Article PRGCOR_2, MML version 4.99.1005
:: PRGCOR_2:th 1
theorem
for b1, b2 being natural set holds
b1 in b2
iff
b1 < b2;
:: PRGCOR_2:exreg 1
registration
let a1 be non empty set;
cluster Relation-like Function-like non empty T-Sequence-like finite T-Sequence of a1;
end;
:: PRGCOR_2:th 2
theorem
for b1 being non empty set
for b2 being non empty finite T-Sequence of b1 holds
0 < len b2;
:: PRGCOR_2:funcnot 1 => PRGCOR_2:func 1
definition
let a1 be set;
let a2 be FinSequence of a1;
func FS2XFS A2 -> finite T-Sequence of a1 means
len it = len a2 &
(for b1 being natural set
st b1 < len a2
holds a2 . (b1 + 1) = it . b1);
end;
:: PRGCOR_2:def 1
theorem
for b1 being set
for b2 being FinSequence of b1
for b3 being finite T-Sequence of b1 holds
b3 = FS2XFS b2
iff
len b3 = len b2 &
(for b4 being natural set
st b4 < len b2
holds b2 . (b4 + 1) = b3 . b4);
:: PRGCOR_2:funcnot 2 => PRGCOR_2:func 2
definition
let a1 be set;
let a2 be finite T-Sequence of a1;
func XFS2FS A2 -> FinSequence of a1 means
len it = len a2 &
(for b1 being natural set
st 1 <= b1 & b1 <= len a2
holds a2 . (b1 -' 1) = it . b1);
end;
:: PRGCOR_2:def 2
theorem
for b1 being set
for b2 being finite T-Sequence of b1
for b3 being FinSequence of b1 holds
b3 = XFS2FS b2
iff
len b3 = len b2 &
(for b4 being natural set
st 1 <= b4 & b4 <= len b2
holds b2 . (b4 -' 1) = b3 . b4);
:: PRGCOR_2:th 3
theorem
for b1 being natural set
for b2 being set holds
b1 --> b2 is Relation-like Function-like T-Sequence-like finite set;
:: PRGCOR_2:th 4
theorem
for b1 being set
for b2 being natural set
for b3 being set
st b3 in b1
holds b2 --> b3 is finite T-Sequence of b1 &
(for b4 being Relation-like Function-like T-Sequence-like finite set
st b4 = b2 --> b3
holds len b4 = b2);
:: PRGCOR_2:funcnot 3 => PRGCOR_2:func 3
definition
let a1 be non empty set;
let a2 be FinSequence of a1;
let a3 be natural set;
assume len a2 < a3 & NAT c= a1;
func FS2XFS*(A2,A3) -> non empty finite T-Sequence of a1 means
len a2 = it . 0 &
len it = a3 &
(for b1 being natural set
st 1 <= b1 & b1 <= len a2
holds it . b1 = a2 . b1) &
(for b1 being natural set
st len a2 < b1 & b1 < a3
holds it . b1 = 0);
end;
:: PRGCOR_2:def 3
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being natural set
st len b2 < b3 & NAT c= b1
for b4 being non empty finite T-Sequence of b1 holds
b4 = FS2XFS*(b2,b3)
iff
len b2 = b4 . 0 &
len b4 = b3 &
(for b5 being natural set
st 1 <= b5 & b5 <= len b2
holds b4 . b5 = b2 . b5) &
(for b5 being natural set
st len b2 < b5 & b5 < b3
holds b4 . b5 = 0);
:: PRGCOR_2:funcnot 4 => PRGCOR_2:func 4
definition
let a1 be non empty set;
let a2 be non empty finite T-Sequence of a1;
assume NAT c= a1 & a2 . 0 is natural set & a2 . 0 in len a2;
func XFS2FS* A2 -> FinSequence of a1 means
for b1 being natural set
st b1 = a2 . 0
holds len it = b1 &
(for b2 being natural set
st 1 <= b2 & b2 <= b1
holds it . b2 = a2 . b2);
end;
:: PRGCOR_2:def 4
theorem
for b1 being non empty set
for b2 being non empty finite T-Sequence of b1
st NAT c= b1 & b2 . 0 is natural set & b2 . 0 in len b2
for b3 being FinSequence of b1 holds
b3 = XFS2FS* b2
iff
for b4 being natural set
st b4 = b2 . 0
holds len b3 = b4 &
(for b5 being natural set
st 1 <= b5 & b5 <= b4
holds b3 . b5 = b2 . b5);
:: PRGCOR_2:th 5
theorem
for b1 being non empty set
for b2 being non empty finite T-Sequence of b1
st NAT c= b1 & b2 . 0 = 0 & 0 < len b2
holds XFS2FS* b2 = {};
:: PRGCOR_2:prednot 1 => PRGCOR_2:pred 1
definition
let a1 be non empty set;
let a2 be finite T-Sequence of a1;
let a3 be FinSequence of a1;
pred A2 is_an_xrep_of A3 means
NAT c= a1 &
a2 . 0 = len a3 &
len a3 < len a2 &
(for b1 being natural set
st 1 <= b1 & b1 <= len a3
holds a2 . b1 = a3 . b1);
end;
:: PRGCOR_2:dfs 5
definiens
let a1 be non empty set;
let a2 be finite T-Sequence of a1;
let a3 be FinSequence of a1;
To prove
a2 is_an_xrep_of a3
it is sufficient to prove
thus NAT c= a1 &
a2 . 0 = len a3 &
len a3 < len a2 &
(for b1 being natural set
st 1 <= b1 & b1 <= len a3
holds a2 . b1 = a3 . b1);
:: PRGCOR_2:def 5
theorem
for b1 being non empty set
for b2 being finite T-Sequence of b1
for b3 being FinSequence of b1 holds
b2 is_an_xrep_of b3
iff
NAT c= b1 &
b2 . 0 = len b3 &
len b3 < len b2 &
(for b4 being natural set
st 1 <= b4 & b4 <= len b3
holds b2 . b4 = b3 . b4);
:: PRGCOR_2:th 6
theorem
for b1 being non empty set
for b2 being non empty finite T-Sequence of b1
st NAT c= b1 & b2 . 0 is natural set & b2 . 0 in len b2
holds b2 is_an_xrep_of XFS2FS* b2;
:: PRGCOR_2:funcnot 5 => PRGCOR_2:func 5
definition
let a1, a2, a3, a4, a5 be set;
func IFLGT(A1,A2,A3,A4,A5) -> set equals
a3
if a1 in a2,
a4
if a1 = a2
otherwise a5;
end;
:: PRGCOR_2:def 6
theorem
for b1, b2, b3, b4, b5 being set holds
(b1 in b2 implies IFLGT(b1,b2,b3,b4,b5) = b3) &
(b1 = b2 implies IFLGT(b1,b2,b3,b4,b5) = b4) &
(not b1 in b2 & b1 <> b2 implies IFLGT(b1,b2,b3,b4,b5) = b5);
:: PRGCOR_2:th 7
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being natural set
st NAT c= b1 & len b2 < b3
holds ex b4 being finite T-Sequence of b1 st
len b4 = b3 & b4 is_an_xrep_of b2;
:: PRGCOR_2:funcnot 6 => PRGCOR_2:func 6
definition
let a1, a2 be finite T-Sequence of REAL;
assume a2 . 0 is natural set & 0 <= a2 . 0 & a2 . 0 < len a1;
func inner_prd_prg(A1,A2) -> Element of REAL means
ex b1 being finite T-Sequence of REAL st
ex b2 being integer set st
len b1 = len a1 &
b1 . 0 = 0 &
b2 = a2 . 0 &
(b2 = 0 or for b3 being natural set
st b3 < b2
holds b1 . (b3 + 1) = (b1 . b3) + ((a1 . (b3 + 1)) * (a2 . (b3 + 1)))) &
it = b1 . b2;
end;
:: PRGCOR_2:def 7
theorem
for b1, b2 being finite T-Sequence of REAL
st b2 . 0 is natural set & 0 <= b2 . 0 & b2 . 0 < len b1
for b3 being Element of REAL holds
b3 = inner_prd_prg(b1,b2)
iff
ex b4 being finite T-Sequence of REAL st
ex b5 being integer set st
len b4 = len b1 &
b4 . 0 = 0 &
b5 = b2 . 0 &
(b5 = 0 or for b6 being natural set
st b6 < b5
holds b4 . (b6 + 1) = (b4 . b6) + ((b1 . (b6 + 1)) * (b2 . (b6 + 1)))) &
b3 = b4 . b5;
:: PRGCOR_2:th 8
theorem
for b1 being FinSequence of REAL
for b2 being finite T-Sequence of REAL
st len b1 < len b2 &
b2 . 0 = 0 &
(for b3 being natural set
st b3 < len b1
holds b2 . (b3 + 1) = (b2 . b3) + (b1 . (b3 + 1)))
holds Sum b1 = b2 . len b1;
:: PRGCOR_2:th 9
theorem
for b1 being FinSequence of REAL holds
ex b2 being finite T-Sequence of REAL st
len b2 = (len b1) + 1 &
b2 . 0 = 0 &
(for b3 being natural set
st b3 < len b1
holds b2 . (b3 + 1) = (b2 . b3) + (b1 . (b3 + 1))) &
Sum b1 = b2 . len b1;
:: PRGCOR_2:th 10
theorem
for b1, b2 being FinSequence of REAL
for b3 being natural set
st len b1 = len b2 & len b1 < b3
holds |(b1,b2)| = inner_prd_prg(FS2XFS*(b1,b3),FS2XFS*(b2,b3));
:: PRGCOR_2:prednot 2 => PRGCOR_2:pred 2
definition
let a1, a2 be finite T-Sequence of REAL;
let a3 be Element of REAL;
let a4 be integer set;
pred A4 scalar_prd_prg A2,A3,A1 means
len a2 = a4 &
len a1 = a4 &
(ex b1 being integer set st
a2 . 0 = a1 . 0 &
b1 = a1 . 0 &
(b1 = 0 or for b2 being natural set
st 1 <= b2 & b2 <= b1
holds a2 . b2 = a3 * (a1 . b2)));
end;
:: PRGCOR_2:dfs 8
definiens
let a1, a2 be finite T-Sequence of REAL;
let a3 be Element of REAL;
let a4 be integer set;
To prove
a4 scalar_prd_prg a2,a3,a1
it is sufficient to prove
thus len a2 = a4 &
len a1 = a4 &
(ex b1 being integer set st
a2 . 0 = a1 . 0 &
b1 = a1 . 0 &
(b1 = 0 or for b2 being natural set
st 1 <= b2 & b2 <= b1
holds a2 . b2 = a3 * (a1 . b2)));
:: PRGCOR_2:def 8
theorem
for b1, b2 being finite T-Sequence of REAL
for b3 being Element of REAL
for b4 being integer set holds
b4 scalar_prd_prg b2,b3,b1
iff
len b2 = b4 &
len b1 = b4 &
(ex b5 being integer set st
b2 . 0 = b1 . 0 &
b5 = b1 . 0 &
(b5 = 0 or for b6 being natural set
st 1 <= b6 & b6 <= b5
holds b2 . b6 = b3 * (b1 . b6)));
:: PRGCOR_2:th 11
theorem
for b1 being non empty finite T-Sequence of REAL
for b2 being Element of REAL
for b3 being natural set
st b1 . 0 is natural set & len b1 = b3 & 0 <= b1 . 0 & b1 . 0 < b3
holds (ex b4 being finite T-Sequence of REAL st
b3 scalar_prd_prg b4,b2,b1) &
(for b4 being non empty finite T-Sequence of REAL
st b3 scalar_prd_prg b4,b2,b1
holds XFS2FS* b4 = b2 * XFS2FS* b1);
:: PRGCOR_2:prednot 3 => PRGCOR_2:pred 3
definition
let a1, a2 be finite T-Sequence of REAL;
let a3 be integer set;
pred A3 vector_minus_prg A2,A1 means
len a2 = a3 &
len a1 = a3 &
(ex b1 being integer set st
a2 . 0 = a1 . 0 &
b1 = a1 . 0 &
(b1 = 0 or for b2 being natural set
st 1 <= b2 & b2 <= b1
holds a2 . b2 = - (a1 . b2)));
end;
:: PRGCOR_2:dfs 9
definiens
let a1, a2 be finite T-Sequence of REAL;
let a3 be integer set;
To prove
a3 vector_minus_prg a2,a1
it is sufficient to prove
thus len a2 = a3 &
len a1 = a3 &
(ex b1 being integer set st
a2 . 0 = a1 . 0 &
b1 = a1 . 0 &
(b1 = 0 or for b2 being natural set
st 1 <= b2 & b2 <= b1
holds a2 . b2 = - (a1 . b2)));
:: PRGCOR_2:def 9
theorem
for b1, b2 being finite T-Sequence of REAL
for b3 being integer set holds
b3 vector_minus_prg b2,b1
iff
len b2 = b3 &
len b1 = b3 &
(ex b4 being integer set st
b2 . 0 = b1 . 0 &
b4 = b1 . 0 &
(b4 = 0 or for b5 being natural set
st 1 <= b5 & b5 <= b4
holds b2 . b5 = - (b1 . b5)));
:: PRGCOR_2:th 12
theorem
for b1 being non empty finite T-Sequence of REAL
for b2 being natural set
st b1 . 0 is natural set & len b1 = b2 & 0 <= b1 . 0 & b1 . 0 < b2
holds (ex b3 being finite T-Sequence of REAL st
b2 vector_minus_prg b3,b1) &
(for b3 being non empty finite T-Sequence of REAL
st b2 vector_minus_prg b3,b1
holds XFS2FS* b3 = - XFS2FS* b1);
:: PRGCOR_2:prednot 4 => PRGCOR_2:pred 4
definition
let a1, a2, a3 be finite T-Sequence of REAL;
let a4 be integer set;
pred A4 vector_add_prg A3,A1,A2 means
len a3 = a4 &
len a1 = a4 &
len a2 = a4 &
(ex b1 being integer set st
a3 . 0 = a2 . 0 &
b1 = a2 . 0 &
(b1 = 0 or for b2 being natural set
st 1 <= b2 & b2 <= b1
holds a3 . b2 = (a1 . b2) + (a2 . b2)));
end;
:: PRGCOR_2:dfs 10
definiens
let a1, a2, a3 be finite T-Sequence of REAL;
let a4 be integer set;
To prove
a4 vector_add_prg a3,a1,a2
it is sufficient to prove
thus len a3 = a4 &
len a1 = a4 &
len a2 = a4 &
(ex b1 being integer set st
a3 . 0 = a2 . 0 &
b1 = a2 . 0 &
(b1 = 0 or for b2 being natural set
st 1 <= b2 & b2 <= b1
holds a3 . b2 = (a1 . b2) + (a2 . b2)));
:: PRGCOR_2:def 10
theorem
for b1, b2, b3 being finite T-Sequence of REAL
for b4 being integer set holds
b4 vector_add_prg b3,b1,b2
iff
len b3 = b4 &
len b1 = b4 &
len b2 = b4 &
(ex b5 being integer set st
b3 . 0 = b2 . 0 &
b5 = b2 . 0 &
(b5 = 0 or for b6 being natural set
st 1 <= b6 & b6 <= b5
holds b3 . b6 = (b1 . b6) + (b2 . b6)));
:: PRGCOR_2:th 13
theorem
for b1, b2 being non empty finite T-Sequence of REAL
for b3 being natural set
st b2 . 0 is natural set & len b1 = b3 & len b2 = b3 & b1 . 0 = b2 . 0 & 0 <= b2 . 0 & b2 . 0 < b3
holds (ex b4 being finite T-Sequence of REAL st
b3 vector_add_prg b4,b1,b2) &
(for b4 being non empty finite T-Sequence of REAL
st b3 vector_add_prg b4,b1,b2
holds XFS2FS* b4 = (XFS2FS* b1) + XFS2FS* b2);
:: PRGCOR_2:prednot 5 => PRGCOR_2:pred 5
definition
let a1, a2, a3 be finite T-Sequence of REAL;
let a4 be integer set;
pred A4 vector_sub_prg A3,A1,A2 means
len a3 = a4 &
len a1 = a4 &
len a2 = a4 &
(ex b1 being integer set st
a3 . 0 = a2 . 0 &
b1 = a2 . 0 &
(b1 = 0 or for b2 being natural set
st 1 <= b2 & b2 <= b1
holds a3 . b2 = (a1 . b2) - (a2 . b2)));
end;
:: PRGCOR_2:dfs 11
definiens
let a1, a2, a3 be finite T-Sequence of REAL;
let a4 be integer set;
To prove
a4 vector_sub_prg a3,a1,a2
it is sufficient to prove
thus len a3 = a4 &
len a1 = a4 &
len a2 = a4 &
(ex b1 being integer set st
a3 . 0 = a2 . 0 &
b1 = a2 . 0 &
(b1 = 0 or for b2 being natural set
st 1 <= b2 & b2 <= b1
holds a3 . b2 = (a1 . b2) - (a2 . b2)));
:: PRGCOR_2:def 11
theorem
for b1, b2, b3 being finite T-Sequence of REAL
for b4 being integer set holds
b4 vector_sub_prg b3,b1,b2
iff
len b3 = b4 &
len b1 = b4 &
len b2 = b4 &
(ex b5 being integer set st
b3 . 0 = b2 . 0 &
b5 = b2 . 0 &
(b5 = 0 or for b6 being natural set
st 1 <= b6 & b6 <= b5
holds b3 . b6 = (b1 . b6) - (b2 . b6)));
:: PRGCOR_2:th 14
theorem
for b1, b2 being non empty finite T-Sequence of REAL
for b3 being natural set
st b2 . 0 is natural set & len b1 = b3 & len b2 = b3 & b1 . 0 = b2 . 0 & 0 <= b2 . 0 & b2 . 0 < b3
holds (ex b4 being finite T-Sequence of REAL st
b3 vector_sub_prg b4,b1,b2) &
(for b4 being non empty finite T-Sequence of REAL
st b3 vector_sub_prg b4,b1,b2
holds XFS2FS* b4 = (XFS2FS* b1) - XFS2FS* b2);