let n2, n3, n4, n5, n1 be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,0 ,n2,n3,n4,n5 = crossover p2,p1,n2,n3,n4,n5 & crossover p1,p2,n1,0 ,n3,n4,n5 = crossover p2,p1,n1,n3,n4,n5 & crossover p1,p2,n1,n2,0 ,n4,n5 = crossover p2,p1,n1,n2,n4,n5 & crossover p1,p2,n1,n2,n3,0 ,n5 = crossover p2,p1,n1,n2,n3,n5 & crossover p1,p2,n1,n2,n3,n4,0 = crossover p2,p1,n1,n2,n3,n4 )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover p1,p2,0 ,n2,n3,n4,n5 = crossover p2,p1,n2,n3,n4,n5 & crossover p1,p2,n1,0 ,n3,n4,n5 = crossover p2,p1,n1,n3,n4,n5 & crossover p1,p2,n1,n2,0 ,n4,n5 = crossover p2,p1,n1,n2,n4,n5 & crossover p1,p2,n1,n2,n3,0 ,n5 = crossover p2,p1,n1,n2,n3,n5 & crossover p1,p2,n1,n2,n3,n4,0 = crossover p2,p1,n1,n2,n3,n4 )
let p1, p2 be Individual of S; :: thesis: ( crossover p1,p2,0 ,n2,n3,n4,n5 = crossover p2,p1,n2,n3,n4,n5 & crossover p1,p2,n1,0 ,n3,n4,n5 = crossover p2,p1,n1,n3,n4,n5 & crossover p1,p2,n1,n2,0 ,n4,n5 = crossover p2,p1,n1,n2,n4,n5 & crossover p1,p2,n1,n2,n3,0 ,n5 = crossover p2,p1,n1,n2,n3,n5 & crossover p1,p2,n1,n2,n3,n4,0 = crossover p2,p1,n1,n2,n3,n4 )
A1:
crossover p1,p2,0 ,n2,n3,n4,n5 = crossover p2,p1,n2,n3,n4,n5
proof
crossover p1,
p2,
0 ,
n2,
n3,
n4,
n5 =
crossover (crossover p2,p1,n2,n3,n4),
(crossover p2,p1,0 ,n2,n3,n4),
n5
by Th29
.=
crossover (crossover p2,p1,n2,n3,n4),
(crossover p1,p2,n2,n3,n4),
n5
by Th29
;
hence
crossover p1,
p2,
0 ,
n2,
n3,
n4,
n5 = crossover p2,
p1,
n2,
n3,
n4,
n5
;
:: thesis: verum
end;
A2:
crossover p1,p2,n1,0 ,n3,n4,n5 = crossover p2,p1,n1,n3,n4,n5
proof
crossover p1,
p2,
n1,
0 ,
n3,
n4,
n5 =
crossover (crossover p2,p1,n1,n3,n4),
(crossover p2,p1,n1,0 ,n3,n4),
n5
by Th29
.=
crossover (crossover p2,p1,n1,n3,n4),
(crossover p1,p2,n1,n3,n4),
n5
by Th29
;
hence
crossover p1,
p2,
n1,
0 ,
n3,
n4,
n5 = crossover p2,
p1,
n1,
n3,
n4,
n5
;
:: thesis: verum
end;
A3:
crossover p1,p2,n1,n2,0 ,n4,n5 = crossover p2,p1,n1,n2,n4,n5
proof
crossover p1,
p2,
n1,
n2,
0 ,
n4,
n5 =
crossover (crossover p2,p1,n1,n2,n4),
(crossover p2,p1,n1,n2,0 ,n4),
n5
by Th29
.=
crossover (crossover p2,p1,n1,n2,n4),
(crossover p1,p2,n1,n2,n4),
n5
by Th29
;
hence
crossover p1,
p2,
n1,
n2,
0 ,
n4,
n5 = crossover p2,
p1,
n1,
n2,
n4,
n5
;
:: thesis: verum
end;
A4:
crossover p1,p2,n1,n2,n3,0 ,n5 = crossover p2,p1,n1,n2,n3,n5
proof
crossover p1,
p2,
n1,
n2,
n3,
0 ,
n5 =
crossover (crossover p2,p1,n1,n2,n3),
(crossover p2,p1,n1,n2,n3,0 ),
n5
by Th29
.=
crossover (crossover p2,p1,n1,n2,n3),
(crossover p1,p2,n1,n2,n3),
n5
by Th29
;
hence
crossover p1,
p2,
n1,
n2,
n3,
0 ,
n5 = crossover p2,
p1,
n1,
n2,
n3,
n5
;
:: thesis: verum
end;
crossover p1,p2,n1,n2,n3,n4,0 = crossover p2,p1,n1,n2,n3,n4
proof
crossover p1,
p2,
n1,
n2,
n3,
n4,
0 = crossover (crossover p1,p2,n1,n2,n3,n4),
(crossover p2,p1,n1,n2,n3,n4),
0
;
hence
crossover p1,
p2,
n1,
n2,
n3,
n4,
0 = crossover p2,
p1,
n1,
n2,
n3,
n4
by Th4;
:: thesis: verum
end;
hence
( crossover p1,p2,0 ,n2,n3,n4,n5 = crossover p2,p1,n2,n3,n4,n5 & crossover p1,p2,n1,0 ,n3,n4,n5 = crossover p2,p1,n1,n3,n4,n5 & crossover p1,p2,n1,n2,0 ,n4,n5 = crossover p2,p1,n1,n2,n4,n5 & crossover p1,p2,n1,n2,n3,0 ,n5 = crossover p2,p1,n1,n2,n3,n5 & crossover p1,p2,n1,n2,n3,n4,0 = crossover p2,p1,n1,n2,n3,n4 )
by A1, A2, A3, A4; :: thesis: verum