Article FIB_NUM, MML version 4.99.1005

:: FIB_NUM:th 1
theorem
for b1, b2 being Element of NAT holds
b1 hcf b2 = b1 hcf (b2 + b1);

:: FIB_NUM:th 2
theorem
for b1, b2, b3 being Element of NAT
      st b1 hcf b2 = 1
   holds b1 hcf (b2 * b3) = b1 hcf b3;

:: FIB_NUM:th 3
theorem
for b1 being real set
      st 0 < b1
   holds ex b2 being Element of NAT st
      0 < b2 & 0 < 1 / b2 & 1 / b2 <= b1;

:: FIB_NUM:sch 1
scheme FIB_NUM:sch 1
for b1 being Element of NAT holds
   P1[b1]
provided
   P1[0]
and
   P1[1]
and
   for b1 being Element of NAT
         st P1[b1] & P1[b1 + 1]
      holds P1[b1 + 2];


:: FIB_NUM:sch 2
scheme FIB_NUM:sch 2
for b1, b2 being Element of NAT holds
P1[b1, b2]
provided
   for b1, b2 being Element of NAT
         st P1[b1, b2]
      holds P1[b2, b1]
and
   for b1 being Element of NAT
      st for b2, b3 being Element of NAT
              st b2 < b1 & b3 < b1
           holds P1[b2, b3]
   for b2 being Element of NAT
         st b2 <= b1
      holds P1[b1, b2];


:: FIB_NUM:th 4
theorem
for b1, b2 being Element of NAT holds
Fib (b1 + (b2 + 1)) = ((Fib b2) * Fib b1) + ((Fib (b2 + 1)) * Fib (b1 + 1));

:: FIB_NUM:th 5
theorem
for b1, b2 being Element of NAT holds
(Fib b1) hcf Fib b2 = Fib (b1 hcf b2);

:: FIB_NUM:th 6
theorem
for b1, b2, b3, b4 being real set
      st b2 <> 0 & 0 <= delta(b2,b3,b4)
   holds    ((b2 * (b1 ^2)) + (b3 * b1)) + b4 = 0
   iff
      (b1 = ((- b3) - sqrt delta(b2,b3,b4)) / (2 * b2) or b1 = ((- b3) + sqrt delta(b2,b3,b4)) / (2 * b2));

:: FIB_NUM:funcnot 1 => FIB_NUM:func 1
definition
  func tau -> real set equals
    (1 + sqrt 5) / 2;
end;

:: FIB_NUM:def 1
theorem
tau = (1 + sqrt 5) / 2;

:: FIB_NUM:funcnot 2 => FIB_NUM:func 2
definition
  func tau_bar -> real set equals
    (1 - sqrt 5) / 2;
end;

:: FIB_NUM:def 2
theorem
tau_bar = (1 - sqrt 5) / 2;

:: FIB_NUM:th 7
theorem
for b1 being Element of NAT holds
   Fib b1 = ((tau to_power b1) - (tau_bar to_power b1)) / sqrt 5;

:: FIB_NUM:th 8
theorem
for b1 being Element of NAT holds
   abs ((Fib b1) - ((tau to_power b1) / sqrt 5)) < 1;

:: FIB_NUM:th 10
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL
      st b2 is non-empty
   holds (b1 /" b2) (#) (b2 /" b3) = b1 /" b3;

:: FIB_NUM:th 11
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Element of NAT holds
   (b1 /" b2) . b3 = (b1 . b3) / (b2 . b3) &
    (b1 /" b2) . b3 = (b1 . b3) * ((b2 . b3) ");

:: FIB_NUM:th 12
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st for b2 being Element of NAT holds
           b1 . b2 = (Fib (b2 + 1)) / Fib b2
   holds b1 is convergent & lim b1 = tau;