:: LAPLACE semantic presentation begin theorem :: LAPLACE:1 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_finseq_3 :::"Del"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1))))) ; theorem :: LAPLACE:2 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "M"))))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k9_matrix_2 :::"Deleting"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Num 1)))))) ; theorem :: LAPLACE:3 (Bool "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "A")) ")" )))) "holds" (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k7_matrix_2 :::"DelCol"::: ) "(" (Set (Var "A")) "," (Set (Var "j")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "A")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1)))))) ; theorem :: LAPLACE:4 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A"))) ($#r1_xxreal_0 :::">"::: ) (Num 1))) "holds" (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k8_matrix_2 :::"DelLine"::: ) "(" (Set (Var "A")) "," (Set (Var "i")) ")" ")" )))))) ; theorem :: LAPLACE:5 (Bool "for" (Set (Var "j")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")) ")" )))) "holds" (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k9_matrix_2 :::"Deleting"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Num 1))))))) ; definitionlet "G" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l3_algstr_0 :::"multMagma"::: ) ; let "B" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "G"))) "," (Set ($#k5_numbers :::"NAT"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "G"))); let "g" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "G")); let "i" be ($#m1_hidden :::"Nat":::); :: original: :::"."::: redefine func "B" :::"."::: "(" "g" "," "i" ")" -> ($#m1_subset_1 :::"Element":::) "of" "G"; end; theorem :: LAPLACE:6 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k9_newton :::"!"::: ) ))) ; theorem :: LAPLACE:7 (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )))) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) "{" (Set (Var "p1")) where p1 "is" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) : (Bool (Set (Set (Var "p1")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "j"))) "}" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k9_newton :::"!"::: ) ))) ; theorem :: LAPLACE:8 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#v12_vectsp_1 :::"Fanoian"::: ) ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "p2")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 2) ")" )) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_finsub_1 :::"Fin"::: ) (Set "(" ($#k2_sgraph1 :::"2Set"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ) ")" )) "st" (Bool (Bool (Set (Var "Y")) ($#r1_hidden :::"="::: ) "{" (Set (Var "s")) where s "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_sgraph1 :::"2Set"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) : (Bool "(" (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Set "(" ($#k1_matrix11 :::"Part_sgn"::: ) "(" (Set (Var "p2")) "," (Set (Var "K")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k1_group_1 :::"1_"::: ) (Set (Var "K")) ")" ))) ")" ) "}" )) "holds" (Bool (Set (Set "the" ($#u2_algstr_0 :::"multF"::: ) "of" (Set (Var "K"))) ($#k7_setwiseo :::"$$"::: ) "(" (Set (Var "X")) "," (Set "(" ($#k1_matrix11 :::"Part_sgn"::: ) "(" (Set (Var "p2")) "," (Set (Var "K")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_group_1 :::"power"::: ) (Set (Var "K")) ")" ) ($#k1_laplace :::"."::: ) "(" (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k1_group_1 :::"1_"::: ) (Set (Var "K")) ")" ) ")" ) "," (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "Y")) ")" ) ")" )))))) ; theorem :: LAPLACE:9 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#v12_vectsp_1 :::"Fanoian"::: ) ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "p2")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 2) ")" )) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 2) ")" ))) & (Bool (Set (Set (Var "p2")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "j")))) "holds" (Bool "ex" (Set (Var "X")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_finsub_1 :::"Fin"::: ) (Set "(" ($#k2_sgraph1 :::"2Set"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ) ")" )) "st" (Bool "(" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set ($#k2_tarski :::"{"::: ) (Set (Var "N")) "," (Set (Var "i")) ($#k2_tarski :::"}"::: ) ) where N "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set ($#k2_tarski :::"{"::: ) (Set (Var "N")) "," (Set (Var "i")) ($#k2_tarski :::"}"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_sgraph1 :::"2Set"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ))) "}" ) & (Bool (Set (Set "the" ($#u2_algstr_0 :::"multF"::: ) "of" (Set (Var "K"))) ($#k7_setwiseo :::"$$"::: ) "(" (Set (Var "X")) "," (Set "(" ($#k1_matrix11 :::"Part_sgn"::: ) "(" (Set (Var "p2")) "," (Set (Var "K")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_group_1 :::"power"::: ) (Set (Var "K")) ")" ) ($#k1_laplace :::"."::: ) "(" (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k1_group_1 :::"1_"::: ) (Set (Var "K")) ")" ) ")" ) "," (Set "(" (Set (Var "i")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "j")) ")" ) ")" )) ")" )))))) ; theorem :: LAPLACE:10 (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 2))) "holds" (Bool "ex" (Set (Var "Proj")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k2_sgraph1 :::"2Set"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) ")" ) "," (Set "(" ($#k2_sgraph1 :::"2Set"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) "st" (Bool "(" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "Proj"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_sgraph1 :::"2Set"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k6_subset_1 :::"\"::: ) "{" (Set ($#k2_tarski :::"{"::: ) (Set (Var "N")) "," (Set (Var "i")) ($#k2_tarski :::"}"::: ) ) where N "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set ($#k2_tarski :::"{"::: ) (Set (Var "N")) "," (Set (Var "i")) ($#k2_tarski :::"}"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_sgraph1 :::"2Set"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))) "}" )) & (Bool (Set (Var "Proj")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool "(" "for" (Set (Var "k")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "m"))) & (Bool (Set ($#k2_tarski :::"{"::: ) (Set (Var "k")) "," (Set (Var "m")) ($#k2_tarski :::"}"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_sgraph1 :::"2Set"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" )))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i")))) "implies" (Bool (Set (Set (Var "Proj")) ($#k1_funct_1 :::"."::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "k")) "," (Set (Var "m")) ($#k2_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "k")) "," (Set (Var "m")) ($#k2_tarski :::"}"::: ) )) ")" & "(" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "i"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i")))) "implies" (Bool (Set (Set (Var "Proj")) ($#k1_funct_1 :::"."::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "k")) "," (Set (Var "m")) ($#k2_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "k")) "," (Set "(" (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k2_tarski :::"}"::: ) )) ")" & "(" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "i"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "i")))) "implies" (Bool (Set (Set (Var "Proj")) ($#k1_funct_1 :::"."::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "k")) "," (Set (Var "m")) ($#k2_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k2_tarski :::"}"::: ) )) ")" ")" ) ")" ) ")" ))) ; theorem :: LAPLACE:11 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Num 2))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))) "holds" (Bool "(" (Bool (Set (Var "p")) "is" ($#v5_matrix_2 :::"even"::: ) ) & (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k1_finseq_2 :::"idseq"::: ) (Set (Var "n")))) ")" ))) ; theorem :: LAPLACE:12 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "," (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "(" ($#k5_finsub_1 :::"Fin"::: ) (Set (Var "Y")) ")" ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k5_finsub_1 :::"Fin"::: ) (Set (Var "Y")) ")" ) "," (Set (Var "D")) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"BinOp":::) "of" (Set (Var "D")) "st" (Bool (Bool "(" "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_finsub_1 :::"Fin"::: ) (Set (Var "Y"))) "st" (Bool (Bool (Set (Var "A")) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "B")))) "holds" (Bool (Set (Set (Var "F")) ($#k5_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "A")) ")" ) "," (Set "(" (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "B")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "A")) ($#k1_finsub_1 :::"\/"::: ) (Set (Var "B")) ")" ))) ")" ) & (Bool (Set (Var "F")) "is" ($#v1_binop_1 :::"commutative"::: ) ) & (Bool (Set (Var "F")) "is" ($#v2_binop_1 :::"associative"::: ) ) & (Bool (Set (Var "F")) "is" ($#v1_setwiseo :::"having_a_unity"::: ) ) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_binop_1 :::"the_unity_wrt"::: ) (Set (Var "F"))))) "holds" (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_finsub_1 :::"Fin"::: ) (Set (Var "X"))) "st" (Bool (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_xboole_0 :::"meets"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))))) "holds" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "y"))) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "F")) ($#k7_setwiseo :::"$$"::: ) "(" (Set (Var "I")) "," (Set "(" (Set (Var "g")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k7_setwiseo :::"$$"::: ) "(" (Set "(" (Set (Var "f")) ($#k8_setwiseo :::".:"::: ) (Set (Var "I")) ")" ) "," (Set (Var "g")) ")" )) & (Bool (Set (Set (Var "F")) ($#k7_setwiseo :::"$$"::: ) "(" (Set "(" (Set (Var "f")) ($#k8_setwiseo :::".:"::: ) (Set (Var "I")) ")" ) "," (Set (Var "g")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_tarski :::"union"::: ) (Set "(" (Set (Var "f")) ($#k8_setwiseo :::".:"::: ) (Set (Var "I")) ")" ) ")" ))) & (Bool (Set ($#k3_tarski :::"union"::: ) (Set "(" (Set (Var "f")) ($#k8_setwiseo :::".:"::: ) (Set (Var "I")) ")" )) "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_finsub_1 :::"Fin"::: ) (Set (Var "Y")))) ")" )))))) ; begin definitionlet "i", "j", "n" be ($#m1_hidden :::"Nat":::); let "K" be ($#l6_algstr_0 :::"Field":::); let "M" be ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Const "n")) "," (Set (Const "K")); assume that (Bool (Set (Const "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Const "n")))) and (Bool (Set (Const "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Const "n")))) ; func :::"Delete"::: "(" "M" "," "i" "," "j" ")" -> ($#m1_matrix_1 :::"Matrix":::) "of" (Set "n" ($#k7_nat_d :::"-'"::: ) (Num 1)) "," "K" equals :: LAPLACE:def 1 (Set ($#k9_matrix_2 :::"Deleting"::: ) "(" "M" "," "i" "," "j" ")" ); end; :: deftheorem defines :::"Delete"::: LAPLACE:def 1 : (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set ($#k2_laplace :::"Delete"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_matrix_2 :::"Deleting"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ))))); theorem :: LAPLACE:13 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool "for" (Set (Var "k")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ))) & (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j")))) "implies" (Bool (Set (Set "(" ($#k2_laplace :::"Delete"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "k")) "," (Set (Var "m")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "k")) "," (Set (Var "m")) ")" )) ")" & "(" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "j")))) "implies" (Bool (Set (Set "(" ($#k2_laplace :::"Delete"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "k")) "," (Set (Var "m")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "k")) "," (Set "(" (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" & "(" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "i"))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j")))) "implies" (Bool (Set (Set "(" ($#k2_laplace :::"Delete"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "k")) "," (Set (Var "m")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "m")) ")" )) ")" & "(" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "i"))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "j")))) "implies" (Bool (Set (Set "(" ($#k2_laplace :::"Delete"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "k")) "," (Set (Var "m")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ")" )))))) ; theorem :: LAPLACE:14 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" ($#k2_laplace :::"Delete"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k5_matrix_1 :::"@"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k2_laplace :::"Delete"::: ) "(" (Set "(" (Set (Var "M")) ($#k5_matrix_1 :::"@"::: ) ")" ) "," (Set (Var "j")) "," (Set (Var "i")) ")" )))))) ; theorem :: LAPLACE:15 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set ($#k2_laplace :::"Delete"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_laplace :::"Delete"::: ) "(" (Set "(" ($#k3_matrix11 :::"RLine"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "f")) ")" ")" ) "," (Set (Var "i")) "," (Set (Var "j")) ")" ))))))) ; definitionlet "c", "n", "m" be ($#m1_hidden :::"Nat":::); let "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "M" be ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Const "n")) "," (Set (Const "m")) "," (Set (Const "D")); let "pD" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); func :::"ReplaceCol"::: "(" "M" "," "c" "," "pD" ")" -> ($#m1_matrix_1 :::"Matrix"::: ) "of" "n" "," "m" "," "D" means :: LAPLACE:def 2 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "M")) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) "M")) & (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 "(" "(" (Bool (Bool (Set (Var "j")) ($#r1_hidden :::"<>"::: ) "c")) "implies" (Bool (Set it ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set "M" ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" & "(" (Bool (Bool (Set (Var "j")) ($#r1_hidden :::"="::: ) "c")) "implies" (Bool (Set it ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," "c" ")" ) ($#r1_hidden :::"="::: ) (Set "pD" ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) ")" ")" ) ")" ) ")" ) if (Bool (Set ($#k3_finseq_1 :::"len"::: ) "pD") ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "M")) otherwise (Bool it ($#r1_hidden :::"="::: ) "M"); end; :: deftheorem defines :::"ReplaceCol"::: LAPLACE:def 2 : (Bool "for" (Set (Var "c")) "," (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "D")) (Bool "for" (Set (Var "pD")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "b7")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "D")) "holds" (Bool "(" "(" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "pD"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M"))))) "implies" (Bool "(" (Bool (Set (Var "b7")) ($#r1_hidden :::"="::: ) (Set ($#k3_laplace :::"ReplaceCol"::: ) "(" (Set (Var "M")) "," (Set (Var "c")) "," (Set (Var "pD")) ")" )) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b7"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "b7"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")))) & (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 "(" "(" (Bool (Bool (Set (Var "j")) ($#r1_hidden :::"<>"::: ) (Set (Var "c")))) "implies" (Bool (Set (Set (Var "b7")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" & "(" (Bool (Bool (Set (Var "j")) ($#r1_hidden :::"="::: ) (Set (Var "c")))) "implies" (Bool (Set (Set (Var "b7")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "c")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "pD")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) ")" ")" ) ")" ) ")" ) ")" ) ")" & "(" (Bool (Bool (Bool "not" (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "pD"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")))))) "implies" (Bool "(" (Bool (Set (Var "b7")) ($#r1_hidden :::"="::: ) (Set ($#k3_laplace :::"ReplaceCol"::: ) "(" (Set (Var "M")) "," (Set (Var "c")) "," (Set (Var "pD")) ")" )) "iff" (Bool (Set (Var "b7")) ($#r1_hidden :::"="::: ) (Set (Var "M"))) ")" ) ")" ")" )))))); notationlet "c", "n", "m" be ($#m1_hidden :::"Nat":::); let "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "M" be ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Const "n")) "," (Set (Const "m")) "," (Set (Const "D")); let "pD" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); synonym :::"RCol"::: "(" "M" "," "c" "," "pD" ")" for :::"ReplaceCol"::: "(" "M" "," "c" "," "pD" ")" ; end; theorem :: LAPLACE:16 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "c")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "AD")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "D")) (Bool "for" (Set (Var "pD")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "AD")) ")" )))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "pD"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "AD"))))) "implies" (Bool (Set ($#k9_matrix_1 :::"Col"::: ) "(" (Set "(" ($#k3_laplace :::"RCol"::: ) "(" (Set (Var "AD")) "," (Set (Var "c")) "," (Set (Var "pD")) ")" ")" ) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "pD"))) ")" & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Set (Var "c")))) "implies" (Bool (Set ($#k9_matrix_1 :::"Col"::: ) "(" (Set "(" ($#k3_laplace :::"RCol"::: ) "(" (Set (Var "AD")) "," (Set (Var "c")) "," (Set (Var "pD")) ")" ")" ) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "AD")) "," (Set (Var "i")) ")" )) ")" ")" )))))) ; theorem :: LAPLACE:17 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "c")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "AD")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "D")) (Bool "for" (Set (Var "pD")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Bool "not" (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "AD")) ")" ))))) "holds" (Bool (Set ($#k3_laplace :::"RCol"::: ) "(" (Set (Var "AD")) "," (Set (Var "c")) "," (Set (Var "pD")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "AD"))))))) ; theorem :: LAPLACE:18 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "c")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "AD")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "D")) "holds" (Bool (Set ($#k3_laplace :::"RCol"::: ) "(" (Set (Var "AD")) "," (Set (Var "c")) "," (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "AD")) "," (Set (Var "c")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "AD")))))) ; theorem :: LAPLACE:19 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "c")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "pD")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "A")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "D")) (Bool "for" (Set (Var "A9")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "D")) "st" (Bool (Bool (Set (Var "A9")) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k4_matrix_1 :::"@"::: ) )) & "(" (Bool (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"0"::: ) ))) "implies" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) ")" ) "holds" (Bool (Set ($#k3_laplace :::"ReplaceCol"::: ) "(" (Set (Var "A")) "," (Set (Var "c")) "," (Set (Var "pD")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_matrix11 :::"ReplaceLine"::: ) "(" (Set (Var "A9")) "," (Set (Var "c")) "," (Set (Var "pD")) ")" ")" ) ($#k4_matrix_1 :::"@"::: ) ))))))) ; begin definitionlet "i", "n" be ($#m1_hidden :::"Nat":::); let "perm" be ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set "(" (Set (Const "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )); assume (Bool (Set (Const "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Const "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) ; func :::"Rem"::: "(" "perm" "," "i" ")" -> ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) "n") means :: LAPLACE:def 3 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) "n"))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) "i")) "implies" (Bool "(" "(" (Bool (Bool (Set "perm" ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_xxreal_0 :::"<"::: ) (Set "perm" ($#k1_funct_1 :::"."::: ) "i"))) "implies" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set "perm" ($#k1_funct_1 :::"."::: ) (Set (Var "k")))) ")" & "(" (Bool (Bool (Set "perm" ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_xxreal_0 :::">="::: ) (Set "perm" ($#k1_funct_1 :::"."::: ) "i"))) "implies" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "perm" ($#k1_funct_1 :::"."::: ) (Set (Var "k")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1))) ")" ")" ) ")" & "(" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::">="::: ) "i")) "implies" (Bool "(" "(" (Bool (Bool (Set "perm" ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set "perm" ($#k1_funct_1 :::"."::: ) "i"))) "implies" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set "perm" ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) ")" & "(" (Bool (Bool (Set "perm" ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_xxreal_0 :::">="::: ) (Set "perm" ($#k1_funct_1 :::"."::: ) "i"))) "implies" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "perm" ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1))) ")" ")" ) ")" ")" )); end; :: deftheorem defines :::"Rem"::: LAPLACE:def 3 : (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "perm")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )))) "holds" (Bool "for" (Set (Var "b4")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k4_laplace :::"Rem"::: ) "(" (Set (Var "perm")) "," (Set (Var "i")) ")" )) "iff" (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i")))) "implies" (Bool "(" "(" (Bool (Bool (Set (Set (Var "perm")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "perm")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))) "implies" (Bool (Set (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "perm")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")))) ")" & "(" (Bool (Bool (Set (Set (Var "perm")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "perm")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))) "implies" (Bool (Set (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "perm")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1))) ")" ")" ) ")" & "(" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "i")))) "implies" (Bool "(" "(" (Bool (Bool (Set (Set (Var "perm")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "perm")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))) "implies" (Bool (Set (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "perm")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) ")" & "(" (Bool (Bool (Set (Set (Var "perm")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "perm")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))) "implies" (Bool (Set (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "perm")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1))) ")" ")" ) ")" ")" )) ")" )))); theorem :: LAPLACE:20 (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )))) "holds" (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p1")) where p1 "is" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) : (Bool (Set (Set (Var "p1")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "j"))) "}" )) "holds" (Bool "ex" (Set (Var "Proj")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "P")) "," (Set "(" ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n")) ")" ) "st" (Bool "(" (Bool (Set (Var "Proj")) "is" ($#v3_funct_2 :::"bijective"::: ) ) & (Bool "(" "for" (Set (Var "q1")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) "st" (Bool (Bool (Set (Set (Var "q1")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "j")))) "holds" (Bool (Set (Set (Var "Proj")) ($#k1_funct_1 :::"."::: ) (Set (Var "q1"))) ($#r1_hidden :::"="::: ) (Set ($#k4_laplace :::"Rem"::: ) "(" (Set (Var "q1")) "," (Set (Var "i")) ")" )) ")" ) ")" )))) ; theorem :: LAPLACE:21 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p1")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) & (Bool (Set (Set (Var "p1")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "j")))) "holds" (Bool (Set ($#k14_matrix_2 :::"-"::: ) "(" (Set (Var "a")) "," (Set (Var "p1")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k4_group_1 :::"power"::: ) (Set (Var "K")) ")" ) ($#k1_laplace :::"."::: ) "(" (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k1_group_1 :::"1_"::: ) (Set (Var "K")) ")" ) ")" ) "," (Set "(" (Set (Var "i")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "j")) ")" ) ")" ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" ($#k14_matrix_2 :::"-"::: ) "(" (Set (Var "a")) "," (Set "(" ($#k4_laplace :::"Rem"::: ) "(" (Set (Var "p1")) "," (Set (Var "i")) ")" ")" ) ")" ")" )))))))) ; theorem :: LAPLACE:22 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p1")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) & (Bool (Set (Set (Var "p1")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "j")))) "holds" (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1)) "," (Set (Var "K")) (Bool "for" (Set (Var "DM")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "DM")) ($#r1_hidden :::"="::: ) (Set ($#k2_laplace :::"Delete"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ))) "holds" (Bool (Set (Set "(" ($#k11_matrix_3 :::"Path_product"::: ) (Set (Var "M")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "p1"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k4_group_1 :::"power"::: ) (Set (Var "K")) ")" ) ($#k1_laplace :::"."::: ) "(" (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k1_group_1 :::"1_"::: ) (Set (Var "K")) ")" ) ")" ) "," (Set "(" (Set (Var "i")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "j")) ")" ) ")" ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set "(" ($#k11_matrix_3 :::"Path_product"::: ) (Set (Var "DM")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" ($#k4_laplace :::"Rem"::: ) "(" (Set (Var "p1")) "," (Set (Var "i")) ")" ")" ) ")" ))))))))) ; begin definitionlet "i", "j", "n" be ($#m1_hidden :::"Nat":::); let "K" be ($#l6_algstr_0 :::"Field":::); let "M" be ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Const "n")) "," (Set (Const "K")); func :::"Minor"::: "(" "M" "," "i" "," "j" ")" -> ($#m1_subset_1 :::"Element":::) "of" "K" equals :: LAPLACE:def 4 (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k2_laplace :::"Delete"::: ) "(" "M" "," "i" "," "j" ")" ")" )); end; :: deftheorem defines :::"Minor"::: LAPLACE:def 4 : (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "holds" (Bool (Set ($#k5_laplace :::"Minor"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k2_laplace :::"Delete"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" )))))); definitionlet "i", "j", "n" be ($#m1_hidden :::"Nat":::); let "K" be ($#l6_algstr_0 :::"Field":::); let "M" be ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Const "n")) "," (Set (Const "K")); func :::"Cofactor"::: "(" "M" "," "i" "," "j" ")" -> ($#m1_subset_1 :::"Element":::) "of" "K" equals :: LAPLACE:def 5 (Set (Set "(" (Set "(" ($#k4_group_1 :::"power"::: ) "K" ")" ) ($#k1_laplace :::"."::: ) "(" (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k1_group_1 :::"1_"::: ) "K" ")" ) ")" ) "," (Set "(" "i" ($#k2_xcmplx_0 :::"+"::: ) "j" ")" ) ")" ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" ($#k5_laplace :::"Minor"::: ) "(" "M" "," "i" "," "j" ")" ")" )); end; :: deftheorem defines :::"Cofactor"::: LAPLACE:def 5 : (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "holds" (Bool (Set ($#k6_laplace :::"Cofactor"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k4_group_1 :::"power"::: ) (Set (Var "K")) ")" ) ($#k1_laplace :::"."::: ) "(" (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k1_group_1 :::"1_"::: ) (Set (Var "K")) ")" ) ")" ) "," (Set "(" (Set (Var "i")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "j")) ")" ) ")" ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" ($#k5_laplace :::"Minor"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" )))))); theorem :: LAPLACE:23 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_finsub_1 :::"Fin"::: ) (Set "(" ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n")) ")" )) "st" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))) : (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "j"))) "}" )) "holds" (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "holds" (Bool (Set (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "K"))) ($#k7_setwiseo :::"$$"::: ) "(" (Set (Var "P")) "," (Set "(" ($#k11_matrix_3 :::"Path_product"::: ) (Set (Var "M")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" ($#k6_laplace :::"Cofactor"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" )))))))) ; theorem :: LAPLACE:24 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set ($#k5_laplace :::"Minor"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k5_laplace :::"Minor"::: ) "(" (Set "(" (Set (Var "M")) ($#k5_matrix_1 :::"@"::: ) ")" ) "," (Set (Var "j")) "," (Set (Var "i")) ")" )))))) ; definitionlet "n" be ($#m1_hidden :::"Nat":::); let "K" be ($#l6_algstr_0 :::"Field":::); let "M" be ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Const "n")) "," (Set (Const "K")); func :::"Matrix_of_Cofactor"::: "M" -> ($#m1_matrix_1 :::"Matrix":::) "of" "n" "," "K" means :: LAPLACE:def 6 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#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"::: ) it))) "holds" (Bool (Set it ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_laplace :::"Cofactor"::: ) "(" "M" "," (Set (Var "i")) "," (Set (Var "j")) ")" ))); end; :: deftheorem defines :::"Matrix_of_Cofactor"::: LAPLACE:def 6 : (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "," (Set (Var "b4")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k7_laplace :::"Matrix_of_Cofactor"::: ) (Set (Var "M")))) "iff" (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 ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "b4"))))) "holds" (Bool (Set (Set (Var "b4")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_laplace :::"Cofactor"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ))) ")" )))); begin definitionlet "n", "i" be ($#m1_hidden :::"Nat":::); let "K" be ($#l6_algstr_0 :::"Field":::); let "M" be ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Const "n")) "," (Set (Const "K")); func :::"LaplaceExpL"::: "(" "M" "," "i" ")" -> ($#m2_finseq_1 :::"FinSequence":::) "of" "K" means :: LAPLACE:def 7 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) "n") & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "M" ($#k3_matrix_1 :::"*"::: ) "(" "i" "," (Set (Var "j")) ")" ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" ($#k6_laplace :::"Cofactor"::: ) "(" "M" "," "i" "," (Set (Var "j")) ")" ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"LaplaceExpL"::: LAPLACE:def 7 : (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "b5")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "K")) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set ($#k8_laplace :::"LaplaceExpL"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" )) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b5"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "b5"))))) "holds" (Bool (Set (Set (Var "b5")) ($#k1_funct_1 :::"."::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" ($#k6_laplace :::"Cofactor"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" ))) ")" ) ")" ) ")" ))))); definitionlet "n", "j" be ($#m1_hidden :::"Nat":::); let "K" be ($#l6_algstr_0 :::"Field":::); let "M" be ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Const "n")) "," (Set (Const "K")); func :::"LaplaceExpC"::: "(" "M" "," "j" ")" -> ($#m2_finseq_1 :::"FinSequence":::) "of" "K" means :: LAPLACE:def 8 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) "n") & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "M" ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," "j" ")" ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" ($#k6_laplace :::"Cofactor"::: ) "(" "M" "," (Set (Var "i")) "," "j" ")" ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"LaplaceExpC"::: LAPLACE:def 8 : (Bool "for" (Set (Var "n")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "b5")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "K")) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set ($#k9_laplace :::"LaplaceExpC"::: ) "(" (Set (Var "M")) "," (Set (Var "j")) ")" )) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b5"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "b5"))))) "holds" (Bool (Set (Set (Var "b5")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" ($#k6_laplace :::"Cofactor"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" ))) ")" ) ")" ) ")" ))))); theorem :: LAPLACE:25 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" ($#k8_laplace :::"LaplaceExpL"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ")" ))))))) ; theorem :: LAPLACE:26 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set ($#k9_laplace :::"LaplaceExpC"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_laplace :::"LaplaceExpL"::: ) "(" (Set "(" (Set (Var "M")) ($#k5_matrix_1 :::"@"::: ) ")" ) "," (Set (Var "i")) ")" )))))) ; theorem :: LAPLACE:27 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" ($#k9_laplace :::"LaplaceExpC"::: ) "(" (Set (Var "M")) "," (Set (Var "j")) ")" ")" ))))))) ; theorem :: LAPLACE:28 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "p")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set ($#k11_fvsum_1 :::"mlt"::: ) "(" (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set "(" ($#k7_laplace :::"Matrix_of_Cofactor"::: ) (Set (Var "M")) ")" ) "," (Set (Var "i")) ")" ")" ) "," (Set (Var "f")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_laplace :::"LaplaceExpL"::: ) "(" (Set "(" ($#k3_matrix11 :::"RLine"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "f")) ")" ")" ) "," (Set (Var "i")) ")" )))))))) ; theorem :: LAPLACE:29 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Set (Var "j")) ")" ")" ) ($#k13_fvsum_1 :::""*""::: ) (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set "(" (Set "(" ($#k7_laplace :::"Matrix_of_Cofactor"::: ) (Set (Var "M")) ")" ) ($#k5_matrix_1 :::"@"::: ) ")" ) "," (Set (Var "i")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k3_matrix11 :::"RLine"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Set (Var "j")) ")" ")" ) ")" ")" )))))) ; theorem :: LAPLACE:30 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set (Var "M"))) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool (Set (Set (Var "M")) ($#k4_matrix_6 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k12_matrix_3 :::"Det"::: ) (Set (Var "M")) ")" ) ($#k11_algstr_0 :::"""::: ) ")" ) ($#k1_matrix_7 :::"*"::: ) (Set "(" (Set "(" ($#k7_laplace :::"Matrix_of_Cofactor"::: ) (Set (Var "M")) ")" ) ($#k5_matrix_1 :::"@"::: ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k12_matrix_1 :::"1."::: ) "(" (Set (Var "K")) "," (Set (Var "n")) ")" ))))) ; theorem :: LAPLACE:31 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set ($#k11_fvsum_1 :::"mlt"::: ) "(" (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set "(" ($#k7_laplace :::"Matrix_of_Cofactor"::: ) (Set (Var "M")) ")" ) "," (Set (Var "i")) ")" ")" ) "," (Set (Var "f")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_laplace :::"LaplaceExpL"::: ) "(" (Set "(" ($#k3_matrix11 :::"RLine"::: ) "(" (Set "(" (Set (Var "M")) ($#k5_matrix_1 :::"@"::: ) ")" ) "," (Set (Var "i")) "," (Set (Var "f")) ")" ")" ) "," (Set (Var "i")) ")" ))))))) ; theorem :: LAPLACE:32 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set "(" (Set "(" ($#k7_laplace :::"Matrix_of_Cofactor"::: ) (Set (Var "M")) ")" ) ($#k5_matrix_1 :::"@"::: ) ")" ) "," (Set (Var "i")) ")" ")" ) ($#k13_fvsum_1 :::""*""::: ) (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "M")) "," (Set (Var "j")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k3_matrix11 :::"RLine"::: ) "(" (Set "(" (Set (Var "M")) ($#k5_matrix_1 :::"@"::: ) ")" ) "," (Set (Var "i")) "," (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set "(" (Set (Var "M")) ($#k5_matrix_1 :::"@"::: ) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ")" )))))) ; theorem :: LAPLACE:33 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set (Var "M"))) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" ($#k12_matrix_3 :::"Det"::: ) (Set (Var "M")) ")" ) ($#k11_algstr_0 :::"""::: ) ")" ) ($#k1_matrix_7 :::"*"::: ) (Set "(" (Set "(" ($#k7_laplace :::"Matrix_of_Cofactor"::: ) (Set (Var "M")) ")" ) ($#k5_matrix_1 :::"@"::: ) ")" ) ")" ) ($#k4_matrix_6 :::"*"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k12_matrix_1 :::"1."::: ) "(" (Set (Var "K")) "," (Set (Var "n")) ")" ))))) ; theorem :: LAPLACE:34 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#v1_matrix_6 :::"invertible"::: ) ) "iff" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set (Var "M"))) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" )))) ; theorem :: LAPLACE:35 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set (Var "M"))) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool (Set (Set (Var "M")) ($#k5_matrix_6 :::"~"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k12_matrix_3 :::"Det"::: ) (Set (Var "M")) ")" ) ($#k11_algstr_0 :::"""::: ) ")" ) ($#k1_matrix_7 :::"*"::: ) (Set "(" (Set "(" ($#k7_laplace :::"Matrix_of_Cofactor"::: ) (Set (Var "M")) ")" ) ($#k5_matrix_1 :::"@"::: ) ")" )))))) ; theorem :: LAPLACE:36 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "M")) "is" ($#v1_matrix_6 :::"invertible"::: ) )) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#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 "(" (Set (Var "M")) ($#k5_matrix_6 :::"~"::: ) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "M")) ($#k5_matrix_6 :::"~"::: ) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k12_matrix_3 :::"Det"::: ) (Set (Var "M")) ")" ) ($#k11_algstr_0 :::"""::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set "(" ($#k4_group_1 :::"power"::: ) (Set (Var "K")) ")" ) ($#k1_laplace :::"."::: ) "(" (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k1_group_1 :::"1_"::: ) (Set (Var "K")) ")" ) ")" ) "," (Set "(" (Set (Var "i")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "j")) ")" ) ")" ")" ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" ($#k5_laplace :::"Minor"::: ) "(" (Set (Var "M")) "," (Set (Var "j")) "," (Set (Var "i")) ")" ")" ))))))) ; theorem :: LAPLACE:37 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "A")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set (Var "A"))) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "b")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "A")) ($#k4_matrix_3 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "b")))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "A")) ($#k5_matrix_6 :::"~"::: ) ")" ) ($#k4_matrix_3 :::"*"::: ) (Set (Var "b")))) & (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 ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "x"))))) "holds" (Bool (Set (Set (Var "x")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k12_matrix_3 :::"Det"::: ) (Set (Var "A")) ")" ) ($#k11_algstr_0 :::"""::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k3_laplace :::"ReplaceCol"::: ) "(" (Set (Var "A")) "," (Set (Var "i")) "," (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "b")) "," (Set (Var "j")) ")" ")" ) ")" ")" ) ")" ))) ")" ) ")" ))))) ; theorem :: LAPLACE:38 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "A")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set (Var "A"))) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "b")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "x")) ($#k4_matrix_3 :::"*"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "b")))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k4_matrix_3 :::"*"::: ) (Set "(" (Set (Var "A")) ($#k5_matrix_6 :::"~"::: ) ")" ))) & (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#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 "x"))))) "holds" (Bool (Set (Set (Var "x")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k12_matrix_3 :::"Det"::: ) (Set (Var "A")) ")" ) ($#k11_algstr_0 :::"""::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k3_matrix11 :::"ReplaceLine"::: ) "(" (Set (Var "A")) "," (Set (Var "j")) "," (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "b")) "," (Set (Var "i")) ")" ")" ) ")" ")" ) ")" ))) ")" ) ")" ))))) ; begin definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); :: original: :::"<*"::: redefine func :::"<*":::"f":::"*>"::: -> ($#m1_matrix_1 :::"Matrix"::: ) "of" (Num 1) "," (Set ($#k3_finseq_1 :::"len"::: ) "f") "," "D"; end; definitionlet "K" be ($#l6_algstr_0 :::"Field":::); let "M" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Const "K")); let "f" be ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Const "K")); func "M" :::"*"::: "f" -> ($#m2_finseq_1 :::"Matrix":::) "of" "K" equals :: LAPLACE:def 9 (Set "M" ($#k4_matrix_3 :::"*"::: ) (Set "(" (Set ($#k10_laplace :::"<*"::: ) "f" ($#k10_laplace :::"*>"::: ) ) ($#k4_matrix_1 :::"@"::: ) ")" )); func "f" :::"*"::: "M" -> ($#m2_finseq_1 :::"Matrix":::) "of" "K" equals :: LAPLACE:def 10 (Set (Set ($#k10_laplace :::"<*"::: ) "f" ($#k10_laplace :::"*>"::: ) ) ($#k4_matrix_3 :::"*"::: ) "M"); end; :: deftheorem defines :::"*"::: LAPLACE:def 9 : (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "K")) "holds" (Bool (Set (Set (Var "M")) ($#k11_laplace :::"*"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "M")) ($#k4_matrix_3 :::"*"::: ) (Set "(" (Set ($#k10_laplace :::"<*"::: ) (Set (Var "f")) ($#k10_laplace :::"*>"::: ) ) ($#k4_matrix_1 :::"@"::: ) ")" )))))); :: deftheorem defines :::"*"::: LAPLACE:def 10 : (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "K")) "holds" (Bool (Set (Set (Var "f")) ($#k12_laplace :::"*"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k10_laplace :::"<*"::: ) (Set (Var "f")) ($#k10_laplace :::"*>"::: ) ) ($#k4_matrix_3 :::"*"::: ) (Set (Var "M"))))))); theorem :: LAPLACE:39 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "A")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set (Var "A"))) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "b")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "A")) ($#k11_laplace :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k10_laplace :::"<*"::: ) (Set (Var "b")) ($#k10_laplace :::"*>"::: ) ) ($#k4_matrix_1 :::"@"::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k10_laplace :::"<*"::: ) (Set (Var "x")) ($#k10_laplace :::"*>"::: ) ) ($#k4_matrix_1 :::"@"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "A")) ($#k5_matrix_6 :::"~"::: ) ")" ) ($#k11_laplace :::"*"::: ) (Set (Var "b")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set (Var "x")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k12_matrix_3 :::"Det"::: ) (Set (Var "A")) ")" ) ($#k11_algstr_0 :::"""::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k3_laplace :::"ReplaceCol"::: ) "(" (Set (Var "A")) "," (Set (Var "i")) "," (Set (Var "b")) ")" ")" ) ")" ))) ")" ) ")" ))))) ; theorem :: LAPLACE:40 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "A")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set (Var "A"))) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "b")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "x")) ($#k12_laplace :::"*"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k10_laplace :::"<*"::: ) (Set (Var "b")) ($#k10_laplace :::"*>"::: ) ))) "holds" (Bool "(" (Bool (Set ($#k10_laplace :::"<*"::: ) (Set (Var "x")) ($#k10_laplace :::"*>"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k12_laplace :::"*"::: ) (Set "(" (Set (Var "A")) ($#k5_matrix_6 :::"~"::: ) ")" ))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set (Var "x")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k12_matrix_3 :::"Det"::: ) (Set (Var "A")) ")" ) ($#k11_algstr_0 :::"""::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k3_matrix11 :::"ReplaceLine"::: ) "(" (Set (Var "A")) "," (Set (Var "i")) "," (Set (Var "b")) ")" ")" ) ")" ))) ")" ) ")" ))))) ;