Article RFINSEQ, MML version 4.99.1005
:: RFINSEQ:prednot 1 => RFINSEQ:pred 1
definition
let a1, a2 be Relation-like set;
pred A1,A2 are_fiberwise_equipotent means
for b1 being set holds
Card (a1 " {b1}) = Card (a2 " {b1});
symmetry;
:: for a1, a2 being Relation-like set
:: st a1,a2 are_fiberwise_equipotent
:: holds a2,a1 are_fiberwise_equipotent;
reflexivity;
:: for a1 being Relation-like set holds
:: a1,a1 are_fiberwise_equipotent;
end;
:: RFINSEQ:dfs 1
definiens
let a1, a2 be Relation-like set;
To prove
a1,a2 are_fiberwise_equipotent
it is sufficient to prove
thus for b1 being set holds
Card (a1 " {b1}) = Card (a2 " {b1});
:: RFINSEQ:def 1
theorem
for b1, b2 being Relation-like set holds
b1,b2 are_fiberwise_equipotent
iff
for b3 being set holds
Card (b1 " {b3}) = Card (b2 " {b3});
:: RFINSEQ:th 1
theorem
for b1, b2 being Relation-like Function-like set
st b1,b2 are_fiberwise_equipotent
holds proj2 b1 = proj2 b2;
:: RFINSEQ:th 2
theorem
for b1, b2, b3 being Relation-like Function-like set
st b1,b2 are_fiberwise_equipotent & b1,b3 are_fiberwise_equipotent
holds b2,b3 are_fiberwise_equipotent;
:: RFINSEQ:th 3
theorem
for b1, b2 being Relation-like Function-like set holds
b1,b2 are_fiberwise_equipotent
iff
ex b3 being Relation-like Function-like set st
proj1 b3 = proj1 b1 & proj2 b3 = proj1 b2 & b3 is one-to-one & b1 = b3 * b2;
:: RFINSEQ:th 4
theorem
for b1, b2 being Relation-like Function-like set holds
b1,b2 are_fiberwise_equipotent
iff
for b3 being set holds
Card (b1 " b3) = Card (b2 " b3);
:: RFINSEQ:th 5
theorem
for b1 being non empty set
for b2, b3 being Relation-like Function-like set
st proj2 b2 c= b1 & proj2 b3 c= b1
holds b2,b3 are_fiberwise_equipotent
iff
for b4 being Element of b1 holds
Card (b2 " {b4}) = Card (b3 " {b4});
:: RFINSEQ:th 6
theorem
for b1, b2 being Relation-like Function-like set
st proj1 b1 = proj1 b2
holds b1,b2 are_fiberwise_equipotent
iff
ex b3 being Function-like quasi_total bijective Relation of proj1 b1,proj1 b1 st
b1 = b3 * b2;
:: RFINSEQ:th 7
theorem
for b1, b2 being Relation-like Function-like set
st b1,b2 are_fiberwise_equipotent
holds Card proj1 b1 = Card proj1 b2;
:: RFINSEQ:th 9
theorem
for b1, b2 being Relation-like Function-like finite set holds
b1,b2 are_fiberwise_equipotent
iff
for b3 being set holds
card (b1 " b3) = card (b2 " b3);
:: RFINSEQ:th 10
theorem
for b1, b2 being Relation-like Function-like finite set
st b1,b2 are_fiberwise_equipotent
holds card proj1 b1 = card proj1 b2;
:: RFINSEQ:th 11
theorem
for b1 being non empty set
for b2, b3 being Relation-like Function-like finite set
st proj2 b2 c= b1 & proj2 b3 c= b1
holds b2,b3 are_fiberwise_equipotent
iff
for b4 being Element of b1 holds
card (b2 " {b4}) = card (b3 " {b4});
:: RFINSEQ:th 13
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set holds
b1,b2 are_fiberwise_equipotent
iff
for b3 being set holds
card (b1 " b3) = card (b2 " b3);
:: RFINSEQ:th 14
theorem
for b1, b2, b3 being Relation-like Function-like FinSequence-like set holds
b1,b2 are_fiberwise_equipotent
iff
b1 ^ b3,b2 ^ b3 are_fiberwise_equipotent;
:: RFINSEQ:th 15
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set holds
b1 ^ b2,b2 ^ b1 are_fiberwise_equipotent;
:: RFINSEQ:th 16
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set
st b1,b2 are_fiberwise_equipotent
holds len b1 = len b2 & dom b1 = dom b2;
:: RFINSEQ:th 17
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set holds
b1,b2 are_fiberwise_equipotent
iff
ex b3 being Function-like quasi_total bijective Relation of dom b2,dom b2 st
b1 = b3 * b2;
:: RFINSEQ:funcreg 1
registration
let a1 be Relation-like Function-like set;
let a2 be finite set;
cluster a1 | a2 -> Relation-like finite;
end;
:: RFINSEQ:th 18
theorem
for b1 being Relation-like Function-like set
for b2 being finite set holds
ex b3 being Relation-like Function-like FinSequence-like set st
b1 | b2,b3 are_fiberwise_equipotent;
:: RFINSEQ:funcnot 1 => RFINSEQ:func 1
definition
let a1 be Relation-like Function-like FinSequence-like set;
let a2 be natural set;
func A1 /^ A2 -> Relation-like Function-like FinSequence-like set means
len it = (len a1) - a2 &
(for b1 being natural set
st b1 in dom it
holds it . b1 = a1 . (b1 + a2))
if a2 <= len a1
otherwise it = {};
end;
:: RFINSEQ:def 2
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being natural set
for b3 being Relation-like Function-like FinSequence-like set holds
(b2 <= len b1 implies (b3 = b1 /^ b2
iff
len b3 = (len b1) - b2 &
(for b4 being natural set
st b4 in dom b3
holds b3 . b4 = b1 . (b4 + b2)))) &
(b2 <= len b1 or (b3 = b1 /^ b2
iff
b3 = {}));
:: RFINSEQ:funcnot 2 => RFINSEQ:func 2
definition
let a1 be set;
let a2 be FinSequence of a1;
let a3 be natural set;
redefine func a2 /^ a3 -> FinSequence of a1;
end;
:: RFINSEQ:th 19
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being natural set
st b3 in dom b2 & b4 in Seg b3
holds (b2 | b3) . b4 = b2 . b4 & b4 in dom b2;
:: RFINSEQ:th 20
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being natural set
for b4 being set
st len b2 = b3 + 1 & b4 = b2 . (b3 + 1)
holds b2 = (b2 | b3) ^ <*b4*>;
:: RFINSEQ:th 21
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being natural set holds
(b2 | b3) ^ (b2 /^ b3) = b2;
:: RFINSEQ:th 22
theorem
for b1, b2 being FinSequence of REAL
st b1,b2 are_fiberwise_equipotent
holds Sum b1 = Sum b2;
:: RFINSEQ:funcnot 3 => RFINSEQ:func 3
definition
let a1 be FinSequence of REAL;
func MIM A1 -> FinSequence of REAL means
len it = len a1 &
it . len it = a1 . len a1 &
(for b1 being natural set
st 1 <= b1 & b1 <= (len it) - 1
holds it . b1 = (a1 . b1) - (a1 . (b1 + 1)));
end;
:: RFINSEQ:def 3
theorem
for b1, b2 being FinSequence of REAL holds
b2 = MIM b1
iff
len b2 = len b1 &
b2 . len b2 = b1 . len b1 &
(for b3 being natural set
st 1 <= b3 & b3 <= (len b2) - 1
holds b2 . b3 = (b1 . b3) - (b1 . (b3 + 1)));
:: RFINSEQ:th 23
theorem
for b1 being FinSequence of REAL
for b2 being Element of REAL
for b3 being Element of NAT
st len b1 = b3 + 2 & b1 . (b3 + 1) = b2
holds MIM (b1 | (b3 + 1)) = ((MIM b1) | b3) ^ <*b2*>;
:: RFINSEQ:th 24
theorem
for b1 being FinSequence of REAL
for b2, b3 being Element of REAL
for b4 being Element of NAT
st len b1 = b4 + 2 & b1 . (b4 + 1) = b2 & b1 . (b4 + 2) = b3
holds MIM b1 = ((MIM b1) | b4) ^ <*b2 - b3,b3*>;
:: RFINSEQ:th 25
theorem
MIM <*> REAL = <*> REAL;
:: RFINSEQ:th 26
theorem
for b1 being Element of REAL holds
MIM <*b1*> = <*b1*>;
:: RFINSEQ:th 27
theorem
for b1, b2 being Element of REAL holds
MIM <*b1,b2*> = <*b1 - b2,b2*>;
:: RFINSEQ:th 28
theorem
for b1 being FinSequence of REAL
for b2 being Element of NAT holds
(MIM b1) /^ b2 = MIM (b1 /^ b2);
:: RFINSEQ:th 29
theorem
for b1 being FinSequence of REAL
st len b1 <> 0
holds Sum MIM b1 = b1 . 1;
:: RFINSEQ:th 30
theorem
for b1 being FinSequence of REAL
for b2 being Element of NAT
st 1 <= b2 & b2 < len b1
holds Sum MIM (b1 /^ b2) = b1 . (b2 + 1);
:: RFINSEQ:attrnot 1 => SEQM_3:attr 4
definition
let a1 be Function-like Relation of NAT,REAL;
attr a1 is non-increasing means
for b1 being Element of NAT
st b1 in dom a1 & b1 + 1 in dom a1
holds a1 . (b1 + 1) <= a1 . b1;
end;
:: RFINSEQ:dfs 4
definiens
let a1 be FinSequence of REAL;
To prove
a1 is non-increasing
it is sufficient to prove
thus for b1 being Element of NAT
st b1 in dom a1 & b1 + 1 in dom a1
holds a1 . (b1 + 1) <= a1 . b1;
:: RFINSEQ:def 4
theorem
for b1 being FinSequence of REAL holds
b1 is non-increasing
iff
for b2 being Element of NAT
st b2 in dom b1 & b2 + 1 in dom b1
holds b1 . (b2 + 1) <= b1 . b2;
:: RFINSEQ:exreg 1
registration
cluster Relation-like Function-like complex-valued ext-real-valued real-valued non-increasing finite FinSequence-like FinSequence of REAL;
end;
:: RFINSEQ:th 31
theorem
for b1 being FinSequence of REAL
st (len b1 = 0 or len b1 = 1)
holds b1 is non-increasing;
:: RFINSEQ:th 32
theorem
for b1 being FinSequence of REAL holds
b1 is non-increasing
iff
for b2, b3 being Element of NAT
st b2 in dom b1 & b3 in dom b1 & b2 < b3
holds b1 . b3 <= b1 . b2;
:: RFINSEQ:th 33
theorem
for b1 being non-increasing FinSequence of REAL
for b2 being Element of NAT holds
b1 | b2 is non-increasing FinSequence of REAL;
:: RFINSEQ:th 34
theorem
for b1 being non-increasing FinSequence of REAL
for b2 being Element of NAT holds
b1 /^ b2 is non-increasing FinSequence of REAL;
:: RFINSEQ:th 35
theorem
for b1 being FinSequence of REAL holds
ex b2 being non-increasing FinSequence of REAL st
b1,b2 are_fiberwise_equipotent;
:: RFINSEQ:th 36
theorem
for b1, b2 being non-increasing FinSequence of REAL
st b1,b2 are_fiberwise_equipotent
holds b1 = b2;
:: RFINSEQ:th 37
theorem
for b1 being FinSequence of REAL
for b2, b3 being Element of REAL
st b2 <> 0
holds b1 " {b3 / b2} = (b2 * b1) " {b3};
:: RFINSEQ:th 38
theorem
for b1 being FinSequence of REAL holds
(0 * b1) " {0} = dom b1;
:: RFINSEQ:th 39
theorem
for b1, b2 being Relation-like Function-like set
st proj2 b1 = proj2 b2 & b1 is one-to-one & b2 is one-to-one
holds b1,b2 are_fiberwise_equipotent;
:: RFINSEQ:th 40
theorem
for b1 being non empty set
for b2 being FinSequence of b1 holds
b2 /^ len b2 = {};
:: RFINSEQ:th 41
theorem
for b1, b2 being Relation-like Function-like set
for b3, b4 being set
st b1 . b3 = b2 . b4 &
b1 . b4 = b2 . b3 &
b3 in proj1 b1 &
b4 in proj1 b1 &
proj1 b1 = proj1 b2 &
(for b5 being set
st b5 <> b3 & b5 <> b4 & b5 in proj1 b1
holds b1 . b5 = b2 . b5)
holds b1,b2 are_fiberwise_equipotent;