:: POLYFORM semantic presentation begin theorem :: POLYFORM:1 (Bool "for" (Set (Var "X")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b"))) & (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k2_tarski :::"}"::: ) )) ")" )) & (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "c")) ($#r1_hidden :::"<>"::: ) (Set (Var "d")))) "holds" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k2_tarski :::"}"::: ) ))) ; begin theorem :: POLYFORM:2 canceled; theorem :: POLYFORM:3 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k")))) "holds" (Bool (Set (Var "k")) "is" ($#m1_hidden :::"Nat":::))) ; definitionlet "a" be ($#m1_hidden :::"Integer":::); let "b" be ($#m1_hidden :::"Nat":::); :: original: :::"*"::: redefine func "a" :::"*"::: "b" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ); end; theorem :: POLYFORM:4 (Bool (Num 1) "is" ($#v1_abian :::"odd"::: ) ) ; theorem :: POLYFORM:5 (Bool (Num 2) "is" ($#v1_abian :::"even"::: ) ) ; theorem :: POLYFORM:6 (Bool (Num 3) "is" ($#v1_abian :::"odd"::: ) ) ; theorem :: POLYFORM:7 (Bool (Num 4) "is" ($#v1_abian :::"even"::: ) ) ; theorem :: POLYFORM:8 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) "is" ($#v1_abian :::"even"::: ) )) "holds" (Bool (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k1_newton :::"|^"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Num 1))) ; theorem :: POLYFORM:9 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) "is" ($#v1_abian :::"odd"::: ) )) "holds" (Bool (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k1_newton :::"|^"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)))) ; theorem :: POLYFORM:10 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k1_newton :::"|^"::: ) (Set (Var "n"))) "is" ($#m1_hidden :::"Integer":::))) ; definitionlet "a" be ($#m1_hidden :::"Integer":::); let "n" be ($#m1_hidden :::"Nat":::); :: original: :::"|^"::: redefine func "a" :::"|^"::: "n" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ); end; theorem :: POLYFORM:11 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set "(" (Set (Var "q")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "r")) ")" ) ")" )))) ; theorem :: POLYFORM:12 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "n")) ($#k2_xcmplx_0 :::"+"::: ) (Num 2)))) ; theorem :: POLYFORM:13 (Bool (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k2_polyform :::"|^"::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Num 1)) ; theorem :: POLYFORM:14 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k2_polyform :::"|^"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k2_polyform :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k2_xcmplx_0 :::"+"::: ) (Num 2) ")" )))) ; begin theorem :: POLYFORM:15 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "s"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "s")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "s")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "s"))))) "holds" (Bool (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set (Var "b")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "s"))))) "holds" (Bool (Set (Set (Var "b")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set (Var "a")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ) ")" ))) ")" )) "holds" (Bool (Set ($#k16_rvsum_1 :::"Sum"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k1_funct_1 :::"."::: ) (Num 1) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set (Var "b")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "s")) ")" ) ")" )))) ; theorem :: POLYFORM:16 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "r")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "q")) ")" ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "r")) ")" )))) ; theorem :: POLYFORM:17 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "p")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" ) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "x"))))) ; theorem :: POLYFORM:18 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "q")) ")" ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "x"))))) ; theorem :: POLYFORM:19 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" )))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "r")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ")" ))))) ; definitionlet "a" be ($#m1_hidden :::"Integer":::); :: original: :::"<*"::: redefine func :::"<*":::"a":::"*>"::: -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ); end; definitionlet "a", "b" be ($#m1_hidden :::"Integer":::); :: original: :::"<*"::: redefine func :::"<*":::"a" "," "b":::"*>"::: -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ); end; definitionlet "a", "b", "c" be ($#m1_hidden :::"Integer":::); :: original: :::"<*"::: redefine func :::"<*":::"a" "," "b" "," "c":::"*>"::: -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ); end; definitionlet "p", "q" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ); :: original: :::"^"::: redefine func "p" :::"^"::: "q" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ); end; theorem :: POLYFORM:20 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) "holds" (Bool (Set ($#k16_rvsum_1 :::"Sum"::: ) (Set "(" (Set ($#k3_polyform :::"<*"::: ) (Set (Var "k")) ($#k3_polyform :::"*>"::: ) ) ($#k6_polyform :::"^"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "k")) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" ($#k16_rvsum_1 :::"Sum"::: ) (Set (Var "p")) ")" ))))) ; theorem :: POLYFORM:21 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) "holds" (Bool (Set ($#k16_rvsum_1 :::"Sum"::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k6_polyform :::"^"::: ) (Set (Var "q")) ")" ) ($#k6_polyform :::"^"::: ) (Set (Var "r")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k16_rvsum_1 :::"Sum"::: ) (Set (Var "p")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" ($#k16_rvsum_1 :::"Sum"::: ) (Set (Var "q")) ")" ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" ($#k16_rvsum_1 :::"Sum"::: ) (Set (Var "r")) ")" )))) ; theorem :: POLYFORM:22 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) "holds" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set ($#k3_pre_poly :::"<*"::: ) (Set (Var "a")) ($#k3_pre_poly :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "a")))) ; begin definitionlet "X", "Y" be ($#m1_hidden :::"set"::: ) ; mode incidence-matrix of "X" "," "Y" is ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k2_zfmisc_1 :::"[:"::: ) "X" "," "Y" ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set ($#k2_tarski :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ) ")" ) "," (Set "(" ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ) ")" ) ($#k2_tarski :::"}"::: ) ) ")" ); end; theorem :: POLYFORM:23 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#k8_funcop_1 :::"-->"::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ) ")" )) "is" ($#m2_funct_2 :::"incidence-matrix":::) "of" (Set (Var "X")) "," (Set (Var "Y")))) ; definitionattr "c1" is :::"strict"::: ; struct :::"PolyhedronStr"::: -> ; aggr :::"PolyhedronStr":::(# :::"PolytopsF":::, :::"IncidenceF"::: #) -> ($#l1_polyform :::"PolyhedronStr"::: ) ; sel :::"PolytopsF"::: "c1" -> ($#v1_pre_poly :::"FinSequence-yielding"::: ) ($#m1_hidden :::"FinSequence":::); sel :::"IncidenceF"::: "c1" -> ($#v1_funcop_1 :::"Function-yielding"::: ) ($#m1_hidden :::"FinSequence":::); end; definitionlet "p" be ($#l1_polyform :::"PolyhedronStr"::: ) ; attr "p" is :::"polyhedron_1"::: means :: POLYFORM:def 1 (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "the" ($#u2_polyform :::"IncidenceF"::: ) "of" "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "the" ($#u1_polyform :::"PolytopsF"::: ) "of" "p") ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1))); attr "p" is :::"polyhedron_2"::: means :: POLYFORM:def 2 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "the" ($#u1_polyform :::"PolytopsF"::: ) "of" "p")))) "holds" (Bool (Set (Set "the" ($#u2_polyform :::"IncidenceF"::: ) "of" "p") ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) "is" ($#m2_funct_2 :::"incidence-matrix":::) "of" (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set "the" ($#u1_polyform :::"PolytopsF"::: ) "of" "p") ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set "the" ($#u1_polyform :::"PolytopsF"::: ) "of" "p") ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ))); attr "p" is :::"polyhedron_3"::: means :: POLYFORM:def 3 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "the" ($#u1_polyform :::"PolytopsF"::: ) "of" "p")))) "holds" (Bool "(" (Bool (Bool "not" (Set (Set "the" ($#u1_polyform :::"PolytopsF"::: ) "of" "p") ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) "is" ($#v1_xboole_0 :::"empty"::: ) )) & (Bool (Set (Set "the" ($#u1_polyform :::"PolytopsF"::: ) "of" "p") ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" )); end; :: deftheorem defines :::"polyhedron_1"::: POLYFORM:def 1 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"PolyhedronStr"::: ) "holds" (Bool "(" (Bool (Set (Var "p")) "is" ($#v2_polyform :::"polyhedron_1"::: ) ) "iff" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "the" ($#u2_polyform :::"IncidenceF"::: ) "of" (Set (Var "p")))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "the" ($#u1_polyform :::"PolytopsF"::: ) "of" (Set (Var "p"))) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1))) ")" )); :: deftheorem defines :::"polyhedron_2"::: POLYFORM:def 2 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"PolyhedronStr"::: ) "holds" (Bool "(" (Bool (Set (Var "p")) "is" ($#v3_polyform :::"polyhedron_2"::: ) ) "iff" (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "the" ($#u1_polyform :::"PolytopsF"::: ) "of" (Set (Var "p")))))) "holds" (Bool (Set (Set "the" ($#u2_polyform :::"IncidenceF"::: ) "of" (Set (Var "p"))) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) "is" ($#m2_funct_2 :::"incidence-matrix":::) "of" (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set "the" ($#u1_polyform :::"PolytopsF"::: ) "of" (Set (Var "p"))) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set "the" ($#u1_polyform :::"PolytopsF"::: ) "of" (Set (Var "p"))) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ))) ")" )); :: deftheorem defines :::"polyhedron_3"::: POLYFORM:def 3 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"PolyhedronStr"::: ) "holds" (Bool "(" (Bool (Set (Var "p")) "is" ($#v4_polyform :::"polyhedron_3"::: ) ) "iff" (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "the" ($#u1_polyform :::"PolytopsF"::: ) "of" (Set (Var "p")))))) "holds" (Bool "(" (Bool (Bool "not" (Set (Set "the" ($#u1_polyform :::"PolytopsF"::: ) "of" (Set (Var "p"))) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) "is" ($#v1_xboole_0 :::"empty"::: ) )) & (Bool (Set (Set "the" ($#u1_polyform :::"PolytopsF"::: ) "of" (Set (Var "p"))) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" )) ")" )); registration cluster ($#v2_polyform :::"polyhedron_1"::: ) ($#v3_polyform :::"polyhedron_2"::: ) ($#v4_polyform :::"polyhedron_3"::: ) for ($#l1_polyform :::"PolyhedronStr"::: ) ; end; definitionmode polyhedron is ($#v2_polyform :::"polyhedron_1"::: ) ($#v3_polyform :::"polyhedron_2"::: ) ($#v4_polyform :::"polyhedron_3"::: ) ($#l1_polyform :::"PolyhedronStr"::: ) ; end; definitionlet "p" be ($#l1_polyform :::"polyhedron":::); func :::"dim"::: "p" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: POLYFORM:def 4 (Set ($#k3_finseq_1 :::"len"::: ) (Set "the" ($#u1_polyform :::"PolytopsF"::: ) "of" "p")); end; :: deftheorem defines :::"dim"::: POLYFORM:def 4 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "the" ($#u1_polyform :::"PolytopsF"::: ) "of" (Set (Var "p")))))); definitionlet "p" be ($#l1_polyform :::"polyhedron":::); let "k" be ($#m1_hidden :::"Integer":::); func "k" :::"-polytopes"::: "p" -> ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) means :: POLYFORM:def 5 (Bool "(" "(" (Bool (Bool "k" ($#r1_xxreal_0 :::"<"::: ) (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)))) "implies" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" & "(" (Bool (Bool "k" ($#r1_hidden :::"="::: ) (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)))) "implies" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) ")" & "(" (Bool (Bool (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) "k") & (Bool "k" ($#r1_xxreal_0 :::"<"::: ) (Set ($#k7_polyform :::"dim"::: ) "p"))) "implies" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set "the" ($#u1_polyform :::"PolytopsF"::: ) "of" "p") ($#k1_funct_1 :::"."::: ) (Set "(" "k" ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ) ")" ))) ")" & "(" (Bool (Bool "k" ($#r1_hidden :::"="::: ) (Set ($#k7_polyform :::"dim"::: ) "p"))) "implies" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) "p" ($#k1_tarski :::"}"::: ) )) ")" & "(" (Bool (Bool "k" ($#r1_xxreal_0 :::">"::: ) (Set ($#k7_polyform :::"dim"::: ) "p"))) "implies" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ")" ); end; :: deftheorem defines :::"-polytopes"::: POLYFORM:def 5 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "b3")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "k")) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p")))) "iff" (Bool "(" "(" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)))) "implies" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" & "(" (Bool (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)))) "implies" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) ")" & "(" (Bool (Bool (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))))) "implies" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set "the" ($#u1_polyform :::"PolytopsF"::: ) "of" (Set (Var "p"))) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))))) "implies" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )) ")" & "(" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))))) "implies" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ")" ) ")" )))); theorem :: POLYFORM:24 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))))) "holds" (Bool "(" (Bool (Set (Set (Var "k")) ($#k2_xcmplx_0 :::"+"::: ) (Num 1)) "is" ($#m1_hidden :::"Nat":::)) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "k")) ($#k2_xcmplx_0 :::"+"::: ) (Num 1))) & (Bool (Set (Set (Var "k")) ($#k2_xcmplx_0 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p")))) ")" ))) ; theorem :: POLYFORM:25 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool "(" (Bool (Bool "not" (Set (Set (Var "k")) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "is" ($#v1_xboole_0 :::"empty"::: ) )) "iff" (Bool "(" (Bool (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p")))) ")" ) ")" ))) ; theorem :: POLYFORM:26 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p")))))) ; definitionlet "p" be ($#l1_polyform :::"polyhedron":::); let "k" be ($#m1_hidden :::"Integer":::); func :::"eta"::: "(" "p" "," "k" ")" -> ($#m2_funct_2 :::"incidence-matrix":::) "of" (Set "(" (Set "(" "k" ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) "p" ")" ) "," (Set "(" "k" ($#k8_polyform :::"-polytopes"::: ) "p" ")" ) means :: POLYFORM:def 6 (Bool "(" "(" (Bool (Bool "k" ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" & "(" (Bool (Bool "k" ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool it ($#r1_funct_2 :::"="::: ) (Set (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) "," (Set "(" (Set ($#k6_numbers :::"0"::: ) ) ($#k8_polyform :::"-polytopes"::: ) "p" ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#k8_funcop_1 :::"-->"::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ) ")" ))) ")" & "(" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) "k") & (Bool "k" ($#r1_xxreal_0 :::"<"::: ) (Set ($#k7_polyform :::"dim"::: ) "p"))) "implies" (Bool it ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u2_polyform :::"IncidenceF"::: ) "of" "p") ($#k1_funct_1 :::"."::: ) "k")) ")" & "(" (Bool (Bool "k" ($#r1_hidden :::"="::: ) (Set ($#k7_polyform :::"dim"::: ) "p"))) "implies" (Bool it ($#r1_funct_2 :::"="::: ) (Set (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) "p" ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) "p" ")" ) "," (Set ($#k1_tarski :::"{"::: ) "p" ($#k1_tarski :::"}"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#k8_funcop_1 :::"-->"::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ) ")" ))) ")" & "(" (Bool (Bool "k" ($#r1_xxreal_0 :::">"::: ) (Set ($#k7_polyform :::"dim"::: ) "p"))) "implies" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ")" ); end; :: deftheorem defines :::"eta"::: POLYFORM:def 6 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "b3")) "being" ($#m2_funct_2 :::"incidence-matrix":::) "of" (Set "(" (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p")) ")" ) "," (Set "(" (Set (Var "k")) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k9_polyform :::"eta"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" )) "iff" (Bool "(" "(" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" & "(" (Bool (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Var "b3")) ($#r1_funct_2 :::"="::: ) (Set (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) "," (Set "(" (Set ($#k6_numbers :::"0"::: ) ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#k8_funcop_1 :::"-->"::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ) ")" ))) ")" & "(" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))))) "implies" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u2_polyform :::"IncidenceF"::: ) "of" (Set (Var "p"))) ($#k1_funct_1 :::"."::: ) (Set (Var "k")))) ")" & "(" (Bool (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))))) "implies" (Bool (Set (Var "b3")) ($#r1_funct_2 :::"="::: ) (Set (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p")) ")" ) "," (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#k8_funcop_1 :::"-->"::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))))) "implies" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ")" ) ")" )))); definitionlet "p" be ($#l1_polyform :::"polyhedron":::); let "k" be ($#m1_hidden :::"Integer":::); func "k" :::"-polytope-seq"::: "p" -> ($#m1_hidden :::"FinSequence":::) means :: POLYFORM:def 7 (Bool "(" "(" (Bool (Bool "k" ($#r1_xxreal_0 :::"<"::: ) (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)))) "implies" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k2_pre_poly :::"<*>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ")" & "(" (Bool (Bool "k" ($#r1_hidden :::"="::: ) (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)))) "implies" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k3_polyform :::"<*"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k3_polyform :::"*>"::: ) )) ")" & "(" (Bool (Bool (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) "k") & (Bool "k" ($#r1_xxreal_0 :::"<"::: ) (Set ($#k7_polyform :::"dim"::: ) "p"))) "implies" (Bool it ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_polyform :::"PolytopsF"::: ) "of" "p") ($#k1_funct_1 :::"."::: ) (Set "(" "k" ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ))) ")" & "(" (Bool (Bool "k" ($#r1_hidden :::"="::: ) (Set ($#k7_polyform :::"dim"::: ) "p"))) "implies" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) "p" ($#k9_finseq_1 :::"*>"::: ) )) ")" & "(" (Bool (Bool "k" ($#r1_xxreal_0 :::">"::: ) (Set ($#k7_polyform :::"dim"::: ) "p"))) "implies" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k2_pre_poly :::"<*>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ")" ")" ); end; :: deftheorem defines :::"-polytope-seq"::: POLYFORM:def 7 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "b3")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "k")) ($#k10_polyform :::"-polytope-seq"::: ) (Set (Var "p")))) "iff" (Bool "(" "(" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)))) "implies" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k2_pre_poly :::"<*>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ")" & "(" (Bool (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)))) "implies" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k3_polyform :::"<*"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k3_polyform :::"*>"::: ) )) ")" & "(" (Bool (Bool (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))))) "implies" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_polyform :::"PolytopsF"::: ) "of" (Set (Var "p"))) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ))) ")" & "(" (Bool (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))))) "implies" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k9_finseq_1 :::"*>"::: ) )) ")" & "(" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))))) "implies" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k2_pre_poly :::"<*>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ")" ")" ) ")" )))); definitionlet "p" be ($#l1_polyform :::"polyhedron":::); let "k" be ($#m1_hidden :::"Integer":::); func :::"num-polytopes"::: "(" "p" "," "k" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: POLYFORM:def 8 (Set ($#k5_card_1 :::"card"::: ) (Set "(" "k" ($#k8_polyform :::"-polytopes"::: ) "p" ")" )); end; :: deftheorem defines :::"num-polytopes"::: POLYFORM:def 8 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set "(" (Set (Var "k")) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p")) ")" ))))); definitionlet "p" be ($#l1_polyform :::"polyhedron":::); func :::"num-vertices"::: "p" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: POLYFORM:def 9 (Set ($#k11_polyform :::"num-polytopes"::: ) "(" "p" "," (Set ($#k6_numbers :::"0"::: ) ) ")" ); func :::"num-edges"::: "p" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: POLYFORM:def 10 (Set ($#k11_polyform :::"num-polytopes"::: ) "(" "p" "," (Num 1) ")" ); func :::"num-faces"::: "p" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: POLYFORM:def 11 (Set ($#k11_polyform :::"num-polytopes"::: ) "(" "p" "," (Num 2) ")" ); end; :: deftheorem defines :::"num-vertices"::: POLYFORM:def 9 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Set ($#k12_polyform :::"num-vertices"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ))); :: deftheorem defines :::"num-edges"::: POLYFORM:def 10 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Set ($#k13_polyform :::"num-edges"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Num 1) ")" ))); :: deftheorem defines :::"num-faces"::: POLYFORM:def 11 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Set ($#k14_polyform :::"num-faces"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Num 2) ")" ))); theorem :: POLYFORM:27 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "k")) ($#k10_polyform :::"-polytope-seq"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" ")" ))))) ; theorem :: POLYFORM:28 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "k")) ($#k10_polyform :::"-polytope-seq"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" )))) ; theorem :: POLYFORM:29 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "k")) ($#k10_polyform :::"-polytope-seq"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "k")) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p")))))) ; theorem :: POLYFORM:30 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Set ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Num 1))) ; theorem :: POLYFORM:31 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Set ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Num 1))) ; definitionlet "p" be ($#l1_polyform :::"polyhedron":::); let "k" be ($#m1_hidden :::"Integer":::); let "n" be ($#m1_hidden :::"Nat":::); assume (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Const "n"))) & (Bool (Set (Const "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Const "p")) "," (Set (Const "k")) ")" )) ")" ) ; func "n" :::"-th-polytope"::: "(" "p" "," "k" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set "k" ($#k8_polyform :::"-polytopes"::: ) "p") equals :: POLYFORM:def 12 (Set (Set "(" "k" ($#k10_polyform :::"-polytope-seq"::: ) "p" ")" ) ($#k1_funct_1 :::"."::: ) "n"); end; :: deftheorem defines :::"-th-polytope"::: POLYFORM:def 12 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" ))) "holds" (Bool (Set (Set (Var "n")) ($#k15_polyform :::"-th-polytope"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "k")) ($#k10_polyform :::"-polytope-seq"::: ) (Set (Var "p")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))))))); theorem :: POLYFORM:32 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))))) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "k")) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) (Bool "ex" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k15_polyform :::"-th-polytope"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" )) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" )) ")" ))))) ; theorem :: POLYFORM:33 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set (Var "k")) ($#k10_polyform :::"-polytope-seq"::: ) (Set (Var "p"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) ))) ; theorem :: POLYFORM:34 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" )) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" )) & (Bool (Set (Set (Var "n")) ($#k15_polyform :::"-th-polytope"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "m")) ($#k15_polyform :::"-th-polytope"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" ))) "holds" (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set (Var "n")))))) ; definitionlet "p" be ($#l1_polyform :::"polyhedron":::); let "k" be ($#m1_hidden :::"Integer":::); let "x" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set (Const "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Const "p"))); let "y" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Const "k")) ($#k8_polyform :::"-polytopes"::: ) (Set (Const "p"))); assume that (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Const "k"))) and (Bool (Set (Const "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k7_polyform :::"dim"::: ) (Set (Const "p")))) ; func :::"incidence-value"::: "(" "x" "," "y" ")" -> ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) equals :: POLYFORM:def 13 (Set (Set "(" ($#k9_polyform :::"eta"::: ) "(" "p" "," "k" ")" ")" ) ($#k1_binop_1 :::"."::: ) "(" "x" "," "y" ")" ); end; :: deftheorem defines :::"incidence-value"::: POLYFORM:def 13 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "k")) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))))) "holds" (Bool (Set ($#k16_polyform :::"incidence-value"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_polyform :::"eta"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" )))))); begin definitionlet "p" be ($#l1_polyform :::"polyhedron":::); let "k" be ($#m1_hidden :::"Integer":::); func "k" :::"-chain-space"::: "p" -> ($#v1_matrlin :::"finite-dimensional"::: ) ($#l1_vectsp_1 :::"VectSp":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) equals :: POLYFORM:def 14 (Set ($#k7_bspace :::"bspace"::: ) (Set "(" "k" ($#k8_polyform :::"-polytopes"::: ) "p" ")" )); end; :: deftheorem defines :::"-chain-space"::: POLYFORM:def 14 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k7_bspace :::"bspace"::: ) (Set "(" (Set (Var "k")) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p")) ")" ))))); theorem :: POLYFORM:35 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "k")) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "holds" (Bool (Set (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) ")" ) ($#k3_bspace :::"@"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set ($#k2_bspace :::"Z_2"::: ) )))))) ; theorem :: POLYFORM:36 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_vectsp_9 :::"dim"::: ) (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ))))) ; definitionlet "p" be ($#l1_polyform :::"polyhedron":::); let "k" be ($#m1_hidden :::"Integer":::); func "k" :::"-chains"::: "p" -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) equals :: POLYFORM:def 15 (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" "k" ($#k8_polyform :::"-polytopes"::: ) "p" ")" )); end; :: deftheorem defines :::"-chains"::: POLYFORM:def 15 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set (Var "k")) ($#k18_polyform :::"-chains"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" (Set (Var "k")) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p")) ")" ))))); definitionlet "p" be ($#l1_polyform :::"polyhedron":::); let "k" be ($#m1_hidden :::"Integer":::); let "x" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set (Const "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Const "p"))); let "v" be ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Const "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Const "p")) ")" ); func :::"incidence-sequence"::: "(" "x" "," "v" ")" -> ($#m2_finseq_1 :::"FinSequence":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) means :: POLYFORM:def 16 (Bool "(" "(" (Bool (Bool (Set (Set "(" "k" ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) "p") "is" ($#v1_xboole_0 :::"empty"::: ) )) "implies" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k2_pre_poly :::"<*>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ")" & "(" (Bool (Bool (Bool "not" (Set (Set "(" "k" ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) "p") "is" ($#v1_xboole_0 :::"empty"::: ) ))) "implies" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k11_polyform :::"num-polytopes"::: ) "(" "p" "," "k" ")" )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k11_polyform :::"num-polytopes"::: ) "(" "p" "," "k" ")" ))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "v" ($#k3_bspace :::"@"::: ) (Set "(" (Set (Var "n")) ($#k15_polyform :::"-th-polytope"::: ) "(" "p" "," "k" ")" ")" ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" ($#k16_polyform :::"incidence-value"::: ) "(" "x" "," (Set "(" (Set (Var "n")) ($#k15_polyform :::"-th-polytope"::: ) "(" "p" "," "k" ")" ")" ) ")" ")" ))) ")" ) ")" ) ")" ")" ); end; :: deftheorem defines :::"incidence-sequence"::: POLYFORM:def 16 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) (Bool "for" (Set (Var "b5")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set ($#k19_polyform :::"incidence-sequence"::: ) "(" (Set (Var "x")) "," (Set (Var "v")) ")" )) "iff" (Bool "(" "(" (Bool (Bool (Set (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "is" ($#v1_xboole_0 :::"empty"::: ) )) "implies" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set ($#k2_pre_poly :::"<*>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ")" & "(" (Bool (Bool (Bool "not" (Set (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "is" ($#v1_xboole_0 :::"empty"::: ) ))) "implies" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b5"))) ($#r1_hidden :::"="::: ) (Set ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" ))) "holds" (Bool (Set (Set (Var "b5")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "v")) ($#k3_bspace :::"@"::: ) (Set "(" (Set (Var "n")) ($#k15_polyform :::"-th-polytope"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" ")" ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" ($#k16_polyform :::"incidence-value"::: ) "(" (Set (Var "x")) "," (Set "(" (Set (Var "n")) ($#k15_polyform :::"-th-polytope"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" ")" ) ")" ")" ))) ")" ) ")" ) ")" ")" ) ")" )))))); theorem :: POLYFORM:37 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "k")) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "holds" (Bool (Set (Set "(" (Set (Var "c")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "d")) ")" ) ($#k3_bspace :::"@"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "c")) ($#k3_bspace :::"@"::: ) (Set (Var "x")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "d")) ($#k3_bspace :::"@"::: ) (Set (Var "x")) ")" ))))))) ; theorem :: POLYFORM:38 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "holds" (Bool (Set ($#k19_polyform :::"incidence-sequence"::: ) "(" (Set (Var "x")) "," (Set "(" (Set (Var "c")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "d")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k19_polyform :::"incidence-sequence"::: ) "(" (Set (Var "x")) "," (Set (Var "c")) ")" ")" ) ($#k3_fvsum_1 :::"+"::: ) (Set "(" ($#k19_polyform :::"incidence-sequence"::: ) "(" (Set (Var "x")) "," (Set (Var "d")) ")" ")" ))))))) ; theorem :: POLYFORM:39 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "holds" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" (Set "(" ($#k19_polyform :::"incidence-sequence"::: ) "(" (Set (Var "x")) "," (Set (Var "c")) ")" ")" ) ($#k3_fvsum_1 :::"+"::: ) (Set "(" ($#k19_polyform :::"incidence-sequence"::: ) "(" (Set (Var "x")) "," (Set (Var "d")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" ($#k19_polyform :::"incidence-sequence"::: ) "(" (Set (Var "x")) "," (Set (Var "c")) ")" ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" ($#k19_polyform :::"incidence-sequence"::: ) "(" (Set (Var "x")) "," (Set (Var "d")) ")" ")" ) ")" ))))))) ; theorem :: POLYFORM:40 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "holds" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" ($#k19_polyform :::"incidence-sequence"::: ) "(" (Set (Var "x")) "," (Set "(" (Set (Var "c")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "d")) ")" ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" ($#k19_polyform :::"incidence-sequence"::: ) "(" (Set (Var "x")) "," (Set (Var "c")) ")" ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" ($#k19_polyform :::"incidence-sequence"::: ) "(" (Set (Var "x")) "," (Set (Var "d")) ")" ")" ) ")" ))))))) ; theorem :: POLYFORM:41 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "k")) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "c")) ")" ) ($#k3_bspace :::"@"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "c")) ($#k3_bspace :::"@"::: ) (Set (Var "x")) ")" )))))))) ; theorem :: POLYFORM:42 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "holds" (Bool (Set ($#k19_polyform :::"incidence-sequence"::: ) "(" (Set (Var "x")) "," (Set "(" (Set (Var "a")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "c")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k9_fvsum_1 :::"*"::: ) (Set "(" ($#k19_polyform :::"incidence-sequence"::: ) "(" (Set (Var "x")) "," (Set (Var "c")) ")" ")" )))))))) ; theorem :: POLYFORM:43 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) "holds" (Bool "(" (Bool (Set (Var "c")) ($#r1_hidden :::"="::: ) (Set (Var "d"))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "k")) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "holds" (Bool (Set (Set (Var "c")) ($#k3_bspace :::"@"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "d")) ($#k3_bspace :::"@"::: ) (Set (Var "x"))))) ")" )))) ; theorem :: POLYFORM:44 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) "holds" (Bool "(" (Bool (Set (Var "c")) ($#r1_hidden :::"="::: ) (Set (Var "d"))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "k")) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "c"))) "iff" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "d"))) ")" )) ")" )))) ; scheme :: POLYFORM:sch 1 ChainEx{ F1() -> ($#l1_polyform :::"polyhedron":::), F2() -> ($#m1_hidden :::"Integer":::), P1[ ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set F2 "(" ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set F1 "(" ")" ))] } : (Bool "ex" (Set (Var "c")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set F2 "(" ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set F1 "(" ")" ) ")" ) "st" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set F2 "(" ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set F1 "(" ")" )) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "c"))) "iff" (Bool "(" (Bool P1[(Set (Var "x"))]) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set F2 "(" ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set F1 "(" ")" ))) ")" ) ")" ))) proof end; definitionlet "p" be ($#l1_polyform :::"polyhedron":::); let "k" be ($#m1_hidden :::"Integer":::); let "v" be ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Const "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Const "p")) ")" ); func :::"Boundary"::: "v" -> ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set "(" "k" ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k17_polyform :::"-chain-space"::: ) "p" ")" ) means :: POLYFORM:def 17 (Bool "(" "(" (Bool (Bool (Set (Set "(" "k" ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) "p") "is" ($#v1_xboole_0 :::"empty"::: ) )) "implies" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" (Set "(" "k" ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k17_polyform :::"-chain-space"::: ) "p" ")" ))) ")" & "(" (Bool (Bool (Bool "not" (Set (Set "(" "k" ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) "p") "is" ($#v1_xboole_0 :::"empty"::: ) ))) "implies" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" "k" ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) "p") "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" ($#k19_polyform :::"incidence-sequence"::: ) "(" (Set (Var "x")) "," "v" ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ))) ")" )) ")" ")" ); end; :: deftheorem defines :::"Boundary"::: POLYFORM:def 17 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k20_polyform :::"Boundary"::: ) (Set (Var "v")))) "iff" (Bool "(" "(" (Bool (Bool (Set (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "is" ($#v1_xboole_0 :::"empty"::: ) )) "implies" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ))) ")" & "(" (Bool (Bool (Bool "not" (Set (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "is" ($#v1_xboole_0 :::"empty"::: ) ))) "implies" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b4"))) "iff" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" ($#k19_polyform :::"incidence-sequence"::: ) "(" (Set (Var "x")) "," (Set (Var "v")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ))) ")" )) ")" ")" ) ")" ))))); theorem :: POLYFORM:45 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "holds" (Bool (Set (Set "(" ($#k20_polyform :::"Boundary"::: ) (Set (Var "c")) ")" ) ($#k3_bspace :::"@"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" ($#k19_polyform :::"incidence-sequence"::: ) "(" (Set (Var "x")) "," (Set (Var "c")) ")" ")" ))))))) ; definitionlet "p" be ($#l1_polyform :::"polyhedron":::); let "k" be ($#m1_hidden :::"Integer":::); func "k" :::"-boundary"::: "p" -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" "k" ($#k17_polyform :::"-chain-space"::: ) "p" ")" ) "," (Set "(" (Set "(" "k" ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k17_polyform :::"-chain-space"::: ) "p" ")" ) means :: POLYFORM:def 18 (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" "k" ($#k17_polyform :::"-chain-space"::: ) "p" ")" ) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k20_polyform :::"Boundary"::: ) (Set (Var "c"))))); end; :: deftheorem defines :::"-boundary"::: POLYFORM:def 18 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) "," (Set "(" (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "k")) ($#k21_polyform :::"-boundary"::: ) (Set (Var "p")))) "iff" (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) "holds" (Bool (Set (Set (Var "b3")) ($#k3_funct_2 :::"."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k20_polyform :::"Boundary"::: ) (Set (Var "c"))))) ")" )))); theorem :: POLYFORM:46 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) "holds" (Bool (Set ($#k20_polyform :::"Boundary"::: ) (Set "(" (Set (Var "c")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "d")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k20_polyform :::"Boundary"::: ) (Set (Var "c")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k20_polyform :::"Boundary"::: ) (Set (Var "d")) ")" )))))) ; theorem :: POLYFORM:47 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) "holds" (Bool (Set ($#k20_polyform :::"Boundary"::: ) (Set "(" (Set (Var "a")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "c")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k4_vectsp_1 :::"*"::: ) (Set "(" ($#k20_polyform :::"Boundary"::: ) (Set (Var "c")) ")" ))))))) ; theorem :: POLYFORM:48 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set (Var "k")) ($#k21_polyform :::"-boundary"::: ) (Set (Var "p"))) "is" ($#m1_subset_1 :::"linear-transformation":::) "of" (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) "," (Set "(" (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" )))) ; registrationlet "p" be ($#l1_polyform :::"polyhedron":::); let "k" be ($#m1_hidden :::"Integer":::); cluster (Set "k" ($#k21_polyform :::"-boundary"::: ) "p") -> ($#v13_vectsp_1 :::"additive"::: ) ($#v1_mod_2 :::"homogeneous"::: ) ; end; definitionlet "p" be ($#l1_polyform :::"polyhedron":::); let "k" be ($#m1_hidden :::"Integer":::); func "k" :::"-circuit-space"::: "p" -> ($#m1_vectsp_4 :::"Subspace"::: ) "of" (Set "k" ($#k17_polyform :::"-chain-space"::: ) "p") equals :: POLYFORM:def 19 (Set ($#k1_ranknull :::"ker"::: ) (Set "(" "k" ($#k21_polyform :::"-boundary"::: ) "p" ")" )); end; :: deftheorem defines :::"-circuit-space"::: POLYFORM:def 19 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set (Var "k")) ($#k22_polyform :::"-circuit-space"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k1_ranknull :::"ker"::: ) (Set "(" (Set (Var "k")) ($#k21_polyform :::"-boundary"::: ) (Set (Var "p")) ")" ))))); definitionlet "p" be ($#l1_polyform :::"polyhedron":::); let "k" be ($#m1_hidden :::"Integer":::); func "k" :::"-circuits"::: "p" -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" "k" ($#k18_polyform :::"-chains"::: ) "p" ")" ) equals :: POLYFORM:def 20 (Set ($#k2_struct_0 :::"[#]"::: ) (Set "(" "k" ($#k22_polyform :::"-circuit-space"::: ) "p" ")" )); end; :: deftheorem defines :::"-circuits"::: POLYFORM:def 20 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set (Var "k")) ($#k23_polyform :::"-circuits"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set "(" (Set (Var "k")) ($#k22_polyform :::"-circuit-space"::: ) (Set (Var "p")) ")" ))))); definitionlet "p" be ($#l1_polyform :::"polyhedron":::); let "k" be ($#m1_hidden :::"Integer":::); func "k" :::"-bounding-chain-space"::: "p" -> ($#m1_vectsp_4 :::"Subspace"::: ) "of" (Set "k" ($#k17_polyform :::"-chain-space"::: ) "p") equals :: POLYFORM:def 21 (Set ($#k3_ranknull :::"im"::: ) (Set "(" (Set "(" "k" ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ) ($#k21_polyform :::"-boundary"::: ) "p" ")" )); end; :: deftheorem defines :::"-bounding-chain-space"::: POLYFORM:def 21 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set (Var "k")) ($#k24_polyform :::"-bounding-chain-space"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_ranknull :::"im"::: ) (Set "(" (Set "(" (Set (Var "k")) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ) ($#k21_polyform :::"-boundary"::: ) (Set (Var "p")) ")" ))))); definitionlet "p" be ($#l1_polyform :::"polyhedron":::); let "k" be ($#m1_hidden :::"Integer":::); func "k" :::"-bounding-chains"::: "p" -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" "k" ($#k18_polyform :::"-chains"::: ) "p" ")" ) equals :: POLYFORM:def 22 (Set ($#k2_struct_0 :::"[#]"::: ) (Set "(" "k" ($#k24_polyform :::"-bounding-chain-space"::: ) "p" ")" )); end; :: deftheorem defines :::"-bounding-chains"::: POLYFORM:def 22 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set (Var "k")) ($#k25_polyform :::"-bounding-chains"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set "(" (Set (Var "k")) ($#k24_polyform :::"-bounding-chain-space"::: ) (Set (Var "p")) ")" ))))); definitionlet "p" be ($#l1_polyform :::"polyhedron":::); let "k" be ($#m1_hidden :::"Integer":::); func "k" :::"-bounding-circuit-space"::: "p" -> ($#m1_vectsp_4 :::"Subspace"::: ) "of" (Set "k" ($#k17_polyform :::"-chain-space"::: ) "p") equals :: POLYFORM:def 23 (Set (Set "(" "k" ($#k24_polyform :::"-bounding-chain-space"::: ) "p" ")" ) ($#k2_vectsp_5 :::"/\"::: ) (Set "(" "k" ($#k22_polyform :::"-circuit-space"::: ) "p" ")" )); end; :: deftheorem defines :::"-bounding-circuit-space"::: POLYFORM:def 23 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set (Var "k")) ($#k26_polyform :::"-bounding-circuit-space"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "k")) ($#k24_polyform :::"-bounding-chain-space"::: ) (Set (Var "p")) ")" ) ($#k2_vectsp_5 :::"/\"::: ) (Set "(" (Set (Var "k")) ($#k22_polyform :::"-circuit-space"::: ) (Set (Var "p")) ")" ))))); definitionlet "p" be ($#l1_polyform :::"polyhedron":::); let "k" be ($#m1_hidden :::"Integer":::); func "k" :::"-bounding-circuits"::: "p" -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" "k" ($#k18_polyform :::"-chains"::: ) "p" ")" ) equals :: POLYFORM:def 24 (Set ($#k2_struct_0 :::"[#]"::: ) (Set "(" "k" ($#k26_polyform :::"-bounding-circuit-space"::: ) "p" ")" )); end; :: deftheorem defines :::"-bounding-circuits"::: POLYFORM:def 24 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set (Var "k")) ($#k27_polyform :::"-bounding-circuits"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set "(" (Set (Var "k")) ($#k26_polyform :::"-bounding-circuit-space"::: ) (Set (Var "p")) ")" ))))); theorem :: POLYFORM:49 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set ($#k1_vectsp_9 :::"dim"::: ) (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_ranknull :::"rank"::: ) (Set "(" (Set (Var "k")) ($#k21_polyform :::"-boundary"::: ) (Set (Var "p")) ")" ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" ($#k11_ranknull :::"nullity"::: ) (Set "(" (Set (Var "k")) ($#k21_polyform :::"-boundary"::: ) (Set (Var "p")) ")" ) ")" ))))) ; begin definitionlet "p" be ($#l1_polyform :::"polyhedron":::); attr "p" is :::"simply-connected"::: means :: POLYFORM:def 25 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set (Var "k")) ($#k23_polyform :::"-circuits"::: ) "p") ($#r1_hidden :::"="::: ) (Set (Set (Var "k")) ($#k25_polyform :::"-bounding-chains"::: ) "p"))); end; :: deftheorem defines :::"simply-connected"::: POLYFORM:def 25 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool "(" (Bool (Set (Var "p")) "is" ($#v5_polyform :::"simply-connected"::: ) ) "iff" (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set (Var "k")) ($#k23_polyform :::"-circuits"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "k")) ($#k25_polyform :::"-bounding-chains"::: ) (Set (Var "p"))))) ")" )); theorem :: POLYFORM:50 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool "(" (Bool (Set (Var "p")) "is" ($#v5_polyform :::"simply-connected"::: ) ) "iff" (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set (Var "n")) ($#k22_polyform :::"-circuit-space"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k24_polyform :::"-bounding-chain-space"::: ) (Set (Var "p"))))) ")" )) ; definitionlet "p" be ($#l1_polyform :::"polyhedron":::); func :::"alternating-f-vector"::: "p" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) means :: POLYFORM:def 26 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_polyform :::"dim"::: ) "p" ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 2))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k7_polyform :::"dim"::: ) "p" ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 2)))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k2_polyform :::"|^"::: ) (Set (Var "k")) ")" ) ($#k1_polyform :::"*"::: ) (Set "(" ($#k11_polyform :::"num-polytopes"::: ) "(" "p" "," (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 2) ")" ) ")" ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"alternating-f-vector"::: POLYFORM:def 26 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k28_polyform :::"alternating-f-vector"::: ) (Set (Var "p")))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 2))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 2)))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k2_polyform :::"|^"::: ) (Set (Var "k")) ")" ) ($#k1_polyform :::"*"::: ) (Set "(" ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 2) ")" ) ")" ")" ))) ")" ) ")" ) ")" ))); definitionlet "p" be ($#l1_polyform :::"polyhedron":::); func :::"alternating-proper-f-vector"::: "p" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) means :: POLYFORM:def 27 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k7_polyform :::"dim"::: ) "p")) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k7_polyform :::"dim"::: ) "p"))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k2_polyform :::"|^"::: ) (Set "(" (Set (Var "k")) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_polyform :::"*"::: ) (Set "(" ($#k11_polyform :::"num-polytopes"::: ) "(" "p" "," (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ")" ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"alternating-proper-f-vector"::: POLYFORM:def 27 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k29_polyform :::"alternating-proper-f-vector"::: ) (Set (Var "p")))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k2_polyform :::"|^"::: ) (Set "(" (Set (Var "k")) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_polyform :::"*"::: ) (Set "(" ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ")" ")" ))) ")" ) ")" ) ")" ))); definitionlet "p" be ($#l1_polyform :::"polyhedron":::); func :::"alternating-semi-proper-f-vector"::: "p" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) means :: POLYFORM:def 28 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_polyform :::"dim"::: ) "p" ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 1))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k7_polyform :::"dim"::: ) "p" ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 1)))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k2_polyform :::"|^"::: ) (Set "(" (Set (Var "k")) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_polyform :::"*"::: ) (Set "(" ($#k11_polyform :::"num-polytopes"::: ) "(" "p" "," (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ")" ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"alternating-semi-proper-f-vector"::: POLYFORM:def 28 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k30_polyform :::"alternating-semi-proper-f-vector"::: ) (Set (Var "p")))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 1))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 1)))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k2_polyform :::"|^"::: ) (Set "(" (Set (Var "k")) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_polyform :::"*"::: ) (Set "(" ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ")" ")" ))) ")" ) ")" ) ")" ))); theorem :: POLYFORM:51 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k29_polyform :::"alternating-proper-f-vector"::: ) (Set (Var "p")) ")" )))) "holds" (Bool (Set (Set "(" ($#k29_polyform :::"alternating-proper-f-vector"::: ) (Set (Var "p")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k2_polyform :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_polyform :::"*"::: ) (Set "(" ($#k1_vectsp_9 :::"dim"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k6_xcmplx_0 :::"-"::: ) (Num 2) ")" ) ($#k24_polyform :::"-bounding-chain-space"::: ) (Set (Var "p")) ")" ) ")" ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k2_polyform :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_polyform :::"*"::: ) (Set "(" ($#k1_vectsp_9 :::"dim"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k22_polyform :::"-circuit-space"::: ) (Set (Var "p")) ")" ) ")" ) ")" ))))) ; definitionlet "p" be ($#l1_polyform :::"polyhedron":::); attr "p" is :::"eulerian"::: means :: POLYFORM:def 29 (Bool (Set ($#k16_rvsum_1 :::"Sum"::: ) (Set "(" ($#k29_polyform :::"alternating-proper-f-vector"::: ) "p" ")" )) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k2_polyform :::"|^"::: ) (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) "p" ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ) ")" ))); end; :: deftheorem defines :::"eulerian"::: POLYFORM:def 29 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool "(" (Bool (Set (Var "p")) "is" ($#v6_polyform :::"eulerian"::: ) ) "iff" (Bool (Set ($#k16_rvsum_1 :::"Sum"::: ) (Set "(" ($#k29_polyform :::"alternating-proper-f-vector"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k2_polyform :::"|^"::: ) (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ) ")" ))) ")" )); theorem :: POLYFORM:52 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Set ($#k30_polyform :::"alternating-semi-proper-f-vector"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k29_polyform :::"alternating-proper-f-vector"::: ) (Set (Var "p")) ")" ) ($#k6_polyform :::"^"::: ) (Set ($#k3_polyform :::"<*"::: ) (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k2_polyform :::"|^"::: ) (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ")" ) ($#k3_polyform :::"*>"::: ) )))) ; definitionlet "p" be ($#l1_polyform :::"polyhedron":::); redefine attr "p" is :::"eulerian"::: means :: POLYFORM:def 30 (Bool (Set ($#k16_rvsum_1 :::"Sum"::: ) (Set "(" ($#k30_polyform :::"alternating-semi-proper-f-vector"::: ) "p" ")" )) ($#r1_hidden :::"="::: ) (Num 1)); end; :: deftheorem defines :::"eulerian"::: POLYFORM:def 30 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool "(" (Bool (Set (Var "p")) "is" ($#v6_polyform :::"eulerian"::: ) ) "iff" (Bool (Set ($#k16_rvsum_1 :::"Sum"::: ) (Set "(" ($#k30_polyform :::"alternating-semi-proper-f-vector"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Num 1)) ")" )); theorem :: POLYFORM:53 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Set ($#k28_polyform :::"alternating-f-vector"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_polyform :::"<*"::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k3_polyform :::"*>"::: ) ) ($#k6_polyform :::"^"::: ) (Set "(" ($#k30_polyform :::"alternating-semi-proper-f-vector"::: ) (Set (Var "p")) ")" )))) ; definitionlet "p" be ($#l1_polyform :::"polyhedron":::); redefine attr "p" is :::"eulerian"::: means :: POLYFORM:def 31 (Bool (Set ($#k16_rvsum_1 :::"Sum"::: ) (Set "(" ($#k28_polyform :::"alternating-f-vector"::: ) "p" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )); end; :: deftheorem defines :::"eulerian"::: POLYFORM:def 31 : (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool "(" (Bool (Set (Var "p")) "is" ($#v6_polyform :::"eulerian"::: ) ) "iff" (Bool (Set ($#k16_rvsum_1 :::"Sum"::: ) (Set "(" ($#k28_polyform :::"alternating-f-vector"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )); begin theorem :: POLYFORM:54 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Bool "not" (Set (Set ($#k6_numbers :::"0"::: ) ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "is" ($#v1_xboole_0 :::"empty"::: ) ))) ; theorem :: POLYFORM:55 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Num 2))) ; theorem :: POLYFORM:56 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Set ($#k2_struct_0 :::"[#]"::: ) (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k2_tarski :::"}"::: ) ))) ; theorem :: POLYFORM:57 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "k")) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "st" (Bool (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "e")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set ($#k16_polyform :::"incidence-value"::: ) "(" (Set (Var "e")) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ))))))) ; theorem :: POLYFORM:58 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "k")) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "v")) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Var "e")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k15_polyform :::"-th-polytope"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" )) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" ))) "holds" (Bool (Set (Set "(" ($#k19_polyform :::"incidence-sequence"::: ) "(" (Set (Var "e")) "," (Set (Var "v")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ))))))))) ; theorem :: POLYFORM:59 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "k")) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "v")) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k15_polyform :::"-th-polytope"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" )) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" )) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set (Var "k")) ")" )) & (Bool (Set (Var "m")) ($#r1_hidden :::"<>"::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" ($#k19_polyform :::"incidence-sequence"::: ) "(" (Set (Var "e")) "," (Set (Var "v")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ))))))))) ; theorem :: POLYFORM:60 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "k")) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Var "k")) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "st" (Bool (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "v")) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Var "e")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" ($#k19_polyform :::"incidence-sequence"::: ) "(" (Set (Var "e")) "," (Set (Var "v")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) )))))))) ; theorem :: POLYFORM:61 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set ($#k6_numbers :::"0"::: ) ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "holds" (Bool (Set (Set "(" (Set ($#k6_numbers :::"0"::: ) ) ($#k21_polyform :::"-boundary"::: ) (Set (Var "p")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: POLYFORM:62 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)))) "holds" (Bool (Set ($#k1_vectsp_9 :::"dim"::: ) (Set "(" (Set (Var "k")) ($#k24_polyform :::"-bounding-chain-space"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Num 1)))) ; theorem :: POLYFORM:63 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Num 2))) ; theorem :: POLYFORM:64 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ))) ; theorem :: POLYFORM:65 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" )))) ; theorem :: POLYFORM:66 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Bool "not" (Set (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "is" ($#v1_xboole_0 :::"empty"::: ) ))) ; registrationlet "p" be ($#l1_polyform :::"polyhedron":::); cluster (Set (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) "p" ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) "p") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ; end; theorem :: POLYFORM:67 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Set ($#k2_struct_0 :::"[#]"::: ) (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) ")" ) "," (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ) ($#k2_tarski :::"}"::: ) ))) ; theorem :: POLYFORM:68 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ))) "or" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )) ")" ))) ; theorem :: POLYFORM:69 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) "holds" (Bool "(" "not" (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y"))) "or" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ))) "or" (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ))) ")" ))) ; theorem :: POLYFORM:70 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Set (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k10_polyform :::"-polytope-seq"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k9_finseq_1 :::"*>"::: ) ))) ; theorem :: POLYFORM:71 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Set (Num 1) ($#k15_polyform :::"-th-polytope"::: ) "(" (Set (Var "p")) "," (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "p")))) ; theorem :: POLYFORM:72 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "st" (Bool (Bool (Set (Var "c")) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set (Set (Var "c")) ($#k3_bspace :::"@"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) )))))) ; theorem :: POLYFORM:73 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) "st" (Bool (Bool (Set (Var "c")) ($#r1_hidden :::"="::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k16_polyform :::"incidence-value"::: ) "(" (Set (Var "x")) "," (Set (Var "c")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) )))))) ; theorem :: POLYFORM:74 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) "st" (Bool (Bool (Set (Var "c")) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set ($#k19_polyform :::"incidence-sequence"::: ) "(" (Set (Var "x")) "," (Set (Var "c")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_pre_poly :::"<*"::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ) ")" ) ($#k3_pre_poly :::"*>"::: ) ))))) ; theorem :: POLYFORM:75 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k17_polyform :::"-chain-space"::: ) (Set (Var "p")) ")" ) "st" (Bool (Bool (Set (Var "c")) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" ($#k19_polyform :::"incidence-sequence"::: ) "(" (Set (Var "x")) "," (Set (Var "c")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) )))))) ; theorem :: POLYFORM:76 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Set (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k21_polyform :::"-boundary"::: ) (Set (Var "p")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k8_polyform :::"-polytopes"::: ) (Set (Var "p"))))) ; theorem :: POLYFORM:77 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Set (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k21_polyform :::"-boundary"::: ) (Set (Var "p"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) ; theorem :: POLYFORM:78 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Set ($#k1_vectsp_9 :::"dim"::: ) (Set "(" (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k24_polyform :::"-bounding-chain-space"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Num 1))) ; theorem :: POLYFORM:79 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "st" (Bool (Bool (Set (Var "p")) "is" ($#v5_polyform :::"simply-connected"::: ) )) "holds" (Bool (Set ($#k1_vectsp_9 :::"dim"::: ) (Set "(" (Set "(" (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k22_polyform :::"-circuit-space"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Num 1))) ; theorem :: POLYFORM:80 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 2)))) "holds" (Bool (Set (Set "(" ($#k28_polyform :::"alternating-f-vector"::: ) (Set (Var "p")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k29_polyform :::"alternating-proper-f-vector"::: ) (Set (Var "p")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ))))) ; theorem :: POLYFORM:81 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "holds" (Bool (Set ($#k28_polyform :::"alternating-f-vector"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k3_polyform :::"<*"::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k3_polyform :::"*>"::: ) ) ($#k6_polyform :::"^"::: ) (Set "(" ($#k29_polyform :::"alternating-proper-f-vector"::: ) (Set (Var "p")) ")" ) ")" ) ($#k6_polyform :::"^"::: ) (Set ($#k3_polyform :::"<*"::: ) (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k2_polyform :::"|^"::: ) (Set "(" ($#k7_polyform :::"dim"::: ) (Set (Var "p")) ")" ) ")" ) ($#k3_polyform :::"*>"::: ) )))) ; begin theorem :: POLYFORM:82 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "st" (Bool (Bool (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))) "is" ($#v1_abian :::"odd"::: ) )) "holds" (Bool (Set ($#k16_rvsum_1 :::"Sum"::: ) (Set "(" ($#k28_polyform :::"alternating-f-vector"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k16_rvsum_1 :::"Sum"::: ) (Set "(" ($#k29_polyform :::"alternating-proper-f-vector"::: ) (Set (Var "p")) ")" ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 2)))) ; theorem :: POLYFORM:83 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "st" (Bool (Bool (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))) "is" ($#v1_abian :::"even"::: ) )) "holds" (Bool (Set ($#k16_rvsum_1 :::"Sum"::: ) (Set "(" ($#k28_polyform :::"alternating-f-vector"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k16_rvsum_1 :::"Sum"::: ) (Set "(" ($#k29_polyform :::"alternating-proper-f-vector"::: ) (Set (Var "p")) ")" )))) ; theorem :: POLYFORM:84 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "st" (Bool (Bool (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set ($#k16_rvsum_1 :::"Sum"::: ) (Set "(" ($#k29_polyform :::"alternating-proper-f-vector"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ))) ; theorem :: POLYFORM:85 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "st" (Bool (Bool (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 2))) "holds" (Bool (Set ($#k16_rvsum_1 :::"Sum"::: ) (Set "(" ($#k29_polyform :::"alternating-proper-f-vector"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Num 1) ")" ")" )))) ; theorem :: POLYFORM:86 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "st" (Bool (Bool (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 3))) "holds" (Bool (Set ($#k16_rvsum_1 :::"Sum"::: ) (Set "(" ($#k29_polyform :::"alternating-proper-f-vector"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Num 1) ")" ")" ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" ($#k11_polyform :::"num-polytopes"::: ) "(" (Set (Var "p")) "," (Num 2) ")" ")" )))) ; theorem :: POLYFORM:87 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "st" (Bool (Bool (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Var "p")) "is" ($#v6_polyform :::"eulerian"::: ) )) ; theorem :: POLYFORM:88 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "st" (Bool (Bool (Set (Var "p")) "is" ($#v5_polyform :::"simply-connected"::: ) )) "holds" (Bool (Set (Var "p")) "is" ($#v6_polyform :::"eulerian"::: ) )) ; theorem :: POLYFORM:89 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "st" (Bool (Bool (Set (Var "p")) "is" ($#v5_polyform :::"simply-connected"::: ) ) & (Bool (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set ($#k12_polyform :::"num-vertices"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 2))) ; theorem :: POLYFORM:90 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "st" (Bool (Bool (Set (Var "p")) "is" ($#v5_polyform :::"simply-connected"::: ) ) & (Bool (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 2))) "holds" (Bool (Set ($#k12_polyform :::"num-vertices"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k13_polyform :::"num-edges"::: ) (Set (Var "p"))))) ; theorem :: POLYFORM:91 (Bool "for" (Set (Var "p")) "being" ($#l1_polyform :::"polyhedron":::) "st" (Bool (Bool (Set (Var "p")) "is" ($#v5_polyform :::"simply-connected"::: ) ) & (Bool (Set ($#k7_polyform :::"dim"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 3))) "holds" (Bool (Set (Set "(" (Set "(" ($#k12_polyform :::"num-vertices"::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" ($#k13_polyform :::"num-edges"::: ) (Set (Var "p")) ")" ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" ($#k14_polyform :::"num-faces"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Num 2))) ;