:: FIB_NUM semantic presentation
theorem Th1: :: FIB_NUM:1
for b
1, b
2 being
Nat holds b
1 hcf b
2 = b
1 hcf (b2 + b1)
theorem Th2: :: FIB_NUM:2
for b
1, b
2, b
3 being
Nat holds
( b
1 hcf b
2 = 1 implies b
1 hcf (b2 * b3) = b
1 hcf b
3 )
theorem Th3: :: FIB_NUM:3
for b
1 being
real number holds
not ( b
1 > 0 & ( for b
2 being
Nat holds
not ( b
2 > 0 & 0
< 1
/ b
2 & 1
/ b
2 <= b
1 ) ) )
scheme :: FIB_NUM:sch 1
s1{ P
1[
Nat] } :
for b
1 being
Nat holds P
1[b
1]
provided
E4:
P
1[0]
and
E5:
P
1[1]
and
E6:
for b
1 being
Nat holds
( P
1[b
1] & P
1[b
1 + 1] implies P
1[b
1 + 2] )
scheme :: FIB_NUM:sch 2
s2{ P
1[
Nat,
Nat] } :
for b
1, b
2 being
Nat holds P
1[b
1,b
2]
provided
E4:
for b
1, b
2 being
Nat holds
( P
1[b
1,b
2] implies P
1[b
2,b
1] )
and
E5:
for b
1 being
Nat holds
( ( for b
2, b
3 being
Nat holds
( b
2 < b
1 & b
3 < b
1 implies P
1[b
2,b
3] ) ) implies for b
2 being
Nat holds
( b
2 <= b
1 implies P
1[b
1,b
2] ) )
(0 + 1) + 1 = 2
;
then Lemma4:
Fib 2 = 1
by PRE_FF:1;
Lemma5:
(1 + 1) + 1 = 3
;
Lemma6:
for b1 being Nat holds Fib (b1 + 1) >= b1
Lemma7:
for b1 being Nat holds Fib (b1 + 1) >= Fib b1
Lemma8:
for b1, b2 being Nat holds
( b1 >= b2 implies Fib b1 >= Fib b2 )
Lemma9:
for b1 being Nat holds
Fib (b1 + 1) <> 0
theorem Th4: :: FIB_NUM:4
Lemma11:
for b1 being Nat holds (Fib b1) hcf (Fib (b1 + 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 :
Lemma13:
( tau ^2 = tau + 1 & tau_bar ^2 = tau_bar + 1 )
Lemma14:
2 < sqrt 5
by SQUARE_1:85, SQUARE_1:95;
Lemma15:
sqrt 5 <> 0
by SQUARE_1:85, SQUARE_1:95;
Lemma16:
sqrt 5 < 3
1 < tau
then Lemma17:
0 < tau
by XXREAL_0:2;
Lemma18:
tau_bar < 0
Lemma19:
abs tau_bar < 1
theorem Th7: :: FIB_NUM:7
Lemma21:
for b1 being Nat
for b2 being real number holds
( abs b2 <= 1 implies abs (b2 |^ b1) <= 1 )
Lemma22:
for b1 being Nat holds
abs ((tau_bar to_power b1) / (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