:: MATRPROB semantic presentation begin definitionlet "d" be ($#m1_hidden :::"set"::: ) ; let "g" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Set (Const "d")) ($#k3_finseq_2 :::"*"::: ) ); let "n" be ($#m1_hidden :::"Nat":::); :: original: :::"."::: redefine func "g" :::"."::: "n" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" "d"; end; definitionlet "x" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; :: original: :::"<*"::: redefine func :::"<*":::"x":::"*>"::: -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ); end; theorem :: MATRPROB:1 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool "(" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Var "m"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "a"))) ")" ) ")" ) "iff" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set (Var "m")) ($#k5_finseq_2 :::"|->"::: ) (Set (Var "a")))) ")" ))))) ; theorem :: MATRPROB:2 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "ex" (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k"))))) "implies" (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "a"))) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")))))) "implies" (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "b"))) ")" ")" ) ")" ) ")" )))))) ; theorem :: MATRPROB:3 (Bool "for" (Set (Var "e")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "e"))))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "e")) ($#k1_seq_1 :::"."::: ) (Set (Var "i")))) ")" )) "holds" (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "e"))))) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set (Var "e")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ")" ))) ")" )) "holds" (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "e")))) & (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "e")))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "m"))))))) ; theorem :: MATRPROB:4 (Bool "for" (Set (Var "e")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "e"))) ($#r1_xxreal_0 :::">="::: ) (Num 1)) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "e"))))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "e")) ($#k1_seq_1 :::"."::: ) (Set (Var "i")))) ")" )) "holds" (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "e")) ($#k1_seq_1 :::"."::: ) (Num 1))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "e"))))) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set (Var "e")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ")" ))) ")" )) "holds" (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "e"))))) "holds" (Bool (Set (Set (Var "e")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))))))) ; theorem :: MATRPROB:5 (Bool "for" (Set (Var "e")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "e"))))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "e")) ($#k1_seq_1 :::"."::: ) (Set (Var "i")))) ")" )) "holds" (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "e"))))) "holds" (Bool (Set (Set (Var "e")) ($#k1_seq_1 :::"."::: ) (Set (Var "k"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "e")))))) ; theorem :: MATRPROB:6 (Bool "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "seq1")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "ex" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Set (Var "seq")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "r1"))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k")))) "implies" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "seq1")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) ")" & "(" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "k")))) "implies" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Var "r2"))) ")" ")" ) ")" ) ")" ))))) ; theorem :: MATRPROB:7 (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F"))))) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "i")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" (Set (Var "F")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ")" ))) ")" ) & (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "F")) ")" ))) ")" ))) ; theorem :: MATRPROB:8 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "e1")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set (Var "n")) ($#k2_finseq_2 :::"|->"::: ) (Set (Var "e1"))) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Set (Var "D")) ($#k3_finseq_2 :::"*"::: ) ))))) ; theorem :: MATRPROB:9 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "e1")) "," (Set (Var "e2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "ex" (Set (Var "e")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Set (Var "D")) ($#k3_finseq_2 :::"*"::: ) ) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "e"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k"))))) "implies" (Bool (Set (Set (Var "e")) ($#k1_matrprob :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "e1"))) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")))))) "implies" (Bool (Set (Set (Var "e")) ($#k1_matrprob :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "e2"))) ")" ")" ) ")" ) ")" )))))) ; theorem :: MATRPROB:10 (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "s")) "is" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D"))) "iff" (Bool "ex" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "s"))))) "holds" (Bool "ex" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool "(" (Bool (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "p"))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) ")" )))) ")" ))) ; theorem :: MATRPROB:11 (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "e")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Set (Var "D")) ($#k3_finseq_2 :::"*"::: ) ) "holds" (Bool "(" (Bool "ex" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "e"))))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "e")) ($#k1_matrprob :::"."::: ) (Set (Var "i")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "n"))))) "iff" (Bool (Set (Var "e")) "is" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D"))) ")" ))) ; theorem :: MATRPROB:12 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "M")) "being" ($#v1_matrix_1 :::"tabular"::: ) ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M")))) "iff" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")) ")" ))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")) ")" ))) ")" ) ")" ))) ; theorem :: MATRPROB:13 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M")))) "iff" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "M")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "M")) ($#k1_matrprob :::"."::: ) (Set (Var "i")) ")" ))) ")" ) ")" )))) ; theorem :: MATRPROB:14 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))) "holds" (Bool (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "M")) ($#k1_matrprob :::"."::: ) (Set (Var "i")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "j"))))))) ; theorem :: MATRPROB:15 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M")))) "iff" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "M")) "," (Set (Var "j")) ")" ")" ))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ")" ))) ")" ) ")" )))) ; theorem :: MATRPROB:16 (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M1")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D1")) (Bool "for" (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D2")) "st" (Bool (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Var "M2")))) "holds" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "M1"))))) "holds" (Bool (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M1")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M2")) "," (Set (Var "i")) ")" )))))) ; theorem :: MATRPROB:17 (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M1")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D1")) (Bool "for" (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D2")) "st" (Bool (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Var "M2")))) "holds" (Bool "for" (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1")) ")" )))) "holds" (Bool (Set ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "M1")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "M2")) "," (Set (Var "j")) ")" )))))) ; theorem :: MATRPROB:18 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "e1")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "e1"))) ($#r1_hidden :::"="::: ) (Set (Var "m")))) "holds" (Bool (Set (Set (Var "n")) ($#k2_finseq_2 :::"|->"::: ) (Set (Var "e1"))) "is" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "D")))))) ; theorem :: MATRPROB:19 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "e1")) "," (Set (Var "e2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "e1"))) ($#r1_hidden :::"="::: ) (Set (Var "m"))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "e2"))) ($#r1_hidden :::"="::: ) (Set (Var "m")))) "holds" (Bool "ex" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "D")) "st" (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k"))))) "implies" (Bool (Set (Set (Var "M")) ($#k1_matrprob :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "e1"))) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")))))) "implies" (Bool (Set (Set (Var "M")) ($#k1_matrprob :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "e2"))) ")" ")" ))))))) ; definitionlet "e" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Set ($#k1_numbers :::"REAL"::: ) ) ($#k3_finseq_2 :::"*"::: ) ); func :::"Sum"::: "e" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) means :: MATRPROB:def 1 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "e")) & (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" "e" ($#k1_matrprob :::"."::: ) (Set (Var "k")) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"Sum"::: MATRPROB:def 1 : (Bool "for" (Set (Var "e")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Set ($#k1_numbers :::"REAL"::: ) ) ($#k3_finseq_2 :::"*"::: ) ) (Bool "for" (Set (Var "b2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k3_matrprob :::"Sum"::: ) (Set (Var "e")))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "e")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "b2"))))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_seq_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" (Set (Var "e")) ($#k1_matrprob :::"."::: ) (Set (Var "k")) ")" ))) ")" ) ")" ) ")" ))); notationlet "m" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); synonym :::"LineSum"::: "m" for :::"Sum"::: "m"; end; theorem :: MATRPROB:20 (Bool "for" (Set (Var "m")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k3_matrprob :::"Sum"::: ) (Set (Var "m")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "m")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "m")) ")" )))) "holds" (Bool (Set (Set "(" ($#k3_matrprob :::"Sum"::: ) (Set (Var "m")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "m")) "," (Set (Var "i")) ")" ")" ))) ")" ) ")" )) ; definitionlet "m" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); func :::"ColSum"::: "m" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) means :: MATRPROB:def 2 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) "m")) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) "m" ")" )))) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" "m" "," (Set (Var "j")) ")" ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"ColSum"::: MATRPROB:def 2 : (Bool "for" (Set (Var "m")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "b2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k4_matrprob :::"ColSum"::: ) (Set (Var "m")))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "m")))) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "m")) ")" )))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_seq_1 :::"."::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "m")) "," (Set (Var "j")) ")" ")" ))) ")" ) ")" ) ")" ))); theorem :: MATRPROB:21 (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k3_matrprob :::"LineSum"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k4_matrprob :::"ColSum"::: ) (Set "(" (Set (Var "M")) ($#k4_matrix_1 :::"@"::: ) ")" )))) ; theorem :: MATRPROB:22 (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k4_matrprob :::"ColSum"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k3_matrprob :::"LineSum"::: ) (Set "(" (Set (Var "M")) ($#k4_matrix_1 :::"@"::: ) ")" )))) ; definitionlet "M" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); func :::"SumAll"::: "M" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) equals :: MATRPROB:def 3 (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" ($#k3_matrprob :::"Sum"::: ) "M" ")" )); end; :: deftheorem defines :::"SumAll"::: MATRPROB:def 3 : (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k5_matrprob :::"SumAll"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" ($#k3_matrprob :::"Sum"::: ) (Set (Var "M")) ")" )))); theorem :: MATRPROB:23 (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k5_matrprob :::"SumAll"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: MATRPROB:24 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "m")) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k5_matrprob :::"SumAll"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: MATRPROB:25 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "M1")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n")) "," (Set (Var "k")) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "M2")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "m")) "," (Set (Var "k")) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k3_matrprob :::"Sum"::: ) (Set "(" (Set (Var "M1")) ($#k8_matrlin :::"^"::: ) (Set (Var "M2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_matrprob :::"Sum"::: ) (Set (Var "M1")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set "(" ($#k3_matrprob :::"Sum"::: ) (Set (Var "M2")) ")" ))))))) ; theorem :: MATRPROB:26 (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "(" ($#k3_matrprob :::"Sum"::: ) (Set (Var "M1")) ")" ) ($#k4_rvsum_1 :::"+"::: ) (Set "(" ($#k3_matrprob :::"Sum"::: ) (Set (Var "M2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_matrprob :::"Sum"::: ) (Set "(" (Set (Var "M1")) ($#k7_matrlin :::"^^"::: ) (Set (Var "M2")) ")" )))) ; theorem :: MATRPROB:27 (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))))) "holds" (Bool (Set (Set "(" ($#k5_matrprob :::"SumAll"::: ) (Set (Var "M1")) ")" ) ($#k9_binop_2 :::"+"::: ) (Set "(" ($#k5_matrprob :::"SumAll"::: ) (Set (Var "M2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_matrprob :::"SumAll"::: ) (Set "(" (Set (Var "M1")) ($#k7_matrlin :::"^^"::: ) (Set (Var "M2")) ")" )))) ; theorem :: MATRPROB:28 (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k5_matrprob :::"SumAll"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k5_matrprob :::"SumAll"::: ) (Set "(" (Set (Var "M")) ($#k4_matrix_1 :::"@"::: ) ")" )))) ; theorem :: MATRPROB:29 (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k5_matrprob :::"SumAll"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" ($#k4_matrprob :::"ColSum"::: ) (Set (Var "M")) ")" )))) ; theorem :: MATRPROB:30 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "y"))))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k14_rvsum_1 :::"mlt"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))))) ; theorem :: MATRPROB:31 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "R")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "i")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) "holds" (Bool (Set ($#k14_rvsum_1 :::"mlt"::: ) "(" (Set "(" (Set (Var "i")) ($#k5_finseq_2 :::"|->"::: ) (Num 1) ")" ) "," (Set (Var "R")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "R"))))) ; theorem :: MATRPROB:32 (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k14_rvsum_1 :::"mlt"::: ) "(" (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "x")) ")" ) ($#k5_finseq_2 :::"|->"::: ) (Num 1) ")" ) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "x")))) ; theorem :: MATRPROB:33 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "x"))))) "holds" (Bool (Set (Set (Var "x")) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) & (Bool "(" "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "y"))))) "holds" (Bool (Set (Set (Var "y")) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" ($#k14_rvsum_1 :::"mlt"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ")" )))) "holds" (Bool (Set (Set "(" ($#k14_rvsum_1 :::"mlt"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "k"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: MATRPROB:34 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "e1")) "," (Set (Var "e2")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "i")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "i")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k2_vectsp_1 :::"F_Real"::: ) ))) "st" (Bool (Bool (Set (Var "e1")) ($#r1_hidden :::"="::: ) (Set (Var "f1"))) & (Bool (Set (Var "e2")) ($#r1_hidden :::"="::: ) (Set (Var "f2")))) "holds" (Bool (Set ($#k15_rvsum_1 :::"mlt"::: ) "(" (Set (Var "e1")) "," (Set (Var "e2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k12_fvsum_1 :::"mlt"::: ) "(" (Set (Var "f1")) "," (Set (Var "f2")) ")" ))))) ; theorem :: MATRPROB:35 (Bool "for" (Set (Var "e1")) "," (Set (Var "e2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "e1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "e2")))) & (Bool (Set (Var "e1")) ($#r1_hidden :::"="::: ) (Set (Var "f1"))) & (Bool (Set (Var "e2")) ($#r1_hidden :::"="::: ) (Set (Var "f2")))) "holds" (Bool (Set ($#k14_rvsum_1 :::"mlt"::: ) "(" (Set (Var "e1")) "," (Set (Var "e2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k11_fvsum_1 :::"mlt"::: ) "(" (Set (Var "f1")) "," (Set (Var "f2")) ")" )))) ; theorem :: MATRPROB:36 (Bool "for" (Set (Var "e")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) "st" (Bool (Bool (Set (Var "e")) ($#r1_hidden :::"="::: ) (Set (Var "f")))) "holds" (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "e"))) ($#r1_hidden :::"="::: ) (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "f")))))) ; notationlet "e1", "e2" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ); synonym "e1" :::""*""::: "e2" for :::"|(":::"e1" "," "e2":::")|":::; end; theorem :: MATRPROB:37 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "e1")) "," (Set (Var "e2")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "i")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "i")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k2_vectsp_1 :::"F_Real"::: ) ))) "st" (Bool (Bool (Set (Var "e1")) ($#r1_hidden :::"="::: ) (Set (Var "f1"))) & (Bool (Set (Var "e2")) ($#r1_hidden :::"="::: ) (Set (Var "f2")))) "holds" (Bool (Set (Set (Var "e1")) ($#k22_rvsum_1 :::""*""::: ) (Set (Var "e2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f1")) ($#k13_fvsum_1 :::""*""::: ) (Set (Var "f2"))))))) ; theorem :: MATRPROB:38 (Bool "for" (Set (Var "e1")) "," (Set (Var "e2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "e1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "e2")))) & (Bool (Set (Var "e1")) ($#r1_hidden :::"="::: ) (Set (Var "f1"))) & (Bool (Set (Var "e2")) ($#r1_hidden :::"="::: ) (Set (Var "f2")))) "holds" (Bool (Set (Set (Var "e1")) ($#k22_rvsum_1 :::""*""::: ) (Set (Var "e2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f1")) ($#k13_fvsum_1 :::""*""::: ) (Set (Var "f2")))))) ; theorem :: MATRPROB:39 (Bool "for" (Set (Var "M")) "," (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))))) "holds" (Bool "(" (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set (Set (Var "M1")) ($#k6_matrixr1 :::"*"::: ) (Set (Var "M2")))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))) "holds" (Bool (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M1")) "," (Set (Var "i")) ")" ")" ) ($#k22_rvsum_1 :::""*""::: ) (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "M2")) "," (Set (Var "j")) ")" ")" ))) ")" ) ")" ) ")" )) ; theorem :: MATRPROB:40 (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))))) "holds" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "p")) ($#k12_matrixr1 :::"*"::: ) (Set (Var "M")) ")" ) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k12_matrixr1 :::"*"::: ) (Set (Var "M")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k22_rvsum_1 :::""*""::: ) (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ")" )))))) ; theorem :: MATRPROB:41 (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "M")) ($#k11_matrixr1 :::"*"::: ) (Set (Var "p")) ")" ) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "M")) ($#k11_matrixr1 :::"*"::: ) (Set (Var "p")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ")" ) ($#k22_rvsum_1 :::""*""::: ) (Set (Var "p"))))))) ; theorem :: MATRPROB:42 (Bool "for" (Set (Var "M")) "," (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))))) "holds" (Bool "(" (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set (Set (Var "M1")) ($#k6_matrixr1 :::"*"::: ) (Set (Var "M2")))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")) ")" )))) "holds" (Bool (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M1")) "," (Set (Var "i")) ")" ")" ) ($#k12_matrixr1 :::"*"::: ) (Set (Var "M2")))) ")" ) ")" ) ")" )) ; definitionlet "x", "y" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "M" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); assume that (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Const "x"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Const "M")))) and (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Const "y"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Const "M")))) ; func :::"QuadraticForm"::: "(" "x" "," "M" "," "y" ")" -> ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) means :: MATRPROB:def 4 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "x")) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "y")) & (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) "M"))) "holds" (Bool (Set it ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" "x" ($#k1_seq_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" "M" ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" "y" ($#k1_seq_1 :::"."::: ) (Set (Var "j")) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"QuadraticForm"::: MATRPROB:def 4 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M"))))) "holds" (Bool "for" (Set (Var "b4")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k6_matrprob :::"QuadraticForm"::: ) "(" (Set (Var "x")) "," (Set (Var "M")) "," (Set (Var "y")) ")" )) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "y")))) & (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))) "holds" (Bool (Set (Set (Var "b4")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "x")) ($#k1_seq_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" (Set (Var "y")) ($#k1_seq_1 :::"."::: ) (Set (Var "j")) ")" ))) ")" ) ")" ) ")" )))); theorem :: MATRPROB:43 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "y"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" ($#k6_matrprob :::"QuadraticForm"::: ) "(" (Set (Var "x")) "," (Set (Var "M")) "," (Set (Var "y")) ")" ")" ) ($#k4_matrix_1 :::"@"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_matrprob :::"QuadraticForm"::: ) "(" (Set (Var "y")) "," (Set "(" (Set (Var "M")) ($#k4_matrix_1 :::"@"::: ) ")" ) "," (Set (Var "x")) ")" )))) ; theorem :: MATRPROB:44 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "y"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k23_rvsum_1 :::"|("::: ) (Set (Var "x")) "," (Set "(" (Set (Var "M")) ($#k11_matrixr1 :::"*"::: ) (Set (Var "y")) ")" ) ($#k23_rvsum_1 :::")|"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k5_matrprob :::"SumAll"::: ) (Set "(" ($#k6_matrprob :::"QuadraticForm"::: ) "(" (Set (Var "x")) "," (Set (Var "M")) "," (Set (Var "y")) ")" ")" ))))) ; theorem :: MATRPROB:45 (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k23_rvsum_1 :::"|("::: ) (Set (Var "x")) "," (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "x")) ")" ) ($#k5_finseq_2 :::"|->"::: ) (Num 1) ")" ) ($#k23_rvsum_1 :::")|"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "x"))))) ; theorem :: MATRPROB:46 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M"))))) "holds" (Bool (Set ($#k23_rvsum_1 :::"|("::: ) (Set "(" (Set (Var "x")) ($#k12_matrixr1 :::"*"::: ) (Set (Var "M")) ")" ) "," (Set (Var "y")) ($#k23_rvsum_1 :::")|"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k5_matrprob :::"SumAll"::: ) (Set "(" ($#k6_matrprob :::"QuadraticForm"::: ) "(" (Set (Var "x")) "," (Set (Var "M")) "," (Set (Var "y")) ")" ")" ))))) ; theorem :: MATRPROB:47 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "y"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k23_rvsum_1 :::"|("::: ) (Set "(" (Set (Var "x")) ($#k12_matrixr1 :::"*"::: ) (Set (Var "M")) ")" ) "," (Set (Var "y")) ($#k23_rvsum_1 :::")|"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k23_rvsum_1 :::"|("::: ) (Set (Var "x")) "," (Set "(" (Set (Var "M")) ($#k11_matrixr1 :::"*"::: ) (Set (Var "y")) ")" ) ($#k23_rvsum_1 :::")|"::: ) )))) ; theorem :: MATRPROB:48 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "y"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k23_rvsum_1 :::"|("::: ) (Set "(" (Set (Var "M")) ($#k11_matrixr1 :::"*"::: ) (Set (Var "x")) ")" ) "," (Set (Var "y")) ($#k23_rvsum_1 :::")|"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k23_rvsum_1 :::"|("::: ) (Set (Var "x")) "," (Set "(" (Set "(" (Set (Var "M")) ($#k4_matrix_1 :::"@"::: ) ")" ) ($#k11_matrixr1 :::"*"::: ) (Set (Var "y")) ")" ) ($#k23_rvsum_1 :::")|"::: ) )))) ; theorem :: MATRPROB:49 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "y"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k23_rvsum_1 :::"|("::: ) (Set (Var "x")) "," (Set "(" (Set (Var "y")) ($#k12_matrixr1 :::"*"::: ) (Set (Var "M")) ")" ) ($#k23_rvsum_1 :::")|"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k23_rvsum_1 :::"|("::: ) (Set "(" (Set (Var "x")) ($#k12_matrixr1 :::"*"::: ) (Set "(" (Set (Var "M")) ($#k4_matrix_1 :::"@"::: ) ")" ) ")" ) "," (Set (Var "y")) ($#k23_rvsum_1 :::")|"::: ) )))) ; theorem :: MATRPROB:50 (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")))) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "x")) ")" ) ($#k5_finseq_2 :::"|->"::: ) (Num 1)))) "holds" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "x")) ($#k12_matrixr1 :::"*"::: ) (Set (Var "M")) ")" ) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k12_matrixr1 :::"*"::: ) (Set (Var "M")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "M")) "," (Set (Var "k")) ")" ")" )))))) ; theorem :: MATRPROB:51 (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "x")) ")" ) ($#k5_finseq_2 :::"|->"::: ) (Num 1)))) "holds" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "M")) ($#k11_matrixr1 :::"*"::: ) (Set (Var "x")) ")" ) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "M")) ($#k11_matrixr1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Set (Var "k")) ")" ")" )))))) ; theorem :: MATRPROB:52 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"Nat":::) (Bool "ex" (Set (Var "P")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "P"))))) "holds" (Bool (Set (Set (Var "P")) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) & (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ))) ; definitionlet "p" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ); attr "p" is :::"ProbFinS"::: means :: MATRPROB:def 5 (Bool "(" (Bool "(" "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "p"))) "holds" (Bool (Set "p" ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) & (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) "p") ($#r1_hidden :::"="::: ) (Num 1)) ")" ); end; :: deftheorem defines :::"ProbFinS"::: MATRPROB:def 5 : (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "p")) "is" ($#v1_matrprob :::"ProbFinS"::: ) ) "iff" (Bool "(" (Bool "(" "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "p")) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) & (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) ")" )); registration cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_FINSET_1() ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ($#v1_valued_0 :::"complex-yielding"::: ) bbbadV2_VALUED_0() bbbadV3_VALUED_0() ($#v1_matrprob :::"ProbFinS"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ); end; theorem :: MATRPROB:53 (Bool "for" (Set (Var "p")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_matrprob :::"ProbFinS"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "p")) ($#k1_seq_1 :::"."::: ) (Set (Var "k"))) ($#r1_xxreal_0 :::"<="::: ) (Num 1)))) ; theorem :: MATRPROB:54 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#~v3_relat_1 "non" ($#v3_relat_1 :::"empty-yielding"::: ) ) ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")))) ")" ))) ; definitionlet "M" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); attr "M" is :::"m-nonnegative"::: means :: MATRPROB:def 6 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) "M"))) "holds" (Bool (Set "M" ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) ))); end; :: deftheorem defines :::"m-nonnegative"::: MATRPROB:def 6 : (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#v2_matrprob :::"m-nonnegative"::: ) ) "iff" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))) "holds" (Bool (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ")" )); definitionlet "M" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); attr "M" is :::"with_sum=1"::: means :: MATRPROB:def 7 (Bool (Set ($#k5_matrprob :::"SumAll"::: ) "M") ($#r1_hidden :::"="::: ) (Num 1)); end; :: deftheorem defines :::"with_sum=1"::: MATRPROB:def 7 : (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#v3_matrprob :::"with_sum=1"::: ) ) "iff" (Bool (Set ($#k5_matrprob :::"SumAll"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" )); definitionlet "M" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); attr "M" is :::"Joint_Probability"::: means :: MATRPROB:def 8 (Bool "(" (Bool "M" "is" ($#v2_matrprob :::"m-nonnegative"::: ) ) & (Bool "M" "is" ($#v3_matrprob :::"with_sum=1"::: ) ) ")" ); end; :: deftheorem defines :::"Joint_Probability"::: MATRPROB:def 8 : (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#v4_matrprob :::"Joint_Probability"::: ) ) "iff" (Bool "(" (Bool (Set (Var "M")) "is" ($#v2_matrprob :::"m-nonnegative"::: ) ) & (Bool (Set (Var "M")) "is" ($#v3_matrprob :::"with_sum=1"::: ) ) ")" ) ")" )); registration cluster ($#v1_matrix_1 :::"tabular"::: ) ($#v4_matrprob :::"Joint_Probability"::: ) -> ($#v2_matrprob :::"m-nonnegative"::: ) ($#v3_matrprob :::"with_sum=1"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set (Set ($#k1_numbers :::"REAL"::: ) ) ($#k3_finseq_2 :::"*"::: ) ); cluster ($#v1_matrix_1 :::"tabular"::: ) ($#v2_matrprob :::"m-nonnegative"::: ) ($#v3_matrprob :::"with_sum=1"::: ) -> ($#v4_matrprob :::"Joint_Probability"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set (Set ($#k1_numbers :::"REAL"::: ) ) ($#k3_finseq_2 :::"*"::: ) ); end; theorem :: MATRPROB:55 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"Nat":::) (Bool "ex" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n")) "," (Set (Var "m")) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool "(" (Bool (Set (Var "M")) "is" ($#v2_matrprob :::"m-nonnegative"::: ) ) & (Bool (Set ($#k5_matrprob :::"SumAll"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ))) ; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#~v3_relat_1 "non" ($#v3_relat_1 :::"empty-yielding"::: ) ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set (Set ($#k1_numbers :::"REAL"::: ) ) ($#k3_finseq_2 :::"*"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_FINSET_1() ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ($#v1_matrix_1 :::"tabular"::: ) bbbadV1_PRE_POLY() ($#v4_matrprob :::"Joint_Probability"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set (Set ($#k1_numbers :::"REAL"::: ) ) ($#k3_finseq_2 :::"*"::: ) ); end; theorem :: MATRPROB:56 (Bool "for" (Set (Var "M")) "being" ($#~v3_relat_1 "non" ($#v3_relat_1 :::"empty-yielding"::: ) ) ($#v4_matrprob :::"Joint_Probability"::: ) ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set (Var "M")) ($#k4_matrix_1 :::"@"::: ) ) "is" ($#~v3_relat_1 "non" ($#v3_relat_1 :::"empty-yielding"::: ) ) ($#v4_matrprob :::"Joint_Probability"::: ) ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ))) ; theorem :: MATRPROB:57 (Bool "for" (Set (Var "M")) "being" ($#~v3_relat_1 "non" ($#v3_relat_1 :::"empty-yielding"::: ) ) ($#v4_matrprob :::"Joint_Probability"::: ) ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))) "holds" (Bool (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)))) ; definitionlet "M" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); attr "M" is :::"with_line_sum=1"::: means :: MATRPROB:def 9 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "M"))) "holds" (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" "M" ($#k1_matrprob :::"."::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) (Num 1))); end; :: deftheorem defines :::"with_line_sum=1"::: MATRPROB:def 9 : (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#v5_matrprob :::"with_line_sum=1"::: ) ) "iff" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "M"))))) "holds" (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" (Set (Var "M")) ($#k1_matrprob :::"."::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) (Num 1))) ")" )); theorem :: MATRPROB:58 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"Nat":::) (Bool "ex" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n")) "," (Set (Var "m")) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool "(" (Bool (Set (Var "M")) "is" ($#v2_matrprob :::"m-nonnegative"::: ) ) & (Bool (Set (Var "M")) "is" ($#v5_matrprob :::"with_line_sum=1"::: ) ) ")" ))) ; definitionlet "M" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); attr "M" is :::"Conditional_Probability"::: means :: MATRPROB:def 10 (Bool "(" (Bool "M" "is" ($#v2_matrprob :::"m-nonnegative"::: ) ) & (Bool "M" "is" ($#v5_matrprob :::"with_line_sum=1"::: ) ) ")" ); end; :: deftheorem defines :::"Conditional_Probability"::: MATRPROB:def 10 : (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#v6_matrprob :::"Conditional_Probability"::: ) ) "iff" (Bool "(" (Bool (Set (Var "M")) "is" ($#v2_matrprob :::"m-nonnegative"::: ) ) & (Bool (Set (Var "M")) "is" ($#v5_matrprob :::"with_line_sum=1"::: ) ) ")" ) ")" )); registration cluster ($#v1_matrix_1 :::"tabular"::: ) ($#v6_matrprob :::"Conditional_Probability"::: ) -> ($#v2_matrprob :::"m-nonnegative"::: ) ($#v5_matrprob :::"with_line_sum=1"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set (Set ($#k1_numbers :::"REAL"::: ) ) ($#k3_finseq_2 :::"*"::: ) ); cluster ($#v1_matrix_1 :::"tabular"::: ) ($#v2_matrprob :::"m-nonnegative"::: ) ($#v5_matrprob :::"with_line_sum=1"::: ) -> ($#v6_matrprob :::"Conditional_Probability"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set (Set ($#k1_numbers :::"REAL"::: ) ) ($#k3_finseq_2 :::"*"::: ) ); end; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#~v3_relat_1 "non" ($#v3_relat_1 :::"empty-yielding"::: ) ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set (Set ($#k1_numbers :::"REAL"::: ) ) ($#k3_finseq_2 :::"*"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_FINSET_1() ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ($#v1_matrix_1 :::"tabular"::: ) bbbadV1_PRE_POLY() ($#v6_matrprob :::"Conditional_Probability"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set (Set ($#k1_numbers :::"REAL"::: ) ) ($#k3_finseq_2 :::"*"::: ) ); end; theorem :: MATRPROB:59 (Bool "for" (Set (Var "M")) "being" ($#~v3_relat_1 "non" ($#v3_relat_1 :::"empty-yielding"::: ) ) ($#v6_matrprob :::"Conditional_Probability"::: ) ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))) "holds" (Bool (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_xxreal_0 :::"<="::: ) (Num 1)))) ; theorem :: MATRPROB:60 (Bool "for" (Set (Var "M")) "being" ($#~v3_relat_1 "non" ($#v3_relat_1 :::"empty-yielding"::: ) ) ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#~v3_relat_1 "non" ($#v3_relat_1 :::"empty-yielding"::: ) ) ($#v6_matrprob :::"Conditional_Probability"::: ) ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) )) "iff" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "M"))))) "holds" (Bool (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ) "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_matrprob :::"ProbFinS"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ))) ")" )) ; theorem :: MATRPROB:61 (Bool "for" (Set (Var "M")) "being" ($#~v3_relat_1 "non" ($#v3_relat_1 :::"empty-yielding"::: ) ) ($#v5_matrprob :::"with_line_sum=1"::: ) ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k5_matrprob :::"SumAll"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M"))))) ; notationlet "M" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); synonym :::"Row_Marginal"::: "M" for :::"LineSum"::: "M"; synonym :::"Column_Marginal"::: "M" for :::"ColSum"::: "M"; end; registrationlet "M" be ($#~v3_relat_1 "non" ($#v3_relat_1 :::"empty-yielding"::: ) ) ($#v4_matrprob :::"Joint_Probability"::: ) ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); cluster (Set ($#k3_matrprob :::"Sum"::: ) "M") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_matrprob :::"ProbFinS"::: ) ; cluster (Set ($#k4_matrprob :::"ColSum"::: ) "M") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_matrprob :::"ProbFinS"::: ) ; end; registrationlet "M" be ($#~v3_relat_1 "non" ($#v3_relat_1 :::"empty-yielding"::: ) ) ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); cluster (Set "M" ($#k4_matrix_1 :::"@"::: ) ) -> ($#~v3_relat_1 "non" ($#v3_relat_1 :::"empty-yielding"::: ) ) ; end; registrationlet "M" be ($#~v3_relat_1 "non" ($#v3_relat_1 :::"empty-yielding"::: ) ) ($#v4_matrprob :::"Joint_Probability"::: ) ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); cluster (Set "M" ($#k4_matrix_1 :::"@"::: ) ) -> ($#v4_matrprob :::"Joint_Probability"::: ) ; end; theorem :: MATRPROB:62 (Bool "for" (Set (Var "p")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_matrprob :::"ProbFinS"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "P")) "being" ($#~v3_relat_1 "non" ($#v3_relat_1 :::"empty-yielding"::: ) ) ($#v6_matrprob :::"Conditional_Probability"::: ) ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "P"))))) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k12_matrixr1 :::"*"::: ) (Set (Var "P"))) "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_matrprob :::"ProbFinS"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "p")) ($#k12_matrixr1 :::"*"::: ) (Set (Var "P")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "P")))) ")" ))) ; theorem :: MATRPROB:63 (Bool "for" (Set (Var "P1")) "," (Set (Var "P2")) "being" ($#~v3_relat_1 "non" ($#v3_relat_1 :::"empty-yielding"::: ) ) ($#v6_matrprob :::"Conditional_Probability"::: ) ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "P1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "P2"))))) "holds" (Bool "(" (Bool (Set (Set (Var "P1")) ($#k6_matrixr1 :::"*"::: ) (Set (Var "P2"))) "is" ($#~v3_relat_1 "non" ($#v3_relat_1 :::"empty-yielding"::: ) ) ($#v6_matrprob :::"Conditional_Probability"::: ) ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "P1")) ($#k6_matrixr1 :::"*"::: ) (Set (Var "P2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "P1")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" (Set (Var "P1")) ($#k6_matrixr1 :::"*"::: ) (Set (Var "P2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "P2")))) ")" )) ;