Article RFUNCT_4, MML version 4.99.1005
:: RFUNCT_4:th 1
theorem
for b1, b2 being real set holds
min(b1,b2) <= max(b1,b2);
:: RFUNCT_4:th 2
theorem
for b1 being Element of NAT
for b2, b3 being Element of b1 -tuples_on REAL
for b4, b5 being Element of REAL holds
mlt(b2 ^ <*b4*>,b3 ^ <*b5*>) = (mlt(b2,b3)) ^ <*b4 * b5*>;
:: RFUNCT_4:th 3
theorem
for b1 being Element of NAT
for b2 being Element of b1 -tuples_on REAL
st Sum b2 = 0 &
(for b3 being Element of NAT
st b3 in dom b2
holds 0 <= b2 . b3)
for b3 being Element of NAT
st b3 in dom b2
holds b2 . b3 = 0;
:: RFUNCT_4:th 4
theorem
for b1 being Element of NAT
for b2 being Element of b1 -tuples_on REAL
st for b3 being Element of NAT
st b3 in dom b2
holds 0 = b2 . b3
holds b2 = b1 |-> 0;
:: RFUNCT_4:th 5
theorem
for b1 being Element of NAT
for b2 being Element of b1 -tuples_on REAL holds
mlt(b1 |-> 0,b2) = b1 |-> 0;
:: RFUNCT_4:prednot 1 => RFUNCT_4:pred 1
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
pred A1 is_strictly_convex_on A2 means
a2 c= proj1 a1 &
(for b1 being Element of REAL
st 0 < b1 & b1 < 1
for b2, b3 being Element of REAL
st b2 in a2 &
b3 in a2 &
(b1 * b2) + ((1 - b1) * b3) in a2 &
b2 <> b3
holds a1 . ((b1 * b2) + ((1 - b1) * b3)) < (b1 * (a1 . b2)) + ((1 - b1) * (a1 . b3)));
end;
:: RFUNCT_4:dfs 1
definiens
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
To prove
a1 is_strictly_convex_on a2
it is sufficient to prove
thus a2 c= proj1 a1 &
(for b1 being Element of REAL
st 0 < b1 & b1 < 1
for b2, b3 being Element of REAL
st b2 in a2 &
b3 in a2 &
(b1 * b2) + ((1 - b1) * b3) in a2 &
b2 <> b3
holds a1 . ((b1 * b2) + ((1 - b1) * b3)) < (b1 * (a1 . b2)) + ((1 - b1) * (a1 . b3)));
:: RFUNCT_4:def 1
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set holds
b1 is_strictly_convex_on b2
iff
b2 c= proj1 b1 &
(for b3 being Element of REAL
st 0 < b3 & b3 < 1
for b4, b5 being Element of REAL
st b4 in b2 &
b5 in b2 &
(b3 * b4) + ((1 - b3) * b5) in b2 &
b4 <> b5
holds b1 . ((b3 * b4) + ((1 - b3) * b5)) < (b3 * (b1 . b4)) + ((1 - b3) * (b1 . b5)));
:: RFUNCT_4:th 6
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set
st b1 is_strictly_convex_on b2
holds b1 is_convex_on b2;
:: RFUNCT_4:th 7
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL holds
b3 is_strictly_convex_on [.b1,b2.]
iff
[.b1,b2.] c= proj1 b3 &
(for b4 being Element of REAL
st 0 < b4 & b4 < 1
for b5, b6 being Element of REAL
st b5 in [.b1,b2.] & b6 in [.b1,b2.] & b5 <> b6
holds b3 . ((b4 * b5) + ((1 - b4) * b6)) < (b4 * (b3 . b5)) + ((1 - b4) * (b3 . b6)));
:: RFUNCT_4:th 8
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL holds
b2 is_convex_on b1
iff
b1 c= proj1 b2 &
(for b3, b4, b5 being Element of REAL
st b3 in b1 & b4 in b1 & b5 in b1 & b3 < b4 & b4 < b5
holds b2 . b4 <= (((b5 - b4) / (b5 - b3)) * (b2 . b3)) + (((b4 - b3) / (b5 - b3)) * (b2 . b5)));
:: RFUNCT_4:th 9
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL holds
b2 is_strictly_convex_on b1
iff
b1 c= proj1 b2 &
(for b3, b4, b5 being Element of REAL
st b3 in b1 & b4 in b1 & b5 in b1 & b3 < b4 & b4 < b5
holds b2 . b4 < (((b5 - b4) / (b5 - b3)) * (b2 . b3)) + (((b4 - b3) / (b5 - b3)) * (b2 . b5)));
:: RFUNCT_4:th 10
theorem
for b1 being Function-like Relation of REAL,REAL
for b2, b3 being set
st b1 is_strictly_convex_on b2 & b3 c= b2
holds b1 is_strictly_convex_on b3;
:: RFUNCT_4:th 11
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
for b3 being set holds
b2 is_strictly_convex_on b3
iff
b2 - b1 is_strictly_convex_on b3;
:: RFUNCT_4:th 12
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
for b3 being set
st 0 < b1
holds b2 is_strictly_convex_on b3
iff
b1 (#) b2 is_strictly_convex_on b3;
:: RFUNCT_4:th 13
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being set
st b1 is_strictly_convex_on b3 & b2 is_strictly_convex_on b3
holds b1 + b2 is_strictly_convex_on b3;
:: RFUNCT_4:th 14
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being set
st b1 is_strictly_convex_on b3 & b2 is_convex_on b3
holds b1 + b2 is_strictly_convex_on b3;
:: RFUNCT_4:th 15
theorem
for b1, b2 being Element of REAL
for b3, b4 being Function-like Relation of REAL,REAL
for b5 being set
st b3 is_strictly_convex_on b5 &
b4 is_strictly_convex_on b5 &
(0 < b1 & 0 <= b2 or 0 <= b1 & 0 < b2)
holds (b1 (#) b3) + (b2 (#) b4) is_strictly_convex_on b5;
:: RFUNCT_4:th 16
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set holds
b1 is_convex_on b2
iff
b2 c= proj1 b1 &
(for b3, b4, b5 being Element of REAL
st b3 in b2 & b4 in b2 & b5 in b2 & b3 < b5 & b5 < b4
holds ((b1 . b5) - (b1 . b3)) / (b5 - b3) <= ((b1 . b4) - (b1 . b3)) / (b4 - b3) &
((b1 . b4) - (b1 . b3)) / (b4 - b3) <= ((b1 . b4) - (b1 . b5)) / (b4 - b5));
:: RFUNCT_4:th 17
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set holds
b1 is_strictly_convex_on b2
iff
b2 c= proj1 b1 &
(for b3, b4, b5 being Element of REAL
st b3 in b2 & b4 in b2 & b5 in b2 & b3 < b5 & b5 < b4
holds ((b1 . b5) - (b1 . b3)) / (b5 - b3) < ((b1 . b4) - (b1 . b3)) / (b4 - b3) &
((b1 . b4) - (b1 . b3)) / (b4 - b3) < ((b1 . b4) - (b1 . b5)) / (b4 - b5));
:: RFUNCT_4:th 18
theorem
for b1 being Function-like Relation of REAL,REAL
st b1 is total(REAL, REAL)
holds for b2 being Element of NAT
for b3, b4, b5 being Element of b2 -tuples_on REAL
st Sum b3 = 1 &
(for b6 being Element of NAT
st b6 in dom b3
holds 0 <= b3 . b6 & b5 . b6 = b1 . (b4 . b6))
holds b1 . Sum mlt(b3,b4) <= Sum mlt(b3,b5)
iff
b1 is_convex_on REAL;
:: RFUNCT_4:th 19
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being interval Element of bool REAL
for b3 being Element of REAL
st (ex b4, b5 being Element of REAL st
b4 in b2 & b5 in b2 & b4 < b3 & b3 < b5) &
b1 is_convex_on b2
holds b1 is_continuous_in b3;
:: RFUNCT_4:prednot 2 => RFUNCT_4:pred 2
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
pred A1 is_quasiconvex_on A2 means
a2 c= proj1 a1 &
(for b1 being Element of REAL
st 0 < b1 & b1 < 1
for b2, b3 being Element of REAL
st b2 in a2 &
b3 in a2 &
(b1 * b2) + ((1 - b1) * b3) in a2
holds a1 . ((b1 * b2) + ((1 - b1) * b3)) <= max(a1 . b2,a1 . b3));
end;
:: RFUNCT_4:dfs 2
definiens
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
To prove
a1 is_quasiconvex_on a2
it is sufficient to prove
thus a2 c= proj1 a1 &
(for b1 being Element of REAL
st 0 < b1 & b1 < 1
for b2, b3 being Element of REAL
st b2 in a2 &
b3 in a2 &
(b1 * b2) + ((1 - b1) * b3) in a2
holds a1 . ((b1 * b2) + ((1 - b1) * b3)) <= max(a1 . b2,a1 . b3));
:: RFUNCT_4:def 2
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set holds
b1 is_quasiconvex_on b2
iff
b2 c= proj1 b1 &
(for b3 being Element of REAL
st 0 < b3 & b3 < 1
for b4, b5 being Element of REAL
st b4 in b2 &
b5 in b2 &
(b3 * b4) + ((1 - b3) * b5) in b2
holds b1 . ((b3 * b4) + ((1 - b3) * b5)) <= max(b1 . b4,b1 . b5));
:: RFUNCT_4:prednot 3 => RFUNCT_4:pred 3
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
pred A1 is_strictly_quasiconvex_on A2 means
a2 c= proj1 a1 &
(for b1 being Element of REAL
st 0 < b1 & b1 < 1
for b2, b3 being Element of REAL
st b2 in a2 &
b3 in a2 &
(b1 * b2) + ((1 - b1) * b3) in a2 &
a1 . b2 <> a1 . b3
holds a1 . ((b1 * b2) + ((1 - b1) * b3)) < max(a1 . b2,a1 . b3));
end;
:: RFUNCT_4:dfs 3
definiens
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
To prove
a1 is_strictly_quasiconvex_on a2
it is sufficient to prove
thus a2 c= proj1 a1 &
(for b1 being Element of REAL
st 0 < b1 & b1 < 1
for b2, b3 being Element of REAL
st b2 in a2 &
b3 in a2 &
(b1 * b2) + ((1 - b1) * b3) in a2 &
a1 . b2 <> a1 . b3
holds a1 . ((b1 * b2) + ((1 - b1) * b3)) < max(a1 . b2,a1 . b3));
:: RFUNCT_4:def 3
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set holds
b1 is_strictly_quasiconvex_on b2
iff
b2 c= proj1 b1 &
(for b3 being Element of REAL
st 0 < b3 & b3 < 1
for b4, b5 being Element of REAL
st b4 in b2 &
b5 in b2 &
(b3 * b4) + ((1 - b3) * b5) in b2 &
b1 . b4 <> b1 . b5
holds b1 . ((b3 * b4) + ((1 - b3) * b5)) < max(b1 . b4,b1 . b5));
:: RFUNCT_4:prednot 4 => RFUNCT_4:pred 4
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
pred A1 is_strongly_quasiconvex_on A2 means
a2 c= proj1 a1 &
(for b1 being Element of REAL
st 0 < b1 & b1 < 1
for b2, b3 being Element of REAL
st b2 in a2 &
b3 in a2 &
(b1 * b2) + ((1 - b1) * b3) in a2 &
b2 <> b3
holds a1 . ((b1 * b2) + ((1 - b1) * b3)) < max(a1 . b2,a1 . b3));
end;
:: RFUNCT_4:dfs 4
definiens
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
To prove
a1 is_strongly_quasiconvex_on a2
it is sufficient to prove
thus a2 c= proj1 a1 &
(for b1 being Element of REAL
st 0 < b1 & b1 < 1
for b2, b3 being Element of REAL
st b2 in a2 &
b3 in a2 &
(b1 * b2) + ((1 - b1) * b3) in a2 &
b2 <> b3
holds a1 . ((b1 * b2) + ((1 - b1) * b3)) < max(a1 . b2,a1 . b3));
:: RFUNCT_4:def 4
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set holds
b1 is_strongly_quasiconvex_on b2
iff
b2 c= proj1 b1 &
(for b3 being Element of REAL
st 0 < b3 & b3 < 1
for b4, b5 being Element of REAL
st b4 in b2 &
b5 in b2 &
(b3 * b4) + ((1 - b3) * b5) in b2 &
b4 <> b5
holds b1 . ((b3 * b4) + ((1 - b3) * b5)) < max(b1 . b4,b1 . b5));
:: RFUNCT_4:prednot 5 => RFUNCT_4:pred 5
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be real set;
pred A1 is_upper_semicontinuous_in A2 means
a2 in proj1 a1 &
(for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of REAL st
0 < b2 &
(for b3 being Element of REAL
st b3 in proj1 a1 & abs (b3 - a2) < b2
holds (a1 . a2) - (a1 . b3) < b1));
end;
:: RFUNCT_4:dfs 5
definiens
let a1 be Function-like Relation of REAL,REAL;
let a2 be real set;
To prove
a1 is_upper_semicontinuous_in a2
it is sufficient to prove
thus a2 in proj1 a1 &
(for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of REAL st
0 < b2 &
(for b3 being Element of REAL
st b3 in proj1 a1 & abs (b3 - a2) < b2
holds (a1 . a2) - (a1 . b3) < b1));
:: RFUNCT_4:def 5
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being real set holds
b1 is_upper_semicontinuous_in b2
iff
b2 in proj1 b1 &
(for b3 being Element of REAL
st 0 < b3
holds ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in proj1 b1 & abs (b5 - b2) < b4
holds (b1 . b2) - (b1 . b5) < b3));
:: RFUNCT_4:prednot 6 => RFUNCT_4:pred 6
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
pred A1 is_upper_semicontinuous_on A2 means
a2 c= proj1 a1 &
(for b1 being Element of REAL
st b1 in a2
holds a1 | a2 is_upper_semicontinuous_in b1);
end;
:: RFUNCT_4:dfs 6
definiens
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
To prove
a1 is_upper_semicontinuous_on a2
it is sufficient to prove
thus a2 c= proj1 a1 &
(for b1 being Element of REAL
st b1 in a2
holds a1 | a2 is_upper_semicontinuous_in b1);
:: RFUNCT_4:def 6
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set holds
b1 is_upper_semicontinuous_on b2
iff
b2 c= proj1 b1 &
(for b3 being Element of REAL
st b3 in b2
holds b1 | b2 is_upper_semicontinuous_in b3);
:: RFUNCT_4:prednot 7 => RFUNCT_4:pred 7
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be real set;
pred A1 is_lower_semicontinuous_in A2 means
a2 in proj1 a1 &
(for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of REAL st
0 < b2 &
(for b3 being Element of REAL
st b3 in proj1 a1 & abs (b3 - a2) < b2
holds (a1 . b3) - (a1 . a2) < b1));
end;
:: RFUNCT_4:dfs 7
definiens
let a1 be Function-like Relation of REAL,REAL;
let a2 be real set;
To prove
a1 is_lower_semicontinuous_in a2
it is sufficient to prove
thus a2 in proj1 a1 &
(for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of REAL st
0 < b2 &
(for b3 being Element of REAL
st b3 in proj1 a1 & abs (b3 - a2) < b2
holds (a1 . b3) - (a1 . a2) < b1));
:: RFUNCT_4:def 7
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being real set holds
b1 is_lower_semicontinuous_in b2
iff
b2 in proj1 b1 &
(for b3 being Element of REAL
st 0 < b3
holds ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in proj1 b1 & abs (b5 - b2) < b4
holds (b1 . b5) - (b1 . b2) < b3));
:: RFUNCT_4:prednot 8 => RFUNCT_4:pred 8
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
pred A1 is_lower_semicontinuous_on A2 means
a2 c= proj1 a1 &
(for b1 being Element of REAL
st b1 in a2
holds a1 | a2 is_lower_semicontinuous_in b1);
end;
:: RFUNCT_4:dfs 8
definiens
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
To prove
a1 is_lower_semicontinuous_on a2
it is sufficient to prove
thus a2 c= proj1 a1 &
(for b1 being Element of REAL
st b1 in a2
holds a1 | a2 is_lower_semicontinuous_in b1);
:: RFUNCT_4:def 8
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set holds
b1 is_lower_semicontinuous_on b2
iff
b2 c= proj1 b1 &
(for b3 being Element of REAL
st b3 in b2
holds b1 | b2 is_lower_semicontinuous_in b3);
:: RFUNCT_4:th 20
theorem
for b1 being real set
for b2 being Function-like Relation of REAL,REAL holds
b2 is_upper_semicontinuous_in b1 & b2 is_lower_semicontinuous_in b1
iff
b2 is_continuous_in b1;
:: RFUNCT_4:th 21
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL holds
b2 is_upper_semicontinuous_on b1 & b2 is_lower_semicontinuous_on b1
iff
b2 is_continuous_on b1;
:: RFUNCT_4:th 22
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
st b2 is_strictly_convex_on b1
holds b2 is_strongly_quasiconvex_on b1;
:: RFUNCT_4:th 23
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
st b2 is_strongly_quasiconvex_on b1
holds b2 is_quasiconvex_on b1;
:: RFUNCT_4:th 24
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
st b2 is_convex_on b1
holds b2 is_strictly_quasiconvex_on b1;
:: RFUNCT_4:th 25
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
st b2 is_strongly_quasiconvex_on b1
holds b2 is_strictly_quasiconvex_on b1;
:: RFUNCT_4:th 26
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
st b2 is_strictly_quasiconvex_on b1 & b2 is one-to-one
holds b2 is_strongly_quasiconvex_on b1;