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 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4,n5) ) & ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n5) ) & ( n1 >= len p1 & n2 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n4) ) & ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n5) ) & ( n1 >= len p1 & n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n4) ) & ( n1 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3) ) & ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n5) ) & ( n2 >= len p1 & n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n4) ) & ( n2 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3) ) & ( n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2) ) )
let S be Gene-Set; for p1, p2 being Individual of S holds
( ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4,n5) ) & ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n5) ) & ( n1 >= len p1 & n2 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n4) ) & ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n5) ) & ( n1 >= len p1 & n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n4) ) & ( n1 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3) ) & ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n5) ) & ( n2 >= len p1 & n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n4) ) & ( n2 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3) ) & ( n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2) ) )
let p1, p2 be Individual of S; ( ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4,n5) ) & ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n5) ) & ( n1 >= len p1 & n2 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n4) ) & ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n5) ) & ( n1 >= len p1 & n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n4) ) & ( n1 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3) ) & ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n5) ) & ( n2 >= len p1 & n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n4) ) & ( n2 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3) ) & ( n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2) ) )
A1:
( n1 >= len p1 & n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n4) )
proof
assume that A2:
(
n1 >= len p1 &
n3 >= len p1 )
and A3:
n5 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n4)
n5 >= len S
by A3, Def1;
then A4:
n5 >= len (crossover (p1,p2,n2,n4))
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 A2, Th34;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
p1,
p2,
n2,
n4)
by A4, Th5;
verum
end;
A5:
( n1 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3) )
proof
assume that A6:
(
n1 >= len p1 &
n4 >= len p1 )
and A7:
n5 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3)
n5 >= len S
by A7, Def1;
then A8:
n5 >= len (crossover (p1,p2,n2,n3))
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 A6, Th34;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
p1,
p2,
n2,
n3)
by A8, Th5;
verum
end;
A9:
( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n5) )
proof
assume that A10:
n1 >= len p1
and A11:
n2 >= len p1
and A12:
n4 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n5)
n1 >= len S
by A10, Def1;
then A13:
n1 >= len p2
by Def1;
n4 >= len S
by A12, Def1;
then A14:
n4 >= len p2
by Def1;
n2 >= len S
by A11, Def1;
then A15:
n2 >= len p2
by Def1;
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5) =
crossover (
(crossover (p1,p2,n3)),
(crossover (p2,p1,n1,n2,n3,n4)),
n5)
by A10, A11, A12, Th35
.=
crossover (
(crossover (p1,p2,n3)),
(crossover (p2,p1,n3)),
n5)
by A13, A15, A14, Th35
;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
p1,
p2,
n3,
n5)
;
verum
end;
A16:
( n2 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3) )
proof
assume that A17:
(
n2 >= len p1 &
n4 >= len p1 )
and A18:
n5 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3)
n5 >= len S
by A18, Def1;
then A19:
n5 >= len (crossover (p1,p2,n1,n3))
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 A17, Th34;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
p1,
p2,
n1,
n3)
by A19, Th5;
verum
end;
A20:
( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n5) )
proof
assume that A21:
n2 >= len p1
and A22:
n3 >= len p1
and A23:
n4 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n5)
n2 >= len S
by A21, Def1;
then A24:
n2 >= len p2
by Def1;
n4 >= len S
by A23, Def1;
then A25:
n4 >= len p2
by Def1;
n3 >= len S
by A22, Def1;
then A26:
n3 >= len p2
by Def1;
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5) =
crossover (
(crossover (p1,p2,n1)),
(crossover (p2,p1,n1,n2,n3,n4)),
n5)
by A21, A22, A23, Th35
.=
crossover (
(crossover (p1,p2,n1)),
(crossover (p2,p1,n1)),
n5)
by A24, A26, A25, Th35
;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
p1,
p2,
n1,
n5)
;
verum
end;
A27:
( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4,n5) )
proof
assume that A28:
n1 >= len p1
and A29:
n2 >= len p1
and A30:
n3 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4,n5)
n1 >= len S
by A28, Def1;
then A31:
n1 >= len p2
by Def1;
n3 >= len S
by A30, Def1;
then A32:
n3 >= len p2
by Def1;
n2 >= len S
by A29, Def1;
then A33:
n2 >= len p2
by Def1;
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5) =
crossover (
(crossover (p1,p2,n4)),
(crossover (p2,p1,n1,n2,n3,n4)),
n5)
by A28, A29, A30, Th35
.=
crossover (
(crossover (p1,p2,n4)),
(crossover (p2,p1,n4)),
n5)
by A31, A33, A32, Th35
;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
p1,
p2,
n4,
n5)
;
verum
end;
A34:
( n2 >= len p1 & n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n4) )
proof
assume that A35:
(
n2 >= len p1 &
n3 >= len p1 )
and A36:
n5 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n4)
n5 >= len S
by A36, Def1;
then A37:
n5 >= len (crossover (p1,p2,n1,n4))
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 A35, Th34;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
p1,
p2,
n1,
n4)
by A37, Th5;
verum
end;
A38:
( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n5) )
proof
assume that A39:
n1 >= len p1
and A40:
n3 >= len p1
and A41:
n4 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n5)
n1 >= len S
by A39, Def1;
then A42:
n1 >= len p2
by Def1;
n4 >= len S
by A41, Def1;
then A43:
n4 >= len p2
by Def1;
n3 >= len S
by A40, Def1;
then A44:
n3 >= len p2
by Def1;
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5) =
crossover (
(crossover (p1,p2,n2)),
(crossover (p2,p1,n1,n2,n3,n4)),
n5)
by A39, A40, A41, Th35
.=
crossover (
(crossover (p1,p2,n2)),
(crossover (p2,p1,n2)),
n5)
by A42, A44, A43, Th35
;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
p1,
p2,
n2,
n5)
;
verum
end;
A45:
( n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2) )
proof
assume that A46:
(
n3 >= len p1 &
n4 >= len p1 )
and A47:
n5 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2)
n5 >= len S
by A47, Def1;
then A48:
n5 >= len (crossover (p1,p2,n1,n2))
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 A46, Th34;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
p1,
p2,
n1,
n2)
by A48, Th5;
verum
end;
( n1 >= len p1 & n2 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n4) )
proof
assume that A49:
(
n1 >= len p1 &
n2 >= len p1 )
and A50:
n5 >= len p1
;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n4)
n5 >= len S
by A50, Def1;
then A51:
n5 >= len (crossover (p1,p2,n3,n4))
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 A49, Th34;
hence
crossover (
p1,
p2,
n1,
n2,
n3,
n4,
n5)
= crossover (
p1,
p2,
n3,
n4)
by A51, Th5;
verum
end;
hence
( ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4,n5) ) & ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n5) ) & ( n1 >= len p1 & n2 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n4) ) & ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n5) ) & ( n1 >= len p1 & n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n4) ) & ( n1 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3) ) & ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n5) ) & ( n2 >= len p1 & n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n4) ) & ( n2 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3) ) & ( n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2) ) )
by A27, A9, A38, A1, A5, A20, A34, A16, A45; verum