let n1, n2, n3, n4, n5 be Element of NAT ; for S being Gene-Set
for p1, p2 being Individual of S holds
( ( n1 >= len p1 & n2 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n3,n4,n5 ) & ( n1 >= len p1 & n3 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n4,n5 ) & ( n1 >= len p1 & n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n3,n5 ) & ( n1 >= len p1 & n5 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n3,n4 ) & ( n2 >= len p1 & n3 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n4,n5 ) & ( n2 >= len p1 & n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n3,n5 ) & ( n2 >= len p1 & n5 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n3,n4 ) & ( n3 >= len p1 & n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n2,n5 ) & ( n3 >= len p1 & n5 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n2,n4 ) & ( n4 >= len p1 & n5 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n2,n3 ) )
let S be Gene-Set; for p1, p2 being Individual of S holds
( ( n1 >= len p1 & n2 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n3,n4,n5 ) & ( n1 >= len p1 & n3 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n4,n5 ) & ( n1 >= len p1 & n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n3,n5 ) & ( n1 >= len p1 & n5 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n3,n4 ) & ( n2 >= len p1 & n3 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n4,n5 ) & ( n2 >= len p1 & n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n3,n5 ) & ( n2 >= len p1 & n5 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n3,n4 ) & ( n3 >= len p1 & n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n2,n5 ) & ( n3 >= len p1 & n5 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n2,n4 ) & ( n4 >= len p1 & n5 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n2,n3 ) )
let p1, p2 be Individual of S; ( ( n1 >= len p1 & n2 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n3,n4,n5 ) & ( n1 >= len p1 & n3 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n4,n5 ) & ( n1 >= len p1 & n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n3,n5 ) & ( n1 >= len p1 & n5 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n3,n4 ) & ( n2 >= len p1 & n3 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n4,n5 ) & ( n2 >= len p1 & n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n3,n5 ) & ( n2 >= len p1 & n5 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n3,n4 ) & ( n3 >= len p1 & n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n2,n5 ) & ( n3 >= len p1 & n5 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n2,n4 ) & ( n4 >= len p1 & n5 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n2,n3 ) )
A1:
( n2 >= len p1 & n5 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n3,n4 )
proof
assume that A2:
n2 >= len p1
and A3:
n5 >= len p1
;
crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n3,n4
n5 >= len S
by A3, Def1;
then A4:
n5 >= len (crossover p1,p2,n1,n3,n4)
by Def1;
crossover p1,
p2,
n1,
n2,
n3,
n4,
n5 = crossover (crossover p1,p2,n1,n3,n4),
(crossover p2,p1,n1,n2,n3,n4),
n5
by A2, Th33;
hence
crossover p1,
p2,
n1,
n2,
n3,
n4,
n5 = crossover p1,
p2,
n1,
n3,
n4
by A4, Th5;
verum
end;
A5:
( n3 >= len p1 & n5 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n2,n4 )
proof
assume that A6:
n3 >= len p1
and A7:
n5 >= len p1
;
crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n2,n4
n5 >= len S
by A7, Def1;
then A8:
n5 >= len (crossover p1,p2,n1,n2,n4)
by Def1;
crossover p1,
p2,
n1,
n2,
n3,
n4,
n5 = crossover (crossover p1,p2,n1,n2,n4),
(crossover p2,p1,n1,n2,n3,n4),
n5
by A6, Th33;
hence
crossover p1,
p2,
n1,
n2,
n3,
n4,
n5 = crossover p1,
p2,
n1,
n2,
n4
by A8, Th5;
verum
end;
A9:
( n1 >= len p1 & n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n3,n5 )
proof
assume that A10:
n1 >= len p1
and A11:
n4 >= len p1
;
crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n3,n5
n1 >= len S
by A10, Def1;
then A12:
n1 >= len p2
by Def1;
n4 >= len S
by A11, Def1;
then A13:
n4 >= len p2
by Def1;
crossover p1,
p2,
n1,
n2,
n3,
n4,
n5 =
crossover (crossover p1,p2,n2,n3),
(crossover p2,p1,n1,n2,n3,n4),
n5
by A10, A11, Th34
.=
crossover (crossover p1,p2,n2,n3),
(crossover p2,p1,n2,n3),
n5
by A12, A13, Th34
;
hence
crossover p1,
p2,
n1,
n2,
n3,
n4,
n5 = crossover p1,
p2,
n2,
n3,
n5
;
verum
end;
A14:
( n1 >= len p1 & n3 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n4,n5 )
proof
assume that A15:
n1 >= len p1
and A16:
n3 >= len p1
;
crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n4,n5
n1 >= len S
by A15, Def1;
then A17:
n1 >= len p2
by Def1;
n3 >= len S
by A16, Def1;
then A18:
n3 >= len p2
by Def1;
crossover p1,
p2,
n1,
n2,
n3,
n4,
n5 =
crossover (crossover p1,p2,n2,n4),
(crossover p2,p1,n1,n2,n3,n4),
n5
by A15, A16, Th34
.=
crossover (crossover p1,p2,n2,n4),
(crossover p2,p1,n2,n4),
n5
by A17, A18, Th34
;
hence
crossover p1,
p2,
n1,
n2,
n3,
n4,
n5 = crossover p1,
p2,
n2,
n4,
n5
;
verum
end;
A19:
( n2 >= len p1 & n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n3,n5 )
proof
assume that A20:
n2 >= len p1
and A21:
n4 >= len p1
;
crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n3,n5
n2 >= len S
by A20, Def1;
then A22:
n2 >= len p2
by Def1;
n4 >= len S
by A21, Def1;
then A23:
n4 >= len p2
by Def1;
crossover p1,
p2,
n1,
n2,
n3,
n4,
n5 =
crossover (crossover p1,p2,n1,n3),
(crossover p2,p1,n1,n2,n3,n4),
n5
by A20, A21, Th34
.=
crossover (crossover p1,p2,n1,n3),
(crossover p2,p1,n1,n3),
n5
by A22, A23, Th34
;
hence
crossover p1,
p2,
n1,
n2,
n3,
n4,
n5 = crossover p1,
p2,
n1,
n3,
n5
;
verum
end;
A24:
( n2 >= len p1 & n3 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n4,n5 )
proof
assume that A25:
n2 >= len p1
and A26:
n3 >= len p1
;
crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n4,n5
n2 >= len S
by A25, Def1;
then A27:
n2 >= len p2
by Def1;
n3 >= len S
by A26, Def1;
then A28:
n3 >= len p2
by Def1;
crossover p1,
p2,
n1,
n2,
n3,
n4,
n5 =
crossover (crossover p1,p2,n1,n4),
(crossover p2,p1,n1,n2,n3,n4),
n5
by A25, A26, Th34
.=
crossover (crossover p1,p2,n1,n4),
(crossover p2,p1,n1,n4),
n5
by A27, A28, Th34
;
hence
crossover p1,
p2,
n1,
n2,
n3,
n4,
n5 = crossover p1,
p2,
n1,
n4,
n5
;
verum
end;
A29:
( n3 >= len p1 & n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n2,n5 )
proof
assume that A30:
n3 >= len p1
and A31:
n4 >= len p1
;
crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n2,n5
n3 >= len S
by A30, Def1;
then A32:
n3 >= len p2
by Def1;
n4 >= len S
by A31, Def1;
then A33:
n4 >= len p2
by Def1;
crossover p1,
p2,
n1,
n2,
n3,
n4,
n5 =
crossover (crossover p1,p2,n1,n2),
(crossover p2,p1,n1,n2,n3,n4),
n5
by A30, A31, Th34
.=
crossover (crossover p1,p2,n1,n2),
(crossover p2,p1,n1,n2),
n5
by A32, A33, Th34
;
hence
crossover p1,
p2,
n1,
n2,
n3,
n4,
n5 = crossover p1,
p2,
n1,
n2,
n5
;
verum
end;
A34:
( n1 >= len p1 & n2 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n3,n4,n5 )
proof
assume that A35:
n1 >= len p1
and A36:
n2 >= len p1
;
crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n3,n4,n5
n1 >= len S
by A35, Def1;
then A37:
n1 >= len p2
by Def1;
n2 >= len S
by A36, Def1;
then A38:
n2 >= len p2
by Def1;
crossover p1,
p2,
n1,
n2,
n3,
n4,
n5 =
crossover (crossover p1,p2,n3,n4),
(crossover p2,p1,n1,n2,n3,n4),
n5
by A35, A36, Th34
.=
crossover (crossover p1,p2,n3,n4),
(crossover p2,p1,n3,n4),
n5
by A37, A38, Th34
;
hence
crossover p1,
p2,
n1,
n2,
n3,
n4,
n5 = crossover p1,
p2,
n3,
n4,
n5
;
verum
end;
A39:
( n4 >= len p1 & n5 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n2,n3 )
proof
assume that A40:
n4 >= len p1
and A41:
n5 >= len p1
;
crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n2,n3
n5 >= len S
by A41, Def1;
then A42:
n5 >= len (crossover p1,p2,n1,n2,n3)
by Def1;
crossover p1,
p2,
n1,
n2,
n3,
n4,
n5 = crossover (crossover p1,p2,n1,n2,n3),
(crossover p2,p1,n1,n2,n3,n4),
n5
by A40, Th33;
hence
crossover p1,
p2,
n1,
n2,
n3,
n4,
n5 = crossover p1,
p2,
n1,
n2,
n3
by A42, Th5;
verum
end;
( n1 >= len p1 & n5 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n3,n4 )
proof
assume that A43:
n1 >= len p1
and A44:
n5 >= len p1
;
crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n3,n4
n5 >= len S
by A44, Def1;
then A45:
n5 >= len (crossover p1,p2,n2,n3,n4)
by Def1;
crossover p1,
p2,
n1,
n2,
n3,
n4,
n5 = crossover (crossover p1,p2,n2,n3,n4),
(crossover p2,p1,n1,n2,n3,n4),
n5
by A43, Th33;
hence
crossover p1,
p2,
n1,
n2,
n3,
n4,
n5 = crossover p1,
p2,
n2,
n3,
n4
by A45, Th5;
verum
end;
hence
( ( n1 >= len p1 & n2 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n3,n4,n5 ) & ( n1 >= len p1 & n3 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n4,n5 ) & ( n1 >= len p1 & n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n3,n5 ) & ( n1 >= len p1 & n5 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n3,n4 ) & ( n2 >= len p1 & n3 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n4,n5 ) & ( n2 >= len p1 & n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n3,n5 ) & ( n2 >= len p1 & n5 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n3,n4 ) & ( n3 >= len p1 & n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n2,n5 ) & ( n3 >= len p1 & n5 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n2,n4 ) & ( n4 >= len p1 & n5 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n1,n2,n3 ) )
by A34, A14, A9, A24, A19, A1, A29, A5, A39; verum