:: IDEA_1 semantic presentation begin theorem :: IDEA_1:1 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "j")) "is" ($#v1_int_2 :::"prime"::: ) ) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j"))) & (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Set "(" (Set (Var "a")) ($#k4_nat_1 :::"*"::: ) (Set (Var "i")) ")" ) ($#k4_nat_d :::"mod"::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set (Var "k"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j"))) ")" ))) ; theorem :: IDEA_1:2 (Bool "for" (Set (Var "n")) "," (Set (Var "k1")) "," (Set (Var "k2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "k1")) ($#k4_nat_d :::"mod"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "k2")) ($#k4_nat_d :::"mod"::: ) (Set (Var "n")))) & (Bool (Set (Var "k1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k2")))) "holds" (Bool "ex" (Set (Var "t")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Set (Var "k2")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "k1"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k4_nat_1 :::"*"::: ) (Set (Var "t")))))) ; theorem :: IDEA_1:3 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "a")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "b"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "a")))) ; theorem :: IDEA_1:4 (Bool "for" (Set (Var "b1")) "," (Set (Var "b2")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "b2")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "c")))) "holds" (Bool (Set (Set (Var "b2")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "b1"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "c")))) ; theorem :: IDEA_1:5 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "a"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "c"))) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) "is" ($#v1_int_2 :::"prime"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k4_nat_1 :::"*"::: ) (Set (Var "b")) ")" ) ($#k4_nat_d :::"mod"::: ) (Set (Var "c"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; begin definitionlet "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"ZERO"::: "n" -> ($#m2_finseq_1 :::"Tuple":::) "of" "n" "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) equals :: IDEA_1:def 1 (Set "n" ($#k5_finseq_2 :::"|->"::: ) (Set ($#k7_margrel1 :::"FALSE"::: ) )); end; :: deftheorem defines :::"ZERO"::: IDEA_1:def 1 : (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k1_idea_1 :::"ZERO"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k5_finseq_2 :::"|->"::: ) (Set ($#k7_margrel1 :::"FALSE"::: ) )))); definitionlet "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "x", "y" be ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Const "n")) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ); func "x" :::"'xor'"::: "y" -> ($#m2_finseq_1 :::"Tuple":::) "of" "n" "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) means :: IDEA_1:def 2 (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"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 (Set "(" "x" ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k2_binarith :::"'xor'"::: ) (Set "(" "y" ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" )))); end; :: deftheorem defines :::"'xor'"::: IDEA_1:def 2 : (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "b4")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k2_idea_1 :::"'xor'"::: ) (Set (Var "y")))) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set (Var "b4")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k2_binarith :::"'xor'"::: ) (Set "(" (Set (Var "y")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" )))) ")" ))); theorem :: IDEA_1:6 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set (Set (Var "x")) ($#k2_idea_1 :::"'xor'"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_idea_1 :::"ZERO"::: ) (Set (Var "n")))))) ; theorem :: IDEA_1:7 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set (Set (Var "x")) ($#k2_idea_1 :::"'xor'"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k2_idea_1 :::"'xor'"::: ) (Set (Var "x")))))) ; definitionlet "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "x", "y" be ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Const "n")) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ); :: original: :::"'xor'"::: redefine func "x" :::"'xor'"::: "y" -> ($#m2_finseq_1 :::"Tuple":::) "of" "n" "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ); commutativity (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Const "n")) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set (Set (Var "x")) ($#k2_idea_1 :::"'xor'"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k2_idea_1 :::"'xor'"::: ) (Set (Var "x"))))) ; end; theorem :: IDEA_1:8 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set (Set "(" ($#k1_idea_1 :::"ZERO"::: ) (Set (Var "n")) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "x"))))) ; theorem :: IDEA_1:9 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m2_finseq_1 :::"Tuple":::) "of" (Set (Var "n")) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k3_idea_1 :::"'xor'"::: ) (Set (Var "y")) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" (Set (Var "y")) ($#k3_idea_1 :::"'xor'"::: ) (Set (Var "z")) ")" ))))) ; definitionlet "n", "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); pred "i" :::"is_expressible_by"::: "n" means :: IDEA_1:def 3 (Bool "i" ($#r1_xxreal_0 :::"<"::: ) (Set (Num 2) ($#k5_series_1 :::"to_power"::: ) "n")); end; :: deftheorem defines :::"is_expressible_by"::: IDEA_1:def 3 : (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "i")) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) "iff" (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")))) ")" )); theorem :: IDEA_1:10 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_idea_1 :::"ZERO"::: ) (Set (Var "n"))))) ; definitionlet "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "i", "j" be ($#m1_hidden :::"Nat":::); func :::"ADD_MOD"::: "(" "i" "," "j" "," "n" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: IDEA_1:def 4 (Set (Set "(" "i" ($#k2_xcmplx_0 :::"+"::: ) "j" ")" ) ($#k4_nat_d :::"mod"::: ) (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) "n" ")" )); end; :: deftheorem defines :::"ADD_MOD"::: IDEA_1:def 4 : (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "i")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "j")) ")" ) ($#k4_nat_d :::"mod"::: ) (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")) ")" ))))); definitionlet "n", "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); assume (Bool (Set (Const "i")) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Const "n"))) ; func :::"NEG_N"::: "(" "i" "," "n" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: IDEA_1:def 5 (Set (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) "n" ")" ) ($#k6_xcmplx_0 :::"-"::: ) "i"); end; :: deftheorem defines :::"NEG_N"::: IDEA_1:def 5 : (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k5_idea_1 :::"NEG_N"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "i"))))); definitionlet "n", "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"NEG_MOD"::: "(" "i" "," "n" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: IDEA_1:def 6 (Set (Set "(" ($#k5_idea_1 :::"NEG_N"::: ) "(" "i" "," "n" ")" ")" ) ($#k4_nat_d :::"mod"::: ) (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) "n" ")" )); end; :: deftheorem defines :::"NEG_MOD"::: IDEA_1:def 6 : (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k6_idea_1 :::"NEG_MOD"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_idea_1 :::"NEG_N"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k4_nat_d :::"mod"::: ) (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")) ")" )))); theorem :: IDEA_1:11 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k6_idea_1 :::"NEG_MOD"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: IDEA_1:12 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set (Var "j")) "," (Set (Var "i")) "," (Set (Var "n")) ")" ))) ; theorem :: IDEA_1:13 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set (Var "i")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "i")))) ; theorem :: IDEA_1:14 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set "(" ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "k")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set (Var "j")) "," (Set (Var "k")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "n")) ")" ))) ; theorem :: IDEA_1:15 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n")))) ; theorem :: IDEA_1:16 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k6_idea_1 :::"NEG_MOD"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n")))) ; definitionlet "n", "i" be ($#m1_hidden :::"Nat":::); func :::"ChangeVal_1"::: "(" "i" "," "n" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: IDEA_1:def 7 (Set (Num 2) ($#k5_series_1 :::"to_power"::: ) "n") if (Bool "i" ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) otherwise "i"; end; :: deftheorem defines :::"ChangeVal_1"::: IDEA_1:def 7 : (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set ($#k7_idea_1 :::"ChangeVal_1"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")))) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )))) "implies" (Bool (Set ($#k7_idea_1 :::"ChangeVal_1"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "i"))) ")" ")" )); theorem :: IDEA_1:17 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n")))) "holds" (Bool "(" (Bool (Set ($#k7_idea_1 :::"ChangeVal_1"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ) ($#r1_xxreal_0 :::"<="::: ) (Set (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")))) & (Bool (Set ($#k7_idea_1 :::"ChangeVal_1"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) ; theorem :: IDEA_1:18 (Bool "for" (Set (Var "n")) "," (Set (Var "a1")) "," (Set (Var "a2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Var "a2")) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set ($#k7_idea_1 :::"ChangeVal_1"::: ) "(" (Set (Var "a1")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k7_idea_1 :::"ChangeVal_1"::: ) "(" (Set (Var "a2")) "," (Set (Var "n")) ")" ))) "holds" (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Var "a2")))) ; definitionlet "n", "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"ChangeVal_2"::: "(" "i" "," "n" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: IDEA_1:def 8 (Set ($#k6_numbers :::"0"::: ) ) if (Bool "i" ($#r1_hidden :::"="::: ) (Set (Num 2) ($#k5_series_1 :::"to_power"::: ) "n")) otherwise "i"; end; :: deftheorem defines :::"ChangeVal_2"::: IDEA_1:def 8 : (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n"))))) "implies" (Bool (Set ($#k8_idea_1 :::"ChangeVal_2"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")))))) "implies" (Bool (Set ($#k8_idea_1 :::"ChangeVal_2"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "i"))) ")" ")" )); theorem :: IDEA_1:19 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k8_idea_1 :::"ChangeVal_2"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n")))) ; theorem :: IDEA_1:20 (Bool "for" (Set (Var "n")) "," (Set (Var "a1")) "," (Set (Var "a2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "a1")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "a2")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k8_idea_1 :::"ChangeVal_2"::: ) "(" (Set (Var "a1")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_idea_1 :::"ChangeVal_2"::: ) "(" (Set (Var "a2")) "," (Set (Var "n")) ")" ))) "holds" (Bool (Set (Var "a1")) ($#r1_hidden :::"="::: ) (Set (Var "a2")))) ; definitionlet "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "i", "j" be ($#m1_hidden :::"Nat":::); func :::"MUL_MOD"::: "(" "i" "," "j" "," "n" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: IDEA_1:def 9 (Set ($#k8_idea_1 :::"ChangeVal_2"::: ) "(" (Set "(" (Set "(" (Set "(" ($#k7_idea_1 :::"ChangeVal_1"::: ) "(" "i" "," "n" ")" ")" ) ($#k4_nat_1 :::"*"::: ) (Set "(" ($#k7_idea_1 :::"ChangeVal_1"::: ) "(" "j" "," "n" ")" ")" ) ")" ) ($#k4_nat_d :::"mod"::: ) (Set "(" (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) "n" ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) "," "n" ")" ); end; :: deftheorem defines :::"MUL_MOD"::: IDEA_1:def 9 : (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_idea_1 :::"ChangeVal_2"::: ) "(" (Set "(" (Set "(" (Set "(" ($#k7_idea_1 :::"ChangeVal_1"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k4_nat_1 :::"*"::: ) (Set "(" ($#k7_idea_1 :::"ChangeVal_1"::: ) "(" (Set (Var "j")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k4_nat_d :::"mod"::: ) (Set "(" (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) "," (Set (Var "n")) ")" )))); definitionlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); assume that (Bool (Set (Const "i")) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Const "n"))) and (Bool (Set (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Const "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)) "is" ($#v1_int_2 :::"prime"::: ) ) ; func :::"INV_MOD"::: "(" "i" "," "n" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: IDEA_1:def 10 (Bool "(" (Bool (Set ($#k9_idea_1 :::"MUL_MOD"::: ) "(" "i" "," it "," "n" ")" ) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool it ($#r1_idea_1 :::"is_expressible_by"::: ) "n") ")" ); end; :: deftheorem defines :::"INV_MOD"::: IDEA_1:def 10 : (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)) "is" ($#v1_int_2 :::"prime"::: ) )) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k10_idea_1 :::"INV_MOD"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" )) "iff" (Bool "(" (Bool (Set ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set (Var "i")) "," (Set (Var "b3")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Var "b3")) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) ")" ) ")" )))); theorem :: IDEA_1:21 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set (Var "j")) "," (Set (Var "i")) "," (Set (Var "n")) ")" ))) ; theorem :: IDEA_1:22 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Num 1) "," (Set (Var "i")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "i")))) ; theorem :: IDEA_1:23 (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)) "is" ($#v1_int_2 :::"prime"::: ) ) & (Bool (Set (Var "i")) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Var "j")) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Var "k")) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "k")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set (Var "j")) "," (Set (Var "k")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "n")) ")" ))) ; theorem :: IDEA_1:24 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n")))) ; theorem :: IDEA_1:25 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k8_idea_1 :::"ChangeVal_2"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Num 1))) ; begin definitionlet "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "m", "k" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"IDEAoperationA"::: "(" "m" "," "k" "," "n" ")" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: IDEA_1:def 11 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "m")) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "m"))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Num 1))) "implies" (Bool (Set it ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) "," (Set "(" "k" ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) "," "n" ")" )) ")" & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Num 2))) "implies" (Bool (Set it ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) "," (Set "(" "k" ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) "," "n" ")" )) ")" & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Num 3))) "implies" (Bool (Set it ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) "," (Set "(" "k" ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) "," "n" ")" )) ")" & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Num 4))) "implies" (Bool (Set it ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 4) ")" ) "," (Set "(" "k" ($#k1_recdef_1 :::"."::: ) (Num 4) ")" ) "," "n" ")" )) ")" & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Num 1)) & (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Num 2)) & (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Num 3)) & (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Num 4))) "implies" (Bool (Set it ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set "m" ($#k1_recdef_1 :::"."::: ) (Set (Var "i")))) ")" ")" ) ")" ) ")" ); end; :: deftheorem defines :::"IDEAoperationA"::: IDEA_1:def 11 : (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "m")) "," (Set (Var "k")) "," (Set (Var "b4")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k11_idea_1 :::"IDEAoperationA"::: ) "(" (Set (Var "m")) "," (Set (Var "k")) "," (Set (Var "n")) ")" )) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "m")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "m"))))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Num 1))) "implies" (Bool (Set (Set (Var "b4")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "k")) ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) "," (Set (Var "n")) ")" )) ")" & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Num 2))) "implies" (Bool (Set (Set (Var "b4")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "k")) ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) "," (Set (Var "n")) ")" )) ")" & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Num 3))) "implies" (Bool (Set (Set (Var "b4")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) "," (Set "(" (Set (Var "k")) ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) "," (Set (Var "n")) ")" )) ")" & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Num 4))) "implies" (Bool (Set (Set (Var "b4")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 4) ")" ) "," (Set "(" (Set (Var "k")) ($#k1_recdef_1 :::"."::: ) (Num 4) ")" ) "," (Set (Var "n")) ")" )) ")" & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Num 1)) & (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Num 2)) & (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Num 3)) & (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Num 4))) "implies" (Bool (Set (Set (Var "b4")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i")))) ")" ")" ) ")" ) ")" ) ")" ))); definitionlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "m", "k" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"IDEAoperationB"::: "(" "m" "," "k" "," "n" ")" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: IDEA_1:def 12 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "m")) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "m"))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Num 1))) "implies" (Bool (Set it ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) ")" ) ")" ) ")" ) "," (Set "(" "k" ($#k1_recdef_1 :::"."::: ) (Num 5) ")" ) "," "n" ")" ")" ) "," (Set "(" ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 4) ")" ) ")" ) ")" ) ")" ) "," "n" ")" ")" ) "," (Set "(" "k" ($#k1_recdef_1 :::"."::: ) (Num 6) ")" ) "," "n" ")" ")" ) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Num 2))) "implies" (Bool (Set it ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) ")" ) ")" ) ")" ) "," (Set "(" "k" ($#k1_recdef_1 :::"."::: ) (Num 5) ")" ) "," "n" ")" ")" ) "," (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) ")" ) ")" ) ")" ) "," (Set "(" "k" ($#k1_recdef_1 :::"."::: ) (Num 5) ")" ) "," "n" ")" ")" ) "," (Set "(" ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 4) ")" ) ")" ) ")" ) ")" ) "," "n" ")" ")" ) "," (Set "(" "k" ($#k1_recdef_1 :::"."::: ) (Num 6) ")" ) "," "n" ")" ")" ) "," "n" ")" ")" ) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Num 3))) "implies" (Bool (Set it ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) ")" ) ")" ) ")" ) "," (Set "(" "k" ($#k1_recdef_1 :::"."::: ) (Num 5) ")" ) "," "n" ")" ")" ) "," (Set "(" ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 4) ")" ) ")" ) ")" ) ")" ) "," "n" ")" ")" ) "," (Set "(" "k" ($#k1_recdef_1 :::"."::: ) (Num 6) ")" ) "," "n" ")" ")" ) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Num 4))) "implies" (Bool (Set it ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 4) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) ")" ) ")" ) ")" ) "," (Set "(" "k" ($#k1_recdef_1 :::"."::: ) (Num 5) ")" ) "," "n" ")" ")" ) "," (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) ")" ) ")" ) ")" ) "," (Set "(" "k" ($#k1_recdef_1 :::"."::: ) (Num 5) ")" ) "," "n" ")" ")" ) "," (Set "(" ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" "n" ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 4) ")" ) ")" ) ")" ) ")" ) "," "n" ")" ")" ) "," (Set "(" "k" ($#k1_recdef_1 :::"."::: ) (Num 6) ")" ) "," "n" ")" ")" ) "," "n" ")" ")" ) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Num 1)) & (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Num 2)) & (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Num 3)) & (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Num 4))) "implies" (Bool (Set it ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set "m" ($#k1_recdef_1 :::"."::: ) (Set (Var "i")))) ")" ")" ) ")" ) ")" ); end; :: deftheorem defines :::"IDEAoperationB"::: IDEA_1:def 12 : (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "m")) "," (Set (Var "k")) "," (Set (Var "b4")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k12_idea_1 :::"IDEAoperationB"::: ) "(" (Set (Var "m")) "," (Set (Var "k")) "," (Set (Var "n")) ")" )) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "m")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "m"))))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Num 1))) "implies" (Bool (Set (Set (Var "b4")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) ")" ) ")" ) ")" ) "," (Set "(" (Set (Var "k")) ($#k1_recdef_1 :::"."::: ) (Num 5) ")" ) "," (Set (Var "n")) ")" ")" ) "," (Set "(" ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 4) ")" ) ")" ) ")" ) ")" ) "," (Set (Var "n")) ")" ")" ) "," (Set "(" (Set (Var "k")) ($#k1_recdef_1 :::"."::: ) (Num 6) ")" ) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Num 2))) "implies" (Bool (Set (Set (Var "b4")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) ")" ) ")" ) ")" ) "," (Set "(" (Set (Var "k")) ($#k1_recdef_1 :::"."::: ) (Num 5) ")" ) "," (Set (Var "n")) ")" ")" ) "," (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) ")" ) ")" ) ")" ) "," (Set "(" (Set (Var "k")) ($#k1_recdef_1 :::"."::: ) (Num 5) ")" ) "," (Set (Var "n")) ")" ")" ) "," (Set "(" ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 4) ")" ) ")" ) ")" ) ")" ) "," (Set (Var "n")) ")" ")" ) "," (Set "(" (Set (Var "k")) ($#k1_recdef_1 :::"."::: ) (Num 6) ")" ) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Num 3))) "implies" (Bool (Set (Set (Var "b4")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) ")" ) ")" ) ")" ) "," (Set "(" (Set (Var "k")) ($#k1_recdef_1 :::"."::: ) (Num 5) ")" ) "," (Set (Var "n")) ")" ")" ) "," (Set "(" ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 4) ")" ) ")" ) ")" ) ")" ) "," (Set (Var "n")) ")" ")" ) "," (Set "(" (Set (Var "k")) ($#k1_recdef_1 :::"."::: ) (Num 6) ")" ) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Num 4))) "implies" (Bool (Set (Set (Var "b4")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 4) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) ")" ) ")" ) ")" ) "," (Set "(" (Set (Var "k")) ($#k1_recdef_1 :::"."::: ) (Num 5) ")" ) "," (Set (Var "n")) ")" ")" ) "," (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" ($#k4_idea_1 :::"ADD_MOD"::: ) "(" (Set "(" ($#k9_idea_1 :::"MUL_MOD"::: ) "(" (Set "(" ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) ")" ) ")" ) ")" ) "," (Set "(" (Set (Var "k")) ($#k1_recdef_1 :::"."::: ) (Num 5) ")" ) "," (Set (Var "n")) ")" ")" ) "," (Set "(" ($#k6_binarith :::"Absval"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) ")" ) ($#k3_idea_1 :::"'xor'"::: ) (Set "(" (Set (Var "n")) ($#k1_binari_3 :::"-BinarySequence"::: ) (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 4) ")" ) ")" ) ")" ) ")" ) "," (Set (Var "n")) ")" ")" ) "," (Set "(" (Set (Var "k")) ($#k1_recdef_1 :::"."::: ) (Num 6) ")" ) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Num 1)) & (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Num 2)) & (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Num 3)) & (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Num 4))) "implies" (Bool (Set (Set (Var "b4")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i")))) ")" ")" ) ")" ) ")" ) ")" ))); definitionlet "m" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"IDEAoperationC"::: "m" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: IDEA_1:def 13 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "m")) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "m"))) "holds" (Bool (Set it ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k15_funcop_1 :::"IFEQ"::: ) "(" (Set (Var "i")) "," (Num 2) "," (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) "," (Set "(" ($#k15_funcop_1 :::"IFEQ"::: ) "(" (Set (Var "i")) "," (Num 3) "," (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) "," (Set "(" "m" ($#k1_recdef_1 :::"."::: ) (Set (Var "i")) ")" ) ")" ")" ) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"IDEAoperationC"::: IDEA_1:def 13 : (Bool "for" (Set (Var "m")) "," (Set (Var "b2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k13_idea_1 :::"IDEAoperationC"::: ) (Set (Var "m")))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "m")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "m"))))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k15_funcop_1 :::"IFEQ"::: ) "(" (Set (Var "i")) "," (Num 2) "," (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) "," (Set "(" ($#k15_funcop_1 :::"IFEQ"::: ) "(" (Set (Var "i")) "," (Num 3) "," (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i")) ")" ) ")" ")" ) ")" )) ")" ) ")" ) ")" )); theorem :: IDEA_1:26 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "m")) "," (Set (Var "k")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "m"))) ($#r1_xxreal_0 :::">="::: ) (Num 4))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k11_idea_1 :::"IDEAoperationA"::: ) "(" (Set (Var "m")) "," (Set (Var "k")) "," (Set (Var "n")) ")" ")" ) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set "(" ($#k11_idea_1 :::"IDEAoperationA"::: ) "(" (Set (Var "m")) "," (Set (Var "k")) "," (Set (Var "n")) ")" ")" ) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set "(" ($#k11_idea_1 :::"IDEAoperationA"::: ) "(" (Set (Var "m")) "," (Set (Var "k")) "," (Set (Var "n")) ")" ")" ) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set "(" ($#k11_idea_1 :::"IDEAoperationA"::: ) "(" (Set (Var "m")) "," (Set (Var "k")) "," (Set (Var "n")) ")" ")" ) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) ")" ))) ; theorem :: IDEA_1:27 (Bool "for" (Set (Var "m")) "," (Set (Var "k")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "m"))) ($#r1_xxreal_0 :::">="::: ) (Num 4))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k12_idea_1 :::"IDEAoperationB"::: ) "(" (Set (Var "m")) "," (Set (Var "k")) "," (Set (Var "n")) ")" ")" ) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set "(" ($#k12_idea_1 :::"IDEAoperationB"::: ) "(" (Set (Var "m")) "," (Set (Var "k")) "," (Set (Var "n")) ")" ")" ) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set "(" ($#k12_idea_1 :::"IDEAoperationB"::: ) "(" (Set (Var "m")) "," (Set (Var "k")) "," (Set (Var "n")) ")" ")" ) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set "(" ($#k12_idea_1 :::"IDEAoperationB"::: ) "(" (Set (Var "m")) "," (Set (Var "k")) "," (Set (Var "n")) ")" ")" ) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) ")" ))) ; theorem :: IDEA_1:28 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "m")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "m"))) ($#r1_xxreal_0 :::">="::: ) (Num 4)) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k13_idea_1 :::"IDEAoperationC"::: ) (Set (Var "m")) ")" ) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set "(" ($#k13_idea_1 :::"IDEAoperationC"::: ) (Set (Var "m")) ")" ) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set "(" ($#k13_idea_1 :::"IDEAoperationC"::: ) (Set (Var "m")) ")" ) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set "(" ($#k13_idea_1 :::"IDEAoperationC"::: ) (Set (Var "m")) ")" ) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) ")" ))) ; theorem :: IDEA_1:29 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "m")) "," (Set (Var "k1")) "," (Set (Var "k2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)) "is" ($#v1_int_2 :::"prime"::: ) ) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "m"))) ($#r1_xxreal_0 :::">="::: ) (Num 4)) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k10_idea_1 :::"INV_MOD"::: ) "(" (Set "(" (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set ($#k6_idea_1 :::"NEG_MOD"::: ) "(" (Set "(" (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set ($#k6_idea_1 :::"NEG_MOD"::: ) "(" (Set "(" (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Set ($#k10_idea_1 :::"INV_MOD"::: ) "(" (Set "(" (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 4) ")" ) "," (Set (Var "n")) ")" ))) "holds" (Bool (Set ($#k11_idea_1 :::"IDEAoperationA"::: ) "(" (Set "(" ($#k11_idea_1 :::"IDEAoperationA"::: ) "(" (Set (Var "m")) "," (Set (Var "k1")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "k2")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "m"))))) ; theorem :: IDEA_1:30 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "m")) "," (Set (Var "k1")) "," (Set (Var "k2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)) "is" ($#v1_int_2 :::"prime"::: ) ) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "m"))) ($#r1_xxreal_0 :::">="::: ) (Num 4)) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k10_idea_1 :::"INV_MOD"::: ) "(" (Set "(" (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set ($#k6_idea_1 :::"NEG_MOD"::: ) "(" (Set "(" (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set ($#k6_idea_1 :::"NEG_MOD"::: ) "(" (Set "(" (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Set ($#k10_idea_1 :::"INV_MOD"::: ) "(" (Set "(" (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 4) ")" ) "," (Set (Var "n")) ")" ))) "holds" (Bool (Set ($#k11_idea_1 :::"IDEAoperationA"::: ) "(" (Set "(" ($#k13_idea_1 :::"IDEAoperationC"::: ) (Set "(" ($#k11_idea_1 :::"IDEAoperationA"::: ) "(" (Set "(" ($#k13_idea_1 :::"IDEAoperationC"::: ) (Set (Var "m")) ")" ) "," (Set (Var "k1")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Set (Var "k2")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "m"))))) ; theorem :: IDEA_1:31 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "m")) "," (Set (Var "k1")) "," (Set (Var "k2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "m"))) ($#r1_xxreal_0 :::">="::: ) (Num 4)) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 5)) ($#r1_hidden :::"="::: ) (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 5))) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 6)) ($#r1_hidden :::"="::: ) (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 6)))) "holds" (Bool (Set ($#k12_idea_1 :::"IDEAoperationB"::: ) "(" (Set "(" ($#k12_idea_1 :::"IDEAoperationB"::: ) "(" (Set (Var "m")) "," (Set (Var "k1")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "k2")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "m"))))) ; theorem :: IDEA_1:32 (Bool "for" (Set (Var "m")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "m"))) ($#r1_xxreal_0 :::">="::: ) (Num 4))) "holds" (Bool (Set ($#k13_idea_1 :::"IDEAoperationC"::: ) (Set "(" ($#k13_idea_1 :::"IDEAoperationC"::: ) (Set (Var "m")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "m")))) ; begin definitionfunc :::"MESSAGES"::: -> ($#m1_hidden :::"set"::: ) equals :: IDEA_1:def 14 (Set (Set ($#k5_numbers :::"NAT"::: ) ) ($#k3_finseq_2 :::"*"::: ) ); end; :: deftheorem defines :::"MESSAGES"::: IDEA_1:def 14 : (Bool (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k5_numbers :::"NAT"::: ) ) ($#k3_finseq_2 :::"*"::: ) )); registration cluster (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; registration cluster (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) -> ($#v4_funct_1 :::"functional"::: ) ; end; registration cluster -> ($#v1_finseq_1 :::"FinSequence-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k14_idea_1 :::"MESSAGES"::: ) ); end; definitionlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "k" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"IDEA_P"::: "(" "k" "," "n" ")" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) means :: IDEA_1:def 15 (Bool "for" (Set (Var "m")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k11_idea_1 :::"IDEAoperationA"::: ) "(" (Set "(" ($#k13_idea_1 :::"IDEAoperationC"::: ) (Set "(" ($#k12_idea_1 :::"IDEAoperationB"::: ) "(" (Set (Var "m")) "," "k" "," "n" ")" ")" ) ")" ) "," "k" "," "n" ")" ))); end; :: deftheorem defines :::"IDEA_P"::: IDEA_1:def 15 : (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "k")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k15_idea_1 :::"IDEA_P"::: ) "(" (Set (Var "k")) "," (Set (Var "n")) ")" )) "iff" (Bool "for" (Set (Var "m")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k11_idea_1 :::"IDEAoperationA"::: ) "(" (Set "(" ($#k13_idea_1 :::"IDEAoperationC"::: ) (Set "(" ($#k12_idea_1 :::"IDEAoperationB"::: ) "(" (Set (Var "m")) "," (Set (Var "k")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Set (Var "k")) "," (Set (Var "n")) ")" ))) ")" )))); definitionlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "k" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"IDEA_Q"::: "(" "k" "," "n" ")" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) means :: IDEA_1:def 16 (Bool "for" (Set (Var "m")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k12_idea_1 :::"IDEAoperationB"::: ) "(" (Set "(" ($#k11_idea_1 :::"IDEAoperationA"::: ) "(" (Set "(" ($#k13_idea_1 :::"IDEAoperationC"::: ) (Set (Var "m")) ")" ) "," "k" "," "n" ")" ")" ) "," "k" "," "n" ")" ))); end; :: deftheorem defines :::"IDEA_Q"::: IDEA_1:def 16 : (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "k")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k16_idea_1 :::"IDEA_Q"::: ) "(" (Set (Var "k")) "," (Set (Var "n")) ")" )) "iff" (Bool "for" (Set (Var "m")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k12_idea_1 :::"IDEAoperationB"::: ) "(" (Set "(" ($#k11_idea_1 :::"IDEAoperationA"::: ) "(" (Set "(" ($#k13_idea_1 :::"IDEAoperationC"::: ) (Set (Var "m")) ")" ) "," (Set (Var "k")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "k")) "," (Set (Var "n")) ")" ))) ")" )))); definitionlet "r", "lk" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "Key" be ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Const "lk")) "," (Num 6) "," (Set ($#k5_numbers :::"NAT"::: ) ); func :::"IDEA_P_F"::: "(" "Key" "," "n" "," "r" ")" -> ($#m1_hidden :::"FinSequence":::) means :: IDEA_1:def 17 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) "r") & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k15_idea_1 :::"IDEA_P"::: ) "(" (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" "Key" "," (Set (Var "i")) ")" ")" ) "," "n" ")" )) ")" ) ")" ); end; :: deftheorem defines :::"IDEA_P_F"::: IDEA_1:def 17 : (Bool "for" (Set (Var "r")) "," (Set (Var "lk")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Key")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "lk")) "," (Num 6) "," (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "b5")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set ($#k17_idea_1 :::"IDEA_P_F"::: ) "(" (Set (Var "Key")) "," (Set (Var "n")) "," (Set (Var "r")) ")" )) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b5"))) ($#r1_hidden :::"="::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"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 ($#k15_idea_1 :::"IDEA_P"::: ) "(" (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "Key")) "," (Set (Var "i")) ")" ")" ) "," (Set (Var "n")) ")" )) ")" ) ")" ) ")" ))))); registrationlet "r", "lk" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "Key" be ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Const "lk")) "," (Num 6) "," (Set ($#k5_numbers :::"NAT"::: ) ); cluster (Set ($#k17_idea_1 :::"IDEA_P_F"::: ) "(" "Key" "," "n" "," "r" ")" ) -> ($#v1_funcop_1 :::"Function-yielding"::: ) ; end; definitionlet "r", "lk" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "Key" be ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Const "lk")) "," (Num 6) "," (Set ($#k5_numbers :::"NAT"::: ) ); func :::"IDEA_Q_F"::: "(" "Key" "," "n" "," "r" ")" -> ($#m1_hidden :::"FinSequence":::) means :: IDEA_1:def 18 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) "r") & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k16_idea_1 :::"IDEA_Q"::: ) "(" (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" "Key" "," (Set "(" (Set "(" "r" ($#k7_nat_d :::"-'"::: ) (Set (Var "i")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," "n" ")" )) ")" ) ")" ); end; :: deftheorem defines :::"IDEA_Q_F"::: IDEA_1:def 18 : (Bool "for" (Set (Var "r")) "," (Set (Var "lk")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Key")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "lk")) "," (Num 6) "," (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "b5")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set ($#k18_idea_1 :::"IDEA_Q_F"::: ) "(" (Set (Var "Key")) "," (Set (Var "n")) "," (Set (Var "r")) ")" )) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b5"))) ($#r1_hidden :::"="::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"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 ($#k16_idea_1 :::"IDEA_Q"::: ) "(" (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "Key")) "," (Set "(" (Set "(" (Set (Var "r")) ($#k7_nat_d :::"-'"::: ) (Set (Var "i")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set (Var "n")) ")" )) ")" ) ")" ) ")" ))))); registrationlet "r", "lk" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "Key" be ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Const "lk")) "," (Num 6) "," (Set ($#k5_numbers :::"NAT"::: ) ); cluster (Set ($#k18_idea_1 :::"IDEA_Q_F"::: ) "(" "Key" "," "n" "," "r" ")" ) -> ($#v1_funcop_1 :::"Function-yielding"::: ) ; end; definitionlet "k" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"IDEA_PS"::: "(" "k" "," "n" ")" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) means :: IDEA_1:def 19 (Bool "for" (Set (Var "m")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k11_idea_1 :::"IDEAoperationA"::: ) "(" (Set (Var "m")) "," "k" "," "n" ")" ))); end; :: deftheorem defines :::"IDEA_PS"::: IDEA_1:def 19 : (Bool "for" (Set (Var "k")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k19_idea_1 :::"IDEA_PS"::: ) "(" (Set (Var "k")) "," (Set (Var "n")) ")" )) "iff" (Bool "for" (Set (Var "m")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k11_idea_1 :::"IDEAoperationA"::: ) "(" (Set (Var "m")) "," (Set (Var "k")) "," (Set (Var "n")) ")" ))) ")" )))); definitionlet "k" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"IDEA_QS"::: "(" "k" "," "n" ")" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) means :: IDEA_1:def 20 (Bool "for" (Set (Var "m")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k11_idea_1 :::"IDEAoperationA"::: ) "(" (Set (Var "m")) "," "k" "," "n" ")" ))); end; :: deftheorem defines :::"IDEA_QS"::: IDEA_1:def 20 : (Bool "for" (Set (Var "k")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k20_idea_1 :::"IDEA_QS"::: ) "(" (Set (Var "k")) "," (Set (Var "n")) ")" )) "iff" (Bool "for" (Set (Var "m")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k11_idea_1 :::"IDEAoperationA"::: ) "(" (Set (Var "m")) "," (Set (Var "k")) "," (Set (Var "n")) ")" ))) ")" )))); definitionlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "k" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"IDEA_PE"::: "(" "k" "," "n" ")" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) means :: IDEA_1:def 21 (Bool "for" (Set (Var "m")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k11_idea_1 :::"IDEAoperationA"::: ) "(" (Set "(" ($#k12_idea_1 :::"IDEAoperationB"::: ) "(" (Set (Var "m")) "," "k" "," "n" ")" ")" ) "," "k" "," "n" ")" ))); end; :: deftheorem defines :::"IDEA_PE"::: IDEA_1:def 21 : (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "k")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k21_idea_1 :::"IDEA_PE"::: ) "(" (Set (Var "k")) "," (Set (Var "n")) ")" )) "iff" (Bool "for" (Set (Var "m")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k11_idea_1 :::"IDEAoperationA"::: ) "(" (Set "(" ($#k12_idea_1 :::"IDEAoperationB"::: ) "(" (Set (Var "m")) "," (Set (Var "k")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "k")) "," (Set (Var "n")) ")" ))) ")" )))); definitionlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "k" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"IDEA_QE"::: "(" "k" "," "n" ")" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) means :: IDEA_1:def 22 (Bool "for" (Set (Var "m")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k12_idea_1 :::"IDEAoperationB"::: ) "(" (Set "(" ($#k11_idea_1 :::"IDEAoperationA"::: ) "(" (Set (Var "m")) "," "k" "," "n" ")" ")" ) "," "k" "," "n" ")" ))); end; :: deftheorem defines :::"IDEA_QE"::: IDEA_1:def 22 : (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "k")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k22_idea_1 :::"IDEA_QE"::: ) "(" (Set (Var "k")) "," (Set (Var "n")) ")" )) "iff" (Bool "for" (Set (Var "m")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k12_idea_1 :::"IDEAoperationB"::: ) "(" (Set "(" ($#k11_idea_1 :::"IDEAoperationA"::: ) "(" (Set (Var "m")) "," (Set (Var "k")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "k")) "," (Set (Var "n")) ")" ))) ")" )))); theorem :: IDEA_1:33 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "m")) "," (Set (Var "k1")) "," (Set (Var "k2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)) "is" ($#v1_int_2 :::"prime"::: ) ) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "m"))) ($#r1_xxreal_0 :::">="::: ) (Num 4)) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k10_idea_1 :::"INV_MOD"::: ) "(" (Set "(" (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set ($#k6_idea_1 :::"NEG_MOD"::: ) "(" (Set "(" (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set ($#k6_idea_1 :::"NEG_MOD"::: ) "(" (Set "(" (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Set ($#k10_idea_1 :::"INV_MOD"::: ) "(" (Set "(" (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 4) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 5)) ($#r1_hidden :::"="::: ) (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 5))) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 6)) ($#r1_hidden :::"="::: ) (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 6)))) "holds" (Bool (Set (Set "(" (Set "(" ($#k16_idea_1 :::"IDEA_Q"::: ) "(" (Set (Var "k2")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k15_idea_1 :::"IDEA_P"::: ) "(" (Set (Var "k1")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set (Var "m"))))) ; theorem :: IDEA_1:34 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "m")) "," (Set (Var "k1")) "," (Set (Var "k2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)) "is" ($#v1_int_2 :::"prime"::: ) ) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "m"))) ($#r1_xxreal_0 :::">="::: ) (Num 4)) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k10_idea_1 :::"INV_MOD"::: ) "(" (Set "(" (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set ($#k6_idea_1 :::"NEG_MOD"::: ) "(" (Set "(" (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set ($#k6_idea_1 :::"NEG_MOD"::: ) "(" (Set "(" (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Set ($#k10_idea_1 :::"INV_MOD"::: ) "(" (Set "(" (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 4) ")" ) "," (Set (Var "n")) ")" ))) "holds" (Bool (Set (Set "(" (Set "(" ($#k20_idea_1 :::"IDEA_QS"::: ) "(" (Set (Var "k2")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k19_idea_1 :::"IDEA_PS"::: ) "(" (Set (Var "k1")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set (Var "m"))))) ; theorem :: IDEA_1:35 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "m")) "," (Set (Var "k1")) "," (Set (Var "k2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)) "is" ($#v1_int_2 :::"prime"::: ) ) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "m"))) ($#r1_xxreal_0 :::">="::: ) (Num 4)) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k10_idea_1 :::"INV_MOD"::: ) "(" (Set "(" (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set ($#k6_idea_1 :::"NEG_MOD"::: ) "(" (Set "(" (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set ($#k6_idea_1 :::"NEG_MOD"::: ) "(" (Set "(" (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Set ($#k10_idea_1 :::"INV_MOD"::: ) "(" (Set "(" (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 4) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 5)) ($#r1_hidden :::"="::: ) (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 5))) & (Bool (Set (Set (Var "k2")) ($#k1_recdef_1 :::"."::: ) (Num 6)) ($#r1_hidden :::"="::: ) (Set (Set (Var "k1")) ($#k1_recdef_1 :::"."::: ) (Num 6)))) "holds" (Bool (Set (Set "(" (Set "(" ($#k22_idea_1 :::"IDEA_QE"::: ) "(" (Set (Var "k2")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k21_idea_1 :::"IDEA_PE"::: ) "(" (Set (Var "k1")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set (Var "m"))))) ; theorem :: IDEA_1:36 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "lk")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Key")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "lk")) "," (Num 6) "," (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k17_idea_1 :::"IDEA_P_F"::: ) "(" (Set (Var "Key")) "," (Set (Var "n")) "," (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k17_idea_1 :::"IDEA_P_F"::: ) "(" (Set (Var "Key")) "," (Set (Var "n")) "," (Set (Var "k")) ")" ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" ($#k15_idea_1 :::"IDEA_P"::: ) "(" (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "Key")) "," (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set (Var "n")) ")" ")" ) ($#k9_finseq_1 :::"*>"::: ) ))))))) ; theorem :: IDEA_1:37 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "lk")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Key")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "lk")) "," (Num 6) "," (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k18_idea_1 :::"IDEA_Q_F"::: ) "(" (Set (Var "Key")) "," (Set (Var "n")) "," (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" ($#k16_idea_1 :::"IDEA_Q"::: ) "(" (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "Key")) "," (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set (Var "n")) ")" ")" ) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set "(" ($#k18_idea_1 :::"IDEA_Q_F"::: ) "(" (Set (Var "Key")) "," (Set (Var "n")) "," (Set (Var "k")) ")" ")" ))))))) ; theorem :: IDEA_1:38 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "lk")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Key")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "lk")) "," (Num 6) "," (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k17_idea_1 :::"IDEA_P_F"::: ) "(" (Set (Var "Key")) "," (Set (Var "n")) "," (Set (Var "k")) ")" ) "is" ($#v1_funct_7 :::"FuncSeq-like"::: ) ($#m1_hidden :::"FinSequence":::)))))) ; theorem :: IDEA_1:39 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "lk")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Key")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "lk")) "," (Num 6) "," (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k18_idea_1 :::"IDEA_Q_F"::: ) "(" (Set (Var "Key")) "," (Set (Var "n")) "," (Set (Var "k")) ")" ) "is" ($#v1_funct_7 :::"FuncSeq-like"::: ) ($#m1_hidden :::"FinSequence":::)))))) ; theorem :: IDEA_1:40 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "lk")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Key")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "lk")) "," (Num 6) "," (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k17_idea_1 :::"IDEA_P_F"::: ) "(" (Set (Var "Key")) "," (Set (Var "n")) "," (Set (Var "k")) ")" ) "is" ($#m1_funct_7 :::"FuncSequence"::: ) "of" (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) )))))) ; theorem :: IDEA_1:41 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "lk")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Key")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "lk")) "," (Num 6) "," (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k18_idea_1 :::"IDEA_Q_F"::: ) "(" (Set (Var "Key")) "," (Set (Var "n")) "," (Set (Var "k")) ")" ) "is" ($#m1_funct_7 :::"FuncSequence"::: ) "of" (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) )))))) ; theorem :: IDEA_1:42 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "M")) "," (Set (Var "m")) "," (Set (Var "k")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k15_idea_1 :::"IDEA_P"::: ) "(" (Set (Var "k")) "," (Set (Var "n")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "m")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "m"))) ($#r1_xxreal_0 :::">="::: ) (Num 4))) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M"))) ($#r1_xxreal_0 :::">="::: ) (Num 4)) & (Bool (Set (Set (Var "M")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "M")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "M")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "M")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) ")" ))) ; theorem :: IDEA_1:43 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "lk")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Key")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "lk")) "," (Num 6) "," (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k4_funct_7 :::"compose"::: ) "(" (Set "(" ($#k17_idea_1 :::"IDEA_P_F"::: ) "(" (Set (Var "Key")) "," (Set (Var "n")) "," (Set (Var "r")) ")" ")" ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k14_idea_1 :::"MESSAGES"::: ) )) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" ($#k4_funct_7 :::"compose"::: ) "(" (Set "(" ($#k17_idea_1 :::"IDEA_P_F"::: ) "(" (Set (Var "Key")) "," (Set (Var "n")) "," (Set (Var "r")) ")" ")" ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k14_idea_1 :::"MESSAGES"::: ) )) ")" ))))) ; theorem :: IDEA_1:44 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "lk")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Key")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "lk")) "," (Num 6) "," (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k4_funct_7 :::"compose"::: ) "(" (Set "(" ($#k18_idea_1 :::"IDEA_Q_F"::: ) "(" (Set (Var "Key")) "," (Set (Var "n")) "," (Set (Var "r")) ")" ")" ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k14_idea_1 :::"MESSAGES"::: ) )) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" ($#k4_funct_7 :::"compose"::: ) "(" (Set "(" ($#k18_idea_1 :::"IDEA_Q_F"::: ) "(" (Set (Var "Key")) "," (Set (Var "n")) "," (Set (Var "r")) ")" ")" ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k14_idea_1 :::"MESSAGES"::: ) )) ")" ))))) ; theorem :: IDEA_1:45 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "lk")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Key")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "lk")) "," (Num 6) "," (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "M")) "," (Set (Var "m")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_funct_7 :::"compose"::: ) "(" (Set "(" ($#k17_idea_1 :::"IDEA_P_F"::: ) "(" (Set (Var "Key")) "," (Set (Var "n")) "," (Set (Var "r")) ")" ")" ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "m")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "m"))) ($#r1_xxreal_0 :::">="::: ) (Num 4)) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n")))) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M"))) ($#r1_xxreal_0 :::">="::: ) (Num 4)) & (Bool (Set (Set (Var "M")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "M")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "M")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "M")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) ")" )))))) ; begin theorem :: IDEA_1:46 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "lk")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Key1")) "," (Set (Var "Key2")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "lk")) "," (Num 6) "," (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "m")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "lk")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "r"))) & (Bool (Set (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)) "is" ($#v1_int_2 :::"prime"::: ) ) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "m"))) ($#r1_xxreal_0 :::">="::: ) (Num 4)) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r")))) "holds" (Bool "(" (Bool (Set (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 1) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 2) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 3) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 4) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 5) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 6) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 1) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 2) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 3) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 4) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 5) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 6) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 1) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k10_idea_1 :::"INV_MOD"::: ) "(" (Set "(" (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 1) ")" ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 2) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_idea_1 :::"NEG_MOD"::: ) "(" (Set "(" (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 3) ")" ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 3) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_idea_1 :::"NEG_MOD"::: ) "(" (Set "(" (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 2) ")" ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 4) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k10_idea_1 :::"INV_MOD"::: ) "(" (Set "(" (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 4) ")" ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 5) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 5) ")" )) & (Bool (Set (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 6) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 6) ")" )) ")" ) ")" )) "holds" (Bool (Set (Set "(" ($#k4_funct_7 :::"compose"::: ) "(" (Set "(" (Set "(" ($#k17_idea_1 :::"IDEA_P_F"::: ) "(" (Set (Var "Key1")) "," (Set (Var "n")) "," (Set (Var "r")) ")" ")" ) ($#k7_finseq_1 :::"^"::: ) (Set "(" ($#k18_idea_1 :::"IDEA_Q_F"::: ) "(" (Set (Var "Key2")) "," (Set (Var "n")) "," (Set (Var "r")) ")" ")" ) ")" ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set (Var "m")))))))) ; theorem :: IDEA_1:47 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "lk")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Key1")) "," (Set (Var "Key2")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "lk")) "," (Num 6) "," (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "ks1")) "," (Set (Var "ks2")) "," (Set (Var "ke1")) "," (Set (Var "ke2")) "," (Set (Var "m")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "lk")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "r"))) & (Bool (Set (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)) "is" ($#v1_int_2 :::"prime"::: ) ) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "m"))) ($#r1_xxreal_0 :::">="::: ) (Num 4)) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "m")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r")))) "holds" (Bool "(" (Bool (Set (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 1) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 2) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 3) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 4) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 5) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 6) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 1) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 2) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 3) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 4) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 5) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 6) ")" ) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 1) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k10_idea_1 :::"INV_MOD"::: ) "(" (Set "(" (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 1) ")" ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 2) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_idea_1 :::"NEG_MOD"::: ) "(" (Set "(" (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 3) ")" ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 3) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_idea_1 :::"NEG_MOD"::: ) "(" (Set "(" (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 2) ")" ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 4) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k10_idea_1 :::"INV_MOD"::: ) "(" (Set "(" (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 4) ")" ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 5) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 5) ")" )) & (Bool (Set (Set (Var "Key1")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 6) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "Key2")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 6) ")" )) ")" ) ")" ) & (Bool (Set (Set (Var "ks1")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "ks1")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "ks1")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "ks1")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "ks2")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k10_idea_1 :::"INV_MOD"::: ) "(" (Set "(" (Set (Var "ks1")) ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "ks2")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set ($#k6_idea_1 :::"NEG_MOD"::: ) "(" (Set "(" (Set (Var "ks1")) ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "ks2")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set ($#k6_idea_1 :::"NEG_MOD"::: ) "(" (Set "(" (Set (Var "ks1")) ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "ks2")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Set ($#k10_idea_1 :::"INV_MOD"::: ) "(" (Set "(" (Set (Var "ks1")) ($#k1_recdef_1 :::"."::: ) (Num 4) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "ke1")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "ke1")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "ke1")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "ke1")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_idea_1 :::"is_expressible_by"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "ke2")) ($#k1_recdef_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k10_idea_1 :::"INV_MOD"::: ) "(" (Set "(" (Set (Var "ke1")) ($#k1_recdef_1 :::"."::: ) (Num 1) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "ke2")) ($#k1_recdef_1 :::"."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set ($#k6_idea_1 :::"NEG_MOD"::: ) "(" (Set "(" (Set (Var "ke1")) ($#k1_recdef_1 :::"."::: ) (Num 2) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "ke2")) ($#k1_recdef_1 :::"."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set ($#k6_idea_1 :::"NEG_MOD"::: ) "(" (Set "(" (Set (Var "ke1")) ($#k1_recdef_1 :::"."::: ) (Num 3) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "ke2")) ($#k1_recdef_1 :::"."::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Set ($#k10_idea_1 :::"INV_MOD"::: ) "(" (Set "(" (Set (Var "ke1")) ($#k1_recdef_1 :::"."::: ) (Num 4) ")" ) "," (Set (Var "n")) ")" )) & (Bool (Set (Set (Var "ke2")) ($#k1_recdef_1 :::"."::: ) (Num 5)) ($#r1_hidden :::"="::: ) (Set (Set (Var "ke1")) ($#k1_recdef_1 :::"."::: ) (Num 5))) & (Bool (Set (Set (Var "ke2")) ($#k1_recdef_1 :::"."::: ) (Num 6)) ($#r1_hidden :::"="::: ) (Set (Set (Var "ke1")) ($#k1_recdef_1 :::"."::: ) (Num 6)))) "holds" (Bool (Set (Set "(" (Set "(" ($#k20_idea_1 :::"IDEA_QS"::: ) "(" (Set (Var "ks2")) "," (Set (Var "n")) ")" ")" ) ($#k3_relat_1 :::"*"::: ) (Set "(" (Set "(" ($#k4_funct_7 :::"compose"::: ) "(" (Set "(" ($#k18_idea_1 :::"IDEA_Q_F"::: ) "(" (Set (Var "Key2")) "," (Set (Var "n")) "," (Set (Var "r")) ")" ")" ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) ")" ")" ) ($#k3_relat_1 :::"*"::: ) (Set "(" (Set "(" ($#k22_idea_1 :::"IDEA_QE"::: ) "(" (Set (Var "ke2")) "," (Set (Var "n")) ")" ")" ) ($#k3_relat_1 :::"*"::: ) (Set "(" (Set "(" ($#k21_idea_1 :::"IDEA_PE"::: ) "(" (Set (Var "ke1")) "," (Set (Var "n")) ")" ")" ) ($#k3_relat_1 :::"*"::: ) (Set "(" (Set "(" ($#k4_funct_7 :::"compose"::: ) "(" (Set "(" ($#k17_idea_1 :::"IDEA_P_F"::: ) "(" (Set (Var "Key1")) "," (Set (Var "n")) "," (Set (Var "r")) ")" ")" ) "," (Set ($#k14_idea_1 :::"MESSAGES"::: ) ) ")" ")" ) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k19_idea_1 :::"IDEA_PS"::: ) "(" (Set (Var "ks1")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ) ")" ) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set (Var "m")))))))) ;