:: CHORD semantic presentation begin theorem :: CHORD:1 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Set (Var "n")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1)) "is" ($#m1_hidden :::"Nat":::)) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) ")" )) ; theorem :: CHORD:2 (Bool "for" (Set (Var "n")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Set (Var "n")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1)) "is" ($#m1_hidden :::"Nat":::)) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) ")" )) ; theorem :: CHORD:3 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "m")))) "holds" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "m")) ($#k6_xcmplx_0 :::"-"::: ) (Num 2)))) ; theorem :: CHORD:4 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool (Set (Set (Var "m")) ($#k2_xcmplx_0 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) ; theorem :: CHORD:5 (Bool "for" (Set (Var "n")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_hidden :::"<>"::: ) (Set (Var "n")))) "holds" (Bool "ex" (Set (Var "m")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "st" (Bool (Set (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Var "n"))))) ; theorem :: CHORD:6 (Bool "for" (Set (Var "n")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Num 2))) "holds" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Num 1))) ; theorem :: CHORD:7 (Bool "for" (Set (Var "n")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "holds" (Bool "(" "not" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Num 4)) "or" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Num 1)) "or" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Num 3)) ")" )) ; theorem :: CHORD:8 (Bool "for" (Set (Var "n")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "holds" (Bool "(" "not" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Num 6)) "or" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Num 1)) "or" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Num 3)) "or" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Num 5)) ")" )) ; theorem :: CHORD:9 (Bool "for" (Set (Var "n")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "holds" (Bool "(" "not" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Num 8)) "or" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Num 1)) "or" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Num 3)) "or" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Num 5)) "or" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Num 7)) ")" )) ; theorem :: CHORD:10 (Bool "for" (Set (Var "n")) "being" ($#v1_abian :::"even"::: ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: CHORD:11 (Bool "for" (Set (Var "n")) "being" ($#v1_abian :::"even"::: ) ($#m1_hidden :::"Nat":::) "holds" (Bool "(" "not" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Num 3)) "or" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) "or" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Num 2)) ")" )) ; theorem :: CHORD:12 (Bool "for" (Set (Var "n")) "being" ($#v1_abian :::"even"::: ) ($#m1_hidden :::"Nat":::) "holds" (Bool "(" "not" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Num 5)) "or" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) "or" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Num 2)) "or" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Num 4)) ")" )) ; theorem :: CHORD:13 (Bool "for" (Set (Var "n")) "being" ($#v1_abian :::"even"::: ) ($#m1_hidden :::"Nat":::) "holds" (Bool "(" "not" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Num 7)) "or" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) "or" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Num 2)) "or" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Num 4)) "or" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Num 6)) ")" )) ; theorem :: CHORD:14 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))))) ; theorem :: CHORD:15 (Bool "for" (Set (Var "p")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "T")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")) ")" ) (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "T"))) & (Bool "(" "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "T")))) "holds" (Bool (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "y")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p")))) ")" ) ")" )))) ; definitionlet "p" be ($#m1_hidden :::"FinSequence":::); let "n" be ($#m1_hidden :::"Nat":::); func "p" :::".followSet"::: "n" -> ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) equals :: CHORD:def 1 (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" "(" "n" "," (Set "(" ($#k3_finseq_1 :::"len"::: ) "p" ")" ) ")" ($#k1_graph_2 :::"-cut"::: ) "p" ")" )); end; :: deftheorem defines :::".followSet"::: CHORD:def 1 : (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "p")) ($#k1_chord :::".followSet"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" "(" (Set (Var "n")) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ")" ($#k1_graph_2 :::"-cut"::: ) (Set (Var "p")) ")" ))))); theorem :: CHORD:16 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))) & (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p")))) & (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "p")) ($#k1_chord :::".followSet"::: ) (Set (Var "n")))) "iff" (Bool (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::">="::: ) (Set (Var "n"))) ")" )))) ; theorem :: CHORD:17 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q"))))) "holds" (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "p")) ($#k1_chord :::".followSet"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k1_chord :::".followSet"::: ) (Set (Var "n"))))))) ; theorem :: CHORD:18 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "f")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k15_finseq_1 :::"Seq"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k15_finseq_1 :::"Seq"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Var "f")))))) ; begin theorem :: CHORD:19 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")) ")" ) (Bool "for" (Set (Var "H")) "being" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "G")) "," (Set (Var "S")) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) & (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) "holds" (Bool "for" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "e")) ($#r1_glib_000 :::"Joins"::: ) (Set (Var "u")) "," (Set (Var "v")) "," (Set (Var "G")))) "holds" (Bool (Set (Var "e")) ($#r1_glib_000 :::"Joins"::: ) (Set (Var "u")) "," (Set (Var "v")) "," (Set (Var "H")))))))) ; theorem :: CHORD:20 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "holds" (Bool "(" (Bool (Set (Var "W")) "is" ($#v4_glib_001 :::"Trail-like"::: ) ) "iff" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set "(" (Set (Var "W")) ($#k14_glib_001 :::".edges()"::: ) ")" ) ")" ) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1))) ")" ))) ; theorem :: CHORD:21 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")) ")" ) (Bool "for" (Set (Var "H")) "being" ($#m2_glib_000 :::"removeVertices":::) "of" (Set (Var "G")) "," (Set (Var "S")) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "W"))))) "holds" (Bool "not" (Bool (Set (Set (Var "W")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) ")" )) "holds" (Bool (Set (Var "W")) "is" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "H"))))))) ; theorem :: CHORD:22 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b")))) "holds" (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Set (Var "W")) ($#k13_glib_001 :::".vertices()"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k2_tarski :::"}"::: ) ))) "holds" (Bool "ex" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Var "e")) ($#r1_glib_000 :::"Joins"::: ) (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "G"))))))) ; theorem :: CHORD:23 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")) ")" ) (Bool "for" (Set (Var "H")) "being" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "G")) "," (Set (Var "S")) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Set (Var "W")) ($#k13_glib_001 :::".vertices()"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "S")))) "holds" (Bool (Set (Var "W")) "is" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "H"))))))) ; theorem :: CHORD:24 (Bool "for" (Set (Var "G1")) "," (Set (Var "G2")) "being" ($#m1_hidden :::"_Graph":::) "st" (Bool (Bool (Set (Var "G1")) ($#r5_glib_000 :::"=="::: ) (Set (Var "G2")))) "holds" (Bool "for" (Set (Var "W1")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G1")) (Bool "for" (Set (Var "W2")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G2")) "st" (Bool (Bool (Set (Var "W1")) ($#r1_hidden :::"="::: ) (Set (Var "W2"))) & (Bool (Set (Var "W1")) "is" ($#v8_glib_001 :::"Cycle-like"::: ) )) "holds" (Bool (Set (Var "W2")) "is" ($#v8_glib_001 :::"Cycle-like"::: ) )))) ; theorem :: CHORD:25 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "P")) "being" ($#m3_glib_001 :::"Path":::) "of" (Set (Var "G")) (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "P")))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "P")))) & (Bool (Set (Set (Var "P")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "P")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")))) & (Bool (Bool "not" (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set (Var "n")))) & (Bool "not" (Bool "(" (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "P")))) ")" ))) "holds" (Bool "(" (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "P")))) & (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Num 1)) ")" )))) ; theorem :: CHORD:26 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "P")) "being" ($#m3_glib_001 :::"Path":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "P")) "is" ($#v1_glib_001 :::"open"::: ) )) "holds" (Bool "for" (Set (Var "a")) "," (Set (Var "e")) "," (Set (Var "b")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Bool "not" (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "P")) ($#k13_glib_001 :::".vertices()"::: ) ))) & (Bool (Set (Var "b")) ($#r1_hidden :::"="::: ) (Set (Set (Var "P")) ($#k3_glib_001 :::".first()"::: ) )) & (Bool (Set (Var "e")) ($#r1_glib_000 :::"Joins"::: ) (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "G")))) "holds" (Bool (Set (Set "(" (Set (Var "G")) ($#k2_glib_001 :::".walkOf"::: ) "(" (Set (Var "a")) "," (Set (Var "e")) "," (Set (Var "b")) ")" ")" ) ($#k7_glib_001 :::".append"::: ) (Set (Var "P"))) "is" ($#v5_glib_001 :::"Path-like"::: ) )))) ; theorem :: CHORD:27 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "P")) "," (Set (Var "H")) "being" ($#m3_glib_001 :::"Path":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Set (Var "P")) ($#k14_glib_001 :::".edges()"::: ) ) ($#r1_xboole_0 :::"misses"::: ) (Set (Set (Var "H")) ($#k14_glib_001 :::".edges()"::: ) )) & (Bool (Set (Var "P")) "is" ($#v1_glib_001 :::"open"::: ) ) & (Bool (Bool "not" (Set (Var "H")) "is" ($#v3_glib_001 :::"trivial"::: ) )) & (Bool (Set (Var "H")) "is" ($#v1_glib_001 :::"open"::: ) ) & (Bool (Set (Set "(" (Set (Var "P")) ($#k13_glib_001 :::".vertices()"::: ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set (Var "H")) ($#k13_glib_001 :::".vertices()"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set "(" (Set (Var "P")) ($#k3_glib_001 :::".first()"::: ) ")" ) "," (Set "(" (Set (Var "P")) ($#k4_glib_001 :::".last()"::: ) ")" ) ($#k7_domain_1 :::"}"::: ) )) & (Bool (Set (Set (Var "H")) ($#k3_glib_001 :::".first()"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "P")) ($#k4_glib_001 :::".last()"::: ) )) & (Bool (Set (Set (Var "H")) ($#k4_glib_001 :::".last()"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "P")) ($#k3_glib_001 :::".first()"::: ) ))) "holds" (Bool (Set (Set (Var "P")) ($#k7_glib_001 :::".append"::: ) (Set (Var "H"))) "is" ($#v8_glib_001 :::"Cycle-like"::: ) ))) ; theorem :: CHORD:28 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Set (Var "W1")) ($#k4_glib_001 :::".last()"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "W2")) ($#k3_glib_001 :::".first()"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "W1")) ($#k7_glib_001 :::".append"::: ) (Set (Var "W2")) ")" ) ($#k15_glib_001 :::".length()"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "W1")) ($#k15_glib_001 :::".length()"::: ) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" (Set (Var "W2")) ($#k15_glib_001 :::".length()"::: ) ")" ))))) ; theorem :: CHORD:29 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")) ")" ) "st" (Bool (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) "holds" (Bool "for" (Set (Var "H1")) "being" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "G")) "," (Set (Var "A")) (Bool "for" (Set (Var "H2")) "being" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "H1")) "," (Set (Var "B")) "holds" (Bool (Set (Var "H2")) "is" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "G")) "," (Set (Var "B"))))))) ; theorem :: CHORD:30 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")) ")" ) "st" (Bool (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) "holds" (Bool "for" (Set (Var "H1")) "being" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "G")) "," (Set (Var "A")) (Bool "for" (Set (Var "H2")) "being" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "G")) "," (Set (Var "B")) "holds" (Bool (Set (Var "H2")) "is" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "H1")) "," (Set (Var "B"))))))) ; theorem :: CHORD:31 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")) ")" ) "st" (Bool (Bool (Set (Var "T")) ($#r1_tarski :::"c="::: ) (Set (Var "S")))) "holds" (Bool "for" (Set (Var "G2")) "being" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "G")) "," (Set (Var "S")) "holds" (Bool (Set (Set (Var "G2")) ($#k21_glib_000 :::".edgesBetween"::: ) (Set (Var "T"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "G")) ($#k21_glib_000 :::".edgesBetween"::: ) (Set (Var "T"))))))) ; scheme :: CHORD:sch 1 FinGraphOrderCompInd{ P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool "for" (Set (Var "G")) "being" ($#v2_glib_000 :::"finite"::: ) ($#m1_hidden :::"_Graph":::) "holds" (Bool P1[(Set (Var "G"))])) provided (Bool "for" (Set (Var "k")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool "(" "for" (Set (Var "Gk")) "being" ($#v2_glib_000 :::"finite"::: ) ($#m1_hidden :::"_Graph":::) "st" (Bool (Bool (Set (Set (Var "Gk")) ($#k15_glib_000 :::".order()"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k")))) "holds" (Bool P1[(Set (Var "Gk"))]) ")" )) "holds" (Bool "for" (Set (Var "Gk1")) "being" ($#v2_glib_000 :::"finite"::: ) ($#m1_hidden :::"_Graph":::) "st" (Bool (Bool (Set (Set (Var "Gk1")) ($#k15_glib_000 :::".order()"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "k")))) "holds" (Bool P1[(Set (Var "Gk1"))]))) proof end; theorem :: CHORD:32 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "W")) "is" ($#v1_glib_001 :::"open"::: ) ) & (Bool (Set (Var "W")) "is" ($#v5_glib_001 :::"Path-like"::: ) )) "holds" (Bool (Set (Var "W")) "is" ($#v6_glib_001 :::"vertex-distinct"::: ) ))) ; theorem :: CHORD:33 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "P")) "being" ($#m3_glib_001 :::"Path":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "P")) "is" ($#v1_glib_001 :::"open"::: ) ) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "P"))) ($#r1_xxreal_0 :::">"::: ) (Num 3))) "holds" (Bool "for" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "e")) ($#r1_glib_000 :::"Joins"::: ) (Set (Set (Var "P")) ($#k4_glib_001 :::".last()"::: ) ) "," (Set (Set (Var "P")) ($#k3_glib_001 :::".first()"::: ) ) "," (Set (Var "G")))) "holds" (Bool (Set (Set (Var "P")) ($#k10_glib_001 :::".addEdge"::: ) (Set (Var "e"))) "is" ($#v8_glib_001 :::"Cycle-like"::: ) )))) ; begin definitionlet "G" be ($#m1_hidden :::"_Graph":::); let "W" be ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Const "G")); attr "W" is :::"minlength"::: means :: CHORD:def 2 (Bool "for" (Set (Var "W2")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" "G" "st" (Bool (Bool (Set (Var "W2")) ($#r1_glib_001 :::"is_Walk_from"::: ) (Set "W" ($#k3_glib_001 :::".first()"::: ) ) "," (Set "W" ($#k4_glib_001 :::".last()"::: ) ))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "W2"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "W"))); end; :: deftheorem defines :::"minlength"::: CHORD:def 2 : (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "holds" (Bool "(" (Bool (Set (Var "W")) "is" ($#v1_chord :::"minlength"::: ) ) "iff" (Bool "for" (Set (Var "W2")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "W2")) ($#r1_glib_001 :::"is_Walk_from"::: ) (Set (Set (Var "W")) ($#k3_glib_001 :::".first()"::: ) ) "," (Set (Set (Var "W")) ($#k4_glib_001 :::".last()"::: ) ))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "W2"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "W"))))) ")" ))); theorem :: CHORD:34 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) (Bool "for" (Set (Var "S")) "being" ($#m4_glib_001 :::"Subwalk"::: ) "of" (Set (Var "W")) "st" (Bool (Bool (Set (Set (Var "S")) ($#k3_glib_001 :::".first()"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "W")) ($#k3_glib_001 :::".first()"::: ) )) & (Bool (Set (Set (Var "S")) ($#k12_glib_001 :::".edgeSeq()"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "W")) ($#k12_glib_001 :::".edgeSeq()"::: ) ))) "holds" (Bool (Set (Var "S")) ($#r1_hidden :::"="::: ) (Set (Var "W")))))) ; theorem :: CHORD:35 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) (Bool "for" (Set (Var "S")) "being" ($#m4_glib_001 :::"Subwalk"::: ) "of" (Set (Var "W")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "W"))))) "holds" (Bool (Set (Var "S")) ($#r1_hidden :::"="::: ) (Set (Var "W")))))) ; theorem :: CHORD:36 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "W")) "is" ($#v1_chord :::"minlength"::: ) )) "holds" (Bool (Set (Var "W")) "is" ($#v5_glib_001 :::"Path-like"::: ) ))) ; theorem :: CHORD:37 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "W")) "is" ($#v1_chord :::"minlength"::: ) )) "holds" (Bool (Set (Var "W")) "is" ($#v5_glib_001 :::"Path-like"::: ) ))) ; theorem :: CHORD:38 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "st" (Bool (Bool "(" "for" (Set (Var "P")) "being" ($#m3_glib_001 :::"Path":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "P")) ($#r1_glib_001 :::"is_Walk_from"::: ) (Set (Set (Var "W")) ($#k3_glib_001 :::".first()"::: ) ) "," (Set (Set (Var "W")) ($#k4_glib_001 :::".last()"::: ) ))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "P"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "W")))) ")" )) "holds" (Bool (Set (Var "W")) "is" ($#v1_chord :::"minlength"::: ) ))) ; theorem :: CHORD:39 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) (Bool "ex" (Set (Var "P")) "being" ($#m3_glib_001 :::"Path":::) "of" (Set (Var "G")) "st" (Bool "(" (Bool (Set (Var "P")) ($#r1_glib_001 :::"is_Walk_from"::: ) (Set (Set (Var "W")) ($#k3_glib_001 :::".first()"::: ) ) "," (Set (Set (Var "W")) ($#k4_glib_001 :::".last()"::: ) )) & (Bool (Set (Var "P")) "is" ($#v1_chord :::"minlength"::: ) ) ")" )))) ; theorem :: CHORD:40 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "W")) "is" ($#v1_chord :::"minlength"::: ) )) "holds" (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "W"))))) "holds" (Bool "for" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Bool "not" (Set (Var "e")) ($#r1_glib_000 :::"Joins"::: ) (Set (Set (Var "W")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) "," (Set (Set (Var "W")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) "," (Set (Var "G")))))))) ; theorem :: CHORD:41 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")) ")" ) (Bool "for" (Set (Var "H")) "being" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "G")) "," (Set (Var "S")) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "H")) "st" (Bool (Bool (Set (Var "W")) "is" ($#v1_chord :::"minlength"::: ) )) "holds" (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "W"))))) "holds" (Bool "for" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Bool "not" (Set (Var "e")) ($#r1_glib_000 :::"Joins"::: ) (Set (Set (Var "W")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) "," (Set (Set (Var "W")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) "," (Set (Var "G")))))))))) ; theorem :: CHORD:42 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "W")) "is" ($#v1_chord :::"minlength"::: ) )) "holds" (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "W"))))) "holds" (Bool (Set (Set (Var "W")) ($#k8_glib_001 :::".cut"::: ) "(" (Set (Var "m")) "," (Set (Var "n")) ")" ) "is" ($#v1_chord :::"minlength"::: ) )))) ; theorem :: CHORD:43 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) "st" (Bool (Bool (Set (Var "G")) "is" ($#v1_glib_002 :::"connected"::: ) )) "holds" (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")) ")" ) "st" (Bool (Bool (Set (Var "A")) ($#r1_subset_1 :::"misses"::: ) (Set (Var "B")))) "holds" (Bool "ex" (Set (Var "P")) "being" ($#m3_glib_001 :::"Path":::) "of" (Set (Var "G")) "st" (Bool "(" (Bool (Set (Var "P")) "is" ($#v1_chord :::"minlength"::: ) ) & (Bool (Bool "not" (Set (Var "P")) "is" ($#v3_glib_001 :::"trivial"::: ) )) & (Bool (Set (Set (Var "P")) ($#k3_glib_001 :::".first()"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "P")) ($#k4_glib_001 :::".last()"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "B"))) & (Bool "(" "for" (Set (Var "n")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "P"))))) "holds" (Bool "(" (Bool (Bool "not" (Set (Set (Var "P")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) & (Bool (Bool "not" (Set (Set (Var "P")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set (Var "B")))) ")" ) ")" ) ")" )))) ; begin definitionlet "G" be ($#m1_hidden :::"_Graph":::); let "a", "b" be ($#m1_subset_1 :::"Vertex":::) "of" (Set (Const "G")); pred "a" "," "b" :::"are_adjacent"::: means :: CHORD:def 3 (Bool "ex" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Var "e")) ($#r1_glib_000 :::"Joins"::: ) "a" "," "b" "," "G")); symmetry (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Const "G")) "st" (Bool (Bool "ex" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Var "e")) ($#r1_glib_000 :::"Joins"::: ) (Set (Var "a")) "," (Set (Var "b")) "," (Set (Const "G"))))) "holds" (Bool "ex" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Var "e")) ($#r1_glib_000 :::"Joins"::: ) (Set (Var "b")) "," (Set (Var "a")) "," (Set (Const "G"))))) ; end; :: deftheorem defines :::"are_adjacent"::: CHORD:def 3 : (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "holds" (Bool "(" (Bool (Set (Var "a")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) ) "iff" (Bool "ex" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Var "e")) ($#r1_glib_000 :::"Joins"::: ) (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "G")))) ")" ))); theorem :: CHORD:44 (Bool "for" (Set (Var "G1")) "," (Set (Var "G2")) "being" ($#m1_hidden :::"_Graph":::) "st" (Bool (Bool (Set (Var "G1")) ($#r5_glib_000 :::"=="::: ) (Set (Var "G2")))) "holds" (Bool "for" (Set (Var "u1")) "," (Set (Var "v1")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G1")) "st" (Bool (Bool (Set (Var "u1")) "," (Set (Var "v1")) ($#r1_chord :::"are_adjacent"::: ) )) "holds" (Bool "for" (Set (Var "u2")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G2")) "st" (Bool (Bool (Set (Var "u1")) ($#r1_hidden :::"="::: ) (Set (Var "u2"))) & (Bool (Set (Var "v1")) ($#r1_hidden :::"="::: ) (Set (Var "v2")))) "holds" (Bool (Set (Var "u2")) "," (Set (Var "v2")) ($#r1_chord :::"are_adjacent"::: ) )))) ; theorem :: CHORD:45 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")) ")" ) (Bool "for" (Set (Var "H")) "being" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "G")) "," (Set (Var "S")) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) (Bool "for" (Set (Var "t")) "," (Set (Var "w")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "H")) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set (Var "t"))) & (Bool (Set (Var "v")) ($#r1_hidden :::"="::: ) (Set (Var "w")))) "holds" (Bool "(" (Bool (Set (Var "u")) "," (Set (Var "v")) ($#r1_chord :::"are_adjacent"::: ) ) "iff" (Bool (Set (Var "t")) "," (Set (Var "w")) ($#r1_chord :::"are_adjacent"::: ) ) ")" )))))) ; theorem :: CHORD:46 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Set (Var "W")) ($#k3_glib_001 :::".first()"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "W")) ($#k4_glib_001 :::".last()"::: ) )) & (Bool (Bool "not" (Set (Set (Var "W")) ($#k3_glib_001 :::".first()"::: ) ) "," (Set (Set (Var "W")) ($#k4_glib_001 :::".last()"::: ) ) ($#r1_chord :::"are_adjacent"::: ) ))) "holds" (Bool (Set (Set (Var "W")) ($#k15_glib_001 :::".length()"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Num 2)))) ; theorem :: CHORD:47 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "," (Set (Var "v3")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "v1")) ($#r1_hidden :::"<>"::: ) (Set (Var "v2"))) & (Bool (Set (Var "v1")) ($#r1_hidden :::"<>"::: ) (Set (Var "v3"))) & (Bool (Set (Var "v2")) ($#r1_hidden :::"<>"::: ) (Set (Var "v3"))) & (Bool (Set (Var "v1")) "," (Set (Var "v2")) ($#r1_chord :::"are_adjacent"::: ) ) & (Bool (Set (Var "v2")) "," (Set (Var "v3")) ($#r1_chord :::"are_adjacent"::: ) )) "holds" (Bool "ex" (Set (Var "P")) "being" ($#m3_glib_001 :::"Path":::) "of" (Set (Var "G"))(Bool "ex" (Set (Var "e1")) "," (Set (Var "e2")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "P")) "is" ($#v1_glib_001 :::"open"::: ) ) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Num 5)) & (Bool (Set (Set (Var "P")) ($#k15_glib_001 :::".length()"::: ) ) ($#r1_hidden :::"="::: ) (Num 2)) & (Bool (Set (Var "e1")) ($#r1_glib_000 :::"Joins"::: ) (Set (Var "v1")) "," (Set (Var "v2")) "," (Set (Var "G"))) & (Bool (Set (Var "e2")) ($#r1_glib_000 :::"Joins"::: ) (Set (Var "v2")) "," (Set (Var "v3")) "," (Set (Var "G"))) & (Bool (Set (Set (Var "P")) ($#k14_glib_001 :::".edges()"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "e1")) "," (Set (Var "e2")) ($#k2_tarski :::"}"::: ) )) & (Bool (Set (Set (Var "P")) ($#k13_glib_001 :::".vertices()"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) "," (Set (Var "v3")) ($#k1_enumset1 :::"}"::: ) )) & (Bool (Set (Set (Var "P")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "v1"))) & (Bool (Set (Set (Var "P")) ($#k1_funct_1 :::"."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set (Var "v2"))) & (Bool (Set (Set (Var "P")) ($#k1_funct_1 :::"."::: ) (Num 5)) ($#r1_hidden :::"="::: ) (Set (Var "v3"))) ")" ))))) ; theorem :: CHORD:48 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "," (Set (Var "v3")) "," (Set (Var "v4")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "v1")) ($#r1_hidden :::"<>"::: ) (Set (Var "v2"))) & (Bool (Set (Var "v1")) ($#r1_hidden :::"<>"::: ) (Set (Var "v3"))) & (Bool (Set (Var "v2")) ($#r1_hidden :::"<>"::: ) (Set (Var "v3"))) & (Bool (Set (Var "v2")) ($#r1_hidden :::"<>"::: ) (Set (Var "v4"))) & (Bool (Set (Var "v3")) ($#r1_hidden :::"<>"::: ) (Set (Var "v4"))) & (Bool (Set (Var "v1")) "," (Set (Var "v2")) ($#r1_chord :::"are_adjacent"::: ) ) & (Bool (Set (Var "v2")) "," (Set (Var "v3")) ($#r1_chord :::"are_adjacent"::: ) ) & (Bool (Set (Var "v3")) "," (Set (Var "v4")) ($#r1_chord :::"are_adjacent"::: ) )) "holds" (Bool "ex" (Set (Var "P")) "being" ($#m3_glib_001 :::"Path":::) "of" (Set (Var "G")) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Num 7)) & (Bool (Set (Set (Var "P")) ($#k15_glib_001 :::".length()"::: ) ) ($#r1_hidden :::"="::: ) (Num 3)) & (Bool (Set (Set (Var "P")) ($#k13_glib_001 :::".vertices()"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k2_enumset1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) "," (Set (Var "v3")) "," (Set (Var "v4")) ($#k2_enumset1 :::"}"::: ) )) & (Bool (Set (Set (Var "P")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "v1"))) & (Bool (Set (Set (Var "P")) ($#k1_funct_1 :::"."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set (Var "v2"))) & (Bool (Set (Set (Var "P")) ($#k1_funct_1 :::"."::: ) (Num 5)) ($#r1_hidden :::"="::: ) (Set (Var "v3"))) & (Bool (Set (Set (Var "P")) ($#k1_funct_1 :::"."::: ) (Num 7)) ($#r1_hidden :::"="::: ) (Set (Var "v4"))) ")" )))) ; definitionlet "G" be ($#m1_hidden :::"_Graph":::); let "S" be ($#m1_hidden :::"set"::: ) ; func "G" :::".AdjacentSet"::: "S" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) "G" ")" ) equals :: CHORD:def 4 "{" (Set (Var "u")) where u "is" ($#m1_subset_1 :::"Vertex":::) "of" "G" : (Bool "(" (Bool (Bool "not" (Set (Var "u")) ($#r2_hidden :::"in"::: ) "S")) & (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vertex":::) "of" "G" "st" (Bool "(" (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) "S") & (Bool (Set (Var "u")) "," (Set (Var "v")) ($#r1_chord :::"are_adjacent"::: ) ) ")" )) ")" ) "}" ; end; :: deftheorem defines :::".AdjacentSet"::: CHORD:def 4 : (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "S")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set (Var "G")) ($#k2_chord :::".AdjacentSet"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "u")) where u "is" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) : (Bool "(" (Bool (Bool "not" (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) & (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool "(" (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) & (Bool (Set (Var "u")) "," (Set (Var "v")) ($#r1_chord :::"are_adjacent"::: ) ) ")" )) ")" ) "}" ))); theorem :: CHORD:49 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "S")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k2_chord :::".AdjacentSet"::: ) (Set (Var "S"))))) "holds" (Bool "not" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))))) ; theorem :: CHORD:50 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "S")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "holds" (Bool "(" (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k2_chord :::".AdjacentSet"::: ) (Set (Var "S")))) "iff" (Bool "(" (Bool (Bool "not" (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) & (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool "(" (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) & (Bool (Set (Var "u")) "," (Set (Var "v")) ($#r1_chord :::"are_adjacent"::: ) ) ")" )) ")" ) ")" )))) ; theorem :: CHORD:51 (Bool "for" (Set (Var "G1")) "," (Set (Var "G2")) "being" ($#m1_hidden :::"_Graph":::) "st" (Bool (Bool (Set (Var "G1")) ($#r5_glib_000 :::"=="::: ) (Set (Var "G2")))) "holds" (Bool "for" (Set (Var "S")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set (Var "G1")) ($#k2_chord :::".AdjacentSet"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "G2")) ($#k2_chord :::".AdjacentSet"::: ) (Set (Var "S")))))) ; theorem :: CHORD:52 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "holds" (Bool "(" (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k2_chord :::".AdjacentSet"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ))) "iff" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"<>"::: ) (Set (Var "v"))) & (Bool (Set (Var "v")) "," (Set (Var "u")) ($#r1_chord :::"are_adjacent"::: ) ) ")" ) ")" ))) ; theorem :: CHORD:53 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k2_chord :::".AdjacentSet"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) ))) "iff" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k2_chord :::".AdjacentSet"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ))) ")" ))) ; theorem :: CHORD:54 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "C")) "being" ($#m3_glib_001 :::"Path":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "C")) "is" ($#v8_glib_001 :::"Cycle-like"::: ) ) & (Bool (Set (Set (Var "C")) ($#k15_glib_001 :::".length()"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Num 3))) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "C")) ($#k13_glib_001 :::".vertices()"::: ) ))) "holds" (Bool "ex" (Set (Var "m")) "," (Set (Var "n")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "C")))) & (Bool "(" "not" (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Num 1)) "or" "not" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "C")))) ")" ) & (Bool "(" "not" (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Num 1)) "or" "not" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "C")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 2))) ")" ) & (Bool "(" "not" (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Num 3)) "or" "not" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "C")))) ")" ) & (Bool (Set (Set (Var "C")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "C")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "C")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k2_chord :::".AdjacentSet"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ))) & (Bool (Set (Set (Var "C")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k2_chord :::".AdjacentSet"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ))) ")" ))))) ; theorem :: CHORD:55 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "C")) "being" ($#m3_glib_001 :::"Path":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "C")) "is" ($#v8_glib_001 :::"Cycle-like"::: ) ) & (Bool (Set (Set (Var "C")) ($#k15_glib_001 :::".length()"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Num 3))) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "C")) ($#k13_glib_001 :::".vertices()"::: ) ))) "holds" (Bool "ex" (Set (Var "m")) "," (Set (Var "n")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "C")))) & (Bool (Set (Set (Var "C")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "C")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "C")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k2_chord :::".AdjacentSet"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ))) & (Bool (Set (Set (Var "C")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k2_chord :::".AdjacentSet"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ))) & (Bool "(" "for" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "e")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "C")) ($#k14_glib_001 :::".edges()"::: ) ))) "holds" (Bool "not" (Bool (Set (Var "e")) ($#r1_glib_000 :::"Joins"::: ) (Set (Set (Var "C")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) "," (Set (Set (Var "C")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) "," (Set (Var "G")))) ")" ) ")" ))))) ; theorem :: CHORD:56 (Bool "for" (Set (Var "G")) "being" ($#v3_glib_000 :::"loopless"::: ) ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "holds" (Bool "(" (Bool (Set (Set (Var "G")) ($#k2_chord :::".AdjacentSet"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "u")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "iff" (Bool (Set (Var "u")) "is" ($#v10_glib_000 :::"isolated"::: ) ) ")" ))) ; theorem :: CHORD:57 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "G0")) "being" ($#m1_glib_000 :::"Subgraph"::: ) "of" (Set (Var "G")) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) (Bool "for" (Set (Var "G1")) "being" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "G")) "," (Set (Var "S")) (Bool "for" (Set (Var "G2")) "being" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "G")) "," (Set "(" (Set (Var "S")) ($#k4_subset_1 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ")" ) "st" (Bool (Bool (Set (Var "G1")) "is" ($#v1_glib_002 :::"connected"::: ) ) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k2_chord :::".AdjacentSet"::: ) (Set "(" ($#k24_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G1")) ")" )))) "holds" (Bool (Set (Var "G2")) "is" ($#v1_glib_002 :::"connected"::: ) ))))))) ; theorem :: CHORD:58 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")) ")" ) (Bool "for" (Set (Var "H")) "being" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "G")) "," (Set (Var "S")) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) & (Bool (Set (Set (Var "G")) ($#k2_chord :::".AdjacentSet"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "u")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_tarski :::"c="::: ) (Set (Var "S")))) "holds" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "H")) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set (Var "v")))) "holds" (Bool (Set (Set (Var "G")) ($#k2_chord :::".AdjacentSet"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "u")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "H")) ($#k2_chord :::".AdjacentSet"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) )))))))) ; definitionlet "G" be ($#m1_hidden :::"_Graph":::); let "S" be ($#m1_hidden :::"set"::: ) ; mode :::"AdjGraph"::: "of" "G" "," "S" -> ($#m1_glib_000 :::"Subgraph"::: ) "of" "G" means :: CHORD:def 5 (Bool it "is" ($#m2_glib_000 :::"inducedSubgraph":::) "of" "G" "," (Set "(" "G" ($#k2_chord :::".AdjacentSet"::: ) "S" ")" )) if (Bool "S" "is" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) "G" ")" )) ; end; :: deftheorem defines :::"AdjGraph"::: CHORD:def 5 : (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "S")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b3")) "being" ($#m1_glib_000 :::"Subgraph"::: ) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "S")) "is" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")) ")" ))) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m1_chord :::"AdjGraph"::: ) "of" (Set (Var "G")) "," (Set (Var "S"))) "iff" (Bool (Set (Var "b3")) "is" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "G")) "," (Set "(" (Set (Var "G")) ($#k2_chord :::".AdjacentSet"::: ) (Set (Var "S")) ")" )) ")" )))); theorem :: CHORD:59 (Bool "for" (Set (Var "G1")) "," (Set (Var "G2")) "being" ($#m1_hidden :::"_Graph":::) "st" (Bool (Bool (Set (Var "G1")) ($#r5_glib_000 :::"=="::: ) (Set (Var "G2")))) "holds" (Bool "for" (Set (Var "u1")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G1")) (Bool "for" (Set (Var "u2")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G2")) "st" (Bool (Bool (Set (Var "u1")) ($#r1_hidden :::"="::: ) (Set (Var "u2")))) "holds" (Bool "for" (Set (Var "H1")) "being" ($#m1_chord :::"AdjGraph"::: ) "of" (Set (Var "G1")) "," (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "u1")) ($#k6_domain_1 :::"}"::: ) ) (Bool "for" (Set (Var "H2")) "being" ($#m1_chord :::"AdjGraph"::: ) "of" (Set (Var "G2")) "," (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "u2")) ($#k6_domain_1 :::"}"::: ) ) "holds" (Bool (Set (Var "H1")) ($#r5_glib_000 :::"=="::: ) (Set (Var "H2")))))))) ; theorem :: CHORD:60 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")) ")" ) (Bool "for" (Set (Var "H")) "being" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "G")) "," (Set (Var "S")) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) & (Bool (Set (Set (Var "G")) ($#k2_chord :::".AdjacentSet"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "u")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_tarski :::"c="::: ) (Set (Var "S"))) & (Bool (Set (Set (Var "G")) ($#k2_chord :::".AdjacentSet"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "u")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "H")) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set (Var "v")))) "holds" (Bool "for" (Set (Var "Ga")) "being" ($#m1_chord :::"AdjGraph"::: ) "of" (Set (Var "G")) "," (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "u")) ($#k6_domain_1 :::"}"::: ) ) (Bool "for" (Set (Var "Ha")) "being" ($#m1_chord :::"AdjGraph"::: ) "of" (Set (Var "H")) "," (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ) "holds" (Bool (Set (Var "Ga")) ($#r5_glib_000 :::"=="::: ) (Set (Var "Ha")))))))))) ; definitionlet "G" be ($#m1_hidden :::"_Graph":::); attr "G" is :::"complete"::: means :: CHORD:def 6 (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Vertex":::) "of" "G" "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"<>"::: ) (Set (Var "v")))) "holds" (Bool (Set (Var "u")) "," (Set (Var "v")) ($#r1_chord :::"are_adjacent"::: ) )); end; :: deftheorem defines :::"complete"::: CHORD:def 6 : (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) "holds" (Bool "(" (Bool (Set (Var "G")) "is" ($#v2_chord :::"complete"::: ) ) "iff" (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"<>"::: ) (Set (Var "v")))) "holds" (Bool (Set (Var "u")) "," (Set (Var "v")) ($#r1_chord :::"are_adjacent"::: ) )) ")" )); theorem :: CHORD:61 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) "st" (Bool (Bool (Set (Var "G")) "is" ($#v4_glib_000 :::"trivial"::: ) )) "holds" (Bool (Set (Var "G")) "is" ($#v2_chord :::"complete"::: ) )) ; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_glib_000 :::"[Graph-like]"::: ) ($#v4_glib_000 :::"trivial"::: ) -> ($#v2_chord :::"complete"::: ) for ($#m1_hidden :::"set"::: ) ; end; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_glib_000 :::"[Graph-like]"::: ) ($#v4_glib_000 :::"trivial"::: ) ($#v7_glib_000 :::"simple"::: ) ($#v2_chord :::"complete"::: ) for ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_glib_000 :::"[Graph-like]"::: ) ($#v2_glib_000 :::"finite"::: ) ($#~v4_glib_000 "non" ($#v4_glib_000 :::"trivial"::: ) ) ($#v7_glib_000 :::"simple"::: ) ($#v2_chord :::"complete"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: CHORD:62 (Bool "for" (Set (Var "G1")) "," (Set (Var "G2")) "being" ($#m1_hidden :::"_Graph":::) "st" (Bool (Bool (Set (Var "G1")) ($#r5_glib_000 :::"=="::: ) (Set (Var "G2"))) & (Bool (Set (Var "G1")) "is" ($#v2_chord :::"complete"::: ) )) "holds" (Bool (Set (Var "G2")) "is" ($#v2_chord :::"complete"::: ) )) ; theorem :: CHORD:63 (Bool "for" (Set (Var "G")) "being" ($#v2_chord :::"complete"::: ) ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")) ")" ) (Bool "for" (Set (Var "H")) "being" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "G")) "," (Set (Var "S")) "holds" (Bool (Set (Var "H")) "is" ($#v2_chord :::"complete"::: ) )))) ; begin definitionlet "G" be ($#m1_hidden :::"_Graph":::); let "v" be ($#m1_subset_1 :::"Vertex":::) "of" (Set (Const "G")); attr "v" is :::"simplicial"::: means :: CHORD:def 7 "(" (Bool (Bool (Set "G" ($#k2_chord :::".AdjacentSet"::: ) (Set ($#k6_domain_1 :::"{"::: ) "v" ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool "for" (Set (Var "G2")) "being" ($#m1_chord :::"AdjGraph"::: ) "of" "G" "," (Set ($#k6_domain_1 :::"{"::: ) "v" ($#k6_domain_1 :::"}"::: ) ) "holds" (Bool (Set (Var "G2")) "is" ($#v2_chord :::"complete"::: ) )) ")" ; end; :: deftheorem defines :::"simplicial"::: CHORD:def 7 : (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "holds" (Bool "(" (Bool (Set (Var "v")) "is" ($#v3_chord :::"simplicial"::: ) ) "iff" "(" (Bool (Bool (Set (Set (Var "G")) ($#k2_chord :::".AdjacentSet"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool "for" (Set (Var "G2")) "being" ($#m1_chord :::"AdjGraph"::: ) "of" (Set (Var "G")) "," (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ) "holds" (Bool (Set (Var "G2")) "is" ($#v2_chord :::"complete"::: ) )) ")" ")" ))); theorem :: CHORD:64 (Bool "for" (Set (Var "G")) "being" ($#v2_chord :::"complete"::: ) ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "holds" (Bool (Set (Var "v")) "is" ($#v3_chord :::"simplicial"::: ) ))) ; theorem :: CHORD:65 (Bool "for" (Set (Var "G")) "being" ($#v4_glib_000 :::"trivial"::: ) ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "holds" (Bool (Set (Var "v")) "is" ($#v3_chord :::"simplicial"::: ) ))) ; theorem :: CHORD:66 (Bool "for" (Set (Var "G1")) "," (Set (Var "G2")) "being" ($#m1_hidden :::"_Graph":::) "st" (Bool (Bool (Set (Var "G1")) ($#r5_glib_000 :::"=="::: ) (Set (Var "G2")))) "holds" (Bool "for" (Set (Var "u1")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G1")) (Bool "for" (Set (Var "u2")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G2")) "st" (Bool (Bool (Set (Var "u1")) ($#r1_hidden :::"="::: ) (Set (Var "u2"))) & (Bool (Set (Var "u1")) "is" ($#v3_chord :::"simplicial"::: ) )) "holds" (Bool (Set (Var "u2")) "is" ($#v3_chord :::"simplicial"::: ) )))) ; theorem :: CHORD:67 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")) ")" ) (Bool "for" (Set (Var "H")) "being" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "G")) "," (Set (Var "S")) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) & (Bool (Set (Set (Var "G")) ($#k2_chord :::".AdjacentSet"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "u")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_tarski :::"c="::: ) (Set (Var "S")))) "holds" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "H")) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set (Var "v")))) "holds" (Bool "(" (Bool (Set (Var "u")) "is" ($#v3_chord :::"simplicial"::: ) ) "iff" (Bool (Set (Var "v")) "is" ($#v3_chord :::"simplicial"::: ) ) ")" )))))) ; theorem :: CHORD:68 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "v")) "is" ($#v3_chord :::"simplicial"::: ) )) "holds" (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k2_chord :::".AdjacentSet"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "G")) ($#k2_chord :::".AdjacentSet"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) )))) "holds" (Bool "ex" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Var "e")) ($#r1_glib_000 :::"Joins"::: ) (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "G"))))))) ; theorem :: CHORD:69 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Bool "not" (Set (Var "v")) "is" ($#v3_chord :::"simplicial"::: ) ))) "holds" (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool "(" (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Set (Var "v")) ($#r1_hidden :::"<>"::: ) (Set (Var "a"))) & (Bool (Set (Var "v")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Set (Var "v")) "," (Set (Var "a")) ($#r1_chord :::"are_adjacent"::: ) ) & (Bool (Set (Var "v")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) ) & (Bool (Bool "not" (Set (Var "a")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) )) ")" )))) ; begin definitionlet "G" be ($#m1_hidden :::"_Graph":::); let "a", "b" be ($#m1_subset_1 :::"Vertex":::) "of" (Set (Const "G")); assume that (Bool (Set (Const "a")) ($#r1_hidden :::"<>"::: ) (Set (Const "b"))) and (Bool (Bool "not" (Set (Const "a")) "," (Set (Const "b")) ($#r1_chord :::"are_adjacent"::: ) )) ; mode :::"VertexSeparator"::: "of" "a" "," "b" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) "G" ")" ) means :: CHORD:def 8 (Bool "(" (Bool (Bool "not" "a" ($#r2_hidden :::"in"::: ) it)) & (Bool (Bool "not" "b" ($#r2_hidden :::"in"::: ) it)) & (Bool "(" "for" (Set (Var "G2")) "being" ($#m2_glib_000 :::"removeVertices":::) "of" "G" "," it (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G2")) "holds" (Bool (Bool "not" (Set (Var "W")) ($#r1_glib_001 :::"is_Walk_from"::: ) "a" "," "b"))) ")" ) ")" ); end; :: deftheorem defines :::"VertexSeparator"::: CHORD:def 8 : (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Bool "not" (Set (Var "a")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) ))) "holds" (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b4")) "is" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "a")) "," (Set (Var "b"))) "iff" (Bool "(" (Bool (Bool "not" (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "b4")))) & (Bool (Bool "not" (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "b4")))) & (Bool "(" "for" (Set (Var "G2")) "being" ($#m2_glib_000 :::"removeVertices":::) "of" (Set (Var "G")) "," (Set (Var "b4")) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G2")) "holds" (Bool (Bool "not" (Set (Var "W")) ($#r1_glib_001 :::"is_Walk_from"::: ) (Set (Var "a")) "," (Set (Var "b"))))) ")" ) ")" ) ")" )))); theorem :: CHORD:70 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Bool "not" (Set (Var "a")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) ))) "holds" (Bool "for" (Set (Var "S")) "being" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) "holds" (Bool (Set (Var "S")) "is" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "b")) "," (Set (Var "a")))))) ; theorem :: CHORD:71 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Bool "not" (Set (Var "a")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) ))) "holds" (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")) ")" ) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "a")) "," (Set (Var "b"))) "iff" (Bool "(" (Bool (Bool "not" (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) & (Bool (Bool "not" (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) & (Bool "(" "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "W")) ($#r1_glib_001 :::"is_Walk_from"::: ) (Set (Var "a")) "," (Set (Var "b")))) "holds" (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "W")) ($#k13_glib_001 :::".vertices()"::: ) )) ")" )) ")" ) ")" ) ")" )))) ; theorem :: CHORD:72 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Bool "not" (Set (Var "a")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) ))) "holds" (Bool "for" (Set (Var "S")) "being" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "W")) ($#r1_glib_001 :::"is_Walk_from"::: ) (Set (Var "a")) "," (Set (Var "b")))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "W")))) & (Bool (Set (Set (Var "W")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) ")" )))))) ; theorem :: CHORD:73 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Bool "not" (Set (Var "a")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) ))) "holds" (Bool "for" (Set (Var "S")) "being" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) "st" (Bool (Bool (Set (Var "S")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "holds" (Bool (Bool "not" (Set (Var "W")) ($#r1_glib_001 :::"is_Walk_from"::: ) (Set (Var "a")) "," (Set (Var "b")))))))) ; theorem :: CHORD:74 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Bool "not" (Set (Var "a")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) )) & (Bool "(" "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "holds" (Bool (Bool "not" (Set (Var "W")) ($#r1_glib_001 :::"is_Walk_from"::: ) (Set (Var "a")) "," (Set (Var "b")))) ")" )) "holds" (Bool (Set ($#k1_xboole_0 :::"{}"::: ) ) "is" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "a")) "," (Set (Var "b"))))) ; theorem :: CHORD:75 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Bool "not" (Set (Var "a")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) ))) "holds" (Bool "for" (Set (Var "S")) "being" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) (Bool "for" (Set (Var "G2")) "being" ($#m2_glib_000 :::"removeVertices":::) "of" (Set (Var "G")) "," (Set (Var "S")) (Bool "for" (Set (Var "a2")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G2")) "holds" (Bool (Set (Set "(" (Set (Var "G2")) ($#k1_glib_002 :::".reachableFrom"::: ) (Set (Var "a2")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))))))) ; theorem :: CHORD:76 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Bool "not" (Set (Var "a")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) ))) "holds" (Bool "for" (Set (Var "S")) "being" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) (Bool "for" (Set (Var "G2")) "being" ($#m2_glib_000 :::"removeVertices":::) "of" (Set (Var "G")) "," (Set (Var "S")) (Bool "for" (Set (Var "a2")) "," (Set (Var "b2")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G2")) "st" (Bool (Bool (Set (Var "a2")) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set (Var "b")))) "holds" (Bool (Set (Set "(" (Set (Var "G2")) ($#k1_glib_002 :::".reachableFrom"::: ) (Set (Var "a2")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set (Var "G2")) ($#k1_glib_002 :::".reachableFrom"::: ) (Set (Var "b2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))))))) ; theorem :: CHORD:77 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Bool "not" (Set (Var "a")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) ))) "holds" (Bool "for" (Set (Var "S")) "being" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) (Bool "for" (Set (Var "G2")) "being" ($#m2_glib_000 :::"removeVertices":::) "of" (Set (Var "G")) "," (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "a")) "is" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G2"))) & (Bool (Set (Var "b")) "is" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G2"))) ")" ))))) ; definitionlet "G" be ($#m1_hidden :::"_Graph":::); let "a", "b" be ($#m1_subset_1 :::"Vertex":::) "of" (Set (Const "G")); let "S" be ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Const "a")) "," (Set (Const "b")); attr "S" is :::"minimal"::: means :: CHORD:def 9 (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" "S" "st" (Bool (Bool (Set (Var "T")) ($#r1_hidden :::"<>"::: ) "S")) "holds" (Bool "not" (Bool (Set (Var "T")) "is" ($#m2_chord :::"VertexSeparator"::: ) "of" "a" "," "b"))); end; :: deftheorem defines :::"minimal"::: CHORD:def 9 : (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) (Bool "for" (Set (Var "S")) "being" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v4_chord :::"minimal"::: ) ) "iff" (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "T")) ($#r1_hidden :::"<>"::: ) (Set (Var "S")))) "holds" (Bool "not" (Bool (Set (Var "T")) "is" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "a")) "," (Set (Var "b"))))) ")" )))); theorem :: CHORD:78 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) (Bool "for" (Set (Var "S")) "being" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) "st" (Bool (Bool (Set (Var "S")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Var "S")) "is" ($#v4_chord :::"minimal"::: ) )))) ; theorem :: CHORD:79 (Bool "for" (Set (Var "G")) "being" ($#v2_glib_000 :::"finite"::: ) ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) (Bool "ex" (Set (Var "S")) "being" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) "st" (Bool (Set (Var "S")) "is" ($#v4_chord :::"minimal"::: ) )))) ; theorem :: CHORD:80 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Bool "not" (Set (Var "a")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) ))) "holds" (Bool "for" (Set (Var "S")) "being" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v4_chord :::"minimal"::: ) )) "holds" (Bool "for" (Set (Var "T")) "being" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "b")) "," (Set (Var "a")) "st" (Bool (Bool (Set (Var "S")) ($#r1_hidden :::"="::: ) (Set (Var "T")))) "holds" (Bool (Set (Var "T")) "is" ($#v4_chord :::"minimal"::: ) ))))) ; theorem :: CHORD:81 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Bool "not" (Set (Var "a")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) ))) "holds" (Bool "for" (Set (Var "S")) "being" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v4_chord :::"minimal"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) "holds" (Bool "ex" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "st" (Bool "(" (Bool (Set (Var "W")) ($#r1_glib_001 :::"is_Walk_from"::: ) (Set (Var "a")) "," (Set (Var "b"))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "W")) ($#k13_glib_001 :::".vertices()"::: ) )) ")" )))))) ; theorem :: CHORD:82 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Bool "not" (Set (Var "a")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) ))) "holds" (Bool "for" (Set (Var "S")) "being" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v4_chord :::"minimal"::: ) )) "holds" (Bool "for" (Set (Var "H")) "being" ($#m2_glib_000 :::"removeVertices":::) "of" (Set (Var "G")) "," (Set (Var "S")) (Bool "for" (Set (Var "aa")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "H")) "st" (Bool (Bool (Set (Var "aa")) ($#r1_hidden :::"="::: ) (Set (Var "a")))) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) "holds" (Bool "ex" (Set (Var "y")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "H")) ($#k1_glib_002 :::".reachableFrom"::: ) (Set (Var "aa")))) & (Bool (Set (Var "x")) "," (Set (Var "y")) ($#r1_chord :::"are_adjacent"::: ) ) ")" )))))))) ; theorem :: CHORD:83 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Bool "not" (Set (Var "a")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) ))) "holds" (Bool "for" (Set (Var "S")) "being" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v4_chord :::"minimal"::: ) )) "holds" (Bool "for" (Set (Var "H")) "being" ($#m2_glib_000 :::"removeVertices":::) "of" (Set (Var "G")) "," (Set (Var "S")) (Bool "for" (Set (Var "aa")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "H")) "st" (Bool (Bool (Set (Var "aa")) ($#r1_hidden :::"="::: ) (Set (Var "b")))) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) "holds" (Bool "ex" (Set (Var "y")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "H")) ($#k1_glib_002 :::".reachableFrom"::: ) (Set (Var "aa")))) & (Bool (Set (Var "x")) "," (Set (Var "y")) ($#r1_chord :::"are_adjacent"::: ) ) ")" )))))))) ; begin definitionlet "G" be ($#m1_hidden :::"_Graph":::); let "W" be ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Const "G")); attr "W" is :::"chordal"::: means :: CHORD:def 10 (Bool "ex" (Set (Var "m")) "," (Set (Var "n")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "W")) & (Bool (Set "W" ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"<>"::: ) (Set "W" ($#k1_funct_1 :::"."::: ) (Set (Var "n")))) & (Bool "ex" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Var "e")) ($#r1_glib_000 :::"Joins"::: ) (Set "W" ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) "," (Set "W" ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) "," "G")) & (Bool "(" "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set "W" ($#k14_glib_001 :::".edges()"::: ) ))) "holds" (Bool "not" (Bool (Set (Var "f")) ($#r1_glib_000 :::"Joins"::: ) (Set "W" ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) "," (Set "W" ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) "," "G")) ")" ) ")" )); end; :: deftheorem defines :::"chordal"::: CHORD:def 10 : (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "holds" (Bool "(" (Bool (Set (Var "W")) "is" ($#v5_chord :::"chordal"::: ) ) "iff" (Bool "ex" (Set (Var "m")) "," (Set (Var "n")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "W")))) & (Bool (Set (Set (Var "W")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "W")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")))) & (Bool "ex" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Var "e")) ($#r1_glib_000 :::"Joins"::: ) (Set (Set (Var "W")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) "," (Set (Set (Var "W")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) "," (Set (Var "G")))) & (Bool "(" "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "W")) ($#k14_glib_001 :::".edges()"::: ) ))) "holds" (Bool "not" (Bool (Set (Var "f")) ($#r1_glib_000 :::"Joins"::: ) (Set (Set (Var "W")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) "," (Set (Set (Var "W")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) "," (Set (Var "G")))) ")" ) ")" )) ")" ))); notationlet "G" be ($#m1_hidden :::"_Graph":::); let "W" be ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Const "G")); antonym :::"chordless"::: "W" for :::"chordal"::: ; end; theorem :: CHORD:84 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "W")) "is" ($#v5_chord :::"chordal"::: ) )) "holds" (Bool "ex" (Set (Var "m")) "," (Set (Var "n")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "W")))) & (Bool (Set (Set (Var "W")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "W")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")))) & (Bool "ex" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Var "e")) ($#r1_glib_000 :::"Joins"::: ) (Set (Set (Var "W")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) "," (Set (Set (Var "W")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) "," (Set (Var "G")))) & "(" (Bool (Bool (Set (Var "W")) "is" ($#v8_glib_001 :::"Cycle-like"::: ) )) "implies" (Bool "(" (Bool "(" "not" (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Num 1)) "or" "not" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "W")))) ")" ) & (Bool "(" "not" (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Num 1)) "or" "not" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "W")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 2))) ")" ) & (Bool "(" "not" (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Num 3)) "or" "not" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "W")))) ")" ) ")" ) ")" ")" )))) ; theorem :: CHORD:85 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "P")) "being" ($#m3_glib_001 :::"Path":::) "of" (Set (Var "G")) "st" (Bool (Bool "ex" (Set (Var "m")) "," (Set (Var "n")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "P")))) & (Bool "ex" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Var "e")) ($#r1_glib_000 :::"Joins"::: ) (Set (Set (Var "P")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) "," (Set (Set (Var "P")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) "," (Set (Var "G")))) & "(" (Bool (Bool (Set (Var "P")) "is" ($#v8_glib_001 :::"Cycle-like"::: ) )) "implies" (Bool "(" (Bool "(" "not" (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Num 1)) "or" "not" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "P")))) ")" ) & (Bool "(" "not" (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Num 1)) "or" "not" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "P")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 2))) ")" ) & (Bool "(" "not" (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Num 3)) "or" "not" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "P")))) ")" ) ")" ) ")" ")" ))) "holds" (Bool (Set (Var "P")) "is" ($#v5_chord :::"chordal"::: ) ))) ; theorem :: CHORD:86 (Bool "for" (Set (Var "G1")) "," (Set (Var "G2")) "being" ($#m1_hidden :::"_Graph":::) "st" (Bool (Bool (Set (Var "G1")) ($#r5_glib_000 :::"=="::: ) (Set (Var "G2")))) "holds" (Bool "for" (Set (Var "W1")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G1")) (Bool "for" (Set (Var "W2")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G2")) "st" (Bool (Bool (Set (Var "W1")) ($#r1_hidden :::"="::: ) (Set (Var "W2"))) & (Bool (Set (Var "W1")) "is" ($#v5_chord :::"chordal"::: ) )) "holds" (Bool (Set (Var "W2")) "is" ($#v5_chord :::"chordal"::: ) )))) ; theorem :: CHORD:87 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")) ")" ) (Bool "for" (Set (Var "H")) "being" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "G")) "," (Set (Var "S")) (Bool "for" (Set (Var "W1")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) (Bool "for" (Set (Var "W2")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "H")) "st" (Bool (Bool (Set (Var "W1")) ($#r1_hidden :::"="::: ) (Set (Var "W2")))) "holds" (Bool "(" (Bool (Set (Var "W2")) "is" ($#v5_chord :::"chordal"::: ) ) "iff" (Bool (Set (Var "W1")) "is" ($#v5_chord :::"chordal"::: ) ) ")" )))))) ; theorem :: CHORD:88 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "W")) "is" ($#v8_glib_001 :::"Cycle-like"::: ) ) & (Bool (Set (Var "W")) "is" ($#v5_chord :::"chordal"::: ) ) & (Bool (Set (Set (Var "W")) ($#k15_glib_001 :::".length()"::: ) ) ($#r1_hidden :::"="::: ) (Num 4))) "holds" (Bool "ex" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "e")) ($#r1_glib_000 :::"Joins"::: ) (Set (Set (Var "W")) ($#k1_funct_1 :::"."::: ) (Num 1)) "," (Set (Set (Var "W")) ($#k1_funct_1 :::"."::: ) (Num 5)) "," (Set (Var "G"))) "or" (Bool (Set (Var "e")) ($#r1_glib_000 :::"Joins"::: ) (Set (Set (Var "W")) ($#k1_funct_1 :::"."::: ) (Num 3)) "," (Set (Set (Var "W")) ($#k1_funct_1 :::"."::: ) (Num 7)) "," (Set (Var "G"))) ")" )))) ; theorem :: CHORD:89 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "W")) "is" ($#v1_chord :::"minlength"::: ) )) "holds" (Bool (Set (Var "W")) "is" ($#v5_chord :::"chordless"::: ) ))) ; theorem :: CHORD:90 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Num 5)) & (Bool (Bool "not" (Set (Set (Var "W")) ($#k3_glib_001 :::".first()"::: ) ) "," (Set (Set (Var "W")) ($#k4_glib_001 :::".last()"::: ) ) ($#r1_chord :::"are_adjacent"::: ) ))) "holds" (Bool (Set (Var "W")) "is" ($#v5_chord :::"chordless"::: ) ))) ; theorem :: CHORD:91 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "holds" (Bool "(" (Bool (Set (Var "W")) "is" ($#v5_chord :::"chordal"::: ) ) "iff" (Bool (Set (Set (Var "W")) ($#k6_glib_001 :::".reverse()"::: ) ) "is" ($#v5_chord :::"chordal"::: ) ) ")" ))) ; theorem :: CHORD:92 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "P")) "being" ($#m3_glib_001 :::"Path":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "P")) "is" ($#v1_glib_001 :::"open"::: ) ) & (Bool (Set (Var "P")) "is" ($#v5_chord :::"chordless"::: ) )) "holds" (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "P"))))) "holds" (Bool "(" (Bool "ex" (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Var "e")) ($#r1_glib_000 :::"Joins"::: ) (Set (Set (Var "P")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) "," (Set (Set (Var "P")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) "," (Set (Var "G")))) "iff" (Bool (Set (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Var "n"))) ")" )))) ; theorem :: CHORD:93 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "P")) "being" ($#m3_glib_001 :::"Path":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "P")) "is" ($#v1_glib_001 :::"open"::: ) ) & (Bool (Set (Var "P")) "is" ($#v5_chord :::"chordless"::: ) )) "holds" (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "P"))))) "holds" (Bool "(" (Bool (Set (Set (Var "P")) ($#k8_glib_001 :::".cut"::: ) "(" (Set (Var "m")) "," (Set (Var "n")) ")" ) "is" ($#v5_chord :::"chordless"::: ) ) & (Bool (Set (Set (Var "P")) ($#k8_glib_001 :::".cut"::: ) "(" (Set (Var "m")) "," (Set (Var "n")) ")" ) "is" ($#v1_glib_001 :::"open"::: ) ) ")" )))) ; theorem :: CHORD:94 (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")) ")" ) (Bool "for" (Set (Var "H")) "being" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "G")) "," (Set (Var "S")) (Bool "for" (Set (Var "W")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) (Bool "for" (Set (Var "V")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "H")) "st" (Bool (Bool (Set (Var "W")) ($#r1_hidden :::"="::: ) (Set (Var "V")))) "holds" (Bool "(" (Bool (Set (Var "W")) "is" ($#v5_chord :::"chordless"::: ) ) "iff" (Bool (Set (Var "V")) "is" ($#v5_chord :::"chordless"::: ) ) ")" )))))) ; definitionlet "G" be ($#m1_hidden :::"_Graph":::); attr "G" is :::"chordal"::: means :: CHORD:def 11 (Bool "for" (Set (Var "P")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" "G" "st" (Bool (Bool (Set (Set (Var "P")) ($#k15_glib_001 :::".length()"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Num 3)) & (Bool (Set (Var "P")) "is" ($#v8_glib_001 :::"Cycle-like"::: ) )) "holds" (Bool (Set (Var "P")) "is" ($#v5_chord :::"chordal"::: ) )); end; :: deftheorem defines :::"chordal"::: CHORD:def 11 : (Bool "for" (Set (Var "G")) "being" ($#m1_hidden :::"_Graph":::) "holds" (Bool "(" (Bool (Set (Var "G")) "is" ($#v6_chord :::"chordal"::: ) ) "iff" (Bool "for" (Set (Var "P")) "being" ($#m3_glib_001 :::"Walk"::: ) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Set (Var "P")) ($#k15_glib_001 :::".length()"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Num 3)) & (Bool (Set (Var "P")) "is" ($#v8_glib_001 :::"Cycle-like"::: ) )) "holds" (Bool (Set (Var "P")) "is" ($#v5_chord :::"chordal"::: ) )) ")" )); theorem :: CHORD:95 (Bool "for" (Set (Var "G1")) "," (Set (Var "G2")) "being" ($#m1_hidden :::"_Graph":::) "st" (Bool (Bool (Set (Var "G1")) ($#r5_glib_000 :::"=="::: ) (Set (Var "G2"))) & (Bool (Set (Var "G1")) "is" ($#v6_chord :::"chordal"::: ) )) "holds" (Bool (Set (Var "G2")) "is" ($#v6_chord :::"chordal"::: ) )) ; theorem :: CHORD:96 (Bool "for" (Set (Var "G")) "being" ($#v2_glib_000 :::"finite"::: ) ($#m1_hidden :::"_Graph":::) "st" (Bool (Bool (Set ($#k5_card_1 :::"card"::: ) (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Num 3))) "holds" (Bool (Set (Var "G")) "is" ($#v6_chord :::"chordal"::: ) )) ; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_glib_000 :::"[Graph-like]"::: ) ($#v2_glib_000 :::"finite"::: ) ($#v4_glib_000 :::"trivial"::: ) ($#v6_chord :::"chordal"::: ) for ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_glib_000 :::"[Graph-like]"::: ) ($#v2_glib_000 :::"finite"::: ) ($#~v4_glib_000 "non" ($#v4_glib_000 :::"trivial"::: ) ) ($#v7_glib_000 :::"simple"::: ) ($#v6_chord :::"chordal"::: ) for ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_glib_000 :::"[Graph-like]"::: ) ($#v2_chord :::"complete"::: ) -> ($#v6_chord :::"chordal"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "G" be ($#v6_chord :::"chordal"::: ) ($#m1_hidden :::"_Graph":::); let "V" be ($#m1_hidden :::"set"::: ) ; cluster -> ($#v6_chord :::"chordal"::: ) for ($#m2_glib_000 :::"inducedSubgraph"::: ) "of" "G" "," "V" "," (Set "G" ($#k21_glib_000 :::".edgesBetween"::: ) "V"); end; theorem :: CHORD:97 (Bool "for" (Set (Var "G")) "being" ($#v6_chord :::"chordal"::: ) ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "P")) "being" ($#m3_glib_001 :::"Path":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "P")) "is" ($#v1_glib_001 :::"open"::: ) ) & (Bool (Set (Var "P")) "is" ($#v5_chord :::"chordless"::: ) )) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "e")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "P")) ($#k13_glib_001 :::".vertices()"::: ) ))) & (Bool (Set (Var "e")) ($#r1_glib_000 :::"Joins"::: ) (Set (Set (Var "P")) ($#k4_glib_001 :::".last()"::: ) ) "," (Set (Var "x")) "," (Set (Var "G"))) & (Bool "(" "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Bool "not" (Set (Var "f")) ($#r1_glib_000 :::"Joins"::: ) (Set (Set (Var "P")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "P")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 2) ")" )) "," (Set (Var "x")) "," (Set (Var "G")))) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "P")) ($#k10_glib_001 :::".addEdge"::: ) (Set (Var "e"))) "is" ($#v5_glib_001 :::"Path-like"::: ) ) & (Bool (Set (Set (Var "P")) ($#k10_glib_001 :::".addEdge"::: ) (Set (Var "e"))) "is" ($#v1_glib_001 :::"open"::: ) ) & (Bool (Set (Set (Var "P")) ($#k10_glib_001 :::".addEdge"::: ) (Set (Var "e"))) "is" ($#v5_chord :::"chordless"::: ) ) ")" )))) ; theorem :: CHORD:98 (Bool "for" (Set (Var "G")) "being" ($#v6_chord :::"chordal"::: ) ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Bool "not" (Set (Var "a")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) ))) "holds" (Bool "for" (Set (Var "S")) "being" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v4_chord :::"minimal"::: ) ) & (Bool (Bool "not" (Set (Var "S")) "is" ($#v1_xboole_0 :::"empty"::: ) ))) "holds" (Bool "for" (Set (Var "H")) "being" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "G")) "," (Set (Var "S")) "holds" (Bool (Set (Var "H")) "is" ($#v2_chord :::"complete"::: ) ))))) ; theorem :: CHORD:99 (Bool "for" (Set (Var "G")) "being" ($#v2_glib_000 :::"finite"::: ) ($#m1_hidden :::"_Graph":::) "st" (Bool (Bool "(" "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Bool "not" (Set (Var "a")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) ))) "holds" (Bool "for" (Set (Var "S")) "being" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v4_chord :::"minimal"::: ) ) & (Bool (Bool "not" (Set (Var "S")) "is" ($#v1_xboole_0 :::"empty"::: ) ))) "holds" (Bool "for" (Set (Var "G2")) "being" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "G")) "," (Set (Var "S")) "holds" (Bool (Set (Var "G2")) "is" ($#v2_chord :::"complete"::: ) ))) ")" )) "holds" (Bool (Set (Var "G")) "is" ($#v6_chord :::"chordal"::: ) )) ; theorem :: CHORD:100 (Bool "for" (Set (Var "G")) "being" ($#v2_glib_000 :::"finite"::: ) ($#v6_chord :::"chordal"::: ) ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Bool "not" (Set (Var "a")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) ))) "holds" (Bool "for" (Set (Var "S")) "being" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v4_chord :::"minimal"::: ) )) "holds" (Bool "for" (Set (Var "H")) "being" ($#m2_glib_000 :::"removeVertices":::) "of" (Set (Var "G")) "," (Set (Var "S")) (Bool "for" (Set (Var "a1")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "H")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "a1")))) "holds" (Bool "ex" (Set (Var "c")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool "(" (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "H")) ($#k1_glib_002 :::".reachableFrom"::: ) (Set (Var "a1")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) "holds" (Bool (Set (Var "c")) "," (Set (Var "x")) ($#r1_chord :::"are_adjacent"::: ) ) ")" ) ")" ))))))) ; theorem :: CHORD:101 (Bool "for" (Set (Var "G")) "being" ($#v2_glib_000 :::"finite"::: ) ($#v6_chord :::"chordal"::: ) ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Bool "not" (Set (Var "a")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) ))) "holds" (Bool "for" (Set (Var "S")) "being" ($#m2_chord :::"VertexSeparator"::: ) "of" (Set (Var "a")) "," (Set (Var "b")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v4_chord :::"minimal"::: ) )) "holds" (Bool "for" (Set (Var "H")) "being" ($#m2_glib_000 :::"removeVertices":::) "of" (Set (Var "G")) "," (Set (Var "S")) (Bool "for" (Set (Var "a1")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "H")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "a1")))) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) "holds" (Bool "ex" (Set (Var "c")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool "(" (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "H")) ($#k1_glib_002 :::".reachableFrom"::: ) (Set (Var "a1")))) & (Bool (Set (Var "c")) "," (Set (Var "x")) ($#r1_chord :::"are_adjacent"::: ) ) & (Bool (Set (Var "c")) "," (Set (Var "y")) ($#r1_chord :::"are_adjacent"::: ) ) ")" )))))))) ; theorem :: CHORD:102 (Bool "for" (Set (Var "G")) "being" ($#v2_glib_000 :::"finite"::: ) ($#~v4_glib_000 "non" ($#v4_glib_000 :::"trivial"::: ) ) ($#v6_chord :::"chordal"::: ) ($#m1_hidden :::"_Graph":::) "st" (Bool (Bool (Bool "not" (Set (Var "G")) "is" ($#v2_chord :::"complete"::: ) ))) "holds" (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool "(" (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Bool "not" (Set (Var "a")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) )) & (Bool (Set (Var "a")) "is" ($#v3_chord :::"simplicial"::: ) ) & (Bool (Set (Var "b")) "is" ($#v3_chord :::"simplicial"::: ) ) ")" ))) ; theorem :: CHORD:103 (Bool "for" (Set (Var "G")) "being" ($#v2_glib_000 :::"finite"::: ) ($#v6_chord :::"chordal"::: ) ($#m1_hidden :::"_Graph":::) (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Set (Var "v")) "is" ($#v3_chord :::"simplicial"::: ) ))) ; begin definitionlet "G" be ($#v2_glib_000 :::"finite"::: ) ($#m1_hidden :::"_Graph":::); mode :::"VertexScheme"::: "of" "G" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k6_glib_000 :::"the_Vertices_of"::: ) "G") means :: CHORD:def 12 (Bool "(" (Bool it "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k6_glib_000 :::"the_Vertices_of"::: ) "G")) ")" ); end; :: deftheorem defines :::"VertexScheme"::: CHORD:def 12 : (Bool "for" (Set (Var "G")) "being" ($#v2_glib_000 :::"finite"::: ) ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G"))) "holds" (Bool "(" (Bool (Set (Var "b2")) "is" ($#m3_chord :::"VertexScheme"::: ) "of" (Set (Var "G"))) "iff" (Bool "(" (Bool (Set (Var "b2")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")))) ")" ) ")" ))); registrationlet "G" be ($#v2_glib_000 :::"finite"::: ) ($#m1_hidden :::"_Graph":::); cluster -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) for ($#m3_chord :::"VertexScheme"::: ) "of" "G"; end; theorem :: CHORD:104 (Bool "for" (Set (Var "G")) "being" ($#v2_glib_000 :::"finite"::: ) ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "S")) "being" ($#m3_chord :::"VertexScheme"::: ) "of" (Set (Var "G")) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) (Set (Var "G")) ")" ))))) ; theorem :: CHORD:105 (Bool "for" (Set (Var "G")) "being" ($#v2_glib_000 :::"finite"::: ) ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "S")) "being" ($#m3_chord :::"VertexScheme"::: ) "of" (Set (Var "G")) "holds" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "S")))))) ; theorem :: CHORD:106 (Bool "for" (Set (Var "G")) "," (Set (Var "H")) "being" ($#v2_glib_000 :::"finite"::: ) ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "g")) "being" ($#m3_chord :::"VertexScheme"::: ) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "G")) ($#r5_glib_000 :::"=="::: ) (Set (Var "H")))) "holds" (Bool (Set (Var "g")) "is" ($#m3_chord :::"VertexScheme"::: ) "of" (Set (Var "H"))))) ; definitionlet "G" be ($#v2_glib_000 :::"finite"::: ) ($#m1_hidden :::"_Graph":::); let "S" be ($#m3_chord :::"VertexScheme"::: ) "of" (Set (Const "G")); let "x" be ($#m1_subset_1 :::"Vertex":::) "of" (Set (Const "G")); :: original: :::".."::: redefine func "x" :::".."::: "S" -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); end; definitionlet "G" be ($#v2_glib_000 :::"finite"::: ) ($#m1_hidden :::"_Graph":::); let "S" be ($#m3_chord :::"VertexScheme"::: ) "of" (Set (Const "G")); let "n" be ($#m1_hidden :::"Nat":::); :: original: :::".followSet"::: redefine func "S" :::".followSet"::: "n" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_glib_000 :::"the_Vertices_of"::: ) "G" ")" ); end; theorem :: CHORD:107 (Bool "for" (Set (Var "G")) "being" ($#v2_glib_000 :::"finite"::: ) ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "S")) "being" ($#m3_chord :::"VertexScheme"::: ) "of" (Set (Var "G")) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "S"))))) "holds" (Bool "not" (Bool (Set (Set (Var "S")) ($#k4_chord :::".followSet"::: ) (Set (Var "n"))) "is" ($#v1_xboole_0 :::"empty"::: ) ))))) ; definitionlet "G" be ($#v2_glib_000 :::"finite"::: ) ($#m1_hidden :::"_Graph":::); let "S" be ($#m3_chord :::"VertexScheme"::: ) "of" (Set (Const "G")); attr "S" is :::"perfect"::: means :: CHORD:def 13 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "S"))) "holds" (Bool "for" (Set (Var "Gf")) "being" ($#m2_glib_000 :::"inducedSubgraph":::) "of" "G" "," (Set "(" "S" ($#k4_chord :::".followSet"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "Gf")) "st" (Bool (Bool (Set (Var "v")) ($#r1_hidden :::"="::: ) (Set "S" ($#k1_funct_1 :::"."::: ) (Set (Var "n"))))) "holds" (Bool (Set (Var "v")) "is" ($#v3_chord :::"simplicial"::: ) )))); end; :: deftheorem defines :::"perfect"::: CHORD:def 13 : (Bool "for" (Set (Var "G")) "being" ($#v2_glib_000 :::"finite"::: ) ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "S")) "being" ($#m3_chord :::"VertexScheme"::: ) "of" (Set (Var "G")) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v7_chord :::"perfect"::: ) ) "iff" (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "S"))))) "holds" (Bool "for" (Set (Var "Gf")) "being" ($#m2_glib_000 :::"inducedSubgraph":::) "of" (Set (Var "G")) "," (Set "(" (Set (Var "S")) ($#k4_chord :::".followSet"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "Gf")) "st" (Bool (Bool (Set (Var "v")) ($#r1_hidden :::"="::: ) (Set (Set (Var "S")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))))) "holds" (Bool (Set (Var "v")) "is" ($#v3_chord :::"simplicial"::: ) )))) ")" ))); theorem :: CHORD:108 (Bool "for" (Set (Var "G")) "being" ($#v2_glib_000 :::"finite"::: ) ($#v4_glib_000 :::"trivial"::: ) ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) (Bool "ex" (Set (Var "S")) "being" ($#m3_chord :::"VertexScheme"::: ) "of" (Set (Var "G")) "st" (Bool "(" (Bool (Set (Var "S")) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "v")) ($#k12_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "S")) "is" ($#v7_chord :::"perfect"::: ) ) ")" )))) ; theorem :: CHORD:109 (Bool "for" (Set (Var "G")) "being" ($#v2_glib_000 :::"finite"::: ) ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "V")) "being" ($#m3_chord :::"VertexScheme"::: ) "of" (Set (Var "G")) "holds" (Bool "(" (Bool (Set (Var "V")) "is" ($#v7_chord :::"perfect"::: ) ) "iff" (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Vertex":::) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "b")) ($#r1_hidden :::"<>"::: ) (Set (Var "c"))) & (Bool (Set (Var "a")) "," (Set (Var "b")) ($#r1_chord :::"are_adjacent"::: ) ) & (Bool (Set (Var "a")) "," (Set (Var "c")) ($#r1_chord :::"are_adjacent"::: ) )) "holds" (Bool "for" (Set (Var "va")) "," (Set (Var "vb")) "," (Set (Var "vc")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "va")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "V")))) & (Bool (Set (Var "vb")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "V")))) & (Bool (Set (Var "vc")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "V")))) & (Bool (Set (Set (Var "V")) ($#k1_funct_1 :::"."::: ) (Set (Var "va"))) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set (Var "V")) ($#k1_funct_1 :::"."::: ) (Set (Var "vb"))) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set (Var "V")) ($#k1_funct_1 :::"."::: ) (Set (Var "vc"))) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "va")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "vb"))) & (Bool (Set (Var "va")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "vc")))) "holds" (Bool (Set (Var "b")) "," (Set (Var "c")) ($#r1_chord :::"are_adjacent"::: ) ))) ")" ))) ; registrationlet "G" be ($#v2_glib_000 :::"finite"::: ) ($#v6_chord :::"chordal"::: ) ($#m1_hidden :::"_Graph":::); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set ($#k6_glib_000 :::"the_Vertices_of"::: ) "G") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ($#v7_chord :::"perfect"::: ) for ($#m3_chord :::"VertexScheme"::: ) "of" "G"; end; theorem :: CHORD:110 (Bool "for" (Set (Var "G")) "," (Set (Var "H")) "being" ($#v2_glib_000 :::"finite"::: ) ($#v6_chord :::"chordal"::: ) ($#m1_hidden :::"_Graph":::) (Bool "for" (Set (Var "g")) "being" ($#v7_chord :::"perfect"::: ) ($#m3_chord :::"VertexScheme"::: ) "of" (Set (Var "G")) "st" (Bool (Bool (Set (Var "G")) ($#r5_glib_000 :::"=="::: ) (Set (Var "H")))) "holds" (Bool (Set (Var "g")) "is" ($#v7_chord :::"perfect"::: ) ($#m3_chord :::"VertexScheme"::: ) "of" (Set (Var "H"))))) ; theorem :: CHORD:111 (Bool "for" (Set (Var "G")) "being" ($#v2_glib_000 :::"finite"::: ) ($#m1_hidden :::"_Graph":::) "st" (Bool (Bool "ex" (Set (Var "S")) "being" ($#m3_chord :::"VertexScheme"::: ) "of" (Set (Var "G")) "st" (Bool (Set (Var "S")) "is" ($#v7_chord :::"perfect"::: ) ))) "holds" (Bool (Set (Var "G")) "is" ($#v6_chord :::"chordal"::: ) )) ;