Article FINSEQ_3, MML version 4.99.1005

:: FINSEQ_3:th 1
theorem
Seg 3 = {1,2,3};

:: FINSEQ_3:th 2
theorem
Seg 4 = {1,2,3,4};

:: FINSEQ_3:th 3
theorem
Seg 5 = {1,2,3,4,5};

:: FINSEQ_3:th 4
theorem
Seg 6 = {1,2,3,4,5,6};

:: FINSEQ_3:th 5
theorem
Seg 7 = {1,2,3,4,5,6,7};

:: FINSEQ_3:th 6
theorem
Seg 8 = {1,2,3,4,5,6,7,8};

:: FINSEQ_3:th 7
theorem
for b1 being natural set holds
      Seg b1 = {}
   iff
      not b1 in Seg b1;

:: FINSEQ_3:th 9
theorem
for b1 being natural set holds
   not b1 + 1 in Seg b1;

:: FINSEQ_3:th 10
theorem
for b1, b2 being natural set
      st b1 <> 0
   holds b1 in Seg (b1 + b2);

:: FINSEQ_3:th 11
theorem
for b1, b2 being natural set
      st b1 + b2 in Seg b1
   holds b2 = 0;

:: FINSEQ_3:th 12
theorem
for b1, b2 being natural set
      st b1 < b2
   holds b1 + 1 in Seg b2;

:: FINSEQ_3:th 13
theorem
for b1, b2, b3 being natural set
      st b1 in Seg b2 & b3 < b1
   holds b1 - b3 in Seg b2;

:: FINSEQ_3:th 14
theorem
for b1, b2 being natural set holds
   b1 - b2 in Seg b1
iff
   b2 < b1;

:: FINSEQ_3:th 15
theorem
for b1 being natural set holds
   Seg b1 misses {b1 + 1};

:: FINSEQ_3:th 16
theorem
for b1 being natural set holds
   (Seg (b1 + 1)) \ Seg b1 = {b1 + 1};

:: FINSEQ_3:th 17
theorem
for b1 being natural set holds
   Seg b1 <> Seg (b1 + 1);

:: FINSEQ_3:th 18
theorem
for b1, b2 being natural set
      st Seg b1 = Seg (b1 + b2)
   holds b2 = 0;

:: FINSEQ_3:th 19
theorem
for b1, b2 being natural set holds
Seg b1 c= Seg (b1 + b2);

:: FINSEQ_3:th 20
theorem
for b1, b2 being natural set holds
Seg b1,Seg b2 are_c=-comparable;

:: FINSEQ_3:th 22
theorem
for b1 being set
for b2 being natural set
      st Seg b2 = {b1}
   holds b2 = 1 & b1 = 1;

:: FINSEQ_3:th 23
theorem
for b1, b2 being set
for b3 being natural set
      st Seg b3 = {b1,b2} & b1 <> b2
   holds b3 = 2 & {b1,b2} = {1,2};

:: FINSEQ_3:th 24
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set
for b3 being set
      st b3 in dom b1
   holds b3 in dom (b1 ^ b2);

:: FINSEQ_3:th 25
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set
      st b2 in dom b1
   holds b2 is Element of NAT;

:: FINSEQ_3:th 26
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set
      st b2 in dom b1
   holds b2 <> 0;

:: FINSEQ_3:th 27
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being natural set holds
      b2 in dom b1
   iff
      1 <= b2 & b2 <= len b1;

:: FINSEQ_3:th 28
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being natural set holds
      b2 in dom b1
   iff
      b2 - 1 is Element of NAT & (len b1) - b2 is Element of NAT;

:: FINSEQ_3:th 29
theorem
for b1, b2 being set holds
dom <*b1,b2*> = Seg 2;

:: FINSEQ_3:th 30
theorem
for b1, b2, b3 being set holds
dom <*b1,b2,b3*> = Seg 3;

:: FINSEQ_3:th 31
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set holds
   len b1 = len b2
iff
   dom b1 = dom b2;

:: FINSEQ_3:th 32
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set holds
   len b1 <= len b2
iff
   dom b1 c= dom b2;

:: FINSEQ_3:th 33
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set
      st b2 in proj2 b1
   holds 1 in dom b1;

:: FINSEQ_3:th 34
theorem
for b1 being Relation-like Function-like FinSequence-like set
      st proj2 b1 <> {}
   holds 1 in dom b1;

:: FINSEQ_3:th 38
theorem
for b1, b2 being set holds
{} <> <*b1,b2*>;

:: FINSEQ_3:th 39
theorem
for b1, b2, b3 being set holds
{} <> <*b1,b2,b3*>;

:: FINSEQ_3:th 40
theorem
for b1, b2, b3 being set holds
<*b1*> <> <*b2,b3*>;

:: FINSEQ_3:th 41
theorem
for b1, b2, b3, b4 being set holds
<*b1*> <> <*b2,b3,b4*>;

:: FINSEQ_3:th 42
theorem
for b1, b2, b3, b4, b5 being set holds
<*b1,b2*> <> <*b3,b4,b5*>;

:: FINSEQ_3:th 43
theorem
for b1, b2, b3 being Relation-like Function-like FinSequence-like set
      st len b1 = (len b2) + len b3 &
         (for b4 being Element of NAT
               st b4 in dom b2
            holds b1 . b4 = b2 . b4) &
         (for b4 being Element of NAT
               st b4 in dom b3
            holds b1 . ((len b2) + b4) = b3 . b4)
   holds b1 = b2 ^ b3;

:: FINSEQ_3:th 44
theorem
for b1 being natural set
for b2 being finite set
      st b2 c= Seg b1
   holds len Sgm b2 = card b2;

:: FINSEQ_3:th 45
theorem
for b1 being natural set
for b2 being finite set
      st b2 c= Seg b1
   holds dom Sgm b2 = Seg card b2;

:: FINSEQ_3:th 46
theorem
for b1 being set
for b2, b3, b4, b5, b6 being natural set
      st b1 c= Seg b2 & b3 < b4 & 1 <= b5 & b6 <= len Sgm b1 & (Sgm b1) . b6 = b3 & (Sgm b1) . b5 = b4
   holds b6 < b5;

:: FINSEQ_3:th 48
theorem
for b1, b2 being set
for b3, b4 being natural set
      st b1 c= Seg b3 & b2 c= Seg b4
   holds    for b5, b6 being Element of NAT
            st b5 in b1 & b6 in b2
         holds b5 < b6
   iff
      Sgm (b1 \/ b2) = (Sgm b1) ^ Sgm b2;

:: FINSEQ_3:th 49
theorem
Sgm {} = {};

:: FINSEQ_3:th 50
theorem
for b1 being natural set
      st 0 <> b1
   holds Sgm {b1} = <*b1*>;

:: FINSEQ_3:th 51
theorem
for b1, b2 being natural set
      st 0 < b1 & b1 < b2
   holds Sgm {b1,b2} = <*b1,b2*>;

:: FINSEQ_3:th 52
theorem
for b1 being natural set holds
   len Sgm Seg b1 = b1;

:: FINSEQ_3:th 53
theorem
for b1, b2 being natural set holds
(Sgm Seg (b1 + b2)) | Seg b1 = Sgm Seg b1;

:: FINSEQ_3:th 54
theorem
for b1 being natural set holds
   Sgm Seg b1 = idseq b1;

:: FINSEQ_3:th 55
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being natural set holds
      b1 | Seg b2 = b1
   iff
      len b1 <= b2;

:: FINSEQ_3:th 56
theorem
for b1, b2 being natural set holds
(idseq (b1 + b2)) | Seg b1 = idseq b1;

:: FINSEQ_3:th 57
theorem
for b1, b2 being natural set holds
   (idseq b1) | Seg b2 = idseq b2
iff
   b2 <= b1;

:: FINSEQ_3:th 58
theorem
for b1, b2 being natural set holds
   (idseq b1) | Seg b2 = idseq b1
iff
   b1 <= b2;

:: FINSEQ_3:th 59
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set
for b3, b4 being natural set
      st len b1 = b3 + b4 & b2 = b1 | Seg b3
   holds len b2 = b3;

:: FINSEQ_3:th 60
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set
for b3, b4 being natural set
      st len b1 = b3 + b4 & b2 = b1 | Seg b3
   holds dom b2 = Seg b3;

:: FINSEQ_3:th 61
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set
for b3 being natural set
      st len b1 = b3 + 1 & b2 = b1 | Seg b3
   holds b1 = b2 ^ <*b1 . (b3 + 1)*>;

:: FINSEQ_3:th 62
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set holds
      b1 | b2 is Relation-like Function-like FinSequence-like set
   iff
      ex b3 being Element of NAT st
         b2 /\ dom b1 = Seg b3;

:: FINSEQ_3:th 63
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set
for b3 being set holds
   card ((b1 ^ b2) " b3) = (card (b1 " b3)) + card (b2 " b3);

:: FINSEQ_3:th 64
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set
for b3 being set holds
   b1 " b3 c= (b1 ^ b2) " b3;

:: FINSEQ_3:funcnot 1 => FINSEQ_3:func 1
definition
  let a1 be Relation-like Function-like FinSequence-like set;
  let a2 be set;
  func A1 - A2 -> Relation-like Function-like FinSequence-like set equals
    (Sgm ((dom a1) \ (a1 " a2))) * a1;
end;

:: FINSEQ_3:def 1
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set holds
   b1 - b2 = (Sgm ((dom b1) \ (b1 " b2))) * b1;

:: FINSEQ_3:th 66
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set holds
   len (b1 - b2) = (len b1) - card (b1 " b2);

:: FINSEQ_3:th 67
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set holds
   len (b1 - b2) <= len b1;

:: FINSEQ_3:th 68
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set
      st len (b1 - b2) = len b1
   holds b2 misses proj2 b1;

:: FINSEQ_3:th 69
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set
for b3 being natural set
      st b3 = (len b1) - card (b1 " b2)
   holds dom (b1 - b2) = Seg b3;

:: FINSEQ_3:th 70
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set holds
   dom (b1 - b2) c= dom b1;

:: FINSEQ_3:th 71
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set
      st dom (b1 - b2) = dom b1
   holds b2 misses proj2 b1;

:: FINSEQ_3:th 72
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set holds
   proj2 (b1 - b2) = (proj2 b1) \ b2;

:: FINSEQ_3:th 73
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set holds
   proj2 (b1 - b2) c= proj2 b1;

:: FINSEQ_3:th 74
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set
      st proj2 (b1 - b2) = proj2 b1
   holds b2 misses proj2 b1;

:: FINSEQ_3:th 75
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set holds
      b1 - b2 = {}
   iff
      proj2 b1 c= b2;

:: FINSEQ_3:th 76
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set holds
      b1 - b2 = b1
   iff
      b2 misses proj2 b1;

:: FINSEQ_3:th 77
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set holds
      b1 - {b2} = b1
   iff
      not b2 in proj2 b1;

:: FINSEQ_3:th 78
theorem
for b1 being Relation-like Function-like FinSequence-like set holds
   b1 - {} = b1;

:: FINSEQ_3:th 79
theorem
for b1 being Relation-like Function-like FinSequence-like set holds
   b1 - proj2 b1 = {};

:: FINSEQ_3:th 80
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set
for b3 being set holds
   (b1 ^ b2) - b3 = (b1 - b3) ^ (b2 - b3);

:: FINSEQ_3:th 81
theorem
for b1 being set holds
   {} - b1 = {};

:: FINSEQ_3:th 82
theorem
for b1, b2 being set holds
   <*b1*> - b2 = <*b1*>
iff
   not b1 in b2;

:: FINSEQ_3:th 83
theorem
for b1, b2 being set holds
   <*b1*> - b2 = {}
iff
   b1 in b2;

:: FINSEQ_3:th 84
theorem
for b1, b2, b3 being set holds
   <*b1,b2*> - b3 = {}
iff
   b1 in b3 & b2 in b3;

:: FINSEQ_3:th 85
theorem
for b1, b2, b3 being set
      st b1 in b2 & not b3 in b2
   holds <*b1,b3*> - b2 = <*b3*>;

:: FINSEQ_3:th 86
theorem
for b1, b2, b3 being set
      st <*b1,b2*> - b3 = <*b2*> &
         b1 <> b2
   holds b1 in b3 & not b2 in b3;

:: FINSEQ_3:th 87
theorem
for b1, b2, b3 being set
      st not b1 in b2 & b3 in b2
   holds <*b1,b3*> - b2 = <*b1*>;

:: FINSEQ_3:th 88
theorem
for b1, b2, b3 being set
      st <*b1,b2*> - b3 = <*b1*> &
         b1 <> b2
   holds not b1 in b3 & b2 in b3;

:: FINSEQ_3:th 89
theorem
for b1, b2, b3 being set holds
   <*b1,b2*> - b3 = <*b1,b2*>
iff
   not b1 in b3 & not b2 in b3;

:: FINSEQ_3:th 90
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set
for b3 being set
for b4 being natural set
      st len b1 = b4 + 1 & b2 = b1 | Seg b4
   holds    b1 . (b4 + 1) in b3
   iff
      b1 - b3 = b2 - b3;

:: FINSEQ_3:th 91
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set
for b3 being set
for b4 being natural set
      st len b1 = b4 + 1 & b2 = b1 | Seg b4
   holds    not b1 . (b4 + 1) in b3
   iff
      b1 - b3 = (b2 - b3) ^ <*b1 . (b4 + 1)*>;

:: FINSEQ_3:th 92
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set
for b3 being natural set
   st b3 in dom b1
for b4 being finite set
      st b4 = {b5 where b5 is Element of NAT: b5 in dom b1 & b5 <= b3 & b1 . b5 in b2} &
         not b1 . b3 in b2
   holds (b1 - b2) . (b3 - card b4) = b1 . b3;

:: FINSEQ_3:th 93
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2, b3 being set
      st b1 is FinSequence of b2
   holds b1 - b3 is FinSequence of b2;

:: FINSEQ_3:th 94
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set
      st b1 is one-to-one
   holds b1 - b2 is one-to-one;

:: FINSEQ_3:th 95
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set
      st b1 is one-to-one
   holds len (b1 - b2) = (len b1) - card (b2 /\ proj2 b1);

:: FINSEQ_3:th 96
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being finite set
      st b1 is one-to-one & b2 c= proj2 b1
   holds len (b1 - b2) = (len b1) - card b2;

:: FINSEQ_3:th 97
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set
      st b1 is one-to-one & b2 in proj2 b1
   holds len (b1 - {b2}) = (len b1) - 1;

:: FINSEQ_3:th 98
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set holds
   proj2 b1 misses proj2 b2 & b1 is one-to-one & b2 is one-to-one
iff
   b1 ^ b2 is one-to-one;

:: FINSEQ_3:th 99
theorem
for b1 being set
for b2 being natural set
      st b1 c= Seg b2
   holds Sgm b1 is one-to-one;

:: FINSEQ_3:th 102
theorem
for b1 being set holds
   <*b1*> is one-to-one;

:: FINSEQ_3:th 103
theorem
for b1, b2 being set holds
   b1 <> b2
iff
   <*b1,b2*> is one-to-one;

:: FINSEQ_3:th 104
theorem
for b1, b2, b3 being set holds
   b1 <> b2 & b2 <> b3 & b3 <> b1
iff
   <*b1,b2,b3*> is one-to-one;

:: FINSEQ_3:th 105
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set
      st b1 is one-to-one & proj2 b1 = {b2}
   holds len b1 = 1;

:: FINSEQ_3:th 106
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set
      st b1 is one-to-one & proj2 b1 = {b2}
   holds b1 = <*b2*>;

:: FINSEQ_3:th 107
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2, b3 being set
      st b1 is one-to-one & proj2 b1 = {b2,b3} & b2 <> b3
   holds len b1 = 2;

:: FINSEQ_3:th 108
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2, b3 being set
      st b1 is one-to-one & proj2 b1 = {b2,b3} & b2 <> b3 & b1 <> <*b2,b3*>
   holds b1 = <*b3,b2*>;

:: FINSEQ_3:th 109
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2, b3, b4 being set
      st b1 is one-to-one & proj2 b1 = {b2,b3,b4} & <*b2,b3,b4*> is one-to-one
   holds len b1 = 3;

:: FINSEQ_3:th 110
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2, b3, b4 being set
      st b1 is one-to-one & proj2 b1 = {b2,b3,b4} & b2 <> b3 & b3 <> b4 & b2 <> b4
   holds len b1 = 3;

:: FINSEQ_3:th 111
theorem
for b1 being non empty set
for b2 being FinSequence of b1
      st b2 is not empty
   holds ex b3 being Element of b1 st
      ex b4 being FinSequence of b1 st
         b3 = b2 . 1 & b2 = <*b3*> ^ b4;

:: FINSEQ_3:th 112
theorem
for b1 being natural set
for b2 being Relation-like Function-like FinSequence-like set
for b3 being set
      st b1 in dom b2
   holds (<*b3*> ^ b2) . (b1 + 1) = b2 . b1;

:: FINSEQ_3:funcnot 2 => FINSEQ_3:func 2
definition
  let a1 be natural set;
  let a2 be Relation-like Function-like FinSequence-like set;
  func Del(A2,A1) -> Relation-like Function-like FinSequence-like set equals
    (Sgm ((dom a2) \ {a1})) * a2;
end;

:: FINSEQ_3:def 2
theorem
for b1 being natural set
for b2 being Relation-like Function-like FinSequence-like set holds
   Del(b2,b1) = (Sgm ((dom b2) \ {b1})) * b2;

:: FINSEQ_3:th 113
theorem
for b1 being natural set
for b2 being Relation-like Function-like FinSequence-like set holds
   (b1 in dom b2 implies ex b3 being natural set st
       len b2 = b3 + 1 & len Del(b2,b1) = b3) &
    (b1 in dom b2 or Del(b2,b1) = b2);

:: FINSEQ_3:th 114
theorem
for b1 being natural set
for b2 being non empty set
for b3 being FinSequence of b2 holds
   Del(b3,b1) is FinSequence of b2;

:: FINSEQ_3:th 115
theorem
for b1 being natural set
for b2 being Relation-like Function-like FinSequence-like set holds
   proj2 Del(b2,b1) c= proj2 b2;

:: FINSEQ_3:th 116
theorem
for b1, b2, b3 being natural set
      st b1 = b2 + 1 & b3 in Seg b1
   holds len Sgm ((Seg b1) \ {b3}) = b2;

:: FINSEQ_3:th 117
theorem
for b1, b2, b3, b4 being Element of NAT
      st b4 = b3 + 1 & b2 in Seg b4 & b1 in Seg b3
   holds (1 <= b1 & b1 < b2 implies (Sgm ((Seg b4) \ {b2})) . b1 = b1) &
    (b2 <= b1 & b1 <= b3 implies (Sgm ((Seg b4) \ {b2})) . b1 = b1 + 1);

:: FINSEQ_3:th 118
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2, b3 being natural set
      st len b1 = b3 + 1 & b2 in dom b1
   holds len Del(b1,b2) = b3;

:: FINSEQ_3:th 119
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2, b3, b4 being Element of NAT
      st len b1 = b3 + 1 & b4 < b2
   holds (Del(b1,b2)) . b4 = b1 . b4;

:: FINSEQ_3:th 120
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2, b3, b4 being Element of NAT
      st len b1 = b3 + 1 & b2 in dom b1 & b2 <= b4 & b4 <= b3
   holds (Del(b1,b2)) . b4 = b1 . (b4 + 1);

:: FINSEQ_3:th 121
theorem
for b1, b2 being natural set
for b3 being Relation-like Function-like FinSequence-like set
      st b1 <= b2
   holds (b3 | b2) . b1 = b3 . b1;

:: FINSEQ_3:th 122
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set
      st b1 c= b2
   holds b2 | len b1 = b1;

:: FINSEQ_3:th 123
theorem
for b1 being set
for b2 being Relation-like Function-like FinSequence-like set holds
   (Sgm (b2 " b1)) ^ Sgm (b2 " ((proj2 b2) \ b1)) is Function-like quasi_total bijective Relation of dom b2,dom b2;

:: FINSEQ_3:th 124
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being Element of bool proj2 b1
      st b1 is one-to-one
   holds ex b3 being Function-like quasi_total bijective Relation of dom b1,dom b1 st
      (b1 - (b2 `)) ^ (b1 - b2) = b3 * b1;

:: FINSEQ_3:th 125
theorem
for b1 being Relation-like Function-like FinSubsequence-like set
      st b1 is Relation-like Function-like FinSequence-like set
   holds Seq b1 = b1;

:: FINSEQ_3:th 126
theorem
for b1 being FinSequence of INT holds
   b1 is FinSequence of REAL;

:: FINSEQ_3:th 127
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set holds
   len b1 < len b2
iff
   dom b1 c< dom b2;