Article CATALAN1, MML version 4.99.1005

:: CATALAN1:th 1
theorem
for b1 being natural set
      st 1 < b1
   holds b1 -' 1 <= (2 * b1) -' 3;

:: CATALAN1:th 2
theorem
for b1 being natural set
      st 1 <= b1
   holds b1 -' 1 <= (2 * b1) -' 2;

:: CATALAN1:th 3
theorem
for b1 being natural set
      st 1 < b1
   holds b1 < (2 * b1) -' 1;

:: CATALAN1:th 4
theorem
for b1 being natural set
      st 1 < b1
   holds (b1 -' 2) + 1 = b1 -' 1;

:: CATALAN1:th 5
theorem
for b1 being natural set
      st 1 < b1
   holds 1 < (((4 * b1) * b1) - (2 * b1)) / (b1 + 1);

:: CATALAN1:th 6
theorem
for b1 being natural set
      st 1 < b1
   holds (((2 * b1) -' 2) ! * b1) * (b1 + 1) < (2 * b1) !;

:: CATALAN1:th 7
theorem
for b1 being natural set holds
   2 * (2 - (3 / (b1 + 1))) < 4;

:: CATALAN1:funcnot 1 => CATALAN1:func 1
definition
  let a1 be natural set;
  func Catalan A1 -> Element of REAL equals
    (((2 * a1) -' 2) choose (a1 -' 1)) / a1;
end;

:: CATALAN1:def 1
theorem
for b1 being natural set holds
   Catalan b1 = (((2 * b1) -' 2) choose (b1 -' 1)) / b1;

:: CATALAN1:th 8
theorem
for b1 being natural set
      st 1 < b1
   holds Catalan b1 = ((2 * b1) -' 2) ! / ((b1 -' 1) ! * (b1 !));

:: CATALAN1:th 9
theorem
for b1 being natural set
      st 1 < b1
   holds Catalan b1 = (4 * (((2 * b1) -' 3) choose (b1 -' 1))) - (((2 * b1) -' 1) choose (b1 -' 1));

:: CATALAN1:th 10
theorem
Catalan 0 = 0;

:: CATALAN1:th 11
theorem
Catalan 1 = 1;

:: CATALAN1:th 12
theorem
Catalan 2 = 1;

:: CATALAN1:th 13
theorem
for b1 being natural set holds
   Catalan b1 is integer set;

:: CATALAN1:th 14
theorem
for b1 being natural set holds
   Catalan (b1 + 1) = (2 * b1) ! / (b1 ! * ((b1 + 1) !));

:: CATALAN1:th 15
theorem
for b1 being natural set
      st 1 < b1
   holds Catalan b1 < Catalan (b1 + 1);

:: CATALAN1:th 16
theorem
for b1 being natural set holds
   Catalan b1 <= Catalan (b1 + 1);

:: CATALAN1:th 17
theorem
for b1 being natural set holds
   0 <= Catalan b1;

:: CATALAN1:th 18
theorem
for b1 being natural set holds
   Catalan b1 is Element of NAT;

:: CATALAN1:th 19
theorem
for b1 being natural set
      st 0 < b1
   holds Catalan (b1 + 1) = (2 * (2 - (3 / (b1 + 1)))) * Catalan b1;

:: CATALAN1:funcreg 1
registration
  let a1 be natural set;
  cluster Catalan a1 -> natural;
end;

:: CATALAN1:th 20
theorem
for b1 being natural set
      st 0 < b1
   holds 0 < Catalan b1;

:: CATALAN1:funcreg 2
registration
  let a1 be non empty natural set;
  cluster Catalan a1 -> non empty;
end;

:: CATALAN1:th 21
theorem
for b1 being natural set
      st 0 < b1
   holds Catalan (b1 + 1) < 4 * Catalan b1;