:: RADIX_2 semantic presentation begin theorem :: RADIX_2:1 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "a")) ($#k4_nat_d :::"mod"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: RADIX_2:2 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "a")) ($#k6_int_1 :::"mod"::: ) (Set (Var "n")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set (Var "b")) ($#k6_int_1 :::"mod"::: ) (Set (Var "n")) ")" ) ")" ) ($#k6_int_1 :::"mod"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set (Var "b")) ($#k6_int_1 :::"mod"::: ) (Set (Var "n")) ")" ) ")" ) ($#k6_int_1 :::"mod"::: ) (Set (Var "n")))))) ; theorem :: RADIX_2:3 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "b")) ")" ) ($#k6_int_1 :::"mod"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set (Var "b")) ($#k6_int_1 :::"mod"::: ) (Set (Var "n")) ")" ) ")" ) ($#k6_int_1 :::"mod"::: ) (Set (Var "n")))))) ; theorem :: RADIX_2:4 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b")))) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k4_nat_d :::"mod"::: ) (Set "(" (Set (Var "b")) ($#k1_newton :::"|^"::: ) (Set (Var "i")) ")" ) ")" ) ($#k3_nat_d :::"div"::: ) (Set "(" (Set (Var "b")) ($#k1_newton :::"|^"::: ) (Set "(" (Set (Var "i")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k3_nat_d :::"div"::: ) (Set "(" (Set (Var "b")) ($#k1_newton :::"|^"::: ) (Set "(" (Set (Var "i")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k4_nat_d :::"mod"::: ) (Set (Var "b"))))) ; theorem :: RADIX_2:5 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )))) ; begin theorem :: RADIX_2:6 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k1_radix_1 :::"Radix"::: ) (Set (Var "k"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: RADIX_2:7 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Num 1) "," (Set (Set (Var "k")) ($#k3_radix_1 :::"-SD"::: ) ) "holds" (Bool (Set ($#k8_radix_1 :::"SDDec"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_radix_1 :::"DigA"::: ) "(" (Set (Var "x")) "," (Num 1) ")" )))) ; theorem :: RADIX_2:8 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set "(" ($#k12_radix_1 :::"SD_Add_Data"::: ) "(" (Set (Var "x")) "," (Set (Var "k")) ")" ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set "(" ($#k11_radix_1 :::"SD_Add_Carry"::: ) (Set (Var "x")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" ($#k1_radix_1 :::"Radix"::: ) (Set (Var "k")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "x"))))) ; theorem :: RADIX_2:9 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1)) "," (Set (Set (Var "k")) ($#k3_radix_1 :::"-SD"::: ) ) (Bool "for" (Set (Var "xn")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set (Set (Var "k")) ($#k3_radix_1 :::"-SD"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#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")) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "xn")) ($#k1_seq_1 :::"."::: ) (Set (Var "i")))) ")" )) "holds" (Bool (Set ($#k1_gr_cy_1 :::"Sum"::: ) (Set "(" ($#k7_radix_1 :::"DigitSD"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_gr_cy_1 :::"Sum"::: ) (Set "(" (Set "(" ($#k7_radix_1 :::"DigitSD"::: ) (Set (Var "xn")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k6_radix_1 :::"SubDigit"::: ) "(" (Set (Var "x")) "," (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "k")) ")" ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ")" )))))) ; theorem :: RADIX_2:10 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1)) "," (Set (Set (Var "k")) ($#k3_radix_1 :::"-SD"::: ) ) (Bool "for" (Set (Var "xn")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set (Set (Var "k")) ($#k3_radix_1 :::"-SD"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#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")) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "xn")) ($#k1_seq_1 :::"."::: ) (Set (Var "i")))) ")" )) "holds" (Bool (Set (Set "(" ($#k8_radix_1 :::"SDDec"::: ) (Set (Var "xn")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set "(" (Set "(" ($#k1_radix_1 :::"Radix"::: ) (Set (Var "k")) ")" ) ($#k1_newton :::"|^"::: ) (Set (Var "n")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" ($#k4_radix_1 :::"DigA"::: ) "(" (Set (Var "x")) "," (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_radix_1 :::"SDDec"::: ) (Set (Var "x"))))))) ; theorem :: RADIX_2:11 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set (Set (Var "k")) ($#k3_radix_1 :::"-SD"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::">="::: ) (Num 2))) "holds" (Bool (Set (Set "(" ($#k8_radix_1 :::"SDDec"::: ) (Set "(" (Set (Var "x")) ($#k14_radix_1 :::"'+'"::: ) (Set (Var "y")) ")" ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set "(" ($#k11_radix_1 :::"SD_Add_Carry"::: ) (Set "(" (Set "(" ($#k4_radix_1 :::"DigA"::: ) "(" (Set (Var "x")) "," (Set (Var "n")) ")" ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" ($#k4_radix_1 :::"DigA"::: ) "(" (Set (Var "y")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set "(" ($#k1_radix_1 :::"Radix"::: ) (Set (Var "k")) ")" ) ($#k1_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k8_radix_1 :::"SDDec"::: ) (Set (Var "x")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" ($#k8_radix_1 :::"SDDec"::: ) (Set (Var "y")) ")" ))))) ; begin definitionlet "i", "k", "n" be ($#m1_hidden :::"Nat":::); let "x" be ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Const "n")) "," (Set ($#k5_numbers :::"NAT"::: ) ); func :::"SubDigit2"::: "(" "x" "," "i" "," "k" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: RADIX_2:def 1 (Set (Set "(" (Set "(" ($#k1_radix_1 :::"Radix"::: ) "k" ")" ) ($#k13_newton :::"|^"::: ) (Set "(" "i" ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ) ($#k4_nat_1 :::"*"::: ) (Set "(" "x" ($#k1_recdef_1 :::"."::: ) "i" ")" )); end; :: deftheorem defines :::"SubDigit2"::: RADIX_2:def 1 : (Bool "for" (Set (Var "i")) "," (Set (Var "k")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k1_radix_2 :::"SubDigit2"::: ) "(" (Set (Var "x")) "," (Set (Var "i")) "," (Set (Var "k")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k1_radix_1 :::"Radix"::: ) (Set (Var "k")) ")" ) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "i")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ) ($#k4_nat_1 :::"*"::: ) (Set "(" (Set (Var "x")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i")) ")" ))))); definitionlet "n", "k" be ($#m1_hidden :::"Nat":::); let "x" be ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Const "n")) "," (Set ($#k5_numbers :::"NAT"::: ) ); func :::"DigitSD2"::: "(" "x" "," "k" ")" -> ($#m2_finseq_1 :::"Tuple":::) "of" "n" "," (Set ($#k5_numbers :::"NAT"::: ) ) means :: RADIX_2:def 2 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) "n"))) "holds" (Bool (Set it ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k1_radix_2 :::"SubDigit2"::: ) "(" "x" "," (Set (Var "i")) "," "k" ")" ))); end; :: deftheorem defines :::"DigitSD2"::: RADIX_2:def 2 : (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "x")) "," (Set (Var "b4")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k2_radix_2 :::"DigitSD2"::: ) "(" (Set (Var "x")) "," (Set (Var "k")) ")" )) "iff" (Bool "for" (Set (Var "i")) "being" ($#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 "b4")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k1_radix_2 :::"SubDigit2"::: ) "(" (Set (Var "x")) "," (Set (Var "i")) "," (Set (Var "k")) ")" ))) ")" ))); definitionlet "n", "k" be ($#m1_hidden :::"Nat":::); let "x" be ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Const "n")) "," (Set ($#k5_numbers :::"NAT"::: ) ); func :::"SDDec2"::: "(" "x" "," "k" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: RADIX_2:def 3 (Set ($#k2_wsierp_1 :::"Sum"::: ) (Set "(" ($#k2_radix_2 :::"DigitSD2"::: ) "(" "x" "," "k" ")" ")" )); end; :: deftheorem defines :::"SDDec2"::: RADIX_2:def 3 : (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k3_radix_2 :::"SDDec2"::: ) "(" (Set (Var "x")) "," (Set (Var "k")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_wsierp_1 :::"Sum"::: ) (Set "(" ($#k2_radix_2 :::"DigitSD2"::: ) "(" (Set (Var "x")) "," (Set (Var "k")) ")" ")" ))))); definitionlet "i", "k", "x" be ($#m1_hidden :::"Nat":::); func :::"DigitDC2"::: "(" "x" "," "i" "," "k" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: RADIX_2:def 4 (Set (Set "(" "x" ($#k4_nat_d :::"mod"::: ) (Set "(" (Set "(" ($#k1_radix_1 :::"Radix"::: ) "k" ")" ) ($#k1_newton :::"|^"::: ) "i" ")" ) ")" ) ($#k3_nat_d :::"div"::: ) (Set "(" (Set "(" ($#k1_radix_1 :::"Radix"::: ) "k" ")" ) ($#k13_newton :::"|^"::: ) (Set "(" "i" ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" )); end; :: deftheorem defines :::"DigitDC2"::: RADIX_2:def 4 : (Bool "for" (Set (Var "i")) "," (Set (Var "k")) "," (Set (Var "x")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k4_radix_2 :::"DigitDC2"::: ) "(" (Set (Var "x")) "," (Set (Var "i")) "," (Set (Var "k")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k4_nat_d :::"mod"::: ) (Set "(" (Set "(" ($#k1_radix_1 :::"Radix"::: ) (Set (Var "k")) ")" ) ($#k1_newton :::"|^"::: ) (Set (Var "i")) ")" ) ")" ) ($#k3_nat_d :::"div"::: ) (Set "(" (Set "(" ($#k1_radix_1 :::"Radix"::: ) (Set (Var "k")) ")" ) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "i")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" )))); definitionlet "k", "n", "x" be ($#m1_hidden :::"Nat":::); func :::"DecSD2"::: "(" "x" "," "n" "," "k" ")" -> ($#m2_finseq_1 :::"Tuple":::) "of" "n" "," (Set ($#k5_numbers :::"NAT"::: ) ) means :: RADIX_2:def 5 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) "n"))) "holds" (Bool (Set it ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k4_radix_2 :::"DigitDC2"::: ) "(" "x" "," (Set (Var "i")) "," "k" ")" ))); end; :: deftheorem defines :::"DecSD2"::: RADIX_2:def 5 : (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "," (Set (Var "x")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "b4")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k5_radix_2 :::"DecSD2"::: ) "(" (Set (Var "x")) "," (Set (Var "n")) "," (Set (Var "k")) ")" )) "iff" (Bool "for" (Set (Var "i")) "being" ($#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 "b4")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k4_radix_2 :::"DigitDC2"::: ) "(" (Set (Var "x")) "," (Set (Var "i")) "," (Set (Var "k")) ")" ))) ")" ))); theorem :: RADIX_2:12 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "y")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set (Set (Var "k")) ($#k3_radix_1 :::"-SD"::: ) ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "y")))) "holds" (Bool (Set ($#k2_radix_2 :::"DigitSD2"::: ) "(" (Set (Var "x")) "," (Set (Var "k")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k7_radix_1 :::"DigitSD"::: ) (Set (Var "y"))))))) ; theorem :: RADIX_2:13 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "y")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set (Set (Var "k")) ($#k3_radix_1 :::"-SD"::: ) ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "y")))) "holds" (Bool (Set ($#k3_radix_2 :::"SDDec2"::: ) "(" (Set (Var "x")) "," (Set (Var "k")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_radix_1 :::"SDDec"::: ) (Set (Var "y"))))))) ; theorem :: RADIX_2:14 (Bool "for" (Set (Var "x")) "," (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k5_radix_2 :::"DecSD2"::: ) "(" (Set (Var "x")) "," (Set (Var "n")) "," (Set (Var "k")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k10_radix_1 :::"DecSD"::: ) "(" (Set (Var "x")) "," (Set (Var "n")) "," (Set (Var "k")) ")" ))) ; theorem :: RADIX_2:15 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool "for" (Set (Var "m")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "m")) ($#r1_radix_1 :::"is_represented_by"::: ) (Set (Var "n")) "," (Set (Var "k")))) "holds" (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set ($#k3_radix_2 :::"SDDec2"::: ) "(" (Set "(" ($#k5_radix_2 :::"DecSD2"::: ) "(" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "k")) ")" ")" ) "," (Set (Var "k")) ")" )))) ; begin definitionlet "q" be ($#m1_hidden :::"Integer":::); let "f", "j", "k", "n" be ($#m1_hidden :::"Nat":::); let "c" be ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Const "n")) "," (Set (Set (Const "k")) ($#k3_radix_1 :::"-SD"::: ) ); func :::"Table1"::: "(" "q" "," "c" "," "f" "," "j" ")" -> ($#m1_hidden :::"Integer":::) equals :: RADIX_2:def 6 (Set (Set "(" "q" ($#k3_xcmplx_0 :::"*"::: ) (Set "(" ($#k4_radix_1 :::"DigA"::: ) "(" "c" "," "j" ")" ")" ) ")" ) ($#k6_int_1 :::"mod"::: ) "f"); end; :: deftheorem defines :::"Table1"::: RADIX_2:def 6 : (Bool "for" (Set (Var "q")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "f")) "," (Set (Var "j")) "," (Set (Var "k")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "c")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set (Set (Var "k")) ($#k3_radix_1 :::"-SD"::: ) ) "holds" (Bool (Set ($#k6_radix_2 :::"Table1"::: ) "(" (Set (Var "q")) "," (Set (Var "c")) "," (Set (Var "f")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "q")) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" ($#k4_radix_1 :::"DigA"::: ) "(" (Set (Var "c")) "," (Set (Var "j")) ")" ")" ) ")" ) ($#k6_int_1 :::"mod"::: ) (Set (Var "f"))))))); definitionlet "q" be ($#m1_hidden :::"Integer":::); let "k", "f", "n" be ($#m1_hidden :::"Nat":::); let "c" be ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Const "n")) "," (Set (Set (Const "k")) ($#k3_radix_1 :::"-SD"::: ) ); assume (Bool (Set (Const "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1)) ; func :::"Mul_mod"::: "(" "q" "," "c" "," "f" "," "k" ")" -> ($#m2_finseq_1 :::"Tuple":::) "of" "n" "," (Set ($#k4_numbers :::"INT"::: ) ) means :: RADIX_2:def 7 (Bool "(" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k6_radix_2 :::"Table1"::: ) "(" "q" "," "c" "," "f" "," "n" ")" )) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set "n" ($#k5_real_1 :::"-"::: ) (Num 1)))) "holds" (Bool "ex" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool "(" (Bool (Set (Var "I1")) ($#r1_hidden :::"="::: ) (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "i")))) & (Bool (Set (Var "I2")) ($#r1_hidden :::"="::: ) (Set it ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) & (Bool (Set (Var "I2")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_radix_1 :::"Radix"::: ) "k" ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "I1")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" ($#k6_radix_2 :::"Table1"::: ) "(" "q" "," "c" "," "f" "," (Set "(" "n" ($#k7_nat_d :::"-'"::: ) (Set (Var "i")) ")" ) ")" ")" ) ")" ) ($#k6_int_1 :::"mod"::: ) "f")) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"Mul_mod"::: RADIX_2:def 7 : (Bool "for" (Set (Var "q")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "k")) "," (Set (Var "f")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "c")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set (Set (Var "k")) ($#k3_radix_1 :::"-SD"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool "for" (Set (Var "b6")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set ($#k4_numbers :::"INT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b6")) ($#r1_hidden :::"="::: ) (Set ($#k7_radix_2 :::"Mul_mod"::: ) "(" (Set (Var "q")) "," (Set (Var "c")) "," (Set (Var "f")) "," (Set (Var "k")) ")" )) "iff" (Bool "(" (Bool (Set (Set (Var "b6")) ($#k1_seq_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k6_radix_2 :::"Table1"::: ) "(" (Set (Var "q")) "," (Set (Var "c")) "," (Set (Var "f")) "," (Set (Var "n")) ")" )) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "n")) ($#k5_real_1 :::"-"::: ) (Num 1)))) "holds" (Bool "ex" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool "(" (Bool (Set (Var "I1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "b6")) ($#k1_seq_1 :::"."::: ) (Set (Var "i")))) & (Bool (Set (Var "I2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "b6")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) & (Bool (Set (Var "I2")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_radix_1 :::"Radix"::: ) (Set (Var "k")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "I1")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" ($#k6_radix_2 :::"Table1"::: ) "(" (Set (Var "q")) "," (Set (Var "c")) "," (Set (Var "f")) "," (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "i")) ")" ) ")" ")" ) ")" ) ($#k6_int_1 :::"mod"::: ) (Set (Var "f")))) ")" )) ")" ) ")" ) ")" ))))); theorem :: RADIX_2:16 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool "for" (Set (Var "q")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "ic")) "," (Set (Var "f")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "ic")) ($#r1_radix_1 :::"is_represented_by"::: ) (Set (Var "n")) "," (Set (Var "k"))) & (Bool (Set (Var "f")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "for" (Set (Var "c")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set (Set (Var "k")) ($#k3_radix_1 :::"-SD"::: ) ) "st" (Bool (Bool (Set (Var "c")) ($#r1_hidden :::"="::: ) (Set ($#k10_radix_1 :::"DecSD"::: ) "(" (Set (Var "ic")) "," (Set (Var "n")) "," (Set (Var "k")) ")" ))) "holds" (Bool (Set (Set "(" ($#k7_radix_2 :::"Mul_mod"::: ) "(" (Set (Var "q")) "," (Set (Var "c")) "," (Set (Var "f")) "," (Set (Var "k")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "q")) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "ic")) ")" ) ($#k6_int_1 :::"mod"::: ) (Set (Var "f")))))))) ; begin definitionlet "n", "f", "j", "m" be ($#m1_hidden :::"Nat":::); let "e" be ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Const "n")) "," (Set ($#k5_numbers :::"NAT"::: ) ); func :::"Table2"::: "(" "m" "," "e" "," "f" "," "j" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: RADIX_2:def 8 (Set (Set "(" "m" ($#k1_newton :::"|^"::: ) (Set "(" "e" ($#k7_partfun1 :::"/."::: ) "j" ")" ) ")" ) ($#k4_nat_d :::"mod"::: ) "f"); end; :: deftheorem defines :::"Table2"::: RADIX_2:def 8 : (Bool "for" (Set (Var "n")) "," (Set (Var "f")) "," (Set (Var "j")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "e")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k8_radix_2 :::"Table2"::: ) "(" (Set (Var "m")) "," (Set (Var "e")) "," (Set (Var "f")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "m")) ($#k1_newton :::"|^"::: ) (Set "(" (Set (Var "e")) ($#k7_partfun1 :::"/."::: ) (Set (Var "j")) ")" ) ")" ) ($#k4_nat_d :::"mod"::: ) (Set (Var "f")))))); definitionlet "k", "f", "m", "n" be ($#m1_hidden :::"Nat":::); let "e" be ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Const "n")) "," (Set ($#k5_numbers :::"NAT"::: ) ); assume (Bool (Set (Const "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1)) ; func :::"Pow_mod"::: "(" "m" "," "e" "," "f" "," "k" ")" -> ($#m2_finseq_1 :::"Tuple":::) "of" "n" "," (Set ($#k5_numbers :::"NAT"::: ) ) means :: RADIX_2:def 9 (Bool "(" (Bool (Set it ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k8_radix_2 :::"Table2"::: ) "(" "m" "," "e" "," "f" "," "n" ")" )) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set "n" ($#k5_real_1 :::"-"::: ) (Num 1)))) "holds" (Bool "ex" (Set (Var "i1")) "," (Set (Var "i2")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "i1")) ($#r1_hidden :::"="::: ) (Set it ($#k1_recdef_1 :::"."::: ) (Set (Var "i")))) & (Bool (Set (Var "i2")) ($#r1_hidden :::"="::: ) (Set it ($#k1_recdef_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) & (Bool (Set (Var "i2")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set (Var "i1")) ($#k1_newton :::"|^"::: ) (Set "(" ($#k1_radix_1 :::"Radix"::: ) "k" ")" ) ")" ) ($#k4_nat_d :::"mod"::: ) "f" ")" ) ($#k4_nat_1 :::"*"::: ) (Set "(" ($#k8_radix_2 :::"Table2"::: ) "(" "m" "," "e" "," "f" "," (Set "(" "n" ($#k7_nat_d :::"-'"::: ) (Set (Var "i")) ")" ) ")" ")" ) ")" ) ($#k4_nat_d :::"mod"::: ) "f")) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"Pow_mod"::: RADIX_2:def 9 : (Bool "for" (Set (Var "k")) "," (Set (Var "f")) "," (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "e")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool "for" (Set (Var "b6")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b6")) ($#r1_hidden :::"="::: ) (Set ($#k9_radix_2 :::"Pow_mod"::: ) "(" (Set (Var "m")) "," (Set (Var "e")) "," (Set (Var "f")) "," (Set (Var "k")) ")" )) "iff" (Bool "(" (Bool (Set (Set (Var "b6")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k8_radix_2 :::"Table2"::: ) "(" (Set (Var "m")) "," (Set (Var "e")) "," (Set (Var "f")) "," (Set (Var "n")) ")" )) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "n")) ($#k5_real_1 :::"-"::: ) (Num 1)))) "holds" (Bool "ex" (Set (Var "i1")) "," (Set (Var "i2")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "i1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "b6")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i")))) & (Bool (Set (Var "i2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "b6")) ($#k1_recdef_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) & (Bool (Set (Var "i2")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set (Var "i1")) ($#k1_newton :::"|^"::: ) (Set "(" ($#k1_radix_1 :::"Radix"::: ) (Set (Var "k")) ")" ) ")" ) ($#k4_nat_d :::"mod"::: ) (Set (Var "f")) ")" ) ($#k4_nat_1 :::"*"::: ) (Set "(" ($#k8_radix_2 :::"Table2"::: ) "(" (Set (Var "m")) "," (Set (Var "e")) "," (Set (Var "f")) "," (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "i")) ")" ) ")" ")" ) ")" ) ($#k4_nat_d :::"mod"::: ) (Set (Var "f")))) ")" )) ")" ) ")" ) ")" )))); theorem :: RADIX_2:17 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool "for" (Set (Var "m")) "," (Set (Var "k")) "," (Set (Var "f")) "," (Set (Var "ie")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "ie")) ($#r1_radix_1 :::"is_represented_by"::: ) (Set (Var "n")) "," (Set (Var "k"))) & (Bool (Set (Var "f")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "for" (Set (Var "e")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "e")) ($#r1_hidden :::"="::: ) (Set ($#k5_radix_2 :::"DecSD2"::: ) "(" (Set (Var "ie")) "," (Set (Var "n")) "," (Set (Var "k")) ")" ))) "holds" (Bool (Set (Set "(" ($#k9_radix_2 :::"Pow_mod"::: ) "(" (Set (Var "m")) "," (Set (Var "e")) "," (Set (Var "f")) "," (Set (Var "k")) ")" ")" ) ($#k1_recdef_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "m")) ($#k1_newton :::"|^"::: ) (Set (Var "ie")) ")" ) ($#k4_nat_d :::"mod"::: ) (Set (Var "f"))))))) ;