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 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5) ) & ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4) ) & ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3) ) & ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2) ) & ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1) ) )
let S be Gene-Set; for p1, p2 being Individual of S holds
( ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5) ) & ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4) ) & ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3) ) & ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2) ) & ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1) ) )
let p1, p2 be Individual of S; ( ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5) ) & ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4) ) & ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3) ) & ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2) ) & ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1) ) )
A1:
( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3) )
proof
assume that A2:
(
n1 >= len p1 &
n2 >= len p1 &
n4 >= len p1 )
and A3:
n5 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3)
n5 >= len S
by A3, Def1;
then A4:
n5 >= len (crossover (p1,p2,n3))
by Def1;
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
(crossover (p1,p2,n3)),
(crossover (p2,p1,n1,n2,n3,n4)),
n5)
by A2, Th35;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
p1,
p2,
n3)
by A4, Th5;
verum
end;
A5:
( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2) )
proof
assume that A6:
(
n1 >= len p1 &
n3 >= len p1 &
n4 >= len p1 )
and A7:
n5 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2)
n5 >= len S
by A7, Def1;
then A8:
n5 >= len (crossover (p1,p2,n2))
by Def1;
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
(crossover (p1,p2,n2)),
(crossover (p2,p1,n1,n2,n3,n4)),
n5)
by A6, Th35;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
p1,
p2,
n2)
by A8, Th5;
verum
end;
A9:
( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5) )
proof
assume that A10:
n1 >= len p1
and A11:
n2 >= len p1
and A12:
n3 >= len p1
and A13:
n4 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5)
n1 >= len S
by A10, Def1;
then A14:
n1 >= len p2
by Def1;
n3 >= len S
by A12, Def1;
then A15:
n3 >= len p2
by Def1;
n2 >= len S
by A11, Def1;
then A16:
n2 >= len p2
by Def1;
n4 >= len S
by A13, Def1;
then A17:
n4 >= len p2
by Def1;
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
p1,
(crossover (p2,p1,n1,n2,n3,n4)),
n5)
by A10, A11, A12, A13, Th36;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
p1,
p2,
n5)
by A14, A16, A15, A17, Th36;
verum
end;
A18:
( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1) )
proof
assume that A19:
(
n2 >= len p1 &
n3 >= len p1 &
n4 >= len p1 )
and A20:
n5 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1)
n5 >= len S
by A20, Def1;
then A21:
n5 >= len (crossover (p1,p2,n1))
by Def1;
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
(crossover (p1,p2,n1)),
(crossover (p2,p1,n1,n2,n3,n4)),
n5)
by A19, Th35;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
p1,
p2,
n1)
by A21, Th5;
verum
end;
( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4) )
proof
assume that A22:
(
n1 >= len p1 &
n2 >= len p1 &
n3 >= len p1 )
and A23:
n5 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4)
n5 >= len S
by A23, Def1;
then A24:
n5 >= len (crossover (p1,p2,n4))
by Def1;
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
(crossover (p1,p2,n4)),
(crossover (p2,p1,n1,n2,n3,n4)),
n5)
by A22, Th35;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
p1,
p2,
n4)
by A24, Th5;
verum
end;
hence
( ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5) ) & ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4) ) & ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3) ) & ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2) ) & ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1) ) )
by A9, A1, A5, A18; verum