq2d eleqtrrd cpr fveq2d eqidd eqtrd acycgrspthendd | |
| preq12d oveq1d ) ACBIUAZHUBUCUDZIUAAWKCRUEABWLIABDUFUAZUBUCUDWLABIDEDBWMU | |
| GUHUDZFGJWNUIKLPABUJHUKUDZUJWMUKUDABHULZBUJHUMUDUNZUPBWOUNAWPWQAWKHIUAZUL | |
| WPACIUOUAZWKWRACEWSUQUDZUNZWSWTUNVHZUPCWSULAXAXBOXBAWSWTURXBEWSUSWSWTUTVA | |
| VBVCCWSWTVDVERAIHEVFUDZUNZWRWSVIAIHEVGUDZUNXDMXEXCIEHVJVKVEZEHIVLVEVMABHW | |
| KWRBHVIWKWRVIVNABHIVSVBVOVPQVCBHVQVEAWMHUJUKAXDDIEVRUAVTZUPWMHVIAXDXGXFPV | |
| CDEHIWAVEZWBWCSAGFUAWRWKWDWMIUAZWKWDTAWRXIWKWKAHWMIAWMHXHUEWEAWKWFWIWGWHA | |
| WMHUBUCXHWJWGWEWG $. | |
| $} | |
| ${ | |
| acycgrwspthendedgd.g $e |- ( ph -> G e. UPGraph ) $. | |
| acycgrwspthendedgd.a $e |- ( ph -> G e. AcyclicGraph ) $. | |
| acycgrwspthendedgd.w $e |- ( ph -> | |
| W e. ( N WSPathsN G ) ) $. | |
| acycgrwspthendedgd.s $e |- ( ph -> | |
| S e. ( G NeighbVtx ( lastS ` W ) ) ) $. | |
| acycgrwspthendedgd.v $e |- ( ph -> ( W ` B ) = S ) $. | |
| $( A neighbor of the final vertex represented at a path index determines | |
| the corresponding graph edge. (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| acycgrwspthendedgd $p |- ( ph -> | |
| { ( W ` N ) , ( W ` B ) } e. ( Edg ` G ) ) $= | |
| ( cfv cpr clsw cedg cwwlksn co wcel wceq syl cwwspthsn wspthsswwlkn sseli | |
| wwlknlsw preq12d prcom eqtrd cnbgr cusgr wb cupgr cacycgr wa upgracycusgr | |
| a1i jca eqid nbusgreledg mpbid eqeltrd ) AEFLZBFLZMZCFNLZMZDOLZAVCVDCMZVE | |
| AVAVDVBCAFEDPQZRZVAVDSAFEDUAQZRVIIVJVHFDEUBUCTDEFUDTKUEVGVESAVDCUFUOUGACD | |
| VDUHQRZVEVFRZJADUIRZVKVLUJADUKRZDULRZUMVMAVNVOGHUPDUNTVFDVDCVFUQURTUSUT | |
| $. | |
| $} | |
| ${ | |
| $d B i $. $d F i $. $d G i $. $d I i $. $d N i $. $d S i $. $d W i $. | |
| $d ph i $. | |
| acycgrwspthendlem2.i $e |- I = ( iEdg ` G ) $. | |
| acycgrwspthendlem2.g $e |- ( ph -> G e. UPGraph ) $. | |
| acycgrwspthendlem2.a $e |- ( ph -> G e. AcyclicGraph ) $. | |
| acycgrwspthendlem2.w $e |- ( ph -> | |
| W e. ( N WSPathsN G ) ) $. | |
| acycgrwspthendlem2.n $e |- ( ph -> N e. NN ) $. | |
| acycgrwspthendlem2.s $e |- ( ph -> | |
| S e. ( G NeighbVtx ( lastS ` W ) ) ) $. | |
| acycgrwspthendlem2.p $e |- ( ph -> F ( SPaths ` G ) W ) $. | |
| acycgrwspthendlem2.b $e |- ( ph -> B e. ( 0 ... N ) ) $. | |
| acycgrwspthendlem2.v $e |- ( ph -> ( W ` B ) = S ) $. | |
| acycgrwspthendlem2.e $e |- ( ph -> | |
| { ( W ` N ) , ( W ` B ) } e. ( Edg ` G ) ) $. | |
| $( Lemma for ~ acycgrwspthendd . (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| acycgrwspthendlem2 $p |- ( ph -> | |
| S = ( W ` ( N - 1 ) ) ) $= | |
| ( wcel adantr vi cfv cpr cv wceq c1 cmin co cdm cedg cuhgr cupgr upgruhgr | |
| wrex wb syl uhgredgiedgb mpbid wa cacycgr cwwspthsn clsw cnbgr cspths wbr | |
| cn cc0 cfz simprl simprr eqcomd acycgrwspthendlem rexlimddv ) AGHUBBHUBZU | |
| CZUAUDZFUBZUEZCGUFUGUHHUBUEUAFUIZAVOEUJUBSZVRUAVSUNZRAEUKSZVTWAUOAEULSZWB | |
| JEUMUPUAVOEFIUQUPURAVPVSSZVRUSZUSZBCDEFVPGHIAWCWEJTAEUTSWEKTAHGEVAUHSWELT | |
| AGVFSWEMTACEHVBUBVCUHSWENTADHEVDUBVEWEOTABVGGVHUHSWEPTAVNCUEWEQTAWDVRVIWF | |
| VOVQAWDVRVJVKVLVM $. | |
| $} | |
| ${ | |
| acycgrwspthendlem3.i $e |- I = ( iEdg ` G ) $. | |
| acycgrwspthendlem3.g $e |- ( ph -> G e. UPGraph ) $. | |
| acycgrwspthendlem3.a $e |- ( ph -> G e. AcyclicGraph ) $. | |
| acycgrwspthendlem3.w $e |- ( ph -> | |
| W e. ( N WSPathsN G ) ) $. | |
| acycgrwspthendlem3.n $e |- ( ph -> N e. NN ) $. | |
| acycgrwspthendlem3.s $e |- ( ph -> | |
| S e. ( G NeighbVtx ( lastS ` W ) ) ) $. | |
| acycgrwspthendlem3.p $e |- ( ph -> F ( SPaths ` G ) W ) $. | |
| acycgrwspthendlem3.b $e |- ( ph -> B e. ( 0 ... N ) ) $. | |
| acycgrwspthendlem3.v $e |- ( ph -> ( W ` B ) = S ) $. | |
| $( Lemma for ~ acycgrwspthendd . (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| acycgrwspthendlem3 $p |- ( ph -> | |
| S = ( W ` ( N - 1 ) ) ) $= | |
| ( acycgrwspthendedgd acycgrwspthendlem2 ) ABCDEFGHIJKLMNOPQABCEGHJKLNQRS | |
| $. | |
| $} | |
| ${ | |
| $d F b $. $d G b $. $d I b $. $d N b $. $d S b $. $d W b $. | |
| $d ph b $. | |
| acycgrwspthendlem4.i $e |- I = ( iEdg ` G ) $. | |
| acycgrwspthendlem4.g $e |- ( ph -> G e. UPGraph ) $. | |
| acycgrwspthendlem4.a $e |- ( ph -> G e. AcyclicGraph ) $. | |
| acycgrwspthendlem4.w $e |- ( ph -> | |
| W e. ( N WSPathsN G ) ) $. | |
| acycgrwspthendlem4.n $e |- ( ph -> N e. NN ) $. | |
| acycgrwspthendlem4.s $e |- ( ph -> | |
| S e. ( G NeighbVtx ( lastS ` W ) ) ) $. | |
| acycgrwspthendlem4.r $e |- ( ph -> S e. ran W ) $. | |
| acycgrwspthendlem4.p $e |- ( ph -> F ( SPaths ` G ) W ) $. | |
| $( Lemma for ~ acycgrwspthendd . (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| acycgrwspthendlem4 $p |- ( ph -> | |
| S = ( W ` ( N - 1 ) ) ) $= | |
| ( vb cfv co wcel adantr cv wceq c1 cmin cc0 cfz crn wrex wfn wb cwwspthsn | |
| wspthnfn syl fvelrnb mpbid wa cupgr cacycgr cn cnbgr cspths simprl simprr | |
| clsw wbr acycgrwspthendlem3 rexlimddv ) APUAZGQBUBZBFUCUDRGQUBPUEFUFRZABG | |
| UGSZVIPVJUHZNAGVJUIZVKVLUJAGFDUKRSZVMKDFGULUMPVJBGUNUMUOAVHVJSZVIUPZUPVHB | |
| CDEFGHADUQSVPITADURSVPJTAVNVPKTAFUSSVPLTABDGVDQUTRSVPMTACGDVAQVEVPOTAVOVI | |
| VBAVOVIVCVFVG $. | |
| $} | |
| ${ | |
| $d G b f i $. $d N b f i $. $d S b f i $. $d W b f i $. $d b f i $. | |
| $( In an acyclic graph, a neighbor of the final vertex which already | |
| occurs on a positive-length simple path is its predecessor. | |
| (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| acycgrwspthendd $p |- ( | |
| ( ( G e. UPGraph /\ G e. AcyclicGraph /\ | |
| W e. ( N WSPathsN G ) ) /\ | |
| ( N e. NN /\ S e. ( G NeighbVtx ( lastS ` W ) ) /\ | |
| S e. ran W ) ) -> | |
| S = ( W ` ( N - 1 ) ) ) $= | |
| ( vf cupgr wcel cacycgr cwwspthsn co w3a cn clsw cfv cnbgr crn cspths syl | |
| wa cv wbr c1 cmin wceq wex simpl simp3 cn0 cvv cwwlksn wspthnp ciedg eqid | |
| simp3d simpll1 simpll2 simpll3 simplr1 simplr2 simplr3 acycgrwspthendlem4 | |
| simpr exlimddv ) BFGZBHGZDCBIJGZKZCLGZABDMNOJGZADPGZKZSZETZDBQNUAZACUBUCJ | |
| DNUDEVLVFVNEUEZVLVGVFVGVKUFVDVEVFUGRVFCUHGBUIGSDCBUJJGVOEBCDUKUNRVLVNSAVM | |
| BBULNZCDVPUMVDVEVFVKVNUOVDVEVFVKVNUPVDVEVFVKVNUQVHVIVJVGVNURVHVIVJVGVNUSV | |
| HVIVJVGVNUTVLVNVBVAVC $. | |
| $} | |
| ${ | |
| $d G m w $. $d N m w $. $d S m w $. $d W m w $. $d m w $. | |
| $( Every neighbor of the final vertex of a positive-length maximum simple | |
| path in an acyclic graph is its predecessor. (Contributed by Mingli | |
| Yuan, 12-Aug-2026.) $) | |
| acycgrwspthmaxend $p |- ( | |
| ( ( G e. UPGraph /\ G e. AcyclicGraph /\ | |
| W e. ( N WSPathsN G ) ) /\ | |
| ( N e. NN /\ | |
| A. m A. w ( w e. ( m WSPathsN G ) -> m <_ N ) /\ | |
| S e. ( G NeighbVtx ( lastS ` W ) ) ) ) -> | |
| S = ( W ` ( N - 1 ) ) ) $= | |
| ( wcel cwwspthsn co w3a cv wal cfv wa wceq cpr 3jca eqid syl jca cupgr cn | |
| cacycgr cle wbr wi clsw cnbgr crn c1 cmin simpl simpr1 simpr3 cvtx simpl1 | |
| cedg simpl3 simpr2 nbgrisvtx prcom a1i wb simpl2 upgracycusgr nbusgreledg | |
| cusgr mpbid eqeltrd wspthmaxend acycgrwspthendd ) DUAGZDUCGZFEDHIGZJZEUBG | |
| ZAKCKZDHIGVQEUDUEUFALCLZBDFUGMZUHIGZJZNZVOVPVTBFUIGZJZNBEUJUKIFMOWBVOWDVO | |
| WAULWBVPVTWCVOVPVRVTUMVOVPVRVTUNZWBVLVNVRJZBDUOMZGZVSBPZDUQMZGZNZNWCWBWFW | |
| LWBVLVNVRVLVMVNWAUPZVLVMVNWAURVOVPVRVTUSQWBWHWKWBVTWHWEDVSBWGWGRUTSWBWIBV | |
| SPZWJWIWNOWBVSBVAVBWBVTWNWJGZWEWBDVGGZVTWOVCWBVLVMNWPWBVLVMWMVLVMVNWAVDTD | |
| VESWJDVSBWJRVFSVHVITTABCDEFVJSQTBDEFVKS $. | |
| $} | |
| ${ | |
| $d G m s w $. $d N m s w $. $d W m s w $. $d m s w $. | |
| $( The neighbors of the final vertex of a positive-length maximum simple | |
| path in an acyclic graph form the singleton containing its predecessor. | |
| (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| acycgrwspthmaxendnb $p |- ( | |
| ( ( G e. UPGraph /\ G e. AcyclicGraph /\ | |
| W e. ( N WSPathsN G ) ) /\ | |
| ( N e. NN /\ | |
| A. m A. w ( w e. ( m WSPathsN G ) -> m <_ N ) ) ) -> | |
| ( G NeighbVtx ( lastS ` W ) ) = | |
| { ( W ` ( N - 1 ) ) } ) $= | |
| ( vs cupgr wcel cacycgr cwwspthsn co w3a cv wal wa cfv wceq jca syl cpr | |
| cn cle wbr wi clsw cnbgr c1 cmin simpll simplrl simplrr acycgrwspthmaxend | |
| csn simpr 3jca elsn sylibr ex ssrdv cedg prcom simpl3 simprl wspthendpred | |
| vex a1i eqeltrd cusgr simpl1 simpl2 upgracycusgr nbusgreledg mpbird snssd | |
| wb eqid eqssd ) CGHZCIHZEDCJKHZLZDUAHZAMBMZCJKHWCDUBUCUDANBNZOZOZCEUEPZUF | |
| KZDUGUHKEPZUMZWFFWHWJWFFMZWHHZWKWJHZWFWLOZWKWIQZWMWNWAWBWDWLLZOWOWNWAWPWA | |
| WEWLUIWNWBWDWLWAWBWDWLUJWAWBWDWLUKWFWLUNUORAWKBCDEULSWKWIFVEUPUQURUSWFWIW | |
| HWFWIWHHZWIWGTZCUTPZHZWFWRWGWITZWSWRXAQWFWIWGVAVFWFVTWBOXAWSHWFVTWBVRVSVT | |
| WEVBWAWBWDVCRCDEVDSVGWFCVHHZWQWTVOWFVRVSOXBWFVRVSVRVSVTWEVIVRVSVTWEVJRCVK | |
| SWSCWGWIWSVPVLSVMVNVQ $. | |
| $} | |
| ${ | |
| $d G x $. $d N x $. $d W x $. | |
| $( The final symbol of a fixed-length simple-path word is a vertex of the | |
| graph. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| wspthnlswvtx $p |- ( W e. ( N WSPathsN G ) -> | |
| ( lastS ` W ) e. ( Vtx ` G ) ) $= | |
| ( cwwspthsn co wcel cfv clsw cvtx cwwlksn wspthsswwlkn sseli wwlknlsw syl | |
| wceq cc0 wa eqid wspthnevtx simpr eqeltrrd ) CBADEZFZBCGZCHGZAIGZUCCBAJEZ | |
| FUDUEOUBUGCABKLABCMNUCPCGUFFZUDUFFZQUIABUFCUFRSUHUITNUA $. | |
| $} | |
| ${ | |
| $d G s $. $d N s $. $d W s $. $d G m w $. $d N m w $. $d W m w $. | |
| $d m s w $. | |
| $( The neighbors of the final vertex of a maximum simple path in an | |
| acyclic graph occur on that path. (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| acycgrwspthmaxendnbss $p |- ( | |
| ( ( G e. UPGraph /\ G e. AcyclicGraph /\ | |
| W e. ( N WSPathsN G ) ) /\ | |
| A. m A. w ( w e. ( m WSPathsN G ) -> m <_ N ) ) -> | |
| ( G NeighbVtx ( lastS ` W ) ) C_ ran W ) $= | |
| ( vs cupgr wcel cwwspthsn co w3a cv wi wal wa cfv cpr eqid a1i syl simpl1 | |
| cacycgr cle wbr clsw cnbgr cvtx cedg simpl3 simpr 3jca nbgrisvtx cusgr wb | |
| crn simpl2 upgracycusgr nbusgreledg eleq1i biimpi sylbid jcad wspthmaxend | |
| jca prcom syl6an ssrdv ) CGHZCUBHZEDCIJHZKZALBLZCIJHVLDUCUDMANBNZOZFCEUEP | |
| ZUFJZEUOZVNVHVJVMKFLZVPHZVRCUGPZHZVOVRQZCUHPZHZOVRVQHVNVHVJVMVHVIVJVMUAZV | |
| HVIVJVMUIVKVMUJUKVNVSWAWDVSWAMVNCVOVRVTVTRULSVNVSVRVOQZWCHZWDVNCUMHZVSWGU | |
| NVNVHVIOWHVNVHVIWEVHVIVJVMUPVDCUQTWCCVOVRWCRURTWGWDMVNWGWDWFWBWCVRVOVEUSU | |
| TSVAVBAVRBCDEVCVFVG $. | |
| $} | |
| $( A zero-length simple path is the singleton word containing its final | |
| vertex. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| wspth0s1 $p |- ( W e. ( 0 WSPathsN G ) -> | |
| W = <" ( lastS ` W ) "> ) $= | |
| ( cc0 cwwspthsn co wcel cfv cs1 clsw cvtx cword chash c1 wa wspthnwrd caddc | |
| wceq cwwlksn syl eqtrd wspthsswwlkn sseli wwlknlen 0p1e1 a1i jca eqs1 s1eqd | |
| wwlknlsw ) BCADEZFZBCBGZHZBIGZHUKBAJGZKFZBLGZMQZNBUMQUKUPURACBOUKUQCMPEZMUK | |
| BCAREZFZUQUSQUJUTBACUAUBZACBUCSUSMQUKUDUETUFUOBUGSUKULUNUKVAULUNQVBACBUISUH | |
| T $. | |
| $( The range of a zero-length simple path consists of its final vertex. | |
| (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| wspth0rn $p |- ( W e. ( 0 WSPathsN G ) -> | |
| ran W = { ( lastS ` W ) } ) $= | |
| ( cc0 cwwspthsn wcel crn clsw cfv cs1 wspth0s1 rneqd cvtx wceq wspthnlswvtx | |
| co csn s1rn syl eqtrd ) BCADOEZBFBGHZIZFZUAPZTBUBABJKTUAALHZEUCUDMACBNUAUEQ | |
| RS $. | |
| $( A simple path whose length parameter is zero has singleton range. | |
| (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| wspthlen0rn $p |- ( ( W e. ( N WSPathsN G ) /\ N = 0 ) -> | |
| ran W = { ( lastS ` W ) } ) $= | |
| ( cwwspthsn co wcel cc0 wceq wa crn clsw cfv csn simpl simpr oveq1d eleqtrd | |
| wspth0rn syl ) CBADEZFZBGHZIZCGADEZFCJCKLMHUCCTUDUAUBNUCBGADUAUBOPQACRS $. | |
| ${ | |
| $d G m s w $. $d W m s w $. $d m s w $. | |
| $( The final vertex of a zero-length maximum simple path in an acyclic | |
| graph has no neighbors. (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| acycgrwspthmaxendnb0 $p |- ( | |
| ( ( G e. UPGraph /\ G e. AcyclicGraph /\ | |
| W e. ( 0 WSPathsN G ) ) /\ | |
| A. m A. w ( w e. ( m WSPathsN G ) -> m <_ 0 ) ) -> | |
| ( G NeighbVtx ( lastS ` W ) ) = (/) ) $= | |
| ( vs cupgr wcel cacycgr cc0 cwwspthsn co w3a cv cle wbr wal a1i wceq syl6 | |
| wi wa clsw cfv cnbgr wnel nbgrnself2 csn crn acycgrwspthmaxendnbss simpl3 | |
| wn wspth0rn syl sseqtrd sseld wb vex elsn sylibd eqcom biimpi eleq1a mpdd | |
| nnel biimpri mt2d eq0rdv ) CFGZCHGZDICJKGZLAMBMZCJKGVKINOTAPBPZUAZECDUBUC | |
| ZUDKZVMEMZVOGZVNVOUEZVRVMCVNUFQVMVQVNVOGZVRUKZVMVQVNVPRZVSVMVQVPVNRZWAVMV | |
| QVPVNUGZGZWBVMVOWCVPVMVODUHZWCABCIDUIVMVJWEWCRVHVIVJVLUJCDULUMUNUOWDWBUPV | |
| MVPVNEUQURQUSWBWAVPVNUTVASVQWAVSTTVMVPVOVNVBQVCVTVSVNVOVDVESVFVG $. | |
| $} | |
| ${ | |
| $d G m w $. $d W m w $. $d m w $. | |
| $( The final vertex of a zero-length maximum simple path in an acyclic | |
| graph has degree zero. (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| acycgrwspthmaxendvd0 $p |- ( | |
| ( ( G e. UPGraph /\ G e. AcyclicGraph /\ | |
| W e. ( 0 WSPathsN G ) ) /\ | |
| A. m A. w ( w e. ( m WSPathsN G ) -> m <_ 0 ) ) -> | |
| ( ( VtxDeg ` G ) ` ( lastS ` W ) ) = 0 ) $= | |
| ( cupgr wcel cacycgr cc0 cwwspthsn co w3a cv wal wa cfv chash wceq jca c0 | |
| syl cle wi clsw cnbgr cvtxdg cusgr cvtx simpl1 simpl2 upgracycusgr simpl3 | |
| wbr wspthnlswvtx hashnbusgrvd acycgrwspthmaxendnb0 fveq2d hash0 a1i eqtrd | |
| eqid eqtr3d ) CEFZCGFZDHCIJFZKALBLZCIJFVEHUAULUBAMBMZNZCDUCOZUDJZPOZVHCUE | |
| OOZHVGCUFFZVHCUGOZFZNVJVKQVGVLVNVGVBVCNVLVGVBVCVBVCVDVFUHVBVCVDVFUIRCUJTV | |
| GVDVNVBVCVDVFUKCHDUMTRVHCVMVMUTUNTVGVJSPOZHVGVISPABCDUOUPVOHQVGUQURUSVA | |
| $. | |
| $} | |
| ${ | |
| $d G m s w $. $d N m s w $. $d W m s w $. $d m s w $. | |
| $( If a maximum simple path in an acyclic graph has length zero, then its | |
| final vertex has no neighbors. (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| acycgrwspthmaxendnb0g $p |- ( | |
| ( ( G e. UPGraph /\ G e. AcyclicGraph /\ | |
| W e. ( N WSPathsN G ) ) /\ | |
| ( N = 0 /\ | |
| A. m A. w ( w e. ( m WSPathsN G ) -> m <_ N ) ) ) -> | |
| ( G NeighbVtx ( lastS ` W ) ) = (/) ) $= | |
| ( vs cupgr wcel cacycgr cwwspthsn co wceq cv wi wal wa a1i jca syl syl6 | |
| w3a cc0 cle wbr clsw cfv wnel nbgrnself2 wn csn crn acycgrwspthmaxendnbss | |
| cnbgr wss simpl simprr simpl3 simprl wspthlen0rn sseqtrd sseld vex sylibd | |
| wb elsn eqcom biimpi eleq1a mpdd nnel biimpri mt2d eq0rdv ) CGHZCIHZEDCJK | |
| HZUAZDUBLZAMBMZCJKHVSDUCUDNAOBOZPZPZFCEUEUFZUMKZWBFMZWDHZWCWDUGZWGWBCWCUH | |
| QWBWFWCWDHZWGUIZWBWFWCWELZWHWBWFWEWCLZWJWBWFWEWCUJZHZWKWBWDWLWEWBWDEUKZWL | |
| WBVQVTPWDWNUNWBVQVTVQWAUOVQVRVTUPRABCDEULSWBVPVRPWNWLLWBVPVRVNVOVPWAUQVQV | |
| RVTURRCDEUSSUTVAWMWKVDWBWEWCFVBVEQVCWKWJWEWCVFVGTWFWJWHNNWBWEWDWCVHQVIWIW | |
| HWCWDVJVKTVLVM $. | |
| $} | |
| ${ | |
| $d G m w $. $d N m w $. $d W m w $. $d m w $. | |
| $( If a maximum simple path in an acyclic graph has length zero, then its | |
| final vertex has degree zero. (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| acycgrwspthmaxendvd0g $p |- ( | |
| ( ( G e. UPGraph /\ G e. AcyclicGraph /\ | |
| W e. ( N WSPathsN G ) ) /\ | |
| ( N = 0 /\ | |
| A. m A. w ( w e. ( m WSPathsN G ) -> m <_ N ) ) ) -> | |
| ( ( VtxDeg ` G ) ` ( lastS ` W ) ) = 0 ) $= | |
| ( cupgr wcel cacycgr cwwspthsn co cc0 wceq cv wal wa cfv chash jca syl c0 | |
| w3a cle wi clsw cnbgr cvtxdg cusgr cvtx simpl1 simpl2 upgracycusgr simpl3 | |
| wbr wspthnlswvtx eqid hashnbusgrvd acycgrwspthmaxendnb0g fveq2d hash0 a1i | |
| eqtrd eqtr3d ) CFGZCHGZEDCIJGZUADKLAMBMZCIJGVFDUBUMUCANBNOZOZCEUDPZUEJZQP | |
| ZVICUFPPZKVHCUGGZVICUHPZGZOVKVLLVHVMVOVHVCVDOVMVHVCVDVCVDVEVGUIVCVDVEVGUJ | |
| RCUKSVHVEVOVCVDVEVGULCDEUNSRVICVNVNUOUPSVHVKTQPZKVHVJTQABCDEUQURVPKLVHUSU | |
| TVAVB $. | |
| $} | |
| ${ | |
| $d G m w $. $d N m w $. $d V m w $. $d W m w $. $d m w $. | |
| acycgrwspthmaxnn.v $e |- V = ( Vtx ` G ) $. | |
| $( A maximum simple path in a connected acyclic graph with at least two | |
| vertices has positive length. (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| acycgrwspthmaxnn $p |- ( | |
| ( ( ( G e. UPGraph /\ G e. ConnGraph /\ G e. AcyclicGraph ) /\ | |
| ( 1 < ( # ` V ) /\ W e. ( N WSPathsN G ) ) ) /\ | |
| A. m A. w ( w e. ( m WSPathsN G ) -> m <_ N ) ) -> | |
| N e. NN ) $= | |
| ( wcel w3a cfv wbr cwwspthsn co wa cv wal cc0 syl jca a1d cconngr cacycgr | |
| cupgr c1 chash clt cle wi cn0 wne cn simplrr wspthnn0 clsw cvtxdg simpll2 | |
| simpll1 simpll3 upgracycumgr wspthnlswvtx eleq2i simplrl vdn0conngrumgrv2 | |
| cumgr cvtx sylibr wceq 3jcad idd simpr jcad acycgrwspthmaxendvd0g necon3d | |
| syl6 mpd elnnne0 ) CUCHZCUAHZCUBHZIZUDEUEJUFKZFDCLMHZNZNZAOBOZCLMHWEDUGKU | |
| HAPBPZNZDUIHZDQUJZNDUKHWGWHWIWGWBWHVTWAWBWFULZCDFUMRWGFUNJZCUOJJZQUJZWIWG | |
| VRCVDHZNZWKEHZWANZNWMWGWOWQWGVRWNVQVRVSWCWFUPWGVQVSNWNWGVQVSVQVRVSWCWFUQZ | |
| VQVRVSWCWFURZSCUSRSWGWPWAWGWKCVEJZHZWPWGWBXAWJCDFUTREWTWKGVAVFVTWAWBWFVBS | |
| SCWKEGVCRWGDQWLQWGDQVGZVQVSWBIZXBWFNZNWLQVGWGXBXCXDWGXBVQVSWBWGVQXBWRTWGV | |
| SXBWSTWGWBXBWJTVHWGXBXBWFWGXBVIWGWFXBWDWFVJTVKVKABCDFVLVNVMVOSDVPVF $. | |
| $} | |
| ${ | |
| $d G m w $. $d N m w $. $d W m w $. $d m w $. | |
| $( The final vertex of a positive-length maximum simple path in an acyclic | |
| graph has degree one. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| acycgrwspthmaxendvd1 $p |- ( | |
| ( ( G e. UPGraph /\ G e. AcyclicGraph /\ | |
| W e. ( N WSPathsN G ) ) /\ | |
| ( N e. NN /\ | |
| A. m A. w ( w e. ( m WSPathsN G ) -> m <_ N ) ) ) -> | |
| ( ( VtxDeg ` G ) ` ( lastS ` W ) ) = 1 ) $= | |
| ( cupgr wcel cacycgr cwwspthsn co cv wal wa cfv chash c1 wceq jca syl cvv | |
| w3a cn cle wbr wi clsw cvtxdg cnbgr cmin cusgr simpl1 simpl2 upgracycusgr | |
| csn cvtx simpl3 wspthnlswvtx eqid hashnbusgrvd eqcomd acycgrwspthmaxendnb | |
| fveq2d fvexd hashsng 3eqtrd ) CFGZCHGZEDCIJGZUADUBGAKBKZCIJGVIDUCUDUEALBL | |
| MZMZEUFNZCUGNNZCVLUHJZONZDPUIJZENZUNZONZPVKVOVMVKCUJGZVLCUONZGZMVOVMQVKVT | |
| WBVKVFVGMVTVKVFVGVFVGVHVJUKVFVGVHVJULRCUMSVKVHWBVFVGVHVJUPCDEUQSRVLCWAWAU | |
| RUSSUTVKVNVROABCDEVAVBVKVQTGVSPQVKVPEVCVQTVDSVE $. | |
| $} | |
| ${ | |
| $d G m w $. $d N m w $. $d V m w $. $d W m w $. $d m w $. | |
| treewspthmaxendvd1.v $e |- V = ( Vtx ` G ) $. | |
| $( The final vertex of a maximum simple path in a tree with at least two | |
| vertices has degree one. (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| treewspthmaxendvd1 $p |- ( | |
| ( ( G e. Tree /\ 1 < ( # ` V ) /\ | |
| W e. ( N WSPathsN G ) ) /\ | |
| A. m A. w ( w e. ( m WSPathsN G ) -> m <_ N ) ) -> | |
| ( ( VtxDeg ` G ) ` ( lastS ` W ) ) = 1 ) $= | |
| ( wcel c1 cfv wbr cwwspthsn co w3a cv wal wa syl 3jca jca ctree chash clt | |
| cle wi cupgr cacycgr cn clsw cvtxdg wceq simpl1 treeacycgr simpl3 cconngr | |
| treeupgr treeconngr simpl2 simpr acycgrwspthmaxnn acycgrwspthmaxendvd1 ) | |
| CUAHZIEUBJUCKZFDCLMHZNZAOBOZCLMHVFDUDKUEAPBPZQZCUFHZCUGHZVDNZDUHHZVGQZQFU | |
| IJCUJJJIUKVHVKVMVHVIVJVDVHVBVIVBVCVDVGULZCUPRZVHVBVJVNCUMRZVBVCVDVGUNZSVH | |
| VLVGVHVICUOHZVJNZVCVDQZQZVGQVLVHWAVGVHVSVTVHVIVRVJVOVHVBVRVNCUQRVPSVHVCVD | |
| VBVCVDVGURVQTTVEVGUSZTABCDEFGUTRWBTTABCDFVAR $. | |
| $} | |
| ${ | |
| $d G m p v w $. $d n m p v w $. $d V m p v w $. $d m p v w $. | |
| $d p v w $. | |
| treewspthmaxleaf.v $e |- V = ( Vtx ` G ) $. | |
| $( A maximum simple path in a tree with at least two vertices supplies a | |
| vertex of degree one. (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| treewspthmaxleaf $p |- ( | |
| ( ( G e. Tree /\ 1 < ( # ` V ) ) /\ | |
| ( p e. ( n WSPathsN G ) /\ | |
| A. m A. w ( w e. ( m WSPathsN G ) -> m <_ n ) ) ) -> | |
| E. v e. V ( ( VtxDeg ` G ) ` v ) = 1 ) $= | |
| ( ctree wcel c1 cfv wbr wa cv cwwspthsn co wal wceq syl clt cle wi cvtxdg | |
| chash clsw id fveq2d eqeq1d cvtx simprl wspthnlswvtx eleq2i sylibr simpll | |
| w3a simplr 3jca simprr jca treewspthmaxendvd1 rspcedvdw ) EIJZKFUELUAMZNZ | |
| GOZDOZEPQJZAOCOZEPQJVIVGUBMUCARCRZNZNZBOZEUDLZLZKSVFUFLZVNLZKSZBVPFVMVPSZ | |
| VOVQKVSVMVPVNVSUGUHUIVLVPEUJLZJZVPFJVLVHWAVEVHVJUKZEVGVFULTFVTVPHUMUNVLVC | |
| VDVHUPZVJNVRVLWCVJVLVCVDVHVCVDVKUOVCVDVKUQWBURVEVHVJUSUTACEVGFVFHVATVB $. | |
| $( If a tree with at least two vertices has a maximum simple path, then it | |
| has a vertex of degree one. (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| treewspthmaxleafex $p |- ( ( G e. Tree /\ 1 < ( # ` V ) ) -> | |
| ( E. p ( p e. ( n WSPathsN G ) /\ | |
| A. m A. w ( w e. ( m WSPathsN G ) -> m <_ n ) ) -> | |
| E. v e. V ( ( VtxDeg ` G ) ` v ) = 1 ) ) $= | |
| ( ctree wcel c1 chash cfv clt wbr wa cv cwwspthsn co wal cle wi wceq wrex | |
| cvtxdg treewspthmaxleaf ex exlimdv ) EIJKFLMNOPZGQDQZERSJAQCQZERSJUKUJUAO | |
| UBATCTPZBQEUEMMKUCBFUDZGUIULUMABCDEFGHUFUGUH $. | |
| $} | |
| ${ | |
| $d G a k m n p v w $. $d L a m p v w $. $d V a m p v w $. | |
| $d a k n p v w $. $d k L $. $d k n $. $d k V $. $d L n $. | |
| $d m n p v w $. $d V n $. $d n p v $. $d p v w $. | |
| treeleaf.v $e |- V = ( Vtx ` G ) $. | |
| treeleaf.l $e |- L = { n e. ( 0 ... ( # ` V ) ) | | |
| ( n WSPathsN G ) =/= (/) } $. | |
| $( A finite tree with at least two vertices has a vertex of degree one. | |
| This follows Observation in [Diestel] p. 14. | |
| (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| treeleaf $p |- ( ( G e. Tree /\ V e. Fin /\ 1 < ( # ` V ) ) -> | |
| E. v e. V ( ( VtxDeg ` G ) ` v ) = 1 ) $= | |
| ( va vp vw vm ctree wcel c1 cfv wbr cv c0 syl wa cfn chash clt w3a cvtxdg | |
| wceq wrex cvtx wne simp1 treevtxne0 neeq1i sylibr cwwspthsn co cle wi wal | |
| wex cvv simpl1 elex simpl2 simpr 3jca wspthmaxw simpll treewspthmaxleafex | |
| simp3 jca simprr sylc rexlimddv n0limd ) CLMZEUAMZNEUBOUCPZUDZAQCUEOONUFA | |
| EUGZHEVRCUHOZRUIZERUIVRVOWAVOVPVQUJZCUKSEVTRFULUMVRHQZEMZTZIQBQZCUNUOMJQK | |
| QZCUNUOMWGWFUPPUQJURKURTIUSZVSBDWECUTMZVPWDUDWHBDUGWEWIVPWDWEVOWIVOVPVQWD | |
| VACLVBSVOVPVQWDVCVRWDVDVEJWCKBCDEIFGVFSWEWFDMZWHTZTZVOVQTZWHVSWLVRWMVRWDW | |
| KVGVRVOVQWBVOVPVQVIVJSWEWJWHVKJAKBCEIFVHVLVMVN $. | |
| $} | |
| ${ | |
| $d B i $. $d G i $. $d N i $. $d W i $. | |
| $( An internal vertex of a fixed-length simple path in a simple graph has | |
| two distinct neighbors on the path. (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| usgrwspthintnb $p |- ( ( G e. USGraph /\ | |
| W e. ( N WSPathsN G ) /\ B e. ( 1 ... ( N - 1 ) ) ) -> | |
| ( ( W ` ( B - 1 ) ) e. ( G NeighbVtx ( W ` B ) ) /\ | |
| ( W ` ( B + 1 ) ) e. ( G NeighbVtx ( W ` B ) ) /\ | |
| ( W ` ( B - 1 ) ) =/= ( W ` ( B + 1 ) ) ) ) $= | |
| ( vi wcel co c1 cfz w3a cfv caddc cpr cc0 wceq syl cn0 wa 3syl cr cmin cv | |
| cusgr cwwspthsn cnbgr wne cedg cfzo wral simp2 cwwlksn wspthsswwlkn sseli | |
| cvtx cword chash eqid wwlknp simp3 fzelp1 cc wspthnn0 nn0cn npcan1 oveq2d | |
| eleqtrd fz1fzo0m1 simpr fveq2d oveq1d cz simpl elfzelz zcn preq12d eleq1d | |
| eqtrd rspcdv mpd wb simp1 nbusgreledg mpbird prcom a1i cuz elfznn elnn0uz | |
| nnnn0 biimpi nn0z zre elfzel2 nn0re cle wbr elfzle2 ltm1d lelttrd eqeltrd | |
| cn elfzod cdm wf1 wspthnf1m nnm1nn0 peano2rem lem1d letrd elfz2nn0 sylibr | |
| 3jca wfn wspthnfn fndm eleqtrrd peano2nn clt zltp1le bicomd zltlem1 bitrd | |
| jca peano2re ltp1d lttrd ltned dff14i ) BUCFZDCBUDGZFZAHCHUAGZIGFZJZAHUAG | |
| ZDKZBADKZUEGZFZAHLGZDKZYRFZYPUUAUFZYNYSYPYQMZBUGKZFZYNEUBZDKZUUGHLGZDKZMZ | |
| UUEFZENCUHGZUIZUUFYNYKUUNYIYKYMUJZYKDCBUKGZFZUUNYJUUPDBCULUMZUUQDBUNKZUOF | |
| ZDUPKCHLGOZUUNJUUNEUUEBCUUSDUUSUQUUEUQZURUUTUVAUUNUSPZPPYNUULUUFEYOUUMYNA | |
| HCIGZFYOUUMFYNAHYLHLGZIGZUVDYNYMAUVFFYIYKYMUSZAHYLUTPYNUVECHIYNCVAFZUVECO | |
| YNCQFZUVHYNYKUVIUUOBCDVBZPZCVCPCVDPVEVFACVGPYNUUGYOOZRZUUKUUDUUEUVMUUHYPU | |
| UJYQUVMUUGYODYNUVLVHZVIUVMUUIADUVMUUIYOHLGZAUVMUUGYOHLUVNVJUVMAVAFZUVOAOU | |
| VMAVKFZUVPUVMYMUVQUVMYNYMYNUVLVLUVGPAHYLVMZPAVNPAVDPVQVIVOVPVRVSYNYIYSUUF | |
| VTYIYKYMWAZUUEBYQYPUVBWBPWCYNUUBUUAYQMZUUEFZYNUVTYQUUAMZUUEUVTUWBOYNUUAYQ | |
| WDWEYNUUNUWBUUEFZYNUUQUUNYNYKUUQUUOUURPUVCPYNUULUWCEAUUMYNANCYNAQFZANWFKF | |
| ZYNAXAFZUWDYNYMUWFUVGAYLWGZPAWIPUWDUWEAWHWJPYNYKUVICVKFZUUOUVJCWKSZYNAYLC | |
| YNYMUVQATFZUVGUVRAWLSZYNYMYLVKFYLTFUVGAHYLWMYLWLSZYNYKUVICTFUUOUVJCWNSZYN | |
| YMAYLWOWPZUVGAHYLWQPZYNCUWMWRWSXBYNUUGAOZRZUUKUWBUUEUWQUUHYQUUJUUAUWQUUGA | |
| DYNUWPVHZVIUWQUUIYTDUWQUUGAHLUWRVJVIVOVPVRVSWTYNYIUUBUWAVTUVSUUEBYQUUAUVB | |
| WBPWCYNDXCZUUSDXDZYOUWSFZYTUWSFZYOYTUFZJZRUUCYNUWTUXDYNYKUWTUUOBCDXEPYNUX | |
| AUXBUXCYNYONCIGZUWSYNYOQFZUVIYOCWOWPZJYOUXEFYNUXFUVIUXGYNYMUWFUXFUVGUWGAX | |
| FSUVKYNYOACYNUWJYOTFUWKAXGPZUWKUWMYNAUWKXHYNAYLCUWKUWLUWMUWOYNCUWMXHXIXIX | |
| LYOCXJXKYNYKDUXEXMUWSUXEOUUOBCDXNUXEDXOSZXPYNYTUXEUWSYNYTQFZUVIYTCWOWPZJY | |
| TUXEFYNUXJUVIUXKYNYTXAFZUXJYNYMUWFUXLUVGUWGAXQSYTWIPUVKYNUXKUWNUWOYNUXKAC | |
| XRWPZUWNYNUXMUXKYNUVQUWHRZUXMUXKVTYNUVQUWHYNYMUVQUVGUVRPUWIYCZACXSPXTYNUX | |
| NUXMUWNVTUXOACYAPYBWCXLYTCXJXKUXIXPYNYOYTUXHYNYOAYTUXHUWKYNUWJYTTFUWKAYDP | |
| YNAUWKWRYNAUWKYEYFYGXLYCUWSUUSDYOYTYHPXL $. | |
| $} | |
| ${ | |
| $d B i $. $d G i $. $d N i $. $d W i $. | |
| $( An internal vertex of a fixed-length simple path in a simple graph has | |
| degree at least two. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| usgrwspthintvdge2 $p |- ( ( G e. USGraph /\ | |
| W e. ( N WSPathsN G ) /\ B e. ( 1 ... ( N - 1 ) ) ) -> | |
| 2 <_ ( ( VtxDeg ` G ) ` ( W ` B ) ) ) $= | |
| ( cusgr wcel cwwspthsn co c1 cmin cfz w3a c2 cfv cnbgr cvtxdg cle cvv syl | |
| chash caddc ovex a1i wne usgrwspthintnb simp1d simp2d simp3d nehash2 cvtx | |
| wceq simp1 nbgrsym sylibr eqid nbgrisvtx jca hashnbusgrvd eqcomd breqtrrd | |
| wa ) BEFZDCBGHFZAICIJHKHFZLZMBADNZOHZTNZVFBPNNZQVEAIJHDNZAIUAHDNZVGRVGRFV | |
| EBVFOUBUCVEVJVGFZVKVGFZVJVKUDZABCDUEZUFZVEVLVMVNVOUGVEVLVMVNVOUHUIVEVHVIV | |
| EVBVFBUJNZFZVAVHVIUKVEVBVRVBVCVDULVEVFBVJOHFZVRVEVLVSVPBVJVFUMUNBVJVFVQVQ | |
| UOZUPSUQVFBVQVTURSUSUT $. | |
| $} | |
| ${ | |
| $( A degree-one vertex cannot occur internally on a fixed-length simple | |
| path in a simple graph. (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| usgrwspthvd1nint $p |- ( ( G e. USGraph /\ | |
| W e. ( N WSPathsN G ) /\ | |
| ( ( VtxDeg ` G ) ` ( W ` B ) ) = 1 ) -> | |
| -. B e. ( 1 ... ( N - 1 ) ) ) $= | |
| ( cusgr wcel cwwspthsn co cfv cvtxdg c1 wceq w3a cmin cfz wa cle wbr clt | |
| c2 wn 1lt2 1re 2re ltnlei mpbi simpl1 simpl2 simpr 3jca usgrwspthintvdge2 | |
| syl simpl3 eqcomd breqtrrd mto imnani ) BEFZDCBGHFZADIBJIIZKLZMZAKCKNHOHF | |
| ZVBVCPZTKQRZKTSRVEUAUBKTUCUDUEUFVDTUTKQVDURUSVCMTUTQRVDURUSVCURUSVAVCUGUR | |
| USVAVCUHVBVCUIUJABCDUKULVDUTKURUSVAVCUMUNUOUPUQ $. | |
| $} | |
| $( A positive index in a half-open integer range belongs to the corresponding | |
| closed range ending at the predecessor. (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| elfzo1elfzm1 $p |- ( K e. ( 1 ..^ N ) -> | |
| K e. ( 1 ... ( N - 1 ) ) ) $= | |
| ( c1 cfzo co wcel cmin cz 1z a1i elfzoel2 mpdan elfzoelz elfzole1 elfzolem1 | |
| zsubcl syl elfzd ) ACBDEFZACBCGEZCHFZSIJSBHFZTHFZACBKUBUAUCUAUBIJBCPLQACBMA | |
| CBNACBOR $. | |
| $( A degree-one vertex cannot occur internally in the vertex sequence of a | |
| simple path in a simple graph. (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| usgrspthvd1nint $p |- ( ( G e. USGraph /\ F ( SPaths ` G ) P /\ | |
| ( ( VtxDeg ` G ) ` ( P ` B ) ) = 1 ) -> | |
| -. B e. ( 1 ... ( ( # ` F ) - 1 ) ) ) $= | |
| ( cusgr wcel cspths cfv wbr cvtxdg c1 wceq w3a chash cwwspthsn cmin cfz syl | |
| co wn simp1 cupgr usgrupgr simp2 jca spthwspthn simp3 3jca usgrwspthvd1nint | |
| wa ) DEFZCBDGHIZABHDJHHKLZMZUKBCNHZDOSFZUMMAKUOKPSQSFTUNUKUPUMUKULUMUAZUNDU | |
| BFZULUJUPUNURULUNUKURUQDUCRUKULUMUDUEBCDUFRUKULUMUGUHADUOBUIR $. | |
| $( A degree-one vertex cannot occur at a positive nonfinal index in the | |
| vertex sequence of a simple path in a simple graph. (Contributed by | |
| Mingli Yuan, 12-Aug-2026.) $) | |
| usgrspthvd1nintfzo $p |- ( ( G e. USGraph /\ F ( SPaths ` G ) P /\ | |
| ( ( VtxDeg ` G ) ` ( P ` B ) ) = 1 ) -> | |
| -. B e. ( 1 ..^ ( # ` F ) ) ) $= | |
| ( cusgr wcel cspths cfv wbr cvtxdg c1 wceq w3a chash co cfz usgrspthvd1nint | |
| cmin cfzo elfzo1elfzm1 nsyl ) DEFCBDGHIABHDJHHKLMAKCNHZKROPOFAKUBSOFABCDQAU | |
| BTUA $. | |
| $( A degree-one vertex on a simple path in a simple graph occurs only at an | |
| endpoint of the path. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| usgrspthvd1endidx $p |- ( ( ( G e. USGraph /\ F ( SPaths ` G ) P /\ | |
| ( ( VtxDeg ` G ) ` ( P ` B ) ) = 1 ) /\ | |
| B e. ( 0 ... ( # ` F ) ) ) -> | |
| ( B = 0 \/ B = ( # ` F ) ) ) $= | |
| ( cusgr wcel cspths cfv wbr cvtxdg c1 wceq w3a cc0 chash cfz co wa wo cfzo | |
| wn usgrspthvd1nintfzo adantr wb simpr elfznelfzob syl bicomd mpbird ) DEFCB | |
| DGHIABHDJHHKLMZANCOHZPQFZRZANLAUKLSZAKUKTQFUAZUJUOULABCDUBUCUMUOUNUMULUOUNU | |
| DUJULUEUKAUFUGUHUI $. | |
| $( A path between two vertices in an acyclic graph is a simple path between | |
| the same vertices. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| pthonacycspthon $p |- ( ( G e. AcyclicGraph /\ | |
| F ( K ( PathsOn ` G ) M ) P ) -> | |
| F ( K ( SPathsOn ` G ) M ) P ) $= | |
| ( cacycgr wcel cpthson cfv co wbr wa cspthson ctrlson cspths pthontrlon jca | |
| cvv w3a syl adantl cpths pthonispth pthacycspth sylan2 cvtx eqid pthsonprop | |
| wb simpr simp1 3simpc simp2 isspthson mpbird ) CFGZBADECHIJKZLZBADECMIJKZBA | |
| DECNIJKZBACOIKZLZURUTVAUQUTUPDEABCPUAUQUPBACUBIKZVADEABCUCABCUDUEQURDCUFIZG | |
| ZEVDGZLZBRGARGLZLZUSVBUIURUQVIUPUQUJUQCRGZVEVFSZVHUTVCLZSZVIDEABCVDVDUGZUHV | |
| MVGVHVMVKVGVKVHVLUKVJVEVFULTVKVHVLUMQTTDEARBCVDRVNUNTUO $. | |
| ${ | |
| $d G f k m p $. $d K f k m p $. $d M f k m p $. $d V f k m p $. | |
| $d f k m p $. | |
| conngrpthon.v $e |- V = ( Vtx ` G ) $. | |
| $( Any two vertices of a connected graph are joined by a path. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| conngrpthon $p |- ( ( G e. ConnGraph /\ K e. V /\ M e. V ) -> | |
| E. f E. p f ( K ( PathsOn ` G ) M ) p ) $= | |
| ( vk vm cconngr wcel wa cv co wbr wex wral cvv syl wceq w3a cpthson simp1 | |
| cfv 3simpc wb elexd isconngr mpbid id oveq1d breqd 2exbidv oveq2d rspc2va | |
| jca ) BJKZCEKZDEKZUAZURUSLZAMZFMZHMZIMZBUBUDZNZOZFPAPZIEQHEQZLVBVCCDVFNZO | |
| ZFPAPZUTVAVJUQURUSUEUTUQVJUQURUSUCZUTBRKUQVJUFUTBJVNUGAHIBERFGUHSUIUPVIVM | |
| VBVCCVEVFNZOZFPAPHICDEEVDCTZVHVPAFVQVGVOVBVCVQVDCVEVFVQUJUKULUMVEDTZVPVLA | |
| FVRVOVKVBVCVRVEDCVFVRUJUNULUMUOS $. | |
| $} | |
| ${ | |
| $d f p $. $d f F $. $d f G $. $d f K $. $d f M $. $d f P $. $d p F $. | |
| $d p G $. $d p K $. $d p M $. $d p P $. | |
| $( A displayed path supplies the existential witnesses used in the | |
| definition of connectedness. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| pthonex $p |- ( F ( K ( PathsOn ` G ) M ) P -> | |
| E. f E. p f ( K ( PathsOn ` G ) M ) p ) $= | |
| ( cpthson cfv co wbr cvv wcel wa cv wex cvtx w3a ctrlson cpths pthsonprop | |
| eqid simp2 syl id breq12 spc2egv sylc ) CAEFDHIJZKZCLMALMNZUJBOZGOZUIKZGP | |
| BPUJDLMEDQIZMFUOMRZUKCAEFDSIJKCADTIKNZRUKEFACDUOUOUBUAUPUKUQUCUDUJUEUNUJB | |
| GCALLULCUMAUIUFUGUH $. | |
| $} | |
| ${ | |
| $d A m n $. $d F m n $. $d G m n $. $d N m n $. $d P m n $. | |
| $d U m n $. $d V m n $. | |
| pthcutidx.v $e |- V = ( Vtx ` G ) $. | |
| $( A path from a set to a vertex outside the set has two consecutive | |
| vertices on opposite sides of the set. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| pthcutidx $p |- ( ( ( A C_ V /\ U e. A /\ N e. ( V \ A ) ) /\ | |
| F ( U ( PathsOn ` G ) N ) P ) -> | |
| E. m e. ( 0 ..^ ( # ` F ) ) | |
| ( ( P ` m ) e. A /\ ( P ` ( m + 1 ) ) e. ( V \ A ) ) ) $= | |
| ( wcel cfv co wa cc0 wnel cn0 syl wb a1i jca vn wss w3a cpthson wbr chash | |
| cdif cv cfz wn c1 caddc wrex cfzo wceq eleq1 fveq2 eqidd neleq12d anbi12d | |
| weq cwlkson ctrlson simpr pthontrlon trlsonwlkon wlkonendpts simpl simpl2 | |
| 3syl eqeltrd nnel bicomi mpbid intnand wne cwlks wlkoniswlk simpl3 eldifn | |
| wlkcl df-nel mpbird elnelne2 wlkon2n0 elnnne0 nn0fz0 rspcedvdw nn0min nfv | |
| cn nfre1 simprrr cuz simprl nn0uz eleqtrd fzofzp1b simprrl simplr simplrl | |
| simprr elfzofz mtand wi wlkpvtx mpd eldifd 3impb 3exp rspe syl6d rexlimd | |
| ) AHUBZCAJZGHAUGZJZUCZEBCGFUDKLUEZMZDUHZNEUFKZUILZJZYABKZAOZMZUJZYAUKULLZ | |
| YCJZYIBKZAOZMZMZDPUMYEAJZYKXPJZMZDNYBUNLZUMZXTUAUHZYCJZYTBKZAOZMZNYCJZNBK | |
| ZAOZMYGYMDUAYTNUOZUUAUUEUUCUUGYTNYCUPUUHUUBUUFAAYTNBUQUUHAURUSUTUADVAZUUA | |
| YDUUCYFYTYAYCUPUUIUUBYEAAYTYABUQUUIAURUSUTYTYIUOZUUAYJUUCYLYTYIYCUPUUJUUB | |
| YKAAYTYIBUQUUJAURUSUTXTUUGUUEXTUUFAJZUUGUJZXTUUFCAXTEBCGFVBKLUEZUUFCUOZXT | |
| XSEBCGFVCKLUEZUUMXRXSVDCGBEFVEZCGBEFVFZVJZUUMUUNYBBKZGUOZMZUUNBEFCGVGZUUN | |
| UUTVHQQXNXOXQXSVIZVKUUKUULRXTUULUUKUUFAVLVMSVNVOXTUUDYBYCJZUUSAOZMUAYBWKY | |
| TYBUOZUUAUVDUUCUVEYTYBYCUPUVFUUBUUSAAYTYBBUQUVFAURUSUTXTYBWKJZYBPJZYBNVPZ | |
| MZXTUVHUVIXTUUMEBFVQKUEZUVHUURCGBEFVRZBEFWAVJZXTUUMCGVPZMUVIXTUUMUVNUURXT | |
| XOGAOZMUVNXTXOUVOUVCXTUVOGAJUJZXTXQUVPXNXOXQXSVSGHAVTQUVOUVPRXTGAWBSWCZTC | |
| GAWDQTCGBEFWEQTUVGUVJRXTYBWFSWCXTUVDUVEXTUVHUVDUVMUVHUVDRXTYBWGSVNXTUVEUV | |
| OUVQXTUUSGAAXTUUMUUTUURUUMUVAUUTUVBUUNUUTVDQQXTAURUSWCTWHWIXTYNYSDPXTDWJY | |
| QDYRWLXTYAPJZYNYAYRJZYQMZYSXTUVRYNUVTXTUVRYNUVTXTUVRYNMZMZUVSYQUWBUVSYJUW | |
| BYMYJXTUVRYHYMWMZYJYLVHZQZUWBYANWNKZJZUVSYJRZUWBYAPUWFXTUVRYNWOPUWFUOZUWB | |
| WPSWQNYBYAWRZQWCUWBYOYPUWBYFUJZYOUWBYFYGXTUVRYHYMWSUWBYFMZYDYFUWLUVSYDUWL | |
| UVSYJUWLYMYJUWLUWAYMXTUWAYFWTUVRYHYMXBQUWDQUWLUWGUWHUWLYAPUWFXTUVRYNYFXAU | |
| WIUWLWPSWQUWJQWCYANYBXCQUWBYFVDTXDUWKYORUWBYEAVLSVNUWBYKHAUWBYJYKHJZUWEUW | |
| BUVKYJUWMXEUWBXSUUMUVKXRXSUWAWTXSUUOUUMUUPUUQQUVLVJBEFYIHIXFQXGUWBYLYKAJU | |
| JZUWBYMYLUWCYJYLVDQYLUWNRUWBYKAWBSVNXHTTXIXJUVTYSXEXTYQDYRXKSXLXMXG $. | |
| $} | |
| ${ | |
| $d A m $. $d E m $. $d F m $. $d G m $. $d N m $. $d P m $. $d U m $. | |
| $d V m $. $d E k $. $d F k $. $d G k $. $d P k $. $d k m $. | |
| pthcutiedg.v $e |- V = ( Vtx ` G ) $. | |
| pthcutiedg.e $e |- E = ( iEdg ` G ) $. | |
| $( A path crossing a vertex cut contains an indexed edge whose endpoints | |
| lie on opposite sides of the cut. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| pthcutiedg $p |- ( ( G e. UPGraph /\ | |
| ( ( A C_ V /\ U e. A /\ N e. ( V \ A ) ) /\ | |
| F ( U ( PathsOn ` G ) N ) P ) ) -> | |
| E. m e. ( 0 ..^ ( # ` F ) ) | |
| ( ( ( P ` m ) e. A /\ ( P ` ( m + 1 ) ) e. ( V \ A ) ) /\ | |
| ( ( F ` m ) e. dom E /\ | |
| ( E ` ( F ` m ) ) = { ( P ` m ) , ( P ` ( m + 1 ) ) } ) ) ) $= | |
| ( vk wcel cfv co wbr wa c1 syl fveq2d cupgr wss cdif w3a cpthson cv caddc | |
| cc0 chash cfzo wrex cdm cpr wceq simpr pthcutidx ctrlson cwlks simpl 4syl | |
| cwlkson pthontrlon trlsonwlkon wlkoniswlk wlkiedgdomd weq preq12d eqeq12d | |
| simplr id oveq1d wral simplll simpllr 3syl upgrwlkedg rspcdva ex reximdva | |
| jca mpd ) GUAMZAIUBCAMHIAUCZMUDZFBCHGUENOPZQZQZDUFZBNZAMWHRUGOZBNZWCMQZDU | |
| HFUINUJOZUKZWLWHFNZEULMZWOENZWIWKUMZUNZQZQZDWMUKWGWFWNWBWFUOZABCDFGHIJUPS | |
| WGWLXADWMWGWHWMMZQZWLXAXDWLQZWLWTXDWLUOXEWPWSXEBEFGWHKXEWEFBCHGUQNOPZFBCH | |
| GVANOPZFBGURNPZXEXDWGWFWEXDWLUSWGXCUSXBWDWEUOZUTCHBFGVBZCHBFGVCZCHBFGVDZU | |
| TWGXCWLVIZVEXELUFZFNZENZXNBNZXNRUGOZBNZUMZUNZWSLWMWHLDVFZXPWQXTWRYBXOWOEY | |
| BXNWHFYBVJZTTYBXQWIXSWKYBXNWHBYCTYBXRWJBYBXNWHRUGYCVKTVGVHXEWBXHQYALWMVLX | |
| EWBXHWBWFXCWLVMXEWEXHXEWFWEWBWFXCWLVNXISWEXFXGXHXJXKXLVOSVTBLFGEKVPSXMVQV | |
| TVTVRVSWA $. | |
| $} | |
| ${ | |
| $d A j n u $. $d B j n u $. $d E j n u $. $d J j n u $. $d N j n u $. | |
| $d U j n u $. $d j n u $. | |
| $( An indexed edge with specified endpoints supplies the corresponding | |
| restricted existential witnesses. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| cutiedgex $p |- ( ( ( U e. A /\ N e. B ) /\ | |
| ( J e. dom E /\ ( E ` J ) = { U , N } ) ) -> | |
| E. j e. dom E E. u e. A E. n e. B | |
| ( E ` j ) = { u , n } ) $= | |
| ( wcel wa cfv cpr wceq cv wrex simpr eqeq2d rspcedv mpd cdm simprr simplr | |
| preq2d simpll preq1d rexbidv simprl fveq2d eqeq1d 2rexbidv ) DBJZICJZKZHG | |
| UAZJZHGLZDIMZNZKZKZUQAOZFOZMZNZFCPZABPZEOZGLZVDNZFCPABPZEUOPVAUQDVCMZNZFC | |
| PZVGVAUSVNUNUPUSUBVAVMUSFICULUMUTUCVAVCINZKZVLURUQVPVCIDVAVOQUDRSTVAVFVNA | |
| DBULUMUTUEVAVBDNZKZVEVMFCVRVDVLUQVRVBDVCVAVQQUFRUGSTVAVKVGEHUOUNUPUSUHVAV | |
| HHNZKZVJVEAFBCVTVIUQVDVTVHHGVAVSQUIUJUKST $. | |
| $} | |
| ${ | |
| $d A m $. $d E m $. $d F m $. $d G m $. $d N m $. $d P m $. $d U m $. | |
| $d V m $. $d A j n u $. $d E j n u $. $d F j n u $. $d P j n u $. | |
| $d V j n u $. $d j m n u $. | |
| pthcutiedgex.v $e |- V = ( Vtx ` G ) $. | |
| pthcutiedgex.e $e |- E = ( iEdg ` G ) $. | |
| $( A path crossing a vertex cut supplies an edge with one endpoint on | |
| each side of the cut. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| pthcutiedgex $p |- ( ( G e. UPGraph /\ | |
| ( ( A C_ V /\ U e. A /\ N e. ( V \ A ) ) /\ | |
| F ( U ( PathsOn ` G ) N ) P ) ) -> | |
| E. j e. dom E E. u e. A E. n e. ( V \ A ) | |
| ( E ` j ) = { u , n } ) $= | |
| ( vm wcel cfv co wa cv wrex cupgr wss cdif w3a cpthson wbr caddc cdm wceq | |
| c1 cpr cc0 chash cfzo pthcutiedg simp3 cutiedgex syl 3exp rexlimdv mpd ) | |
| IUAOBKUBDBOJKBUCZOUDHCDJIUEPQUFRRZNSZCPZBOVDUJUGQCPZVBORVDHPZGUHZOVGGPVEV | |
| FUKUIRRZNULHUMPUNQZTESGPASFSUKUIFVBTABTEVHTZBCDNGHIJKLMUOVCVIVKNVJVCVDVJO | |
| ZVIVKVCVLVIUDVIVKVCVLVIUPABVBVEEFGVGVFUQURUSUTVA $. | |
| $} | |
| ${ | |
| $d A f p $. $d E f p $. $d G f p $. $d N f p $. $d U f p $. | |
| $d V f p $. $d f j n p u $. $d A j n u $. $d E j n u $. $d V j n u $. | |
| conngrcutiedgpt.v $e |- V = ( Vtx ` G ) $. | |
| conngrcutiedgpt.e $e |- E = ( iEdg ` G ) $. | |
| $( In a connected graph, two specified vertices on opposite sides of a | |
| cut are joined through an edge crossing the cut. (Contributed by | |
| Mingli Yuan, 13-Aug-2026.) $) | |
| conngrcutiedgpt $p |- ( ( ( G e. UPGraph /\ G e. ConnGraph ) /\ | |
| ( A C_ V /\ U e. A /\ N e. ( V \ A ) ) ) -> | |
| E. j e. dom E E. u e. A E. n e. ( V \ A ) | |
| ( E ` j ) = { u , n } ) $= | |
| ( vf vp wcel wa w3a cv cfv wex wrex cupgr cconngr wss cdif cpthson co wbr | |
| cpr wceq cdm simplr simpr1 simpr2 sseldd eldifad 3jca conngrpthon simplll | |
| simpr3 syl simpr jca pthcutiedgex ex exlimdvv mpd ) GUANZGUBNZOZBIUCZCBNZ | |
| HIBUDZNZPZOZLQZMQZCHGUERUFUGZMSLSZDQFRAQEQUHUIEVLTABTDFUJTZVOVHCINZHINZPV | |
| SVOVHWAWBVGVHVNUKVOBICVIVJVKVMULVIVJVKVMUMUNVOHIBVIVJVKVMUSUOUPLGCHIMJUQU | |
| TVOVRVTLMVOVRVTVOVROZVGVNVROZOVTWCVGWDVGVHVNVRURWCVNVRVIVNVRUKVOVRVAVBVBA | |
| BVQCDEFVPGHIJKVCUTVDVEVF $. | |
| $} | |
| ${ | |
| $d A x y $. $d E x y $. $d G x y $. $d V x y $. $d j n u x y $. | |
| $d x y $. $d A j n u $. $d E j n u $. $d V j n u $. | |
| conngrcutiedg.v $e |- V = ( Vtx ` G ) $. | |
| conngrcutiedg.e $e |- E = ( iEdg ` G ) $. | |
| $( Every nontrivial vertex cut of a connected graph is crossed by an | |
| indexed edge. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| conngrcutiedg $p |- ( ( ( G e. UPGraph /\ G e. ConnGraph ) /\ | |
| ( A C_ V /\ A =/= (/) /\ A =/= V ) ) -> | |
| E. j e. dom E E. u e. A E. n e. ( V \ A ) | |
| ( E ` j ) = { u , n } ) $= | |
| ( vx vy wcel wa c0 wne w3a cv wex wrex n0 cupgr cconngr wss cfv wceq cdif | |
| cpr cdm simpr2 sylib wi simpr1 simpr3 jca pssdifn0 syl simp1l simp3 simp2 | |
| simp1r1 3jca conngrcutiedgpt 3exp exlimdv mpd ) FUALFUBLMZBGUCZBNOZBGOZPZ | |
| MZJQZBLZJRZCQEUDAQDQUGUEDGBUFZSABSCEUHSZVKVHVNVFVGVHVIUIJBTUJVKVMVPJVKKQZ | |
| VOLZKRZVMVPUKZVKVONOZVSVKVGVIMWAVKVGVIVFVGVHVIULVFVGVHVIUMUNBGUOUPKVOTUJV | |
| KVRVTKVKVRVMVPVKVRVMPZVFVGVMVRPZMVPWBVFWCVFVJVRVMUQWBVGVMVRVGVHVIVFVRVMUT | |
| VKVRVMURVKVRVMUSVAUNABVLCDEFVQGHIVBUPVCVDVEVDVE $. | |
| $} | |
| $( The vertex set of a subgraph is a subclass of that of the ambient graph. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| subgrvtxss $p |- ( T SubGraph G -> | |
| ( Vtx ` T ) C_ ( Vtx ` G ) ) $= | |
| ( csubgr wbr cvtx cfv wss ciedg cedg cpw w3a eqid subgrprop2 simp1 syl ) AB | |
| CDAEFZBEFZGZAHFZBHFZGZAIFZPJGZKRQTAUBBSPPLQLSLTLUBLMRUAUCNO $. | |
| $( The indexed-edge function of a subgraph is a subclass of that of the | |
| ambient graph. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| subgriedgss $p |- ( T SubGraph G -> | |
| ( iEdg ` T ) C_ ( iEdg ` G ) ) $= | |
| ( csubgr wbr cvtx cfv wss ciedg cedg cpw w3a eqid subgrprop2 simp2 syl ) AB | |
| CDAEFZBEFZGZAHFZBHFZGZAIFZPJGZKUAQTAUBBSPPLQLSLTLUBLMRUAUCNO $. | |
| $( An indexed edge retained by a subgraph has the same value in the | |
| subgraph and the ambient hypergraph. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| subgriedgfv $p |- ( ( G e. UHGraph /\ T SubGraph G /\ | |
| J e. dom ( iEdg ` T ) ) -> | |
| ( ( iEdg ` T ) ` J ) = ( ( iEdg ` G ) ` J ) ) $= | |
| ( cuhgr wcel csubgr wbr ciedg cfv cdm w3a wfun wss wceq simp1 uhgrfun simp2 | |
| eqid syl subgriedgss simp3 3jca funssfv eqcomd ) BDEZABFGZCAHIZJEZKZCBHIZIZ | |
| CUGIZUIUJLZUGUJMZUHKUKULNUIUMUNUHUIUEUMUEUFUHOUJBUJRPSUIUFUNUEUFUHQABTSUEUF | |
| UHUAUBCUJUGUCSUD $. | |
| $( An edge of the ambient graph incident with a vertex outside a tree cannot | |
| already be indexed by the tree. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| trnewedgnin $p |- ( ( ( ( G e. UHGraph /\ T e. Tree ) /\ | |
| T SubGraph G ) /\ ( J e. dom ( iEdg ` G ) /\ | |
| ( ( iEdg ` G ) ` J ) = { U , N } /\ | |
| ( N e. _V /\ N e/ ( Vtx ` T ) ) ) ) -> | |
| J e/ dom ( iEdg ` T ) ) $= | |
| ( cuhgr wcel ctree wa csubgr wbr ciedg cfv cdm wceq cvv wnel w3a syl eqid | |
| cpr cvtx simpr3r cpw simplr simp3l prid2g simp-4l simpllr simpr subgriedgfv | |
| 3jca simplr2 eqtrd eleqtrrd c0 csn subgruhgredgd eldifad elelpwi syl2anc ex | |
| nelcon3d mpd ) CFGZAHGZIZACJKZIZDCLMZNGZDVJMZBEUAZOZEPGZEAUBMZQZIZRZIZVQDAL | |
| MZNZQVOVQVKVNVIUCVTDWBEVPVTDWBGZEVPGZVTWCIZEDWAMZGWFVPUDZGWDWEEVMWFWEVOEVMG | |
| WEVSVOVIVSWCUEVKVNVOVQUFSBEPUGSWEWFVLVMWEVEVHWCRWFVLOWEVEVHWCVEVFVHVSWCUHZV | |
| GVHVSWCUIZVTWCUJZULACDUKSVKVNVRVIWCUMUNUOWEWFWGUPUQWEACWAVPDVPTWATWHWIWJURU | |
| SEWFVPUTVAVBVCVD $. | |
| ${ | |
| $d E x $. $d G x $. $d I x $. $d J x $. $d N x $. | |
| $d T x $. $d U x $. $d V x $. $d W x $. | |
| trleafext.v $e |- V = ( Vtx ` T ) $. | |
| trleafext.e $e |- E = ( iEdg ` T ) $. | |
| trleafext.w $e |- W = ( Vtx ` G ) $. | |
| trleafext.i $e |- I = ( iEdg ` G ) $. | |
| trleafext.a $e |- A = ( V u. { N } ) $. | |
| trleafext.p $e |- P = ( E u. { <. J , { U , N } >. } ) $. | |
| trleafext.s $e |- S = <. A , P >. $. | |
| $( The hypotheses supplied by an ambient crossing edge satisfy the fresh | |
| leaf-construction interface. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| trleafhyp $p |- ( ( ( ( G e. USGraph /\ T e. Tree /\ | |
| T SubGraph G ) /\ ( U e. V /\ N e. ( W \ V ) ) ) /\ | |
| ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> | |
| ( ( T e. Tree /\ U e. V /\ ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) ) $= | |
| ( wcel cusgr ctree csubgr wbr w3a cdif wa cdm cfv cpr cvv simpll2 simplrl | |
| wceq wnel simplrr elex syl wn eldifn df-nel sylibr jca simprl cuhgr ciedg | |
| 3jca cvtx simpll1 usgruhgr simpll3 dmeq ax-mp eqcomi eleq2 fveq1i eqtr3id | |
| wb simprr neleq2 bicomi trnewedgnin ) GUATZDUBTZDGUCUDZUEZEKTZJLKUFZTZUGZ | |
| UGZIHUHZTZIHUIZEJUJZUNZUGZUGZWDWGJUKTZJKUOZUGZUEIUKTZIFUHZUOZUGWRWDWGXAWC | |
| WDWEWJWQULZWFWGWIWQUMWRWSWTWRWIWSWFWGWIWQUPZJWHUQURZWRJKTUSZWTWRWIXHXFJLK | |
| UTURJKVAVBZVCVGWRXBXDWRWMXBWKWMWPVDZIWLUQURWRGVETZWDUGZWEUGZIGVFUIZUHZTZI | |
| XNUIZWOUNZWSJDVHUIZUOZUGZUEZUGZXDWRXMYBWRXLWEWRXKWDWRWCXKWCWDWEWJWQVIGVJU | |
| RXEVCWCWDWEWJWQVKVCWRXPXRYAWRWMXPXJXOWLUNXPWMVRWLXOHXNUNWLXOUNPHXNVLVMVNX | |
| OWLIVOVMVBWRXQWNWOIHXNPVPWKWMWPVSVQWRWSXTXGWRWTXTXIWTXTKXSUNWTXTVRMKXSJVT | |
| VMWAVBVCVGVCYCIDVFUIZUHZUOZXDDEGIJWBXCYEUNZXDYFVRFYDUNYGNFYDVLVMXCYEIVTVM | |
| VBURVCVC $. | |
| $} | |
| ${ | |
| $d G a b f p $. $d K a b f p $. $d M a b f p $. | |
| $d a b f p $. | |
| $( In an acyclic undirected pseudograph, existence of a path is symmetric | |
| in its endpoints. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| acycgrpthonsym $p |- ( ( ( G e. UPGraph /\ G e. AcyclicGraph ) /\ | |
| E. a E. b a ( K ( PathsOn ` G ) M ) b ) -> | |
| E. f E. p f ( M ( PathsOn ` G ) K ) p ) $= | |
| ( cupgr wcel cacycgr wa cv cpthson cfv co wbr wex creverse syl syl6 simpr | |
| cspthson cvtx simpll id simplr jca pthonacycspthon revspthond spthonpthon | |
| eqid ex pthonex exlimdvv imp ) BHIZBJIZKZFLZGLZCDBMNZOPZGQFQALELDCVAOZPEQ | |
| AQZURVBVDFGURVBUSRNZUTRNZVCPZVDURVBVEVFDCBUBNZOPZVGURVBVIURVBKZUTUSBCDBUC | |
| NZVKUKVJUPUPUPUQVBUDUPUESVJUQVBKUSUTCDVHOPVJUQVBUPUQVBUFURVBUAUGUTUSBCDUH | |
| SUIULDCVFVEBUJTVFAVEBDCEUMTUNUO $. | |
| $} | |
| ${ | |
| $d f p $. $d f G $. $d f K $. $d f M $. $d f N $. $d p G $. $d p K $. | |
| $d p M $. $d p N $. | |
| $( Substitution of the first endpoint in a path-existence assertion. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| pthonexeq1 $p |- ( K = N -> | |
| ( E. f E. p f ( K ( PathsOn ` G ) M ) p <-> | |
| E. f E. p f ( N ( PathsOn ` G ) M ) p ) ) $= | |
| ( wceq cv cpthson cfv co wbr id oveq1d breqd 2exbidv ) CEGZAHZFHZCDBIJZKZ | |
| LRSEDTKZLAFQUAUBRSQCEDTQMNOP $. | |
| $( Substitution of the second endpoint in a path-existence assertion. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| pthonexeq2 $p |- ( M = N -> | |
| ( E. f E. p f ( K ( PathsOn ` G ) M ) p <-> | |
| E. f E. p f ( K ( PathsOn ` G ) N ) p ) ) $= | |
| ( wceq cv cpthson cfv co wbr id oveq2d breqd 2exbidv ) DEGZAHZFHZCDBIJZKZ | |
| LRSCETKZLAFQUAUBRSQDECTQMNOP $. | |
| $} | |
| ${ | |
| $d A a b f p $. $d E a b f p $. $d F a b f p $. | |
| $d G a b f p $. $d J a b f p $. $d K a b f p $. | |
| $d M a b f p $. $d N a b f p $. $d P a b f p $. | |
| $d Q a b f p $. $d S a b f p $. $d U a b f p $. | |
| $d V a b f p $. $d a b f p $. | |
| grleafpth.v $e |- V = ( Vtx ` G ) $. | |
| grleafpth.e $e |- E = ( iEdg ` G ) $. | |
| grleafpth.a $e |- A = ( V u. { N } ) $. | |
| grleafpth.p $e |- P = ( E u. { <. J , { U , N } >. } ) $. | |
| grleafpth.s $e |- S = <. A , P >. $. | |
| $( A path whose endpoints are old vertices remains a path after a leaf is | |
| adjoined. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafoldpthond $p |- ( ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) /\ | |
| F ( K ( PathsOn ` G ) M ) Q ) -> | |
| F ( K ( PathsOn ` S ) M ) Q ) $= | |
| ( wcel wa ctree cvv wnel w3a cdm cpthson cfv co wbr cspthson csubgr cusgr | |
| simpll1 treeusgr syl simpll2 simpll3 simplr jca grleafsubgr cacycgr simpr | |
| 3jca treeacycgr pthonacycspthon subgrspthon syl5com mpd spthonpthon ) HUA | |
| SZEMSZLUBSLMUCTZUDZIUBSIFUEUCTZTZGCJKHUFUGUHUIZTZGCJKDUJUGUHUIZGCJKDUFUGU | |
| HUIVQHDUKUIZVRVQHULSZVKVLUDZVNTVSVQWAVNVQVTVKVLVQVJVTVJVKVLVNVPUMZHUNUOVJ | |
| VKVLVNVPUPVJVKVLVNVPUQVCVMVNVPURUSABDEFHILMNOPQRUTUOVQGCJKHUJUGUHUIZVSVRV | |
| QHVASZVPTWCVQWDVPVQVJWDWBHVDUOVOVPVBUSCGHJKVEUOCHGDJKVFVGVHJKCGDVIUO $. | |
| $( Any two old vertices remain joined by a path after a leaf is adjoined. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafoldoldpthon $p |- ( ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) /\ ( K e. V /\ M e. V ) ) -> | |
| E. f E. p f ( K ( PathsOn ` S ) M ) p ) $= | |
| ( wcel wa ctree cvv wnel w3a cdm cv cpthson cfv co wbr cconngr treeconngr | |
| wex simpll1 syl simprl simprr 3jca conngrpthon simpll jca grleafoldpthond | |
| simpr ex 2eximdv mpd ) GUASZDLSZKUBSKLUCTZUDHUBSHFUEUCTZTZILSZJLSZTZTZEUF | |
| ZMUFZIJGUGUHUIUJZMUMEUMZVPVQIJCUGUHUIUJZMUMEUMVOGUKSZVLVMUDVSVOWAVLVMVOVG | |
| WAVGVHVIVJVNUNGULUOVKVLVMUPVKVLVMUQUREGIJLMNUSUOVOVRVTEMVOVRVTVOVRTZVKVRT | |
| VTWBVKVRVKVNVRUTVOVRVCVAABVQCDFVPGHIJKLNOPQRVBUOVDVEVF $. | |
| $( A path from an old vertex to the attachment vertex extends to the new | |
| leaf. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafoldnewpthond $p |- ( ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) /\ | |
| F ( K ( PathsOn ` G ) U ) Q ) -> | |
| ( F ++ <" J "> ) ( K ( PathsOn ` S ) N ) | |
| ( Q ++ <" N "> ) ) $= | |
| ( wcel wa co ctree cvv w3a cdm cpthson cfv wbr cs1 cconcat cspthson simpl | |
| wnel cacycgr simpll1 treeacycgr syl simpr jca pthonacycspthon spthonpthon | |
| grleafspthond ) HUARZELRZKUBRKLULSZUCIUBRIFUDULSZSZGCJEHUEUFTUGZSZGIUHUIT | |
| ZCKUHUITZJKDUJUFTUGZVIVJJKDUEUFTUGVHVFGCJEHUJUFTUGZSVKVHVFVLVFVGUKVHHUMRZ | |
| VGSVLVHVMVGVHVBVMVBVCVDVEVGUNHUOUPVFVGUQURCGHJEUSUPURABCDEFGHIJKLMNOPQVAU | |
| PJKVJVIDUTUP $. | |
| $( Every old vertex is joined to the new leaf after adjoining the leaf. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafoldnewpthon $p |- ( ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) /\ K e. V ) -> | |
| E. f E. p f ( K ( PathsOn ` S ) N ) p ) $= | |
| ( va wcel wa vb ctree cvv wnel w3a cdm cv cpthson cfv wbr cconngr simpll1 | |
| co wex treeconngr syl simpll2 3jca conngrpthon cconcat grleafoldnewpthond | |
| simpr cs1 simpll jca ex pthonex syl6 exlimdvv mpd ) GUBSZDKSZJUCSJKUDTZUE | |
| HUCSHFUFUDTZTZIKSZTZRUGZUAUGZIDGUHUIUMUJZUAUNRUNZEUGLUGIJCUHUIUMZUJLUNEUN | |
| ZVQGUKSZVPVLUEWAVQWDVPVLVQVKWDVKVLVMVNVPULGUOUPVOVPVBVKVLVMVNVPUQURRGIDKU | |
| AMUSUPVQVTWCRUAVQVTVRHVCUTUMZVSJVCUTUMZWBUJZWCVQVTWGVQVTTZVOVTTWGWHVOVTVO | |
| VPVTVDVQVTVBVEABVSCDFVRGHIJKMNOPQVAUPVFWFEWECIJLVGVHVIVJ $. | |
| $( The new leaf is joined to every old vertex after adjoining the leaf. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafnewoldpthon $p |- ( ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) /\ K e. V ) -> | |
| E. f E. p f ( N ( PathsOn ` S ) K ) p ) $= | |
| ( wcel wa syl va vb ctree cvv w3a cdm cupgr cacycgr cv cpthson cfv co wbr | |
| wnel wex simpll1 treeusgr simpll2 simpll3 3jca simplr grleafusgr usgrupgr | |
| cusgr jca simpl grleafacycgr grleafoldnewpthon acycgrpthonsym ) GUCRZDKRZ | |
| JUDRJKUNSZUEZHUDRHFUFUNSZSZIKRZSZCUGRZCUHRZSZUAUIUBUIIJCUJUKZULUMUBUOUAUO | |
| ZSEUILUIJIWAULUMLUOEUOVQVTWBVQVRVSVQCVDRZVRVQGVDRZVKVLUEZVNSWCVQWEVNVQWDV | |
| KVLVQVJWDVJVKVLVNVPUPGUQTVJVKVLVNVPURVJVKVLVNVPUSUTVMVNVPVAVEABCDFGHJKMNO | |
| PQVBTCVCTVQVOVSVOVPVFABCDFGHJKMNOPQVGTVEABCDUAFGHIJKUBMNOPQVHVEECIJLUAUBV | |
| IT $. | |
| $( Membership in the enlarged vertex set is membership in the old vertex | |
| set or equality with the new leaf. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| grleafela $p |- ( K e. A -> ( K e. V \/ K = N ) ) $= | |
| ( wcel csn wo biimpi syl cun wceq eleq2i elun id elsni orim12i ) HAPZHJIQ | |
| ZUAZPZHJPZHIUBZRZUHUKAUJHMUCSUKULHUIPZRZUNUKUPHJUIUDSULULUOUMULUEHIUFUGTT | |
| $. | |
| $( The adjoined leaf belongs to the vertex set of the enlarged graph. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafnewvtx $p |- ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) -> N e. ( Vtx ` S ) ) $= | |
| ( wcel cvv wnel wa w3a syl ctree cdm cfv csn cun simpl3 simpl snidg elun2 | |
| cvtx wceq a1i eleqtrrd cusgr simpl1 treeusgr simpl2 3jca simpr grleafvtx | |
| jca ) FUAOZDIOZHPOZHIQZRZSZGPOGEUBQRZRZHACUJUCZVIHIHUDZUEZAVIVDHVLOZVIVFV | |
| DVBVCVFVHUFZVDVEUGTVDHVKOVMHPUHHVKIUITTAVLUKVILULUMVIFUNOZVCVFSZVHRVJAUKV | |
| IVPVHVIVOVCVFVIVBVOVBVCVFVHUOFUPTVBVCVFVHUQVNURVGVHUSVAABCDEFGHIJKLMNUTTU | |
| M $. | |
| $( The old-to-new endpoint case for paths in the enlarged graph. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafpthcaseon $p |- ( ( ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) /\ K e. V ) /\ M = N ) -> | |
| E. f E. p f ( K ( PathsOn ` S ) M ) p ) $= | |
| ( wcel wa ctree cvv wnel w3a cdm wceq cv cpthson cfv co grleafoldnewpthon | |
| wbr wex simpl syl wb simpr pthonexeq2 mpbird ) GUASDLSKUBSKLUCTUDHUBSHFUE | |
| UCTTILSTZJKUFZTZEUGZMUGZIJCUHUIZUJULMUMEUMZVCVDIKVEUJULMUMEUMZVBUTVGUTVAU | |
| NABCDEFGHIKLMNOPQRUKUOVBVAVFVGUPUTVAUQECIJKMURUOUS $. | |
| $( The new-to-old endpoint case for paths in the enlarged graph. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafpthcaseno $p |- ( ( ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) /\ M e. V ) /\ K = N ) -> | |
| E. f E. p f ( K ( PathsOn ` S ) M ) p ) $= | |
| ( wcel wa ctree cvv wnel w3a cdm wceq cv cpthson cfv co grleafnewoldpthon | |
| wbr wex simpl syl wb simpr pthonexeq1 mpbird ) GUASDLSKUBSKLUCTUDHUBSHFUE | |
| UCTTJLSTZIKUFZTZEUGZMUGZIJCUHUIZUJULMUMEUMZVCVDKJVEUJULMUMEUMZVBUTVGUTVAU | |
| NABCDEFGHJKLMNOPQRUKUOVBVAVFVGUPUTVAUQECIJKMURUOUS $. | |
| $( The new-to-new endpoint case for paths in the enlarged graph. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafpthcasenn $p |- ( ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) /\ ( K = N /\ M = N ) ) -> | |
| E. f E. p f ( K ( PathsOn ` S ) M ) p ) $= | |
| ( wcel wex ctree cvv wnel wa w3a cdm wceq cv cpthson cfv wbr grleafnewvtx | |
| co simpl cvtx eqid 0pthonv syl simprr pthonexeq2 mpbird simprl pthonexeq1 | |
| wb ) GUASDLSKUBSKLUCUDUEHUBSHFUFUCUDUDZIKUGZJKUGZUDZUDZEUHZMUHZIJCUIUJZUM | |
| UKMTETZVJVKKJVLUMUKMTETZVIVNVJVKKKVLUMUKMTETZVIVEVOVEVHUNVEKCUOUJZSVOABCD | |
| FGHKLNOPQRULECKVPMVPUPUQURURVIVGVNVOVDVEVFVGUSECKJKMUTURVAVIVFVMVNVDVEVFV | |
| GVBECIJKMVCURVA $. | |
| $( Every pair of vertices in the enlarged graph is joined by a path. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafallpthon $p |- ( ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) /\ ( K e. A /\ M e. A ) ) -> | |
| E. f E. p f ( K ( PathsOn ` S ) M ) p ) $= | |
| ( wcel wa ctree cvv wnel w3a cdm wceq wo cv cpthson cfv co wbr wex simprl | |
| grleafela syl simprr jca wi simpl grleafoldoldpthon grleafpthcaseno exp31 | |
| ex com23 impd grleafpthcaseon grleafpthcasenn ccased mpd ) GUASDLSKUBSKLU | |
| CTUDHUBSHFUEUCTTZIASZJASZTZTZILSZIKUFZUGZJLSZJKUFZUGZTZEUHMUHIJCUIUJUKULM | |
| UMEUMZVOVRWAVOVLVRVKVLVMUNABCDFGHIKLNOPQRUOUPVOVMWAVKVLVMUQABCDFGHJKLNOPQ | |
| RUOUPURVOVKWBWCUSVKVNUTVKVPVSVQVTWCVKVPVSTWCABCDEFGHIJKLMNOPQRVAVDVKVQVSW | |
| CVKVSVQWCVKVSVQWCABCDEFGHIJKLMNOPQRVBVCVEVFVKVPVTWCVKVPVTWCABCDEFGHIJKLMN | |
| OPQRVGVCVFVKVQVTTWCABCDEFGHIJKLMNOPQRVHVDVIUPVJ $. | |
| $( Adjoining a fresh leaf to a tree produces an undirected simple graph. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleaftreeusgr $p |- ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) -> S e. USGraph ) $= | |
| ( wcel cvv wnel wa w3a cusgr ctree cdm simpl1 treeusgr simpl2 simpl3 3jca | |
| syl simpr jca grleafusgr ) FUAOZDIOZHPOHIQRZSZGPOGEUBQRZRZFTOZUMUNSZUPRCT | |
| OUQUSUPUQURUMUNUQULURULUMUNUPUCFUDUHULUMUNUPUEULUMUNUPUFUGUOUPUIUJABCDEFG | |
| HIJKLMNUKUH $. | |
| $( The vertex set after adjoining a fresh leaf to a tree is the prescribed | |
| enlarged vertex set. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleaftreevtx $p |- ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) -> ( Vtx ` S ) = A ) $= | |
| ( wcel cvv wnel wa w3a syl ctree cdm cvtx cfv wceq simpl1 treeusgr simpl2 | |
| cusgr simpl3 3jca simpr jca grleafvtx ) FUAOZDIOZHPOHIQRZSZGPOGEUBQRZRZFU | |
| IOZUPUQSZUSRCUCUDAUEUTVBUSUTVAUPUQUTUOVAUOUPUQUSUFFUGTUOUPUQUSUHUOUPUQUSU | |
| JUKURUSULUMABCDEFGHIJKLMNUNT $. | |
| ${ | |
| $d A f k m p $. $d E f k m p $. $d G f k m p $. | |
| $d J f k m p $. $d N f k m p $. $d U f k m p $. | |
| $d P f k m p $. $d V f k m p $. $d S f k m p $. | |
| $d f k m p $. | |
| $( Every pair of vertices in a tree enlarged by a fresh leaf is joined | |
| by a path. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafallpthonvtx $p |- ( ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) /\ | |
| ( k e. ( Vtx ` S ) /\ m e. ( Vtx ` S ) ) ) -> | |
| E. f E. p f ( k ( PathsOn ` S ) m ) p ) $= | |
| ( wcel wa ctree cvv wnel w3a cdm cv cfv cpthson co wbr wex simpl simprl | |
| cvtx wceq grleaftreevtx syl eleqtrd simprr jca grleafallpthon ) IUASDLS | |
| KUBSKLUCTUDJUBSJHUEUCTTZFUFZCUNUGZSZGUFZVDSZTZTZVBVCASZVFASZTZTEUFMUFVC | |
| VFCUHUGUIUJMUKEUKVIVBVLVBVHULZVIVJVKVIVCVDAVBVEVGUMVIVBVDAUOVMABCDHIJKL | |
| NOPQRUPUQZURVIVFVDAVBVEVGUSVNURUTUTABCDEHIJVCVFKLMNOPQRVAUQ $. | |
| $( Adjoining a fresh leaf to a tree preserves connectedness. | |
| (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| grleafconngr $p |- ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) -> S e. ConnGraph ) $= | |
| ( vf vp vk vm wcel cvv ctree wnel wa w3a cdm cconngr cv cpthson cfv wbr | |
| co cvtx wral grleafallpthonvtx ralrimivva wb cusgr grleaftreeusgr elexd | |
| wex eqid isconngr syl mpbird ) FUASDISHTSHIUBUCUDGTSGEUEUBUCUCZCUFSZOUG | |
| PUGQUGRUGCUHUIUKUJPUTOUTZRCULUIZUMQVHUMZVEVGQRVHVHABCDOQREFGHIPJKLMNUNU | |
| OVECTSVFVIUPVECUQABCDEFGHIJKLMNURUSOQRCVHTPVHVAVBVCVD $. | |
| $} | |
| $( Adjoining a fresh leaf to a tree produces a tree. This is the standard | |
| leaf-extension step; compare [Diestel] p. 14. (Contributed by Mingli | |
| Yuan, 13-Aug-2026.) $) | |
| grleaftree $p |- ( ( ( G e. Tree /\ U e. V /\ | |
| ( N e. _V /\ N e/ V ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) -> S e. Tree ) $= | |
| ( ctree wcel cvv wnel wa syl w3a cdm cupgr cconngr cacycgr cvtx cfv cusgr | |
| c0 wne grleaftreeusgr usgrupgr grleafconngr grleafacycgr grleafnewvtx jca | |
| 3jca ne0i wb elexd istree mpbird ) FOPDIPHQPHIRSUAGQPGEUBRSSZCOPZCUCPZCUD | |
| PZCUEPZUAZCUFUGZUIUJZSZVCVHVJVCVEVFVGVCCUHPVEABCDEFGHIJKLMNUKZCULTABCDEFG | |
| HIJKLMNUMABCDEFGHIJKLMNUNUQVCHVIPVJABCDEFGHIJKLMNUOVIHURTUPVCCQPVDVKUSVCC | |
| UHVLUTCQVATVB $. | |
| $} | |
| ${ | |
| $d A x $. $d E x $. $d G x $. $d I x $. $d J x $. | |
| $d N x $. $d P x $. $d S x $. $d T x $. $d U x $. | |
| $d V x $. $d W x $. | |
| trleafsubgr.v $e |- V = ( Vtx ` T ) $. | |
| trleafsubgr.e $e |- E = ( iEdg ` T ) $. | |
| trleafsubgr.w $e |- W = ( Vtx ` G ) $. | |
| trleafsubgr.i $e |- I = ( iEdg ` G ) $. | |
| trleafsubgr.a $e |- A = ( V u. { N } ) $. | |
| trleafsubgr.p $e |- P = ( E u. { <. J , { U , N } >. } ) $. | |
| trleafsubgr.s $e |- S = <. A , P >. $. | |
| $( The vertex set of a fresh-leaf extension is a subclass of that of the | |
| ambient graph. (Contributed by Mingli Yuan, 13-Aug-2026.) $) | |
| trleafvtxss $p |- ( ( ( ( G e. USGraph /\ T e. Tree /\ | |
| T SubGraph G ) /\ ( U e. V /\ N e. ( W \ V ) ) ) /\ | |
| ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> | |
| ( Vtx ` S ) C_ W ) $= | |
| ( wcel cusgr csubgr wbr w3a cdif wa cdm cfv cpr wceq cvtx wss csn | |
| simpll3 ctree cun subgrvtxss syl wb pm3.2i sseq12 ax-mp sylibr eldifi | |
| snssd unssd simplrr a1i sseq1 mpbird cvv wnel trleafhyp grleaftreevtx | |
| sseq1d ) | |
| GUATZDUOTZDGUBUCZUDZEKTZJLKUETZUFZUFIHUGTIHUHEJUIUJUFZUFZCUKUHZLULAL | |
| ULZWDWFKJUMZUPZLULZWDKWGLWDDUKUHZGUKUHZULZKLULZWDVRWLVPVQVRWBWCUNDGUQ | |
| URKWJUJZLWKUJZUFWMWLUSWNWOMOUTKWJLWKVAVBVCWDJLWDWAJLTVSVTWAWCVGJLKVDU | |
| RVEVFWDAWHUJZWFWIUSWPWDQVHAWHLVIURVJWDWEALWDVQVTJVKTJKVLUFUDIVKTIFUGV | |
| LUFUFWEAUJABCDEFGHIJKLMNOPQRSVMABCEFDIJKMNQRSVNURVOVJ $. | |
| $( The indexed-edge function of a fresh-leaf extension is a subclass of | |
| that of the ambient graph. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| trleafiedgss $p |- ( ( ( ( G e. USGraph /\ T e. Tree /\ | |
| T SubGraph G ) /\ ( U e. V /\ N e. ( W \ V ) ) ) /\ | |
| ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> | |
| ( iEdg ` S ) C_ I ) $= | |
| ( wa cusgr wcel ctree wbr w3a cdif cdm cfv cpr wceq ciedg wss cop csn cun | |
| csubgr simpll3 subgriedgss syl wb pm3.2i sseq12 ax-mp sylibr simprr opeq2 | |
| wfun simpll1 usgrfun funeq simprl jca funfvop eqeltrrd snssd unssd mpbird | |
| a1i sseq1 cvv wnel trleafhyp simpl1 treeusgr simpl2 3jca simpr grleafiedg | |
| simpl3 sseq1d ) GUAUBZDUCUBZDGUPUDZUEEKUBZJLKUFUBTZTZIHUGUBZIHUHZEJUIZUJZ | |
| TZTZCUKUHZHULBHULZXBXDFIWSUMZUNZUOZHULZXBFXFHXBDUKUHZGUKUHZULZFHULZXBWMXK | |
| WKWLWMWOXAUQDGURUSFXIUJZHXJUJZTXLXKUTXMXNNPVAFXIHXJVBVCVDXBXEHXBIWRUMZXEH | |
| XBWTXOXEUJWPWQWTVEWRWSIVFUSXBHVGZWQTXOHUBXBXPWQXBXJVGZXPXBWKXQWKWLWMWOXAV | |
| HGVIUSXNXPXQUTPHXJVJVCVDWPWQWTVKVLIHVMUSVNVOVPXBBXGUJZXDXHUTXRXBRVRBXGHVS | |
| USVQXBXCBHXBDUAUBZWNJVTUBJKWATZUEZIVTUBIFUGWATZTZXCBUJXBWLWNXTUEZYBTZYCAB | |
| CDEFGHIJKLMNOPQRSWBYEYAYBYEXSWNXTYEWLXSWLWNXTYBWCDWDUSWLWNXTYBWEWLWNXTYBW | |
| IWFYDYBWGVLUSABCEFDIJKMNQRSWHUSWJVQ $. | |
| $( A fresh leaf of a tree, chosen from an ambient simple graph, remains a | |
| subgraph of the ambient graph. (Contributed by Mingli Yuan, | |
| 13-Aug-2026.) $) | |
| trleafsubgr $p |- ( ( ( ( G e. USGraph /\ T e. Tree /\ | |
| T SubGraph G ) /\ ( U e. V /\ N e. ( W \ V ) ) ) /\ | |
| ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> | |
| S SubGraph G ) $= | |
| ( wcel cusgr ctree csubgr wbr w3a cdif wa cdm cfv cpr wceq cvtx wss ciedg | |
| trleafvtxss trleafiedgss jca cvv wfun cuhgr wb simpll1 elex id usgrfun wi | |
| syl funeq ax-mp a1i wnel trleafhyp grleaftree treeusgr usgruhgr 3jca eqid | |
| sylibr uhgrissubgr mpbird ) GUATZDUBTZDGUCUDZUEEKTZJLKUFTUGZUGIHUHTIHUIEJ | |
| UJUKUGZUGZCGUCUDZCULUIZLUMZCUNUIZHUMZUGZWGWJWLABCDEFGHIJKLMNOPQRSUOABCDEF | |
| GHIJKLMNOPQRSUPUQWGGURTZHUSZCUTTZUEWHWMVAWGWNWOWPWGWAWNWAWBWCWEWFVBZGUAVC | |
| VGWGGUNUIZUSZWOWGWAWSWGWAWAWQWAVDVGGVEVGHWRUKZWTVFZWOWSVAZWTVDXBXAWTXBPHW | |
| RVHVIVJVIVRWGCUBTZWPWGWBWDJURTJKVKUGUEIURTIFUHVKUGUGXCABCDEFGHIJKLMNOPQRS | |
| VLABCEFDIJKMNQRSVMVGXCCUATWPCVNCVOVGVGVPLHCGWKWIURWIVQOWKVQPVSVGVT $. | |
| $} | |
| $( The hypotheses used to extend a tree select exactly the ambient | |
| fresh-leaf data needed by ~ trleafhyp . (Contributed by Mingli Yuan, | |
| 14-Aug-2026.) $) | |
| trextleafdata $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| W e. Fin ) /\ ( T e. Tree /\ T SubGraph G /\ A =/= W ) ) /\ | |
| ( U e. A /\ N e. ( W \ A ) ) ) /\ | |
| ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> | |
| ( ( ( G e. USGraph /\ T e. Tree /\ T SubGraph G ) /\ | |
| ( U e. A /\ N e. ( W \ A ) ) ) /\ | |
| ( J e. dom I /\ ( I ` J ) = { U , N } ) ) ) $= | |
| ( cusgr wcel cconngr cfn w3a ctree csubgr wbr wne wa syl jca cdif cdm simpl | |
| cfv cpr wceq simpll1 simplr1 simplr2 3jca simplr simpr ) DIJZDKJZHLJZMZBNJZ | |
| BDOPZAHQZMZRZCAJGHAUAJRZRZFEUBJFEUDCGUEUFRZRZUMUQURMZVBRVDVEVFVBVEUMUQURVEV | |
| CUMVCVDUCZUMUNUOUTVBUGSVEVCUQVGUQURUSUPVBUHSVEVCURVGUQURUSUPVBUISUJVAVBVDUK | |
| TVCVDULT $. | |
| ${ | |
| $d A J N U $. $d E J N U $. $d G J N U $. | |
| $d I J N U $. $d T J N U $. $d W J N U $. | |
| $d J N U $. | |
| trextfresh.a $e |- A = ( Vtx ` T ) $. | |
| trextfresh.e $e |- E = ( iEdg ` T ) $. | |
| trextfresh.w $e |- W = ( Vtx ` G ) $. | |
| trextfresh.i $e |- I = ( iEdg ` G ) $. | |
| trextfresh.x $e |- X = | |
| <. ( A u. { N } ) , ( E u. { <. J , { U , N } >. } ) >. $. | |
| $( Extension data give a genuinely fresh leaf of the old tree. | |
| (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| trextfresh $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| W e. Fin ) /\ ( T e. Tree /\ T SubGraph G /\ A =/= W ) ) /\ | |
| ( U e. A /\ N e. ( W \ A ) ) ) /\ | |
| ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> | |
| ( ( T e. Tree /\ U e. A /\ ( N e. _V /\ N e/ A ) ) /\ | |
| ( J e. _V /\ J e/ dom E ) ) ) $= | |
| ( wcel w3a wa cdm cvv cusgr cconngr cfn ctree csubgr wbr wne cdif cfv cpr | |
| wceq wnel trextleafdata csn cun cop eqid trleafhyp syl ) EUAPZEUBPIUCPQBU | |
| DPZBEUEUFZAIUGQRCAPZHIAUHPRZRGFSPGFUICHUJZUKRZRUTVAVBQVDRVFRVAVCHTPHAULRQ | |
| GTPGDSULRRABCEFGHIUMAHUNUOZDGVEUPUNUOZJBCDEFGHAIKLMNVGUQVHUQOURUS $. | |
| $( The graph obtained from the extension data is a tree. | |
| (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| trexttree $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| W e. Fin ) /\ ( T e. Tree /\ T SubGraph G /\ A =/= W ) ) /\ | |
| ( U e. A /\ N e. ( W \ A ) ) ) /\ | |
| ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> X e. Tree ) $= | |
| ( wcel w3a ctree wa cdm cusgr cconngr cfn csubgr wbr wne cdif cfv cpr cvv | |
| wceq wnel trextfresh csn cun cop eqid grleaftree syl ) EUAPEUBPIUCPQBRPZB | |
| EUDUEAIUFQSCAPZHIAUGPSSGFTPGFUHCHUIZUKSSUTVAHUJPHAULSQGUJPGDTULSSJRPABCDE | |
| FGHIJKLMNOUMAHUNUOZDGVBUPUNUOZJCDBGHAKLVCUQVDUQOURUS $. | |
| $( The graph obtained from the extension data remains a subgraph of the | |
| ambient graph. (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| trextxsubgr $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| W e. Fin ) /\ ( T e. Tree /\ T SubGraph G /\ A =/= W ) ) /\ | |
| ( U e. A /\ N e. ( W \ A ) ) ) /\ | |
| ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> X SubGraph G ) $= | |
| ( wcel w3a csubgr wbr wa cusgr cconngr cfn wne cdif cdm cfv trextleafdata | |
| ctree cpr wceq csn cun cop eqid trleafsubgr syl ) EUAPZEUBPIUCPQBUIPZBERS | |
| ZAIUDQTCAPHIAUEPTZTGFUFPGFUGCHUJZUKTZTURUSUTQVATVCTJERSABCEFGHIUHAHULUMZD | |
| GVBUNULUMZJBCDEFGHAIKLMNVDUOVEUOOUPUQ $. | |
| $( The extension adjoins precisely the selected new vertex. | |
| (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| trextvtx $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| W e. Fin ) /\ ( T e. Tree /\ T SubGraph G /\ A =/= W ) ) /\ | |
| ( U e. A /\ N e. ( W \ A ) ) ) /\ | |
| ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> | |
| ( Vtx ` X ) = ( A u. { N } ) ) $= | |
| ( wcel w3a wa cdm cfv cusgr cconngr cfn ctree csubgr wbr wne cdif cpr cvv | |
| wceq wnel cvtx csn cun trextfresh cop eqid grleaftreevtx syl ) EUAPEUBPIU | |
| CPQBUDPZBEUEUFAIUGQRCAPZHIAUHPRRGFSPGFTCHUIZUKRRVAVBHUJPHAULRQGUJPGDSULRR | |
| JUMTAHUNUOZUKABCDEFGHIJKLMNOUPVDDGVCUQUNUOZJCDBGHAKLVDURVEUROUSUT $. | |
| $} | |
| ${ | |
| trextvtxss.a $e |- A = ( Vtx ` T ) $. | |
| trextvtxss.w $e |- W = ( Vtx ` G ) $. | |
| $( The old tree's vertices form a subclass of the ambient vertex set. | |
| (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| trextvtxss $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| W e. Fin ) /\ ( T e. Tree /\ T SubGraph G /\ A =/= W ) ) /\ | |
| ( U e. A /\ N e. ( W \ A ) ) ) /\ | |
| ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> A C_ W ) $= | |
| ( cusgr wcel cconngr cfn w3a wa cfv cvtx wss syl ctree csubgr wbr wne cdm | |
| cdif cpr wceq simpllr simp2 subgrvtxss sseq12i bicomi biimpi ) DKLDMLHNLO | |
| ZBUALZBDUBUCZAHUDZOZPCALGHAUFLPZPFEUELFEQCGUGUHPZPZBRQZDRQZSZAHSZVBUQVEVB | |
| USUQUOUSUTVAUIUPUQURUJTBDUKTVEVFVFVEAVCHVDIJULUMUNT $. | |
| $( The old tree has a finite vertex set. | |
| (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| trextafin $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| W e. Fin ) /\ ( T e. Tree /\ T SubGraph G /\ A =/= W ) ) /\ | |
| ( U e. A /\ N e. ( W \ A ) ) ) /\ | |
| ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> A e. Fin ) $= | |
| ( cusgr wcel cconngr cfn w3a ctree csubgr wbr wa syl wne cdif cdm cfv cpr | |
| wceq wss simplll simp3 trextvtxss jca ssfi ) DKLZDMLZHNLZOZBPLBDQRAHUAOZS | |
| CALGHAUBLSZSFEUCLFEUDCGUEUFSZSZUOAHUGZSANLUTUOVAUTUPUOUPUQURUSUHUMUNUOUIT | |
| ABCDEFGHIJUJUKHAULT $. | |
| $} | |
| ${ | |
| $d A J N U s $. $d E J N U s $. $d G J N U s $. | |
| $d I J N U s $. $d T J N U s $. $d W J N U s $. | |
| $d X s $. $d J N U s $. | |
| trextsubgrexlem.a $e |- A = ( Vtx ` T ) $. | |
| trextsubgrexlem.e $e |- E = ( iEdg ` T ) $. | |
| trextsubgrexlem.w $e |- W = ( Vtx ` G ) $. | |
| trextsubgrexlem.i $e |- I = ( iEdg ` G ) $. | |
| trextsubgrexlem.x $e |- X = | |
| <. ( A u. { N } ) , ( E u. { <. J , { U , N } >. } ) >. $. | |
| $( A crossing edge extends a finite tree subgraph by exactly one vertex. | |
| (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| trextsubgrexlem $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| W e. Fin ) /\ ( T e. Tree /\ T SubGraph G /\ A =/= W ) ) /\ | |
| ( U e. A /\ N e. ( W \ A ) ) ) /\ | |
| ( J e. dom I /\ ( I ` J ) = { U , N } ) ) -> | |
| E. s ( s e. Tree /\ s SubGraph G /\ | |
| ( # ` ( Vtx ` s ) ) = ( ( # ` A ) + 1 ) ) ) $= | |
| ( wcel wa cfv chash cusgr cconngr cfn w3a ctree wbr wne cdif cdm cpr wceq | |
| csubgr cv cvtx c1 caddc co cvv trexttree syl trextxsubgr csn cun trextvtx | |
| elex fveq2d simplrr trextafin eldifn jca hashunsng eqtrd 3jca eleq1 breq1 | |
| wn imp 2fveq3 eqeq1d 3anbi123d spcedv ) EUAQEUBQIUCQUDBUEQBEULUFAIUGUDRZC | |
| AQZHIAUHZQZRRGFUIQGFSCHUJUKRZRZKUMZUEQZWHEULUFZWHUNSTSZATSUOUPUQZUKZUDJUE | |
| QZJEULUFZJUNSZTSZWLUKZUDKURJWGWNJURQABCDEFGHIJLMNOPUSZJUEVEUTWGWNWOWRWSAB | |
| CDEFGHIJLMNOPVAWGWQAHVBVCZTSZWLWGWPWTTABCDEFGHIJLMNOPVDVFWGHURQZAUCQZHAQV | |
| PZRZRXAWLUKZWGXBXEWGWEXBWBWCWEWFVGZHWDVEUTWGXCXDABCEFGHILNVHWGWEXDXGHIAVI | |
| UTVJVJXBXEXFAHURVKVQUTVLVMWHJUKZWIWNWJWOWMWRWHJUEVNWHJEULVOXHWKWQWLWHJTUN | |
| VRVSVTWA $. | |
| $} | |
| ${ | |
| $d A j n u $. $d G j n u $. $d I j n u $. | |
| $d T j n u $. $d W j n u $. $d j n u $. | |
| trextcut.a $e |- A = ( Vtx ` T ) $. | |
| trextcut.w $e |- W = ( Vtx ` G ) $. | |
| trextcut.i $e |- I = ( iEdg ` G ) $. | |
| $( A proper tree subgraph of a connected simple graph has an ambient | |
| indexed edge crossing its vertex cut. (Contributed by Mingli Yuan, | |
| 14-Aug-2026.) $) | |
| trextcut $p |- ( ( ( G e. USGraph /\ G e. ConnGraph /\ W e. Fin ) /\ | |
| ( T e. Tree /\ T SubGraph G /\ A =/= W ) ) -> | |
| E. j e. dom I E. u e. A E. n e. ( W \ A ) | |
| ( I ` j ) = { u , n } ) $= | |
| ( wcel w3a wne wa c0 cv cfv wrex syl cusgr cconngr cfn ctree csubgr cupgr | |
| wbr wss cpr wceq cdm simpl1 usgrupgr simpl2 jca simpr2 subgrvtxss sseq12i | |
| cdif cvtx bicomi biimpi simpr1 treevtxne0 neeq1i simpr3 conngrcutiedg | |
| 3jca ) FUALZFUBLZHUCLZMZCUDLZCFUEUGZBHNZMZOZFUFLZVJOZBHUHZBPNZVOMZODQGRAQ | |
| EQUIUJEHBUSSABSDGUKSVQVSWBVQVRVJVQVIVRVIVJVKVPULFUMTVIVJVKVPUNUOVQVTWAVOV | |
| QCUTRZFUTRZUHZVTVQVNWEVLVMVNVOUPCFUQTWEVTVTWEBWCHWDIJURVAVBTVQWCPNZWAVQVM | |
| WFVLVMVNVOVCCVDTWFWAWAWFBWCPIVEVAVBTVLVMVNVOVFVHUOABDEGFHJKVGT $. | |
| $} | |
| ${ | |
| $d A j n s u $. $d E j n s u $. $d G j n s u $. | |
| $d I j n s u $. $d T j n s u $. $d W j n s u $. | |
| $d j n s u $. | |
| trextsubgrex.a $e |- A = ( Vtx ` T ) $. | |
| trextsubgrex.e $e |- E = ( iEdg ` T ) $. | |
| trextsubgrex.w $e |- W = ( Vtx ` G ) $. | |
| trextsubgrex.i $e |- I = ( iEdg ` G ) $. | |
| $( A proper finite tree subgraph of a connected simple graph can be | |
| extended to a tree subgraph with one more vertex. (Contributed by | |
| Mingli Yuan, 14-Aug-2026.) $) | |
| trextsubgrex $p |- ( ( ( G e. USGraph /\ G e. ConnGraph /\ W e. Fin ) /\ | |
| ( T e. Tree /\ T SubGraph G /\ A =/= W ) ) -> | |
| E. s ( s e. Tree /\ s SubGraph G /\ | |
| ( # ` ( Vtx ` s ) ) = ( ( # ` A ) + 1 ) ) ) $= | |
| ( vj vu vn wcel w3a wa cv cfv jca cusgr cconngr cfn ctree csubgr wbr wceq | |
| wne cpr cdif wrex cdm cvtx chash c1 caddc co wex trextcut wi simp1 simp22 | |
| simp23 simp21 simp3 csn cun cop eqid trextsubgrexlem syl 3exp rexlimdvvva | |
| imp mpd ) DUAODUBOFUCOPBUDOBDUEUFAFUHPQZLRZESMRZNRZUIZUGZNFAUJZUKMAUKLEUL | |
| ZUKGRZUDOWDDUEUFWDUMSUNSAUNSUOUPUQUGPGURZMABLNDEFHJKUSVPWAWELMNWCAWBVPVQW | |
| COZVRAOZVSWBOZPZWAWEUTVPWIWAWEVPWIWAPZVPWGWHQZQZWFWAQZQWEWJWLWMWJVPWKVPWI | |
| WAVAWJWGWHVPWFWGWHWAVBVPWFWGWHWAVCTTWJWFWAVPWFWGWHWAVDVPWIWAVETTABVRCDEVQ | |
| VSFAVSVFVGCVQVTVHVFVGVHZGHIJKWNVIVJVKVLVNVMVO $. | |
| $} | |
| ${ | |
| $d A Y s $. $d E Y s $. $d G Y s $. | |
| $d I Y s $. $d T Y s $. $d W Y s $. | |
| trextsubgrhash.a $e |- A = ( Vtx ` T ) $. | |
| trextsubgrhash.e $e |- E = ( iEdg ` T ) $. | |
| trextsubgrhash.w $e |- W = ( Vtx ` G ) $. | |
| trextsubgrhash.i $e |- I = ( iEdg ` G ) $. | |
| $( A finite tree subgraph whose vertex count is still below the ambient | |
| count can be extended by one vertex. (Contributed by Mingli Yuan, | |
| 14-Aug-2026.) $) | |
| trextsubgrhash $p |- ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| W e. Fin ) /\ | |
| ( T e. Tree /\ T SubGraph G /\ ( # ` A ) = Y ) ) /\ | |
| ( Y e. NN /\ ( Y + 1 ) <_ ( # ` W ) ) ) -> | |
| E. s ( s e. Tree /\ s SubGraph G /\ | |
| ( # ` ( Vtx ` s ) ) = ( Y + 1 ) ) ) $= | |
| ( wcel w3a wbr chash cfv wceq wa syl cusgr cconngr cfn ctree csubgr cn c1 | |
| caddc co cle cv cvtx wex simpll simplr1 simplr2 cr simplr3 simprl eqeltrd | |
| wne nnre clt simprr cz nnz simpll3 hashcl nn0z 3syl zltp1led mpbird ltned | |
| eqbrtrd fveq2 necon3i 3jca trextsubgrex simpr1 simpr2 simpr3 simpl oveq1d | |
| cn0 jca eqtrd ex eximdv mpd ) DUAMZDUBMZFUCMZNZBUDMZBDUEOZAPQZGRZNZSZGUFM | |
| ZGUGUHUIZFPQZUJOZSZSZHUKZUDMZXFDUEOZXFULQPQZWPUGUHUIZRZNZHUMZXGXHXIXARZNZ | |
| HUMXEWMWNWOAFVAZNZSXMXEWMXQWMWRXDUNXEWNWOXPWNWOWQWMXDUOWNWOWQWMXDUPXEWPXB | |
| VAXPXEWPXBXEWPGUQWNWOWQWMXDURZXEWTGUQMWSWTXCUSZGVBTUTXEWPGXBVCXRXEGXBVCOX | |
| CWSWTXCVDXEGXBXEWTGVEMXSGVFTXEWLXBWDMXBVEMWJWKWLWRXDVGFVHXBVIVJVKVLVNVMAF | |
| WPXBAFPVOVPTVQWEABCDEFHIJKLVRTXEXLXOHXEXLXOXEXLSZXGXHXNXEXGXHXKVSXEXGXHXK | |
| VTXTXIXJXAXEXGXHXKWAXTWPGUGUHXTXEWQXEXLWBXRTWCWFVQWGWHWI $. | |
| $} | |
| ${ | |
| $d G T Y s $. | |
| $( A finite tree subgraph whose vertex count is still below that of the | |
| ambient graph can be extended by one vertex. This version expands the | |
| vertex- and indexed-edge-set abbreviations for use in induction. | |
| (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| trextsubgrhashfv $p |- ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ | |
| ( T e. Tree /\ T SubGraph G /\ | |
| ( # ` ( Vtx ` T ) ) = Y ) ) /\ | |
| ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) -> | |
| E. s ( s e. Tree /\ s SubGraph G /\ | |
| ( # ` ( Vtx ` s ) ) = ( Y + 1 ) ) ) $= | |
| ( cvtx cfv ciedg eqid trextsubgrhash ) AEFZAAGFZBBGFZBEFZCDJHKHMHLHI $. | |
| $} | |
| ${ | |
| $d F b $. $d G b $. $d K b $. $d M b $. $d N b $. $d P b $. | |
| $( A degree-one vertex on a simple path between two vertices is one of its | |
| endpoints. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| usgrspthonvd1inend $p |- ( ( G e. USGraph /\ | |
| F ( K ( SPathsOn ` G ) M ) P /\ | |
| ( ( ( VtxDeg ` G ) ` N ) = 1 /\ N e. ran P ) ) -> | |
| ( N = K \/ N = M ) ) $= | |
| ( vb wcel cfv co wbr c1 wceq wa w3a wo cc0 syl fveq2d simpl cvtxdg crn cv | |
| cusgr cspthson chash cfz wrex simp3r wfn wb wf1 cspths simp2 spthonisspth | |
| cvtx spthdifv fvelrnb mpbid simplrr eqcomd cwlkson simpl2 cpthson ctrlson | |
| f1fn simpr spthonpthon pthontrlon trlsonwlkon 3syl wlkonendpts 3eqtrd orc | |
| ex syl6 simprd olc simpl1 simprr simp3l 3jca simprl jca usgrspthvd1endidx | |
| eqtrd mpjaod rexlimddv ) CUDHZBADECUEIJKZFCUAIZIZLMZFAUBHZNZOZGUCZAIZFMZF | |
| DMZFEMZPZGQBUFIZUGJZWPWNWSGXDUHZWIWJWMWNUIWPAXDUJZWNXEUKWPXDCUPIZAULZXFWP | |
| BACUMIKZXHWPWJXIWIWJWOUNDEABCUOZRABCUQRXDXGAVFRGXDFAURRUSWPWQXDHZWSNZNZWQ | |
| QMZXBWQXCMZXMXNWTXBXMXNWTXMXNNZFWRQAIZDXPWRFWPXKWSXNUTVAXPWQQAXMXNVGSXPXQ | |
| DMZXCAIZEMZNZXRXPBADECVBIJKZYAXPXMYBXMXNTXMWJYBWIWJWOXLVCZWJBADECVDIJKBAD | |
| ECVEIJKYBDEABCVHDEABCVIDEABCVJVKRZRABCDEVLZRXRXTTRVMVOWTXAVNVPXMXOXAXBXMX | |
| OXAXMXONZFWRXSEYFWRFWPXKWSXOUTVAYFWQXCAXMXOVGSYFYBXTYFXMYBXMXOTYDRYBXRXTY | |
| EVQRVMVOXAWTVRVPXMWIXIWRWKIZLMZOZXKNXNXOPXMYIXKXMWIXIYHWIWJWOXLVSXMWJXIYC | |
| XJRXMYGWLLXMWRFWKWPXKWSVTSXMWPWMWPXLTWIWJWMWNWARWFWBWPXKWSWCWDWQABCWERWGW | |
| H $. | |
| $( A degree-one vertex different from both endpoints of a simple path is | |
| not on the path. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| usgrspthonvd1nran $p |- ( ( G e. USGraph /\ | |
| F ( K ( SPathsOn ` G ) M ) P /\ | |
| ( ( ( VtxDeg ` G ) ` N ) = 1 /\ N =/= K /\ N =/= M ) ) -> | |
| N e/ ran P ) $= | |
| ( cusgr wcel cspthson cfv wceq wne w3a wn wa df-ne sylibr sylib jca syl | |
| co wbr cvtxdg c1 crn wnel simp32 bicomi simp33 pm4.56 simpl1 simpl2 simpl | |
| wo simp31 simpr 3jca usgrspthonvd1inend ex mtod df-nel ) CGHZBADECIJUAUBZ | |
| FCUCJJUDKZFDLZFELZMZMZFAUEZHZNFVIUFVHVJFDKZFEKZUNZVHVKNZVLNZOVMNVHVNVOVHV | |
| EVNVBVCVDVEVFUGVEVNFDPUHQVHVFVOVBVCVDVEVFUIFEPRSVKVLUJRVHVJVMVHVJOZVBVCVD | |
| VJOZMVMVPVBVCVQVBVCVGVJUKVBVCVGVJULVPVDVJVPVHVDVHVJUMVBVCVDVEVFUOTVHVJUPS | |
| UQABCDEFURTUSUTFVIVAQ $. | |
| $} | |
| ${ | |
| $d E a b f i k m p $. $d F a b f i k m p $. $d G a b f i k m p $. | |
| $d I a b f i k m p $. $d K a b f i k m p $. $d M a b f i k m p $. | |
| $d N a b f i k m p $. $d P a b f i k m p $. $d S a b f i k m p $. | |
| $d V a b f i k m p $. $d a b f i k m p $. | |
| usgrspthres.v $e |- V = ( Vtx ` G ) $. | |
| usgrspthres.e $e |- E = ( iEdg ` G ) $. | |
| usgrspthres.i $e |- I = { i e. dom E | N e/ ( E ` i ) } $. | |
| usgrspthres.s $e |- S = <. ( V \ { N } ) , ( E |` I ) >. $. | |
| $( A simple path avoiding a removed vertex remains a simple path in the | |
| corresponding induced subgraph. (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| usgrspthres $p |- ( ( ( G e. USGraph /\ N e. V ) /\ | |
| ( F ( SPaths ` G ) P /\ N e/ ran P ) ) -> | |
| F ( SPaths ` S ) P ) $= | |
| ( vk wcel wa cfv wceq syl jca cusgr cspths wbr wnel cupgr cwlks ccnv wfun | |
| crn w3a simpl usgrres usgrupgr 3syl ciedg cdm cword cc0 chash cfz co cvtx | |
| wf cv c1 caddc cpr cfzo wral cvv simprl spthiswlk wlkf simpr wrdsymbcl wn | |
| wrdv wne simplrl eqid wlkp ffun elfzofz fdmd eleqtrrd simplrr 3jca necomd | |
| nelrnfvne fzofzp1 nelprd df-nel sylibr eqidd weq 2fveq3 id fveq2d preq12d | |
| fvoveq1d eqeq12d simpll upgrwlkedg rspcdva neleq12d mpbird wb crab eleq2i | |
| fveq2 neleq2 elrab2 bitri a1i ralrimiva iswrdsymb cres vtxdginducedm1lem2 | |
| csn cdif eqcomi wrdeqi eleqtrd wfn wss ffn frn simprr biimpi ssdifsn df-f | |
| uhgrspan1lem2 feq3 ax-mp sylib simplll vtxdginducedm1lem3 eqtrd upgriswlk | |
| spthf1 upgrwlkdvspth ) FUAOZHIOZPZEAFUBQUCZHAUIZUDZPZPZBUEOZEABUFQUCZAUGU | |
| HZUJEABUBQUCUUIUUJUUKUULUUIUUDBUAOUUJUUDUUHUKBCDGFHIJKLMULBUMUNZUUIUUKEBU | |
| OQZUPZUQZOZUREUSQZUTVAZBVBQZAVCZNVDZEQZUUNQZUVBAQZUVBVEVFVAZAQZVGZRZNURUU | |
| RVHVAZVIZUJZUUIUUQUVAUVKUUIEGUQZUUPUUIEVJUQOZUVCGOZNUVJVIZPEUVMOUUIUVNUVP | |
| UUIEDUPZUQOZUVNUUIUUEEAFUFQUCZUVRUUDUUEUUGVKZAEFVLZAEFDKVMZUNZUVQEVQSUUIU | |
| VONUVJUUIUVBUVJOZPZUVOUVCUVQOZHUVCDQZUDZPZUWEUWFUWHUWEUVRUWDPZUWFUWEUVRUW | |
| DUWEUUIUVRUUIUWDUKZUWCSUUIUWDVNZTUVBUVQEVOZSUWEUWHHUVHUDZUWEHUVHOVPUWNUWE | |
| HUVEUVGUWEUVEHUWEAUHZUVBAUPZOZUUGUJUVEHVRUWEUWOUWQUUGUWEUUSFVBQZAVCZUWOUW | |
| EUVSUWSUWEUUEUVSUUDUUEUUGUWDVSUWASZAEFUWRUWRVTWASZUUSUWRAWBSZUWEUVBUUSUWP | |
| UWEUWDUVBUUSOUWLUVBURUURWCSUWEUUSUWRAUXAWDZWEUUDUUEUUGUWDWFZWGAUVBHWISWHU | |
| WEUVGHUWEUWOUVFUWPOZUUGUJUVGHVRUWEUWOUXEUUGUXBUWEUVFUUSUWPUWEUWDUVFUUSOUW | |
| LURUURUVBWJSUXCWEUXDWGAUVFHWISWHWKHUVHWLWMZUWEHHUWGUVHUWEHWNUWECVDZEQDQZU | |
| XGAQZUXGVEVFVAAQZVGZRZUWGUVHRZCUVJUVBCNWOZUXHUWGUXKUVHUXGUVBDEWPUXNUXIUVE | |
| UXJUVGUXNUXGUVBAUXNWQZWRUXNUXGUVBVEAVFUXOWTWSXAZUWEUUIUXLCUVJVIZUWKUUIFUE | |
| OZUVSPZUXQUUIUXRUVSUUIUUBUXRUUBUUCUUHXBFUMZSUUIUUEUVSUVTUWASTACEFDKXCZSSU | |
| WLXDXEXFTUVOUWIXGUWEUVOUVCHUXGDQZUDZCUVQXHZOZUWIGUYDUVCLXIUYCUWHCUVCUVQUY | |
| DUXGUVCRZUYBUWGRUYCUWHXGUXGUVCDXJUYBUWGHXKSUYDVTZXLXMXNXFXOTNGEXPSUVMUUPR | |
| UUIGUUOUUOGDGXQZBCDFGIHXSXTZHIJKUYIVTZLUYHVTZMXRYAYBXNYCUUIUUSUYIAVCZUVAU | |
| UIUYLAUUSYDZUUFUYIYEZPZUUIUYMUYNUUIUUSIAVCZUYMUUIUUEUVSUYPUVTUWAAEFIJWAUN | |
| ZUUSIAYFSUUIUYNUUFIYEZHUUFOVPZPZUUIUYRUYSUUIUYPUYRUYQUUSIAYGSUUIUUGUYSUUD | |
| UUEUUGYHUUGUYSHUUFWLYISTUYNUYTXGUUIUUFIHYJXNXFTUYLUYOXGUUIUUSUYIAYKXNXFUY | |
| IUUTRUYLUVAXGUUTUYIBCGFDHIJKLMYLYAUYIUUTUUSAYMYNYOUUIUVINUVJUWEUVDUWGUVHU | |
| WEUVOUVDUWGRUWEUVCUYDGUWEUWIUYEUWEUWFUWHUWEUWJUWFUWEUVRUWDUWEUVSUVRUWTUWB | |
| SUWLTUWMSUWEUWHUWNUXFUWEUXMUWHUWNXGUWEUXLUXMCUVJUVBUXPUWEUXSUXQUWEUXRUVSU | |
| WEUUBUXRUUBUUCUUHUWDYPUXTSUWTTUYASUWLXDZUWGUVHHXKSXFTUYCUWHCUVCUVQUYDUYFH | |
| HUYBUWGHHRUYFHVTXNUYFUXGUVCDUYFWQWRXEUYGXLWMGUYDRUWELXNWEUYHBCDFUVCGUYIHI | |
| JKUYJLUYKMYQSVUAYRXOWGUUIUUJUUKUVLXGUUMANEBUUNUUTUUTVTUUNVTYSSXFUUIUUEUUL | |
| UVTAEFYTSWGAEBUUAS $. | |
| $( A simple path between two vertices which avoids a removed vertex remains | |
| a path between the same vertices in the corresponding induced subgraph. | |
| (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| usgrspthonres $p |- ( ( ( G e. USGraph /\ N e. V ) /\ | |
| ( F ( K ( SPathsOn ` G ) M ) P /\ N e/ ran P ) ) -> | |
| F ( K ( PathsOn ` S ) M ) P ) $= | |
| ( wa cfv co wbr 3syl cusgr wcel cspthson crn wnel cc0 cpthson wceq simprl | |
| cwlkson ctrlson spthonpthon trlsonwlkon wlkonendpts simpld simprd oveq12d | |
| chash pthontrlon cspths cpths simpl spthonisspth syl simprr jca spthispth | |
| usgrspthres pthonpth breqdi ) FUAUBJKUBPZEAHIFUCQRSZJAUDUEZPZPZUFAQZEURQA | |
| QZBUGQZRZHIVRREAVOVPHVQIVRVOVPHUHZVQIUHZVOVLEAHIFUJQRSZVTWAPVKVLVMUIZVLEA | |
| HIFUGQRSEAHIFUKQRSWBHIAEFULHIAEFUSHIAEFUMTAEFHIUNTZUOVOVTWAWDUPUQVOEABUTQ | |
| SZEABVAQSEAVSSVOVKEAFUTQSZVMPZPWEVOVKWGVKVNVBVOWFVMVOVLWFWCHIAEFVCVDVKVLV | |
| MVEVFVFABCDEFGJKLMNOVHVDAEBVGAEBVITVJ $. | |
| $( Removing a degree-one vertex from a tree leaves a connected graph. | |
| (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| treeleafresconngr $p |- ( ( G e. Tree /\ N e. V /\ | |
| ( ( VtxDeg ` G ) ` N ) = 1 ) -> S e. ConnGraph ) $= | |
| ( vf vp wcel cfv cv wex wa syl jca vk vm va ctree cvtxdg wceq w3a cconngr | |
| vb c1 cpthson co wbr csn cdif wral simprl eldifi simprr simpl1 treeconngr | |
| cvv wb isconngr mpbid weq id oveq1d breqd 2exbidv oveq2d rspc2va cspthson | |
| elex cusgr crn wnel treeusgr simpl2 adantr cacycgr simpl treeacycgr simpr | |
| pthonacycspthon 3syl simpl3 eldifsni 3jca usgrspthonvd1nran usgrspthonres | |
| wne necomd ex 2eximdv ralrimivva simp1 simp2 usgrres uhgrspan1lem2 eqcomi | |
| mpd cvtx mpbird ) DUDNZFGNZFDUEOOUJUFZUGZAUHNZLPZMPZUAPZUBPZAUKOULUMZMQLQ | |
| ZUBGFUNZUOZUPUAXQUPZXHXOUAUBXQXQXHXLXQNZXMXQNZRZRZXJXKXLXMDUKOZULZUMZMQLQ | |
| ZXOYBXLGNZXMGNZRZXJXKUCPZUIPZYCULZUMZMQLQZUIGUPUCGUPZRYFYBYIYOYBYGYHYBXSY | |
| GXHXSXTUQZXLGXPURSYBXTYHXHXSXTUSZXMGXPURSTYBDUHNZYOYBXEYRXEXFXGYAUTZDVASY | |
| BDVBNZYRYOVCYBXEYTYSDUDVNSLUCUIDGVBMHVDSVETYNYFXJXKXLYKYCULZUMZMQLQUCUIXL | |
| XMGGUCUAVFZYMUUBLMUUCYLUUAXJXKUUCYJXLYKYCUUCVGVHVIVJUIUBVFZUUBYELMUUDUUAY | |
| DXJXKUUDYKXMXLYCUUDVGVKVIVJVLSYBYEXNLMYBYEXNYBYERZDVONZXFRZXJXKXLXMDVMOUL | |
| UMZFXKVPVQZRZRXNUUEUUGUUJYBUUGYEYBUUFXFYBXEUUFYSDVRZSXEXFXGYAVSTVTUUEUUHU | |
| UIUUEDWANZYERZUUHUUEUULYEUUEXEUULUUEYBXEYBYEWBZYSSZDWCZSYBYEWDZTXKXJDXLXM | |
| WEZSUUEUUFUUHXGFXLWLZFXMWLZUGZUGUUIUUEUUFUUHUVAUUEXEUUFUUOUUKSUUEUUMUUHUU | |
| EUULYEUUEYBXEUULUUNYSUUPWFUUQTUURSUUEXGUUSUUTUUEYBXGUUNXEXFXGYAWGSUUEXLFU | |
| UEYBXSXLFWLUUNYPXLGFWHWFWMUUEXMFUUEYBXTXMFWLUUNYQXMGFWHWFWMWIWIXKXJDXLXMF | |
| WJSTTXKABCXJDEXLXMFGHIJKWKSWNWOXBWPXHAVBNZXIXRVCXHAVONZUVBXHUUGUVCXHUUFXF | |
| XHXEUUFXEXFXGWQUUKSXEXFXGWRTABCEDFGHIJKWSSAVOVNSLUAUBAXQVBMAXCOXQABEDCFGH | |
| IJKWTXAVDSXD $. | |
| $( Removing a vertex from a tree leaves an acyclic graph. | |
| (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| treevtxresacycgr $p |- ( ( G e. Tree /\ N e. V ) -> | |
| S e. AcyclicGraph ) $= | |
| ( ctree wcel wa cacycgr csubgr wbr simpl syl jca treeacycgr treeupgr 3syl | |
| cuhgr cupgr upgruhgr simpr uhgrspan1 acycgrsubgr ) DLMZFGMZNZDOMZADPQZNAO | |
| MULUMUNULUJUMUJUKRZDUASULDUDMZUKNUNULUPUKULUJDUEMUPUODUBDUFUCUJUKUGTABEDC | |
| FGHIJKUHSTADUIS $. | |
| $( Removing a leaf from a finite tree with at least two vertices leaves a | |
| tree. See Observation in [Diestel] p. 14. (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| treeleafres $p |- ( ( ( G e. Tree /\ V e. Fin /\ | |
| 1 < ( # ` V ) ) /\ | |
| ( N e. V /\ ( ( VtxDeg ` G ) ` N ) = 1 ) ) -> | |
| S e. Tree ) $= | |
| ( vb wcel cfv w3a wa c0 wne syl jca ctree cfn c1 chash clt wbr wceq cupgr | |
| cvtxdg cconngr cacycgr cvtx cusgr simpl1 treeusgr simprl usgrres usgrupgr | |
| 3syl simprr 3jca treeleafresconngr treevtxresacycgr csn uhgrspan1lem2 a1i | |
| cdif cv wrex simpl2 simpl3 hashgt12el2 simpl simpr necomd eldifsn biimpri | |
| ne0i ex rexlimiv eqnetrd wb istree mpbird ) DUAMZGUBMZUCGUDNUEUFZOZFGMZFD | |
| UINNUCUGZPZPZAUAMZAUHMZAUJMZAUKMZOZAULNZQRZPZWLWQWSWLWNWOWPWLDUMMZWIPAUMM | |
| WNWLXAWIWLWEXAWEWFWGWKUNZDUOSWHWIWJUPZTABCEDFGHIJKUQAURUSZWLWEWIWJOWOWLWE | |
| WIWJXBXCWHWIWJUTVAABCDEFGHIJKVBSWLWEWIPWPWLWEWIXBXCTABCDEFGHIJKVCSVAWLWRG | |
| FVDVGZQWRXEUGWLABEDCFGHIJKVEVFWLFLVHZRZLGVIZXEQRZWLWFWGWIOXHWLWFWGWIWEWFW | |
| GWKVJWEWFWGWKVKXCVAFGUBLVLSXGXILGXFGMZXGXIXJXGPZXFXEMZXIXKXJXFFRZPZXLXKXJ | |
| XMXJXGVMXKFXFXJXGVNVOTXLXNXFGFVPVQSXEXFVRSVSVTSWATWLWNWMWTWBXDAUHWCSWD $. | |
| $} | |
| ${ | |
| $d E i $. $d G i $. $d N i $. $d V i $. | |
| treeleafiedghash.v $e |- V = ( Vtx ` G ) $. | |
| treeleafiedghash.e $e |- E = ( iEdg ` G ) $. | |
| $( A leaf of a finite tree is incident with exactly one indexed edge. | |
| (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| treeleafiedghash $p |- ( ( ( G e. Tree /\ V e. Fin ) /\ | |
| ( N e. V /\ ( ( VtxDeg ` G ) ` N ) = 1 ) ) -> | |
| ( # ` { i e. dom E | N e. ( E ` i ) } ) = 1 ) $= | |
| ( wcel cfn wa cfv c1 wceq crab chash caddc co cc0 syl c0 ctree cvtxdg cdm | |
| cv csn cfusgr cupgr w3a cusgr simpll treeusgr simplr isfusgr fusgrfupgrfs | |
| jca sylibr simp3 3syl dmfi simprl eqid vtxdgfival eqidd usgrnloop0 fveq2d | |
| hash0 a1i eqtrd oveq12d cn0 cc rabfi hashcl addridd 3eqtrd eqcomd simprr | |
| nn0cn ) CUAHZEIHZJZDEHZDCUBKKZLMZJZJZDAUDBKZHZABUCZNZOKZWCLWFWCWKWFWCWKWG | |
| DUEMAWINZOKZPQZWKRPQWKWFWIIHZWBJWCWNMWFWOWBWFBIHZWOWFCUFHZCUGHZVTWPUHWPWF | |
| CUIHZVTJWQWFWSVTWFVSWSVSVTWEUJCUKSZVSVTWEULUOCEFUMUPCBEFGUNWRVTWPUQURBUSS | |
| ZWAWBWDUTUOAWIDCBEFGWIVAVBSWFWKWKWMRPWFWKVCWFWMTOKZRWFWLTOWFWSWLTMWTADBCG | |
| VDSVEXBRMWFVFVGVHVIWFWKWFWKVJHZWKVKHWFWJIHZXCWFWOXDXAWHAWIVLSWJVMSWKVRSVN | |
| VOVPWAWBWDVQVH $. | |
| $} | |
| ${ | |
| $d E i $. $d G i $. $d I i $. $d J i $. $d N i $. $d V i $. | |
| treeleafedgstep.v $e |- V = ( Vtx ` G ) $. | |
| treeleafedgstep.e $e |- E = ( iEdg ` G ) $. | |
| treeleafedgstep.k $e |- K = ( V \ { N } ) $. | |
| treeleafedgstep.i $e |- I = { i e. dom E | N e/ ( E ` i ) } $. | |
| treeleafedgstep.p $e |- P = ( E |` I ) $. | |
| treeleafedgstep.s $e |- S = <. K , P >. $. | |
| treeleafedgstep.j $e |- J = { i e. dom E | N e. ( E ` i ) } $. | |
| $( Removing a leaf from a finite tree removes exactly one indexed edge. | |
| (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| treeleafedgstep $p |- ( ( ( G e. Tree /\ V e. Fin ) /\ | |
| ( N e. V /\ ( ( VtxDeg ` G ) ` N ) = 1 ) ) -> | |
| ( # ` dom E ) = ( ( # ` dom P ) + 1 ) ) $= | |
| ( wcel wa cfv ctree cfn cvtxdg c1 wceq cdm chash caddc co wfun ciedg eqid | |
| cvv fvexi eqeltri a1i cuhgr simpll treeupgr upgruhgr 3syl uhgrfun syl jca | |
| cupgr hashfundm eqcomd simprl simplr cfusgr cusgr treeusgr isfusgr sylibr | |
| w3a fusgrfupgrfs simp3 finsumvtxdg2ssteplem1 cres funres funeqd mpbird cv | |
| resex crab fveq2i treeleafiedghash eqtrd oveq12d 3eqtrd ) EUARZJUBRZSZIJR | |
| ZIEUCTTUDUEZSZSZDUFZUGTZDUGTZAUGTZGUGTZUHUIZAUFUGTZUDUHUIWQWTWSWQDUMRZDUJ | |
| ZSWTWSUEWQXEXFXEWQDEUKTZUMLXGEUKXGULUNUOZUPWQEUQRZXFWQWKEVERZXIWKWLWPURZE | |
| USZEUTVADELVBVCZVDDUMVFVCVGWQXJWNSZWLDUBRZSZSWTXCUEWQXNXPWQXJWNWQWKXJXKXL | |
| VCWMWNWOVHVDWQWLXOWKWLWPVIZWQEVJRZXJWLXOVOXOWQEVKRZWLSXRWQXSWLWQWKXSXKEVL | |
| VCXQVDEJKVMVNEDJKLVPXJWLXOVQVAVDVDABCDEFGHIJKLMNOPQVRVCWQXAXDXBUDUHWQAUMR | |
| ZAUJZSXAXDUEWQXTYAXTWQADFVSZUMODFXHWDUOUPWQYAYBUJZWQXFYCXMFDVTVCWQAYBAYBU | |
| EWQOUPWAWBVDAUMVFVCWQXBICWCDTRCWRWEZUGTZUDXBYEUEWQGYDUGQWFUPCDEIJKLWGWHWI | |
| WJ $. | |
| $} | |
| ${ | |
| tree1v0iedghash.v $e |- V = ( Vtx ` G ) $. | |
| tree1v0iedghash.i $e |- I = ( iEdg ` G ) $. | |
| $( A tree with one vertex has no indexed edges. | |
| (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| tree1v0iedghash $p |- ( ( G e. Tree /\ ( # ` V ) = 1 ) -> | |
| ( # ` dom I ) = 0 ) $= | |
| ( ctree wcel chash cfv c1 wceq wa cdm cedg cc0 cvv ciedg eqid fvexi syl | |
| eqeltri dmex cusgr wf1o simpl treeusgr usgrf1oedg hasheqf1od eqidd anim1i | |
| a1i usgr1v0e 3eqtrd ) AFGZCHIJKZLZBMZHIANIZHIZUSOUPUQURPBUQPGUPBBAQIZPEUT | |
| AQUTRSUAUBUKUPAUCGZUQURBUDUPUNVAUNUOUEAUFZTURABEURRZUGTUHUPUSUIUPVAUOLUSO | |
| KUNVAUOVBUJURACDVCULTUM $. | |
| $} | |
| ${ | |
| $d E g i $. $d G g i $. $d I g i $. $d J g i $. $d K g i $. | |
| $d N g i $. $d P g i $. $d S g i $. $d V g i $. $d Y g i $. | |
| treeiedghashleaf.v $e |- V = ( Vtx ` G ) $. | |
| treeiedghashleaf.e $e |- E = ( iEdg ` G ) $. | |
| treeiedghashleaf.k $e |- K = ( V \ { N } ) $. | |
| treeiedghashleaf.i $e |- I = { i e. dom E | N e/ ( E ` i ) } $. | |
| treeiedghashleaf.p $e |- P = ( E |` I ) $. | |
| treeiedghashleaf.s $e |- S = <. K , P >. $. | |
| treeiedghashleaf.j $e |- J = { i e. dom E | N e. ( E ` i ) } $. | |
| treeiedghashleaf.h $e |- A. g ( ( g e. Tree /\ | |
| ( Vtx ` g ) e. Fin /\ ( # ` ( Vtx ` g ) ) = Y ) -> | |
| ( ( # ` dom ( iEdg ` g ) ) + 1 ) = ( # ` ( Vtx ` g ) ) ) $. | |
| $( The induction hypothesis for a smaller tree proves the indexed-edge | |
| count after removing a leaf. (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| treeiedghashleaf $p |- ( ( ( Y e. NN /\ ( G e. Tree /\ V e. Fin /\ | |
| ( # ` V ) = ( Y + 1 ) ) ) /\ | |
| ( N e. V /\ ( ( VtxDeg ` G ) ` N ) = 1 ) ) -> | |
| ( ( # ` dom E ) + 1 ) = ( # ` V ) ) $= | |
| ( cn wcel ctree cfn chash cfv c1 caddc co wceq w3a cvtxdg cdm simpl simpr | |
| wa 3simpa 3syl jca treeleafedgstep syl ciedg cvtx cop a1i fveq2d cvv cdif | |
| csn eqid fvexi eqeltri difexi cres resex opiedgfv eqtrd eqcomd oveq1d clt | |
| dmeqd wbr simp1 simp2 simpll cc0 nngt0 cr wb 1re ltaddpos2 mpbid breqtrrd | |
| simp3 opeq12i eqtri treeleafres uhgrspan1lem2 diffi eqeltrd eqtr4d fveq2i | |
| nnre 3jca cn0 wi elex simprl nnnn0 hashdifsnp1 mpd wal opex eleq1d eqeq1d | |
| cv id 3anbi123d eqeq12d imbi12d spcv ax-mp 3eqtrd ) LUAUBZFUCUBZKUDUBZKUE | |
| UFZLUGUHUIZUJZUKZUPZJKUBZJFULUFUFUGUJZUPZUPZEUMUEUFZUGUHUIYHYGYOYPLUGUHYO | |
| YPAUMZUEUFZUGUHUIZIUEUFZLYOYEYFUPZYNUPYPYSUJYOUUAYNYOYKYJUUAYKYNUNZYDYJUO | |
| ZYEYFYIUQURYKYNUOZUSABDEFGHIJKMNOPQRSUTVAYOYSBVBUFZUMZUEUFZUGUHUIZBVCUFZU | |
| EUFZYTYOYRUUGUGUHYOYQUUFUEYOAUUEYOUUEAYOUUEIAVDZVBUFZAYOBUUKVBBUUKUJYORVE | |
| VFYOIVGUBZAVGUBZUPUULAUJYOUUMUUNUUMYOIKJVIZVHZVGOKUUOKFVCUFZVGMUUQFVCUUQV | |
| JVKVLVMVLVEUUNYOAEGVNZVGQEGEFVBUFZVGNUUSFVBUUSVJVKVLVOVLVEUSAIVGVGVPVAVQV | |
| RWAVFVSYOBUCUBZUUIUDUBZUUJLUJZUKZUUHUUJUJZYOUUTUVAUVBYOYEYFUGYGVTWBZUKZYN | |
| UPUUTYOUVFYNYOYEYFUVEYOYKYJYEUUBUUCYEYFYIWCURYOYKYJYFUUBUUCYEYFYIWDURZYOU | |
| GYHYGVTYOYDUGYHVTWBZYDYJYNWEZYDWFLVTWBZUVHLWGYDLWHUBZUGWHUBZUPUVJUVHWIYDU | |
| VKUVLLXCUVLYDWJVEUSLUGWKVAWLVAYOYKYJYIUUBUUCYEYFYIWNURZWMXDUUDUSBDEFGJKMN | |
| PBUUKUUPUURVDRIUUPAUUROQWOWPZWQVAYOUUIUUPUDUUIUUPUJYOBDGFEJKMNPUVNWRVEZYO | |
| YFUUPUDUBUVGKUUOWSVAWTYOUUJYTLYOUUIIUEYOUUIUUPIUVOIUUPUJYOOVEXAVFZYOYTUUP | |
| UEUFZLYTUVQUJYOIUUPUEOXBVEYOYIUVQLUJZUVMYOKVGUBZYLLXEUBZUKYIUVRXFYOUVSYLU | |
| VTYOYFUVSUVGKUDXGVAYKYLYMXHYOYDUVTUVILXIVAXDJKVGLXJVAXKVQZVQXDUVCUVDXFZYO | |
| CXPZUCUBZUWCVCUFZUDUBZUWEUEUFZLUJZUKZUWCVBUFZUMZUEUFZUGUHUIZUWGUJZXFZCXLU | |
| WBTUWOUWBCBBUUKVGRIAXMVLUWCBUJZUWIUVCUWNUVDUWPUWDUUTUWFUVAUWHUVBUWPUWCBUC | |
| UWPXQZXNUWPUWEUUIUDUWPUWCBVCUWQVFZXNUWPUWGUUJLUWPUWEUUIUEUWRVFZXOXRUWPUWM | |
| UUHUWGUUJUWPUWLUUGUGUHUWPUWKUUFUEUWPUWJUUEUWPUWCBVBUWQVFWAVFVSUWSXSXTYAYB | |
| VEXKUVPYCUWAYCVSYOYGYHUVMVRVQ $. | |
| $} | |
| ${ | |
| $d E g i $. $d G g i $. $d I g i $. $d J g i $. $d K g i $. | |
| $d N g i $. $d P g i $. $d S g i $. $d V g i $. $d Y g i $. | |
| treeiedghashleafd.v $e |- V = ( Vtx ` G ) $. | |
| treeiedghashleafd.e $e |- E = ( iEdg ` G ) $. | |
| treeiedghashleafd.k $e |- K = ( V \ { N } ) $. | |
| treeiedghashleafd.i $e |- I = { i e. dom E | N e/ ( E ` i ) } $. | |
| treeiedghashleafd.p $e |- P = ( E |` I ) $. | |
| treeiedghashleafd.s $e |- S = <. K , P >. $. | |
| treeiedghashleafd.j $e |- J = { i e. dom E | N e. ( E ` i ) } $. | |
| treeiedghashleafd.h $e |- ( ph -> A. g ( ( g e. Tree /\ | |
| ( Vtx ` g ) e. Fin /\ ( # ` ( Vtx ` g ) ) = Y ) -> | |
| ( ( # ` dom ( iEdg ` g ) ) + 1 ) = ( # ` ( Vtx ` g ) ) ) ) $. | |
| $( Deduction form of the leaf-removal induction step for the indexed-edge | |
| count of a finite tree. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| treeiedghashleafd $p |- ( ( ph /\ | |
| ( ( Y e. NN /\ ( G e. Tree /\ V e. Fin /\ | |
| ( # ` V ) = ( Y + 1 ) ) ) /\ | |
| ( N e. V /\ ( ( VtxDeg ` G ) ` N ) = 1 ) ) ) -> | |
| ( ( # ` dom E ) + 1 ) = ( # ` V ) ) $= | |
| ( cn wcel ctree cfn chash cfv c1 caddc co wceq w3a cvtxdg cdm simpr simpl | |
| 3simpa 3syl jca ancom biimpi treeleafedgstep syl ciedg cvtx cres cop cdif | |
| wa csn opeq12i eqtri uhgrspan1lem3 eqtr4i dmeqi fveq2i a1i oveq1d clt wbr | |
| simp1 simp2 simpll cc0 nngt0 cr wb nnre 1re ltaddpos2 mpbid breqtrrd 3jca | |
| simp3 treeleafres uhgrspan1lem2 diffi eqeltrd fveq2d cvv cn0 simprl nnnn0 | |
| wi elex hashdifsnp1 mpd eqtrd cv wal opex eqeltri eleq1d eqeq1d 3anbi123d | |
| id dmeqd eqeq12d imbi12d spcv 3eqtrd eqcomd ) AMUBUCZGUDUCZLUEUCZLUFUGZMU | |
| HUIUJZUKZULZVIZKLUCZKGUMUGUGUHUKZVIZVIZVIZFUNUFUGZUHUIUJYGYFYOYPMUHUIYOYP | |
| BUNZUFUGZUHUIUJZJUFUGZMYOYNYPYSUKZAYNUOZYNYMYDYEVIZVIZUUAYNYMUUCYJYMUOZYN | |
| YJYIUUCYJYMUPZYCYIUOZYDYEYHUQURUSUUDUUCYMVIZUUAUUDUUHYMUUCUTVABCEFGHIJKLN | |
| OPQRSTVBVCVCVCYOYSCVDUGZUNZUFUGZUHUIUJZCVEUGZUFUGZYTYOYRUUKUHUIYRUUKUKYOY | |
| QUUJUFBUUIBFHVFZUUIRCEHGFKLNOQCJBVGZLKVJZVHZUUOVGSJUURBUUOPRVKVLZVMVNVOVP | |
| VQVRYOCUDUCZUUMUEUCZUUNMUKZULZUULUUNUKZYOYNUVCUUBYNUUTUVAUVBYNYDYEUHYFVSV | |
| TZULZYMVIUUTYNUVFYMYNYDYEUVEYNYJYIYDUUFUUGYDYEYHWAURYNYJYIYEUUFUUGYDYEYHW | |
| BURZYNUHYGYFVSYNYCUHYGVSVTZYCYIYMWCZYCWDMVSVTZUVHMWEYCMWFUCZUHWFUCZVIUVJU | |
| VHWGYCUVKUVLMWHUVLYCWIVQUSMUHWJVCWKVCYNYJYIYHUUFUUGYDYEYHWNURZWLWMUUEUSCE | |
| FGHKLNOQUUSWOVCYNUUMUURUEUUMUURUKYNCEHGFKLNOQUUSWPZVQZYNYEUURUEUCUVGLUUQW | |
| QVCWRYNUUNUURUFUGZMYNUUMUURUFUVOWSYNYHUVPMUKZUVMYNLWTUCZYKMXAUCZULYHUVQXD | |
| YNUVRYKUVSYNYEUVRUVGLUEXEVCYJYKYLXBYNYCUVSUVIMXCVCWMKLWTMXFVCXGZXHWMVCYOA | |
| DXIZUDUCZUWAVEUGZUEUCZUWCUFUGZMUKZULZUWAVDUGZUNZUFUGZUHUIUJZUWEUKZXDZDXJU | |
| VCUVDXDZAYNUPUAUWMUWNDCCUUPWTSJBXKXLUWACUKZUWGUVCUWLUVDUWOUWBUUTUWDUVAUWF | |
| UVBUWOUWACUDUWOXPZXMUWOUWCUUMUEUWOUWACVEUWPWSZXMUWOUWEUUNMUWOUWCUUMUFUWQW | |
| SZXNXOUWOUWKUULUWEUUNUWOUWJUUKUHUIUWOUWIUUJUFUWOUWHUUIUWOUWACVDUWPWSXQWSV | |
| RUWRXRXSXTURXGUUNYTUKYOUUMJUFUUMUURJUVNPVNVPVQYAYOYNYTMUKUUBYNYTUVPMYTUVP | |
| UKYNJUURUFPVPVQUVTXHVCYAVRYOYFYGYOYNYHUUBUVMVCYBXH $. | |
| $} | |
| ${ | |
| $d Y g h $. | |
| $( Change the bound graph variable in the induction predicate for the | |
| indexed-edge count of a finite tree. (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| treeiedghashcbv $p |- ( A. g ( ( g e. Tree /\ ( Vtx ` g ) e. Fin /\ | |
| ( # ` ( Vtx ` g ) ) = Y ) -> | |
| ( ( # ` dom ( iEdg ` g ) ) + 1 ) = ( # ` ( Vtx ` g ) ) ) -> | |
| A. h ( ( h e. Tree /\ ( Vtx ` h ) e. Fin /\ | |
| ( # ` ( Vtx ` h ) ) = Y ) -> | |
| ( ( # ` dom ( iEdg ` h ) ) + 1 ) = ( # ` ( Vtx ` h ) ) ) ) $= | |
| ( cv ctree wcel cvtx cfv cfn chash wceq w3a ciedg cdm caddc eleq1d fveq2d | |
| c1 co wi id eqeq1d 3anbi123d dmeqd oveq1d eqeq12d imbi12d biimpd cbvalivw | |
| weq ) ADZEFZUKGHZIFZUMJHZCKZLZUKMHZNZJHZROSZUOKZTZBDZEFZVDGHZIFZVFJHZCKZL | |
| ZVDMHZNZJHZROSZVHKZTZABABUJZVCVPVQUQVJVBVOVQULVEUNVGUPVIVQUKVDEVQUAZPVQUM | |
| VFIVQUKVDGVRQZPVQUOVHCVQUMVFJVSQZUBUCVQVAVNUOVHVQUTVMROVQUSVLJVQURVKVQUKV | |
| DMVRQUDQUEVTUFUGUHUI $. | |
| $} | |
| ${ | |
| $d G g h i j k n $. $d Y g h i j k n $. | |
| $( The induction hypothesis for trees with ` Y ` vertices proves the | |
| indexed-edge count for a fixed tree with ` Y + 1 ` vertices. | |
| (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| treeiedghashstep $p |- ( Y e. NN -> | |
| ( A. g ( ( g e. Tree /\ ( Vtx ` g ) e. Fin /\ | |
| ( # ` ( Vtx ` g ) ) = Y ) -> | |
| ( ( # ` dom ( iEdg ` g ) ) + 1 ) = ( # ` ( Vtx ` g ) ) ) -> | |
| ( ( G e. Tree /\ ( Vtx ` G ) e. Fin /\ | |
| ( # ` ( Vtx ` G ) ) = ( Y + 1 ) ) -> | |
| ( ( # ` dom ( iEdg ` G ) ) + 1 ) = | |
| ( # ` ( Vtx ` G ) ) ) ) ) $= | |
| ( vn vj vi vh vk wcel cv cfv chash wceq w3a c1 co syl wa jca eqid cn cvtx | |
| ctree cfn ciedg cdm caddc wal cvtxdg wrex clt wbr simp3 simp1 simp2 nngt0 | |
| wi cc0 cr wb nnre 1re a1i ltaddpos2 mpbid breqtrrd 3jca cwwspthsn wne cfz | |
| c0 crab weq id oveq1d neeq1d cbvrabv treeleaf simpll simplr wnel cres csn | |
| simpr cop eqidd fveq2d neleq12d eleq12d treeiedghashcbv treeiedghashleafd | |
| cdif 3ad2ant2 ex rexlimdva mpd 3exp ) CUAIZAJZUCIWSUBKZUDIWTLKZCMNWSUEKUF | |
| LKOUGPXAMUQAUHZBUCIZBUBKZUDIZXDLKZCOUGPZMZNZBUEKZUFZLKOUGPXFMZWRXBXINZDJZ | |
| BUIKKOMZDXDUJZXLXMXCXEOXFUKULZNXPXMXCXEXQXMXIXCWRXBXIUMZXCXEXHUNQXMXIXEXR | |
| XCXEXHUOQXMOXGXFUKXMWROXGUKULZWRXBXIUNZWRURCUKULZXSCUPWRCUSIZOUSIZRYAXSUT | |
| WRYBYCCVAYCWRVBVCSCOVDQVEQXMXIXHXRXCXEXHUMQVFVGDEBFJZBVHPZVKVIZFURXFVJPZV | |
| LXDXDTZYFEJZBVHPZVKVIFEYGFEVMZYEYJVKYKYDYIBVHYKVNVOVPVQVRQXMXOXLDXDXMXNXD | |
| IZRZXOXLYMXORZXMWRXIRZYLXORZRZRXLYNXMYQXMYLXOVSZYNYOYPYNWRXIYNXMWRYRXTQYN | |
| XMXIYRXRQSYNYLXOXMYLXOVTYMXOWDSSSXMXJXNYDXJKZWAZFXKVLZWBZXDXNWCWLZUUBWEZG | |
| HXJBUUAXNYSIZFXKVLUUCXNXDCYHXJTUUCTYTXNHJZXJKZWAFHXKFHVMZXNXNYSUUGUUHXNWF | |
| ZUUHYDUUFXJUUHVNWGZWHVQUUBTUUDTUUEXNUUGIFHXKUUHXNXNYSUUGUUIUUJWIVQXBWRGJZ | |
| UCIUUKUBKZUDIUULLKZCMNUUKUEKUFLKOUGPUUMMUQGUHXIAGCWJWMWKQWNWOWPWQ $. | |
| $} | |
| ${ | |
| $d Y g h $. | |
| $( The indexed-edge count induction predicate for finite trees is closed | |
| under successor. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| treeiedghashsucc $p |- ( Y e. NN -> | |
| ( A. h ( ( h e. Tree /\ ( Vtx ` h ) e. Fin /\ | |
| ( # ` ( Vtx ` h ) ) = Y ) -> | |
| ( ( # ` dom ( iEdg ` h ) ) + 1 ) = ( # ` ( Vtx ` h ) ) ) -> | |
| A. h ( ( h e. Tree /\ ( Vtx ` h ) e. Fin /\ | |
| ( # ` ( Vtx ` h ) ) = ( Y + 1 ) ) -> | |
| ( ( # ` dom ( iEdg ` h ) ) + 1 ) = | |
| ( # ` ( Vtx ` h ) ) ) ) ) $= | |
| ( vg cn wcel cv ctree cvtx cfv cfn chash w3a ciedg cdm c1 caddc co wi wal | |
| wceq treeiedghashstep alrimdv treeiedghashcbv syl6 ) BDEZAFZGEZUFHIZJEZUH | |
| KIZBTLUFMINKIOPQUJTZRASZCFZGEUMHIZJEUNKIZBOPQZTLUMMINKIOPQUOTRZCSUGUIUJUP | |
| TLUKRASUEULUQCAUMBUAUBCAUPUCUD $. | |
| $} | |
| $( The indexed-edge count identity holds for every tree with one vertex. | |
| (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| treeiedghashbase $p |- A. h ( ( h e. Tree /\ | |
| ( Vtx ` h ) e. Fin /\ ( # ` ( Vtx ` h ) ) = 1 ) -> | |
| ( ( # ` dom ( iEdg ` h ) ) + 1 ) = ( # ` ( Vtx ` h ) ) ) $= | |
| ( cv ctree wcel cvtx cfv cfn chash c1 wceq w3a ciedg cdm caddc co wi cc0 wa | |
| simp1 eqid simp3 jca tree1v0iedghash syl oveq1d 0p1e1 eqcomd 3eqtrd ax-gen | |
| a1i ) ABZCDZUKEFZGDZUMHFZIJZKZUKLFZMHFZINOZUOJPAUQUTQINOZIUOUQUSQINUQULUPRU | |
| SQJUQULUPULUNUPSULUNUPUAZUBUKURUMUMTURTUCUDUEVAIJUQUFUJUQUOIVBUGUHUI $. | |
| ${ | |
| $d A h $. $d B h $. | |
| $( Congruence of the indexed-edge count induction predicate in its vertex | |
| count parameter. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| treeiedghashbid $p |- ( A = B -> | |
| ( A. h ( ( h e. Tree /\ ( Vtx ` h ) e. Fin /\ | |
| ( # ` ( Vtx ` h ) ) = A ) -> | |
| ( ( # ` dom ( iEdg ` h ) ) + 1 ) = ( # ` ( Vtx ` h ) ) ) <-> | |
| A. h ( ( h e. Tree /\ ( Vtx ` h ) e. Fin /\ | |
| ( # ` ( Vtx ` h ) ) = B ) -> | |
| ( ( # ` dom ( iEdg ` h ) ) + 1 ) = | |
| ( # ` ( Vtx ` h ) ) ) ) ) $= | |
| ( wceq cv ctree wcel cfv cfn chash w3a ciedg cdm c1 caddc co wi id eqeq2d | |
| cvtx 3anbi3d imbi1d albidv ) ABDZCEZFGZUETHZIGZUGJHZADZKZUELHMJHNOPUIDZQU | |
| FUHUIBDZKZULQCUDUKUNULUDUJUMUFUHUDABUIUDRSUAUBUC $. | |
| $} | |
| ${ | |
| $d h x y $. $d Y h x $. | |
| $( The indexed-edge count identity holds for every positive finite-tree | |
| vertex count. (Contributed by Mingli Yuan, 12-Aug-2026.) $) | |
| treeiedghashn $p |- ( Y e. NN -> | |
| A. h ( ( h e. Tree /\ ( Vtx ` h ) e. Fin /\ | |
| ( # ` ( Vtx ` h ) ) = Y ) -> | |
| ( ( # ` dom ( iEdg ` h ) ) + 1 ) = | |
| ( # ` ( Vtx ` h ) ) ) ) $= | |
| ( vx vy cv ctree wcel cvtx cfv cfn chash wceq ciedg caddc treeiedghashbid | |
| w3a c1 co wi wal cdm treeiedghashbase treeiedghashsucc nnind ) AEZFGZUEHI | |
| ZJGZUGKIZCEZLPUEMIUAKIQNRUILZSATUFUHUIQLPUKSATUFUHUIDEZLPUKSATUFUHUIULQNR | |
| ZLPUKSATUFUHUIBLPUKSATCDBUJQAOUJULAOUJUMAOUJBAOAUBAULUCUD $. | |
| $} | |
| ${ | |
| $d G t u $. $d W t u $. $d t u $. | |
| sptrbase.w $e |- W = ( Vtx ` G ) $. | |
| $( A nonempty simple graph has a one-vertex tree subgraph. | |
| (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| sptrbase $p |- ( ( G e. USGraph /\ W =/= (/) ) -> | |
| E. t ( t e. Tree /\ t SubGraph G /\ | |
| ( # ` ( Vtx ` t ) ) = 1 ) ) $= | |
| ( vu cusgr wcel c0 wa cv ctree csubgr cvtx cfv chash c1 wceq cvv elex syl | |
| wne wex wbr w3a simpr bicomi sylibr csn cop simpl sngrtree simprl eleqtrd | |
| n0 id a1i jca sngrsubgr snexg 0ex opvtxfv fveq2d hashsng eqtrd 3jca eleq1 | |
| breq1 2fveq3 eqeq1d 3anbi123d spcedv expcom exlimdv mpd ) BFGZCHUAZIZEJZC | |
| GZEUBZAJZKGZWABLUCZWAMNONZPQZUDZAUBZVQVPVTVOVPUEVPVTECUNUFUGVQVSWGEVSVQWG | |
| VSVQIZWFVRUHZHUIZKGZWJBLUCZWJMNZONZPQZUDARWJWHWKWJRGWHVRRGZWKWHVSWPVSVQUJ | |
| ZVRCSTZVRUKTZWJKSTWHWKWLWOWSWHBRGZVRBMNZGZIWLWHWTXBWHVOWTVSVOVPULBFSTWHVS | |
| XBWQVSVRCXAVSUOCXAQVSDUPUMTUQVRBURTWHWPWOWRWPWNWIONPWPWMWIOWPWIRGZHRGZIWM | |
| WIQWPXCXDVRRUSXDWPUTUPUQHWIRRVATVBVRRVCVDTVEWAWJQZWBWKWCWLWEWOWAWJKVFWAWJ | |
| BLVGXEWDWNPWAWJOMVHVIVJVKVLVMVN $. | |
| $} | |
| ${ | |
| $d G t $. | |
| $( A nonempty simple graph has a one-vertex tree subgraph, with the vertex | |
| set abbreviation expanded. (Contributed by Mingli Yuan, | |
| 14-Aug-2026.) $) | |
| sptrbasefv $p |- ( ( G e. USGraph /\ ( Vtx ` G ) =/= (/) ) -> | |
| E. t ( t e. Tree /\ t SubGraph G /\ | |
| ( # ` ( Vtx ` t ) ) = 1 ) ) $= | |
| ( cvtx cfv eqid sptrbase ) ABBCDZGEF $. | |
| $} | |
| ${ | |
| $d G Y h t $. | |
| $( Substitution in the tree-subgraph existence predicate. | |
| (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| sptrcbvlem $p |- ( h = t -> | |
| ( ( h e. Tree /\ h SubGraph G /\ | |
| ( # ` ( Vtx ` h ) ) = Y ) <-> | |
| ( t e. Tree /\ t SubGraph G /\ | |
| ( # ` ( Vtx ` t ) ) = Y ) ) ) $= | |
| ( weq ctree wcel csubgr wbr cvtx cfv chash wceq eleq1 breq1 fveq2d eqeq1d | |
| cv id 3anbi123d ) BAEZBRZFGARZFGUBCHIUCCHIUBJKZLKZDMUCJKZLKZDMUBUCFNUBUCC | |
| HOUAUEUGDUAUDUFLUAUBUCJUASPPQT $. | |
| $( Change the bound variable in the tree-subgraph existence predicate. | |
| (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| sptrcbv $p |- ( E. h ( h e. Tree /\ h SubGraph G /\ | |
| ( # ` ( Vtx ` h ) ) = Y ) <-> | |
| E. t ( t e. Tree /\ t SubGraph G /\ | |
| ( # ` ( Vtx ` t ) ) = Y ) ) $= | |
| ( cv ctree wcel csubgr wbr cvtx cfv chash wceq w3a sptrcbvlem cbvexvw ) B | |
| EZFGQCHIQJKLKDMNAEZFGRCHIRJKLKDMNBAABCDOP $. | |
| $} | |
| ${ | |
| $d G Y s t $. | |
| $( An existentially given tree satisfying the extension hypotheses can be | |
| enlarged by one vertex. (Contributed by Mingli Yuan, | |
| 14-Aug-2026.) $) | |
| sptrgrow $p |- ( E. t ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ | |
| ( t e. Tree /\ t SubGraph G /\ | |
| ( # ` ( Vtx ` t ) ) = Y ) ) /\ | |
| ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) -> | |
| E. s ( s e. Tree /\ s SubGraph G /\ | |
| ( # ` ( Vtx ` s ) ) = ( Y + 1 ) ) ) $= | |
| ( cusgr wcel cconngr cvtx cfv cfn w3a cv ctree csubgr chash wceq wa cn c1 | |
| wbr caddc co cle wex trextsubgrhashfv exlimiv ) BEFBGFBHIZJFKALZMFUHBNTUH | |
| HIOICPKQCRFCSUAUBZUGOIUCTQQDLZMFUJBNTUJHIOIUIPKDUDAUHBCDUEUF $. | |
| $} | |
| ${ | |
| $d G Y $. | |
| $( A successor cardinal bound implies the corresponding predecessor | |
| bound for a nonempty finite graph. (Contributed by Mingli Yuan, | |
| 14-Aug-2026.) $) | |
| sptryle $p |- ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) -> | |
| Y <_ ( # ` ( Vtx ` G ) ) ) $= | |
| ( cusgr wcel cconngr cvtx cfv cfn w3a c0 wne wa cn caddc cle wbr nnre syl | |
| c1 cr co chash simprl simplr simpll3 hashnncl mpbird simprr p1le syl3anc | |
| wb ) ACDZAEDZAFGZHDZIZUNJKZLZBMDZBSNUAUNUBGZOPZLZLZBTDZUTTDZVABUTOPVCUSVD | |
| URUSVAUCBQRVCUTMDZVEVCVFUQUPUQVBUDVCUOVFUQUKULUMUOUQVBUEUNUFRUGUTQRURUSVA | |
| UHBUTUIUJ $. | |
| $} | |
| ${ | |
| $d G Y h $. | |
| $( The bounded induction hypothesis supplies a tree at the predecessor | |
| cardinal. (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| sptrih $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) /\ | |
| ( Y <_ ( # ` ( Vtx ` G ) ) -> | |
| E. h ( h e. Tree /\ h SubGraph G /\ | |
| ( # ` ( Vtx ` h ) ) = Y ) ) ) -> | |
| E. h ( h e. Tree /\ h SubGraph G /\ | |
| ( # ` ( Vtx ` h ) ) = Y ) ) $= | |
| ( cusgr wcel cconngr cvtx cfv cfn w3a c0 wne wa cn c1 caddc chash cle wbr | |
| co cv ctree csubgr wceq wex wi sptryle id mpan9 ) BDEBFEBGHZIEJUJKLMCNECO | |
| PTUJQHZRSMMCUKRSZULAUAZUBEUMBUCSUMGHQHCUDJAUEZUFZUNBCUGUOUHUI $. | |
| $} | |
| ${ | |
| $d G Y t $. | |
| $( The induction data and a predecessor tree imply the hypotheses of one | |
| extension step. (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| sptrdatain $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) /\ | |
| ( t e. Tree /\ t SubGraph G /\ | |
| ( # ` ( Vtx ` t ) ) = Y ) ) -> | |
| ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ | |
| ( t e. Tree /\ t SubGraph G /\ | |
| ( # ` ( Vtx ` t ) ) = Y ) ) /\ | |
| ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) ) $= | |
| ( cusgr wcel cconngr cvtx cfv cfn w3a c0 wne wa cn c1 caddc chash wbr jca | |
| co cle cv ctree csubgr wceq simplll simpr simplr ) BDEBFEBGHZIEJZUIKLZMZC | |
| NECOPTUIQHUARMZMZAUBZUCEUOBUDRUOGHQHCUEJZMZUJUPMUMUQUJUPUJUKUMUPUFUNUPUGS | |
| ULUMUPUHS $. | |
| $} | |
| ${ | |
| $d G Y h t $. | |
| $( Rename the bound variable of the predecessor-tree witness under the | |
| fixed induction data. (Contributed by Mingli Yuan, | |
| 14-Aug-2026.) $) | |
| sptrcbvan $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) /\ | |
| E. h ( h e. Tree /\ h SubGraph G /\ | |
| ( # ` ( Vtx ` h ) ) = Y ) ) -> | |
| ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) /\ | |
| E. t ( t e. Tree /\ t SubGraph G /\ | |
| ( # ` ( Vtx ` t ) ) = Y ) ) ) $= | |
| ( cusgr wcel cconngr cvtx cfv cfn w3a c0 wa chash wbr cv ctree csubgr wex | |
| wceq wne cn c1 caddc co cle sptrcbv anbi2i biimpi ) CEFCGFCHIZJFKUJLUAMDU | |
| BFDUCUDUEUJNIUFOMMZBPZQFULCROULHINIDTKBSZMUKAPZQFUNCROUNHINIDTKASZMUMUOUK | |
| ABCDUGUHUI $. | |
| $( Move the fixed induction data inside the existential quantifier and | |
| discard the unused nonemptiness hypothesis. (Contributed by | |
| Mingli Yuan, 14-Aug-2026.) $) | |
| sptrdatat $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) /\ | |
| E. t ( t e. Tree /\ t SubGraph G /\ | |
| ( # ` ( Vtx ` t ) ) = Y ) ) -> | |
| E. t ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ | |
| ( t e. Tree /\ t SubGraph G /\ | |
| ( # ` ( Vtx ` t ) ) = Y ) ) /\ | |
| ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) ) $= | |
| ( cusgr wcel cconngr cvtx cfv cfn w3a c0 wne wa cn c1 caddc chash wbr wex | |
| co cle cv ctree csubgr wceq 19.42v sptrdatain eximi sylbir ) BDEBFEBGHZIE | |
| JZUJKLMCNECOPTUJQHUARMZMZAUBZUCEUNBUDRUNGHQHCUEJZASMUMUOMZASUKUOMULMZASUM | |
| UOAUFUPUQAABCUGUHUI $. | |
| $} | |
| ${ | |
| $d G Y h $. | |
| $( Package the fixed induction data with the predecessor-tree witness. | |
| (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| sptrdatah $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) /\ | |
| ( Y <_ ( # ` ( Vtx ` G ) ) -> | |
| E. h ( h e. Tree /\ h SubGraph G /\ | |
| ( # ` ( Vtx ` h ) ) = Y ) ) ) -> | |
| ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) /\ | |
| E. h ( h e. Tree /\ h SubGraph G /\ | |
| ( # ` ( Vtx ` h ) ) = Y ) ) ) $= | |
| ( cusgr wcel cconngr cvtx cfv cfn w3a c0 wne wa cn c1 caddc chash cle wbr | |
| co cv ctree csubgr wceq wex wi simpl sptrih jca ) BDEBFEBGHZIEJUJKLMCNECO | |
| PTUJQHZRSMMZCUKRSAUAZUBEUMBUCSUMGHQHCUDJAUEZUFZMULUNULUOUGABCUHUI $. | |
| $} | |
| ${ | |
| $d G h $. | |
| $( Under the standing graph hypotheses, the bounded induction predicate | |
| holds at one. (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| sptrbaseh $p |- ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) -> | |
| E. h ( h e. Tree /\ h SubGraph G /\ | |
| ( # ` ( Vtx ` h ) ) = 1 ) ) $= | |
| ( cusgr wcel cconngr cvtx cfv cfn w3a c0 wne wa cv ctree csubgr wbr chash | |
| c1 wceq wex simp1l 3anandirs simpr jca sptrbasefv syl ) BCDZBEDZBFGZHDZIZ | |
| UIJKZLZUGULLAMZNDUNBOPUNFGQGRSIATUMUGULUGUHUJULUGUGULUHULLUJULLUAUBUKULUC | |
| UDABUEUF $. | |
| $( Base case of the bounded vertex-cardinality induction. | |
| (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| sptrindbase $p |- ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| 1 <_ ( # ` ( Vtx ` G ) ) ) -> | |
| E. h ( h e. Tree /\ h SubGraph G /\ | |
| ( # ` ( Vtx ` h ) ) = 1 ) ) $= | |
| ( cusgr wcel cconngr cvtx cfv cfn w3a c0 wne wa cv ctree csubgr wbr chash | |
| c1 wceq wex cle sptrbaseh adantr ) BCDBEDBFGZHDIUDJKLAMZNDUEBOPUEFGQGRSIA | |
| TRUDQGUAPABUBUC $. | |
| $} | |
| ${ | |
| $d G Y h t $. $d G Y s t $. | |
| $( The bounded spanning-tree construction predicate is closed under the | |
| successor step. (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| sptrstep $p |- ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) /\ | |
| ( Y <_ ( # ` ( Vtx ` G ) ) -> | |
| E. h ( h e. Tree /\ h SubGraph G /\ | |
| ( # ` ( Vtx ` h ) ) = Y ) ) ) -> | |
| E. s ( s e. Tree /\ s SubGraph G /\ | |
| ( # ` ( Vtx ` s ) ) = ( Y + 1 ) ) ) $= | |
| ( vt cusgr wcel cconngr cvtx cfv w3a wa chash cle wbr cv ctree csubgr wex | |
| wceq cfn c0 cn c1 caddc co wi sptrdatah sptrcbvan sptrdatat sptrgrow 4syl | |
| wne ) BFGBHGBIJZUAGKZUNUBUMLCUCGCUDUEUFZUNMJZNOLZLZCUQNOAPZQGUTBROUTIJMJC | |
| TKASZUGLUSVALUSEPZQGVBBROVBIJMJCTKZESLUOVCLURLESDPZQGVDBROVDIJMJUPTKDSABC | |
| UHEABCUIEBCUJEBCDUKUL $. | |
| $} | |
| ${ | |
| $d G Y h $. | |
| $( Rearrange the hypotheses of the induction successor step into the form | |
| required by ~ sptrstep . (Contributed by Mingli Yuan, | |
| 14-Aug-2026.) $) | |
| sptrstepant $p |- ( ( Y e. NN /\ | |
| ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| Y <_ ( # ` ( Vtx ` G ) ) ) -> | |
| E. h ( h e. Tree /\ h SubGraph G /\ | |
| ( # ` ( Vtx ` h ) ) = Y ) ) /\ | |
| ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) -> | |
| ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| ( Y e. NN /\ ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) ) /\ | |
| ( Y <_ ( # ` ( Vtx ` G ) ) -> | |
| E. h ( h e. Tree /\ h SubGraph G /\ | |
| ( # ` ( Vtx ` h ) ) = Y ) ) ) ) $= | |
| ( cn wcel cusgr cconngr cvtx cfv cfn w3a c0 wne wa chash cle wbr cv jca | |
| wi ctree csubgr wceq wex c1 caddc co simp3l simp1 simp3r simp2 mpand ) CD | |
| EZBFEBGEBHIZJEKUNLMNZCUNOIZPQZNARZUAEURBUBQURHIOICUCKAUDZTZUOCUEUFUGUPPQZ | |
| NZKZUOUMVANZNUQUSTVCUOVDUMUTUOVAUHZVCUMVAUMUTVBUIUMUTUOVAUJSSVCUOUQUSVEUM | |
| UTVBUKULS $. | |
| $( Successor case of the bounded vertex-cardinality induction. | |
| (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| sptrindsucc $p |- ( Y e. NN -> | |
| ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| Y <_ ( # ` ( Vtx ` G ) ) ) -> | |
| E. h ( h e. Tree /\ h SubGraph G /\ | |
| ( # ` ( Vtx ` h ) ) = Y ) ) -> | |
| ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| ( Y + 1 ) <_ ( # ` ( Vtx ` G ) ) ) -> | |
| E. h ( h e. Tree /\ h SubGraph G /\ | |
| ( # ` ( Vtx ` h ) ) = ( Y + 1 ) ) ) ) ) $= | |
| ( cn wcel cusgr cconngr cvtx cfv cfn w3a c0 wne wa chash cle wbr wceq wex | |
| wi cv ctree csubgr c1 caddc co sptrstepant sptrstep syl 3exp ) CDEZBFEBGE | |
| BHIZJEKULLMNZCULOIZPQZNAUAZUBEZUPBUCQZUPHIOIZCRKASZTZUMCUDUEUFZUNPQZNZUQU | |
| RUSVBRKASZUKVAVDKUMUKVCNNUOUTTNVEABCUGABCAUHUIUJ $. | |
| $} | |
| ${ | |
| $d A h $. $d B h $. | |
| $( Congruence of the bounded spanning-tree construction predicate in its | |
| vertex-count parameter. (Contributed by Mingli Yuan, | |
| 14-Aug-2026.) $) | |
| sptrindbid $p |- ( A = B -> | |
| ( ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| A <_ ( # ` ( Vtx ` G ) ) ) -> | |
| E. h ( h e. Tree /\ h SubGraph G /\ | |
| ( # ` ( Vtx ` h ) ) = A ) ) <-> | |
| ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| B <_ ( # ` ( Vtx ` G ) ) ) -> | |
| E. h ( h e. Tree /\ h SubGraph G /\ | |
| ( # ` ( Vtx ` h ) ) = B ) ) ) ) $= | |
| ( wceq cusgr wcel cconngr cvtx cfv cfn w3a c0 wne wa chash cle wbr cv wex | |
| ctree csubgr breq1 anbi2d id eqeq2d 3anbi3d exbidv imbi12d ) ABEZDFGDHGDI | |
| JZKGLUKMNOZAUKPJZQRZOULBUMQRZOCSZUAGZUPDUBRZUPIJPJZAEZLZCTUQURUSBEZLZCTUJ | |
| UNUOULABUMQUCUDUJVAVCCUJUTVBUQURUJABUSUJUEUFUGUHUI $. | |
| $} | |
| ${ | |
| $d G h x y $. $d Y h x $. | |
| $( Every positive vertex count not exceeding that of the ambient graph is | |
| attained by a tree subgraph. (Contributed by Mingli Yuan, | |
| 14-Aug-2026.) $) | |
| sptrindn $p |- ( Y e. NN -> | |
| ( ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) /\ | |
| Y <_ ( # ` ( Vtx ` G ) ) ) -> | |
| E. h ( h e. Tree /\ h SubGraph G /\ | |
| ( # ` ( Vtx ` h ) ) = Y ) ) ) $= | |
| ( vx vy cusgr wcel cvtx cfv w3a wa cv chash cle wbr wceq wi c1 sptrindbid | |
| wex cconngr cfn c0 wne ctree csubgr caddc sptrindbase sptrindsucc nnind | |
| co ) BFGBUAGBHIZUBGJULUCUDKZDLZULMIZNOKALZUEGZUPBUFOZUPHIMIZUNPJATQUMRUON | |
| OKUQURUSRPJATQUMELZUONOKUQURUSUTPJATQUMUTRUGUKZUONOKUQURUSVAPJATQUMCUONOK | |
| UQURUSCPJATQDECUNRABSUNUTABSUNVAABSUNCABSABUHABUTUIUJ $. | |
| $} | |
| ${ | |
| $d G h $. | |
| $( The bounded construction reaches the full vertex cardinality of the | |
| ambient finite nonempty connected graph. (Contributed by Mingli Yuan, | |
| 14-Aug-2026.) $) | |
| sptrfull $p |- ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) -> | |
| E. h ( h e. Tree /\ h SubGraph G /\ | |
| ( # ` ( Vtx ` h ) ) = ( # ` ( Vtx ` G ) ) ) ) $= | |
| ( cusgr wcel cconngr cvtx cfv cfn w3a c0 wne wa chash cn cle wbr cv ctree | |
| csubgr syl wceq wex simpr wb simpl simp3 hashnncl mpbird id cr nnre leidd | |
| jca sptrindn sylc ) BCDZBEDZBFGZHDZIZURJKZLZURMGZNDZVBVCVCOPZLAQZRDVFBSPV | |
| FFGMGVCUAIAUBVBVDVAUTVAUCVBUSVDVAUDVBUTUSUTVAUEUPUQUSUFTURUGTUHZVBVBVEVBU | |
| IVBVCVBVDVCUJDVGVCUKTULUMABVCUNUO $. | |
| $} | |
| $( A finite set is equal to a subset having the same cardinality. This | |
| lemma keeps ~ sptrvtxeq independent of other mathboxes. (Contributed by | |
| Mingli Yuan, 14-Aug-2026.) $) | |
| sptrvtxeqlem $p |- ( ( B e. Fin /\ A C_ B /\ | |
| ( # ` A ) = ( # ` B ) ) -> A = B ) $= | |
| ( cfn wcel wss chash cfv wceq w3a ssfi 3adant3 wi wa cen wbr hashen biimp3a | |
| pm3.2 3ad2ant2 expcom fisseneq 3expa sylsyld 3expb com23 3impia 3com23 mpd | |
| ) BCDZABEZAFGBFGHZIACDZABHZUIUJULUKBAJKUIUKUJULUMLZUIUKUJUNUIUKMZULUJUMULUO | |
| UJUMLZULUIUKUPULUIUKIABNOZUJUIUJMZUMULUIUKUQABPQUIULUJURLUKUIUJRSURUQUMUIUJ | |
| UQUMABUAUBTUCUDTUEUFUGUH $. | |
| $( A finite ambient vertex set and equality of vertex cardinalities force a | |
| subgraph to contain every ambient vertex. (Contributed by Mingli Yuan, | |
| 14-Aug-2026.) $) | |
| sptrvtxeq $p |- ( ( ( Vtx ` G ) e. Fin /\ T SubGraph G /\ | |
| ( # ` ( Vtx ` T ) ) = ( # ` ( Vtx ` G ) ) ) -> | |
| ( Vtx ` T ) = ( Vtx ` G ) ) $= | |
| ( cvtx cfv cfn wcel wbr chash wceq w3a wss simp1 simp2 subgrvtxss syl simp3 | |
| csubgr 3jca sptrvtxeqlem ) | |
| BCDZEFZABQGZACDZHDTHDIZJZUAUCTKZUDJUCTIUEUAUFUDUAUBUDLUEUBUFUAUBUDMABNOUAUB | |
| UDPRUCTSO $. | |
| $( A tree subgraph of a finite simple graph with the full vertex cardinality | |
| is a spanning tree. (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| sptrwitness $p |- ( ( ( G e. USGraph /\ ( Vtx ` G ) e. Fin ) /\ | |
| ( T e. Tree /\ T SubGraph G /\ | |
| ( # ` ( Vtx ` T ) ) = ( # ` ( Vtx ` G ) ) ) ) -> | |
| T SpanningTree G ) $= | |
| ( cusgr wcel cvtx cfv cfn ctree csubgr wbr chash wceq w3a csptr simpr simp1 | |
| wa syl simp2 jca simplr simp3 3jca sptrvtxeq wb simpll issptr mpbird ) BCDZ | |
| BEFZGDZQZAHDZABIJZAEFZKFUJKFLZMZQZABNJZUMUNQZUOUJLZQZURUTVAURUMUNURUQUMULUQ | |
| OZUMUNUPPRZURUQUNVCUMUNUPSRZTURUKUNUPMVAURUKUNUPUIUKUQUAVEURUQUPVCUMUNUPUBR | |
| UCABUDRTURUMUIQUSVBUEURUMUIVDUIUKUQUFTACBHUGRUH $. | |
| ${ | |
| $d G h t $. | |
| $( Every finite nonempty connected simple graph has a spanning tree. See | |
| Result in [Diestel] p. 14. (Contributed by Mingli Yuan, | |
| 14-Aug-2026.) $) | |
| sptriex $p |- ( ( ( G e. USGraph /\ G e. ConnGraph /\ | |
| ( Vtx ` G ) e. Fin ) /\ ( Vtx ` G ) =/= (/) ) -> | |
| E. h h SpanningTree G ) $= | |
| ( vt cusgr wcel cconngr cvtx cfv cfn w3a c0 wne wa ctree csubgr wbr chash | |
| cv csptr jca wex sptrfull weq simpll1 simpll3 simpr sptrwitness syl breq1 | |
| wceq biimprd syl5 spimevw exlimddv ) BDEZBFEZBGHZIEZJUQKLZMZCRZNEVABOPVAG | |
| HQHUQQHUJJZARZBSPZAUACCBUBUTVBMZVDACVEVABSPZACUCZVDVEUOURMZVBMVFVEVHVBVEU | |
| OURUOUPURUSVBUDUOUPURUSVBUETUTVBUFTVABUGUHVGVDVFVCVABSUIUKULUMUN $. | |
| $} | |
| ${ | |
| $d G h $. | |
| $( A finite tree has one fewer indexed edge than vertices. See Corollary | |
| 1.5.3 in [Diestel] p. 14. (Contributed by Mingli Yuan, | |
| 12-Aug-2026.) $) | |
| treeiedghash $p |- ( ( G e. Tree /\ ( Vtx ` G ) e. Fin ) -> | |
| ( ( # ` dom ( iEdg ` G ) ) + 1 ) = ( # ` ( Vtx ` G ) ) ) $= | |
| ( vh ctree wcel cvtx cfv cfn wa chash wceq w3a ciedg c1 caddc co wi simpr | |
| cdm syl fveq2d cv cn c0 simpl treevtxne0 wb hashnncl mpbird treeiedghashn | |
| wal id eleq1d eqeq1d 3anbi123d dmeqd oveq1d 2fveq3 eqeq12d imbi12d biimpd | |
| wne spcimdv mpd eqidd 3jca syl5com ) ACDZAEFZGDZHZVGVIVHIFZVKJZKZALFZRZIF | |
| ZMNOZVKJZPZVRVJBUAZCDZVTEFZGDZWBIFZVKJZKZVTLFZRZIFZMNOZWDJZPZBUJZVSVJVKUB | |
| DZWMVJWNVHUCVAZVJVGWOVGVIUDZAUESVJVIWNWOUFVGVIQZVHUGSUHBVKUISVJWLVSBACWPV | |
| JVTAJZHWRWLVSPVJWRQWRWLVSWRWFVMWKVRWRWAVGWCVIWEVLWRVTACWRUKZULWRWBVHGWRVT | |
| AEWSTZULWRWDVKVKWRWBVHIWTTUMUNWRWJVQWDVKWRWIVPMNWRWHVOIWRWGVNWRVTALWSTUOT | |
| UPVTAIEUQURUSUTSVBVCVJVMVSVRVJVGVIVLWPWQVJVKVDVEVSUKVFVC $. | |
| $} | |
| $( A finite spanning tree has one fewer indexed edge than the vertices of | |
| the graph it spans. (Contributed by Mingli Yuan, 14-Aug-2026.) $) | |
| sptriedghash $p |- ( ( T SpanningTree G /\ ( Vtx ` G ) e. Fin ) -> | |
| ( ( # ` dom ( iEdg ` T ) ) + 1 ) = ( # ` ( Vtx ` G ) ) ) $= | |
| ( csptr wbr cvtx cfv cfn wcel wa ciedg cdm chash c1 caddc co ctree sptrtree | |
| wceq simpl syl sptrvtx simpr eqeltrd jca treeiedghash fveq2d eqtrd ) ABCDZB | |
| EFZGHZIZAJFKLFMNOZAEFZLFZUILFUKAPHZUMGHZIULUNRUKUOUPUKUHUOUHUJSZABQTUKUMUIG | |
| UKUHUMUIRUQABUATZUHUJUBUCUDAUETUKUMUILURUFUG $. | |
| $( (End of Mingli Yuan's mathbox.) $) |