Article NUMERAL1, MML version 4.99.1005
:: NUMERAL1:th 1
theorem
for b1, b2 being finite T-Sequence of NAT holds
Sum (b1 ^ b2) = (Sum b1) + Sum b2;
:: NUMERAL1:th 2
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being finite T-Sequence of NAT
for b3 being natural set
st b2 = b1 | (b3 + 1)
holds Sum b2 = (Partial_Sums b1) . b3;
:: NUMERAL1:th 3
theorem
for b1, b2, b3 being natural set holds
(b1 (#) (b2 GeoSeq)) | b3 is finite T-Sequence of NAT;
:: NUMERAL1:th 4
theorem
for b1, b2 being finite T-Sequence of NAT
st 1 <= len b1 &
len b1 = len b2 &
(for b3 being natural set
st b3 in dom b1
holds b1 . b3 <= b2 . b3)
holds Sum b1 <= Sum b2;
:: NUMERAL1:th 5
theorem
for b1 being finite T-Sequence of NAT
for b2 being natural set
st for b3 being natural set
st b3 in dom b1
holds b2 divides b1 . b3
holds b2 divides Sum b1;
:: NUMERAL1:th 6
theorem
for b1, b2 being finite T-Sequence of NAT
for b3 being natural set
st dom b1 = dom b2 &
(for b4 being natural set
st b4 in dom b1
holds b2 . b4 = (b1 . b4) mod b3)
holds (Sum b1) mod b3 = (Sum b2) mod b3;
:: NUMERAL1:funcnot 1 => NUMERAL1:func 1
definition
let a1 be finite T-Sequence of NAT;
let a2 be natural set;
func value(A1,A2) -> natural set means
ex b1 being finite T-Sequence of NAT st
dom b1 = dom a1 &
(for b2 being natural set
st b2 in dom b1
holds b1 . b2 = (a1 . b2) * (a2 |^ b2)) &
it = Sum b1;
end;
:: NUMERAL1:def 1
theorem
for b1 being finite T-Sequence of NAT
for b2, b3 being natural set holds
b3 = value(b1,b2)
iff
ex b4 being finite T-Sequence of NAT st
dom b4 = dom b1 &
(for b5 being natural set
st b5 in dom b4
holds b4 . b5 = (b1 . b5) * (b2 |^ b5)) &
b3 = Sum b4;
:: NUMERAL1:funcnot 2 => NUMERAL1:func 2
definition
let a1, a2 be natural set;
assume 1 < a2;
func digits(A1,A2) -> finite T-Sequence of NAT means
value(it,a2) = a1 &
it . ((len it) - 1) <> 0 &
(for b1 being natural set
st b1 in dom it
holds 0 <= it . b1 & it . b1 < a2)
if a1 <> 0
otherwise it = <%0%>;
end;
:: NUMERAL1:def 2
theorem
for b1, b2 being natural set
st 1 < b2
for b3 being finite T-Sequence of NAT holds
(b1 = 0 or (b3 = digits(b1,b2)
iff
value(b3,b2) = b1 &
b3 . ((len b3) - 1) <> 0 &
(for b4 being natural set
st b4 in dom b3
holds 0 <= b3 . b4 & b3 . b4 < b2))) &
(b1 = 0 implies (b3 = digits(b1,b2)
iff
b3 = <%0%>));
:: NUMERAL1:th 7
theorem
for b1, b2 being natural set
st 1 < b2
holds 1 <= len digits(b1,b2);
:: NUMERAL1:th 8
theorem
for b1, b2 being natural set
st 1 < b2
holds value(digits(b1,b2),b2) = b1;
:: NUMERAL1:th 9
theorem
for b1, b2 being natural set
st b2 = (10 |^ b1) - 1
holds 9 divides b2;
:: NUMERAL1:th 10
theorem
for b1, b2 being natural set
st 1 < b2
holds b2 divides b1
iff
(digits(b1,b2)) . 0 = 0;
:: NUMERAL1:th 11
theorem
for b1 being natural set holds
2 divides b1
iff
2 divides (digits(b1,10)) . 0;
:: NUMERAL1:th 12
theorem
for b1 being natural set holds
3 divides b1
iff
3 divides Sum digits(b1,10);
:: NUMERAL1:th 13
theorem
for b1 being natural set holds
5 divides b1
iff
5 divides (digits(b1,10)) . 0;