Article ALGSEQ_1, MML version 4.99.1005
:: ALGSEQ_1:th 10
theorem
for b1, b2 being natural set holds
b1 in Segm b2
iff
b1 < b2;
:: ALGSEQ_1:th 11
theorem
Segm 0 = {} & Segm 1 = {0} & Segm 2 = {0,1};
:: ALGSEQ_1:th 12
theorem
for b1 being natural set holds
b1 in Segm (b1 + 1);
:: ALGSEQ_1:th 13
theorem
for b1, b2 being natural set holds
b1 <= b2
iff
Segm b1 c= Segm b2;
:: ALGSEQ_1:th 14
theorem
for b1, b2 being natural set
st Segm b1 = Segm b2
holds b1 = b2;
:: ALGSEQ_1:th 15
theorem
for b1, b2 being natural set
st b1 <= b2
holds Segm b1 = (Segm b1) /\ Segm b2 & Segm b1 = (Segm b2) /\ Segm b1;
:: ALGSEQ_1:th 16
theorem
for b1, b2 being natural set
st (Segm b1 = (Segm b1) /\ Segm b2 or Segm b1 = (Segm b2) /\ Segm b1)
holds b1 <= b2;
:: ALGSEQ_1:attrnot 1 => ALGSEQ_1:attr 1
definition
let a1 be non empty ZeroStr;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
attr a2 is finite-Support means
ex b1 being natural set st
for b2 being natural set
st b1 <= b2
holds a2 . b2 = 0. a1;
end;
:: ALGSEQ_1:dfs 1
definiens
let a1 be non empty ZeroStr;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
a2 is finite-Support
it is sufficient to prove
thus ex b1 being natural set st
for b2 being natural set
st b1 <= b2
holds a2 . b2 = 0. a1;
:: ALGSEQ_1:def 2
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is finite-Support(b1)
iff
ex b3 being natural set st
for b4 being natural set
st b3 <= b4
holds b2 . b4 = 0. b1;
:: ALGSEQ_1:exreg 1
registration
let a1 be non empty ZeroStr;
cluster Relation-like Function-like quasi_total finite-Support Relation of NAT,the carrier of a1;
end;
:: ALGSEQ_1:modenot 1
definition
let a1 be non empty ZeroStr;
mode AlgSequence of a1 is Function-like quasi_total finite-Support Relation of NAT,the carrier of a1;
end;
:: ALGSEQ_1:prednot 1 => ALGSEQ_1:pred 1
definition
let a1 be non empty ZeroStr;
let a2 be Function-like quasi_total finite-Support Relation of NAT,the carrier of a1;
let a3 be natural set;
pred A3 is_at_least_length_of A2 means
for b1 being natural set
st a3 <= b1
holds a2 . b1 = 0. a1;
end;
:: ALGSEQ_1:dfs 2
definiens
let a1 be non empty ZeroStr;
let a2 be Function-like quasi_total finite-Support Relation of NAT,the carrier of a1;
let a3 be natural set;
To prove
a3 is_at_least_length_of a2
it is sufficient to prove
thus for b1 being natural set
st a3 <= b1
holds a2 . b1 = 0. a1;
:: ALGSEQ_1:def 3
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b3 being natural set holds
b3 is_at_least_length_of b2
iff
for b4 being natural set
st b3 <= b4
holds b2 . b4 = 0. b1;
:: ALGSEQ_1:funcnot 1 => ALGSEQ_1:func 1
definition
let a1 be non empty ZeroStr;
let a2 be Function-like quasi_total finite-Support Relation of NAT,the carrier of a1;
func len A2 -> Element of NAT means
it is_at_least_length_of a2 &
(for b1 being natural set
st b1 is_at_least_length_of a2
holds it <= b1);
end;
:: ALGSEQ_1:def 4
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b3 being Element of NAT holds
b3 = len b2
iff
b3 is_at_least_length_of b2 &
(for b4 being natural set
st b4 is_at_least_length_of b2
holds b3 <= b4);
:: ALGSEQ_1:th 22
theorem
for b1 being natural set
for b2 being non empty ZeroStr
for b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b2
st len b3 <= b1
holds b3 . b1 = 0. b2;
:: ALGSEQ_1:th 24
theorem
for b1 being natural set
for b2 being non empty ZeroStr
for b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b2
st for b4 being natural set
st b4 < b1
holds b3 . b4 <> 0. b2
holds b1 <= len b3;
:: ALGSEQ_1:th 25
theorem
for b1 being natural set
for b2 being non empty ZeroStr
for b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b2
st len b3 = b1 + 1
holds b3 . b1 <> 0. b2;
:: ALGSEQ_1:funcnot 2 => ALGSEQ_1:func 2
definition
let a1 be non empty ZeroStr;
let a2 be Function-like quasi_total finite-Support Relation of NAT,the carrier of a1;
func support A2 -> Element of bool NAT equals
Segm len a2;
end;
:: ALGSEQ_1:def 5
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
support b2 = Segm len b2;
:: ALGSEQ_1:th 27
theorem
for b1 being natural set
for b2 being non empty ZeroStr
for b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b2 holds
b1 = len b3
iff
Segm b1 = support b3;
:: ALGSEQ_1:sch 1
scheme ALGSEQ_1:sch 1
{F1 -> non empty ZeroStr,
F2 -> natural set,
F3 -> Element of the carrier of F1()}:
ex b1 being Function-like quasi_total finite-Support Relation of NAT,the carrier of F1() st
len b1 <= F2() &
(for b2 being natural set
st b2 < F2()
holds b1 . b2 = F3(b2))
:: ALGSEQ_1:th 28
theorem
for b1 being non empty ZeroStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
st len b2 = len b3 &
(for b4 being natural set
st b4 < len b2
holds b2 . b4 = b3 . b4)
holds b2 = b3;
:: ALGSEQ_1:th 29
theorem
for b1 being non empty ZeroStr
st the carrier of b1 <> {0. b1}
for b2 being natural set holds
ex b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 st
len b3 = b2;
:: ALGSEQ_1:funcnot 3 => ALGSEQ_1:func 3
definition
let a1 be non empty ZeroStr;
let a2 be Element of the carrier of a1;
func <%A2%> -> Function-like quasi_total finite-Support Relation of NAT,the carrier of a1 means
len it <= 1 & it . 0 = a2;
end;
:: ALGSEQ_1:def 6
theorem
for b1 being non empty ZeroStr
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
b3 = <%b2%>
iff
len b3 <= 1 & b3 . 0 = b2;
:: ALGSEQ_1:th 31
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
b2 = <%0. b1%>
iff
len b2 = 0;
:: ALGSEQ_1:th 32
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
b2 = <%0. b1%>
iff
support b2 = {};
:: ALGSEQ_1:th 33
theorem
for b1 being natural set
for b2 being non empty ZeroStr holds
<%0. b2%> . b1 = 0. b2;
:: ALGSEQ_1:th 34
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
b2 = <%0. b1%>
iff
proj2 b2 = {0. b1};