let K be Field; for p, q being FinSequence of K st p is first-line-of-circulant & q is first-line-of-circulant & len p = len q holds
p + q is first-line-of-circulant
let p, q be FinSequence of K; ( p is first-line-of-circulant & q is first-line-of-circulant & len p = len q implies p + q is first-line-of-circulant )
set n = len p;
assume that
A1:
p is first-line-of-circulant
and
A2:
q is first-line-of-circulant
and
A3:
len p = len q
; p + q is first-line-of-circulant
consider M2 being Matrix of len p,K such that
A4:
M2 is_line_circulant_about q
by A2, A3;
A5:
dom (p + q) = Seg (len (p + q))
by FINSEQ_1:def 3;
A6:
dom p = Seg (len p)
by FINSEQ_1:def 3;
dom q = Seg (len p)
by A3, FINSEQ_1:def 3;
then A7:
dom (p + q) = dom p
by A6, POLYNOM1:1;
then A8:
len (p + q) = len p
by A6, FINSEQ_1:def 3;
consider M1 being Matrix of len p,K such that
A9:
M1 is_line_circulant_about p
by A1;
A10:
Indices M1 = [:(Seg (len p)),(Seg (len p)):]
by MATRIX_0:24;
set M3 = M1 + M2;
A11:
width (M1 + M2) = len p
by MATRIX_0:24;
A12:
Indices M2 = [:(Seg (len p)),(Seg (len p)):]
by MATRIX_0:24;
for i, j being Nat st [i,j] in Indices (M1 + M2) holds
(M1 + M2) * (i,j) = (p + q) . (((j - i) mod (len (p + q))) + 1)
proof
let i,
j be
Nat;
( [i,j] in Indices (M1 + M2) implies (M1 + M2) * (i,j) = (p + q) . (((j - i) mod (len (p + q))) + 1) )
assume A13:
[i,j] in Indices (M1 + M2)
;
(M1 + M2) * (i,j) = (p + q) . (((j - i) mod (len (p + q))) + 1)
then A14:
[i,j] in Indices M1
by A10, MATRIX_0:24;
then A15:
((j - i) mod (len (p + q))) + 1
in dom (p + q)
by A10, A6, A5, A7, Lm3;
A16:
[i,j] in Indices M2
by A12, A13, MATRIX_0:24;
(M1 + M2) * (
i,
j) =
(M1 * (i,j)) + (M2 * (i,j))
by A14, MATRIX_3:def 3
.=
the
addF of
K . (
(M1 * (i,j)),
(q . (((j - i) mod (len q)) + 1)))
by A4, A16
.=
the
addF of
K . (
(p . (((j - i) mod (len (p + q))) + 1)),
(q . (((j - i) mod (len (p + q))) + 1)))
by A3, A9, A8, A14
.=
(p + q) . (((j - i) mod (len (p + q))) + 1)
by A15, FUNCOP_1:22
;
hence
(M1 + M2) * (
i,
j)
= (p + q) . (((j - i) mod (len (p + q))) + 1)
;
verum
end;
then
M1 + M2 is_line_circulant_about p + q
by A11, A8;
then consider M3 being Matrix of len (p + q),K such that
len (p + q) = width M3
and
A17:
M3 is_line_circulant_about p + q
by A11;
take
M3
; MATRIX16:def 3 M3 is_line_circulant_about p + q
thus
M3 is_line_circulant_about p + q
by A17; verum