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;