:: FIB_NUM2 semantic presentation begin theorem :: FIB_NUM2:1 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1)))) ; theorem :: FIB_NUM2:2 (Bool "for" (Set (Var "n")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k3_power :::"to_power"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)))) ; theorem :: FIB_NUM2:3 (Bool "for" (Set (Var "n")) "being" ($#v1_abian :::"even"::: ) ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k3_power :::"to_power"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Num 1))) ; theorem :: FIB_NUM2:4 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "m")) ")" ) ($#k3_power :::"to_power"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k3_power :::"to_power"::: ) (Set (Var "n")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k3_power :::"to_power"::: ) (Set (Var "n")) ")" ))))) ; theorem :: FIB_NUM2:5 (Bool "for" (Set (Var "k")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "a")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set (Var "a")) ($#k3_power :::"to_power"::: ) (Set "(" (Set (Var "k")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "m")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k3_power :::"to_power"::: ) (Set (Var "k")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set (Var "a")) ($#k3_power :::"to_power"::: ) (Set (Var "m")) ")" ))))) ; theorem :: FIB_NUM2:6 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "k")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "m")) "being" ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Integer":::) "holds" (Bool (Set (Set "(" (Set (Var "k")) ($#k3_power :::"to_power"::: ) (Set (Var "m")) ")" ) ($#k3_power :::"to_power"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "k")) ($#k3_power :::"to_power"::: ) (Set "(" (Set (Var "m")) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "n")) ")" )))))) ; theorem :: FIB_NUM2:7 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k3_power :::"to_power"::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set (Var "n")) ")" ) ")" ) ($#k3_square_1 :::"^2"::: ) ) ($#r1_hidden :::"="::: ) (Num 1))) ; theorem :: FIB_NUM2:8 (Bool "for" (Set (Var "k")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "a")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k3_power :::"to_power"::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set (Var "k")) ")" ) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set (Var "a")) ($#k3_power :::"to_power"::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set (Var "m")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k3_power :::"to_power"::: ) (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set (Var "k")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "m")) ")" ))))) ; theorem :: FIB_NUM2:9 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k3_power :::"to_power"::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Num 1))) ; theorem :: FIB_NUM2:10 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "a")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k3_power :::"to_power"::: ) (Set (Var "k")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set (Var "a")) ($#k3_power :::"to_power"::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set (Var "k")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Num 1)))) ; registrationlet "n" be ($#v1_abian :::"odd"::: ) ($#m1_hidden :::"Integer":::); cluster (Set ($#k4_xcmplx_0 :::"-"::: ) "n") -> ($#v1_abian :::"odd"::: ) ; end; registrationlet "n" be ($#v1_abian :::"even"::: ) ($#m1_hidden :::"Integer":::); cluster (Set ($#k4_xcmplx_0 :::"-"::: ) "n") -> ($#v1_abian :::"even"::: ) ; end; theorem :: FIB_NUM2:11 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k3_power :::"to_power"::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k3_power :::"to_power"::: ) (Set (Var "n"))))) ; theorem :: FIB_NUM2:12 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "k")) "," (Set (Var "m")) "," (Set (Var "m1")) "," (Set (Var "n1")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r1_nat_d :::"divides"::: ) (Set (Var "m"))) & (Bool (Set (Var "k")) ($#r1_nat_d :::"divides"::: ) (Set (Var "n")))) "holds" (Bool (Set (Var "k")) ($#r1_nat_d :::"divides"::: ) (Set (Set "(" (Set (Var "m")) ($#k4_nat_1 :::"*"::: ) (Set (Var "m1")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" (Set (Var "n")) ($#k3_nat_1 :::"*"::: ) (Set (Var "n1")) ")" ))))) ; registration cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ($#v6_membered :::"natural-membered"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "f" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k5_numbers :::"NAT"::: ) ); let "A" be ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ($#v6_membered :::"natural-membered"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set "f" ($#k5_relat_1 :::"|"::: ) "A") -> ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ; end; theorem :: FIB_NUM2:13 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSubsequence":::) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k15_finseq_1 :::"Seq"::: ) (Set (Var "p")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))))) ; definitionlet "f" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k5_numbers :::"NAT"::: ) ); let "A" be ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ($#v6_membered :::"natural-membered"::: ) ($#m1_hidden :::"set"::: ) ; func :::"Prefix"::: "(" "f" "," "A" ")" -> ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: FIB_NUM2:def 1 (Set ($#k15_finseq_1 :::"Seq"::: ) (Set "(" "f" ($#k5_relat_1 :::"|"::: ) "A" ")" )); end; :: deftheorem defines :::"Prefix"::: FIB_NUM2:def 1 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ($#v6_membered :::"natural-membered"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_fib_num2 :::"Prefix"::: ) "(" (Set (Var "f")) "," (Set (Var "A")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k15_finseq_1 :::"Seq"::: ) (Set "(" (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set (Var "A")) ")" ))))); theorem :: FIB_NUM2:14 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Set (Var "m"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))))) ; registration cluster (Set ($#k4_ordinal1 :::"omega"::: ) ) -> ($#v3_xxreal_2 :::"bounded_below"::: ) ; end; theorem :: FIB_NUM2:15 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j")))) "holds" (Bool (Set ($#k2_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "x")) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set (Var "j")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#k2_tarski :::"}"::: ) ) "is" ($#m1_hidden :::"FinSubsequence":::)))) ; theorem :: FIB_NUM2:16 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "q")) "being" ($#m1_hidden :::"FinSubsequence":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j"))) & (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "x")) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set (Var "j")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#k2_tarski :::"}"::: ) ))) "holds" (Bool (Set ($#k15_finseq_1 :::"Seq"::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ))))) ; registrationlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster (Set ($#k1_finseq_1 :::"Seg"::: ) "n") -> ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ; end; registrationlet "A" be ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ($#m1_hidden :::"set"::: ) ; cluster -> ($#v1_setfam_1 :::"with_non-empty_elements"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "A"); end; registrationlet "A" be ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ($#m1_hidden :::"set"::: ) ; let "B" be ($#m1_hidden :::"set"::: ) ; cluster (Set "A" ($#k3_xboole_0 :::"/\"::: ) "B") -> ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ; cluster (Set "B" ($#k3_xboole_0 :::"/\"::: ) "A") -> ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ; end; theorem :: FIB_NUM2:17 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "k")) "," (Set (Var "a")) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) "is" ($#m1_hidden :::"FinSubsequence":::)))) ; theorem :: FIB_NUM2:18 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSubsequence":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Num 1) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set (Set (Var "i")) ($#k12_pnproc_1 :::"Shift"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" (Num 1) ($#k2_nat_1 :::"+"::: ) (Set (Var "i")) ")" ) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ))))) ; theorem :: FIB_NUM2:19 (Bool "for" (Set (Var "q")) "being" ($#m1_hidden :::"FinSubsequence":::) (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "q"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "k")))) "holds" (Bool "ex" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool "(" (Bool (Set (Var "q")) ($#r1_tarski :::"c="::: ) (Set (Var "p"))) & (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) ")" )))) ; theorem :: FIB_NUM2:20 (Bool "for" (Set (Var "q")) "being" ($#m1_hidden :::"FinSubsequence":::) (Bool "ex" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Set (Var "q")) ($#r1_tarski :::"c="::: ) (Set (Var "p"))))) ; begin scheme :: FIB_NUM2:sch 1 FibInd1{ P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool "for" (Set (Var "k")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"Nat":::) "holds" (Bool P1[(Set (Var "k"))])) provided (Bool P1[(Num 1)]) and (Bool P1[(Num 2)]) and (Bool "for" (Set (Var "k")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool P1[(Set (Var "k"))]) & (Bool P1[(Set (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1))])) "holds" (Bool P1[(Set (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 2))])) proof end; scheme :: FIB_NUM2:sch 2 FibInd2{ P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool "for" (Set (Var "k")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m1_hidden :::"Nat":::) "holds" (Bool P1[(Set (Var "k"))])) provided (Bool P1[(Num 2)]) and (Bool P1[(Num 3)]) and (Bool "for" (Set (Var "k")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool P1[(Set (Var "k"))]) & (Bool P1[(Set (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1))])) "holds" (Bool P1[(Set (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 2))])) proof end; theorem :: FIB_NUM2:21 (Bool (Set ($#k1_pre_ff :::"Fib"::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Num 1)) ; theorem :: FIB_NUM2:22 (Bool (Set ($#k1_pre_ff :::"Fib"::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Num 2)) ; theorem :: FIB_NUM2:23 (Bool (Set ($#k1_pre_ff :::"Fib"::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Num 3)) ; theorem :: FIB_NUM2:24 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )))) ; theorem :: FIB_NUM2:25 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 3) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )))) ; theorem :: FIB_NUM2:26 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 4) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 3) ")" ) ")" )))) ; theorem :: FIB_NUM2:27 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 5) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 3) ")" ) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 4) ")" ) ")" )))) ; theorem :: FIB_NUM2:28 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 3) ")" ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )))) ; theorem :: FIB_NUM2:29 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set (Var "n")) ")" )))) ; theorem :: FIB_NUM2:30 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k1_pre_ff :::"Fib"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )))) ; begin theorem :: FIB_NUM2:31 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set (Var "n")) ")" ) ($#k4_nat_1 :::"*"::: ) (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_pepin :::"^2"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k1_newton :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )))) ; theorem :: FIB_NUM2:32 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ) ($#k4_nat_1 :::"*"::: ) (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set (Var "n")) ")" ) ($#k1_pepin :::"^2"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k1_newton :::"|^"::: ) (Set (Var "n"))))) ; theorem :: FIB_NUM2:33 (Bool (Set ($#k1_fib_num :::"tau"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ; theorem :: FIB_NUM2:34 (Bool (Set ($#k2_fib_num :::"tau_bar"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set ($#k1_fib_num :::"tau"::: ) ) ")" ) ($#k3_power :::"to_power"::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ))) ; theorem :: FIB_NUM2:35 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set ($#k1_fib_num :::"tau"::: ) ) ")" ) ($#k3_power :::"to_power"::: ) (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set ($#k1_fib_num :::"tau"::: ) ) ")" ) ($#k3_power :::"to_power"::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ")" ) ($#k3_power :::"to_power"::: ) (Set (Var "n"))))) ; theorem :: FIB_NUM2:36 (Bool (Set ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Num 1) ($#k7_xcmplx_0 :::"/"::: ) (Set ($#k1_fib_num :::"tau"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_fib_num :::"tau_bar"::: ) )) ; theorem :: FIB_NUM2:37 (Bool "for" (Set (Var "r")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set ($#k1_fib_num :::"tau"::: ) ) ($#k3_power :::"to_power"::: ) (Set (Var "r")) ")" ) ($#k3_square_1 :::"^2"::: ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" (Num 2) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k3_power :::"to_power"::: ) (Set (Var "r")) ")" ) ")" ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set "(" (Set ($#k1_fib_num :::"tau"::: ) ) ($#k3_power :::"to_power"::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Set (Var "r")) ")" ) ")" ) ($#k3_square_1 :::"^2"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k1_fib_num :::"tau"::: ) ) ($#k3_power :::"to_power"::: ) (Set (Var "r")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" (Set ($#k2_fib_num :::"tau_bar"::: ) ) ($#k3_power :::"to_power"::: ) (Set (Var "r")) ")" ) ")" ) ($#k3_square_1 :::"^2"::: ) ))) ; theorem :: FIB_NUM2:38 (Bool "for" (Set (Var "n")) "," (Set (Var "r")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set (Var "n")) ")" ) ($#k1_pepin :::"^2"::: ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "r")) ")" ) ")" ) ($#k4_nat_1 :::"*"::: ) (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "r")) ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k1_newton :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "r")) ")" ) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set (Var "r")) ")" ) ($#k1_pepin :::"^2"::: ) ")" )))) ; theorem :: FIB_NUM2:39 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set (Var "n")) ")" ) ($#k1_pepin :::"^2"::: ) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_pepin :::"^2"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )))) ; theorem :: FIB_NUM2:40 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "k")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set (Var "k")) ")" ) ($#k4_nat_1 :::"*"::: ) (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "k")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ) ($#k4_nat_1 :::"*"::: ) (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set (Var "n")) ")" ) ")" ))))) ; theorem :: FIB_NUM2:41 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k1_pre_ff :::"Fib"::: ) (Set (Var "n"))) ($#r1_nat_d :::"divides"::: ) (Set ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k4_nat_1 :::"*"::: ) (Set (Var "k")) ")" ))))) ; theorem :: FIB_NUM2:42 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "k")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r1_nat_d :::"divides"::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k1_pre_ff :::"Fib"::: ) (Set (Var "k"))) ($#r1_nat_d :::"divides"::: ) (Set ($#k1_pre_ff :::"Fib"::: ) (Set (Var "n")))))) ; theorem :: FIB_NUM2:43 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k1_pre_ff :::"Fib"::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )))) ; theorem :: FIB_NUM2:44 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Num 1))) "holds" (Bool (Set ($#k1_pre_ff :::"Fib"::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )))) ; theorem :: FIB_NUM2:45 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k1_pre_ff :::"Fib"::: ) (Set (Var "m"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k1_pre_ff :::"Fib"::: ) (Set (Var "n"))))) ; theorem :: FIB_NUM2:46 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::">"::: ) (Num 1)) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k1_pre_ff :::"Fib"::: ) (Set (Var "k"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_pre_ff :::"Fib"::: ) (Set (Var "n"))))) ; theorem :: FIB_NUM2:47 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set ($#k1_pre_ff :::"Fib"::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Num 1)) "iff" (Bool "(" (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Num 1)) "or" (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Num 2)) ")" ) ")" )) ; theorem :: FIB_NUM2:48 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Num 1)) & (Bool (Set (Var "k")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "k")) ($#r1_hidden :::"<>"::: ) (Num 1))) "holds" (Bool "(" (Bool (Set ($#k1_pre_ff :::"Fib"::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k1_pre_ff :::"Fib"::: ) (Set (Var "n")))) "iff" (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set (Var "n"))) ")" )) ; theorem :: FIB_NUM2:49 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Num 1)) & (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Num 4)) & (Bool (Bool "not" (Set (Var "n")) "is" ($#v1_int_2 :::"prime"::: ) ))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "k")) ($#r1_hidden :::"<>"::: ) (Num 1)) & (Bool (Set (Var "k")) ($#r1_hidden :::"<>"::: ) (Num 2)) & (Bool (Set (Var "k")) ($#r1_hidden :::"<>"::: ) (Set (Var "n"))) & (Bool (Set (Var "k")) ($#r1_nat_d :::"divides"::: ) (Set (Var "n"))) ")" ))) ; theorem :: FIB_NUM2:50 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Num 1)) & (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Num 4)) & (Bool (Set ($#k1_pre_ff :::"Fib"::: ) (Set (Var "n"))) "is" ($#v1_int_2 :::"prime"::: ) )) "holds" (Bool (Set (Var "n")) "is" ($#v1_int_2 :::"prime"::: ) )) ; begin definitionfunc :::"FIB"::: -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k5_numbers :::"NAT"::: ) ) means :: FIB_NUM2:def 2 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k1_pre_ff :::"Fib"::: ) (Set (Var "k"))))); end; :: deftheorem defines :::"FIB"::: FIB_NUM2:def 2 : (Bool "for" (Set (Var "b1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k2_fib_num2 :::"FIB"::: ) )) "iff" (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b1")) ($#k3_funct_2 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k1_pre_ff :::"Fib"::: ) (Set (Var "k"))))) ")" )); definitionfunc :::"EvenNAT"::: -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: FIB_NUM2:def 3 "{" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k")) ")" ) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool verum) "}" ; func :::"OddNAT"::: -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: FIB_NUM2:def 4 "{" (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool verum) "}" ; end; :: deftheorem defines :::"EvenNAT"::: FIB_NUM2:def 3 : (Bool (Set ($#k3_fib_num2 :::"EvenNAT"::: ) ) ($#r1_hidden :::"="::: ) "{" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k")) ")" ) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool verum) "}" ); :: deftheorem defines :::"OddNAT"::: FIB_NUM2:def 4 : (Bool (Set ($#k4_fib_num2 :::"OddNAT"::: ) ) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool verum) "}" ); theorem :: FIB_NUM2:51 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k"))) ($#r2_hidden :::"in"::: ) (Set ($#k3_fib_num2 :::"EvenNAT"::: ) )) & (Bool (Bool "not" (Set (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k3_fib_num2 :::"EvenNAT"::: ) ))) ")" )) ; theorem :: FIB_NUM2:52 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k4_fib_num2 :::"OddNAT"::: ) )) & (Bool (Bool "not" (Set (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_fib_num2 :::"OddNAT"::: ) ))) ")" )) ; definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"EvenFibs"::: "n" -> ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: FIB_NUM2:def 5 (Set ($#k1_fib_num2 :::"Prefix"::: ) "(" (Set ($#k2_fib_num2 :::"FIB"::: ) ) "," (Set "(" (Set ($#k3_fib_num2 :::"EvenNAT"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) "n" ")" ) ")" ) ")" ); func :::"OddFibs"::: "n" -> ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: FIB_NUM2:def 6 (Set ($#k1_fib_num2 :::"Prefix"::: ) "(" (Set ($#k2_fib_num2 :::"FIB"::: ) ) "," (Set "(" (Set ($#k4_fib_num2 :::"OddNAT"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) "n" ")" ) ")" ) ")" ); end; :: deftheorem defines :::"EvenFibs"::: FIB_NUM2:def 5 : (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k5_fib_num2 :::"EvenFibs"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k1_fib_num2 :::"Prefix"::: ) "(" (Set ($#k2_fib_num2 :::"FIB"::: ) ) "," (Set "(" (Set ($#k3_fib_num2 :::"EvenNAT"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) ")" ) ")" ))); :: deftheorem defines :::"OddFibs"::: FIB_NUM2:def 6 : (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k6_fib_num2 :::"OddFibs"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k1_fib_num2 :::"Prefix"::: ) "(" (Set ($#k2_fib_num2 :::"FIB"::: ) ) "," (Set "(" (Set ($#k4_fib_num2 :::"OddNAT"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) ")" ) ")" ))); theorem :: FIB_NUM2:53 (Bool (Set ($#k5_fib_num2 :::"EvenFibs"::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ; theorem :: FIB_NUM2:54 (Bool (Set ($#k15_finseq_1 :::"Seq"::: ) (Set "(" (Set ($#k2_fib_num2 :::"FIB"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Num 2) ($#k6_domain_1 :::"}"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Num 1) ($#k12_finseq_1 :::"*>"::: ) )) ; theorem :: FIB_NUM2:55 (Bool (Set ($#k5_fib_num2 :::"EvenFibs"::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Num 1) ($#k12_finseq_1 :::"*>"::: ) )) ; theorem :: FIB_NUM2:56 (Bool (Set ($#k5_fib_num2 :::"EvenFibs"::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Num 1) "," (Num 3) ($#k10_finseq_1 :::"*>"::: ) )) ; theorem :: FIB_NUM2:57 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" (Set ($#k3_fib_num2 :::"EvenNAT"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 4) ")" ) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_fib_num2 :::"EvenNAT"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 4) ")" ) ")" )))) ; theorem :: FIB_NUM2:58 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" (Set ($#k2_fib_num2 :::"FIB"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set "(" (Set ($#k3_fib_num2 :::"EvenNAT"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ) ")" ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_domain_1 :::"["::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 4) ")" ) "," (Set "(" (Set ($#k2_fib_num2 :::"FIB"::: ) ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 4) ")" ) ")" ) ($#k1_domain_1 :::"]"::: ) ) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_fib_num2 :::"FIB"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set "(" (Set ($#k3_fib_num2 :::"EvenNAT"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 4) ")" ) ")" ) ")" )))) ; theorem :: FIB_NUM2:59 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k5_fib_num2 :::"EvenFibs"::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_fib_num2 :::"EvenFibs"::: ) (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) )))) ; theorem :: FIB_NUM2:60 (Bool (Set ($#k6_fib_num2 :::"OddFibs"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Num 1) ($#k12_finseq_1 :::"*>"::: ) )) ; theorem :: FIB_NUM2:61 (Bool (Set ($#k6_fib_num2 :::"OddFibs"::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Num 1) "," (Num 2) ($#k10_finseq_1 :::"*>"::: ) )) ; theorem :: FIB_NUM2:62 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" (Set ($#k4_fib_num2 :::"OddNAT"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 3) ")" ) ")" ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 5) ")" ) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_fib_num2 :::"OddNAT"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 5) ")" ) ")" )))) ; theorem :: FIB_NUM2:63 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" (Set ($#k2_fib_num2 :::"FIB"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set "(" (Set ($#k4_fib_num2 :::"OddNAT"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 3) ")" ) ")" ) ")" ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_domain_1 :::"["::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 5) ")" ) "," (Set "(" (Set ($#k2_fib_num2 :::"FIB"::: ) ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 5) ")" ) ")" ) ($#k1_domain_1 :::"]"::: ) ) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_fib_num2 :::"FIB"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set "(" (Set ($#k4_fib_num2 :::"OddNAT"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "k")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 5) ")" ) ")" ) ")" )))) ; theorem :: FIB_NUM2:64 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k6_fib_num2 :::"OddFibs"::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 3) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_fib_num2 :::"OddFibs"::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 3) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) )))) ; theorem :: FIB_NUM2:65 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" ($#k5_fib_num2 :::"EvenFibs"::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 3) ")" ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1)))) ; theorem :: FIB_NUM2:66 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" ($#k6_fib_num2 :::"OddFibs"::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )))) ; begin theorem :: FIB_NUM2:67 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k1_pre_ff :::"Fib"::: ) (Set (Var "n"))) "," (Set ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_int_2 :::"are_relative_prime"::: ) )) ; theorem :: FIB_NUM2:68 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "m")) ($#r1_hidden :::"<>"::: ) (Num 1)) & (Bool (Set (Var "m")) ($#r1_nat_d :::"divides"::: ) (Set ($#k1_pre_ff :::"Fib"::: ) (Set (Var "n"))))) "holds" (Bool "not" (Bool (Set (Var "m")) ($#r1_nat_d :::"divides"::: ) (Set ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )))))) ; theorem :: FIB_NUM2:69 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "m")) "is" ($#v1_int_2 :::"prime"::: ) ) & (Bool (Set (Var "n")) "is" ($#v1_int_2 :::"prime"::: ) ) & (Bool (Set (Var "m")) ($#r1_nat_d :::"divides"::: ) (Set ($#k1_pre_ff :::"Fib"::: ) (Set (Var "n"))))) "holds" (Bool "for" (Set (Var "r")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set (Var "r")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "not" (Bool (Set (Var "m")) ($#r1_nat_d :::"divides"::: ) (Set ($#k1_pre_ff :::"Fib"::: ) (Set (Var "r")))))))) ; begin theorem :: FIB_NUM2:70 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k8_domain_1 :::"{"::: ) (Set "(" (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set (Var "n")) ")" ) ($#k4_nat_1 :::"*"::: ) (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 3) ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k4_nat_1 :::"*"::: ) (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_pepin :::"^2"::: ) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" (Set "(" ($#k1_pre_ff :::"Fib"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ) ($#k1_pepin :::"^2"::: ) ")" ) ")" ) ($#k8_domain_1 :::"}"::: ) ) "is" ($#m1_pythtrip :::"Pythagorean_triple"::: ) )) ;