Article RPR_1, MML version 4.99.1005

:: RPR_1:exreg 1
registration
  let a1 be non empty set;
  cluster non empty trivial Element of bool a1;
end;

:: RPR_1:modenot 1
definition
  let a1 be non empty set;
  mode El_ev of a1 is non empty trivial Element of bool a1;
end;

:: RPR_1:th 1
theorem
for b1 being non empty set
for b2 being non empty Element of bool b1 holds
      b2 is non empty trivial Element of bool b1
   iff
      for b3 being set holds
            b3 c= b2
         iff
            (b3 = {} or b3 = b2);

:: RPR_1:condreg 1
registration
  let a1 be non empty set;
  cluster non empty trivial -> finite (Element of bool a1);
end;

:: RPR_1:th 5
theorem
for b1 being non empty set
for b2, b3 being Element of bool b1
for b4 being non empty trivial Element of bool b1
      st b4 = b2 \/ b3 & b2 <> b3 & (b2 = {} implies b3 <> b4)
   holds b2 = b4 & b3 = {};

:: RPR_1:th 6
theorem
for b1 being non empty set
for b2, b3 being Element of bool b1
for b4 being non empty trivial Element of bool b1
      st b4 = b2 \/ b3 & (b2 = b4 implies b3 <> b4) & (b2 = b4 implies b3 <> {})
   holds b2 = {} & b3 = b4;

:: RPR_1:th 7
theorem
for b1 being non empty set
for b2 being Element of b1 holds
   {b2} is non empty trivial Element of bool b1;

:: RPR_1:th 10
theorem
for b1 being non empty set
for b2, b3 being non empty trivial Element of bool b1
      st b2 c= b3
   holds b2 = b3;

:: RPR_1:th 11
theorem
for b1 being non empty set
for b2 being non empty trivial Element of bool b1 holds
   ex b3 being Element of b1 st
      b3 in b1 & b2 = {b3};

:: RPR_1:th 12
theorem
for b1 being non empty set holds
   ex b2 being non empty trivial Element of bool b1 st
      b2 is non empty trivial Element of bool b1;

:: RPR_1:th 14
theorem
for b1 being non empty set
for b2 being non empty trivial Element of bool b1 holds
   ex b3 being Relation-like Function-like FinSequence-like set st
      b3 is FinSequence of b1 & proj2 b3 = b2 & len b3 = 1;

:: RPR_1:modenot 2
definition
  let a1 be set;
  mode Event of a1 is Element of bool a1;
end;

:: RPR_1:th 22
theorem
for b1 being non empty set
for b2 being non empty trivial Element of bool b1
for b3 being Element of bool b1
      st b2 meets b3
   holds b2 /\ b3 = b2;

:: RPR_1:th 25
theorem
for b1 being non empty set
for b2 being Element of bool b1
      st b2 <> {}
   holds ex b3 being non empty trivial Element of bool b1 st
      b3 c= b2;

:: RPR_1:th 26
theorem
for b1 being non empty set
for b2 being non empty trivial Element of bool b1
for b3 being Element of bool b1
      st b2 c= b3 \/ (b3 `) & not b2 c= b3
   holds b2 c= b3 `;

:: RPR_1:th 27
theorem
for b1 being non empty set
for b2, b3 being non empty trivial Element of bool b1
      st b2 <> b3
   holds b2 misses b3;

:: RPR_1:th 34
theorem
for b1 being non empty set
for b2, b3 being Element of bool b1 holds
b2 /\ b3 misses b2 /\ (b3 `);

:: RPR_1:funcnot 1 => RPR_1:func 1
definition
  let a1 be non empty finite set;
  let a2 be Element of bool a1;
  func prob A2 -> Element of REAL equals
    (card a2) / card a1;
end;

:: RPR_1:def 4
theorem
for b1 being non empty finite set
for b2 being Element of bool b1 holds
   prob b2 = (card b2) / card b1;

:: RPR_1:th 38
theorem
for b1 being non empty finite set
for b2 being non empty trivial Element of bool b1 holds
   prob b2 = 1 / card b1;

:: RPR_1:th 39
theorem
for b1 being non empty finite set holds
   prob [#] b1 = 1;

:: RPR_1:th 41
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1
      st b2 misses b3
   holds prob (b2 /\ b3) = 0;

:: RPR_1:th 42
theorem
for b1 being non empty finite set
for b2 being Element of bool b1 holds
   prob b2 <= 1;

:: RPR_1:th 43
theorem
for b1 being non empty finite set
for b2 being Element of bool b1 holds
   0 <= prob b2;

:: RPR_1:th 44
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1
      st b2 c= b3
   holds prob b2 <= prob b3;

:: RPR_1:th 46
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1 holds
prob (b2 \/ b3) = ((prob b2) + prob b3) - prob (b2 /\ b3);

:: RPR_1:th 47
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1
      st b2 misses b3
   holds prob (b2 \/ b3) = (prob b2) + prob b3;

:: RPR_1:th 48
theorem
for b1 being non empty finite set
for b2 being Element of bool b1 holds
   prob b2 = 1 - prob (b2 `) & prob (b2 `) = 1 - prob b2;

:: RPR_1:th 49
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1 holds
prob (b2 \ b3) = (prob b2) - prob (b2 /\ b3);

:: RPR_1:th 50
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1
      st b3 c= b2
   holds prob (b2 \ b3) = (prob b2) - prob b3;

:: RPR_1:th 51
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1 holds
prob (b2 \/ b3) <= (prob b2) + prob b3;

:: RPR_1:th 53
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1 holds
prob b2 = (prob (b2 /\ b3)) + prob (b2 /\ (b3 `));

:: RPR_1:th 54
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1 holds
prob b2 = (prob (b2 \/ b3)) - prob (b3 \ b2);

:: RPR_1:th 55
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1 holds
(prob b2) + prob (b2 ` /\ b3) = (prob b3) + prob (b3 ` /\ b2);

:: RPR_1:th 56
theorem
for b1 being non empty finite set
for b2, b3, b4 being Element of bool b1 holds
prob ((b2 \/ b3) \/ b4) = ((((prob b2) + prob b3) + prob b4) - (((prob (b2 /\ b3)) + prob (b2 /\ b4)) + prob (b3 /\ b4))) + prob ((b2 /\ b3) /\ b4);

:: RPR_1:th 57
theorem
for b1 being non empty finite set
for b2, b3, b4 being Element of bool b1
      st b2 misses b3 & b2 misses b4 & b3 misses b4
   holds prob ((b2 \/ b3) \/ b4) = ((prob b2) + prob b3) + prob b4;

:: RPR_1:th 58
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1 holds
(prob b2) - prob b3 <= prob (b2 \ b3);

:: RPR_1:funcnot 2 => RPR_1:func 2
definition
  let a1 be non empty finite set;
  let a2, a3 be Element of bool a1;
  func prob(A3,A2) -> Element of REAL equals
    (prob (a3 /\ a2)) / prob a2;
end;

:: RPR_1:def 5
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1 holds
prob(b3,b2) = (prob (b3 /\ b2)) / prob b2;

:: RPR_1:th 61
theorem
for b1 being non empty finite set
for b2 being Element of bool b1 holds
   prob(b2,[#] b1) = prob b2;

:: RPR_1:th 62
theorem
for b1 being non empty finite set holds
   prob([#] b1,[#] b1) = 1;

:: RPR_1:th 64
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1
      st 0 < prob b3
   holds prob(b2,b3) <= 1;

:: RPR_1:th 65
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1
      st 0 < prob b3
   holds 0 <= prob(b2,b3);

:: RPR_1:th 66
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1
      st 0 < prob b3
   holds prob(b2,b3) = 1 - ((prob (b3 \ b2)) / prob b3);

:: RPR_1:th 67
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1
      st 0 < prob b3 & b2 c= b3
   holds prob(b2,b3) = (prob b2) / prob b3;

:: RPR_1:th 68
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1
      st b2 misses b3
   holds prob(b2,b3) = 0;

:: RPR_1:th 69
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1
      st 0 < prob b2 & 0 < prob b3
   holds (prob b2) * prob(b3,b2) = (prob b3) * prob(b2,b3);

:: RPR_1:th 70
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1
      st 0 < prob b3
   holds prob(b2,b3) = 1 - prob(b2 `,b3) &
    prob(b2 `,b3) = 1 - prob(b2,b3);

:: RPR_1:th 71
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1
      st 0 < prob b3 & b3 c= b2
   holds prob(b2,b3) = 1;

:: RPR_1:th 72
theorem
for b1 being non empty finite set
for b2 being Element of bool b1
      st 0 < prob b2
   holds prob([#] b1,b2) = 1;

:: RPR_1:th 73
theorem
for b1 being non empty finite set
for b2 being Element of bool b1 holds
   prob(b2 `,b2) = 0;

:: RPR_1:th 74
theorem
for b1 being non empty finite set
for b2 being Element of bool b1 holds
   prob(b2,b2 `) = 0;

:: RPR_1:th 75
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1
      st 0 < prob b3 & b2 misses b3
   holds prob(b2 `,b3) = 1;

:: RPR_1:th 76
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1
      st 0 < prob b2 & prob b3 < 1 & b2 misses b3
   holds prob(b2,b3 `) = (prob b2) / (1 - prob b3);

:: RPR_1:th 77
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1
      st 0 < prob b2 & prob b3 < 1 & b2 misses b3
   holds prob(b2 `,b3 `) = 1 - ((prob b2) / (1 - prob b3));

:: RPR_1:th 78
theorem
for b1 being non empty finite set
for b2, b3, b4 being Element of bool b1
      st 0 < prob (b3 /\ b4) & 0 < prob b4
   holds prob ((b2 /\ b3) /\ b4) = ((prob(b2,b3 /\ b4)) * prob(b3,b4)) * prob b4;

:: RPR_1:th 79
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1
      st 0 < prob b3 & prob b3 < 1
   holds prob b2 = ((prob(b2,b3)) * prob b3) + ((prob(b2,b3 `)) * prob (b3 `));

:: RPR_1:th 80
theorem
for b1 being non empty finite set
for b2, b3, b4 being Element of bool b1
      st 0 < prob b3 & 0 < prob b4 & b3 \/ b4 = b1 & b3 misses b4
   holds prob b2 = ((prob(b2,b3)) * prob b3) + ((prob(b2,b4)) * prob b4);

:: RPR_1:th 81
theorem
for b1 being non empty finite set
for b2, b3, b4, b5 being Element of bool b1
      st 0 < prob b3 & 0 < prob b4 & 0 < prob b5 & (b3 \/ b4) \/ b5 = b1 & b3 misses b4 & b3 misses b5 & b4 misses b5
   holds prob b2 = (((prob(b2,b3)) * prob b3) + ((prob(b2,b4)) * prob b4)) + ((prob(b2,b5)) * prob b5);

:: RPR_1:th 82
theorem
for b1 being non empty finite set
for b2, b3, b4 being Element of bool b1
      st 0 < prob b3 & 0 < prob b4 & b3 \/ b4 = b1 & b3 misses b4
   holds prob(b3,b2) = ((prob(b2,b3)) * prob b3) / (((prob(b2,b3)) * prob b3) + ((prob(b2,b4)) * prob b4));

:: RPR_1:th 83
theorem
for b1 being non empty finite set
for b2, b3, b4, b5 being Element of bool b1
      st 0 < prob b3 & 0 < prob b4 & 0 < prob b5 & (b3 \/ b4) \/ b5 = b1 & b3 misses b4 & b3 misses b5 & b4 misses b5
   holds prob(b3,b2) = ((prob(b2,b3)) * prob b3) / ((((prob(b2,b3)) * prob b3) + ((prob(b2,b4)) * prob b4)) + ((prob(b2,b5)) * prob b5));

:: RPR_1:prednot 1 => RPR_1:pred 1
definition
  let a1 be non empty finite set;
  let a2, a3 be Element of bool a1;
  pred A2,A3 are_independent means
    prob (a2 /\ a3) = (prob a2) * prob a3;
  symmetry;
::  for a1 being non empty finite set
::  for a2, a3 being Element of bool a1
::        st a2,a3 are_independent
::     holds a3,a2 are_independent;
end;

:: RPR_1:dfs 3
definiens
  let a1 be non empty finite set;
  let a2, a3 be Element of bool a1;
To prove
     a2,a3 are_independent
it is sufficient to prove
  thus prob (a2 /\ a3) = (prob a2) * prob a3;

:: RPR_1:def 6
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1 holds
   b2,b3 are_independent
iff
   prob (b2 /\ b3) = (prob b2) * prob b3;

:: RPR_1:th 86
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1
      st 0 < prob b3 & b2,b3 are_independent
   holds prob(b2,b3) = prob b2;

:: RPR_1:th 87
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1
      st prob b3 = 0
   holds b2,b3 are_independent;

:: RPR_1:th 88
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1
      st b2,b3 are_independent
   holds b2 `,b3 are_independent;

:: RPR_1:th 89
theorem
for b1 being non empty finite set
for b2, b3 being Element of bool b1
      st b2 misses b3 & b2,b3 are_independent & prob b2 <> 0
   holds prob b3 = 0;