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 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4,n5) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4,n5) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4,n5) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n5) ) & ( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n4) ) )
let S be Gene-Set; for p1, p2 being Individual of S holds
( ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4,n5) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4,n5) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4,n5) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n5) ) & ( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n4) ) )
let p1, p2 be Individual of S; ( ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4,n5) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4,n5) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4,n5) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n5) ) & ( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n4) ) )
A1:
( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n4) )
proof
assume
n5 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n4)
then
n5 >= len S
by Def1;
then A2:
n5 >= len (crossover (p1,p2,n1,n2,n3,n4))
by Def1;
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
(crossover (p1,p2,n1,n2,n3,n4)),
(crossover (p2,p1,n1,n2,n3,n4)),
n5)
;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
p1,
p2,
n1,
n2,
n3,
n4)
by A2, Th5;
verum
end;
A3:
( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4,n5) )
proof
assume A4:
n2 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4,n5)
then
n2 >= len S
by Def1;
then A5:
n2 >= len p2
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 A4, Th33
.=
crossover (
(crossover (p1,p2,n1,n3,n4)),
(crossover (p2,p1,n1,n3,n4)),
n5)
by A5, Th33
;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
p1,
p2,
n1,
n3,
n4,
n5)
;
verum
end;
A6:
( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n5) )
proof
assume A7:
n4 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n5)
then
n4 >= len S
by Def1;
then A8:
n4 >= len p2
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 A7, Th33
.=
crossover (
(crossover (p1,p2,n1,n2,n3)),
(crossover (p2,p1,n1,n2,n3)),
n5)
by A8, Th33
;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
p1,
p2,
n1,
n2,
n3,
n5)
;
verum
end;
A9:
( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4,n5) )
proof
assume A10:
n3 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4,n5)
then
n3 >= len S
by Def1;
then A11:
n3 >= len p2
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 A10, Th33
.=
crossover (
(crossover (p1,p2,n1,n2,n4)),
(crossover (p2,p1,n1,n2,n4)),
n5)
by A11, Th33
;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
p1,
p2,
n1,
n2,
n4,
n5)
;
verum
end;
( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4,n5) )
proof
assume A12:
n1 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4,n5)
then
n1 >= len S
by Def1;
then A13:
n1 >= len p2
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 A12, Th33
.=
crossover (
(crossover (p1,p2,n2,n3,n4)),
(crossover (p2,p1,n2,n3,n4)),
n5)
by A13, Th33
;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
p1,
p2,
n2,
n3,
n4,
n5)
;
verum
end;
hence
( ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4,n5) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4,n5) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4,n5) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n5) ) & ( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n4) ) )
by A3, A9, A6, A1; verum