:: About Supergraphs, {P}art {III} :: by Sebastian Koch :: :: Received May 27, 2019 :: Copyright (c) 2019 Association of Mizar Users :: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland). :: This code can be distributed under the GNU General Public Licence :: version 3.0 or later, or the Creative Commons Attribution-ShareAlike :: License version 3.0 or later, subject to the binding interpretation :: detailed in file COPYING.interpretation. :: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these :: licenses, or see http://www.gnu.org/licenses/gpl.html and :: http://creativecommons.org/licenses/by-sa/3.0/. environ vocabularies NUMBERS, FINSEQ_1, SUBSET_1, RELAT_1, FUNCT_1, XXREAL_0, TARSKI, ORDINAL4, ARYTM_3, CARD_1, XBOOLE_0, NAT_1, ARYTM_1, GLIB_000, PBOOLE, FINSET_1, ZFMISC_1, RELAT_2, GLIB_002, FUNCOP_1, TREES_1, GLIB_001, FUNCT_4, ABIAN, REWRITE1, CHORD, HELLY, FINSEQ_8, GRAPH_1, RFINSEQ, XCMPLX_0, CARD_2, GRAPH_2, GLIB_006, GLIB_007, SCMYCIEL, GLIB_003, EQREL_1; notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, EQREL_1, FUNCT_1, ORDINAL1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1, DOMAIN_1, FUNCT_3, FUNCOP_1, FUNCT_4, FINSET_1, CARD_1, PBOOLE, NUMBERS, XCMPLX_0, XXREAL_0, XREAL_0, NAT_1, INT_1, VALUED_0, NAT_D, CARD_2, FINSEQ_1, RVSUM_1, RFINSEQ, FUNCT_7, ABIAN, FINSEQ_6, GLIB_000, FINSEQ_8, GRAPH_2, GRAPH_5, GLIB_001, GLIB_002, HELLY, GLIB_003, CHORD, GLIB_006, GLIB_007; constructors DOMAIN_1, BINOP_1, BINOP_2, FINSOP_1, RVSUM_1, FINSEQ_5, GRAPH_5, ABIAN, WELLORD2, FUNCT_2, FIB_NUM2, FINSEQ_8, GLIB_001, GLIB_002, HELLY, RELSET_1, FUNCT_3, CHORD, NAT_D, GRAPH_2, RFINSEQ, FINSEQ_6, CARD_2, FUNCT_7, FINSEQ_1, SUBSET_1, GLIB_006, GLIB_007, RELAT_1, GLIB_003, EQREL_1; registrations XBOOLE_0, RELAT_1, FUNCT_1, ORDINAL1, FUNCOP_1, FINSET_1, ABIAN, NUMBERS, XREAL_0, NAT_1, MEMBERED, FINSEQ_1, GLIB_000, GLIB_001, HELLY, GLIB_002, CHORD, INT_1, VALUED_0, CARD_1, FUNCT_2, PARTFUN1, HURWITZ2, RELSET_1, RAMSEY_1, FUNCT_4, SUBSET_1, GLIB_006, GLIB_007, GLIB_003, MSAFREE5, JORDAN1J, CARD_5, EQREL_1; requirements ARITHM, BOOLE, NUMERALS, REAL, SUBSET; definitions GLIB_000, GLIB_002; theorems TARSKI, XBOOLE_0, XBOOLE_1, ORDINAL1, XXREAL_0, RELAT_1, CARD_1, CARD_2, FUNCT_2, FUNCOP_1, FUNCT_4, ZFMISC_1, FUNCT_1, GLIB_000, GLIB_001, GLIB_002, CHORD, POLYFORM, NAT_1, XREAL_1, ABIAN, FINSEQ_6, FINSEQ_8, FINSEQ_1, FINSEQ_3, FINSEQ_5, RFINSEQ, INT_1, HELLY, PARTFUN1, GLIB_006, GLIB_007, GLIB_003, FUNCT_7, EQREL_1, WELLORD2; schemes NAT_1, FUNCT_1; begin :: Preliminaries Lm12: for p being non empty FinSequence, x being object for n being Element of dom(p^<*x*>) st n <= len(p^<*x*>) - 1 holds n = len(p^<*x*>) - 1 or n <= len p - 1 proof let p be non empty FinSequence, x be object, n be Element of dom(p^<*x*>); A1: len(p^<*x*>) = len p + len <*x*> by FINSEQ_1:22 .= len p + 1 by FINSEQ_1:39; assume A2: n <= len(p^<*x*>) - 1; assume n <> len(p^<*x*>) - 1; then A3: n < len p - 1 + 1 by A1, A2, XXREAL_0:1; len p - 1 is Nat by INT_1:74; hence n <= len p - 1 by A3, NAT_1:13; end; Lm13: for p being non empty FinSequence, x being object for n being Element of dom(<*x*>^p) st n <= len(<*x*>^p) - 1 holds n = 1 or (n-1 in dom p & n-1 <= len p - 1) proof let p be non empty FinSequence, x be object, n be Element of dom(<*x*>^p); A1: len(<*x*>^p) = len <*x*> + len p by FINSEQ_1:22 .= len p + 1 by FINSEQ_1:39; assume n <= len(<*x*>^p) - 1; then A2: n-1 <= len p - 1 by A1, XREAL_1:9; then A3: n-1 + 0 <= len p - 1 + 1 by XREAL_1:7; assume A4: n <> 1; n is non zero by FINSEQ_3:25; then A5: n-1 is Nat by INT_1:74; 1 <= n-1 proof assume not 1 <= n-1; then n-1 = 0 by A5, NAT_1:14; hence contradiction by A4; end; hence n-1 in dom p by A3, A5, FINSEQ_3:25; thus n-1 <= len p - 1 by A2; end; ::: START into GLIB_000 ? registration let G be _Graph, v be Vertex of G; cluster -> trivial for inducedSubgraph of G, {v}; coherence proof G.edgesBetween({v}) c= G.edgesBetween({v}); hence thesis; end; end; theorem Th1: for G being _Graph, X being set, v being Vertex of G holds G.edgesBetween(X \ {v}) = G.edgesBetween(X) \ v.edgesInOut() proof let G be _Graph, X be set, v be Vertex of G; for e being object holds e in G.edgesBetween(X \ {v}) iff e in G.edgesBetween(X) \ v.edgesInOut() proof let e be object; set u = (the_Source_of G).e, w = (the_Target_of G).e; hereby assume e in G.edgesBetween(X \ {v}); then A1: e in the_Edges_of G & u in X\{v} & w in X\{v} by GLIB_000:31; then A2: e in G.edgesBetween(X) by GLIB_000:31; not e in v.edgesInOut() proof assume e in v.edgesInOut(); then u = v or w = v by GLIB_000:61; hence contradiction by A1, ZFMISC_1:56; end; hence e in G.edgesBetween(X) \ v.edgesInOut() by A2, XBOOLE_0:def 5; end; assume e in G.edgesBetween(X) \ v.edgesInOut(); then A3: e in G.edgesBetween(X) & not e in v.edgesInOut() by XBOOLE_0:def 5; then A4: e in the_Edges_of G & u in X & w in X by GLIB_000:31; u <> v & w <> v by A3, GLIB_000:61; then not u in {v} & not w in {v} by TARSKI:def 1; then u in X\{v} & w in X\{v} by A4, XBOOLE_0:def 5; hence thesis by A3, GLIB_000:31; end; hence thesis by TARSKI:2; end; theorem Th2: for G being _Graph, X being set, v being Vertex of G st v is isolated holds G.edgesBetween(X \ {v}) = G.edgesBetween(X) proof let G be _Graph, X be set, v be Vertex of G; assume A1: v is isolated; thus G.edgesBetween(X \ {v}) = G.edgesBetween(X) \ v.edgesInOut() by Th1 .= G.edgesBetween(X) \ {} by A1, GLIB_000:def 49 .= G.edgesBetween(X); end; theorem Th3: for G being non-Dmulti _Graph, v being Vertex of G holds v.inDegree() = card v.inNeighbors() proof let G be non-Dmulti _Graph, v be Vertex of G; defpred P[object,object] means $2 DJoins $1,v,G; A1: for x,y1,y2 being object st x in v.inNeighbors() & P[x,y1] & P[x,y2] holds y1 = y2 by GLIB_000:def 21; A2: for x being object st x in v.inNeighbors() ex y being object st P[x,y] by GLIB_000:69; consider f being Function such that A3: dom f = v.inNeighbors() and A4: for x being object st x in v.inNeighbors() holds P[x,f.x] from FUNCT_1:sch 2(A1,A2); for y being object holds y in rng f iff y in v.edgesIn() proof let y be object; hereby assume y in rng f; then consider x being object such that A5: x in dom f & f.x = y by FUNCT_1:def 3; reconsider x as set by TARSKI:1; thus y in v.edgesIn() by A3, A4, A5, GLIB_000:57; end; assume y in v.edgesIn(); then consider x being set such that A6: y DJoins x,v,G by GLIB_000:57; a7: x in v.inNeighbors() by A6, GLIB_000:69; then x in dom f & P[x,f.x] by A3, A4; then y = f.x by A1, A3, A6; hence thesis by a7, FUNCT_1:def 3, A3; end; then A8: rng f = v.edgesIn() by TARSKI:2; for x1,x2 being object st x1 in dom f & x2 in dom f & f.x1 = f.x2 holds x1 = x2 proof let x1,x2 be object; assume x1 in dom f & x2 in dom f & f.x1 = f.x2; then P[x1,f.x1] & P[x2,f.x1] by A3, A4; then (the_Source_of G).(f.x1) = x1 & (the_Source_of G).(f.x1) = x2 by GLIB_000:def 14; hence x1 = x2; end; then f is one-to-one by FUNCT_1:def 4; then card v.inNeighbors() = card v.edgesIn() by A3, A8, WELLORD2:def 4, CARD_1:5; hence thesis by GLIB_000:def 42; end; theorem Th4: for G being non-Dmulti _Graph, v being Vertex of G holds v.outDegree() = card v.outNeighbors() proof let G be non-Dmulti _Graph, v be Vertex of G; defpred P[object,object] means $2 DJoins v,$1,G; A1: for x,y1,y2 being object st x in v.outNeighbors() & P[x,y1] & P[x,y2] holds y1 = y2 by GLIB_000:def 21; A2: for x being object st x in v.outNeighbors() ex y being object st P[x,y] by GLIB_000:70; consider f being Function such that A3: dom f = v.outNeighbors() and A4: for x being object st x in v.outNeighbors() holds P[x,f.x] from FUNCT_1:sch 2(A1,A2); for y being object holds y in rng f iff y in v.edgesOut() proof let y be object; hereby assume y in rng f; then consider x being object such that A5: x in dom f & f.x = y by FUNCT_1:def 3; reconsider x as set by TARSKI:1; thus y in v.edgesOut() by A3, A4, A5, GLIB_000:59; end; assume y in v.edgesOut(); then consider x being set such that A6: y DJoins v,x,G by GLIB_000:59; a7: x in v.outNeighbors() by A6, GLIB_000:70; then x in dom f & P[x,f.x] by A3, A4; then y = f.x by A1, A3, A6; hence thesis by a7, A3, FUNCT_1:def 3; end; then A8: rng f = v.edgesOut() by TARSKI:2; for x1,x2 being object st x1 in dom f & x2 in dom f & f.x1 = f.x2 holds x1 = x2 proof let x1,x2 be object; assume x1 in dom f & x2 in dom f & f.x1 = f.x2; then P[x1,f.x1] & P[x2,f.x1] by A3, A4; then (the_Target_of G).(f.x1) = x1 & (the_Target_of G).(f.x1) = x2 by GLIB_000:def 14; hence x1 = x2; end; then f is one-to-one by FUNCT_1:def 4; then card v.outNeighbors() = card v.edgesOut() by A3, A8, WELLORD2:def 4, CARD_1:5; hence thesis by GLIB_000:def 43; end; theorem Th5: for G being simple _Graph, v being Vertex of G holds v.degree() = card v.allNeighbors() proof let G be simple _Graph, v be Vertex of G; v.inNeighbors() /\ v.outNeighbors() = {} proof assume v.inNeighbors() /\ v.outNeighbors() <> {}; then consider w being object such that A1: w in v.inNeighbors() /\ v.outNeighbors() by XBOOLE_0:def 1; A2: w in v.inNeighbors() & w in v.outNeighbors() by A1, XBOOLE_0:def 4; consider e1 being object such that A3: e1 DJoins w,v,G by A2, GLIB_000:69; consider e2 being object such that A4: e2 DJoins v,w,G by A2, GLIB_000:70; e1 Joins v,w,G & e2 Joins v,w,G by A3, A4, GLIB_000:16; then e1 = e2 by GLIB_000:def 20; then e1 in the_Edges_of G & (the_Source_of G).e1 = v & (the_Target_of G).e1 = v by A3, A4, GLIB_000:def 14; hence contradiction by GLIB_000:def 18; end; then card(v.inNeighbors() \/ v.outNeighbors()) = card v.inNeighbors() +` card v.outNeighbors() by CARD_2:35,XBOOLE_0:def 7 .= v.inDegree() +` card v.outNeighbors() by Th3 .= v.inDegree() +` v.outDegree() by Th4 .= v.degree() by GLIB_000:def 44; hence thesis by GLIB_000:def 48; end; theorem Th6: for G being _Graph holds G is loopless iff for v being Vertex of G holds not v in v.allNeighbors() proof let G be _Graph; hereby assume A1: G is loopless; let v be Vertex of G; thus not v in v.allNeighbors() proof assume v in v.allNeighbors(); then consider e being object such that A2: e Joins v,v,G by GLIB_000:71; thus contradiction by A1, A2, GLIB_000:18; end; end; assume A3: for v being Vertex of G holds not v in v.allNeighbors(); for v being object holds not ex e being object st e Joins v,v,G proof given v being object such that A4: ex e being object st e Joins v,v,G; reconsider v0 = v as Vertex of G by A4, GLIB_000:13; not v0 in v0.allNeighbors() by A3; hence contradiction by A4, GLIB_000:71; end; hence thesis by GLIB_000:18; end; theorem Th7: for G being _Graph, v being Vertex of G holds v is isolated iff v.allNeighbors() = {} proof let G be _Graph, v be Vertex of G; hereby assume v is isolated; then v.edgesInOut() = {} by GLIB_000:def 49; then v.edgesIn() \/ v.edgesOut() = {} by GLIB_000:60; then v.edgesIn() = {} & v.edgesOut() = {}; then (the_Source_of G).:v.edgesIn() = {} & (the_Target_of G).:v.edgesOut() = {}; then v.inNeighbors() = {} & v.outNeighbors() = {} by GLIB_000:def 46, GLIB_000:def 47; then v.inNeighbors() \/ v.outNeighbors() = {} \/ {}; hence v.allNeighbors() = {} by GLIB_000:def 48; end; assume A1: v.allNeighbors() = {}; assume not v is isolated; then A2: v.edgesInOut() <> {} by GLIB_000:def 49; set e = the Element of v.edgesInOut(); consider v0 being Vertex of G such that A3: e Joins v,v0,G by A2, GLIB_000:64; thus contradiction by A1, A3, GLIB_000:71; end; Lm2: for G1 being _Graph, v being set, G2 being removeVertex of G1, v st not v in the_Vertices_of G1 holds G1 == G2 proof let G1 be _Graph, v be set, G2 be removeVertex of G1, v; set V = the_Vertices_of G1 \ {v}; assume not v in the_Vertices_of G1; then A1: V = the_Vertices_of G1 by ZFMISC_1:57; then the_Vertices_of G2 = V & the_Edges_of G2 = G1.edgesBetween(V) by GLIB_000:def 37; then A2: the_Vertices_of G2 = the_Vertices_of G1 & the_Edges_of G2 = the_Edges_of G1 by A1, GLIB_000:34; G1 is Subgraph of G1 by GLIB_000:40; hence thesis by A2, GLIB_000:86; end; theorem Th8: for G1 being _Graph, v being set, G2 being removeVertex of G1, v st G1 is trivial or not v in the_Vertices_of G1 holds G1 == G2 proof let G1 be _Graph, v be set, G2 be removeVertex of G1, v; assume G1 is trivial or not v in the_Vertices_of G1; then per cases; suppose G1 is trivial; then consider v1 being Vertex of G1 such that A1: the_Vertices_of G1 = {v1} by GLIB_000:22; set V = the_Vertices_of G1 \ {v}, E = G1.edgesBetween(V); per cases; suppose V is non empty Subset of the_Vertices_of G1 & E c= G1.edgesBetween(V); then V = {v1} by A1, ZFMISC_1:33; hence thesis by A1, Lm2, ZFMISC_1:57; end; suppose not (V is non empty Subset of the_Vertices_of G1 & E c= G1.edgesBetween(V)); hence thesis by GLIB_000:def 37; end; end; suppose not v in the_Vertices_of G1; hence thesis by Lm2; end; end; Lm3: for G1, G2 being _Graph, v being set st G1 == G2 & not v in the_Vertices_of G1 holds G2 is removeVertex of G1, v proof let G1, G2 be _Graph, v be set; assume A1: G1 == G2 & not v in the_Vertices_of G1; then A2: G2 is Subgraph of G1 by GLIB_000:87; set V = the_Vertices_of G1 \ {v}; A3: V = the_Vertices_of G1 by A1, ZFMISC_1:57; A4: G1.edgesBetween(V) = the_Edges_of G1 by A3, GLIB_000:34 .= the_Edges_of G2 by A1, GLIB_000:def 34; V = the_Vertices_of G2 by A1, A3, GLIB_000:def 34; hence thesis by A2, A4, GLIB_000:def 37; end; theorem Th9: for G1, G2 being _Graph, v being set st G1 == G2 & (G1 is trivial or not v in the_Vertices_of G1) holds G2 is removeVertex of G1, v proof let G1, G2 be _Graph, v be set; assume that A1: G1 == G2 and A2: G1 is trivial or not v in the_Vertices_of G1; A3: G2 is Subgraph of G1 by A1, GLIB_000:87; set V = the_Vertices_of G1 \ {v}; per cases by A2; suppose G1 is trivial; then consider v1 being Vertex of G1 such that A4: the_Vertices_of G1 = {v1} by GLIB_000:22; per cases; suppose v = v1; then V = {} by A4, XBOOLE_1:37; hence thesis by A1, A3, GLIB_000:def 37; end; suppose v <> v1; then not v in the_Vertices_of G1 by A4, TARSKI:def 1; hence thesis by A1, Lm3; end; end; suppose not v in the_Vertices_of G1; hence thesis by A1, Lm3; end; end; :: converse of GLIB_000:21 theorem Th10: for G being _Graph holds (ex v1, v2 being Vertex of G st v1 <> v2) implies G is non trivial proof let G be _Graph; given v1, v2 being Vertex of G such that A1: v1 <> v2; card the_Vertices_of G <> 1 proof assume card the_Vertices_of G = 1; then card {v1,v2} c= 1 by CARD_1:11; then {0,1} c= 1 by A1, CARD_2:57, CARD_1:50; then 1 in 1 by ZFMISC_1:32; hence contradiction; end; hence thesis by GLIB_000:def 19; end; registration let G be non trivial _Graph, X be set; cluster -> non trivial for removeEdges of G, X; coherence proof let G2 be removeEdges of G, X; ex v1, v2 being Vertex of G2 st v1 <> v2 proof consider v1, v2 being Vertex of G such that A1: v1 <> v2 by GLIB_000:21; reconsider v1, v2 as Vertex of G2 by GLIB_000:53; take v1, v2; thus thesis by A1; end; hence thesis by Th10; end; end; theorem Th11: for G1 being finite _Graph, G2 being Subgraph of G1 holds G2 is spanning iff G1.order() = G2.order() proof let G1 be finite _Graph, G2 be Subgraph of G1; hereby assume A1: G2 is spanning; thus G1.order() = card the_Vertices_of G1 by GLIB_000:def 24 .= card the_Vertices_of G2 by A1, GLIB_000:def 33 .= G2.order() by GLIB_000:def 24; end; assume A2: G1.order() = G2.order(); card the_Vertices_of G1 = G1.order() by GLIB_000:def 24 .= card the_Vertices_of G2 by A2, GLIB_000:def 24; hence thesis by GLIB_000:def 33, CARD_2:102; end; theorem Th12: for G1 being _Graph, G2 being spanning Subgraph of G1 st the_Edges_of G1 = the_Edges_of G2 holds G1 == G2 proof let G1 be _Graph, G2 be spanning Subgraph of G1; assume A1: the_Edges_of G1 = the_Edges_of G2; A2: G1 is Subgraph of G1 by GLIB_000:40; the_Vertices_of G1 = the_Vertices_of G2 by GLIB_000:def 33; hence thesis by A1, A2, GLIB_000:86; end; theorem Th13: for G1 being finite _Graph, G2 being spanning Subgraph of G1 st G1.size() = G2.size() holds G1 == G2 proof let G1 be finite _Graph, G2 be spanning Subgraph of G1; assume A1: G1.size() = G2.size(); card the_Edges_of G1 = G1.size() by GLIB_000:def 25 .= card the_Edges_of G2 by A1, GLIB_000:def 25; then the_Edges_of G1 = the_Edges_of G2 by CARD_2:102; hence thesis by Th12; end; theorem Th14: for G1 being _Graph, V being set, G2 being inducedSubgraph of G1, V st G2 is spanning holds G1 == G2 proof let G1 be _Graph, V be set, G2 be inducedSubgraph of G1, V; assume G2 is spanning; then A1: the_Vertices_of G1 = the_Vertices_of G2 by GLIB_000:def 33; per cases; suppose V is non empty Subset of the_Vertices_of G1; then the_Vertices_of G2 = V & the_Edges_of G2 = G1.edgesBetween(V) by GLIB_000:def 37; then A2: the_Edges_of G2 = the_Edges_of G1 by A1, GLIB_000:34; G1 is Subgraph of G1 by GLIB_000:40; hence thesis by A1, A2, GLIB_000:86; end; suppose not V is non empty Subset of the_Vertices_of G1; hence thesis by GLIB_000:def 37; end; end; theorem Th15: for G being _Graph holds G is non trivial iff ex H being Subgraph of G st H is non spanning proof let G be _Graph; hereby assume A1: G is non trivial; set v1 = the Vertex of G, v2 = the Vertex of G; reconsider H = the removeVertex of G, v1 as Subgraph of G; take H; A2: the_Vertices_of H = the_Vertices_of G \ {v1} by A1, GLIB_000:47; v1 in {v1} by TARSKI:def 1; then the_Vertices_of H <> the_Vertices_of G by A2, XBOOLE_0:def 5; hence H is non spanning by GLIB_000:def 33; end; given H being Subgraph of G such that A3: H is non spanning; A4: the_Vertices_of G <> the_Vertices_of H by A3, GLIB_000:def 33; the_Vertices_of H c= the_Vertices_of G; then reconsider v1 = the Vertex of H as Vertex of G by TARSKI:def 3; not the_Vertices_of G c= the_Vertices_of H by A4, XBOOLE_0:def 10; then A5: the_Vertices_of G \ the_Vertices_of H <> {} by XBOOLE_1:37; set v2 = the Element of the_Vertices_of G \ the_Vertices_of H; reconsider v2 as Vertex of G by A5, TARSKI:def 3; A6: v1 <> v2 by A5, XBOOLE_0:def 5; card {v1, v2} c= card the_Vertices_of G by CARD_1:11; then A7: 2 c= card the_Vertices_of G by A6, CARD_2:57; assume G is trivial; then A8: 2 c= 1 by A7, GLIB_000:def 19; 1 in {0,1} by TARSKI:def 2; then 1 in 1 by A8, CARD_1:50; hence contradiction; end; theorem Th16: for G being _Graph holds (ex v being Vertex of G st v is endvertex) implies G is non trivial proof let G be _Graph; given v being Vertex of G such that A1: v is endvertex; set G2 = the removeVertex of G, v; consider e being object such that A2: v.edgesInOut() = {e} & not e Joins v,v,G by A1, GLIB_000:def 51; set w = v.adj(e); A3: e in v.edgesInOut() by A2, TARSKI:def 1; for u being Vertex of G holds the_Vertices_of G <> {u} proof let u be Vertex of G; assume the_Vertices_of G = {u}; then v = u & w = u by TARSKI:def 1; hence contradiction by A2, A3, GLIB_000:67; end; hence G is non trivial by GLIB_000:22; end; :: note that induced subgraphs generally commute ::theorem :: for G1 being _Graph, V1, E1, V2, E2 being set :: for G2 being inducedSubgraph of G1, V1, E1 :: for G3 being inducedSubgraph of G1, V2, E2 :: for G4 being inducedSubgraph of G2, V2, E2 :: holds G4 is inducedSubgraph of G3, V1, E1; :: but the form above doesn't translate to removeEdge/Vertex exactly Lm4: for G1, G2 being _Graph, e being set for G3 being removeEdge of G1, e, G4 being removeEdge of G2, e st G1 == G2 holds G3 == G4 proof let G1, G2 be _Graph, e be set; let G3 be removeEdge of G1, e, G4 be removeEdge of G2, e; assume A1: G1 == G2; then the_Vertices_of G1 = the_Vertices_of G2 & the_Edges_of G1 = the_Edges_of G2 by GLIB_000:def 34; then G4 is removeEdge of G1, e by A1, GLIB_000:95; hence thesis by GLIB_000:93; end; :: removeEdge and removeVertex commute, Part I theorem for G1 being _Graph, v,e being set, G2 being removeVertex of G1, v for G3 being removeEdge of G1, e, G4 being removeEdge of G2, e holds G4 is removeVertex of G3, v proof let G1 be _Graph, v,e be set, G2 be removeVertex of G1, v; let G3 be removeEdge of G1, e, G4 be removeEdge of G2, e; A1: the_Vertices_of G3 = the_Vertices_of G1 & the_Edges_of G3 = the_Edges_of G1 \ {e} by GLIB_000:51; A2: the_Vertices_of G4 = the_Vertices_of G2 & the_Edges_of G4 = the_Edges_of G2 \ {e} by GLIB_000:51; per cases; suppose A3: G1 is non trivial & v in the_Vertices_of G1; then reconsider v1 = v as Vertex of G1; :: basic relations reconsider v3 = v1 as Vertex of G3 by A1; A4: the_Vertices_of G2 = the_Vertices_of G1 \ {v} & the_Edges_of G2 = G1.edgesBetween(the_Vertices_of G1 \ {v}) by A3, GLIB_000:47; then A5: the_Edges_of G2 = G1.edgesBetween(the_Vertices_of G1) \ v1.edgesInOut() by Th1 .= the_Edges_of G1 \ v1.edgesInOut() by GLIB_000:34; :: G4 is subgraph of G3 A6: the_Vertices_of G4 c= the_Vertices_of G3 by A1, A2; A7: G4 is Subgraph of G1 by GLIB_000:43; for x being object holds x in the_Edges_of G4 implies x in the_Edges_of G3 proof let x be object; assume x in the_Edges_of G4; then x in the_Edges_of G2 & not x in {e} by A2, XBOOLE_0:def 5; hence thesis by A1, XBOOLE_0:def 5; end; then the_Edges_of G4 c= the_Edges_of G3 by TARSKI:def 3; then A8: G4 is Subgraph of G3 by A6, A7, GLIB_000:44; :: properties of removeVertex now thus the_Vertices_of G4 = the_Vertices_of G3 \ {v} by A1, A2, A4; for x being object holds x in the_Edges_of G4 iff x in G3.edgesBetween(the_Vertices_of G3 \ {v}) proof let x be object; A9: G3.edgesBetween(the_Vertices_of G3 \ {v}) = G3.edgesBetween(the_Vertices_of G3) \ v3.edgesInOut() by Th1 .= the_Edges_of G3 \ v3.edgesInOut() by GLIB_000:34; hereby assume x in the_Edges_of G4; then A10: x in the_Edges_of G2 & not x in {e} by A2, XBOOLE_0:def 5; then x in the_Edges_of G1 & not x in v1.edgesInOut() by A5, XBOOLE_0:def 5; then A11: not x in v3.edgesInOut() by GLIB_000:78, TARSKI:def 3; x in the_Edges_of G3 by A1, A10, XBOOLE_0:def 5; hence x in G3.edgesBetween(the_Vertices_of G3 \ {v}) by A9, A11, XBOOLE_0:def 5; end; assume x in G3.edgesBetween(the_Vertices_of G3 \ {v}); then A12: x in the_Edges_of G3 & not x in v3.edgesInOut() by A9, XBOOLE_0:def 5; then A13: x in the_Edges_of G1 & not x in {e} by A1, XBOOLE_0:def 5; not x in v1.edgesInOut() proof assume x in v1.edgesInOut(); then x in v1.edgesInOut() /\ (the_Edges_of G3) by A12, XBOOLE_0:def 4; hence contradiction by A12, GLIB_000:79; end; then x in the_Edges_of G2 by A5, A13, XBOOLE_0:def 5; hence thesis by A2, A13, XBOOLE_0:def 5; end; hence the_Edges_of G4 = G3.edgesBetween(the_Vertices_of G3 \ {v}) by TARSKI:2; end; hence thesis by A8, GLIB_000:def 37; end; suppose A14: G1 is trivial or not v in the_Vertices_of G1; then G1 == G2 by Th8; then A15: G3 == G4 by Lm4; G3 is trivial or not v in the_Vertices_of G3 by A14; hence thesis by A15, Th9; end; end; Lm5: for G1,G3,G4 being _Graph, e being set, G2 being removeEdge of G1, e st G1 == G3 & G2 == G4 holds G4 is removeEdge of G3, e proof let G1,G3,G4 be _Graph, e be set; let G2 be removeEdge of G1, e; assume A1: G1 == G3 & G2 == G4; then the_Vertices_of G1 = the_Vertices_of G3 & the_Edges_of G1 = the_Edges_of G3 by GLIB_000:def 34; then G2 is removeEdge of G3, e by A1, GLIB_000:95; hence thesis by A1, GLIB_006:16; end; :: removeEdge and removeVertex commute, Part II theorem Th18: for G1 being _Graph, v,e being set for G2 being removeEdge of G1, e for G3 being removeVertex of G1, v, G4 being removeVertex of G2, v holds G4 is removeEdge of G3, e proof let G1 be _Graph, v,e be set; let G2 be removeEdge of G1, e; let G3 be removeVertex of G1, v, G4 be removeVertex of G2, v; A1: the_Vertices_of G2 = the_Vertices_of G1 & the_Edges_of G2 = the_Edges_of G1 \ {e} by GLIB_000:51; per cases; suppose A2: G1 is non trivial & v in the_Vertices_of G1; then reconsider v1 = v as Vertex of G1; reconsider v2 = v1 as Vertex of G2 by A1; :: basic relations the_Vertices_of G3 = the_Vertices_of G1 \ {v} & the_Edges_of G3 = G1.edgesBetween(the_Vertices_of G1 \ {v}) by A2, GLIB_000:47; then A3: the_Edges_of G3 = G1.edgesBetween(the_Vertices_of G1) \ v1.edgesInOut() by Th1 .= the_Edges_of G1 \ v1.edgesInOut() by GLIB_000:34; the_Vertices_of G4 = the_Vertices_of G2 \ {v} & the_Edges_of G4 = G2.edgesBetween(the_Vertices_of G2 \ {v}) by A1, A2, GLIB_000:47; then A4: the_Edges_of G4 = G2.edgesBetween(the_Vertices_of G2) \ v2.edgesInOut() by Th1 .= the_Edges_of G2 \ v2.edgesInOut() by GLIB_000:34; :: G4 is subgraph of G3 A5: the_Vertices_of G4 = the_Vertices_of G1 \ {v} by A1, A2, GLIB_000:47 .= the_Vertices_of G3 by A2, GLIB_000:47; for x being object holds x in the_Edges_of G4 implies x in the_Edges_of G3 \ {e} proof let x be object; assume x in the_Edges_of G4; then A6: x in the_Edges_of G2 & not x in v2.edgesInOut() by A4, XBOOLE_0:def 5; then A7: x in the_Edges_of G1 & not x in {e} by A1, XBOOLE_0:def 5; not x in v1.edgesInOut() proof assume x in v1.edgesInOut(); then per cases by GLIB_000:61; suppose (the_Source_of G1).x = v1; then (the_Source_of G2).x = v1 by A6, GLIB_000:def 32; hence contradiction by A6, GLIB_000:61; end; suppose (the_Target_of G1).x = v1; then (the_Target_of G2).x = v1 by A6, GLIB_000:def 32; hence contradiction by A6, GLIB_000:61; end; end; then x in the_Edges_of G3 by A3, A6, XBOOLE_0:def 5; hence thesis by A7, XBOOLE_0:def 5; end; then A8: the_Edges_of G4 c= the_Edges_of G3 \ {e} by TARSKI:def 3; the_Edges_of G3 \ {e} c= the_Edges_of G3 by XBOOLE_1:36; then A9: the_Edges_of G4 c= the_Edges_of G3 by A8, XBOOLE_1:1; G4 is Subgraph of G1 by GLIB_000:43; then A10: G4 is Subgraph of G3 by A5, A9, GLIB_000:44; :: property assumptions for induced subgraph A11: now the_Vertices_of G3 c= the_Vertices_of G3; hence the_Vertices_of G3 is non empty Subset of the_Vertices_of G3; the_Edges_of G3 \ {e} c= the_Edges_of G3 by XBOOLE_1:36; hence the_Edges_of G3 \ {e} c= G3.edgesBetween(the_Vertices_of G3) by GLIB_000:34; end; :: properties of removeEdge now thus the_Vertices_of G4 = the_Vertices_of G3 by A5; for x being object holds x in the_Edges_of G3 \ {e} implies x in the_Edges_of G4 proof let x be object; assume x in the_Edges_of G3 \ {e}; then A12: x in the_Edges_of G3 & not x in {e} by XBOOLE_0:def 5; then A13: x in the_Edges_of G1 & not x in v1.edgesInOut() by A3, XBOOLE_0:def 5; A14: x in the_Edges_of G2 by A1, A12, XBOOLE_0:def 5; not x in v2.edgesInOut() by A13, GLIB_000:78, TARSKI:def 3; hence thesis by A4, A14, XBOOLE_0:def 5; end; then the_Edges_of G3 \ {e} c= the_Edges_of G4 by TARSKI:def 3; hence the_Edges_of G4 = the_Edges_of G3 \ {e} by A8, XBOOLE_0:def 10; end; hence thesis by A10, A11, GLIB_000:def 37; end; suppose A15: G1 is trivial or not v in the_Vertices_of G1; then A16: G1 == G3 by Th8; G2 is trivial or not v in the_Vertices_of G2 proof per cases by A15; suppose G1 is trivial; hence thesis; end; suppose not v in the_Vertices_of G1; hence thesis; end; end; then G2 == G4 by Th8; hence thesis by A16, Lm5; end; end; :: END into GLIB_000 ? :: START into GLIB_002 ? Lm6: for G being finite connected _Graph ex H being Subgraph of G st H is spanning Tree-like connected acyclic proof defpred P[Nat] means for G being finite connected _Graph st G.order() + $1 = G.size() + 1 ex H being Subgraph of G st H is spanning Tree-like connected acyclic; A1: P[0] proof let G be finite connected _Graph; assume A2: G.order() + 0 = G.size() + 1; reconsider H = G as Subgraph of G by GLIB_000:40; take H; the_Vertices_of H = the_Vertices_of G; hence H is spanning by GLIB_000:def 33; thus H is Tree-like by A2, GLIB_002:47; hence thesis; end; A3: for k being Nat st P[k] holds P[k+1] proof let k be Nat; assume A4: P[k]; let G be finite connected _Graph; assume A5: G.order() + (k+1) = G.size() + 1; :: we want to remove an edge that is part of a cycle :: so that the resulting subgraph is still connected k+1 <> 0; then G.order() <> G.size() + 1 by A5; then G is non acyclic by GLIB_002:47; then consider C being Walk of G such that A6: C is Cycle-like by GLIB_002:def 2; A7: C.edges() is non empty by A6, GLIB_001:136; set e = the Element of C.edges(); set G2 = the removeEdge of G, e; A8: G2 is connected by A6, A7, GLIB_002:5; G.order() + k + 1 = G2.size() + 1 + 1 by A5, A7, TARSKI:def 3, GLIB_000:52; then G2.order() + k = G2.size() + 1 by GLIB_000:52; then consider H being Subgraph of G2 such that A9: H is spanning Tree-like connected acyclic by A4, A8; reconsider H as spanning Subgraph of G by A9, GLIB_000:74; take H; thus thesis by A9; end; A10: for k being Nat holds P[k] from NAT_1:sch 2(A1,A3); let G be finite connected _Graph; ex k being Nat st G.size() + 1 = G.order() + k by GLIB_002:40, NAT_1:10; hence thesis by A10; end; registration let G be finite connected _Graph; cluster spanning Tree-like connected acyclic for Subgraph of G; existence by Lm6; end; theorem Th19: for G1 being connected _Graph, G2 being Subgraph of G1 st the_Edges_of G1 c= the_Edges_of G2 holds G1 == G2 proof let G1 be connected _Graph, G2 be Subgraph of G1; assume A1: the_Edges_of G1 c= the_Edges_of G2; A2: the_Edges_of G1 = the_Edges_of G2 by A1, XBOOLE_0:def 10; A3: G1 is Subgraph of G1 by GLIB_000:40; the_Vertices_of G1 = the_Vertices_of G2 proof per cases; suppose A4: G1 is non trivial; assume the_Vertices_of G1 <> the_Vertices_of G2; then not the_Vertices_of G1 c= the_Vertices_of G2 by XBOOLE_0:def 10; then A5: the_Vertices_of G1 \ the_Vertices_of G2 <> {} by XBOOLE_1:37; set v = the Element of the_Vertices_of G1 \ the_Vertices_of G2; reconsider v as Vertex of G1 by A5, TARSKI:def 3; per cases; suppose v.edgesInOut() = {}; hence contradiction by A4, GLIB_000:def 49, GLIB_002:2; end; suppose A6: v.edgesInOut() <> {}; set e = the Element of v.edgesInOut(); per cases by A6, GLIB_000:61; suppose A7: (the_Source_of G1).e = v; A8: e in the_Edges_of G2 by A2, A6, GLIB_000:61; then (the_Source_of G2).e = v by A7, GLIB_000:def 32; then v in the_Vertices_of G2 by A8, FUNCT_2:5; hence contradiction by A5, XBOOLE_0:def 5; end; suppose A9: (the_Target_of G1).e = v; A10: e in the_Edges_of G2 by A2, A6, GLIB_000:61; then (the_Target_of G2).e = v by A9, GLIB_000:def 32; then v in the_Vertices_of G2 by A10, FUNCT_2:5; hence contradiction by A5, XBOOLE_0:def 5; end; end; end; suppose A11: G1 is trivial; then consider v1 being Vertex of G1 such that A12: the_Vertices_of G1 = {v1} by GLIB_000:22; consider v2 being Vertex of G2 such that A13: the_Vertices_of G2 = {v2} by A11, GLIB_000:22; thus thesis by A12, A13, ZFMISC_1:3; end; end; hence thesis by A2, A3, GLIB_000:86; end; theorem for G1 being finite connected _Graph, G2 being Subgraph of G1 st G1.size() = G2.size() holds G1 == G2 proof let G1 be finite connected _Graph, G2 be Subgraph of G1; assume A1: G1.size() = G2.size(); card the_Edges_of G2 = G2.size() by GLIB_000:def 25 .= card the_Edges_of G1 by A1, GLIB_000:def 25; then the_Edges_of G1 = the_Edges_of G2 by CARD_2:102; hence thesis by Th19; end; theorem Th21: for G1 being finite Tree-like _Graph for G2 being spanning Tree-like Subgraph of G1 holds G1 == G2 proof let G1 be finite Tree-like _Graph, G2 be spanning Tree-like Subgraph of G1; G1.order() = G2.order() by Th11; then G1.size() + 1 = G2.order() by GLIB_002:47; then G1.size() + 1 = G2.size() + 1 by GLIB_002:47; hence thesis by Th13; end; registration let G be non trivial _Graph; cluster non spanning trivial connected for Subgraph of G; existence proof set v = the Vertex of G; set H = the inducedSubgraph of G, {v}; the_Vertices_of H <> the_Vertices_of G proof assume A1: the_Vertices_of H = the_Vertices_of G; 1 = card the_Vertices_of G by A1, GLIB_000:def 19; hence contradiction by GLIB_000:def 19; end; hence thesis by GLIB_000:def 33; end; end; :: counterpart of GLIB_002:12 theorem Th22: for G being _Graph, v1, v2 being Vertex of G st not v1 in G.reachableFrom(v2) holds G.reachableFrom(v1) misses G.reachableFrom(v2) proof let G be _Graph, v1, v2 be Vertex of G; assume A1: not v1 in G.reachableFrom(v2); assume not G.reachableFrom(v1) misses G.reachableFrom(v2); then G.reachableFrom(v1) /\ G.reachableFrom(v2) <> {} by XBOOLE_0:def 7; then consider w being object such that A2: w in G.reachableFrom(v1) /\ G.reachableFrom(v2) by XBOOLE_0:def 1; A3: w in G.reachableFrom(v1) & w in G.reachableFrom(v2) by A2, XBOOLE_0:def 4; then consider W1 being Walk of G such that A4: W1 is_Walk_from v1,w by GLIB_002:def 5; consider W2 being Walk of G such that A5: W2 is_Walk_from v2,w by A3, GLIB_002:def 5; W1.reverse() is_Walk_from w,v1 by A4, GLIB_001:23; then W2.append(W1.reverse()) is_Walk_from v2,v1 by A5, GLIB_001:31; hence contradiction by A1, GLIB_002:def 5; end; theorem Th23: for G being _Graph holds G.componentSet() is a_partition of the_Vertices_of G proof let G be _Graph; set V = the_Vertices_of G; A1: union G.componentSet() = V by GLIB_002:24; for A being Subset of V st A in G.componentSet() holds A <> {} & for B being Subset of V st B in G.componentSet() holds A = B or A misses B proof let A be Subset of V; assume A in G.componentSet(); then consider v being Vertex of G such that A2: A = G.reachableFrom(v) by GLIB_002:def 8; thus A <> {} by A2; let B be Subset of V; assume B in G.componentSet(); then consider w being Vertex of G such that A3: B = G.reachableFrom(w) by GLIB_002:def 8; per cases; suppose v in G.reachableFrom(w); hence thesis by A2, A3, GLIB_002:12; end; suppose not v in G.reachableFrom(w); hence thesis by A2, A3, Th22; end; end; hence thesis by A1, EQREL_1:def 4; end; theorem Th24: for G being _Graph, C being a_partition of the_Vertices_of G, v being Vertex of G st C = G.componentSet() holds EqClass(v,C) = G.reachableFrom(v) proof let G be _Graph, C be a_partition of the_Vertices_of G, v be Vertex of G; assume A1: C = G.componentSet(); EqClass(v,C) in C by EQREL_1:def 6; then consider w being Vertex of G such that A2: EqClass(v,C) = G.reachableFrom(w) by A1, GLIB_002:def 8; v in EqClass(v,C) by EQREL_1:def 6; hence thesis by A2, GLIB_002:12; end; :: holds more generally when G is non trivial and v0 is non cut-vertex; :: however, this theorem is used to show that endvertex implies non cut-vertex theorem Th25: for G1 being _Graph, v0, v1 being Vertex of G1 for G2 being removeVertex of G1,v0, v2 being Vertex of G2 st v0 is endvertex & v1 = v2 & v1 in G1.reachableFrom(v0) holds G2.reachableFrom(v2) = G1.reachableFrom(v1) \ {v0} proof let G1 be _Graph, v0,v1 be Vertex of G1; let G2 be removeVertex of G1, v0, v2 be Vertex of G2; assume A1: v0 is endvertex & v1 = v2 & v1 in G1.reachableFrom(v0); then A2: G1 is non trivial by Th16; then A3: the_Vertices_of G2 = the_Vertices_of G1 \ {v0} by GLIB_000:47; for w being object holds w in G2.reachableFrom(v2) iff w in G1.reachableFrom(v1) & not w in {v0} proof let w be object; thus w in G2.reachableFrom(v2) implies w in G1.reachableFrom(v1) & not w in {v0} by A1, A3, GLIB_002:14, TARSKI:def 3, XBOOLE_0:def 5; assume A4: w in G1.reachableFrom(v1) & not w in {v0}; then consider W being Walk of G1 such that A5: W is_Walk_from v1, w by GLIB_002:def 5; set P = the Path of W; v1 is set & w is set by TARSKI:1; then A6: P is_Walk_from v1, w by A5, GLIB_001:160; not v0 in P.vertices() proof assume v0 in P.vertices(); then per cases by A1, GLIB_001:143; suppose v0 = P.first(); then v0 = v2 by A1, A6, GLIB_001:def 23; then not v0 in {v0} by A3, XBOOLE_0:def 5; hence contradiction by TARSKI:def 1; end; suppose v0 = P.last(); then v0 = w by A6, GLIB_001:def 23; hence contradiction by A4, TARSKI:def 1; end; end; then reconsider P as Walk of G2 by A2, GLIB_001:171; P is_Walk_from v2, w by A1, A6, GLIB_001:19; hence w in G2.reachableFrom(v2) by GLIB_002:def 5; end; hence thesis by XBOOLE_0:def 5; end; theorem Th26: for G1 being non trivial _Graph, v0,v1 being Vertex of G1 for G2 being removeVertex of G1, v0, v2 being Vertex of G2 st v1 = v2 & not v1 in G1.reachableFrom(v0) holds G2.reachableFrom(v2) = G1.reachableFrom(v1) proof let G1 be non trivial _Graph, v0,v1 be Vertex of G1; let G2 be removeVertex of G1, v0, v2 be Vertex of G2; assume A1: v1 = v2 & not v1 in G1.reachableFrom(v0); then A2: G2.reachableFrom(v2) c= G1.reachableFrom(v1) by GLIB_002:14; for w being object holds w in G1.reachableFrom(v1) implies w in G2.reachableFrom(v2) proof let w be object; assume w in G1.reachableFrom(v1); then consider W being Walk of G1 such that A3: W is_Walk_from v1, w by GLIB_002:def 5; not v0 in W.vertices() proof assume A4: v0 in W.vertices(); reconsider m = 1 as odd Element of NAT by POLYFORM:4; reconsider n = W.find(v0) as odd Element of NAT; set U = W.cut(m,n); m <= n & n <= len W by A4, GLIB_001:def 19, CHORD:2; then U is_Walk_from W.m, W.n by GLIB_001:37; then U is_Walk_from W.first(), W.n by GLIB_001:def 6; then U is_Walk_from v1, W.n by A3, GLIB_001:def 23; then U is_Walk_from v1, v0 by A4, GLIB_001:def 19; then U.reverse() is_Walk_from v0, v1 by GLIB_001:23; hence contradiction by A1, GLIB_002:def 5; end; then reconsider W2 = W as Walk of G2 by GLIB_001:171; W2 is_Walk_from v2, w by A1, A3, GLIB_001:19; hence w in G2.reachableFrom(v2) by GLIB_002:def 5; end; then G1.reachableFrom(v1) c= G2.reachableFrom(v2) by TARSKI:def 3; hence thesis by A2, XBOOLE_0:def 10; end; theorem Th27: for G being non trivial finite Tree-like _Graph, v being Vertex of G st G.order() = 2 holds v is endvertex proof let G be non trivial finite Tree-like _Graph, v be Vertex of G; assume G.order() = 2; then card the_Vertices_of G = 2 by GLIB_000:def 24; then consider v1, v2 being object such that A1: v1 <> v2 & the_Vertices_of G = {v1,v2} by CARD_2:60; consider w1,w2 being Vertex of G such that A2: w1 <> w2 & w1 is endvertex & w2 is endvertex by GLIB_002:45; (w1 = v1 or w1 = v2) & (w2 = v1 or w2 = v2) by A1, TARSKI:def 2; then per cases by A2; suppose w1 = v1 & w2 = v2; hence thesis by A1, A2, TARSKI:def 2; end; suppose w1= v2 & w2 = v1; hence thesis by A1, A2, TARSKI:def 2; end; end; registration let G be non trivial connected _Graph, v be Vertex of G; cluster v.allNeighbors() -> non empty; coherence by Th7, GLIB_002:2; end; :: END into GLIB_002 ? :: into HELLY ? theorem for T being _Tree, a being Vertex of T holds T.pathBetween(a,a) = T.walkOf(a) proof let T be _Tree, a be Vertex of T; consider b being Vertex of T such that A1: T.pathBetween(a,a) = T.walkOf(b) by GLIB_001:128; {a} = T.pathBetween(a,a).vertices() by HELLY:30 .= {b} by A1, GLIB_001:90; hence thesis by A1, ZFMISC_1:3; end; :: into HELLY ? theorem Th29: for T being _Tree, a,b being Vertex of T, e being object st e Joins a,b,T holds T.pathBetween(a,b) = T.walkOf(a,e,b) by GLIB_001:15, HELLY:def 2; :: into HELLY ? theorem Th30: for T being non trivial finite _Tree, v being Vertex of T ex v1, v2 being Vertex of T st v1 <> v2 & v1 is endvertex & v2 is endvertex & v in T.pathBetween(v1,v2).vertices() proof defpred P[Nat] means for T being non trivial finite _Tree for v being Vertex of T st T.order() = $1 + 2 holds ex v1, v2 being Vertex of T st v1 <> v2 & v1 is endvertex & v2 is endvertex & v in T.pathBetween(v1,v2).vertices(); A1: P[0] proof let T be non trivial finite _Tree, v be Vertex of T; assume A2: T.order() = 0 + 2; then card the_Vertices_of T = 2 by GLIB_000:def 24; then consider v1, v2 being object such that A3: v1 <> v2 & the_Vertices_of T = {v1,v2} by CARD_2:60; reconsider v1, v2 as Vertex of T by A3, TARSKI:def 2; take v1, v2; thus v1 <> v2 by A3; thus v1 is endvertex & v2 is endvertex by A2, Th27; v = v1 or v = v2 by A3, TARSKI:def 2; hence thesis by HELLY:29; end; A4: for k being Nat st P[k] holds P[k+1] proof let k be Nat; assume A5: P[k]; let T be non trivial finite _Tree, v be Vertex of T; assume A6: T.order() = (k+1) + 2; set v0 = the endvertex Vertex of T; per cases; suppose A7: v = v0; consider v1, v2 being Vertex of T such that A8: v1 <> v2 & v1 is endvertex & v2 is endvertex by GLIB_002:45; per cases; suppose A9: v <> v1; take v, v1; thus v <> v1 & v is endvertex & v1 is endvertex by A7, A8, A9; thus thesis by HELLY:29; end; suppose A10: v = v1; take v, v2; thus v <> v2 & v is endvertex & v2 is endvertex by A8, A10; thus thesis by HELLY:29; end; end; suppose A11: v <> v0; set T0 = the removeVertex of T, v0; a12: T0.order() + 1 = k + 2 + 1 by A6, GLIB_000:48; T0.order() <> 1 proof assume T0.order() = 1; then 0+1 = k + 1 + 1 by a12; hence contradiction; end; then reconsider T0 as non trivial finite _Tree by GLIB_000:26; not v in {v0} by A11, TARSKI:def 1; then v in the_Vertices_of T \ {v0} by XBOOLE_0:def 5; then v in the_Vertices_of T0 by GLIB_000:47; then consider v1,v2 being Vertex of T0 such that A13: v1 <> v2 & v1 is endvertex & v2 is endvertex and A14: v in T0.pathBetween(v1,v2).vertices() by A5, a12; the_Vertices_of T0 c= the_Vertices_of T by GLIB_000:def 32; then reconsider w1 = v1, w2 = v2 as Vertex of T by TARSKI:def 3; consider e0 being object such that A15: v0.edgesInOut() = {e0} & not e0 Joins v0,v0,T by GLIB_000:def 51; A16: the_Edges_of T0 = T.edgesBetween(the_Vertices_of T \ {v0}) by GLIB_000:47 .= T.edgesBetween(the_Vertices_of T) \ v0.edgesInOut() by Th1 .= the_Edges_of T \ {e0} by A15, GLIB_000:34; consider e1 being object such that A17: v1.edgesInOut() = {e1} & not e1 Joins v1,v1,T0 by A13, GLIB_000:def 51; consider e2 being object such that A18: v2.edgesInOut() = {e2} & not e2 Joins v2,v2,T0 by A13, GLIB_000:def 51; v0 in {v0} by TARSKI:def 1; then not v0 in the_Vertices_of T \ {v0} by XBOOLE_0:def 5; then A19: not v0 in the_Vertices_of T0 by GLIB_000:47; A20: w1 <> v0.adj(e0) implies w1 is endvertex proof assume A21: w1 <> v0.adj(e0); A22: v1.edgesInOut() c= w1.edgesInOut() by GLIB_000:78; for e being object st e in w1.edgesInOut() holds e in v1.edgesInOut() proof let e be object; assume A23: e in w1.edgesInOut(); e <> e0 proof assume e = e0; then A24: e0 Joins w1,w1.adj(e0),T by A23, GLIB_000:67; e0 in v0.edgesInOut() by A15, TARSKI:def 1; then e0 Joins v0,v0.adj(e0),T by GLIB_000:67; then w1 = v0 or w1 = v0.adj(e0) by A24, GLIB_000:15; hence contradiction by A19, A21; end; then not e in {e0} by TARSKI:def 1; then A25: e in the_Edges_of T0 by A16, A23, XBOOLE_0:def 5; then (the_Source_of T0).e = (the_Source_of T).e & (the_Target_of T0).e = (the_Target_of T).e by GLIB_000:def 32; then (the_Source_of T0).e = w1 or (the_Target_of T0).e = w1 by A23, GLIB_000:61; hence thesis by A25, GLIB_000:61; end; then w1.edgesInOut() c= v1.edgesInOut() by TARSKI:def 3; then v1.edgesInOut() = w1.edgesInOut() by A22, XBOOLE_0:def 10; hence w1 is endvertex by A17, GLIB_000:18, GLIB_000:def 51; end; A26: w2 <> v0.adj(e0) implies w2 is endvertex proof assume A27: w2 <> v0.adj(e0); A28: v2.edgesInOut() c= w2.edgesInOut() by GLIB_000:78; for e being object st e in w2.edgesInOut() holds e in v2.edgesInOut() proof let e be object; assume A29: e in w2.edgesInOut(); e <> e0 proof assume e = e0; then A30: e0 Joins w2,w2.adj(e0),T by A29, GLIB_000:67; e0 in v0.edgesInOut() by A15, TARSKI:def 1; then e0 Joins v0,v0.adj(e0),T by GLIB_000:67; then w2 = v0 or w2 = v0.adj(e0) by A30, GLIB_000:15; hence contradiction by A19, A27; end; then not e in {e0} by TARSKI:def 1; then A31: e in the_Edges_of T0 by A16, A29, XBOOLE_0:def 5; then (the_Source_of T0).e = (the_Source_of T).e & (the_Target_of T0).e = (the_Target_of T).e by GLIB_000:def 32; then (the_Source_of T0).e = w2 or (the_Target_of T0).e = w2 by A29, GLIB_000:61; hence thesis by A31, GLIB_000:61; end; then w2.edgesInOut() c= v2.edgesInOut() by TARSKI:def 3; then v2.edgesInOut() = w2.edgesInOut() by A28, XBOOLE_0:def 10; hence w2 is endvertex by A18, GLIB_000:18, GLIB_000:def 51; end; A32: T0.pathBetween(v1,v2) = T.pathBetween(w1,w2) by HELLY:33; per cases; suppose A33: w1 = v0.adj(e0); then A34: w2 <> v0.adj(e0) by A13; take v0, w2; thus v0 <> w2 & v0 is endvertex & w2 is endvertex by A19, A26, A34; e0 in v0.edgesInOut() by A15, TARSKI:def 1; then A35: e0 Joins v0, w1, T by A33, GLIB_000:67; then A36: T.walkOf(v0,e0,w1) = T.pathBetween(v0,w1) by Th29; set P1 = T.pathBetween(v0,w1), P2 = T.pathBetween(w1,w2); set P = P1.append(P2); A37: P1.last() = w1 & P2.first() = w1 by HELLY:28; for x being object holds x in P1.vertices() /\ P2.vertices() iff x = w1 proof let x be object; hereby assume x in P1.vertices() /\ P2.vertices(); then A38: x in P1.vertices() & x in P2.vertices() by XBOOLE_0:def 4; then A39: x in {v0,w1} by A35, A36, GLIB_001:91; x in T0.pathBetween(v1,v2).vertices() by A32, A38, GLIB_001:98; then x in the_Vertices_of T0; hence x = w1 by A19, A39, TARSKI:def 2; end; assume x = w1; then x = P1.last() & x = P2.first() by HELLY:28; then x in P1.vertices() & x in P2.vertices() by GLIB_001:88; hence thesis by XBOOLE_0:def 4; end; then P1.vertices() /\ P2.vertices() = {P1.last()} by A37,TARSKI:def 1; then reconsider P as Path of T by A37, HELLY:38; P1 is_Walk_from v0,w1 & P2 is_Walk_from w1,w2 by HELLY:def 2; then A40: P = T.pathBetween(v0,w2) by GLIB_001:31, HELLY:def 2; T0.pathBetween(v1,v2).vertices() = P2.vertices() by HELLY:33, GLIB_001:98; then v in P1.vertices() \/ P2.vertices() by A14, XBOOLE_0:def 3; hence thesis by A37, A40, GLIB_001:93; end; suppose A41: w2 = v0.adj(e0); then A42: w1 <> v0.adj(e0) by A13; take w1, v0; thus w1 <> v0 & w1 is endvertex & v0 is endvertex by A19, A20, A42; e0 in v0.edgesInOut() by A15, TARSKI:def 1; then A43: e0 Joins w2, v0, T by A41, GLIB_000:67, GLIB_000:14; then A44: T.walkOf(w2,e0,v0) = T.pathBetween(w2,v0) by Th29; set P1 = T.pathBetween(w1,w2), P2 = T.pathBetween(w2,v0); set P = P1.append(P2); A45: P1.last() = w2 & P2.first() = w2 by HELLY:28; for x being object holds x in P1.vertices() /\ P2.vertices() iff x = w2 proof let x be object; hereby assume x in P1.vertices() /\ P2.vertices(); then A46: x in P1.vertices() & x in P2.vertices() by XBOOLE_0:def 4; then A47: x in {w2,v0} by A43, A44, GLIB_001:91; x in T0.pathBetween(v1,v2).vertices() by A32, A46, GLIB_001:98; then x in the_Vertices_of T0; hence x = w2 by A19, A47, TARSKI:def 2; end; assume x = w2; then x in P1.vertices() & x in P2.vertices() by GLIB_001:88, A45; hence thesis by XBOOLE_0:def 4; end; then P1.vertices() /\ P2.vertices() = {P1.last()} by A45,TARSKI:def 1; then reconsider P as Path of T by A45, HELLY:38; P1 is_Walk_from w1,w2 & P2 is_Walk_from w2,v0 by HELLY:def 2; then A48: P = T.pathBetween(w1,v0) by GLIB_001:31, HELLY:def 2; T0.pathBetween(v1,v2).vertices() = P1.vertices() by HELLY:33, GLIB_001:98; then v in P1.vertices() \/ P2.vertices() by A14, XBOOLE_0:def 3; hence thesis by A45, A48, GLIB_001:93; end; suppose A49: w1 <> v0.adj(e0) & w2 <> v0.adj(e0); take w1, w2; thus w1<>w2 & w1 is endvertex & w2 is endvertex by A13, A20, A26, A49; thus thesis by A14, GLIB_001:98, HELLY:33; end; end; end; A50: for k being Nat holds P[k] from NAT_1:sch 2(A1,A4); let T be non trivial finite _Tree, v be Vertex of T; T.order() >= 1 & T.order() <> 1 by GLIB_000:25, GLIB_000:26; then T.order() > 1 by XXREAL_0:1; then T.order() >= 1+1 by INT_1:7; then consider k being Nat such that A51: T.order() = 2 + k by NAT_1:10; thus thesis by A50, A51; end; :: into HELLY ? theorem Th31: for G1 being non trivial finite Tree-like _Graph for G2 being non spanning connected Subgraph of G1 ex v being Vertex of G1 st v is endvertex & not v in the_Vertices_of G2 proof let G1 be non trivial finite Tree-like _Graph; let G2 be non spanning connected Subgraph of G1; assume A1: for v being Vertex of G1 holds not v is endvertex or v in the_Vertices_of G2; A2: not the_Vertices_of G1 c= the_Vertices_of G2 by GLIB_000:def 33, XBOOLE_0:def 10; :: Now we show the negation for x being object st x in the_Vertices_of G1 holds x in the_Vertices_of G2 proof let x be object; assume x in the_Vertices_of G1; then reconsider v = x as Vertex of G1; consider v1, v2 being Vertex of G1 such that A3: v1 <> v2 & v1 is endvertex & v2 is endvertex and A4: v in G1.pathBetween(v1,v2).vertices() by Th30; reconsider w1 = v1, w2 = v2 as Vertex of G2 by A1, A3; v in G2.pathBetween(w1,w2).vertices() by A4, HELLY:33, GLIB_001:98; hence thesis; end; hence contradiction by A2, TARSKI:def 3; end; :: START into GLIB_006 ? theorem Th32: for G2, G3 being _Graph, V being set, G1 being addVertices of G2, V st G2 == G3 holds G1 is addVertices of G3, V proof let G2, G3 be _Graph, V be set, G1 be addVertices of G2, V; assume A1: G2 == G3; then the_Vertices_of G3 = the_Vertices_of G2 & the_Edges_of G3 = the_Edges_of G2 & the_Target_of G3 = the_Target_of G2 & the_Source_of G3 = the_Source_of G2 by GLIB_000:def 34; then A2: the_Vertices_of G1 = the_Vertices_of G3 \/ V & the_Edges_of G1 = the_Edges_of G3 & the_Source_of G1 = the_Source_of G3 & the_Target_of G1 = the_Target_of G3 by GLIB_006:def 10; G2 is Supergraph of G3 by A1, GLIB_006:58; then G1 is Supergraph of G3 by GLIB_006:62; hence thesis by A2, GLIB_006:def 10; end; theorem Th33: for G2 being _Graph, G1 being Supergraph of G2 st the_Edges_of G1 = the_Edges_of G2 holds G1 is addVertices of G2, the_Vertices_of G1 \ the_Vertices_of G2 proof let G2 be _Graph, G1 be Supergraph of G2; assume A1: the_Edges_of G1 = the_Edges_of G2; the_Vertices_of G2 c= the_Vertices_of G1 by GLIB_006:def 9; then A2: the_Vertices_of G1 = the_Vertices_of G2 \/ (the_Vertices_of G1 \ the_Vertices_of G2) by XBOOLE_1:45; A3: the_Source_of G2 = (the_Source_of G1)|the_Edges_of G2 by GLIB_006:69 .= the_Source_of G1 by A1; the_Target_of G2 = (the_Target_of G1)|the_Edges_of G2 by GLIB_006:69 .= the_Target_of G1 by A1; hence thesis by A1, A2, A3, GLIB_006:def 10; end; theorem Th34: for G1 being finite _Graph, G2 being Subgraph of G1 st G1.size() = G2.size() holds G1 is addVertices of G2, the_Vertices_of G1 \ the_Vertices_of G2 proof let G1 be finite _Graph, G2 be Subgraph of G1; assume A1: G1.size() = G2.size(); card the_Edges_of G2 = G2.size() by GLIB_000:def 25 .= card the_Edges_of G1 by A1, GLIB_000:def 25; then A2: the_Edges_of G1 = the_Edges_of G2 by CARD_2:102; G1 is Supergraph of G2 by GLIB_006:57; hence thesis by A2, Th33; end; theorem for G1 being non trivial _Graph, v being Vertex of G1 for G2 being removeVertex of G1, v st v is isolated holds G1 is addVertex of G2, v proof let G1 be non trivial _Graph, v be Vertex of G1; let G2 be removeVertex of G1, v; assume A1: v is isolated; A2: G1 is Supergraph of G2 by GLIB_006:57; A3: the_Vertices_of G1 = (the_Vertices_of G1 \ {v}) \/ {v} by ZFMISC_1:116 .= the_Vertices_of G2 \/ {v} by GLIB_000:47; A4: the_Edges_of G1 = G1.edgesBetween(the_Vertices_of G1) by GLIB_000:34 .= G1.edgesBetween(the_Vertices_of G1 \ {v}) by A1, Th2 .= the_Edges_of G2 by GLIB_000:47; A5: dom the_Source_of G1 = the_Edges_of G1 by FUNCT_2:def 1 .= dom the_Source_of G2 by A4, FUNCT_2:def 1; for e being object st e in dom the_Source_of G2 holds (the_Source_of G2).e = (the_Source_of G1).e by GLIB_000:def 32; then A6: the_Source_of G1 = the_Source_of G2 by A5, FUNCT_1:2; A7: dom the_Target_of G1 = the_Edges_of G1 by FUNCT_2:def 1 .= dom the_Target_of G2 by A4, FUNCT_2:def 1; for e being object st e in dom the_Target_of G2 holds (the_Target_of G2).e = (the_Target_of G1).e by GLIB_000:def 32; then the_Target_of G1 = the_Target_of G2 by A7, FUNCT_1:2; hence thesis by A2, A3, A4, A6, GLIB_006:def 10; end; theorem Th36: for G2, G3 being _Graph, v1,e,v2 being object, G1 being addEdge of G2,v1,e,v2 st G2 == G3 holds G1 is addEdge of G3,v1,e,v2 proof let G2, G3 be _Graph, v1,e,v2 be object; let G1 be addEdge of G2,v1,e,v2; assume A1: G2 == G3; then A2: the_Vertices_of G3 = the_Vertices_of G2 & the_Edges_of G3 = the_Edges_of G2 & the_Target_of G3 = the_Target_of G2 & the_Source_of G3 = the_Source_of G2 by GLIB_000:def 34; per cases; suppose A3: v1 in the_Vertices_of G2 & v2 in the_Vertices_of G2 & not e in the_Edges_of G2; then A4: the_Vertices_of G1 = the_Vertices_of G3 & the_Edges_of G1 = the_Edges_of G3 \/ {e} & the_Source_of G1 = the_Source_of G3 +* (e .--> v1) & the_Target_of G1 = the_Target_of G3 +* (e .--> v2) by A2, GLIB_006:def 11; G2 is Supergraph of G3 by A1, GLIB_006:58; then G1 is Supergraph of G3 by GLIB_006:62; hence thesis by A2, A3, A4, GLIB_006:def 11; end; suppose A5: not (v1 in the_Vertices_of G2 & v2 in the_Vertices_of G2 & not e in the_Edges_of G2); then G1 == G2 by GLIB_006:def 11; then A6: G1 == G3 by A1, GLIB_000:85; then G1 is Supergraph of G3 by GLIB_006:58; hence thesis by A2, A5, A6, GLIB_006:def 11; end; end; theorem Th37: for G1 being _Graph, e being set, G2 being removeEdge of G1, e st e in the_Edges_of G1 holds G1 is addEdge of G2, (the_Source_of G1).e, e, (the_Target_of G1).e proof let G1 be _Graph, e be set, G2 be removeEdge of G1, e; assume A1: e in the_Edges_of G1; set u = (the_Source_of G1).e, w = (the_Target_of G1).e; A2: G1 is Supergraph of G2 by GLIB_006:57; A3: the_Vertices_of G1 = the_Vertices_of G2 by GLIB_000:51; A4: the_Edges_of G1 = (the_Edges_of G1 \ {e}) \/ {e} by A1, ZFMISC_1:116 .= the_Edges_of G2 \/ {e} by GLIB_000:51; A5: dom the_Source_of G1 = the_Edges_of G1 by FUNCT_2:def 1 .= dom the_Source_of G2 \/ dom({e} --> u) by A4, FUNCT_2:def 1 .= dom the_Source_of G2 \/ dom(e .--> u) by FUNCOP_1:def 9 .= dom(the_Source_of G2 +* (e .--> u)) by FUNCT_4:def 1; for e0 being object st e0 in dom the_Source_of G1 holds (the_Source_of G1).e0 = (the_Source_of G2 +* (e .--> u)).e0 proof let e0 be object; assume A6: e0 in dom the_Source_of G1; per cases; suppose e0 = e; hence (the_Source_of G1).e0 = (the_Source_of G2 +* (e .--> u)).e0 by FUNCT_4:113; end; suppose A7: e0 <> e; then not e0 in {e} by TARSKI:def 1; then e0 in the_Edges_of G1 \ {e} by A6, XBOOLE_0:def 5; then e0 in the_Edges_of G2 by GLIB_000:51; hence (the_Source_of G1).e0 = (the_Source_of G2).e0 by GLIB_000:def 32 .= (the_Source_of G2 +* (e .--> u)).e0 by A7, FUNCT_4:83; end; end; then A8: the_Source_of G1 = the_Source_of G2 +* (e .--> u) by A5, FUNCT_1:2; A9: dom the_Target_of G1 = the_Edges_of G1 by FUNCT_2:def 1 .= dom the_Target_of G2 \/ dom({e} --> w) by A4, FUNCT_2:def 1 .= dom the_Target_of G2 \/ dom(e .--> w) by FUNCOP_1:def 9 .= dom(the_Target_of G2 +* (e .--> w)) by FUNCT_4:def 1; for e0 being object st e0 in dom the_Target_of G1 holds (the_Target_of G1).e0 = (the_Target_of G2 +* (e .--> w)).e0 proof let e0 be object; assume A10: e0 in dom the_Target_of G1; per cases; suppose e0 = e; hence (the_Target_of G1).e0 = (the_Target_of G2 +* (e .--> w)).e0 by FUNCT_4:113; end; suppose A11: e0 <> e; then not e0 in {e} by TARSKI:def 1; then e0 in the_Edges_of G1 \ {e} by A10, XBOOLE_0:def 5; then e0 in the_Edges_of G2 by GLIB_000:51; hence (the_Target_of G1).e0 = (the_Target_of G2).e0 by GLIB_000:def 32 .= (the_Target_of G2 +* (e .--> w)).e0 by A11, FUNCT_4:83; end; end; then A12: the_Target_of G1 = the_Target_of G2 +*(e .--> w) by A9, FUNCT_1:2; A13: u in the_Vertices_of G2 & w in the_Vertices_of G2 by A1, A3, FUNCT_2:5; e in {e} by TARSKI:def 1; then not e in the_Edges_of G1 \ {e} by XBOOLE_0:def 5; then not e in the_Edges_of G2 by GLIB_000:51; hence thesis by A2, A3, A4, A8, A12, A13, GLIB_006:def 11; end; theorem Th38: for G1 being non trivial _Graph, v being Vertex of G1, e being object for G2 being removeVertex of G1, v st {e} = v.edgesInOut() & not e Joins v,v,G1 :: i.e. v is endvertex holds G1 is addAdjVertex of G2, v.adj(e), e, v or G1 is addAdjVertex of G2, v, e, v.adj(e) proof let G1 be non trivial _Graph, v be Vertex of G1, e be object; let G2 be removeVertex of G1, v; assume A1: {e} = v.edgesInOut() & not e Joins v,v,G1; A2: G1 is Supergraph of G2 by GLIB_006:57; A3: the_Vertices_of G1 = (the_Vertices_of G1 \ {v}) \/ {v} by ZFMISC_1:116 .= the_Vertices_of G2 \/ {v} by GLIB_000:47; A4: e in {e} by TARSKI:def 1; A5: the_Edges_of G1 = (the_Edges_of G1 \ {e}) \/ {e} by A1, A4, ZFMISC_1:116 .= (G1.edgesBetween(the_Vertices_of G1) \ v.edgesInOut()) \/ {e} by A1, GLIB_000:34 .= G1.edgesBetween(the_Vertices_of G1 \ {v}) \/ {e} by Th1 .= the_Edges_of G2 \/ {e} by GLIB_000:47; set u = (the_Source_of G1).e, w = (the_Target_of G1).e; v in {v} by TARSKI:def 1; then not v in the_Vertices_of G1 \ {v} by XBOOLE_0:def 5; then A6: not v in the_Vertices_of G2 by GLIB_000:47; not e in G1.edgesBetween(the_Vertices_of G1) \ v.edgesInOut() by A1, A4, XBOOLE_0:def 5; then not e in G1.edgesBetween(the_Vertices_of G1 \ {v}) by Th1; then A7: not e in the_Edges_of G2 by GLIB_000:47; e in v.edgesInOut() by A1, TARSKI:def 1; then A8: v <> v.adj(e) by A1, GLIB_000:67; then not v.adj(e) in {v} by TARSKI:def 1; then v.adj(e) in the_Vertices_of G1 \ {v} by XBOOLE_0:def 5; then A9: v.adj(e) in the_Vertices_of G2 by GLIB_000:47; A10: dom the_Source_of G1 = the_Edges_of G1 by FUNCT_2:def 1 .= dom the_Source_of G2 \/ {e} by A5, FUNCT_2:def 1; A11: dom the_Target_of G1 = the_Edges_of G1 by FUNCT_2:def 1 .= dom the_Target_of G2 \/ {e} by A5, FUNCT_2:def 1; e in v.edgesInOut() by A1, TARSKI:def 1; then per cases by GLIB_000:61; suppose A12: u = v; then A13: w = v.adj(e) by A1, A4, A8, GLIB_000:def 41; :: we show G1 is addAdjVertex of G2, v, e, v.adj(e) :: using the second part of the definition A14: dom the_Source_of G1 = dom the_Source_of G2 \/ dom({e} --> v) by A10 .= dom(the_Source_of G2 +* ({e} --> v)) by FUNCT_4:def 1 .= dom(the_Source_of G2 +* (e .--> v)) by FUNCOP_1:def 9; for e0 being object st e0 in dom the_Source_of G1 holds (the_Source_of G1).e0 = (the_Source_of G2 +* (e .--> v)).e0 proof let e0 be object; assume A15: e0 in dom the_Source_of G1; per cases; suppose e0 = e; hence (the_Source_of G1).e0 = (the_Source_of G2 +* (e .--> v)).e0 by A12, FUNCT_4:113; end; suppose A16: e0 <> e; then not e0 in {e} by TARSKI:def 1; then e0 in the_Edges_of G2 by A5, A15, XBOOLE_0:def 3; hence (the_Source_of G1).e0 = (the_Source_of G2).e0 by GLIB_000:def 32 .= (the_Source_of G2 +* (e .--> v)).e0 by A16, FUNCT_4:83; end; end; then A17: the_Source_of G1 = the_Source_of G2 +* (e .--> v) by A14, FUNCT_1:2; A18: dom the_Target_of G1 = dom the_Target_of G2 \/ dom({e} --> v.adj(e)) by A11 .= dom(the_Target_of G2 +* ({e} --> v.adj(e))) by FUNCT_4:def 1 .= dom(the_Target_of G2 +* (e .--> v.adj(e))) by FUNCOP_1:def 9; for e0 being object st e0 in dom the_Target_of G1 holds (the_Target_of G1).e0 = (the_Target_of G2 +* (e .--> v.adj(e))).e0 proof let e0 be object; assume A19: e0 in dom the_Target_of G1; per cases; suppose e0 = e; hence (the_Target_of G1).e0 = (the_Target_of G2 +* (e .--> v.adj(e))).e0 by A13, FUNCT_4:113; end; suppose A20: e0 <> e; then not e0 in {e} by TARSKI:def 1; then e0 in the_Edges_of G2 by A5, A19, XBOOLE_0:def 3; hence (the_Target_of G1).e0 = (the_Target_of G2).e0 by GLIB_000:def 32 .= (the_Target_of G2 +* (e .--> v.adj(e))).e0 by A20, FUNCT_4:83; end; end; then the_Target_of G1 = the_Target_of G2 +* (e .--> v.adj(e)) by A18, FUNCT_1:2; hence thesis by A2, A3, A5, A6, A7, A9, A17, GLIB_006:def 12; end; suppose A21: w = v; then A22: u = v.adj(e) by A1, A4, GLIB_000:def 41; :: we show G1 is addAdjVertex of G2, v.adj(e), e, v :: using the first part of the definition A23: dom the_Target_of G1 = dom the_Target_of G2 \/ dom({e} --> v) by A11 .= dom(the_Target_of G2 +* ({e} --> v)) by FUNCT_4:def 1 .= dom(the_Target_of G2 +* (e .--> v)) by FUNCOP_1:def 9; for e0 being object st e0 in dom the_Target_of G1 holds (the_Target_of G1).e0 = (the_Target_of G2 +* (e .--> v)).e0 proof let e0 be object; assume A24: e0 in dom the_Target_of G1; per cases; suppose e0 = e; hence (the_Target_of G1).e0 = (the_Target_of G2 +* (e .--> v)).e0 by A21, FUNCT_4:113; end; suppose A25: e0 <> e; then not e0 in {e} by TARSKI:def 1; then e0 in the_Edges_of G2 by A5, A24, XBOOLE_0:def 3; hence (the_Target_of G1).e0 = (the_Target_of G2).e0 by GLIB_000:def 32 .= (the_Target_of G2 +* (e .--> v)).e0 by A25, FUNCT_4:83; end; end; then A26: the_Target_of G1 = the_Target_of G2 +* (e .--> v) by A23, FUNCT_1:2; A27: dom the_Source_of G1 = dom the_Source_of G2 \/ dom({e} --> v.adj(e)) by A10 .= dom(the_Source_of G2 +* ({e} --> v.adj(e))) by FUNCT_4:def 1 .= dom(the_Source_of G2 +* (e .--> v.adj(e))) by FUNCOP_1:def 9; for e0 being object st e0 in dom the_Source_of G1 holds (the_Source_of G1).e0 = (the_Source_of G2 +* (e .--> v.adj(e))).e0 proof let e0 be object; assume A28: e0 in dom the_Source_of G1; per cases; suppose e0 = e; hence (the_Source_of G1).e0 = (the_Source_of G2 +* (e .--> v.adj(e))).e0 by A22, FUNCT_4:113; end; suppose A29: e0 <> e; then not e0 in {e} by TARSKI:def 1; then e0 in the_Edges_of G2 by A5, A28, XBOOLE_0:def 3; hence (the_Source_of G1).e0 = (the_Source_of G2).e0 by GLIB_000:def 32 .= (the_Source_of G2 +* (e .--> v.adj(e))).e0 by A29, FUNCT_4:83; end; end; then the_Source_of G1 = the_Source_of G2 +* (e .--> v.adj(e)) by A27, FUNCT_1:2; hence thesis by A2, A3, A5, A6, A7, A9, A26, GLIB_006:def 12; end; end; theorem Th39: for G2 being _Graph, v1,v2 being Vertex of G2, e being object for G1 being addEdge of G2,v1,e,v2 for w being Vertex of G1, v being Vertex of G2 st v2 in G2.reachableFrom(v1) & v = w holds G1.reachableFrom(w) = G2.reachableFrom(v) proof let G2 be _Graph, v1,v2 be Vertex of G2, e be object; let G1 be addEdge of G2,v1,e,v2; let w be Vertex of G1, v be Vertex of G2; assume that A1: v2 in G2.reachableFrom(v1) and A2: w = v; per cases; suppose A3: not e in the_Edges_of G2; G2 is Subgraph of G1 by GLIB_006:57; then A4: G2.reachableFrom(v) c= G1.reachableFrom(w) by A2, GLIB_002:14; for y being object holds y in G1.reachableFrom(w) implies y in G2.reachableFrom(v) proof let y be object; assume y in G1.reachableFrom(w); then consider W1 being Walk of G1 such that A5: W1 is_Walk_from w,y by GLIB_002:def 5; set T = the Trail of W1; y is set by TARSKI:1; then A6: T is_Walk_from w,y by A5, GLIB_001:160; per cases; suppose not e in T.edges(); then reconsider W2 = T as Walk of G2 by GLIB_006:109; W2 is_Walk_from v,y by A6, A2, GLIB_001:19; hence y in G2.reachableFrom(v) by GLIB_002:def 5; end; suppose e in T.edges(); then consider w1, w2 being Vertex of G1, n being odd Element of NAT such that A7: n+2 <= len T and A8: w1 = T.n & e = T.(n+1) & w2 = T.(n+2) and A9: e Joins w1,w2,G1 by GLIB_001:103; set E = G1.walkOf(w1,e,w2); A10: E is_odd_substring_of T,0 by A7, A8, GLIB_006:27; e DJoins v1,v2,G1 by A3, GLIB_006:105; then A11: e Joins v1,v2,G1 by GLIB_000:16; then per cases by A9, GLIB_000:15; suppose A12: v1 = w1 & v2 = w2; consider W2 being Walk of G2 such that A13: W2 is_Walk_from v1,v2 by A1, GLIB_002:def 5; reconsider W4 = W2 as Walk of G1 by GLIB_006:75; not e in W2.edges() by A3; then A14: not e in W4.edges() by GLIB_001:110; set W5 = T.replaceEdgeWith(e,W4); A15: W5 is_Walk_from v,y by A2, A6, GLIB_006:47; W2.first() = v1 & W2.last() = v2 by A13, GLIB_001:def 23; then W4.first() = v1 & W4.last() = v2 by GLIB_001:16; then not e in W5.edges() by A11, A14, GLIB_006:40, A10, A12; then reconsider W = W5 as Walk of G2 by GLIB_006:109; W is_Walk_from v,y by A15, GLIB_001:19; hence y in G2.reachableFrom(v) by GLIB_002:def 5; end; suppose A17: v1 = w2 & v2 = w1; consider W3 being Walk of G2 such that A18: W3 is_Walk_from v1,v2 by A1, GLIB_002:def 5; A19: W3.reverse() is_Walk_from v2,v1 by A18, GLIB_001:23; reconsider W4 = W3.reverse() as Walk of G1 by GLIB_006:75; not e in W3.reverse().edges() by A3; then A20: not e in W4.edges() by GLIB_001:110; set W5 = T.replaceEdgeWith(e,W4); A21: W5 is_Walk_from v,y by A2, A6, GLIB_006:47; W3.reverse().first() = v2 & W3.reverse().last() = v1 by A19, GLIB_001:def 23; then W4.first() = v2 & W4.last() = v1 by GLIB_001:16; then not e in W5.edges() by A11, A20, GLIB_006:40, GLIB_000:14, A10, A17; then reconsider W = W5 as Walk of G2 by GLIB_006:109; W is_Walk_from v,y by A21, GLIB_001:19; hence y in G2.reachableFrom(v) by GLIB_002:def 5; end; end; end; then G1.reachableFrom(w) c= G2.reachableFrom(v) by TARSKI:def 3; hence G1.reachableFrom(w) = G2.reachableFrom(v) by A4, XBOOLE_0:def 10; end; suppose e in the_Edges_of G2; then G1 == G2 by GLIB_006:def 11; hence thesis by A2, GLIB_002:17; end; end; theorem Th40: for G2 being _Graph, v1, v2 being Vertex of G2, e being object for G1 being addEdge of G2,v1,e,v2 st v2 in G2.reachableFrom(v1) holds G1.componentSet() = G2.componentSet() proof let G2 be _Graph, v1,v2 be Vertex of G2, e be object; let G1 be addEdge of G2,v1,e,v2; assume A1: v2 in G2.reachableFrom(v1); per cases; suppose A2: not e in the_Edges_of G2; then A3: the_Vertices_of G1 = the_Vertices_of G2 by GLIB_006:def 11; for x being set holds x in G2.componentSet() iff ex v being Vertex of G1 st x = G1.reachableFrom(v) proof let x be set; hereby assume x in G2.componentSet(); then consider v0 being Vertex of G2 such that A4: x = G2.reachableFrom(v0) by GLIB_002:def 8; reconsider v = v0 as Vertex of G1 by A2, GLIB_006:def 11; take v; thus x = G1.reachableFrom(v) by A1, A4, Th39; end; given v being Vertex of G1 such that A5: x = G1.reachableFrom(v); ex v0 being Vertex of G2 st x = G2.reachableFrom(v0) proof reconsider v0 = v as Vertex of G2 by A2, GLIB_006:def 11; take v0; thus thesis by A1, A5, Th39; end; hence x in G2.componentSet() by GLIB_002:def 8; end; hence G1.componentSet() = G2.componentSet() by A3, GLIB_002:def 8; end; suppose e in the_Edges_of G2; then G1 == G2 by GLIB_006:def 11; hence thesis by GLIB_002:26; end; end; :: v1 and v2 are connected in supergraph theorem Th41: for G2 being _Graph, v1,v2 being Vertex of G2, e being object for G1 being addEdge of G2,v1,e,v2 for w1, w2 being Vertex of G1 st not e in the_Edges_of G2 & w1 = v1 & w2 = v2 holds w2 in G1.reachableFrom(w1) proof let G2 be _Graph, v1,v2 be Vertex of G2, e be object; let G1 be addEdge of G2,v1,e,v2; let w1, w2 be Vertex of G1; assume that A1: not e in the_Edges_of G2 and A2: w1 = v1 & w2 = v2; e DJoins v1,v2,G1 by A1, GLIB_006:105; then e Joins w1,v2,G1 by A2, GLIB_000:16; then G1.walkOf(w1,e,w2) is_Walk_from w1,w2 by A2, GLIB_001:15; hence thesis by GLIB_002:def 5; end; :: their components are connected as well theorem Th42: for G2 being _Graph, v1,v2 being Vertex of G2, e being object for G1 being addEdge of G2,v1,e,v2 for w1 being Vertex of G1 st not e in the_Edges_of G2 & w1 = v1 holds G1.reachableFrom(w1) = G2.reachableFrom(v1) \/ G2.reachableFrom(v2) proof let G2 be _Graph, v1,v2 be Vertex of G2, e be object; let G1 be addEdge of G2,v1,e,v2, w1 be Vertex of G1; assume A1: not e in the_Edges_of G2 & w1 = v1; A2: G2 is Subgraph of G1 by GLIB_006:57; e DJoins v1,v2,G1 by A1, GLIB_006:105; then A3: e Joins v1,v2,G1 by GLIB_000:16; for x being object holds x in G1.reachableFrom(w1) implies x in G2.reachableFrom(v1) \/ G2.reachableFrom(v2) proof let x be object; assume A4: x in G1.reachableFrom(w1); per cases; suppose x = w1; then x in G2.reachableFrom(v1) by GLIB_002:9, A1; hence thesis by XBOOLE_0:def 3; end; suppose A5: x <> w1; consider W1 being Walk of G1 such that A6: W1 is_Walk_from w1, x by A4, GLIB_002:def 5; set P1 = the Path of W1.reverse(); A7: W1.reverse() is_Walk_from x, w1 by A6, GLIB_001:23; x is set by TARSKI:1; then A8: P1 is_Walk_from x, w1 by A7, GLIB_001:160; per cases; suppose not e in P1.edges(); then reconsider W2 = P1 as Walk of G2 by GLIB_006:109; W2 is_Walk_from x, w1 by A8, GLIB_001:19; then W2.reverse() is_Walk_from v1, x by A1, GLIB_001:23; then x in G2.reachableFrom(v1) by GLIB_002:def 5; hence thesis by XBOOLE_0:def 3; end; suppose e in P1.edges(); then consider u1, u2 being Vertex of G1, n being odd Element of NAT such that A9: n+2 <= len P1 & u1 = P1.n & e = P1.(n+1) & u2 = P1.(n+2) and A10: e Joins u1,u2,G1 by GLIB_001:103; A11: v2 = u1 & v1 = u2 proof assume not (v2 = u1 & v1 = u2); then A12: v1 = P1.n by A3, A9, A10, GLIB_000:15; A13: P1.len P1 = P1.last() by GLIB_001:def 7 .= v1 by A1, A8, GLIB_001:def 23; n+2-2 < len P1 - 0 by A9, XREAL_1:15; then n = 1 by A12, A13, GLIB_001:def 28; then v1 = P1.first() by A12, GLIB_001:def 6 .= x by A8, GLIB_001:def 23; hence contradiction by A1, A5; end; set P2 = P1.cut(1,n); not e in P2.edges() proof assume e in P2.edges(); then consider u3, u4 being Vertex of G1, m being odd Element of NAT such that A14: m+2 <= len P2 & u3 = P2.m & e = P2.(m+1) & u4 = P2.(m+2) and e Joins u3,u4,G1 by GLIB_001:103; n+2-2 <= len P1 - 0 by A9, XREAL_1:13; then A15: 1 <= n & n <= len P1 by ABIAN:12; A16: m+2-2 < len P2 - 0 by A14, XREAL_1:15; then A17: P1.(m+1) = P1.(n+1) by A9, A14, A15, POLYFORM:4, GLIB_001:36; A18: 1+0 <= m+1 & m < n by A15, A16, GLIB_001:45, XREAL_1:7; then A19: m+1 < n+1 by XREAL_1:6; n+2-1 <= len P1 - 0 by A9, XREAL_1:13; hence contradiction by A17, A18, A19, GLIB_001:138; end; then reconsider W2 = P2 as Walk of G2 by GLIB_006:109; n+2-2 <= len P1 - 0 by A9, XREAL_1:13; then 1 <= n & n <= len P1 by ABIAN:12; then P2 is_Walk_from P1.1, P1.n by POLYFORM:4, GLIB_001:37; then W2 is_Walk_from P1.1, P1.n by GLIB_001:19; then W2 is_Walk_from P1.first(), P1.n by GLIB_001:def 6; then W2 is_Walk_from x, v2 by A8, A9, A11, GLIB_001:def 23; then W2.reverse() is_Walk_from v2, x by GLIB_001:23; then x in G2.reachableFrom(v2) by GLIB_002:def 5; hence thesis by XBOOLE_0:def 3; end; end; end; then A20: G1.reachableFrom(w1) c= G2.reachableFrom(v1) \/ G2.reachableFrom(v2) by TARSKI:def 3; A21: G2.reachableFrom(v1) c= G1.reachableFrom(w1) by A1, A2, GLIB_002:14; reconsider w2 = v2 as Vertex of G1 by GLIB_006:68; G2.reachableFrom(v2) c= G1.reachableFrom(w2) by A2, GLIB_002:14; then G2.reachableFrom(v2) c= G1.reachableFrom(w1) by A1, Th41, GLIB_002:12; then G2.reachableFrom(v1) \/ G2.reachableFrom(v2) c= G1.reachableFrom(w1) by A21, XBOOLE_1:8; hence thesis by A20, XBOOLE_0:def 10; end; :: the other components stay unaffected theorem Th43: for G2 being _Graph, v1,v2 being Vertex of G2, e being object for G1 being addEdge of G2,v1,e,v2 for w being Vertex of G1, v being Vertex of G2 st not e in the_Edges_of G2 & v = w & not v in G2.reachableFrom(v1) & not v in G2.reachableFrom(v2) holds G1.reachableFrom(w) = G2.reachableFrom(v) proof let G2 be _Graph, v1,v2 be Vertex of G2, e be object; let G1 be addEdge of G2,v1,e,v2; let w be Vertex of G1, v be Vertex of G2; assume that A1: not e in the_Edges_of G2 & v = w and A2: not v in G2.reachableFrom(v1) & not v in G2.reachableFrom(v2); e DJoins v1,v2,G1 by A1, GLIB_006:105; then A3: e Joins v1,v2,G1 by GLIB_000:16; for x being object holds x in G1.reachableFrom(w) implies x in G2.reachableFrom(v) proof let x be object; assume x in G1.reachableFrom(w); then consider W1 being Walk of G1 such that A4: W1 is_Walk_from w,x by GLIB_002:def 5; set T1 = the Trail of W1; x is set by TARSKI:1; then A5: T1 is_Walk_from w,x by A4, GLIB_001:160; not e in T1.edges() proof assume e in T1.edges(); then consider u1, u2 being Vertex of G1, n being odd Element of NAT such that A6: n+2 <= len T1 & u1 = T1.n & e = T1.(n+1) & u2 = T1.(n+2) and A7: e Joins u1,u2,G1 by GLIB_001:103; set T2 = T1.cut(1,n); n+2-2 <= len T1 - 0 by A6, XREAL_1:13; then A8: 1 <= n & n <= len T1 by ABIAN:12; not e in T2.edges() proof assume e in T2.edges(); then consider u3, u4 being Vertex of G1, m being odd Element of NAT such that A9: m+2 <= len T2 & u3 = T2.m & e = T2.(m+1) & u4 = T2.(m+2) and e Joins u3,u4,G1 by GLIB_001:103; A10: m+2-2 < len T2 - 0 by A9, XREAL_1:15; then A11: T1.(m+1) = T1.(n+1) by A6, A9, A8, POLYFORM:4, GLIB_001:36; A12: 1+0 <= m+1 & m < n by A8, A10, XREAL_1:7, GLIB_001:45; then A13: m+1 < n+1 by XREAL_1:6; n+2-1 <= len T1 - 0 by A6, XREAL_1:13; hence contradiction by A11, A12, A13, GLIB_001:138; end; then reconsider W2 = T2 as Walk of G2 by GLIB_006:109; T2 is_Walk_from T1.1, T1.n by A8, POLYFORM:4, GLIB_001:37; then T2 is_Walk_from T1.first(), T1.n by GLIB_001:def 6; then T2 is_Walk_from w, T1.n by A5, GLIB_001:def 23; then A14: W2 is_Walk_from v, u1 by A1, A6, GLIB_001:19; per cases by A3, A7, GLIB_000:15; suppose v1 = u1 & v2 = u2; then W2.reverse() is_Walk_from v1, v by A14, GLIB_001:23; hence contradiction by A2, GLIB_002:def 5; end; suppose v1 = u2 & v2 = u1; then W2.reverse() is_Walk_from v2, v by A14, GLIB_001:23; hence contradiction by A2, GLIB_002:def 5; end; end; then reconsider W = T1 as Walk of G2 by GLIB_006:109; W is_Walk_from w,x by A5, GLIB_001:19; hence x in G2.reachableFrom(v) by A1, GLIB_002:def 5; end; then A15: G1.reachableFrom(w) c= G2.reachableFrom(v) by TARSKI:def 3; G2 is Subgraph of G1 by GLIB_006:57; then G2.reachableFrom(v) c= G1.reachableFrom(w) by A1, GLIB_002:14; hence thesis by A15, XBOOLE_0:def 10; end; :: composition of component set in graph with added edge theorem Th44: for G2 being _Graph, v1,v2 being Vertex of G2, e being object for G1 being addEdge of G2,v1,e,v2 st not e in the_Edges_of G2 holds G1.componentSet() = (G2.componentSet() \ {G2.reachableFrom(v1), G2.reachableFrom(v2)}) \/ {G2.reachableFrom(v1) \/ G2.reachableFrom(v2)} proof let G2 be _Graph, v1,v2 be Vertex of G2, e be object; let G1 be addEdge of G2,v1,e,v2; assume A1: not e in the_Edges_of G2; set V1 = G2.reachableFrom(v1), V2 = G2.reachableFrom(v2); A2: G2 is Subgraph of G1 by GLIB_006:57; for x being object holds x in G1.componentSet() iff x in (G2.componentSet() \ {V1, V2}) \/ {V1 \/ V2} proof let x be object; reconsider w1 = v1, w2 = v2 as Vertex of G1 by GLIB_006:68; thus x in G1.componentSet() implies x in (G2.componentSet() \ {V1, V2}) \/ {V1 \/ V2} proof assume x in G1.componentSet(); then consider w being Vertex of G1 such that A3: x = G1.reachableFrom(w) by GLIB_002:def 8; reconsider v = w as Vertex of G2 by GLIB_006:102; per cases; suppose A4: not v in G2.reachableFrom(v1) & not v in G2.reachableFrom(v2); then A5: G1.reachableFrom(w) = G2.reachableFrom(v) by A1, Th43; G2.reachableFrom(v) <> V1 & G2.reachableFrom(v) <> V2 by A4, Th22; then A6: not G1.reachableFrom(w) in {V1, V2} by A5, TARSKI:def 2; G2.reachableFrom(v) in G2.componentSet() by GLIB_002:def 8; then G1.reachableFrom(w) in G2.componentSet() \ {V1, V2} by A5, A6, XBOOLE_0:def 5; hence thesis by A3, XBOOLE_0:def 3; end; suppose v in G2.reachableFrom(v1); then w in G1.reachableFrom(w1) by A2, TARSKI:def 3, GLIB_002:14; then x = G1.reachableFrom(w1) by A3, GLIB_002:12 .= V1 \/ V2 by A1, Th42; then x in {V1 \/ V2} by TARSKI:def 1; hence thesis by XBOOLE_0:def 3; end; suppose v in G2.reachableFrom(v2); then w in G1.reachableFrom(w2) by A2, TARSKI:def 3, GLIB_002:14; then w in G1.reachableFrom(w1) by A1, Th41, GLIB_002:12; then x = G1.reachableFrom(w1) by A3, GLIB_002:12 .= V1 \/ V2 by A1, Th42; then x in {V1 \/ V2} by TARSKI:def 1; hence thesis by XBOOLE_0:def 3; end; end; assume x in (G2.componentSet() \ {V1, V2}) \/ {V1 \/ V2}; then per cases by XBOOLE_0:def 3; suppose x in G2.componentSet() \ {V1, V2}; then A7: x in G2.componentSet() & not x in {V1, V2} by XBOOLE_0:def 5; then A8: x <> V1 & x <> V2 by TARSKI:def 2; consider v being Vertex of G2 such that A9: x = G2.reachableFrom(v) by A7, GLIB_002:def 8; reconsider w = v as Vertex of G1 by GLIB_006:68; not v in V1 & not v in V2 by A8, A9, GLIB_002:12; then G1.reachableFrom(w) = x by A1, A9, Th43; hence thesis by GLIB_002:def 8; end; suppose x in {V1 \/ V2}; then x = V1 \/ V2 by TARSKI:def 1; then x = G1.reachableFrom(w1) by A1, Th42; hence thesis by GLIB_002:def 8; end; end; hence thesis by TARSKI:2; end; Lm7: for G2 being _Graph, v1,v2 being Vertex of G2 st not v2 in G2.reachableFrom(v1) holds not G2.reachableFrom(v1) \/ G2.reachableFrom(v2) in G2.componentSet() proof let G2 be _Graph, v1,v2 be Vertex of G2; assume A1: not v2 in G2.reachableFrom(v1); set V1 = G2.reachableFrom(v1), V2 = G2.reachableFrom(v2); reconsider C = G2.componentSet() as a_partition of the_Vertices_of G2 by Th23; A2: V1 in C & V2 in C by GLIB_002:def 8; A3: V1 misses V2 by A1, Th22; assume V1 \/ V2 in G2.componentSet(); then per cases by A2, EQREL_1:def 4; suppose V1 \/ V2 = V1; then V2 c= V1 by XBOOLE_1:11; hence contradiction by A3, XBOOLE_1:63, XBOOLE_1:66; end; suppose V1 \/ V2 misses V1; then V1 misses V1 by XBOOLE_1:70; hence contradiction; end; end; :: the finite restriction is not neccessary, but all that's needed here Lm8: for G1 being finite _Graph, e being set, G2 being removeEdge of G1, e holds G1.numComponents() c= G2.numComponents() proof let G1 be finite _Graph, e be set, G2 be removeEdge of G1, e; per cases; suppose A1: e in the_Edges_of G1; set w1 = (the_Source_of G1).e, w2 = (the_Target_of G1).e; reconsider w1, w2 as Vertex of G1 by A1, FUNCT_2:5; A2: G1 is addEdge of G2,w1,e,w2 by A1, Th37; reconsider v1 = w1, v2 = w2 as Vertex of G2 by GLIB_000:51; reconsider G1 as addEdge of G2,v1,e,v2 by A2; A3: the_Edges_of G2 = the_Edges_of G1 \ {e} by GLIB_000:51; e in {e} by TARSKI:def 1; then A4: not e in the_Edges_of G2 by A3, XBOOLE_0:def 5; set V1 = G2.reachableFrom(v1), V2 = G2.reachableFrom(v2); per cases; suppose A5: not v2 in G2.reachableFrom(v1); then not V1 \/ V2 in G2.componentSet() by Lm7; then A6: not V1 \/ V2 in G2.componentSet() \ {V1,V2} by XBOOLE_1:36, TARSKI:def 3; V1 in G2.componentSet() & V2 in G2.componentSet() by GLIB_002:def 8; then A7: {V1, V2} c= G2.componentSet() by ZFMISC_1:32; A8: V1 <> V2 by A5, Th22; G1.numComponents() = card G1.componentSet() by GLIB_002:def 9 .= card((G2.componentSet()\{V1,V2})\/{V1 \/ V2}) by A4, Th44 .= card(G2.componentSet()\{V1,V2}) + 1 by A6, CARD_2:41 .= card G2.componentSet() - card {V1,V2} + 1 by A7, CARD_2:44 .= card G2.componentSet() - 2 + 1 by A8, CARD_2:57 .= G2.numComponents() - 2 + 1 by GLIB_002:def 9 .= G2.numComponents() - 1; then G2.numComponents() = G1.numComponents() +` 1; hence thesis by CARD_2:94; end; suppose A9: v2 in G2.reachableFrom(v1); G1.numComponents() = card G1.componentSet() by GLIB_002:def 9 .= card G2.componentSet() by A9, Th40 .= G2.numComponents() by GLIB_002:def 9; hence thesis; end; end; suppose not e in the_Edges_of G1; then the_Edges_of G1 \ {e} = the_Edges_of G1 by ZFMISC_1:57 .= G1.edgesBetween(the_Vertices_of G1) by GLIB_000:34; hence thesis by GLIB_002:29, GLIB_000:94; end; end; :: removing an endvertex doesn't change number of components theorem Th45: for G1 being _Graph, v being Vertex of G1, G2 being removeVertex of G1, v st v is endvertex holds G1.numComponents() = G2.numComponents() proof let G1 be _Graph, v be Vertex of G1, G2 be removeVertex of G1, v; assume A1: v is endvertex; then A2: G1 is non trivial by Th16; ex f being Function st f is one-to-one & dom f = G1.componentSet() & rng f = G2.componentSet() proof reconsider C = G1.componentSet() as a_partition of the_Vertices_of G1 by Th23; set V = EqClass(v,C); set f = id C +* (V, V\{v}); take f; for x1,x2 being object st x1 in dom f & x2 in dom f & f.x1 = f.x2 holds x1 = x2 proof let x1,x2 be object; assume A3: x1 in dom f & x2 in dom f & f.x1 = f.x2; then A4: x1 in dom id C & x2 in dom id C by FUNCT_7:30; A5: not V\{v} in C proof assume A6: V\{v} in C; set w = the Element of V\{v}; w in V by A6, XBOOLE_1:36, TARSKI:def 3; then w in V /\ (V\{v}) by A6, XBOOLE_0:def 4; then A8: not V misses V\{v} by XBOOLE_0:def 7; v in {v} by TARSKI:def 1; then not v in V\{v} by XBOOLE_0:def 5; then A9: V <> V \{v} by EQREL_1:def 6; V in C by EQREL_1:def 6; hence contradiction by A6, A8, A9, EQREL_1:def 4; end; per cases; suppose x1 <> V & x2 <> V; then f.x1 = (id C).x1 & f.x2 = (id C).x2 by FUNCT_7:32; then f.x1 = x1 & f.x2 = x2 by A4, FUNCT_1:18; hence thesis by A3; end; suppose x1 = V & x2 <> V; then f.x1 = V\{v} & f.x2 = (id C).x2 by A4, FUNCT_7:31, FUNCT_7:32; hence thesis by A5, A3, A4, FUNCT_1:18; :: by contradiction end; suppose x1 <> V & x2 = V; then f.x2 = V\{v} & f.x1 = (id C).x1 by A4, FUNCT_7:31, FUNCT_7:32; hence thesis by A5, A3, A4, FUNCT_1:18; :: by contradiction end; suppose x1 = V & x2 = V; hence thesis; end; end; hence f is one-to-one by FUNCT_1:def 4; thus A10: dom f = dom id C by FUNCT_7:30 .= G1.componentSet(); for y being object holds y in rng f iff y in G2.componentSet() proof let y be object; hereby assume y in rng f; then consider x being object such that A11: x in dom f & y = f.x by FUNCT_1:def 3; reconsider x as set by TARSKI:1; per cases; suppose A12: x = V; V in dom id C by EQREL_1:def 6; then A13: y = V\{v} by A11, A12, FUNCT_7:31; ex w being Vertex of G2 st V\{v} = G2.reachableFrom(w) proof consider e being object such that A14: v.edgesInOut() = {e} & not e Joins v,v,G1 by A1, GLIB_000:def 51; set w = v.adj(e); A15: e in v.edgesInOut() by A14, TARSKI:def 1; then A16: v <> w by A14, GLIB_000:67; A17: the_Vertices_of G2 = the_Vertices_of G1 \ {v} by A2, GLIB_000:47; v in G1.reachableFrom(v) & e Joins v,w,G1 by A15, GLIB_000:67, GLIB_002:9; then A18: w in G1.reachableFrom(v) by GLIB_002:10; not w in {v} by A16, TARSKI:def 1; then reconsider w0 = w as Vertex of G2 by A17, XBOOLE_0:def 5; take w0; thus V\{v} = G1.reachableFrom(v)\{v} by Th24 .= G1.reachableFrom(w)\{v} by A18, GLIB_002:12 .= G2.reachableFrom(w0) by A1, A18, Th25; end; hence y in G2.componentSet() by A13, GLIB_002:def 8; end; suppose A19: x <> V; consider w being Vertex of G1 such that A20: x = G1.reachableFrom(w) by A10, A11, GLIB_002:def 8; A21: x in C & V in C by A10, A11, EQREL_1:def 6; x misses V by A19, A21, EQREL_1:def 4; then A22: x misses G1.reachableFrom(v) by Th24; G1.reachableFrom(w) <> G1.reachableFrom(v) by A20, A22, XBOOLE_1:66; then A23: not w in G1.reachableFrom(v) by GLIB_002:12; then A24: v <> w by GLIB_002:9; A25: the_Vertices_of G2 = the_Vertices_of G1 \ {v} by A2, GLIB_000:47; not w in {v} by A24, TARSKI:def 1; then reconsider w0 = w as Vertex of G2 by A25, XBOOLE_0:def 5; G1.reachableFrom(w) = G2.reachableFrom(w0) by A2, A23, Th26; then A26: x in G2.componentSet() by A20, GLIB_002:def 8; x = (id C).x by A10, A11, FUNCT_1:18 .= y by A11, A19, FUNCT_7:32; hence y in G2.componentSet() by A26; end; end; assume y in G2.componentSet(); then consider w0 being Vertex of G2 such that A27: y = G2.reachableFrom(w0) by GLIB_002:def 8; the_Vertices_of G2 c= the_Vertices_of G1; then reconsider w = w0 as Vertex of G1 by TARSKI:def 3; ex x being object st x in dom f & y = f.x proof per cases; suppose A28: w in G1.reachableFrom(v); then G2.reachableFrom(w0) = G1.reachableFrom(w) \ {v} by A1, Th25; then y = G1.reachableFrom(v) \ {v} by A27, A28, GLIB_002:12; then A29: y = V\{v} by Th24; take V; V = G1.reachableFrom(v) by Th24; then A30: V in C by GLIB_002:def 8; hence V in dom f by A10; V in dom id C by A30; hence y = f.V by A29, FUNCT_7:31; end; suppose A31: not w in G1.reachableFrom(v); then A32: G2.reachableFrom(w0) = G1.reachableFrom(w) by A2, Th26; set x = G1.reachableFrom(w); take x; A33: x in G1.componentSet() by GLIB_002:def 8; hence x in dom f by A10; x <> V proof assume x = V; then x = G1.reachableFrom(v) by Th24; hence contradiction by A31, GLIB_002:9; end; then f.x = (id C).x by FUNCT_7:32 .= x by A33, FUNCT_1:18; hence y = f.x by A27, A32; end; end; hence thesis by FUNCT_1:def 3; end; hence rng f = G2.componentSet() by TARSKI:2; end; then card G1.componentSet() = card G2.componentSet() by WELLORD2:def 4, CARD_1:5; then G1.numComponents() = card G2.componentSet() by GLIB_002:def 9; hence thesis by GLIB_002:def 9; end; registration let G be _Graph; cluster endvertex -> non cut-vertex for Vertex of G; coherence proof let v be Vertex of G; assume A1: v is endvertex; set H = the removeVertex of G, v; G.numComponents() = H.numComponents() by A1, Th45; then not G.numComponents() in H.numComponents(); hence thesis by GLIB_002:def 10; end; end; theorem for G1 being non trivial finite connected _Graph for G2 being non spanning connected Subgraph of G1 ex v being Vertex of G1 st v is non cut-vertex & not v in the_Vertices_of G2 proof defpred P[Nat] means for G1 being non trivial finite connected _Graph for G2 being non spanning connected Subgraph of G1 st G1.order() + $1 = G1.size() + 1 ex v being Vertex of G1 st v is non cut-vertex & not v in the_Vertices_of G2; A1: P[0] proof let G1 be non trivial finite connected _Graph; let G2 be non spanning connected Subgraph of G1; assume G1.order() + 0 = G1.size() + 1; then G1 is Tree-like by GLIB_002:47; then consider v being Vertex of G1 such that A2: v is endvertex & not v in the_Vertices_of G2 by Th31; thus thesis by A2; end; A3: for k being Nat st P[k] holds P[k+1] :: remove an edge proof let k be Nat; assume A4: P[k]; let G1 be non trivial finite connected _Graph; let G2 be non spanning connected Subgraph of G1; assume A5: G1.order() + (k+1) = G1.size() + 1; G1 is non acyclic proof assume G1 is acyclic; then k + 1 + (G1.size() + 1) = 0 + (G1.size() + 1) by A5, GLIB_002:46; hence contradiction; end; then consider C being Walk of G1 such that A6: C is Cycle-like by GLIB_002:def 2; ex e being set st e in C.edges() & (C.edges() c= the_Edges_of G2 or not e in the_Edges_of G2) proof per cases; suppose A7: C.edges() c= the_Edges_of G2; set e = the Element of C.edges(); C.edges() <> {} by A6, GLIB_001:136; then e in C.edges(); hence thesis by A7; end; suppose not C.edges() c= the_Edges_of G2; then A8: C.edges() \ the_Edges_of G2 <> {} by XBOOLE_1:37; set e = the Element of C.edges() \ the_Edges_of G2; e in C.edges() & not e in the_Edges_of G2 by A8, XBOOLE_0:def 5; hence thesis; end; end; then consider e being set such that A9: e in C.edges() and A10: C.edges() c= the_Edges_of G2 or not e in the_Edges_of G2; set G3 = the removeEdge of G1, e; reconsider G3 as non trivial _Graph; reconsider G3 as non trivial finite connected _Graph by A6, A9, GLIB_002:5; A11: the_Vertices_of G3 = the_Vertices_of G1 by GLIB_000:51; A12: the_Edges_of G3 = the_Edges_of G1 \ {e} by GLIB_000:51; A13: the_Edges_of G2 \ {e} c= the_Edges_of G3 by A12, XBOOLE_1:33; set G4 = the removeEdge of G2, e; reconsider G4 as Subgraph of G1 by GLIB_000:43; A14: the_Vertices_of G4 = the_Vertices_of G2 & the_Edges_of G4 = the_Edges_of G2 \ {e} by GLIB_000:51; then reconsider G4 as Subgraph of G3 by A11, A13, GLIB_000:44; G4 is connected proof per cases by A10; suppose A15: C.edges() c= the_Edges_of G2; reconsider C2 = C as Walk of G2 by A6, A15, GLIB_001:169; A16: C2 is Cycle-like by A6, GLIB_006:24; e in C2.edges() by A9, GLIB_001:110; hence thesis by A16, GLIB_002:5; end; suppose not e in the_Edges_of G2; then the_Edges_of G4 = the_Edges_of G2 by A14, ZFMISC_1:57; hence thesis by A14, GLIB_000:86, GLIB_002:8; end; end; then reconsider G4 as connected Subgraph of G3; G4 is non spanning proof assume G4 is spanning; then the_Vertices_of G2 = the_Vertices_of G1 by A11, A14, GLIB_000:def 33; hence contradiction by GLIB_000:def 33; end; then reconsider G4 as non spanning connected Subgraph of G3; G1.order() + k + 1 = G3.size() + 1 + 1 by A5, A9, GLIB_000:52; then G3.order() + k = G3.size() + 1 by GLIB_000:52; then consider v being Vertex of G3 such that A17: v is non cut-vertex & not v in the_Vertices_of G4 by A4; the_Vertices_of G3 c= the_Vertices_of G1 by GLIB_000:def 32; then reconsider v0 = v as Vertex of G1 by TARSKI:def 3; take v0; now let H1 be removeVertex of G1, v0; set H2 = the removeVertex of G3, v; A18: H2.numComponents() c= G3.numComponents() by A17, GLIB_002:35; H2 is removeEdge of H1, e by Th18; then A19: H1.numComponents() c= H2.numComponents() by Lm8; G1.numComponents() = 1 & G3.numComponents() = 1 by GLIB_002:28; hence H1.numComponents() c= G1.numComponents() by A18, A19, XBOOLE_1:1; end; hence v0 is non cut-vertex by GLIB_002:35; thus not v0 in the_Vertices_of G2 by A14, A17; end; A20: for k being Nat holds P[k] from NAT_1:sch 2(A1,A3); let G1 be non trivial finite connected _Graph; let G2 be non spanning connected Subgraph of G1; ex k being Nat st G1.order() + k = G1.size() + 1 by GLIB_002:40, NAT_1:10; hence thesis by A20; end; :: improved in GLIB_007:61 :: not e in the_Edges_of G3 can then be followed by GLIB_006:def 9 Lm9: for G3 being _Graph, v being object, V being set, v1 being Vertex of G3 for G1 being addAdjVertexAll of G3,v,V \/ {v1} st V c= the_Vertices_of G3 & not v in the_Vertices_of G3 & not v1 in V ex G2 being addAdjVertexAll of G3,v,V, e being object st not e in the_Edges_of G2 & (G1 is addEdge of G2,v,e,v1 or G1 is addEdge of G2,v1,e,v) by GLIB_007:61; :: into GLIB_007 ? theorem Th47: for G1 being non trivial simple _Graph, v being Vertex of G1 for G2 being removeVertex of G1, v holds G1 is addAdjVertexAll of G2, v, v.allNeighbors() proof let G1 be non trivial simple _Graph, v be Vertex of G1; let G2 be removeVertex of G1, v; A1: G1 is Supergraph of G2 by GLIB_006:57; A2: the_Vertices_of G2 = the_Vertices_of G1 \ {v} by GLIB_000:47; A3: the_Edges_of G2 = G1.edgesBetween(the_Vertices_of G1 \ {v}) by GLIB_000:47 .= G1.edgesBetween(the_Vertices_of G1) \ v.edgesInOut() by Th1 .= the_Edges_of G1 \ v.edgesInOut() by GLIB_000:34; v in {v} by TARSKI:def 1; then A4: not v in the_Vertices_of G2 by A2, XBOOLE_0:def 5; for x being object holds x in v.allNeighbors() implies x in the_Vertices_of G2 proof let x be object; assume A5: x in v.allNeighbors(); x <> v proof assume x = v; then consider e being object such that A6: e Joins v,v,G1 by A5, GLIB_000:71; thus contradiction by A6, GLIB_000:18; end; then not x in {v} by TARSKI:def 1; hence thesis by A2, A5, XBOOLE_0:def 5; end; then A7: v.allNeighbors() c= the_Vertices_of G2 by TARSKI:def 3; :: show the properties of addAdjVertexAll now thus the_Vertices_of G1 = the_Vertices_of G2 \/ {v} by A2, ZFMISC_1:116; hereby let e be object; thus not e Joins v,v,G1 by GLIB_000:18; let v1 be object; hereby assume A8: not v1 in v.allNeighbors(); v1 is set by TARSKI:1; hence not e Joins v1,v,G1 by A8, GLIB_000:71, GLIB_000:14; end; let v2 be object; assume A9: v1 <> v & v2 <> v & e DJoins v1,v2,G1; thus e DJoins v1,v2,G2 proof assume not e DJoins v1,v2,G2; then A10: not e in the_Edges_of G2 by A1, A9, GLIB_006:71; e in the_Edges_of G1 by A9, GLIB_000:def 14; then e in v.edgesInOut() by A3, A10, XBOOLE_0:def 5; then per cases by GLIB_000:61; suppose (the_Source_of G1).e = v; hence contradiction by A9, GLIB_000:def 14; end; suppose (the_Target_of G1).e = v; hence contradiction by A9, GLIB_000:def 14; end; end; end; reconsider E = v.edgesInOut() as set; take E; card E = v.degree() by GLIB_000:19; hence card v.allNeighbors() = card E by Th5; E /\ the_Edges_of G2 = {} proof assume E /\ the_Edges_of G2 <> {}; then consider e being object such that A11: e in E /\ the_Edges_of G2 by XBOOLE_0:def 1; A12: e in E & e in the_Edges_of G2 by A11, XBOOLE_0:def 4; then (the_Source_of G1).e = v or (the_Target_of G1).e = v by GLIB_000:61; then (the_Source_of G2).e = v or (the_Target_of G2).e = v by A12, GLIB_000:def 32; then (the_Source_of G2).e in {v} or (the_Target_of G2).e in {v} by TARSKI:def 1; then not (the_Source_of G2).e in the_Vertices_of G2 or not (the_Target_of G2).e in the_Vertices_of G2 by A2, XBOOLE_0:def 5; hence contradiction by A12, FUNCT_2:5; end; hence E misses the_Edges_of G2 by XBOOLE_0:def 7; A13: the_Edges_of G2 \/ E c= the_Edges_of G1 by XBOOLE_1:8; for e being object holds e in the_Edges_of G1 implies e in the_Edges_of G2 \/ E proof let e be object; assume A14: e in the_Edges_of G1; e in the_Edges_of G2 or e in E by A3, A14, XBOOLE_0:def 5; hence thesis by XBOOLE_0:def 3; end; then the_Edges_of G1 c= the_Edges_of G2 \/ E by TARSKI:def 3; hence the_Edges_of G1 = the_Edges_of G2 \/ E by A13, XBOOLE_0:def 10; let v1 be object; assume A15: v1 in v.allNeighbors(); then consider e1 being object such that A16: e1 Joins v,v1,G1 by GLIB_000:71; take e1; v1 in the_Vertices_of G1 & e1 is set by A15, TARSKI:1; hence e1 in E by A16, GLIB_000:64; thus A17: e1 Joins v1,v,G1 by A16, GLIB_000:14; let e2 be object; assume e2 Joins v1,v,G1; hence e1 = e2 by A17, GLIB_000:def 20; end; hence thesis by A1, A4, A7, GLIB_007:def 4; end; begin :: Edgeless and non edgeless graphs definition let G be _Graph; attr G is edgeless means :Def1: the_Edges_of G = {}; end; theorem for G being _Graph holds G is edgeless iff card the_Edges_of G = 0; theorem Th49: for G being _Graph holds G is edgeless iff G.size() = 0 proof let G be _Graph; hereby assume G is edgeless; then card the_Edges_of G = 0; hence G.size() = 0 by GLIB_000:def 25; end; assume G.size() = 0; then card the_Edges_of G = 0 by GLIB_000:def 25; hence thesis; end; registration let G be _Graph; cluster -> edgeless for removeEdges of G, the_Edges_of G; coherence proof let G2 be removeEdges of G, the_Edges_of G; the_Edges_of G2 = the_Edges_of G \ the_Edges_of G by GLIB_000:53 .= {} by XBOOLE_1:37; hence thesis; end; end; registration cluster edgeless for _Graph; existence proof take the removeEdges of the _Graph, the_Edges_of the _Graph; thus thesis; end; let G be _Graph; cluster edgeless spanning for Subgraph of G; existence proof take the removeEdges of G, the_Edges_of G; thus thesis; end; cluster edgeless trivial for Subgraph of G; existence proof set G2 = the inducedSubgraph of G, {the Vertex of G}; reconsider G3 = the removeEdges of G2, the_Edges_of G2 as Subgraph of G by GLIB_000:43; take G3; thus thesis; end; end; registration let G be edgeless _Graph; cluster the_Edges_of G -> empty; coherence by Def1; end; registration cluster edgeless -> non-multi non-Dmulti loopless simple Dsimple for _Graph; coherence proof let G be _Graph; assume A1: G is edgeless; then for e1,e2,v1,v2 being object holds e1 Joins v1,v2,G & e2 Joins v1,v2,G implies e1 = e2 by GLIB_000:def 13; hence A2: G is non-multi by GLIB_000:def 20; for e1,e2,v1,v2 being object holds e1 DJoins v1,v2,G & e2 DJoins v1,v2,G implies e1 = e2 by A1, GLIB_000:def 14; hence G is non-Dmulti by GLIB_000:def 21; not ex e being object st e in the_Edges_of G & (the_Source_of G).e = (the_Target_of G).e by A1; hence G is loopless by GLIB_000:def 18; hence thesis by A2; end; cluster trivial loopless -> edgeless for _Graph; coherence by GLIB_000:23; let V be non empty set, S,T be Function of {},V; cluster createGraph(V,{},S,T) -> edgeless; coherence by GLIB_000:6; end; theorem for G being edgeless_Graph, e,v1,v2 being object holds not e Joins v1,v2,G & not e DJoins v1,v2,G proof let G be edgeless _Graph; let e,v1,v2 be object; not e Joins v1,v2,G proof assume e Joins v1,v2,G; then e in the_Edges_of G by GLIB_000:def 13; hence contradiction; end; hence thesis by GLIB_000:16; end; theorem for G being edgeless_Graph, e being object, X, Y being set holds not e SJoins X,Y,G & not e DSJoins X,Y,G proof let G be edgeless _Graph; let e be object, X, Y be set; thus not e SJoins X,Y,G proof assume e SJoins X,Y,G; then e in the_Edges_of G by GLIB_000:def 15; hence contradiction; end; thus not e DSJoins X,Y,G proof assume e DSJoins X,Y,G; then e in the_Edges_of G by GLIB_000:def 16; hence contradiction; end; end; theorem Th52: for G1, G2 being _Graph st G1 == G2 holds G1 is edgeless implies G2 is edgeless by GLIB_000:def 34; registration let G be edgeless _Graph; cluster -> trivial for Walk of G; coherence proof let W be Walk of G; W.edges() = {}; hence thesis by GLIB_001:136; end; cluster -> edgeless for Subgraph of G; coherence proof let G2 be Subgraph of G; the_Edges_of G2 c= the_Edges_of G; hence thesis; end; let X be set; cluster G.edgesInto(X) -> empty; coherence; cluster G.edgesOutOf(X) -> empty; coherence; cluster G.edgesInOut(X) -> empty; coherence; cluster G.edgesBetween(X) -> empty; coherence; cluster G.set(WeightSelector, X) -> edgeless; coherence by GLIB_003:7, Th52; cluster G.set(ELabelSelector, X) -> edgeless; coherence by GLIB_003:7, Th52; cluster G.set(VLabelSelector, X) -> edgeless; coherence by GLIB_003:7, Th52; cluster -> edgeless for addVertices of G, X; coherence proof let G1 be addVertices of G, X; the_Edges_of G1 = the_Edges_of G by GLIB_006:def 10; hence thesis; end; cluster -> edgeless for reverseEdgeDirections of G, X; coherence proof let G1 be reverseEdgeDirections of G, X; per cases; suppose X c= the_Edges_of G; hence thesis by GLIB_007:def 1; end; suppose not X c= the_Edges_of G; then G1 == G by GLIB_007:def 1; then the_Edges_of G1 = the_Edges_of G by GLIB_000:def 34; hence thesis; end; end; let Y be set; cluster G.edgesBetween(X,Y) -> empty; coherence; cluster G.edgesDBetween(X,Y) -> empty; coherence; end; registration cluster edgeless -> acyclic chordal for _Graph; coherence proof let G be _Graph; assume A1: G is edgeless; then not ex W being Walk of G st W is Cycle-like; hence G is acyclic by GLIB_002:def 2; for W being Walk of G st W.length()>3 & W is Cycle-like holds W is chordal by A1; hence thesis by CHORD:def 11; end; cluster trivial edgeless -> Tree-like for _Graph; coherence; cluster non trivial edgeless -> non connected non Tree-like non complete for _Graph; coherence proof let G be _Graph; assume A2: G is non trivial edgeless; then consider v1, v2 being Vertex of G such that A3: v1 <> v2 by GLIB_000:21; not ex W being Walk of G st W is_Walk_from v1, v2 proof given W being Walk of G such that A4: W is_Walk_from v1,v2; W.first() = v1 & W.last() = v2 by A4, GLIB_001:def 23; hence contradiction by A2, A3, GLIB_001:127; end; hence thesis by GLIB_002:def 1; end; cluster connected edgeless -> trivial for _Graph; coherence; end; theorem for G1 being edgeless _Graph, G2 being Subgraph of G1 holds G1 is addVertices of G2, the_Vertices_of G1 \ the_Vertices_of G2 proof let G1 be edgeless _Graph, G2 be Subgraph of G1; A1: the_Edges_of G1 = the_Edges_of G2; G1 is Supergraph of G2 by GLIB_006:57; hence thesis by A1, Th33; end; theorem Th54: for G2 being _Graph, v1, v2 being Vertex of G2, e being object for G1 being addEdge of G2,v1,e,v2 st not e in the_Edges_of G2 holds G1 is non edgeless proof let G2 be _Graph, v1, v2 be Vertex of G2, e be object; let G1 be addEdge of G2, v1, e, v2; assume not e in the_Edges_of G2; then the_Edges_of G1 = the_Edges_of G2 \/ {e} by GLIB_006:def 11; hence thesis; end; theorem Th55: for G2 being _Graph, v1 being Vertex of G2, e, v2 being object for G1 being addAdjVertex of G2,v1,e,v2 st not v2 in the_Vertices_of G2 & not e in the_Edges_of G2 holds G1 is non edgeless proof let G2 be _Graph, v1 be Vertex of G2, e, v2 be object; let G1 be addAdjVertex of G2, v1, e, v2; assume not v2 in the_Vertices_of G2 & not e in the_Edges_of G2; then the_Edges_of G1 = the_Edges_of G2 \/ {e} by GLIB_006:def 13; hence thesis; end; theorem Th56: for G2 being _Graph, v1, e being object, v2 being Vertex of G2 for G1 being addAdjVertex of G2,v1,e,v2 st not v1 in the_Vertices_of G2 & not e in the_Edges_of G2 holds G1 is non edgeless proof let G2 be _Graph, v1, e be object, v2 be Vertex of G2; let G1 be addAdjVertex of G2, v1, e, v2; assume not v1 in the_Vertices_of G2 & not e in the_Edges_of G2; then the_Edges_of G1 = the_Edges_of G2 \/ {e} by GLIB_006:def 14; hence thesis; end; theorem Th57: for G2 being _Graph, v being object for V being non empty Subset of the_Vertices_of G2 for G1 being addAdjVertexAll of G2,v,V st not v in the_Vertices_of G2 holds G1 is non edgeless by GLIB_007:47; registration let G be _Graph; cluster -> non edgeless for addAdjVertexToAll of G, the_Vertices_of G; coherence proof the_Vertices_of G is non empty & the_Vertices_of G c= the_Vertices_of G; then A1: not the_Vertices_of G in the_Vertices_of G & the_Vertices_of G is non empty Subset of the_Vertices_of G; let G1 be addAdjVertexToAll of G, the_Vertices_of G; G1 is addAdjVertexToAll of G, the_Vertices_of G, the_Vertices_of G; hence thesis by A1, Th57; end; cluster -> non edgeless for addAdjVertexFromAll of G, the_Vertices_of G; coherence proof the_Vertices_of G is non empty & the_Vertices_of G c= the_Vertices_of G; then A2: not the_Vertices_of G in the_Vertices_of G & the_Vertices_of G is non empty Subset of the_Vertices_of G; let G1 be addAdjVertexFromAll of G, the_Vertices_of G; G1 is addAdjVertexFromAll of G, the_Vertices_of G, the_Vertices_of G; hence thesis by A2, Th57; end; cluster -> non edgeless for addAdjVertexAll of G, the_Vertices_of G; coherence proof the_Vertices_of G is non empty & the_Vertices_of G c= the_Vertices_of G; then not the_Vertices_of G in the_Vertices_of G & the_Vertices_of G is non empty Subset of the_Vertices_of G; hence thesis by Th57; end; let v be Vertex of G; cluster -> non edgeless for addAdjVertex of G, v, the_Edges_of G, the_Vertices_of G; coherence proof not the_Vertices_of G in the_Vertices_of G & not the_Edges_of G in the_Edges_of G; hence thesis by Th55; end; cluster -> non edgeless for addAdjVertex of G, the_Vertices_of G, the_Edges_of G, v; coherence proof not the_Vertices_of G in the_Vertices_of G & not the_Edges_of G in the_Edges_of G; hence thesis by Th56; end; let w be Vertex of G; cluster -> non edgeless for addEdge of G, v, the_Edges_of G, w; coherence proof not the_Edges_of G in the_Edges_of G; hence thesis by Th54; end; end; registration let G be edgeless _Graph; cluster -> trivial for Component of G; coherence; let v be Vertex of G; cluster v.edgesIn() -> empty; coherence; cluster v.edgesOut() -> empty; coherence; cluster v.edgesInOut() -> empty; coherence; end; registration let G be edgeless _Graph; cluster -> isolated non cut-vertex non endvertex for Vertex of G; coherence proof let v be Vertex of G; v.edgesInOut() = {}; hence v is isolated by GLIB_000:def 49; hence v is non cut-vertex; thus v is non endvertex proof assume v is endvertex; then consider e being object such that A1: v.edgesInOut() = {e} & not e Joins v,v,G by GLIB_000:def 51; thus contradiction by A1; end; end; let v be Vertex of G; cluster v.inDegree() -> empty; coherence proof v.edgesIn() = {}; hence thesis by GLIB_000:def 42, CARD_1:27; end; cluster v.outDegree() -> empty; coherence proof v.edgesOut() = {}; hence thesis by GLIB_000:def 43, CARD_1:27; end; cluster v.inNeighbors() -> empty; coherence proof (the_Source_of G).:(v.edgesIn()) = {}; hence thesis by GLIB_000:def 46; end; cluster v.outNeighbors() -> empty; coherence proof (the_Target_of G).:(v.edgesOut()) = {}; hence thesis by GLIB_000:def 47; end; end; registration let G be edgeless _Graph, v be Vertex of G; cluster v.degree() -> empty; coherence proof v.inDegree() +` v.outDegree() = 0 + 0; hence thesis by GLIB_000:def 44; end; cluster v.allNeighbors() -> empty; coherence proof v.inNeighbors() \/ v.outNeighbors() = {}; hence thesis by GLIB_000:def 48; end; end; registration cluster trivial finite edgeless for _Graph; existence proof set G = the trivial finite _Graph; take the removeEdges of G, the_Edges_of G; thus thesis; end; cluster non trivial finite edgeless for _Graph; existence proof set G = the non trivial finite _Graph; take the removeEdges of G, the_Edges_of G; thus thesis; end; cluster trivial finite non edgeless for _Graph; existence proof set G = the trivial finite _Graph; set v = the Vertex of G; take the addEdge of G, v, the_Edges_of G, v; thus thesis; end; cluster non trivial finite non edgeless for _Graph; existence proof set G = the non trivial finite _Graph; set v = the Vertex of G; take the addEdge of G, v, the_Edges_of G, v; thus thesis; end; end; registration let G be non edgeless _Graph; cluster the_Edges_of G -> non empty; coherence by Def1; cluster -> non edgeless for Supergraph of G; coherence proof let G1 be Supergraph of G; assume A1: G1 is edgeless; G is Subgraph of G1 by GLIB_006:57; hence contradiction by A1; end; let X be set; cluster -> non edgeless for reverseEdgeDirections of G, X; coherence proof let G1 be reverseEdgeDirections of G, X; per cases; suppose X c= the_Edges_of G; hence thesis by GLIB_007:def 1; end; suppose not X c= the_Edges_of G; then G1 == G by GLIB_007:def 1; then the_Edges_of G1 = the_Edges_of G by GLIB_000:def 34; hence thesis; end; end; cluster G.set(WeightSelector, X) -> non edgeless; coherence proof G.set(WeightSelector, X) == G by GLIB_003:7; hence thesis by Th52; end; cluster G.set(ELabelSelector, X) -> non edgeless; coherence proof G.set(ELabelSelector, X) == G by GLIB_003:7; hence thesis by Th52; end; cluster G.set(VLabelSelector, X) -> non edgeless; coherence proof G.set(VLabelSelector, X) == G by GLIB_003:7; hence thesis by Th52; end; end; definition let G be non edgeless _Graph; mode Edge of G is Element of the_Edges_of G; end; theorem for G1 being finite edgeless _Graph, G2 being Subgraph of G1 st G1.order() = G2.order() holds G1 == G2 proof let G1 be finite edgeless _Graph, G2 be Subgraph of G1; assume A1: G1.order() = G2.order(); A2: card the_Vertices_of G1 = G1.order() by GLIB_000:def 24 .= card the_Vertices_of G2 by A1, GLIB_000:def 24; A3: the_Vertices_of G2 = the_Vertices_of G1 by A2, CARD_2:102; A4: the_Edges_of G1 = the_Edges_of G2; G1 is Subgraph of G1 by GLIB_000:40; hence thesis by A3, A4, GLIB_000:86; end; definition let GF be Graph-yielding Function; attr GF is edgeless means :Def2: for x being object st x in dom GF ex G being _Graph st GF.x = G & G is edgeless; end; definition let GF be non empty Graph-yielding Function; redefine attr GF is edgeless means :Def3: for x being Element of dom GF holds GF.x is edgeless; compatibility proof hereby assume A1: GF is edgeless; let x be Element of dom GF; consider G being _Graph such that A2: GF.x = G & G is edgeless by A1; thus GF.x is edgeless by A2; end; assume A3: for x being Element of dom GF holds GF.x is edgeless; let x be object; assume x in dom GF; then reconsider y = x as Element of dom GF; take GF.y; thus thesis by A3; end; end; Lm10: :: copied from GLIB_000 for F being ManySortedSet of NAT, n being object holds n is Nat iff n in dom F proof let F be ManySortedSet of NAT, n being object; hereby assume n is Nat; then n in NAT by ORDINAL1:def 12; hence n in dom F by PARTFUN1:def 2; end; assume n in dom F; hence n is Nat; end; definition let GSq be GraphSeq; redefine attr GSq is edgeless means :Def4: for n being Nat holds GSq.n is edgeless; compatibility proof hereby assume A1: GSq is edgeless; let x be Nat; x in dom GSq by Lm10; hence GSq.x is edgeless by A1; end; assume A2: for x being Nat holds GSq.x is edgeless; let x be Element of dom GSq; thus thesis by A2; end; end; registration cluster trivial loopless -> edgeless for Graph-yielding Function; coherence proof let GF be Graph-yielding Function; assume A1: GF is trivial loopless; let x be object; assume A2: x in dom GF; consider G1 being _Graph such that A3: GF.x = G1 & G1 is trivial by A1, A2, GLIB_000:def 60; consider G2 being _Graph such that A4: GF.x = G2 & G2 is loopless by A1, A2, GLIB_000:def 59; take G1; thus thesis by A3, A4; end; cluster edgeless -> non-multi non-Dmulti loopless simple Dsimple acyclic for Graph-yielding Function; coherence proof let GF be Graph-yielding Function; assume A5: GF is edgeless; now let x be object; assume x in dom GF; then consider G being _Graph such that A6: GF.x = G & G is edgeless by A5; take G; thus GF.x = G by A6; thus G is non-multi by A6; end; hence GF is non-multi by GLIB_000:def 62; now let x be object; assume x in dom GF; then consider G being _Graph such that A7: GF.x = G & G is edgeless by A5; take G; thus GF.x = G by A7; thus G is non-Dmulti by A7; end; hence GF is non-Dmulti by GLIB_000:def 63; now let x be object; assume x in dom GF; then consider G being _Graph such that A8: GF.x = G & G is edgeless by A5; take G; thus GF.x = G by A8; thus G is loopless by A8; end; hence GF is loopless by GLIB_000:def 59; now let x be object; assume x in dom GF; then consider G being _Graph such that A9: GF.x = G & G is edgeless by A5; take G; thus GF.x = G by A9; thus G is simple by A9; end; hence GF is simple by GLIB_000:def 64; now let x be object; assume x in dom GF; then consider G being _Graph such that A10: GF.x = G & G is edgeless by A5; take G; thus GF.x = G by A10; thus G is Dsimple by A10; end; hence GF is Dsimple by GLIB_000:def 65; now let x be object; assume x in dom GF; then consider G being _Graph such that A11: GF.x = G & G is edgeless by A5; take G; thus GF.x = G by A11; thus G is acyclic by A11; end; hence GF is acyclic by GLIB_002:def 13; end; end; registration let GF be edgeless non empty Graph-yielding Function, x be Element of dom GF; cluster GF.x -> edgeless; coherence by Def3; end; registration let GSq be edgeless GraphSeq, x be Nat; cluster GSq.x -> edgeless; coherence by Def4; end; begin :: Finite Graph sequences registration let G be _Graph; cluster <* G *> -> Graph-yielding; coherence proof let x be object; assume x in dom <* G *>; then x in Seg 1 by FINSEQ_1:def 8; then x = 1 by FINSEQ_1:2, TARSKI:def 1; hence <* G *>.x is _Graph by FINSEQ_1:def 8; end; end; registration let G be finite _Graph; cluster <* G *> -> finite; coherence proof let x be Element of dom <* G *>; x in dom <* G *>; then x in Seg 1 by FINSEQ_1:def 8; then x = 1 by FINSEQ_1:2, TARSKI:def 1; hence <* G *>.x is finite by FINSEQ_1:def 8; end; end; registration let G be loopless _Graph; cluster <* G *> -> loopless; coherence proof let x be Element of dom <* G *>; x in dom <* G *>; then x in Seg 1 by FINSEQ_1:def 8; then x = 1 by FINSEQ_1:2, TARSKI:def 1; hence <* G *>.x is loopless by FINSEQ_1:def 8; end; end; registration let G be trivial _Graph; cluster <* G *> -> trivial; coherence proof let x be Element of dom <* G *>; x in dom <* G *>; then x in Seg 1 by FINSEQ_1:def 8; then x = 1 by FINSEQ_1:2, TARSKI:def 1; hence <* G *>.x is trivial by FINSEQ_1:def 8; end; end; registration let G be non trivial _Graph; cluster <* G *> -> non-trivial; coherence proof let x be Element of dom <* G *>; x in dom <* G *>; then x in Seg 1 by FINSEQ_1:def 8; then x = 1 by FINSEQ_1:2, TARSKI:def 1; hence <* G *>.x is non trivial by FINSEQ_1:def 8; end; end; registration let G be non-multi _Graph; cluster <* G *> -> non-multi; coherence proof let x be Element of dom <* G *>; x in dom <* G *>; then x in Seg 1 by FINSEQ_1:def 8; then x = 1 by FINSEQ_1:2, TARSKI:def 1; hence <* G *>.x is non-multi by FINSEQ_1:def 8; end; end; registration let G be non-Dmulti _Graph; cluster <* G *> -> non-Dmulti; coherence proof let x be Element of dom <* G *>; x in dom <* G *>; then x in Seg 1 by FINSEQ_1:def 8; then x = 1 by FINSEQ_1:2, TARSKI:def 1; hence <* G *>.x is non-Dmulti by FINSEQ_1:def 8; end; end; registration let G be simple _Graph; cluster <* G *> -> simple; coherence; end; registration let G be Dsimple _Graph; cluster <* G *> -> Dsimple; coherence; end; registration let G be connected _Graph; cluster <* G *> -> connected; coherence proof let x be Element of dom <* G *>; x in dom <* G *>; then x in Seg 1 by FINSEQ_1:def 8; then x = 1 by FINSEQ_1:2, TARSKI:def 1; hence <* G *>.x is connected by FINSEQ_1:def 8; end; end; registration let G be acyclic _Graph; cluster <* G *> -> acyclic; coherence proof let x be Element of dom <* G *>; x in dom <* G *>; then x in Seg 1 by FINSEQ_1:def 8; then x = 1 by FINSEQ_1:2, TARSKI:def 1; hence <* G *>.x is acyclic by FINSEQ_1:def 8; end; end; registration let G be Tree-like _Graph; cluster <* G *> -> Tree-like; coherence; end; registration let G be edgeless _Graph; cluster <* G *> -> edgeless; coherence proof let x be Element of dom <* G *>; x in dom <* G *>; then x in Seg 1 by FINSEQ_1:def 8; then x = 1 by FINSEQ_1:2, TARSKI:def 1; hence <* G *>.x is edgeless by FINSEQ_1:def 8; end; end; registration cluster empty Graph-yielding for FinSequence; existence proof take the empty FinSequence; thus thesis; end; cluster non empty Graph-yielding for FinSequence; existence proof take <* the _Graph *>; thus thesis; end; end; registration let p be non empty Graph-yielding FinSequence; cluster p.1 -> Function-like Relation-like; coherence proof 1 in dom p by FINSEQ_5:6; hence thesis; end; cluster p.len p -> Function-like Relation-like; coherence proof len p in dom p by FINSEQ_5:6; hence thesis; end; end; registration let p be non empty Graph-yielding FinSequence; cluster p.1 -> finite NAT-defined; coherence proof 1 in dom p by FINSEQ_5:6; hence thesis; end; cluster p.len p -> finite NAT-defined; coherence proof len p in dom p by FINSEQ_5:6; hence thesis; end; end; registration let p be non empty Graph-yielding FinSequence; cluster p.1 -> [Graph-like]; coherence proof 1 in dom p by FINSEQ_5:6; hence thesis; end; cluster p.len p -> [Graph-like]; coherence proof len p in dom p by FINSEQ_5:6; hence thesis; end; end; registration cluster non empty finite loopless trivial non-multi non-Dmulti simple Dsimple connected acyclic Tree-like edgeless for Graph-yielding FinSequence; existence proof take <* the finite trivial edgeless _Graph *>; thus thesis; end; cluster non empty finite loopless non-trivial non-multi non-Dmulti simple Dsimple connected acyclic Tree-like for Graph-yielding FinSequence; existence proof take <* the finite non trivial Tree-like _Graph *>; thus thesis; end; end; registration let p be Graph-yielding FinSequence, n be Nat; cluster p | n -> Graph-yielding; coherence proof A1: p|n = p|Seg n by FINSEQ_1:def 15; now let x be object; assume A2: x in dom(p|n); then x in dom p by A1, RELAT_1:60, TARSKI:def 3; then p.x is _Graph by GLIB_000:def 53; hence (p|n).x is _Graph by A1, A2, FUNCT_1:47; end; hence thesis by GLIB_000:def 53; end; cluster p /^ n -> Graph-yielding; coherence proof per cases; suppose A3: n <= len p; now let x be object; assume A4: x in dom(p /^ n); then reconsider i = x as Nat; n+i in dom p by A4, FINSEQ_5:26; then p.(n+i) is _Graph by GLIB_000:def 53; hence (p /^ n).x is _Graph by A3, A4, RFINSEQ:def 1; end; hence thesis by GLIB_000:def 53; end; suppose not n <= len p; hence thesis by RFINSEQ:def 1; end; end; let m be Nat; cluster smid(p,m,n) -> Graph-yielding; coherence proof smid(p,m,n) = (p/^(m-'1))|(n+1-'m) by FINSEQ_8:def 1; hence thesis; end; cluster (m,n)-cut p -> Graph-yielding; coherence proof per cases; suppose 1 <= m & m <= n & n <= len p; then 1 <= m & m <= len p & 1 <= n & n <= len p by XXREAL_0:2; then m in dom p & n in dom p by FINSEQ_3:25; then smid(p,m,n) = (m,n)-cut p by FINSEQ_8:29; hence thesis; end; suppose not(1 <= m & m <= n & n <= len p); hence thesis by FINSEQ_6:def 4; end; end; end; registration let p be finite Graph-yielding FinSequence, n be Nat; cluster p | n -> finite; coherence proof A1: p|n = p|Seg n by FINSEQ_1:def 15; now let x be object; assume A2: x in dom(p|n); then x in dom p by A1, RELAT_1:60, TARSKI:def 3; then consider G being _Graph such that A3: p.x = G & G is finite by GLIB_000:def 58; take G; thus (p|n).x = G & G is finite by A1, A3, A2, FUNCT_1:47; end; hence thesis by GLIB_000:def 58; end; cluster p /^ n -> finite; coherence proof per cases; suppose A4: n <= len p; now let x be object; assume A5: x in dom(p /^ n); then reconsider i = x as Nat; n+i in dom p by A5, FINSEQ_5:26; then consider G being _Graph such that A6: p.(n+i) = G & G is finite by GLIB_000:def 58; take G; thus (p /^ n).x = G & G is finite by A4, A5, A6, RFINSEQ:def 1; end; hence thesis by GLIB_000:def 58; end; suppose not n <= len p; hence thesis by RFINSEQ:def 1; end; end; let m be Nat; cluster smid(p,m,n) -> finite; coherence proof smid(p,m,n) = (p/^(m-'1))|(n+1-'m) by FINSEQ_8:def 1; hence thesis; end; cluster (m,n)-cut p -> finite; coherence proof per cases; suppose 1 <= m & m <= n & n <= len p; then 1 <= m & m <= len p & 1 <= n & n <= len p by XXREAL_0:2; then m in dom p & n in dom p by FINSEQ_3:25; then smid(p,m,n) = (m,n)-cut p by FINSEQ_8:29; hence thesis; end; suppose not(1 <= m & m <= n & n <= len p); hence thesis by FINSEQ_6:def 4; end; end; end; registration let p be loopless Graph-yielding FinSequence, n be Nat; cluster p | n -> loopless; coherence proof A1: p|n = p|Seg n by FINSEQ_1:def 15; now let x be object; assume A2: x in dom(p|n); then x in dom p by A1, RELAT_1:60, TARSKI:def 3; then consider G being _Graph such that A3: p.x = G & G is loopless by GLIB_000:def 59; take G; thus (p|n).x = G & G is loopless by A1, A3, A2, FUNCT_1:47; end; hence thesis by GLIB_000:def 59; end; cluster p /^ n -> loopless; coherence proof per cases; suppose A4: n <= len p; now let x be object; assume A5: x in dom(p /^ n); then reconsider i = x as Nat; n+i in dom p by A5, FINSEQ_5:26; then consider G being _Graph such that A6: p.(n+i) = G & G is loopless by GLIB_000:def 59; take G; thus (p /^ n).x = G & G is loopless by A4, A5, A6, RFINSEQ:def 1; end; hence thesis by GLIB_000:def 59; end; suppose not n <= len p; hence thesis by RFINSEQ:def 1; end; end; let m be Nat; cluster smid(p,m,n) -> loopless; coherence proof smid(p,m,n) = (p/^(m-'1))|(n+1-'m) by FINSEQ_8:def 1; hence thesis; end; cluster (m,n)-cut p -> loopless; coherence proof per cases; suppose 1 <= m & m <= n & n <= len p; then 1 <= m & m <= len p & 1 <= n & n <= len p by XXREAL_0:2; then m in dom p & n in dom p by FINSEQ_3:25; then smid(p,m,n) = (m,n)-cut p by FINSEQ_8:29; hence thesis; end; suppose not(1 <= m & m <= n & n <= len p); hence thesis by FINSEQ_6:def 4; end; end; end; registration let p be trivial Graph-yielding FinSequence, n be Nat; cluster p | n -> trivial; coherence proof A1: p|n = p|Seg n by FINSEQ_1:def 15; now let x be object; assume A2: x in dom(p|n); then x in dom p by A1, RELAT_1:60, TARSKI:def 3; then consider G being _Graph such that A3: p.x = G & G is trivial by GLIB_000:def 60; take G; thus (p|n).x = G & G is trivial by A1, A3, A2, FUNCT_1:47; end; hence thesis by GLIB_000:def 60; end; cluster p /^ n -> trivial; coherence proof per cases; suppose A4: n <= len p; now let x be object; assume A5: x in dom(p /^ n); then reconsider i = x as Nat; n+i in dom p by A5, FINSEQ_5:26; then consider G being _Graph such that A6: p.(n+i) = G & G is trivial by GLIB_000:def 60; take G; thus (p /^ n).x = G & G is trivial by A4, A5, A6, RFINSEQ:def 1; end; hence thesis by GLIB_000:def 60; end; suppose not n <= len p; hence thesis by RFINSEQ:def 1; end; end; let m be Nat; cluster smid(p,m,n) -> trivial; coherence proof smid(p,m,n) = (p/^(m-'1))|(n+1-'m) by FINSEQ_8:def 1; hence thesis; end; cluster (m,n)-cut p -> trivial; coherence proof per cases; suppose 1 <= m & m <= n & n <= len p; then 1 <= m & m <= len p & 1 <= n & n <= len p by XXREAL_0:2; then m in dom p & n in dom p by FINSEQ_3:25; then smid(p,m,n) = (m,n)-cut p by FINSEQ_8:29; hence thesis; end; suppose not(1 <= m & m <= n & n <= len p); hence thesis by FINSEQ_6:def 4; end; end; end; registration let p be non-trivial Graph-yielding FinSequence, n be Nat; cluster p | n -> non-trivial; coherence proof A1: p|n = p|Seg n by FINSEQ_1:def 15; now let x be object; assume A2: x in dom(p|n); then x in dom p by A1, RELAT_1:60, TARSKI:def 3; then consider G being _Graph such that A3: p.x = G & G is non trivial by GLIB_000:def 61; take G; thus (p|n).x = G & G is non trivial by A1, A3, A2, FUNCT_1:47; end; hence thesis by GLIB_000:def 61; end; cluster p /^ n -> non-trivial; coherence proof per cases; suppose A4: n <= len p; now let x be object; assume A5: x in dom(p /^ n); then reconsider i = x as Nat; n+i in dom p by A5, FINSEQ_5:26; then consider G being _Graph such that A6: p.(n+i) = G & G is non trivial by GLIB_000:def 61; take G; thus (p /^ n).x = G & G is non trivial by A4, A5, A6, RFINSEQ:def 1; end; hence thesis by GLIB_000:def 61; end; suppose not n <= len p; hence thesis by RFINSEQ:def 1; end; end; let m be Nat; cluster smid(p,m,n) -> non-trivial; coherence proof smid(p,m,n) = (p/^(m-'1))|(n+1-'m) by FINSEQ_8:def 1; hence thesis; end; cluster (m,n)-cut p -> non-trivial; coherence proof per cases; suppose 1 <= m & m <= n & n <= len p; then 1 <= m & m <= len p & 1 <= n & n <= len p by XXREAL_0:2; then m in dom p & n in dom p by FINSEQ_3:25; then smid(p,m,n) = (m,n)-cut p by FINSEQ_8:29; hence thesis; end; suppose not(1 <= m & m <= n & n <= len p); hence thesis by FINSEQ_6:def 4; end; end; end; registration let p be non-multi Graph-yielding FinSequence, n be Nat; cluster p | n -> non-multi; coherence proof A1: p|n = p|Seg n by FINSEQ_1:def 15; now let x be object; assume A2: x in dom(p|n); then x in dom p by A1, RELAT_1:60, TARSKI:def 3; then consider G being _Graph such that A3: p.x = G & G is non-multi by GLIB_000:def 62; take G; thus (p|n).x = G & G is non-multi by A1, A3, A2, FUNCT_1:47; end; hence thesis by GLIB_000:def 62; end; cluster p /^ n -> non-multi; coherence proof per cases; suppose A4: n <= len p; now let x be object; assume A5: x in dom(p /^ n); then reconsider i = x as Nat; n+i in dom p by A5, FINSEQ_5:26; then consider G being _Graph such that A6: p.(n+i) = G & G is non-multi by GLIB_000:def 62; take G; thus (p /^ n).x = G & G is non-multi by A4, A5, A6, RFINSEQ:def 1; end; hence thesis by GLIB_000:def 62; end; suppose not n <= len p; hence thesis by RFINSEQ:def 1; end; end; let m be Nat; cluster smid(p,m,n) -> non-multi; coherence proof smid(p,m,n) = (p/^(m-'1))|(n+1-'m) by FINSEQ_8:def 1; hence thesis; end; cluster (m,n)-cut p -> non-multi; coherence proof per cases; suppose 1 <= m & m <= n & n <= len p; then 1 <= m & m <= len p & 1 <= n & n <= len p by XXREAL_0:2; then m in dom p & n in dom p by FINSEQ_3:25; then smid(p,m,n) = (m,n)-cut p by FINSEQ_8:29; hence thesis; end; suppose not(1 <= m & m <= n & n <= len p); hence thesis by FINSEQ_6:def 4; end; end; end; registration let p be non-Dmulti Graph-yielding FinSequence, n be Nat; cluster p | n -> non-Dmulti; coherence proof A1: p|n = p|Seg n by FINSEQ_1:def 15; now let x be object; assume A2: x in dom(p|n); then x in dom p by A1, RELAT_1:60, TARSKI:def 3; then consider G being _Graph such that A3: p.x = G & G is non-Dmulti by GLIB_000:def 63; take G; thus (p|n).x = G & G is non-Dmulti by A1, A3, A2, FUNCT_1:47; end; hence thesis by GLIB_000:def 63; end; cluster p /^ n -> non-Dmulti; coherence proof per cases; suppose A4: n <= len p; now let x be object; assume A5: x in dom(p /^ n); then reconsider i = x as Nat; n+i in dom p by A5, FINSEQ_5:26; then consider G being _Graph such that A6: p.(n+i) = G & G is non-Dmulti by GLIB_000:def 63; take G; thus (p /^ n).x = G & G is non-Dmulti by A4, A5, A6, RFINSEQ:def 1; end; hence thesis by GLIB_000:def 63; end; suppose not n <= len p; hence thesis by RFINSEQ:def 1; end; end; let m be Nat; cluster smid(p,m,n) -> non-Dmulti; coherence proof smid(p,m,n) = (p/^(m-'1))|(n+1-'m) by FINSEQ_8:def 1; hence thesis; end; cluster (m,n)-cut p -> non-Dmulti; coherence proof per cases; suppose 1 <= m & m <= n & n <= len p; then 1 <= m & m <= len p & 1 <= n & n <= len p by XXREAL_0:2; then m in dom p & n in dom p by FINSEQ_3:25; then smid(p,m,n) = (m,n)-cut p by FINSEQ_8:29; hence thesis; end; suppose not(1 <= m & m <= n & n <= len p); hence thesis by FINSEQ_6:def 4; end; end; end; registration let p be simple Graph-yielding FinSequence, n be Nat; cluster p | n -> simple; coherence; cluster p /^ n -> simple; coherence; let m be Nat; cluster smid(p,m,n) -> simple; coherence; cluster (m,n)-cut p -> simple; coherence; end; registration let p be Dsimple Graph-yielding FinSequence, n be Nat; cluster p | n -> Dsimple; coherence; cluster p /^ n -> Dsimple; coherence; let m be Nat; cluster smid(p,m,n) -> Dsimple; coherence; cluster (m,n)-cut p -> Dsimple; coherence; end; registration let p be connected Graph-yielding FinSequence, n be Nat; cluster p | n -> connected; coherence proof A1: p|n = p|Seg n by FINSEQ_1:def 15; now let x be object; assume A2: x in dom(p|n); then x in dom p by A1, RELAT_1:60, TARSKI:def 3; then consider G being _Graph such that A3: p.x = G & G is connected by GLIB_002:def 12; take G; thus (p|n).x = G & G is connected by A1, A3, A2, FUNCT_1:47; end; hence thesis by GLIB_002:def 12; end; cluster p /^ n -> connected; coherence proof per cases; suppose A4: n <= len p; now let x be object; assume A5: x in dom(p /^ n); then reconsider i = x as Nat; n+i in dom p by A5, FINSEQ_5:26; then consider G being _Graph such that A6: p.(n+i) = G & G is connected by GLIB_002:def 12; take G; thus (p /^ n).x = G & G is connected by A4, A5, A6, RFINSEQ:def 1; end; hence thesis by GLIB_002:def 12; end; suppose not n <= len p; hence thesis by RFINSEQ:def 1; end; end; let m be Nat; cluster smid(p,m,n) -> connected; coherence proof smid(p,m,n) = (p/^(m-'1))|(n+1-'m) by FINSEQ_8:def 1; hence thesis; end; cluster (m,n)-cut p -> connected; coherence proof per cases; suppose 1 <= m & m <= n & n <= len p; then 1 <= m & m <= len p & 1 <= n & n <= len p by XXREAL_0:2; then m in dom p & n in dom p by FINSEQ_3:25; then smid(p,m,n) = (m,n)-cut p by FINSEQ_8:29; hence thesis; end; suppose not(1 <= m & m <= n & n <= len p); hence thesis by FINSEQ_6:def 4; end; end; end; registration let p be acyclic Graph-yielding FinSequence, n be Nat; cluster p | n -> acyclic; coherence proof A1: p|n = p|Seg n by FINSEQ_1:def 15; now let x be object; assume A2: x in dom(p|n); then x in dom p by A1, RELAT_1:60, TARSKI:def 3; then consider G being _Graph such that A3: p.x = G & G is acyclic by GLIB_002:def 13; take G; thus (p|n).x = G & G is acyclic by A1, A3, A2, FUNCT_1:47; end; hence thesis by GLIB_002:def 13; end; cluster p /^ n -> acyclic; coherence proof per cases; suppose A4: n <= len p; now let x be object; assume A5: x in dom(p /^ n); then reconsider i = x as Nat; n+i in dom p by A5, FINSEQ_5:26; then consider G being _Graph such that A6: p.(n+i) = G & G is acyclic by GLIB_002:def 13; take G; thus (p /^ n).x = G & G is acyclic by A4, A5, A6, RFINSEQ:def 1; end; hence thesis by GLIB_002:def 13; end; suppose not (n <= len p); hence thesis by RFINSEQ:def 1; end; end; let m be Nat; cluster smid(p,m,n) -> acyclic; coherence proof smid(p,m,n) = (p/^(m-'1))|(n+1-'m) by FINSEQ_8:def 1; hence thesis; end; cluster (m,n)-cut p -> acyclic; coherence proof per cases; suppose 1 <= m & m <= n & n <= len p; then 1 <= m & m <= len p & 1 <= n & n <= len p by XXREAL_0:2; then m in dom p & n in dom p by FINSEQ_3:25; then smid(p,m,n) = (m,n)-cut p by FINSEQ_8:29; hence thesis; end; suppose not(1 <= m & m <= n & n <= len p); hence thesis by FINSEQ_6:def 4; end; end; end; registration let p be Tree-like Graph-yielding FinSequence, n be Nat; cluster p | n -> Tree-like; coherence; cluster p /^ n -> Tree-like; coherence; let m be Nat; cluster smid(p,m,n) -> Tree-like; coherence; cluster (m,n)-cut p -> Tree-like; coherence; end; registration let p be edgeless Graph-yielding FinSequence, n be Nat; cluster p | n -> edgeless; coherence proof A1: p|n = p|Seg n by FINSEQ_1:def 15; now let x be object; assume A2: x in dom(p|n); then x in dom p by A1, RELAT_1:60, TARSKI:def 3; then consider G being _Graph such that A3: p.x = G & G is edgeless by Def2; take G; thus (p|n).x = G & G is edgeless by A1, A3, A2, FUNCT_1:47; end; hence thesis; end; cluster p /^ n -> edgeless; coherence proof per cases; suppose A4: n <= len p; now let x be object; assume A5: x in dom(p /^ n); then reconsider i = x as Nat; n+i in dom p by A5, FINSEQ_5:26; then consider G being _Graph such that A6: p.(n+i) = G & G is edgeless by Def2; take G; thus (p /^ n).x = G & G is edgeless by A4, A5, A6, RFINSEQ:def 1; end; hence thesis; end; suppose not (n <= len p); hence thesis by RFINSEQ:def 1; end; end; let m be Nat; cluster smid(p,m,n) -> edgeless; coherence proof smid(p,m,n) = (p/^(m-'1))|(n+1-'m) by FINSEQ_8:def 1; hence thesis; end; cluster (m,n)-cut p -> edgeless; coherence proof per cases; suppose 1 <= m & m <= n & n <= len p; then 1 <= m & m <= len p & 1 <= n & n <= len p by XXREAL_0:2; then m in dom p & n in dom p by FINSEQ_3:25; then smid(p,m,n) = (m,n)-cut p by FINSEQ_8:29; hence thesis; end; suppose not(1 <= m & m <= n & n <= len p); hence thesis by FINSEQ_6:def 4; end; end; end; registration let p, q be Graph-yielding FinSequence; cluster p ^ q -> Graph-yielding; coherence proof now let x be object; assume A1: x in dom (p^q); then reconsider k = x as Nat; per cases by A1, FINSEQ_1:25; suppose k in dom p; then (p^q).k = p.k & p.k is _Graph by FINSEQ_1:def 7, GLIB_000:def 53; hence (p^q).x is _Graph; end; suppose ex n being Nat st n in dom q & k = len p + n; then consider n being Nat such that A2: n in dom q & k = len p + n; (p^q).(len p + n) = q.n & q.n is _Graph by A2, FINSEQ_1:def 7, GLIB_000:def 53; hence (p^q).x is _Graph by A2; end; end; hence thesis by GLIB_000:def 53; end; cluster p ^' q -> Graph-yielding; coherence proof p ^' q = p^(2,len q)-cut q by FINSEQ_6:def 5; hence thesis; end; end; registration let p, q be finite Graph-yielding FinSequence; cluster p ^ q -> finite; coherence proof for x being object st x in dom (p^q) ex G being _Graph st G = (p^q).x & G is finite proof let x be object; assume A1: x in dom (p^q); then reconsider k = x as Nat; per cases by A1, FINSEQ_1:25; suppose k in dom p; then (p^q).k = p.k & ex G being _Graph st p.k = G & G is finite by FINSEQ_1:def 7, GLIB_000:def 58; hence ex G being _Graph st (p^q).x = G & G is finite; end; suppose ex n being Nat st n in dom q & k = len p + n; then consider n being Nat such that A2: n in dom q & k = len p + n; (p^q).(len p + n) = q.n & ex G being _Graph st q.n = G & G is finite by A2, FINSEQ_1:def 7, GLIB_000:def 58; hence ex G being _Graph st (p^q).x = G & G is finite by A2; end; end; hence thesis by GLIB_000:def 58; end; cluster p ^' q -> finite; coherence proof p ^' q = p^(2,len q)-cut q by FINSEQ_6:def 5; hence thesis; end; end; registration let p, q be loopless Graph-yielding FinSequence; cluster p ^ q -> loopless; coherence proof for x being object st x in dom (p^q) ex G being _Graph st G = (p^q).x & G is loopless proof let x be object; assume A1: x in dom (p^q); then reconsider k = x as Nat; per cases by A1, FINSEQ_1:25; suppose k in dom p; then (p^q).k = p.k & ex G being _Graph st p.k = G & G is loopless by FINSEQ_1:def 7, GLIB_000:def 59; hence ex G being _Graph st (p^q).x = G & G is loopless; end; suppose ex n being Nat st n in dom q & k = len p + n; then consider n being Nat such that A2: n in dom q & k = len p + n; (p^q).(len p + n) = q.n & ex G being _Graph st q.n = G & G is loopless by A2, FINSEQ_1:def 7, GLIB_000:def 59; hence ex G being _Graph st (p^q).x = G & G is loopless by A2; end; end; hence thesis by GLIB_000:def 59; end; cluster p ^' q -> loopless; coherence proof p ^' q = p^(2,len q)-cut q by FINSEQ_6:def 5; hence thesis; end; end; registration let p, q be trivial Graph-yielding FinSequence; cluster p ^ q -> trivial; coherence proof for x being object st x in dom (p^q) ex G being _Graph st G = (p^q).x & G is trivial proof let x be object; assume A1: x in dom (p^q); then reconsider k = x as Nat; per cases by A1, FINSEQ_1:25; suppose k in dom p; then (p^q).k = p.k & ex G being _Graph st p.k = G & G is trivial by FINSEQ_1:def 7, GLIB_000:def 60; hence ex G being _Graph st (p^q).x = G & G is trivial; end; suppose ex n being Nat st n in dom q & k = len p + n; then consider n being Nat such that A2: n in dom q & k = len p + n; (p^q).(len p + n) = q.n & ex G being _Graph st q.n = G & G is trivial by A2, FINSEQ_1:def 7, GLIB_000:def 60; hence ex G being _Graph st (p^q).x = G & G is trivial by A2; end; end; hence thesis by GLIB_000:def 60; end; cluster p ^' q -> trivial; coherence proof p ^' q = p^(2,len q)-cut q by FINSEQ_6:def 5; hence thesis; end; end; registration let p, q be non-trivial Graph-yielding FinSequence; cluster p ^ q -> non-trivial; coherence proof for x being object st x in dom (p^q) ex G being _Graph st G = (p^q).x & G is non trivial proof let x be object; assume A1: x in dom (p^q); then reconsider k = x as Nat; per cases by A1, FINSEQ_1:25; suppose k in dom p; then (p^q).k = p.k & ex G being _Graph st p.k = G & G is non trivial by FINSEQ_1:def 7, GLIB_000:def 61; hence ex G being _Graph st (p^q).x = G & G is non trivial; end; suppose ex n being Nat st n in dom q & k = len p + n; then consider n being Nat such that A2: n in dom q & k = len p + n; (p^q).(len p + n) = q.n & ex G being _Graph st q.n = G & G is non trivial by A2, FINSEQ_1:def 7, GLIB_000:def 61; hence ex G being _Graph st (p^q).x = G & G is non trivial by A2; end; end; hence thesis by GLIB_000:def 61; end; cluster p ^' q -> non-trivial; coherence proof p ^' q = p^(2,len q)-cut q by FINSEQ_6:def 5; hence thesis; end; end; registration let p, q be non-multi Graph-yielding FinSequence; cluster p ^ q -> non-multi; coherence proof for x being object st x in dom (p^q) ex G being _Graph st G = (p^q).x & G is non-multi proof let x be object; assume A1: x in dom (p^q); then reconsider k = x as Nat; per cases by A1, FINSEQ_1:25; suppose k in dom p; then (p^q).k = p.k & ex G being _Graph st p.k = G & G is non-multi by FINSEQ_1:def 7, GLIB_000:def 62; hence ex G being _Graph st (p^q).x = G & G is non-multi; end; suppose ex n being Nat st n in dom q & k = len p + n; then consider n being Nat such that A2: n in dom q & k = len p + n; (p^q).(len p + n) = q.n & ex G being _Graph st q.n = G & G is non-multi by A2, FINSEQ_1:def 7, GLIB_000:def 62; hence ex G being _Graph st (p^q).x = G & G is non-multi by A2; end; end; hence thesis by GLIB_000:def 62; end; cluster p ^' q -> non-multi; coherence proof p ^' q = p^(2,len q)-cut q by FINSEQ_6:def 5; hence thesis; end; end; registration let p, q be non-Dmulti Graph-yielding FinSequence; cluster p ^ q -> non-Dmulti; coherence proof for x being object st x in dom (p^q) ex G being _Graph st G = (p^q).x & G is non-Dmulti proof let x be object; assume A1: x in dom (p^q); then reconsider k = x as Nat; per cases by A1, FINSEQ_1:25; suppose k in dom p; then (p^q).k = p.k & ex G being _Graph st p.k = G & G is non-Dmulti by FINSEQ_1:def 7, GLIB_000:def 63; hence ex G being _Graph st (p^q).x = G & G is non-Dmulti; end; suppose ex n being Nat st n in dom q & k = len p + n; then consider n being Nat such that A2: n in dom q & k = len p + n; (p^q).(len p + n) = q.n & ex G being _Graph st q.n = G & G is non-Dmulti by A2, FINSEQ_1:def 7, GLIB_000:def 63; hence ex G being _Graph st (p^q).x = G & G is non-Dmulti by A2; end; end; hence thesis by GLIB_000:def 63; end; cluster p ^' q -> non-Dmulti; coherence proof p ^' q = p^(2,len q)-cut q by FINSEQ_6:def 5; hence thesis; end; end; registration let p, q be simple Graph-yielding FinSequence; cluster p ^ q -> simple; coherence; cluster p ^' q -> simple; coherence; end; registration let p, q be Dsimple Graph-yielding FinSequence; cluster p ^ q -> Dsimple; coherence; cluster p ^' q -> Dsimple; coherence; end; registration let p, q be connected Graph-yielding FinSequence; cluster p ^ q -> connected; coherence proof for x being object st x in dom (p^q) ex G being _Graph st G = (p^q).x & G is connected proof let x be object; assume A1: x in dom (p^q); then reconsider k = x as Nat; per cases by A1, FINSEQ_1:25; suppose k in dom p; then (p^q).k = p.k & ex G being _Graph st p.k = G & G is connected by FINSEQ_1:def 7, GLIB_002:def 12; hence ex G being _Graph st (p^q).x = G & G is connected; end; suppose ex n being Nat st n in dom q & k = len p + n; then consider n being Nat such that A2: n in dom q & k = len p + n; (p^q).(len p + n) = q.n & ex G being _Graph st q.n = G & G is connected by A2, FINSEQ_1:def 7, GLIB_002:def 12; hence ex G being _Graph st (p^q).x = G & G is connected by A2; end; end; hence thesis by GLIB_002:def 12; end; cluster p ^' q -> connected; coherence proof p ^' q = p^(2,len q)-cut q by FINSEQ_6:def 5; hence thesis; end; end; registration let p, q be acyclic Graph-yielding FinSequence; cluster p ^ q -> acyclic; coherence proof for x being object st x in dom (p^q) ex G being _Graph st G = (p^q).x & G is acyclic proof let x be object; assume A1: x in dom (p^q); then reconsider k = x as Nat; per cases by A1, FINSEQ_1:25; suppose k in dom p; then (p^q).k = p.k & ex G being _Graph st p.k = G & G is acyclic by FINSEQ_1:def 7, GLIB_002:def 13; hence ex G being _Graph st (p^q).x = G & G is acyclic; end; suppose ex n being Nat st n in dom q & k = len p + n; then consider n being Nat such that A2: n in dom q & k = len p + n; (p^q).(len p + n) = q.n & ex G being _Graph st q.n = G & G is acyclic by A2, FINSEQ_1:def 7, GLIB_002:def 13; hence ex G being _Graph st (p^q).x = G & G is acyclic by A2; end; end; hence thesis by GLIB_002:def 13; end; cluster p ^' q -> acyclic; coherence proof p ^' q = p^(2,len q)-cut q by FINSEQ_6:def 5; hence thesis; end; end; registration let p, q be Tree-like Graph-yielding FinSequence; cluster p ^ q -> Tree-like; coherence; cluster p ^' q -> Tree-like; coherence; end; registration let p, q be edgeless Graph-yielding FinSequence; cluster p ^ q -> edgeless; coherence proof for x being object st x in dom (p^q) ex G being _Graph st G = (p^q).x & G is edgeless proof let x be object; assume A1: x in dom (p^q); then reconsider k = x as Nat; per cases by A1, FINSEQ_1:25; suppose k in dom p; then (p^q).k = p.k & ex G being _Graph st p.k = G & G is edgeless by FINSEQ_1:def 7, Def2; hence ex G being _Graph st (p^q).x = G & G is edgeless; end; suppose ex n being Nat st n in dom q & k = len p + n; then consider n being Nat such that A2: n in dom q & k = len p + n; (p^q).(len p + n) = q.n & ex G being _Graph st q.n = G & G is edgeless by A2, FINSEQ_1:def 7, Def2; hence ex G being _Graph st (p^q).x = G & G is edgeless by A2; end; end; hence thesis; end; cluster p ^' q -> edgeless; coherence proof p ^' q = p^(2,len q)-cut q by FINSEQ_6:def 5; hence thesis; end; end; registration let G1, G2 be _Graph; cluster <* G1, G2 *> -> Graph-yielding; coherence proof <* G1, G2 *> = <* G1 *> ^ <* G2 *> by FINSEQ_1:def 9; hence thesis; end; let G3 be _Graph; cluster <* G1, G2, G3 *> -> Graph-yielding; coherence proof <* G1, G2, G3 *> = <* G1 *> ^ <* G2 *> ^ <* G3 *> by FINSEQ_1:def 10; hence thesis; end; end; registration let G1, G2 be finite _Graph; cluster <* G1, G2 *> -> finite; coherence proof <* G1, G2 *> = <* G1 *> ^ <* G2 *> by FINSEQ_1:def 9; hence thesis; end; let G3 be finite _Graph; cluster <* G1, G2, G3 *> -> finite; coherence proof <* G1, G2, G3 *> = <* G1 *> ^ <* G2 *> ^ <* G3 *> by FINSEQ_1:def 10; hence thesis; end; end; registration let G1, G2 be loopless _Graph; cluster <* G1, G2 *> -> loopless; coherence proof <* G1, G2 *> = <* G1 *> ^ <* G2 *> by FINSEQ_1:def 9; hence thesis; end; let G3 be loopless _Graph; cluster <* G1, G2, G3 *> -> loopless; coherence proof <* G1, G2, G3 *> = <* G1 *> ^ <* G2 *> ^ <* G3 *> by FINSEQ_1:def 10; hence thesis; end; end; registration let G1, G2 be trivial _Graph; cluster <* G1, G2 *> -> trivial; coherence proof <* G1, G2 *> = <* G1 *> ^ <* G2 *> by FINSEQ_1:def 9; hence thesis; end; let G3 be trivial _Graph; cluster <* G1, G2, G3 *> -> trivial; coherence proof <* G1, G2, G3 *> = <* G1 *> ^ <* G2 *> ^ <* G3 *> by FINSEQ_1:def 10; hence thesis; end; end; registration let G1, G2 be non trivial _Graph; cluster <* G1, G2 *> -> non-trivial; coherence proof <* G1, G2 *> = <* G1 *> ^ <* G2 *> by FINSEQ_1:def 9; hence thesis; end; let G3 be non trivial _Graph; cluster <* G1, G2, G3 *> -> non-trivial; coherence proof <* G1, G2, G3 *> = <* G1 *> ^ <* G2 *> ^ <* G3 *> by FINSEQ_1:def 10; hence thesis; end; end; registration let G1, G2 be non-multi _Graph; cluster <* G1, G2 *> -> non-multi; coherence proof <* G1, G2 *> = <* G1 *> ^ <* G2 *> by FINSEQ_1:def 9; hence thesis; end; let G3 be non-multi _Graph; cluster <* G1, G2, G3 *> -> non-multi; coherence proof <* G1, G2, G3 *> = <* G1 *> ^ <* G2 *> ^ <* G3 *> by FINSEQ_1:def 10; hence thesis; end; end; registration let G1, G2 be non-Dmulti _Graph; cluster <* G1, G2 *> -> non-Dmulti; coherence proof <* G1, G2 *> = <* G1 *> ^ <* G2 *> by FINSEQ_1:def 9; hence thesis; end; let G3 be non-Dmulti _Graph; cluster <* G1, G2, G3 *> -> non-Dmulti; coherence proof <* G1, G2, G3 *> = <* G1 *> ^ <* G2 *> ^ <* G3 *> by FINSEQ_1:def 10; hence thesis; end; end; registration let G1, G2 be simple _Graph; cluster <* G1, G2 *> -> simple; coherence; let G3 be simple _Graph; cluster <* G1, G2, G3 *> -> simple; coherence; end; registration let G1, G2 be Dsimple _Graph; cluster <* G1, G2 *> -> Dsimple; coherence; let G3 be Dsimple _Graph; cluster <* G1, G2, G3 *> -> Dsimple; coherence; end; registration let G1, G2 be connected _Graph; cluster <* G1, G2 *> -> connected; coherence proof <* G1, G2 *> = <* G1 *> ^ <* G2 *> by FINSEQ_1:def 9; hence thesis; end; let G3 be connected _Graph; cluster <* G1, G2, G3 *> -> connected; coherence proof <* G1, G2, G3 *> = <* G1 *> ^ <* G2 *> ^ <* G3 *> by FINSEQ_1:def 10; hence thesis; end; end; registration let G1, G2 be acyclic _Graph; cluster <* G1, G2 *> -> acyclic; coherence proof <* G1, G2 *> = <* G1 *> ^ <* G2 *> by FINSEQ_1:def 9; hence thesis; end; let G3 be acyclic _Graph; cluster <* G1, G2, G3 *> -> acyclic; coherence proof <* G1, G2, G3 *> = <* G1 *> ^ <* G2 *> ^ <* G3 *> by FINSEQ_1:def 10; hence thesis; end; end; registration let G1, G2 be Tree-like _Graph; cluster <* G1, G2 *> -> Tree-like; coherence; let G3 be Tree-like _Graph; cluster <* G1, G2, G3 *> -> Tree-like; coherence; end; registration let G1, G2 be edgeless _Graph; cluster <* G1, G2 *> -> edgeless; coherence proof <* G1, G2 *> = <* G1 *> ^ <* G2 *> by FINSEQ_1:def 9; hence thesis; end; let G3 be edgeless _Graph; cluster <* G1, G2, G3 *> -> edgeless; coherence proof <* G1, G2, G3 *> = <* G1 *> ^ <* G2 *> ^ <* G3 *> by FINSEQ_1:def 10; hence thesis; end; end; begin :: Construction of finite Graphs :: finite addition of vertices can be finitely constructed with addVertex only theorem Th59: for G2 being _Graph, V being finite set, G1 being addVertices of G2, V ex p being non empty Graph-yielding FinSequence st p.1 == G2 & p.len p = G1 & len p = card (V \ the_Vertices_of G2) + 1 & for n being Element of dom p st n <= len p - 1 holds ex v being Vertex of G1 st p.(n+1) is addVertex of p.n, v & not v in the_Vertices_of p.n proof let G2 be _Graph; defpred P[Nat] means for V being finite set, G1 being addVertices of G2, V st card (V \ the_Vertices_of G2) = $1 holds ex p being non empty Graph-yielding FinSequence st p.1 == G2 & p.len p = G1 & len p = card (V \ the_Vertices_of G2) + 1 & for n being Element of dom p st n <= len p - 1 holds ex v being Vertex of G1 st p.(n+1) is addVertex of p.n, v & not v in the_Vertices_of p.n; A1: P[0] proof let V be finite set, G1 be addVertices of G2, V; assume A2: card (V \ the_Vertices_of G2) = 0; then V \ the_Vertices_of G2 = {}; then A3: G1 == G2 by XBOOLE_1:37, GLIB_006:78; reconsider p = <* G1 *> as non empty Graph-yielding FinSequence; take p; thus p.1 == G2 by A3, FINSEQ_1:40; thus p.len p = p.1 by FINSEQ_1:40 .= G1 by FINSEQ_1:40; thus len p = card (V \ the_Vertices_of G2) + 1 by A2, FINSEQ_1:40; let n be Element of dom p; 1 <= n & n <= len p by FINSEQ_3:25; then 1 <= n & n <= 1 by FINSEQ_1:40; then A4: n = 1 by XXREAL_0:1; assume n <= len p - 1; then n <= 1 - 1 by FINSEQ_1:40; hence thesis by A4; :: by contradiction end; A5: for k being Nat st P[k] holds P[k+1] proof let k be Nat; assume A6: P[k]; let V be finite set, G1 be addVertices of G2, V; assume A7: card (V \ the_Vertices_of G2) = k+1; then A8: V \ the_Vertices_of G2 <> {}; set v0 = the Element of V \ the_Vertices_of G2; set V0 = V \ {v0}; set G3 = the addVertices of G2, V0; V0 \ the_Vertices_of G2 = V0 \ (V0 /\ the_Vertices_of G2) by XBOOLE_1:47; then A9: card(V0 \ the_Vertices_of G2) = card V0 - card(V0 /\ the_Vertices_of G2) by CARD_2:44, XBOOLE_1:17; A10: v0 in V by A8, XBOOLE_0:def 5; then A11: card V0 = card V - card {v0} by CARD_2:44, ZFMISC_1:31 .= card V - 1 by CARD_1:30; A12: not v0 in the_Vertices_of G2 by A8, XBOOLE_0:def 5; then the_Vertices_of G2 misses {v0} by ZFMISC_1:50; then the_Vertices_of G2 /\ ({v0} \/ V0) = the_Vertices_of G2 /\ V0 by XBOOLE_1:78; then the_Vertices_of G2 /\ V0 = the_Vertices_of G2 /\ V by A10, ZFMISC_1:116; then A13: card(V0 \ the_Vertices_of G2) = card V - card(V /\ the_Vertices_of G2) - 1 by A9, A11 .= card(V \ (V /\ the_Vertices_of G2)) - 1 by CARD_2:44, XBOOLE_1:17 .= card(V \ the_Vertices_of G2) - 1 by XBOOLE_1:47 .= k by A7; then consider p being non empty Graph-yielding FinSequence such that A14: p.1 == G2 & p.len p = G3 & len p = card (V0 \ the_Vertices_of G2) + 1 and A15: for n being Element of dom p st n <= len p - 1 holds ex v being Vertex of G3 st p.(n+1) is addVertex of p.n, v & not v in the_Vertices_of p.n by A6; reconsider q = p ^ <* G1 *> as non empty Graph-yielding FinSequence; take q; 1 in dom p by FINSEQ_5:6; hence q.1 == G2 by A14, FINSEQ_1:def 7; A16: len q = len p + len <* G1 *> by FINSEQ_1:22 .= len p + 1 by FINSEQ_1:40; hence A17: q.len q = G1 by FINSEQ_1:42; thus len q = card (V \ the_Vertices_of G2) + 1 by A7, A13, A14, A16; G1 is addVertices of G2, {v0} \/ V0 by A10, ZFMISC_1:116; then consider G4 being addVertices of G2, V0 such that A18: G1 is addVertices of G4, {v0} by GLIB_006:83; A19: G1 is addVertices of G3, {v0} by A18, Th32, GLIB_006:77; let n be Element of dom q; assume n <= len q - 1; then per cases by Lm12; suppose A20: n = len q - 1; then A21: q.(n+1) = G1 by A17; 1 <= n by FINSEQ_3:25; then n in dom p by A16, A20, FINSEQ_3:25; then A24: q.n = G3 by A14, A16, A20, FINSEQ_1:def 7; the_Vertices_of G1 = the_Vertices_of G2 \/ V by GLIB_006:def 10; then reconsider v0 as Vertex of G1 by A10, TARSKI:def 3, XBOOLE_1:7; take v0; thus q.(n+1) is addVertex of q.n, v0 by A19, A21, A24; v0 in {v0} by TARSKI:def 1; then not v0 in V0 by XBOOLE_0:def 5; then not v0 in the_Vertices_of G2 \/ V0 by A12, XBOOLE_0:def 3; hence not v0 in the_Vertices_of q.n by A24, GLIB_006:def 10; end; suppose A25: n <= len p - 1; then A26: n+0 <= len p - 1 + 1 by XREAL_1:7; 1 <= n by FINSEQ_3:25; then reconsider m = n as Element of dom p by A26, FINSEQ_3:25; consider v being Vertex of G3 such that A27: p.(m+1) is addVertex of p.m, v & not v in the_Vertices_of p.m by A15, A25; 1+0 <= n+1 & n+1 <= len p - 1 + 1 by A25, XREAL_1:6; then n+1 in dom p by FINSEQ_3:25; then A28: q.(n+1) = p.(m+1) by FINSEQ_1:def 7; A29: q.n = p.m by FINSEQ_1:def 7; the_Vertices_of G3 c= the_Vertices_of G1 by A19, GLIB_006:def 9; then reconsider v as Vertex of G1 by TARSKI:def 3; take v; thus thesis by A27, A28, A29; end; end; A30: for k being Nat holds P[k] from NAT_1:sch 2(A1,A5); thus thesis by A30; end; theorem Th60: for G being finite _Graph, H being Subgraph of G st G.size() = H.size() ex p being non empty finite Graph-yielding FinSequence st p.1 == H & p.len p = G & len p = G.order() - H.order() + 1 & for n being Element of dom p st n <= len p - 1 holds ex v being Vertex of G st p.(n+1) is addVertex of p.n, v & not v in the_Vertices_of p.n proof let G be finite _Graph, H be Subgraph of G; assume A1: G.size() = H.size(); set V = the_Vertices_of G \ the_Vertices_of H; G is addVertices of H, V by A1, Th34; then consider p being non empty Graph-yielding FinSequence such that A2: p.1 == H & p.len p = G & len p = card (V \ the_Vertices_of H) + 1 and A3: for n being Element of dom p st n <= len p - 1 holds ex v being Vertex of G st p.(n+1) is addVertex of p.n, v & not v in the_Vertices_of p.n by Th59; defpred P[Nat] means for n being Element of dom p st $1 = n holds p.n is finite; A4: P[1] by A2, GLIB_000:89; A5: for k being non zero Nat st P[k] holds P[k+1] proof let k be non zero Nat; assume A6: P[k]; let m be Element of dom p; assume A7: k+1 = m; then A8: k+1 <= len p by FINSEQ_3:25; then A9: k+1-1 <= len p - 0 by XREAL_1:13; 1 <= k by NAT_1:14; then reconsider n = k as Element of dom p by A9, FINSEQ_3:25; k+1-1 <= len p - 1 by A8, XREAL_1:9; then consider v being Vertex of G such that A10: p.(n+1) is addVertex of p.n, v & not v in the_Vertices_of p.n by A3; p.n is finite by A6; hence thesis by A7, A10; end; A11: for k being non zero Nat holds P[k] from NAT_1:sch 10(A4,A5); for x being Element of dom p holds p.x is finite proof let x be Element of dom p; x is non zero Nat by FINSEQ_3:25; hence thesis by A11; end; then reconsider p as non empty finite Graph-yielding FinSequence by GLIB_000:def 66; take p; thus p.1 == H & p.len p = G by A2; V \ the_Vertices_of H = the_Vertices_of G \ (the_Vertices_of H \/ the_Vertices_of H) by XBOOLE_1:41 .= the_Vertices_of G \ the_Vertices_of H; hence len p = card (the_Vertices_of G) - card(the_Vertices_of H) + 1 by A2, CARD_2:44 .= G.order() - card(the_Vertices_of H) + 1 by GLIB_000:def 24 .= G.order() - H.order() + 1 by GLIB_000:def 24; thus thesis by A3; end; theorem Th61: for G being finite edgeless _Graph, H being Subgraph of G ex p being non empty finite edgeless Graph-yielding FinSequence st p.1 == H & p.len p = G & len p = G.order() - H.order() + 1 & for n being Element of dom p st n <= len p - 1 holds ex v being Vertex of G st p.(n+1) is addVertex of p.n,v & not v in the_Vertices_of p.n proof let G be finite edgeless _Graph, H be Subgraph of G; G.size() = 0 by Th49 .= H.size() by Th49; then consider p being non empty finite Graph-yielding FinSequence such that A1: p.1 == H & p.len p = G & len p = G.order() - H.order() + 1 and A2: for n being Element of dom p st n <= len p - 1 holds ex v being Vertex of G st p.(n+1) is addVertex of p.n, v & not v in the_Vertices_of p.n by Th60; defpred P[Nat] means for n being Element of dom p st $1 = n holds p.n is edgeless; A3: P[1] by A1, Th52; A4: for k being non zero Nat st P[k] holds P[k+1] proof let k be non zero Nat; assume A5: P[k]; let m be Element of dom p; assume A6: k+1 = m; then A7: k+1 <= len p by FINSEQ_3:25; then A8: k+1-1 <= len p - 0 by XREAL_1:13; 1 <= k by NAT_1:14; then reconsider n = k as Element of dom p by A8, FINSEQ_3:25; k+1-1 <= len p - 1 by A7, XREAL_1:9; then consider v being Vertex of G such that A9: p.(n+1) is addVertex of p.n, v & not v in the_Vertices_of p.n by A2; p.n is edgeless by A5; hence thesis by A6, A9; end; A10: for k being non zero Nat holds P[k] from NAT_1:sch 10(A3,A4); for x being Element of dom p holds p.x is edgeless proof let x be Element of dom p; x is non zero Nat by FINSEQ_3:25; hence thesis by A10; end; then p is edgeless; then reconsider p as non empty finite edgeless Graph-yielding FinSequence; take p; thus thesis by A1, A2; end; :: finite edgeless graphs can be finitely constructed with addVertex only theorem Th62: for G being finite edgeless _Graph ex p being non empty finite edgeless Graph-yielding FinSequence st p.1 is trivial edgeless & p.len p = G & len p = G.order() & for n being Element of dom p st n <= len p - 1 holds ex v being Vertex of G st p.(n+1) is addVertex of p.n,v & not v in the_Vertices_of p.n proof let G be finite edgeless _Graph; set v0 = the Vertex of G; set H = the inducedSubgraph of G, {v0}; consider p being non empty finite edgeless Graph-yielding FinSequence such that A1: p.1 == H & p.len p = G & len p = G.order() - H.order() + 1 and A2: for n being Element of dom p st n <= len p - 1 holds ex v being Vertex of G st p.(n+1) is addVertex of p.n, v & not v in the_Vertices_of p.n by Th61; take p; thus p.1 is trivial edgeless by A1, Th52, GLIB_000:89; thus p.len p = G by A1; H.order() = 1 by GLIB_000:26; hence len p = G.order() by A1; thus thesis by A2; end; scheme FinEdgelessGraphs { P[finite _Graph] } : for G being finite edgeless _Graph holds P[G] provided A1: for G being trivial edgeless _Graph holds P[G] and A2: for G2 being finite edgeless _Graph, v being object for G1 being addVertex of G2,v st not v in the_Vertices_of G2 & P[G2] holds P[G1] proof let G be finite edgeless _Graph; consider p being non empty finite edgeless Graph-yielding FinSequence such that A3: p.1 is trivial edgeless & p.len p = G & len p = G.order() and A4: for n being Element of dom p st n <= len p - 1 holds ex v being Vertex of G st p.(n+1) is addVertex of p.n,v & not v in the_Vertices_of p.n by Th62; defpred Q[Nat] means $1 + 1 <= len p implies ex k being Element of dom p st k = $1 + 1 & P[p.k]; A5: Q[0] proof assume 0 + 1 <= len p; then reconsider k = 1 as Element of dom p by FINSEQ_3:25; take k; thus k = 0 + 1; thus P[p.k] by A1, A3; end; A6: for m being Nat st Q[m] holds Q[m+1] proof let m be Nat; assume A7: Q[m]; assume A8: m+1 + 1 <= len p; then m+1 +1 -1 <= len p - 0 by XREAL_1:13; then consider k0 being Element of dom p such that A9: k0 = m + 1 & P[p.k0] by A7; set k = k0+1; 0+1 <= k by XREAL_1:6; then reconsider k as Element of dom p by A8, A9, FINSEQ_3:25; take k; thus k = m+1 + 1 by A9; k0 +1-1 <= len p - 1 by A8, A9, XREAL_1:9; then consider v being Vertex of G such that A10: p.k is addVertex of p.k0,v & not v in the_Vertices_of p.k0 by A4; thus P[p.k] by A2, A9, A10; end; A11: for m being Nat holds Q[m] from NAT_1:sch 2(A5,A6); len p - 1 is Nat & len p - 1 + 1 <= len p by INT_1:74; then ex k being Element of dom p st k = len p - 1 + 1 & P[p.k] by A11; hence thesis by A3; end; theorem for p being non empty Graph-yielding FinSequence st p.1 is edgeless & for n being Element of dom p st n <= len p - 1 holds ex v being object st p.(n+1) is addVertex of p.n,v holds p.len p is edgeless proof defpred P[Nat] means for p being non empty Graph-yielding FinSequence st len p = $1 & p.1 is edgeless & for n being Element of dom p st n <= len p - 1 holds ex v being object st p.(n+1) is addVertex of p.n,v holds p.len p is edgeless; A1: P[1]; A2: for m being non zero Nat st P[m] holds P[m+1] proof let m be non zero Nat; assume A3: P[m]; let p be non empty Graph-yielding FinSequence; assume that A4: len p = m + 1 & p.1 is edgeless and A5: for n being Element of dom p st n <= len p - 1 holds ex v being object st p.(n+1) is addVertex of p.n,v; reconsider q = p|m as non empty Graph-yielding FinSequence; m+1-1 <= len p - 0 by A4, XREAL_1:10; then A6: len q = m by FINSEQ_1:59; A7: 1 <= m by INT_1:74; then A8: q.1 is edgeless by A4, FINSEQ_3:112; now let n be Element of dom q; assume A9: n <= len q - 1; then n + 0 <= m - 1 + 1 by A6, XREAL_1:7; then A10: n <= len p - 1 by A4; A11: 1 <= n & n <= len q by FINSEQ_3:25; then 1 <= n & n + 0 <= m + 1 by A6, XREAL_1:7; then reconsider k = n as Element of dom p by A4, FINSEQ_3:25; consider v being object such that A12: p.(k+1) is addVertex of p.k,v by A5, A10; take v; n+1 <= len q - 1 + 1 by A9, XREAL_1:6; then p.(k+1) = q.(n+1) & p.k = q.k by A6, A11, FINSEQ_3:112; hence q.(n+1) is addVertex of q.n,v by A12; end; then A13: q.len q is edgeless by A3, A6, A8; m+0 <= len p - 1 + 1 by A4, XREAL_1:6; then reconsider k=m as Element of dom p by A7, FINSEQ_3:25; consider v be object such that A14: p.(k+1) is addVertex of p.k,v by A5, A4; p.k = q.len q by A6, FINSEQ_3:112; hence thesis by A13, A4, A14; end; A15: for m being non zero Nat holds P[m] from NAT_1:sch 10(A1,A2); let p be non empty Graph-yielding FinSequence; assume that A16: p.1 is edgeless & for n being Element of dom p st n <= len p - 1 holds ex v being object st p.(n+1) is addVertex of p.n,v; thus thesis by A15, A16; end; theorem Th64: for G being finite _Graph, H being spanning Subgraph of G ex p being non empty finite Graph-yielding FinSequence st p.1 == H & p.len p = G & len p = G.size() - H.size() + 1 & for n being Element of dom p st n <= len p - 1 holds ex v1,v2 being Vertex of G, e being object st p.(n+1) is addEdge of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n proof let G be finite _Graph; defpred P[Nat] means for H being spanning Subgraph of G st G.size() - H.size() = $1 ex p being non empty finite Graph-yielding FinSequence st p.1 == H & p.len p = G & len p = G.size() - H.size() + 1 & for n being Element of dom p st n <= len p - 1 holds ex v1,v2 being Vertex of G, e being object st p.(n+1) is addEdge of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n; A1: P[0] proof let H be spanning Subgraph of G; assume A2: G.size() - H.size() = 0; then A3: G == H by Th13; reconsider p = <* G *> as non empty finite Graph-yielding FinSequence; take p; thus p.1 == H by A3, FINSEQ_1:40; thus p.len p = p.1 by FINSEQ_1:40 .= G by FINSEQ_1:40; thus len p = G.size() - H.size() + 1 by A2, FINSEQ_1:40; let n be Element of dom p; 1 <= n & n <= len p by FINSEQ_3:25; then 1 <= n & n <= 1 by FINSEQ_1:40; then A4: n = 1 by XXREAL_0:1; assume n <= len p - 1; then n <= 1 - 1 by FINSEQ_1:40; hence thesis by A4; :: by contradiction end; A5: for k being Nat st P[k] holds P[k+1] proof let k be Nat; assume A6: P[k]; let H be spanning Subgraph of G; assume A7: G.size() - H.size() = k+1; A8: the_Edges_of G \ the_Edges_of H <> {} proof assume A9: the_Edges_of G \ the_Edges_of H = {}; 0 = card(the_Edges_of G \ the_Edges_of H) by A9 .= card the_Edges_of G - card the_Edges_of H by CARD_2:44 .= G.size() - card the_Edges_of H by GLIB_000:def 25 .= G.size() - H.size() by GLIB_000:def 25; hence contradiction by A7; end; set e0 = the Element of the_Edges_of G \ the_Edges_of H; A10: e0 in the_Edges_of G by A8, TARSKI:def 3; then reconsider e0 as Element of the_Edges_of G; set u = (the_Source_of G).e0, w = (the_Target_of G).e0; reconsider u,w as Vertex of G by A10, FUNCT_2:5; A11: the_Vertices_of G = the_Vertices_of H by GLIB_000:def 33; set H1 = the addEdge of H, u, e0, w; A12: not e0 in the_Edges_of H by A8, XBOOLE_0:def 5; A13: k = G.size() - (H.size() + 1) by A7 .= G.size() - H1.size() by A12, A11, GLIB_006:111; A14: the_Vertices_of H1 = the_Vertices_of H by A11, GLIB_006:102; H1 is Subgraph of G proof for e being object holds e in the_Edges_of H1 implies e in the_Edges_of G proof let e be object; assume A15: e in the_Edges_of H1; the_Edges_of H1 = the_Edges_of H \/ {e0} by A11, A12, GLIB_006:def 11; then per cases by A15, XBOOLE_0:def 3; suppose e in the_Edges_of H; hence thesis; end; suppose e in {e0}; hence thesis by A10, TARSKI:def 1; end; end; then A16: the_Edges_of H1 c= the_Edges_of G by TARSKI:def 3; now let e be set; assume e in the_Edges_of H1; then A17: e in the_Edges_of H \/ {e0} by A11, A12, GLIB_006:def 11; per cases; suppose e <> e0; then not e in {e0} by TARSKI:def 1; then A18: e in the_Edges_of H by A17, XBOOLE_0:def 3; thus (the_Source_of H1).e = (the_Source_of H).e by A18 , GLIB_006:def 9 .= (the_Source_of G).e by A18, GLIB_000:def 32; thus (the_Target_of H1).e = (the_Target_of H).e by A18 , GLIB_006:def 9 .= (the_Target_of G).e by A18, GLIB_000:def 32; end; suppose A19: e = e0; then e DJoins u,w,H1 by A11, A12, GLIB_006:105; hence (the_Source_of H1).e = (the_Source_of G).e & (the_Target_of H1).e = (the_Target_of G).e by A19, GLIB_000:def 14; end; end; hence thesis by A14, A16, GLIB_000:def 32; end; then H1 is spanning Subgraph of G by A11, A14, GLIB_000:def 33; then consider p being non empty finite Graph-yielding FinSequence such that A20: p.1 == H1 & p.len p = G & len p = G.size() - H1.size() + 1 and A21: for n being Element of dom p st n <= len p - 1 holds ex v1,v2 being Vertex of G, e being object st p.(n+1) is addEdge of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n by A6, A13; reconsider q = <* H *> ^ p as non empty finite Graph-yielding FinSequence; take q; A22: q.1 = H by FINSEQ_1:41; thus q.1 == H by FINSEQ_1:41; A23: len q = len <* H *> + len p by FINSEQ_1:22 .= len p + 1 by FINSEQ_1:40; len p in dom p by FINSEQ_5:6; hence q.len q = G by A20, A23, FINSEQ_3:103; A24: H.size() + 1 - 1 = H1.size() - 1 by A12, A11, GLIB_006:111; thus len q = G.size() - H.size() + 1 by A20, A23, A24; let n be Element of dom q; assume n <= len q - 1; then per cases by Lm13; suppose A25: n = 1; take u,w,e0; 1 in dom p by FINSEQ_5:6; then A26: p.1 = q.(len <* H *> +1) by FINSEQ_1:def 7 .= q.(n+1) by A25, FINSEQ_1:40; H1 is addEdge of q.n, u, e0, w by A25, FINSEQ_1:41; hence q.(n+1) is addEdge of q.n, u, e0, w by A26, A20, GLIB_006:101; thus e0 in the_Edges_of G \ the_Edges_of q.n by A10, A12, A22, A25, XBOOLE_0:def 5; thus u in the_Vertices_of q.n & w in the_Vertices_of q.n by A11, A22, A25; end; suppose A27: n-1 in dom p & n-1 <= len p - 1; then reconsider m = n-1 as Element of dom p; consider v1,v2 being Vertex of G, e being object such that A28: p.(m+1) is addEdge of p.m,v1,e,v2 and A29: e in the_Edges_of G \ the_Edges_of p.m & v1 in the_Vertices_of p.m & v2 in the_Vertices_of p.m by A21, A27; take v1,v2,e; A30: 1 <= n by FINSEQ_3:25; n <= len p by A27, XREAL_1:9; then n in dom p by A30, FINSEQ_3:25; then A31: p.n = q.(len <* H *> + n) by FINSEQ_1:def 7 .= q.(n+1) by FINSEQ_1:40; p.m = q.(len <* H *> + m) by FINSEQ_1:def 7 .= q.(m+1) by FINSEQ_1:40 .= q.n; hence thesis by A28, A29, A31; end; end; A32: for k being Nat holds P[k] from NAT_1:sch 2(A1,A5); let H be spanning Subgraph of G; G.size() - H.size() is Nat by GLIB_000:75, NAT_1:21; hence thesis by A32; end; theorem for G being finite _Graph ex p being non empty finite Graph-yielding FinSequence st p.1 is edgeless & p.len p = G & len p = G.size() + 1 & for n being Element of dom p st n <= len p - 1 holds ex v1,v2 being Vertex of G, e being object st p.(n+1) is addEdge of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n proof let G be finite _Graph; set H = the edgeless spanning Subgraph of G; consider p being non empty finite Graph-yielding FinSequence such that A1: p.1 == H & p.len p = G & len p = G.size() - H.size() + 1 and A2: for n being Element of dom p st n <= len p - 1 holds ex v1,v2 being Vertex of G, e being object st p.(n+1) is addEdge of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n by Th64; take p; thus p.1 is edgeless & p.len p = G by A1, Th52; thus len p = G.size() - 0 + 1 by A1, Th49 .= G.size() + 1; thus thesis by A2; end; theorem Th66: for G being finite connected _Graph, H being spanning connected Subgraph of G ex p being non empty finite connected Graph-yielding FinSequence st p.1 == H & p.len p = G & len p = G.size() - H.size() + 1 & for n being Element of dom p st n <= len p - 1 holds ex v1,v2 being Vertex of G, e being object st p.(n+1) is addEdge of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n proof let G be finite connected _Graph, H be spanning connected Subgraph of G; consider p being non empty finite Graph-yielding FinSequence such that A1: p.1 == H & p.len p = G & len p = G.size() - H.size() + 1 and A2: for n being Element of dom p st n <= len p - 1 holds ex v1,v2 being Vertex of G, e being object st p.(n+1) is addEdge of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n by Th64; defpred P[Nat] means for n being Element of dom p st $1 = n holds p.n is connected; A3: P[1] by A1, GLIB_002:8; A4: for k being non zero Nat st P[k] holds P[k+1] proof let k be non zero Nat; assume A5: P[k]; let m be Element of dom p; assume A6: k+1 = m; then A7: k+1 <= len p by FINSEQ_3:25; then A8: k+1-1 <= len p - 0 by XREAL_1:13; 1 <= k by NAT_1:14; then reconsider n = k as Element of dom p by A8, FINSEQ_3:25; k+1-1 <= len p - 1 by A7, XREAL_1:9; then consider v1,v2 being Vertex of G, e being object such that A9: p.(n+1) is addEdge of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n by A2; p.n is connected by A5; hence thesis by A6, A9; end; A10: for k being non zero Nat holds P[k] from NAT_1:sch 10(A3,A4); for x being Element of dom p holds p.x is connected proof let x be Element of dom p; x is non zero Nat by FINSEQ_3:25; hence thesis by A10; end; then reconsider p as non empty finite connected Graph-yielding FinSequence by GLIB_002:def 15; take p; thus thesis by A1, A2; end; theorem Th67: for G1 being finite _Graph, H being Subgraph of G1 ex G2 being spanning Subgraph of G1, p being non empty finite Graph-yielding FinSequence st H.size() = G2.size() & p.1 == H & p.len p = G2 & len p = G1.order() - H.order() + 1 & for n being Element of dom p st n <= len p - 1 holds ex v being Vertex of G1 st p.(n+1) is addVertex of p.n, v & not v in the_Vertices_of p.n proof let G1 be finite _Graph, H be Subgraph of G1; set V = the_Vertices_of G1 \ the_Vertices_of H; set G2 = the addVertices of H, V; consider p being non empty Graph-yielding FinSequence such that A1: p.1 == H & p.len p = G2 & len p = card (V \ the_Vertices_of H) + 1 and A2: for n being Element of dom p st n <= len p - 1 holds ex v being Vertex of G2 st p.(n+1) is addVertex of p.n, v & not v in the_Vertices_of p.n by Th59; defpred P[Nat] means for n being Element of dom p st $1 = n holds p.n is finite; A3: P[1] by A1, GLIB_000:89; A4: for k being non zero Nat st P[k] holds P[k+1] proof let k be non zero Nat; assume A5: P[k]; let m be Element of dom p; assume A6: k+1 = m; then A7: k+1 <= len p by FINSEQ_3:25; then A8: k+1-1 <= len p - 0 by XREAL_1:13; 1 <= k by NAT_1:14; then reconsider n = k as Element of dom p by A8, FINSEQ_3:25; k+1-1 <= len p - 1 by A7, XREAL_1:9; then consider v being Vertex of G2 such that A9: p.(n+1) is addVertex of p.n, v & not v in the_Vertices_of p.n by A2; p.n is finite by A5; hence thesis by A6, A9; end; A10: for k being non zero Nat holds P[k] from NAT_1:sch 10(A3,A4); for x being Element of dom p holds p.x is finite proof let x be Element of dom p; x is non zero Nat by FINSEQ_3:25; hence thesis by A10; end; then reconsider p as non empty finite Graph-yielding FinSequence by GLIB_000:def 66; A11: the_Vertices_of G2 = the_Vertices_of H \/ (the_Vertices_of G1 \ the_Vertices_of H) by GLIB_006:def 10 .= the_Vertices_of G1 by XBOOLE_1:45; G2 is Subgraph of G1 proof the_Edges_of G2 = the_Edges_of H & the_Source_of G2 = the_Source_of H & the_Target_of G2 = the_Target_of H by GLIB_006:def 10; then the_Edges_of G2 c= the_Edges_of G1 & for e being set st e in the_Edges_of G2 holds (the_Source_of G2).e = (the_Source_of G1).e & (the_Target_of G2).e = (the_Target_of G1).e by GLIB_000:def 32; hence thesis by A11, GLIB_000:def 32; end; then reconsider G2 as spanning Subgraph of G1 by A11, GLIB_000:def 33; take G2, p; thus H.size() = G2.size() by GLIB_006:95; thus p.1 == H & p.len p = G2 by A1; V \ the_Vertices_of H = the_Vertices_of G1 \ (the_Vertices_of H \/ the_Vertices_of H) by XBOOLE_1:41 .= the_Vertices_of G1 \ the_Vertices_of H; hence len p = card (the_Vertices_of G1) - card(the_Vertices_of H) + 1 by A1, CARD_2:44 .= G1.order() - card(the_Vertices_of H) + 1 by GLIB_000:def 24 .= G1.order() - H.order() + 1 by GLIB_000:def 24; let n be Element of dom p; assume n <= len p - 1; then consider v being Vertex of G2 such that A12: p.(n+1) is addVertex of p.n, v & not v in the_Vertices_of p.n by A2; the_Vertices_of G2 c= the_Vertices_of G1; then reconsider v as Vertex of G1 by TARSKI:def 3; take v; thus thesis by A12; end; theorem Th68: for G being finite _Graph, H being Subgraph of G ex p being non empty finite Graph-yielding FinSequence st p.1 == H & p.len p = G & len p = G.order() + G.size() - (H.order() + H.size()) + 1 & for n being Element of dom p st n <= len p - 1 holds (ex v1,v2 being Vertex of G, e being object st p.(n+1) is addEdge of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n) or (ex v being Vertex of G st p.(n+1) is addVertex of p.n, v & not v in the_Vertices_of p.n) proof let G be finite _Graph, H be Subgraph of G; per cases; suppose A1: G.size() <> H.size(); consider G2 being spanning Subgraph of G, p being non empty finite Graph-yielding FinSequence such that A2: H.size() = G2.size() and A3: p.1 == H & p.len p = G2 & len p = G.order() - H.order() + 1 and A4:for n being Element of dom p st n <= len p - 1 holds ex v being Vertex of G st p.(n+1) is addVertex of p.n, v & not v in the_Vertices_of p.n by Th67; consider q being non empty finite Graph-yielding FinSequence such that A5: q.1 == G2 & q.len q = G & len q = G.size() - G2.size() + 1 and A6: for n being Element of dom q st n <= len q - 1 holds ex v1,v2 being Vertex of G, e being object st q.(n+1) is addEdge of q.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of q.n & v1 in the_Vertices_of q.n & v2 in the_Vertices_of q.n by Th64; reconsider r = p ^' q as non empty finite Graph-yielding FinSequence; take r; 1 <= len p by FINSEQ_1:20; hence r.1 == H by A3, FINSEQ_6:140; A7: 1 < len q proof assume not 1 < len q; then len q <= 1 & 1 <= len q by FINSEQ_1:20; then len q = 1 by XXREAL_0:1; hence contradiction by A1, A2, A5; end; hence r.len r = G by A5, FINSEQ_6:142; len r + 1 = len p + len q by FINSEQ_6:139 .= G.order() + G.size() - (H.order() + H.size()) + 1 + 1 by A2, A3, A5; hence len r = G.order() + G.size() - (H.order() + H.size()) + 1; let n be Element of dom r; assume A8: n <= len r - 1; n < len p or n = len p or n > len p by XXREAL_0:1; then n+1 <= len p or n = len p or n > len p by INT_1:7; then n+1-1 <= len p - 1 or n = len p or n > len p by XREAL_1:9; then per cases; suppose A9: n <= len p - 1; then A10: n+0 <= len p - 1+1 & n+1 <= len p -1+1 by XREAL_1:7; A11: 1 <= n by FINSEQ_3:25; then reconsider m = n as Element of dom p by A10, FINSEQ_3:25; A12: r.n = p.m by A10, A11, FINSEQ_6:140; 1+0 <= n+1 by XREAL_1:7; then r.(n+1) = p.(m+1) by A10, FINSEQ_6:140; hence thesis by A4, A9, A12; end; suppose A13: n = len p; then 1 <= n & n <= len p by FINSEQ_1:20; then A14: r.n = G2 by A3, A13, FINSEQ_6:140; reconsider m = 1 as Element of dom q by A7, FINSEQ_3:25; A15: r.(n+1) = q.(m+1) by A13, A7, FINSEQ_6:141; now m+1 <= len q by A7, INT_1:7; then m+1-1 <= len q - 1 by XREAL_1:9; then consider v1,v2 being Vertex of G, e being object such that A16: q.(m+1) is addEdge of q.m,v1,e,v2 & e in the_Edges_of G \ the_Edges_of q.m & v1 in the_Vertices_of q.m & v2 in the_Vertices_of q.m by A6; take v1,v2,e; thus r.(n+1) is addEdge of r.n,v1,e,v2 by A14, A15, A5, A16, Th36; thus e in the_Edges_of G \ the_Edges_of r.n & v1 in the_Vertices_of r.n & v2 in the_Vertices_of r.n by A14, A16, A5, GLIB_000:def 34; end; hence thesis; end; suppose A17: n > len p; then reconsider n1 = n-1 as Nat by NAT_1:20; n1+1 = n; then len p <= n1 by A17, NAT_1:13; then reconsider k1 = n1 - len p as Nat by NAT_1:21; set k = k1+1; A18: k+1 < len q proof assume len q <= k + 1; then len q + len p <= k + 1 + len p by XREAL_1:6; then len r + 1 <= n + 1 by FINSEQ_6:139; then len r - 1 <= n - 1 by XREAL_1:6, XREAL_1:9; then n <= n-1 by A8, XXREAL_0:2; then n-n <= n-1-n by XREAL_1:9; then 0 <= -1; hence contradiction; end; then k+1-1 < len q - 0 by XREAL_1:14; then r.(len p + k) = q.(k+1) by NAT_1:14, FINSEQ_6:141; then A19: r.n = q.(k+1); 1+0 <= k+1 by XREAL_1:7; then r.(len p +(k+1)) = q.((k+1)+1) by A18, FINSEQ_6:141; then A20: r.(n+1) = q.((k+1)+1); 1+0 <= k+1 & k+1 <= len q by A18, XREAL_1:7; then reconsider m=k+1 as Element of dom q by FINSEQ_3:25; now m+1 <= len q by A18, INT_1:7; then m+1-1 <= len q - 1 by XREAL_1:9; then consider v1,v2 being Vertex of G, e being object such that A21: q.(m+1) is addEdge of q.m,v1,e,v2 & e in the_Edges_of G \ the_Edges_of q.m & v1 in the_Vertices_of q.m & v2 in the_Vertices_of q.m by A6; take v1,v2,e; thus r.(n+1) is addEdge of r.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of r.n & v1 in the_Vertices_of r.n & v2 in the_Vertices_of r.n by A19, A20, A21; end; hence thesis; end; end; suppose A22: G.size() = H.size(); then consider p being non empty finite Graph-yielding FinSequence such that A23: p.1 == H & p.len p = G & len p = G.order() - H.order() + 1 and A24: for n being Element of dom p st n <= len p - 1 holds ex v being Vertex of G st p.(n+1) is addVertex of p.n, v & not v in the_Vertices_of p.n by Th60; take p; thus p.1 == H & p.len p = G by A23; thus len p = G.order() + G.size() - (H.order() + H.size()) + 1 by A22, A23; let n be Element of dom p; assume n <= len p - 1; hence thesis by A24; end; end; :: finite graphs can be finitely constructed with addVertex and addEdge only theorem Th69: for G being finite _Graph ex p being non empty finite Graph-yielding FinSequence st p.1 is trivial edgeless & p.len p = G & len p = G.order() + G.size() & for n being Element of dom p st n <= len p - 1 holds (ex v1,v2 being Vertex of G, e being object st p.(n+1) is addEdge of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n) or (ex v being Vertex of G st p.(n+1) is addVertex of p.n, v & not v in the_Vertices_of p.n) proof let G be finite _Graph; set H = the trivial edgeless Subgraph of G; consider p being non empty finite Graph-yielding FinSequence such that A1: p.1 == H & p.len p = G and A2: len p = G.order() + G.size() - (H.order() + H.size()) + 1 and A3: for n being Element of dom p st n <= len p - 1 holds (ex v1,v2 being Vertex of G, e being object st p.(n+1) is addEdge of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n) or (ex v being Vertex of G st p.(n+1) is addVertex of p.n, v & not v in the_Vertices_of p.n) by Th68; take p; thus p.1 is trivial edgeless by A1, Th52, GLIB_000:89; thus p.len p = G by A1; thus len p = G.order() + G.size() - (H.order() + 0) + 1 by A2, Th49 .= G.order() + G.size() - 1 + 1 by GLIB_000:26 .= G.order() + G.size(); thus thesis by A3; end; scheme FinGraphs { P[finite _Graph] } : for G being finite _Graph holds P[G] provided A1: for G being trivial edgeless _Graph holds P[G] and A2: for G2 being finite _Graph, v being object for G1 being addVertex of G2,v st not v in the_Vertices_of G2 & P[G2] holds P[G1] and A3: for G2 being finite _Graph, v1, v2 being Vertex of G2, e being object for G1 being addEdge of G2,v1,e,v2 st not e in the_Edges_of G2 & P[G2] holds P[G1] proof let G be finite _Graph; consider p being non empty finite Graph-yielding FinSequence such that A4: p.1 is trivial edgeless & p.len p = G & len p = G.order() + G.size() and A5: for n being Element of dom p st n <= len p - 1 holds (ex v1,v2 being Vertex of G, e being object st p.(n+1) is addEdge of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n) or (ex v being Vertex of G st p.(n+1) is addVertex of p.n, v & not v in the_Vertices_of p.n) by Th69; defpred Q[Nat] means $1 <= len p implies ex k being Element of dom p st $1 = k & P[p.k]; A6: Q[1] proof assume 1 <= len p; then reconsider k = 1 as Element of dom p by FINSEQ_3:25; take k; thus 1 = k; thus P[p.k] by A1, A4; end; A7: for m being non zero Nat st Q[m] holds Q[m+1] proof let m be non zero Nat; assume A8: Q[m]; assume A9: m+1 <= len p; 0+1 <= m+1 by XREAL_1:6; then reconsider k = m+1 as Element of dom p by A9, FINSEQ_3:25; take k; thus m+1 = k; m+1-1 <= len p - 0 by A9, XREAL_1:13; then consider k0 being Element of dom p such that A10: m = k0 & P[p.k0] by A8; m+1-1 <= len p - 1 by A9, XREAL_1:9; then per cases by A5, A10; suppose ex v1,v2 being Vertex of G, e being object st p.(k0+1) is addEdge of p.k0,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.k0 & v1 in the_Vertices_of p.k0 & v2 in the_Vertices_of p.k0; then consider v1,v2 being Vertex of G, e being object such that A11: p.(k0+1) is addEdge of p.k0,v1,e,v2 and A12: e in the_Edges_of G \ the_Edges_of p.k0 & v1 in the_Vertices_of p.k0 & v2 in the_Vertices_of p.k0; reconsider v1, v2 as Vertex of p.k0 by A12; A13: not e in the_Edges_of p.k0 by A12, XBOOLE_0:def 5; A14: p.k is addEdge of p.k0,v1,e,v2 by A10, A11; thus thesis by A3, A10, A13, A14; end; suppose ex v being Vertex of G st p.(k0+1) is addVertex of p.k0, v & not v in the_Vertices_of p.k0; then consider v being Vertex of G such that A15: p.(k0+1) is addVertex of p.k0, v & not v in the_Vertices_of p.k0; thus thesis by A2, A10, A15; end; end; A16: for m being non zero Nat holds Q[m] from NAT_1:sch 10(A6,A7); consider k being Element of dom p such that A17: len p = k & P[p.k] by A16; thus thesis by A4, A17; end; theorem for p being non empty Graph-yielding FinSequence st p.1 is finite & for n being Element of dom p st n <= len p - 1 holds (ex v being object st p.(n+1) is addVertex of p.n,v) or (ex v1,e,v2 being object st p.(n+1) is addEdge of p.n,v1,e,v2) holds p.len p is finite proof let p be non empty Graph-yielding FinSequence; assume that A1: p.1 is finite and A2: for n being Element of dom p st n <= len p - 1 holds (ex v being object st p.(n+1) is addVertex of p.n,v) or (ex v1,e,v2 being object st p.(n+1) is addEdge of p.n,v1,e,v2); defpred Q[Nat] means $1 <= len p implies ex k being Element of dom p st $1 = k & p.k is finite; A3: Q[1] proof assume 1 <= len p; then reconsider k = 1 as Element of dom p by FINSEQ_3:25; take k; thus thesis by A1; end; A4: for m being non zero Nat st Q[m] holds Q[m+1] proof let m be non zero Nat; assume A5: Q[m]; assume A6: m+1 <= len p; 0+1 <= m+1 by XREAL_1:6; then reconsider k = m+1 as Element of dom p by A6, FINSEQ_3:25; take k; thus m+1 = k; m+1-1 <= len p - 0 by A6, XREAL_1:13; then consider k0 being Element of dom p such that A7: m = k0 & p.k0 is finite by A5; m+1-1 <= len p - 1 by A6, XREAL_1:9; then per cases by A2, A7; suppose ex v being object st p.(k0+1) is addVertex of p.k0,v; hence thesis by A7; end; suppose ex v1,e,v2 being object st p.(k0+1) is addEdge of p.k0,v1,e,v2; hence thesis by A7; end; end; A8: for m being non zero Nat holds Q[m] from NAT_1:sch 10(A3,A4); consider k being Element of dom p such that A9: len p = k & p.k is finite by A8; thus thesis by A9; end; theorem Th71: for G being finite Tree-like _Graph, H being connected Subgraph of G ex p being non empty finite Tree-like Graph-yielding FinSequence st p.1 == H & p.len p = G & len p = G.order() - H.order() + 1 & for n being Element of dom p st n <= len p - 1 ex v1,v2 being Vertex of G, e being object st p.(n+1) is addAdjVertex of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & ((v1 in the_Vertices_of p.n & not v2 in the_Vertices_of p.n) or (not v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n)) proof defpred P[Nat] means for G being finite Tree-like _Graph for H being connected Subgraph of G st $1 = G.order() - H.order() ex p being non empty finite Tree-like Graph-yielding FinSequence st p.1 == H & p.len p = G & len p = G.order() - H.order() + 1 & for n being Element of dom p st n <= len p - 1 ex v1,v2 being Vertex of G, e being object st p.(n+1) is addAdjVertex of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & ((v1 in the_Vertices_of p.n & not v2 in the_Vertices_of p.n) or (not v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n)); A1: P[0] proof let G be finite Tree-like _Graph, H be connected Subgraph of G; assume A2: 0 = G.order() - H.order(); then H is spanning by Th11; then A3: G == H by Th21; reconsider p = <* G *> as non empty finite Tree-like Graph-yielding FinSequence; take p; thus p.1 == H by A3, FINSEQ_1:40; thus p.len p = p.1 by FINSEQ_1:40 .= G by FINSEQ_1:40; thus len p = G.order() - H.order() + 1 by A2, FINSEQ_1:40; let n be Element of dom p; 1 <= n & n <= len p by FINSEQ_3:25; then 1 <= n & n <= 1 by FINSEQ_1:40; then A4: n = 1 by XXREAL_0:1; assume n <= len p - 1; then n <= 1 - 1 by FINSEQ_1:40; hence thesis by A4; :: by contradiction end; A5: for k being Nat st P[k] holds P[k+1] proof let k be Nat; assume A6: P[k]; let G be finite Tree-like _Graph, H be connected Subgraph of G; assume A7: k+1 = G.order() - H.order(); then A8: G.order() = H.order() + k + 1; G.order() <> H.order() proof assume G.order() = H.order(); then 0 + H.order() = k+1 + H.order() by A8; hence contradiction; end; then A9: H is non spanning by Th11; G.order() <> 1 by A8, NAT_1:7; then A10: G is non trivial by GLIB_000:26; then consider v being Vertex of G such that A11: v is endvertex & not v in the_Vertices_of H by A9, Th31; consider e0 being object such that A12: v.edgesInOut() = {e0} & not e0 Joins v,v,G by A11, GLIB_000:def 51; set G2 = the removeVertex of G, v; A13: G2.order() + 1 = G.order() by A10, GLIB_000:48; then A14: k = G2.order() - H.order() by A7; A15: the_Edges_of G2 = G.edgesBetween(the_Vertices_of G \ {v}) by A10, GLIB_000:47 .= G.edgesBetween(the_Vertices_of G) \ v.edgesInOut() by Th1 .= the_Edges_of G \ {e0} by A12, GLIB_000:34; A16: H is Subgraph of G2 proof the_Vertices_of H misses {v} by A11, ZFMISC_1:50; then the_Vertices_of H c= the_Vertices_of G \ {v} by XBOOLE_1:86; then A17: the_Vertices_of H c= the_Vertices_of G2 by A10, GLIB_000:47; not e0 in the_Edges_of H proof assume A18: e0 in the_Edges_of H; e0 in v.edgesInOut() by A12, TARSKI:def 1; then per cases by GLIB_000:61; suppose (the_Source_of G).e0 = v; then (the_Source_of H).e0 = v by A18, GLIB_000:def 32; hence contradiction by A11, A18, FUNCT_2:5; end; suppose (the_Target_of G).e0 = v; then (the_Target_of H).e0 = v by A18, GLIB_000:def 32; hence contradiction by A11, A18, FUNCT_2:5; end; end; then the_Edges_of H misses {e0} by ZFMISC_1:50; hence thesis by A15, A17, GLIB_000:44, XBOOLE_1:86; end; consider p being non empty finite Tree-like Graph-yielding FinSequence such that A19: p.1 == H & p.len p = G2 & len p = G2.order() - H.order() + 1 and A20: for n being Element of dom p st n <= len p - 1 ex v1,v2 being Vertex of G2, e being object st p.(n+1) is addAdjVertex of p.n,v1,e,v2 & e in the_Edges_of G2 \ the_Edges_of p.n & ((v1 in the_Vertices_of p.n & not v2 in the_Vertices_of p.n) or (not v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n)) by A6, A14, A16, A10, A11; reconsider q = p ^ <* G *> as non empty finite Tree-like Graph-yielding FinSequence; take q; 1 in dom p by FINSEQ_5:6; hence q.1 == H by A19, FINSEQ_1:def 7; A21: len q = len p + len <* G *> by FINSEQ_1:22 .= len p + 1 by FINSEQ_1:40; hence A22: q.len q = G by FINSEQ_1:42; thus len q = G.order() - H.order() + 1 by A13, A19, A21; let n be Element of dom q; assume n <= len q - 1; then per cases by Lm12; suppose A23: n = len q - 1; then A24: q.(n+1) = G by A22; 1 <= n by FINSEQ_3:25; then n in dom p by A21, A23, FINSEQ_3:25; then A27: q.n = G2 by A19, A21, A23, FINSEQ_1:def 7; A28: e0 in v.edgesInOut() by A12, TARSKI:def 1; then A29: e0 in the_Edges_of G; e0 in {e0} by TARSKI:def 1; then A30: not e0 in the_Edges_of q.n by A15, A27, XBOOLE_0:def 5; v in {v} by TARSKI:def 1; then not v in the_Vertices_of G \ {v} by XBOOLE_0:def 5; then A31: not v in the_Vertices_of q.n by A10, A27, GLIB_000:47; v.adj(e0) <> v by A12, A28, GLIB_000:67; then not v.adj(e0) in {v} by TARSKI:def 1; then v.adj(e0) in the_Vertices_of G \ {v} by XBOOLE_0:def 5; then A32: v.adj(e0) in the_Vertices_of q.n by A10, A27, GLIB_000:47; per cases by A10, A12, Th38; suppose A33: G is addAdjVertex of G2, v.adj(e0), e0, v; take v.adj(e0), v, e0; thus q.(n+1) is addAdjVertex of q.n, v.adj(e0), e0, v by A24, A27, A33; thus thesis by A29, A30, A31, A32, XBOOLE_0:def 5; end; suppose A34: G is addAdjVertex of G2, v, e0, v.adj(e0); take v, v.adj(e0), e0; thus q.(n+1) is addAdjVertex of q.n, v, e0, v.adj(e0) by A24, A27, A34; thus thesis by A29, A30, A31, A32, XBOOLE_0:def 5; end; end; suppose A35: n <= len p - 1; then A36: n+0 <= len p - 1 + 1 by XREAL_1:7; 1 <= n by FINSEQ_3:25; then reconsider m = n as Element of dom p by A36, FINSEQ_3:25; consider v1,v2 being Vertex of G2, e being object such that A37: p.(m+1) is addAdjVertex of p.m,v1,e,v2 & e in the_Edges_of G2 \ the_Edges_of p.m & ((v1 in the_Vertices_of p.m & not v2 in the_Vertices_of p.m) or (not v1 in the_Vertices_of p.m & v2 in the_Vertices_of p.m)) by A20, A35; 1+0 <= n+1 & n+1 <= len p - 1 + 1 by A35, XREAL_1:6; then n+1 in dom p by FINSEQ_3:25; then A38: q.(n+1) = p.(m+1) by FINSEQ_1:def 7; A39: q.n = p.m by FINSEQ_1:def 7; the_Vertices_of G2 c= the_Vertices_of G; then reconsider v1,v2 as Vertex of G by TARSKI:def 3; take v1,v2,e; the_Edges_of G2 \ the_Edges_of p.m c= the_Edges_of G \ the_Edges_of p.m by XBOOLE_1:33; hence thesis by A37, A38, A39; end; end; A40: for k being Nat holds P[k] from NAT_1:sch 2(A1,A5); let G be finite Tree-like _Graph; let H be connected Subgraph of G; G.order() - H.order() is Nat by GLIB_000:75, NAT_1:21; hence thesis by A40; end; :: finite trees can be finitely constructed with addAdjVertex only theorem Th72: for G being finite Tree-like _Graph ex p being non empty finite Tree-like Graph-yielding FinSequence st p.1 is trivial edgeless & p.len p = G & len p = G.order() & for n being Element of dom p st n <= len p - 1 ex v1,v2 being Vertex of G, e being object st p.(n+1) is addAdjVertex of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & ((v1 in the_Vertices_of p.n & not v2 in the_Vertices_of p.n) or (not v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n)) proof let G be finite Tree-like _Graph; set H = the trivial Subgraph of G; consider p being non empty finite Tree-like Graph-yielding FinSequence such that A1: p.1 == H & p.len p = G & len p = G.order() - H.order() +1 and A2: for n being Element of dom p st n <= len p - 1 ex v1,v2 being Vertex of G, e being object st p.(n+1) is addAdjVertex of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & ((v1 in the_Vertices_of p.n & not v2 in the_Vertices_of p.n) or (not v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n)) by Th71; take p; thus p.1 is trivial edgeless by A1, Th52, GLIB_000:89; thus p.len p = G by A1; thus len p = G.order() - 1 + 1 by A1, GLIB_000:26 .= G.order(); thus thesis by A2; end; scheme FinTrees { P[finite _Graph] } : for G being finite Tree-like _Graph holds P[G] provided A1: for G being trivial edgeless _Graph holds P[G] and A2: for G2 being finite Tree-like _Graph for v being Vertex of G2, e, w being object st not e in the_Edges_of G2 & not w in the_Vertices_of G2 & P[G2] holds (for G1 being addAdjVertex of G2,v,e,w holds P[G1]) & (for G1 being addAdjVertex of G2,w,e,v holds P[G1]) proof let G be finite Tree-like _Graph; consider p being non empty finite Tree-like Graph-yielding FinSequence such that A3: p.1 is trivial edgeless & p.len p = G & len p = G.order() and A4: for n being Element of dom p st n <= len p - 1 ex v1,v2 being Vertex of G, e being object st p.(n+1) is addAdjVertex of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & ((v1 in the_Vertices_of p.n & not v2 in the_Vertices_of p.n) or (not v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n)) by Th72; defpred Q[Nat] means $1 <= len p implies ex k being Element of dom p st $1 = k & P[p.k]; A5: Q[1] proof assume 1 <= len p; then reconsider k = 1 as Element of dom p by FINSEQ_3:25; take k; thus 1 = k; thus P[p.k] by A1, A3; end; A6: for m being non zero Nat st Q[m] holds Q[m+1] proof let m be non zero Nat; assume A7: Q[m]; assume A8: m+1 <= len p; 0+1 <= m+1 by XREAL_1:6; then reconsider k = m+1 as Element of dom p by A8, FINSEQ_3:25; take k; thus m+1 = k; m+1-1 <= len p - 0 by A8, XREAL_1:13; then consider k0 being Element of dom p such that A9: m = k0 & P[p.k0] by A7; m+1-1 <= len p - 1 by A8, XREAL_1:9; then consider v1,v2 being Vertex of G, e being object such that A10: p.(k0+1) is addAdjVertex of p.k0,v1,e,v2 and A11: e in the_Edges_of G \ the_Edges_of p.k0 and A12: (v1 in the_Vertices_of p.k0 & not v2 in the_Vertices_of p.k0) or (not v1 in the_Vertices_of p.k0 & v2 in the_Vertices_of p.k0) by A4, A9; A13: not e in the_Edges_of p.k0 by A11, XBOOLE_0:def 5; per cases by A12; suppose v1 in the_Vertices_of p.k0 & not v2 in the_Vertices_of p.k0; hence thesis by A2, A9, A10, A13; end; suppose not v1 in the_Vertices_of p.k0 & v2 in the_Vertices_of p.k0; hence thesis by A2, A9, A10, A13; end; end; A14: for m being non zero Nat holds Q[m] from NAT_1:sch 10(A5,A6); consider k being Element of dom p such that A15: len p = k & P[p.k] by A14; thus thesis by A3, A15; end; theorem for p being non empty Graph-yielding FinSequence st p.1 is Tree-like & for n being Element of dom p st n <= len p - 1 ex v1,e,v2 being object st p.(n+1) is addAdjVertex of p.n,v1,e,v2 holds p.len p is Tree-like proof let p be non empty Graph-yielding FinSequence; assume that A1: p.1 is Tree-like and A2: for n being Element of dom p st n <= len p - 1 holds ex v1,e,v2 being object st p.(n+1) is addAdjVertex of p.n,v1,e,v2; defpred Q[Nat] means $1 <= len p implies ex k being Element of dom p st $1 = k & p.k is Tree-like; A3: Q[1] proof assume 1 <= len p; then reconsider k = 1 as Element of dom p by FINSEQ_3:25; take k; thus thesis by A1; end; A4: for m being non zero Nat st Q[m] holds Q[m+1] proof let m be non zero Nat; assume A5: Q[m]; assume A6: m+1 <= len p; 0+1 <= m+1 by XREAL_1:6; then reconsider k = m+1 as Element of dom p by A6, FINSEQ_3:25; take k; thus m+1 = k; m+1-1 <= len p - 0 by A6, XREAL_1:13; then consider k0 being Element of dom p such that A7: m = k0 & p.k0 is Tree-like by A5; m+1-1 <= len p - 1 by A6, XREAL_1:9; then consider v1,e,v2 being object such that A8: p.(k0+1) is addAdjVertex of p.k0,v1,e,v2 by A2, A7; thus thesis by A7, A8; end; A9: for m being non zero Nat holds Q[m] from NAT_1:sch 10(A3,A4); consider k being Element of dom p such that A10: len p = k & p.k is Tree-like by A9; thus thesis by A10; end; :: finite connected graphs can be constructed with addAdjVertex and addEdge theorem Th74: for G being finite connected _Graph ex p being non empty finite connected Graph-yielding FinSequence st p.1 is trivial edgeless & p.len p = G & len p = G.size() + 1 & for n being Element of dom p st n <= len p - 1 holds (ex v1,v2 being Vertex of G, e being object st p.(n+1) is addAdjVertex of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & ((v1 in the_Vertices_of p.n & not v2 in the_Vertices_of p.n) or (not v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n)) ) or (ex v1,v2 being Vertex of G, e being object st p.(n+1) is addEdge of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n) proof let G be finite connected _Graph; per cases; suppose A1: G is non acyclic; set H = the spanning Tree-like Subgraph of G; consider p being non empty finite Tree-like Graph-yielding FinSequence such that A2: p.1 is trivial edgeless & p.len p = H & len p = H.order() and A3: for n being Element of dom p st n <= len p - 1 ex v1,v2 being Vertex of H, e being object st p.(n+1) is addAdjVertex of p.n,v1,e,v2 & e in the_Edges_of H \ the_Edges_of p.n & ((v1 in the_Vertices_of p.n & not v2 in the_Vertices_of p.n) or (not v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n)) by Th72; consider q being non empty finite connected Graph-yielding FinSequence such that A4: q.1 == H & q.len q = G & len q = G.size() - H.size() + 1 and A5: for n being Element of dom q st n <= len q - 1 holds ex v1,v2 being Vertex of G, e being object st q.(n+1) is addEdge of q.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of q.n & v1 in the_Vertices_of q.n & v2 in the_Vertices_of q.n by Th66; reconsider r = p ^' q as non empty finite connected Graph-yielding FinSequence; take r;1 <= len p by FINSEQ_1:20; hence r.1 is trivial edgeless by A2, FINSEQ_6:140; A6: 1 < len q proof assume not 1 < len q; then len q <= 1 & 1 <= len q by FINSEQ_1:20; then G == H by A4, XXREAL_0:1; hence contradiction by A1, GLIB_002:44; end; hence r.len r = G by A4, FINSEQ_6:142; len r + 1 = len p + len q by FINSEQ_6:139 .= H.order() + G.size() - H.size() + 1 by A2, A4 .= H.size() + 1 + G.size() - H.size() + 1 by GLIB_002:46 .= G.size() + 1 + 1; hence len r = G.size() + 1; let n be Element of dom r; assume A7: n <= len r - 1; n < len p or n = len p or n > len p by XXREAL_0:1; then n+1 <= len p or n = len p or n > len p by INT_1:7; then n+1-1 <= len p - 1 or n = len p or n > len p by XREAL_1:9; then per cases; suppose A8: n <= len p - 1; then A9: n+0 <= len p - 1+1 & n+1 <= len p -1+1 by XREAL_1:7; A10: 1 <= n by FINSEQ_3:25; then reconsider m = n as Element of dom p by A9, FINSEQ_3:25; A11: r.n = p.m by A9, A10, FINSEQ_6:140; 1+0 <= n+1 by XREAL_1:7; then A12: r.(n+1) = p.(m+1) by A9, FINSEQ_6:140; now consider v1,v2 being Vertex of H, e being object such that A13: p.(m+1) is addAdjVertex of p.m,v1,e,v2 & e in the_Edges_of H \ the_Edges_of p.m & ((v1 in the_Vertices_of p.m & not v2 in the_Vertices_of p.m) or (not v1 in the_Vertices_of p.m & v2 in the_Vertices_of p.m)) by A3, A8; the_Vertices_of H c= the_Vertices_of G; then reconsider v1, v2 as Vertex of G by TARSKI:def 3; take v1,v2,e; the_Edges_of H \ the_Edges_of p.m c= the_Edges_of G \ the_Edges_of p.m by XBOOLE_1:33; hence r.(n+1) is addAdjVertex of r.n, v1,e,v2 & e in the_Edges_of G \ the_Edges_of r.n & ((v1 in the_Vertices_of r.n & not v2 in the_Vertices_of r.n) or (not v1 in the_Vertices_of r.n & v2 in the_Vertices_of r.n)) by A11, A12, A13; end; hence thesis; end; suppose A14: n = len p; then 1 <= n & n <= len p by FINSEQ_1:20; then A15: r.n = H by A2, A14, FINSEQ_6:140; reconsider m = 1 as Element of dom q by A6, FINSEQ_3:25; A16: r.(n+1) = q.(m+1) by A14, A6, FINSEQ_6:141; now m+1 <= len q by A6, INT_1:7; then m+1-1 <= len q - 1 by XREAL_1:9; then consider v1,v2 being Vertex of G, e being object such that A17: q.(m+1) is addEdge of q.m,v1,e,v2 & e in the_Edges_of G \ the_Edges_of q.m & v1 in the_Vertices_of q.m & v2 in the_Vertices_of q.m by A5; take v1,v2,e; thus r.(n+1) is addEdge of r.n,v1,e,v2 by A4, A15, A16, A17, Th36; thus e in the_Edges_of G \ the_Edges_of r.n & v1 in the_Vertices_of r.n & v2 in the_Vertices_of r.n by A4, A15, A17, GLIB_000:def 34; end; hence thesis; end; suppose A18: n > len p; then reconsider n1 = n-1 as Nat by NAT_1:20; n1+1 = n; then len p <= n1 by A18, NAT_1:13; then reconsider k1 = n1 - len p as Nat by NAT_1:21; set k = k1+1; A19: k+1 < len q proof assume len q <= k + 1; then len q + len p <= k + 1 + len p by XREAL_1:6; then len r + 1 <= n + 1 by FINSEQ_6:139; then len r - 1 <= n - 1 by XREAL_1:6, XREAL_1:9; then n <= n-1 by A7, XXREAL_0:2; then n-n <= n-1-n by XREAL_1:9; then 0 <= -1; hence contradiction; end; then k+1-1 < len q - 0 by XREAL_1:14; then r.(len p + k) = q.(k+1) by NAT_1:14, FINSEQ_6:141; then A20: r.n = q.(k+1); 1+0 <= k+1 by XREAL_1:7; then r.(len p +(k+1)) = q.((k+1)+1) by A19, FINSEQ_6:141; then A21: r.(n+1) = q.((k+1)+1); 1+0 <= k+1 & k+1 <= len q by A19, XREAL_1:7; then reconsider m=k+1 as Element of dom q by FINSEQ_3:25; now m+1 <= len q by A19, INT_1:7; then m+1-1 <= len q - 1 by XREAL_1:9; then consider v1,v2 being Vertex of G, e being object such that A22: q.(m+1) is addEdge of q.m,v1,e,v2 & e in the_Edges_of G \ the_Edges_of q.m & v1 in the_Vertices_of q.m & v2 in the_Vertices_of q.m by A5; take v1,v2,e; thus r.(n+1) is addEdge of r.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of r.n & v1 in the_Vertices_of r.n & v2 in the_Vertices_of r.n by A20, A21, A22; end; hence thesis; end; end; suppose A23: G is acyclic; then consider p being non empty finite Tree-like Graph-yielding FinSequence such that A24: p.1 is trivial edgeless & p.len p = G & len p = G.order() and A25: for n being Element of dom p st n <= len p - 1 ex v1,v2 being Vertex of G, e being object st p.(n+1) is addAdjVertex of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & ((v1 in the_Vertices_of p.n & not v2 in the_Vertices_of p.n) or (not v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n)) by Th72; take p; thus thesis by A24, A25, A23, GLIB_002:46; end; end; scheme FinConnectedGraphs { P[finite _Graph] } : for G being finite connected _Graph holds P[G] provided A1: for G being trivial edgeless _Graph holds P[G] and A2: for G2 being finite connected _Graph for v being Vertex of G2, e, w being object st not e in the_Edges_of G2 & not w in the_Vertices_of G2 & P[G2] holds (for G1 being addAdjVertex of G2,v,e,w holds P[G1]) & (for G1 being addAdjVertex of G2,w,e,v holds P[G1]) and A3: for G2 being finite connected _Graph for v1,v2 being Vertex of G2, e being object for G1 being addEdge of G2,v1,e,v2 st not e in the_Edges_of G2 & P[G2] holds P[G1] proof let G be finite connected _Graph; consider p being non empty finite connected Graph-yielding FinSequence such that A4: p.1 is trivial edgeless & p.len p = G & len p = G.size() + 1 and A5: for n being Element of dom p st n <= len p - 1 holds (ex v1,v2 being Vertex of G, e being object st p.(n+1) is addAdjVertex of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & ((v1 in the_Vertices_of p.n & not v2 in the_Vertices_of p.n) or (not v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n)) ) or (ex v1,v2 being Vertex of G, e being object st p.(n+1) is addEdge of p.n,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.n & v1 in the_Vertices_of p.n & v2 in the_Vertices_of p.n) by Th74; defpred Q[Nat] means $1 <= len p implies ex k being Element of dom p st $1 = k & P[p.k]; A6: Q[1] proof assume 1 <= len p; then reconsider k = 1 as Element of dom p by FINSEQ_3:25; take k; thus 1 = k; thus P[p.k] by A1, A4; end; A7: for m being non zero Nat st Q[m] holds Q[m+1] proof let m be non zero Nat; assume A8: Q[m]; assume A9: m+1 <= len p; 0+1 <= m+1 by XREAL_1:6; then reconsider k = m+1 as Element of dom p by A9, FINSEQ_3:25; take k; thus m+1 = k; m+1-1 <= len p - 0 by A9, XREAL_1:13; then consider k0 being Element of dom p such that A10: m = k0 & P[p.k0] by A8; m+1-1 <= len p - 1 by A9, XREAL_1:9; then per cases by A5, A10; suppose ex v1,v2 being Vertex of G, e being object st p.(k0+1) is addAdjVertex of p.k0,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.k0 & ((v1 in the_Vertices_of p.k0 & not v2 in the_Vertices_of p.k0) or (not v1 in the_Vertices_of p.k0 & v2 in the_Vertices_of p.k0)); then consider v1,v2 being Vertex of G, e being object such that A11: p.(k0+1) is addAdjVertex of p.k0,v1,e,v2 and A12: e in the_Edges_of G \ the_Edges_of p.k0 and A13: (v1 in the_Vertices_of p.k0 & not v2 in the_Vertices_of p.k0) or (not v1 in the_Vertices_of p.k0 & v2 in the_Vertices_of p.k0); A14: not e in the_Edges_of p.k0 by A12, XBOOLE_0:def 5; per cases by A13; suppose v1 in the_Vertices_of p.k0 & not v2 in the_Vertices_of p.k0; hence thesis by A2, A10, A11, A14; end; suppose not v1 in the_Vertices_of p.k0 & v2 in the_Vertices_of p.k0; hence thesis by A2, A10, A11, A14; end; end; suppose ex v1,v2 being Vertex of G, e being object st p.(k0+1) is addEdge of p.k0,v1,e,v2 & e in the_Edges_of G \ the_Edges_of p.k0 & v1 in the_Vertices_of p.k0 & v2 in the_Vertices_of p.k0; then consider v1,v2 being Vertex of G, e being object such that A15: p.(k0+1) is addEdge of p.k0,v1,e,v2 and A16: e in the_Edges_of G \ the_Edges_of p.k0 & v1 in the_Vertices_of p.k0 & v2 in the_Vertices_of p.k0; reconsider v1, v2 as Vertex of p.k0 by A16; A17: not e in the_Edges_of p.k0 by A16, XBOOLE_0:def 5; p.k is addEdge of p.k0,v1,e,v2 by A10, A15; hence thesis by A3, A10, A17; end; end; for m being non zero Nat holds Q[m] from NAT_1:sch 10(A6,A7); then ex k being Element of dom p st len p = k & P[p.k]; hence thesis by A4; end; theorem for p being non empty Graph-yielding FinSequence st p.1 is connected & for n being Element of dom p st n <= len p - 1 ex v1,e,v2 being object st p.(n+1) is addAdjVertex of p.n,v1,e,v2 or p.(n+1) is addEdge of p.n,v1,e,v2 holds p.len p is connected proof let p be non empty Graph-yielding FinSequence; assume that A1: p.1 is connected and A2: for n being Element of dom p st n <= len p - 1 holds ex v1,e,v2 being object st p.(n+1) is addAdjVertex of p.n,v1,e,v2 or p.(n+1) is addEdge of p.n,v1,e,v2; defpred Q[Nat] means $1 <= len p implies ex k being Element of dom p st $1 = k & p.k is connected; A3: Q[1] proof assume 1 <= len p; then reconsider k = 1 as Element of dom p by FINSEQ_3:25; take k; thus thesis by A1; end; A4: for m being non zero Nat st Q[m] holds Q[m+1] proof let m be non zero Nat; assume A5: Q[m]; assume A6: m+1 <= len p; 0+1 <= m+1 by XREAL_1:6; then reconsider k = m+1 as Element of dom p by A6, FINSEQ_3:25; take k; thus m+1 = k; m+1-1 <= len p - 0 by A6, XREAL_1:13; then consider k0 being Element of dom p such that A7: m = k0 & p.k0 is connected by A5; m+1-1 <= len p - 1 by A6, XREAL_1:9; then consider v1,e,v2 being object such that A8: p.(k0+1) is addAdjVertex of p.k0,v1,e,v2 or p.(k0+1) is addEdge of p.k0,v1,e,v2 by A2, A7; thus thesis by A7, A8; end; for m being non zero Nat holds Q[m] from NAT_1:sch 10(A3,A4); then ex k being Element of dom p st len p = k & p.k is connected; hence thesis; end; theorem Th76: for G2 being _Graph, v being object for V1 being set, V2 being finite set for G1 being addAdjVertexAll of G2, v, V1 \/ V2 st V1 \/ V2 c= the_Vertices_of G2 & not v in the_Vertices_of G2 & V1 misses V2 ex p being non empty Graph-yielding FinSequence st p.1 = G2 & p.len p = G1 & len p = card V2 + 2 & p.2 is addAdjVertexAll of G2, v, V1 & for n being Element of dom p st 2 <= n & n <= len p - 1 holds ex w being Vertex of G2, e being object st e in the_Edges_of G1 \ the_Edges_of p.n & (p.(n+1) is addEdge of p.n,v,e,w or p.(n+1) is addEdge of p.n,w,e,v) proof let G2 be _Graph, v be object, V1 be set; defpred P[Nat] means for V2 being finite set for G1 being addAdjVertexAll of G2, v, V1 \/ V2 st V1 \/ V2 c= the_Vertices_of G2 & not v in the_Vertices_of G2 & V1 misses V2 & card V2 = $1 ex p being non empty Graph-yielding FinSequence st p.1 = G2 & p.len p = G1 & len p = card V2 + 2 & p.2 is addAdjVertexAll of G2, v, V1 & for n being Element of dom p st 2 <= n & n <= len p - 1 holds ex w being Vertex of G2, e being object st e in the_Edges_of G1 \ the_Edges_of p.n & (p.(n+1) is addEdge of p.n,v,e,w or p.(n+1) is addEdge of p.n,w,e,v); A1: P[0] proof let V2 be finite set; let G1 be addAdjVertexAll of G2, v, V1 \/ V2; assume that V1 \/ V2 c= the_Vertices_of G2 & not v in the_Vertices_of G2 and A2: V1 misses V2 & card V2 = 0; reconsider p = <* G2, G1 *> as non empty Graph-yielding FinSequence; take p; thus p.1 = G2 by FINSEQ_1:44; thus p.len p = p.2 by FINSEQ_1:44 .= G1 by FINSEQ_1:44; thus len p = card V2 + 2 by A2, FINSEQ_1:44; V2 = {} by A2; hence p.2 is addAdjVertexAll of G2, v, V1 by FINSEQ_1:44; let n be Element of dom p; assume 2 <= n & n <= len p - 1; then 2 <= n & n <= 2 - 1 by FINSEQ_1:44; hence thesis by XXREAL_0:2; :: by contradiction end; A3: for k being Nat st P[k] holds P[k+1] proof let k be Nat; assume A4: P[k]; let V2 be finite set, G1 be addAdjVertexAll of G2, v, V1 \/ V2; assume that A5: V1 \/ V2 c= the_Vertices_of G2 & not v in the_Vertices_of G2 and A6: V1 misses V2 & card V2 = k+1; set v0 = the Element of V2; set V3 = V2 \ {v0}; A7: V2 is non empty by A6; A8: V2 = V3 \/ {v0} by A7, ZFMISC_1:116; V1 \/ V3 c= V1 \/ V2 by XBOOLE_1:9; then A9: V1 \/ V3 c= the_Vertices_of G2 by A5, XBOOLE_1:1; A10: V1 misses V3 by A6, XBOOLE_1:64; v0 in {v0} by TARSKI:def 1; then A11: not v0 in V3 by XBOOLE_0:def 5; then A12: card V2 = card V3 + 1 by A8, CARD_2:41; then A13: card V3 = k by A6; v0 in V1 \/ V2 by A7, XBOOLE_1:7, TARSKI:def 3; then reconsider v0 as Vertex of G2 by A5; V1 /\ V2 = {} by A6, XBOOLE_0:def 7; then not v0 in V1 by A7, XBOOLE_0:def 4; then A14: not v0 in V1 \/ V3 by A11, XBOOLE_0:def 3; G1 is addAdjVertexAll of G2,v,(V1 \/ V3) \/ {v0} by A8, XBOOLE_1:4; then consider G3 being addAdjVertexAll of G2,v,V1 \/ V3, e being object such that A15: not e in the_Edges_of G3 and A16: G1 is addEdge of G3,v,e,v0 or G1 is addEdge of G3,v0,e,v by A5, A9, A14, Lm9; consider p being non empty Graph-yielding FinSequence such that A17: p.1 = G2 & p.len p = G3 & len p = card V3 + 2 & p.2 is addAdjVertexAll of G2, v, V1 and A18: for n being Element of dom p st 2 <= n & n <= len p - 1 holds ex w being Vertex of G2, e being object st e in the_Edges_of G3 \ the_Edges_of p.n & (p.(n+1) is addEdge of p.n,v,e,w or p.(n+1) is addEdge of p.n,w,e,v) by A4, A5, A9, A10, A13; reconsider q = p ^ <* G1 *> as non empty Graph-yielding FinSequence; take q; 1 in dom p by FINSEQ_5:6; hence q.1 = G2 by A17, FINSEQ_1:def 7; A19: len q = len p + len <* G1 *> by FINSEQ_1:22 .= len p + 1 by FINSEQ_1:40; hence A20: q.len q = G1 by FINSEQ_1:42; thus len q = card V2 + 2 by A12, A17, A19; 2 + card V3 - card V3 <= len p - 0 by A17, XREAL_1:13; then 2 in dom p by FINSEQ_3:25; hence q.2 is addAdjVertexAll of G2, v, V1 by A17, FINSEQ_1:def 7; let n be Element of dom q; assume A21: 2 <= n & n <= len q - 1; then per cases by Lm12; suppose A22: n = len q - 1; then A23: q.(n+1) = G1 by A20; 1 <= n by FINSEQ_3:25; then n in dom p by A19, A22, FINSEQ_3:25; then A26: q.n = G3 by A17, A19, A22, FINSEQ_1:def 7; v0 in the_Vertices_of G3 & v in the_Vertices_of G3 proof A27: v0 in the_Vertices_of G2; the_Vertices_of G2 c= the_Vertices_of G3 by GLIB_006:def 9; hence v0 in the_Vertices_of G3 by A27; v is Vertex of G3 by A5, A9, GLIB_007:50; hence v in the_Vertices_of G3; end; then e DJoins v,v0,G1 or e DJoins v0,v,G1 by A15, A16, GLIB_006:105; then A28: e in the_Edges_of G1 by GLIB_000:def 14; take v0, e; thus e in the_Edges_of G1 \ the_Edges_of q.n by A15, A26, A28, XBOOLE_0:def 5; thus q.(n+1) is addEdge of q.n,v,e,v0 or q.(n+1) is addEdge of q.n,v0,e,v by A16, A23, A26; end; suppose A29: n <= len p - 1; then A30: n+0 <= len p - 1 + 1 by XREAL_1:7; 1 <= n by FINSEQ_3:25; then reconsider m = n as Element of dom p by A30, FINSEQ_3:25; consider w being Vertex of G2, e being object such that A31: e in the_Edges_of G3 \ the_Edges_of p.m & (p.(m+1) is addEdge of p.m,v,e,w or p.(m+1) is addEdge of p.m,w,e,v) by A18, A21, A29; 1+0 <= n+1 & n+1 <= len p - 1 + 1 by A29, XREAL_1:6; then n+1 in dom p by FINSEQ_3:25; then A32: q.(n+1) = p.(m+1) by FINSEQ_1:def 7; A33: q.n = p.m by FINSEQ_1:def 7; the_Edges_of G3 c= the_Edges_of G1 by A16, GLIB_006:def 9; then A34: the_Edges_of G3 \ the_Edges_of p.m c= the_Edges_of G1 \ the_Edges_of p.m by XBOOLE_1:33; take w,e; thus thesis by A31, A32, A33, A34; end; end; for k being Nat holds P[k] from NAT_1:sch 2(A1,A3); hence thesis; end; theorem for G2 being _Graph, v being object for V being finite set, G1 being addAdjVertexAll of G2, v, V st V c= the_Vertices_of G2 & not v in the_Vertices_of G2 ex p being non empty Graph-yielding FinSequence st p.1 = G2 & p.len p = G1 & len p = card V + 2 & p.2 is addVertex of G2,v & for n being Element of dom p st 2 <= n & n <= len p - 1 holds ex w being Vertex of G2, e being object st e in the_Edges_of G1 \ the_Edges_of p.n & (p.(n+1) is addEdge of p.n,v,e,w or p.(n+1) is addEdge of p.n,w,e,v) proof let G2 be _Graph, v be object, V be finite set; let G1 be addAdjVertexAll of G2, v, V; assume A1: V c= the_Vertices_of G2 & not v in the_Vertices_of G2; V = V \/ {} & {} misses V by XBOOLE_1:65; then consider p being non empty Graph-yielding FinSequence such that A2: p.1 = G2 & p.len p = G1 & len p = card V + 2 and A3: p.2 is addAdjVertexAll of G2, v, {} and A4: for n being Element of dom p st 2 <= n & n <= len p - 1 holds ex w being Vertex of G2, e being object st e in the_Edges_of G1 \ the_Edges_of p.n & (p.(n+1) is addEdge of p.n,v,e,w or p.(n+1) is addEdge of p.n,w,e,v) by A1, Th76; take p; thus p.1 = G2 & p.len p = G1 & len p = card V + 2 by A2; thus p.2 is addVertex of G2, v by A3, GLIB_007:55; thus thesis by A4; end; theorem for G2 being _Graph, v being object for V being non empty finite set, G1 being addAdjVertexAll of G2, v, V st V c= the_Vertices_of G2 & not v in the_Vertices_of G2 ex p being non empty Graph-yielding FinSequence st p.1 = G2 & p.len p = G1 & len p = card V + 1 & (ex w being Vertex of G2, e being object st e in the_Edges_of G1 \ the_Edges_of G2 & (p.2 is addAdjVertex of G2,v,e,w or p.2 is addAdjVertex of G2,w,e,v)) & for n being Element of dom p st 2 <= n & n <= len p - 1 holds ex w being Vertex of G2, e being object st e in the_Edges_of G1 \ the_Edges_of p.n & (p.(n+1) is addEdge of p.n,v,e,w or p.(n+1) is addEdge of p.n,w,e,v) proof let G2 be _Graph, v be object, V be non empty finite set; let G1 be addAdjVertexAll of G2, v, V; assume A1: V c= the_Vertices_of G2 & not v in the_Vertices_of G2; set v0 = the Element of V; set V0 = V \ {v0}; v0 in {v0} by TARSKI:def 1; then not v0 in V0 by XBOOLE_0:def 5; then V = V0 \/ {v0} & {v0} misses V0 by ZFMISC_1:50, ZFMISC_1:116; then consider p being non empty Graph-yielding FinSequence such that A2: p.1 = G2 & p.len p = G1 & len p = card V0 + 2 and A3: p.2 is addAdjVertexAll of G2, v, {v0} and A4: for n being Element of dom p st 2 <= n & n <= len p - 1 holds ex w being Vertex of G2, e being object st e in the_Edges_of G1 \ the_Edges_of p.n & (p.(n+1) is addEdge of p.n,v,e,w or p.(n+1) is addEdge of p.n,w,e,v) by A1, Th76; take p; thus p.1 = G2 & p.len p = G1 by A2; thus len p = card V - card {v0} + 2 by A2, CARD_2:44 .= card V - 1 + 2 by CARD_1:30 .= card V + 1; hereby reconsider w = v0 as Vertex of G2 by A1, TARSKI:def 3; consider e being object such that A5: not e in the_Edges_of G2 and A6: p.2 is addAdjVertex of G2,v,e,w or p.2 is addAdjVertex of G2,w,e,v by A1, A3, GLIB_007:56; A7: e in the_Edges_of G1 proof defpred P[Nat] means $1 + 2 <= len p implies ex k being Element of dom p st k = $1 + 2 & e in the_Edges_of p.k; A8: P[0] proof assume 0 + 2 <= len p; card V0 + 2 - card V0 <= len p - 0 by A2, XREAL_1:10; then reconsider k = 2 as Element of dom p by FINSEQ_3:25; take k; thus k = 0 + 2; per cases by A6; suppose p.k is addAdjVertex of G2,v,e,w; then e Joins v,w,p.k by A1, A5, GLIB_006:132; hence e in the_Edges_of p.k by GLIB_000:def 13; end; suppose p.k is addAdjVertex of G2,w,e,v; then e Joins w,v,p.k by A1, A5, GLIB_006:131; hence e in the_Edges_of p.k by GLIB_000:def 13; end; end; A9: for m being Nat st P[m] holds P[m+1] proof let m be Nat; assume A10: P[m]; assume A11: m+1 + 2 <= len p; then m+1 + 2 - 1 <= len p - 0 by XREAL_1:13; then consider k0 being Element of dom p such that A12: k0 = m + 2 & e in the_Edges_of p.k0 by A10; 1+0 <= m+1+2 by XREAL_1:7; then reconsider k = m+1 + 2 as Element of dom p by A11, FINSEQ_3:25; take k; thus k = m+1 + 2; A13: 0+(1+1) <= m+(1+1) by XREAL_1:7; m+1 + 2 - 1 <= len p - 1 by A11, XREAL_1:9; then consider u being Vertex of G2, f being object such that f in the_Edges_of G1 \ the_Edges_of p.k0 and A14: (p.(k0+1) is addEdge of p.k0,v,f,u or p.(k0+1) is addEdge of p.k0,u,f,v) by A4, A12, A13; the_Edges_of p.k0 c= the_Edges_of p.k by A12, A14, GLIB_006:def 9; hence thesis by A12; end; A15: for m being Nat holds P[m] from NAT_1:sch 2(A8,A9); reconsider m = len p - 2 as Nat by A2; consider k being Element of dom p such that A16: k = m + 2 & e in the_Edges_of p.k by A15; thus thesis by A2, A16; end; take w,e; thus e in the_Edges_of G1 \ the_Edges_of G2 & (p.2 is addAdjVertex of G2,v,e,w or p.2 is addAdjVertex of G2,w,e,v) by A5, A6, A7, XBOOLE_0:def 5; end; thus thesis by A4; end; theorem Th79: for G being finite simple _Graph for W being set, H being inducedSubgraph of G, W ex p being non empty finite simple Graph-yielding FinSequence st p.1 == H & p.len p = G & len p = G.order() - H.order() + 1 & for n being Element of dom p st n <= len p - 1 ex v being object, V being finite set st v in the_Vertices_of G \ the_Vertices_of p.n & V c= the_Vertices_of p.n & p.(n+1) is addAdjVertexAll of p.n,v,V proof defpred P[Nat] means for G being finite simple _Graph for W being set, H being inducedSubgraph of G, W st G.order() - H.order() = $1 ex p being non empty finite simple Graph-yielding FinSequence st p.1 == H & p.len p = G & len p = G.order() - H.order() + 1 & for n being Element of dom p st n <= len p - 1 ex v being object, V being finite set st v in the_Vertices_of G \ the_Vertices_of p.n & V c= the_Vertices_of p.n & p.(n+1) is addAdjVertexAll of p.n,v,V; A1: P[0] proof let G be finite simple _Graph; let W be set, H be inducedSubgraph of G, W; assume A2: G.order() - H.order() = 0; then H is spanning by Th11; then A3: G == H by Th14; set p = <* G *>; take p; thus p.1 == H by A3, FINSEQ_1:40; thus p.len p = p.1 by FINSEQ_1:40 .= G by FINSEQ_1:40; thus len p = G.order() - H.order() + 1 by A2, FINSEQ_1:40; let n be Element of dom p; 1 <= n & n <= len p by FINSEQ_3:25; then 1 <= n & n <= 1 by FINSEQ_1:40; then A4: n = 1 by XXREAL_0:1; assume n <= len p - 1; then n <= 1 - 1 by FINSEQ_1:40; hence thesis by A4; :: by contradiction end; A5: for k being Nat st P[k] holds P[k+1] proof let k be Nat; assume A6: P[k]; let G be finite simple _Graph; let W be set, H be inducedSubgraph of G, W; assume A7: G.order() - H.order() = k+1; A8: the_Vertices_of G \ the_Vertices_of H <> {} proof assume the_Vertices_of G \ the_Vertices_of H = {}; then the_Vertices_of G c= the_Vertices_of H by XBOOLE_1:37; then A9: the_Vertices_of G = the_Vertices_of H by XBOOLE_0:def 10; G.order() = card the_Vertices_of G by GLIB_000:def 24 .= H.order() by A9, GLIB_000:def 24; then 0 = k+1 by A7; hence contradiction; end; set v0 = the Element of the_Vertices_of G \ the_Vertices_of H; v0 in the_Vertices_of G \ the_Vertices_of H by A8; then reconsider v0 as Vertex of G; set G2 = the removeVertex of G,v0; A10: the_Vertices_of G <> the_Vertices_of H by A8, XBOOLE_1:37; then A11: G is non trivial by Th15, GLIB_000:def 33; then A12: G2.order() + 1 = G.order() by GLIB_000:48; then A13: G2.order() - H.order() = k by A7; A14: the_Edges_of G2 = G.edgesBetween(the_Vertices_of G \ {v0}) by A11, GLIB_000:47 .= G.edgesBetween(the_Vertices_of G) \ v0.edgesInOut() by Th1 .= the_Edges_of G \ v0.edgesInOut() by GLIB_000:34; the_Vertices_of H c= the_Vertices_of G2 & the_Edges_of H c= the_Edges_of G2 proof A15: not v0 in the_Vertices_of H by A8, XBOOLE_0:def 5; the_Vertices_of H c= the_Vertices_of G \ {v0} by A15, ZFMISC_1:34; hence the_Vertices_of H c= the_Vertices_of G2 by A11, GLIB_000:47; the_Edges_of H /\ v0.edgesInOut() = {} proof assume A16: the_Edges_of H /\ v0.edgesInOut() <> {}; set e = the Element of the_Edges_of H /\ v0.edgesInOut(); A17: e in the_Edges_of H & e in v0.edgesInOut() by A16, XBOOLE_0:def 4; then per cases by GLIB_000:61; suppose (the_Source_of G).e = v0; then (the_Source_of H).e = v0 by A17, GLIB_000:def 32; hence contradiction by A15, A17, FUNCT_2:5; end; suppose (the_Target_of G).e = v0; then (the_Target_of H).e = v0 by A17, GLIB_000:def 32; hence contradiction by A15, A17, FUNCT_2:5; end; end; then the_Edges_of H = the_Edges_of H \ v0.edgesInOut() by XBOOLE_0:def 7, XBOOLE_1:83; hence thesis by A14, XBOOLE_1:33; end; then A18: H is Subgraph of G2 by GLIB_000:44; H is inducedSubgraph of G2, W proof H != G by A10, GLIB_000:def 34; then W is non empty Subset of the_Vertices_of G by GLIB_000:def 37; then A19: the_Vertices_of H = W & the_Edges_of H = G.edgesBetween(W) by GLIB_000:def 37; then A20: W is non empty Subset of the_Vertices_of G2 by A18, GLIB_000:def 32; for x being object st x in G.edgesBetween(W) holds x in G2.edgesBetween(W) proof let x be object; assume A21: x in G.edgesBetween(W); reconsider e = x as set by TARSKI:1; A22: (the_Source_of G).e in W & (the_Target_of G).e in W by A21, GLIB_000:31; A23: e in the_Edges_of G2 proof assume not e in the_Edges_of G2; then e in v0.edgesInOut() by A14, A21, XBOOLE_0:def 5; then per cases by GLIB_000:61; suppose (the_Source_of G).e = v0; hence contradiction by A8, A19, A22, XBOOLE_0:def 5; end; suppose (the_Target_of G).e = v0; hence contradiction by A8, A19, A22, XBOOLE_0:def 5; end; end; then (the_Source_of G2).e = (the_Source_of G).e & (the_Target_of G2).e = (the_Target_of G).e by GLIB_000:def 32; hence thesis by A22, A23, GLIB_000:31; end; then A24: G.edgesBetween(W) c= G2.edgesBetween(W) by TARSKI:def 3; G2.edgesBetween(W) c= G.edgesBetween(W) by GLIB_000:76; then G2.edgesBetween(W) = the_Edges_of H by A19, A24, XBOOLE_0:def 10; hence thesis by A18, A19, A20, GLIB_000:def 37; end; then consider p being non empty finite simple Graph-yielding FinSequence such that A25: p.1 == H & p.len p = G2 & len p = G2.order() - H.order() + 1 and A26: for n being Element of dom p st n <= len p - 1 ex v being object, V being finite set st v in the_Vertices_of G2 \ the_Vertices_of p.n & V c= the_Vertices_of p.n & p.(n+1) is addAdjVertexAll of p.n,v,V by A6, A13; set q = p ^ <* G *>; take q; 1 in dom p by FINSEQ_5:6; hence q.1 == H by A25, FINSEQ_1:def 7; A27: len q = len p + len <* G *> by FINSEQ_1:22 .= len p + 1 by FINSEQ_1:40; hence A28: q.len q = G by FINSEQ_1:42; thus len q = G.order() - H.order() + 1 by A12, A25, A27; let n be Element of dom q; assume n <= len q - 1; then per cases by Lm12; suppose A29: n = len q - 1; then A30: q.(n+1) = G by A28; A31: 1 <= n by FINSEQ_3:25; A32: n = len p by A27, A29; then n in dom p by A31, FINSEQ_3:25; then A33: q.n = G2 by A25, A32, FINSEQ_1:def 7; reconsider V = v0.allNeighbors() as finite set; take v0,V; v0 in {v0} by TARSKI:def 1; then not v0 in the_Vertices_of G \ {v0} by XBOOLE_0:def 5; then not v0 in the_Vertices_of G2 by A11, GLIB_000:47; hence v0 in the_Vertices_of G \ the_Vertices_of q.n by A33, XBOOLE_0:def 5; for x being object st x in V holds x in the_Vertices_of G2 proof let x be object; assume A34: x in V; then x <> v0 by Th6; then A35: not x in {v0} by TARSKI:def 1; x in the_Vertices_of G \ {v0} by A34, A35, XBOOLE_0:def 5; hence thesis by A11, GLIB_000:47; end; hence V c= the_Vertices_of q.n by A33, TARSKI:def 3; thus q.(n+1) is addAdjVertexAll of q.n,v0,V by A11, A30, A33, Th47; end; suppose A36: n <= len p - 1; then A37: n+0 <= len p - 1 + 1 by XREAL_1:7; 1 <= n by FINSEQ_3:25; then reconsider m = n as Element of dom p by A37, FINSEQ_3:25; consider v being object, V being finite set such that A38: v in the_Vertices_of G2 \ the_Vertices_of p.m & V c= the_Vertices_of p.m & p.(m+1) is addAdjVertexAll of p.m,v,V by A26, A36; 1+0 <= n+1 & n+1 <= len p - 1 + 1 by A36, XREAL_1:6; then n+1 in dom p by FINSEQ_3:25; then A39: q.(n+1) = p.(m+1) by FINSEQ_1:def 7; A40: q.n = p.m by FINSEQ_1:def 7; take v,V; the_Vertices_of G2 \ the_Vertices_of p.m c= the_Vertices_of G \ the_Vertices_of p.m by XBOOLE_1:33; hence v in the_Vertices_of G \ the_Vertices_of q.n by A38, A40; thus thesis by A38, A39, A40; end; end; A41: for k being Nat holds P[k] from NAT_1:sch 2(A1,A5); let G be finite simple _Graph; let W be set, H be inducedSubgraph of G, W; G.order() - H.order() is Nat by GLIB_000:75, NAT_1:21; hence thesis by A41; end; theorem Th80: for G being finite simple _Graph ex p being non empty finite simple Graph-yielding FinSequence st p.1 is trivial edgeless & p.len p = G & len p = G.order() & for n being Element of dom p st n <= len p - 1 ex v being object, V being finite set st v in the_Vertices_of G \ the_Vertices_of p.n & V c= the_Vertices_of p.n & p.(n+1) is addAdjVertexAll of p.n,v,V proof let G be finite simple _Graph; set v0 = the Vertex of G; set H = the inducedSubgraph of G,{v0}; consider p being non empty finite simple Graph-yielding FinSequence such that A1: p.1 == H & p.len p = G & len p = G.order() - H.order() + 1 and A2: for n being Element of dom p st n <= len p - 1 ex v being object, V being finite set st v in the_Vertices_of G \ the_Vertices_of p.n & V c= the_Vertices_of p.n & p.(n+1) is addAdjVertexAll of p.n,v,V by Th79; take p; thus p.1 is trivial edgeless by A1, Th52, GLIB_000:89; thus p.len p = G by A1; H.order() = 1 by GLIB_000:26; hence len p = G.order() by A1; thus thesis by A2; end; scheme FinSimpleGraphs { P[finite _Graph] } : for G being finite simple _Graph holds P[G] provided A1: for G being trivial edgeless _Graph holds P[G] and A2: for G2 being finite simple _Graph, v being object, V being finite set for G1 being addAdjVertexAll of G2,v,V st not v in the_Vertices_of G2 & V c= the_Vertices_of G2 & P[G2] holds P[G1] proof let G be finite simple _Graph; consider p being non empty finite simple Graph-yielding FinSequence such that A3: p.1 is trivial edgeless & p.len p = G & len p = G.order() and A4: for n being Element of dom p st n <= len p - 1 ex v being object, V being finite set st v in the_Vertices_of G \ the_Vertices_of p.n & V c= the_Vertices_of p.n & p.(n+1) is addAdjVertexAll of p.n,v,V by Th80; defpred Q[Nat] means $1 <= len p implies ex k being Element of dom p st $1 = k & P[p.k]; A5: Q[1] proof assume 1 <= len p; then reconsider k = 1 as Element of dom p by FINSEQ_3:25; take k; thus 1 = k; thus P[p.k] by A1, A3; end; A6: for m being non zero Nat st Q[m] holds Q[m+1] proof let m be non zero Nat; assume A7: Q[m]; assume A8: m+1 <= len p; 0+1 <= m+1 by XREAL_1:6; then reconsider k = m+1 as Element of dom p by A8, FINSEQ_3:25; take k; thus m+1 = k; m+1-1 <= len p - 0 by A8, XREAL_1:13; then consider k0 being Element of dom p such that A9: m = k0 & P[p.k0] by A7; m+1-1 <= len p - 1 by A8, XREAL_1:9; then consider v being object, V being finite set such that A10: v in the_Vertices_of G \ the_Vertices_of p.k0 and A11: V c= the_Vertices_of p.k0 & p.(k0+1) is addAdjVertexAll of p.k0,v,V by A4, A9; not v in the_Vertices_of p.k0 by A10, XBOOLE_0:def 5; hence thesis by A2, A9, A11; end; for m being non zero Nat holds Q[m] from NAT_1:sch 10(A5,A6); then ex k being Element of dom p st len p = k & P[p.k]; hence thesis by A3; end; theorem for p being non empty Graph-yielding FinSequence st p.1 is simple & for n being Element of dom p st n <= len p - 1 ex v being object, V being set st p.(n+1) is addAdjVertexAll of p.n,v,V holds p.len p is simple proof let p be non empty Graph-yielding FinSequence; assume that A1: p.1 is simple and A2: for n being Element of dom p st n <= len p - 1 holds ex v being object, V being set st p.(n+1) is addAdjVertexAll of p.n,v,V; defpred Q[Nat] means $1 <= len p implies ex k being Element of dom p st $1 = k & p.k is simple; A3: Q[1] proof assume 1 <= len p; then reconsider k = 1 as Element of dom p by FINSEQ_3:25; take k; thus thesis by A1; end; A4: for m being non zero Nat st Q[m] holds Q[m+1] proof let m be non zero Nat; assume A5: Q[m]; assume A6: m+1 <= len p; 0+1 <= m+1 by XREAL_1:6; then reconsider k = m+1 as Element of dom p by A6, FINSEQ_3:25; take k; thus m+1 = k; m+1-1 <= len p - 0 by A6, XREAL_1:13; then consider k0 being Element of dom p such that A7: m = k0 & p.k0 is simple by A5; m+1-1 <= len p - 1 by A6, XREAL_1:9; then consider v being object, V being set such that A8: p.(k0+1) is addAdjVertexAll of p.k0,v,V by A2, A7; thus thesis by A7, A8; end; for m being non zero Nat holds Q[m] from NAT_1:sch 10(A3,A4); then ex k being Element of dom p st len p = k & p.k is simple; hence thesis; end; theorem Th82: for G being finite simple connected _Graph ex p being non empty finite simple connected Graph-yielding FinSequence st p.1 is trivial edgeless & p.len p = G & len p = G.order() & for n being Element of dom p st n <= len p - 1 ex v being object, V being non empty finite set st v in the_Vertices_of G \ the_Vertices_of p.n & V c= the_Vertices_of p.n & p.(n+1) is addAdjVertexAll of p.n,v,V proof defpred P[Nat] means for G being finite simple connected _Graph st G.order() = $1 ex p being non empty finite simple connected Graph-yielding FinSequence st p.1 is trivial edgeless & p.len p = G & len p = G.order() & for n being Element of dom p st n <= len p - 1 ex v being object, V being non empty finite set st v in the_Vertices_of G \ the_Vertices_of p.n & V c= the_Vertices_of p.n & p.(n+1) is addAdjVertexAll of p.n,v,V; A1: P[1] proof let G be finite simple connected _Graph; assume A2: G.order() = 1; set p = <* G *>; take p; G is trivial by A2, GLIB_000:26; hence p.1 is trivial edgeless by FINSEQ_1:40; thus p.len p = p.1 by FINSEQ_1:40 .= G by FINSEQ_1:40; thus len p = G.order() by A2, FINSEQ_1:40; let n be Element of dom p; 1 <= n & n <= len p by FINSEQ_3:25; then 1 <= n & n <= 1 by FINSEQ_1:40; then A3: n = 1 by XXREAL_0:1; assume n <= len p - 1; then n <= 1 - 1 by FINSEQ_1:40; hence thesis by A3; :: by contradiction end; A4: for k being non zero Nat st P[k] holds P[k+1] proof let k be non zero Nat; assume A5: P[k]; let G be finite simple connected _Graph; assume A6: G.order() = k+1; A7: G is non trivial proof assume G is trivial; then G.order() = 0 + 1 by GLIB_000:26; hence contradiction by A6; end; then consider v9, v0 being Vertex of G such that v9 <> v0 & v9 is non cut-vertex and A8: v0 is non cut-vertex by GLIB_002:37; set G2 = the removeVertex of G,v0; A9: G2.order() + 1 = G.order() by A7, GLIB_000:48; then A10: G2.order() = k by A6; G2 is connected by A8, GLIB_002:36; then consider p being non empty finite simple connected Graph-yielding FinSequence such that A11: p.1 is trivial edgeless & p.len p = G2 & len p = G2.order() and A12: for n being Element of dom p st n <= len p - 1 ex v being object, V being non empty finite set st v in the_Vertices_of G2 \ the_Vertices_of p.n & V c= the_Vertices_of p.n & p.(n+1) is addAdjVertexAll of p.n,v,V by A5, A10; set q = p ^ <* G *>; take q; 1 in dom p by FINSEQ_5:6; hence q.1 is trivial edgeless by A11, FINSEQ_1:def 7; A13: len q = len p + len <* G *> by FINSEQ_1:22 .= len p + 1 by FINSEQ_1:40; hence A14: q.len q = G by FINSEQ_1:42; thus len q = G.order() by A9, A11, A13; let n be Element of dom q; assume n <= len q - 1; then per cases by Lm12; suppose A15: n = len q - 1; then A16: q.(n+1) = G by A14; A17: 1 <= n by FINSEQ_3:25; A18: n = len p by A13, A15; then n in dom p by A17, FINSEQ_3:25; then A19: q.n = G2 by A11, A18, FINSEQ_1:def 7; reconsider V = v0.allNeighbors() as non empty finite set by A7; take v0,V; v0 in {v0} by TARSKI:def 1; then not v0 in the_Vertices_of G \ {v0} by XBOOLE_0:def 5; then not v0 in the_Vertices_of G2 by A7, GLIB_000:47; hence v0 in the_Vertices_of G \ the_Vertices_of q.n by A19, XBOOLE_0:def 5; for x being object st x in V holds x in the_Vertices_of G2 proof let x be object; assume A20: x in V; then x <> v0 by Th6; then A21: not x in {v0} by TARSKI:def 1; x in the_Vertices_of G \ {v0} by A20, A21, XBOOLE_0:def 5; hence thesis by A7, GLIB_000:47; end; hence V c= the_Vertices_of q.n by A19, TARSKI:def 3; thus q.(n+1) is addAdjVertexAll of q.n,v0,V by A7, A16, A19, Th47; end; suppose A22: n <= len p - 1; then A23: n+0 <= len p - 1 + 1 by XREAL_1:7; 1 <= n by FINSEQ_3:25; then reconsider m = n as Element of dom p by A23, FINSEQ_3:25; consider v being object, V being non empty finite set such that A24: v in the_Vertices_of G2 \ the_Vertices_of p.m & V c= the_Vertices_of p.m & p.(m+1) is addAdjVertexAll of p.m,v,V by A12, A22; 1+0 <= n+1 & n+1 <= len p - 1 + 1 by A22, XREAL_1:6; then n+1 in dom p by FINSEQ_3:25; then A25: q.(n+1) = p.(m+1) by FINSEQ_1:def 7; A26: q.n = p.m by FINSEQ_1:def 7; take v,V; the_Vertices_of G2 \ the_Vertices_of p.m c= the_Vertices_of G \ the_Vertices_of p.m by XBOOLE_1:33; hence v in the_Vertices_of G \ the_Vertices_of q.n by A24, A26; thus thesis by A24, A25, A26; end; end; for k being non zero Nat holds P[k] from NAT_1:sch 10(A1,A4); hence thesis; end; scheme FinSimpleConnectedGraphs { P[finite _Graph] } : for G being finite simple connected _Graph holds P[G] provided A1: for G being trivial edgeless _Graph holds P[G] and A2: for G2 being finite simple connected _Graph for v being object, V being non empty finite set for G1 being addAdjVertexAll of G2,v,V st not v in the_Vertices_of G2 & V c= the_Vertices_of G2 & P[G2] holds P[G1] proof let G be finite simple connected _Graph; consider p being non empty finite simple connected Graph-yielding FinSequence such that A3: p.1 is trivial edgeless & p.len p = G & len p = G.order() and A4: for n being Element of dom p st n <= len p - 1 ex v being object, V being non empty finite set st v in the_Vertices_of G \ the_Vertices_of p.n & V c= the_Vertices_of p.n & p.(n+1) is addAdjVertexAll of p.n,v,V by Th82; defpred Q[Nat] means $1 <= len p implies ex k being Element of dom p st $1 = k & P[p.k]; A5: Q[1] proof assume 1 <= len p; then reconsider k = 1 as Element of dom p by FINSEQ_3:25; take k; thus 1 = k; thus P[p.k] by A1, A3; end; A6: for m being non zero Nat st Q[m] holds Q[m+1] proof let m be non zero Nat; assume A7: Q[m]; assume A8: m+1 <= len p; 0+1 <= m+1 by XREAL_1:6; then reconsider k = m+1 as Element of dom p by A8, FINSEQ_3:25; take k; thus m+1 = k; m+1-1 <= len p - 0 by A8, XREAL_1:13; then consider k0 being Element of dom p such that A9: m = k0 & P[p.k0] by A7; m+1-1 <= len p - 1 by A8, XREAL_1:9; then consider v being object, V being non empty finite set such that A10: v in the_Vertices_of G \ the_Vertices_of p.k0 and A11: V c= the_Vertices_of p.k0 & p.(k0+1) is addAdjVertexAll of p.k0,v,V by A4, A9; not v in the_Vertices_of p.k0 by A10, XBOOLE_0:def 5; hence thesis by A2, A9, A11; end; for m being non zero Nat holds Q[m] from NAT_1:sch 10(A5,A6); then ex k being Element of dom p st len p = k & P[p.k]; hence thesis by A3; end; theorem for p being non empty Graph-yielding FinSequence st p.1 is simple connected & for n being Element of dom p st n <= len p - 1 ex v being object, V being non empty set st p.(n+1) is addAdjVertexAll of p.n,v,V holds p.len p is simple connected proof let p be non empty Graph-yielding FinSequence; assume that A1: p.1 is simple connected and A2: for n being Element of dom p st n <= len p - 1 holds ex v being object, V being non empty set st p.(n+1) is addAdjVertexAll of p.n,v,V; defpred Q[Nat] means $1 <= len p implies ex k being Element of dom p st $1 = k & p.k is simple connected; A3: Q[1] proof assume 1 <= len p; then reconsider k = 1 as Element of dom p by FINSEQ_3:25; take k; thus thesis by A1; end; A4: for m being non zero Nat st Q[m] holds Q[m+1] proof let m be non zero Nat; assume A5: Q[m]; assume A6: m+1 <= len p; 0+1 <= m+1 by XREAL_1:6; then reconsider k = m+1 as Element of dom p by A6, FINSEQ_3:25; take k; thus m+1 = k; m+1-1 <= len p - 0 by A6, XREAL_1:13; then consider k0 being Element of dom p such that A7: m = k0 & p.k0 is simple connected by A5; m+1-1 <= len p - 1 by A6, XREAL_1:9; then consider v being object, V being non empty set such that A8: p.(k0+1) is addAdjVertexAll of p.k0,v,V by A2, A7; thus thesis by A7, A8; end; for m being non zero Nat holds Q[m] from NAT_1:sch 10(A3,A4); then ex k being Element of dom p st len p = k & p.k is simple connected; hence thesis; end;