:: FIB_NUM semantic presentation
theorem Th1: :: FIB_NUM:1
theorem Th2: :: FIB_NUM:2
theorem Th3: :: FIB_NUM:3
(0 + 1) + 1 = 2
;
then Lemma57:
Fib 2 = 1
by PRE_FF:1;
Lemma58:
(1 + 1) + 1 = 3
;
Lemma59:
for k being Element of NAT holds Fib (k + 1) >= k
Lemma60:
for m being Element of NAT holds Fib (m + 1) >= Fib m
Lemma61:
for m, n being Element of NAT st m >= n holds
Fib m >= Fib n
Lemma62:
for m being Element of NAT holds Fib (m + 1) <> 0
theorem Th4: :: FIB_NUM:4
Lemma65:
for n being Element of NAT holds (Fib n) hcf (Fib (n + 1)) = 1
theorem Th5: :: FIB_NUM:5
theorem Th6: :: FIB_NUM:6
:: deftheorem Def1 defines tau FIB_NUM:def 1 :
:: deftheorem Def2 defines tau_bar FIB_NUM:def 2 :
Lemma72:
( tau ^2 = tau + 1 & tau_bar ^2 = tau_bar + 1 )
Lemma73:
2 < sqrt 5
by SQUARE_1:85, SQUARE_1:95;
Lemma74:
sqrt 5 <> 0
by SQUARE_1:85, SQUARE_1:95;
Lemma75:
sqrt 5 < 3
1 < tau
then Lemma76:
0 < tau
by XXREAL_0:2;
Lemma77:
tau_bar < 0
Lemma78:
abs tau_bar < 1
theorem Th7: :: FIB_NUM:7
Lemma83:
for n being Element of NAT
for x being real number st abs x <= 1 holds
abs (x |^ n) <= 1
Lemma84:
for n being Element of NAT holds abs ((tau_bar to_power n) / (sqrt 5)) < 1
theorem Th8: :: FIB_NUM:8
theorem Th9: :: FIB_NUM:9
theorem Th10: :: FIB_NUM:10
theorem Th11: :: FIB_NUM:11
theorem Th12: :: FIB_NUM:12