:: Basic Properties of Genetic Algorithm
:: by Akihiko Uchibori and Noboru Endou
::
:: Received April 24, 1999
:: Copyright (c) 1999 Association of Mizar Users

theorem Th1: :: GENEALG1:1

for D being non empty set
for f1, f2 being FinSequence of D
for n being Nat st n <= len f1 holds
(f1 ^ f2) /^ n = (f1 /^ n) ^ f2

proof end;

theorem Th2: :: GENEALG1:2

for D being non empty set
for f1, f2 being FinSequence of D
for i being Element of NAT holds (f1 ^ f2) | ((len f1) + i) = f1 ^ (f2 | i)

proof end;

definition

mode Gene-Set is non empty non-empty FinSequence;

end;

notation

let S be Gene-Set;
synonym GA-Space S for Union S;

end;

registration

let f be non empty non-empty Function;
cluster GA-Space f -> non empty ;
coherence
not Union f is empty

proof end;

end;

definition

let S be Gene-Set;
mode Individual of S -> FinSequence of GA-Space S means :Def1: :: GENEALG1:def 1
( len it = len S & ( for i being Element of NAT st i in dom it holds
it . i in S . i ) );
existence
ex b₁ being FinSequence of GA-Space S st
( len b₁ = len S & ( for i being Element of NAT st i in dom b₁ holds
b₁ . i in S . i ) )

proof end;

end;

:: deftheorem Def1 defines Individual GENEALG1:def 1 :

for S being Gene-Set
for b₂ being FinSequence of GA-Space S holds
( b₂ is Individual of S iff ( len b₂ = len S & ( for i being Element of NAT st i in dom b₂ holds
b₂ . i in S . i ) ) );

definition

let S be Gene-Set;
let p1, p2 be FinSequence of GA-Space S;
let n be Element of NAT ;
func crossover p1,p2,n -> FinSequence of GA-Space S equals :: GENEALG1:def 2
(p1 | n) ^ (p2 /^ n);
correctness
coherence
(p1 | n) ^ (p2 /^ n) is FinSequence of GA-Space S;
;

end;

:: deftheorem defines crossover GENEALG1:def 2 :

for S being Gene-Set
for p1, p2 being FinSequence of GA-Space S
for n being Element of NAT holds crossover p1,p2,n = (p1 | n) ^ (p2 /^ n);

definition

let S be Gene-Set;
let p1, p2 be FinSequence of GA-Space S;
let n1, n2 be Element of NAT ;
func crossover p1,p2,n1,n2 -> FinSequence of GA-Space S equals :: GENEALG1:def 3
crossover (crossover p1,p2,n1),(crossover p2,p1,n1),n2;
correctness
coherence
crossover (crossover p1,p2,n1),(crossover p2,p1,n1),n2 is FinSequence of GA-Space S;
;

end;

:: deftheorem defines crossover GENEALG1:def 3 :

for S being Gene-Set
for p1, p2 being FinSequence of GA-Space S
for n1, n2 being Element of NAT holds crossover p1,p2,n1,n2 = crossover (crossover p1,p2,n1),(crossover p2,p1,n1),n2;

definition

let S be Gene-Set;
let p1, p2 be FinSequence of GA-Space S;
let n1, n2, n3 be Element of NAT ;
func crossover p1,p2,n1,n2,n3 -> FinSequence of GA-Space S equals :: GENEALG1:def 4
crossover (crossover p1,p2,n1,n2),(crossover p2,p1,n1,n2),n3;
correctness
coherence
crossover (crossover p1,p2,n1,n2),(crossover p2,p1,n1,n2),n3 is FinSequence of GA-Space S;
;

end;

:: deftheorem defines crossover GENEALG1:def 4 :

for S being Gene-Set
for p1, p2 being FinSequence of GA-Space S
for n1, n2, n3 being Element of NAT holds crossover p1,p2,n1,n2,n3 = crossover (crossover p1,p2,n1,n2),(crossover p2,p1,n1,n2),n3;

definition

let S be Gene-Set;
let p1, p2 be FinSequence of GA-Space S;
let n1, n2, n3, n4 be Element of NAT ;
func crossover p1,p2,n1,n2,n3,n4 -> FinSequence of GA-Space S equals :: GENEALG1:def 5
crossover (crossover p1,p2,n1,n2,n3),(crossover p2,p1,n1,n2,n3),n4;
correctness
coherence
crossover (crossover p1,p2,n1,n2,n3),(crossover p2,p1,n1,n2,n3),n4 is FinSequence of GA-Space S;
;

end;

:: deftheorem defines crossover GENEALG1:def 5 :

for S being Gene-Set
for p1, p2 being FinSequence of GA-Space S
for n1, n2, n3, n4 being Element of NAT holds crossover p1,p2,n1,n2,n3,n4 = crossover (crossover p1,p2,n1,n2,n3),(crossover p2,p1,n1,n2,n3),n4;

definition

let S be Gene-Set;
let p1, p2 be FinSequence of GA-Space S;
let n1, n2, n3, n4, n5 be Element of NAT ;
func crossover p1,p2,n1,n2,n3,n4,n5 -> FinSequence of GA-Space S equals :: GENEALG1:def 6
crossover (crossover p1,p2,n1,n2,n3,n4),(crossover p2,p1,n1,n2,n3,n4),n5;
correctness
coherence
crossover (crossover p1,p2,n1,n2,n3,n4),(crossover p2,p1,n1,n2,n3,n4),n5 is FinSequence of GA-Space S;
;

end;

:: deftheorem defines crossover GENEALG1:def 6 :

for S being Gene-Set
for p1, p2 being FinSequence of GA-Space S
for n1, n2, n3, n4, n5 being Element of NAT holds crossover p1,p2,n1,n2,n3,n4,n5 = crossover (crossover p1,p2,n1,n2,n3,n4),(crossover p2,p1,n1,n2,n3,n4),n5;

definition

let S be Gene-Set;
let p1, p2 be FinSequence of GA-Space S;
let n1, n2, n3, n4, n5, n6 be Element of NAT ;
func crossover p1,p2,n1,n2,n3,n4,n5,n6 -> FinSequence of GA-Space S equals :: GENEALG1:def 7
crossover (crossover p1,p2,n1,n2,n3,n4,n5),(crossover p2,p1,n1,n2,n3,n4,n5),n6;
correctness
coherence
crossover (crossover p1,p2,n1,n2,n3,n4,n5),(crossover p2,p1,n1,n2,n3,n4,n5),n6 is FinSequence of GA-Space S;
;

end;

:: deftheorem defines crossover GENEALG1:def 7 :

for S being Gene-Set
for p1, p2 being FinSequence of GA-Space S
for n1, n2, n3, n4, n5, n6 being Element of NAT holds crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover (crossover p1,p2,n1,n2,n3,n4,n5),(crossover p2,p1,n1,n2,n3,n4,n5),n6;

theorem Th3: :: GENEALG1:3

for n being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover p1,p2,n is Individual of S

proof end;

definition

let S be Gene-Set;
let p1, p2 be Individual of S;
let n be Element of NAT ;
:: original: crossover
redefine func crossover p1,p2,n -> Individual of S;
correctness
coherence
crossover p1,p2,n is Individual of S;
by Th3;

end;

theorem Th4: :: GENEALG1:4

for S being Gene-Set
for p1, p2 being Individual of S holds crossover p1,p2,0 = p2

proof end;

theorem Th5: :: GENEALG1:5

for n being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n >= len p1 holds
crossover p1,p2,n = p1

proof end;

theorem Th6: :: GENEALG1:6

for n1, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover p1,p2,n1,n2 is Individual of S

proof end;

definition

let S be Gene-Set;
let p1, p2 be Individual of S;
let n1, n2 be Element of NAT ;
:: original: crossover
redefine func crossover p1,p2,n1,n2 -> Individual of S;
correctness
coherence
crossover p1,p2,n1,n2 is Individual of S;
by Th6;

end;

theorem Th7: :: GENEALG1:7

for n being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover p1,p2,0 ,n = crossover p2,p1,n

proof end;

theorem Th8: :: GENEALG1:8

for n being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover p1,p2,n,0 = crossover p2,p1,n

proof end;

theorem Th9: :: GENEALG1:9

for n1, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 holds
crossover p1,p2,n1,n2 = crossover p1,p2,n2

proof end;

theorem Th10: :: GENEALG1:10

for n2, n1 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n2 >= len p1 holds
crossover p1,p2,n1,n2 = crossover p1,p2,n1

proof end;

theorem :: GENEALG1:11

for n1, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 holds
crossover p1,p2,n1,n2 = p1

proof end;

theorem Th12: :: GENEALG1:12

for n1 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover p1,p2,n1,n1 = p1

proof end;

theorem Th13: :: GENEALG1:13

for n1, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover p1,p2,n1,n2 = crossover p1,p2,n2,n1

proof end;

theorem Th14: :: GENEALG1:14

for n1, n2, n3 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover p1,p2,n1,n2,n3 is Individual of S

proof end;

definition

let S be Gene-Set;
let p1, p2 be Individual of S;
let n1, n2, n3 be Element of NAT ;
:: original: crossover
redefine func crossover p1,p2,n1,n2,n3 -> Individual of S;
correctness
coherence
crossover p1,p2,n1,n2,n3 is Individual of S;
by Th14;

end;

theorem Th15: :: GENEALG1:15

for n2, n3, n1 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,0 ,n2,n3 = crossover p2,p1,n2,n3 & crossover p1,p2,n1,0 ,n3 = crossover p2,p1,n1,n3 & crossover p1,p2,n1,n2,0 = crossover p2,p1,n1,n2 )

proof end;

theorem :: GENEALG1:16

for n3, n1, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,0 ,0 ,n3 = crossover p1,p2,n3 & crossover p1,p2,n1,0 ,0 = crossover p1,p2,n1 & crossover p1,p2,0 ,n2,0 = crossover p1,p2,n2 )

proof end;

theorem :: GENEALG1:17

for S being Gene-Set
for p1, p2 being Individual of S holds crossover p1,p2,0 ,0 ,0 = p2

proof end;

theorem Th18: :: GENEALG1:18

for n1, n2, n3 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 holds
crossover p1,p2,n1,n2,n3 = crossover p1,p2,n2,n3

proof end;

theorem Th19: :: GENEALG1:19

for n2, n1, n3 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n2 >= len p1 holds
crossover p1,p2,n1,n2,n3 = crossover p1,p2,n1,n3

proof end;

theorem Th20: :: GENEALG1:20

for n3, n1, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n3 >= len p1 holds
crossover p1,p2,n1,n2,n3 = crossover p1,p2,n1,n2

proof end;

theorem Th21: :: GENEALG1:21

for n1, n2, n3 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 holds
crossover p1,p2,n1,n2,n3 = crossover p1,p2,n3

proof end;

theorem :: GENEALG1:22

for n1, n3, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 & n3 >= len p1 holds
crossover p1,p2,n1,n2,n3 = crossover p1,p2,n2

proof end;

theorem :: GENEALG1:23

for n2, n3, n1 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n2 >= len p1 & n3 >= len p1 holds
crossover p1,p2,n1,n2,n3 = crossover p1,p2,n1

proof end;

theorem :: GENEALG1:24

for n1, n2, n3 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 & n3 >= len p1 holds
crossover p1,p2,n1,n2,n3 = p1

proof end;

theorem Th25: :: GENEALG1:25

for n1, n2, n3 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,n1,n2,n3 = crossover p1,p2,n2,n1,n3 & crossover p1,p2,n1,n2,n3 = crossover p1,p2,n1,n3,n2 )

proof end;

theorem Th26: :: GENEALG1:26

for n1, n2, n3 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover p1,p2,n1,n2,n3 = crossover p1,p2,n3,n1,n2

proof end;

theorem Th27: :: GENEALG1:27

for n1, n3, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,n1,n1,n3 = crossover p1,p2,n3 & crossover p1,p2,n1,n2,n1 = crossover p1,p2,n2 & crossover p1,p2,n1,n2,n2 = crossover p1,p2,n1 )

proof end;

theorem Th28: :: GENEALG1:28

for n1, n2, n3, n4 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover p1,p2,n1,n2,n3,n4 is Individual of S

proof end;

definition

let S be Gene-Set;
let p1, p2 be Individual of S;
let n1, n2, n3, n4 be Element of NAT ;
:: original: crossover
redefine func crossover p1,p2,n1,n2,n3,n4 -> Individual of S;
correctness
coherence
crossover p1,p2,n1,n2,n3,n4 is Individual of S;
by Th28;

end;

theorem Th29: :: GENEALG1:29

for n2, n3, n4, n1 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,0 ,n2,n3,n4 = crossover p2,p1,n2,n3,n4 & crossover p1,p2,n1,0 ,n3,n4 = crossover p2,p1,n1,n3,n4 & crossover p1,p2,n1,n2,0 ,n4 = crossover p2,p1,n1,n2,n4 & crossover p1,p2,n1,n2,n3,0 = crossover p2,p1,n1,n2,n3 )

proof end;

theorem Th30: :: GENEALG1:30

for n3, n4, n2, n1 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,0 ,0 ,n3,n4 = crossover p1,p2,n3,n4 & crossover p1,p2,0 ,n2,0 ,n4 = crossover p1,p2,n2,n4 & crossover p1,p2,0 ,n2,n3,0 = crossover p1,p2,n2,n3 & crossover p1,p2,n1,0 ,n3,0 = crossover p1,p2,n1,n3 & crossover p1,p2,n1,0 ,0 ,n4 = crossover p1,p2,n1,n4 & crossover p1,p2,n1,n2,0 ,0 = crossover p1,p2,n1,n2 )

proof end;

theorem Th31: :: GENEALG1:31

for n1, n2, n3, n4 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,n1,0 ,0 ,0 = crossover p2,p1,n1 & crossover p1,p2,0 ,n2,0 ,0 = crossover p2,p1,n2 & crossover p1,p2,0 ,0 ,n3,0 = crossover p2,p1,n3 & crossover p1,p2,0 ,0 ,0 ,n4 = crossover p2,p1,n4 )

proof end;

theorem Th32: :: GENEALG1:32

for S being Gene-Set
for p1, p2 being Individual of S holds crossover p1,p2,0 ,0 ,0 ,0 = p1

proof end;

theorem Th33: :: GENEALG1:33

for n1, n2, n3, n4 being 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 = crossover p1,p2,n2,n3,n4 ) & ( n2 >= len p1 implies crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n1,n3,n4 ) & ( n3 >= len p1 implies crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n1,n2,n4 ) & ( n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n1,n2,n3 ) )

proof end;

theorem Th34: :: GENEALG1:34

for n1, n2, n3, n4 being 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 = crossover p1,p2,n3,n4 ) & ( n1 >= len p1 & n3 >= len p1 implies crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n2,n4 ) & ( n1 >= len p1 & n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n2,n3 ) & ( n2 >= len p1 & n3 >= len p1 implies crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n1,n4 ) & ( n2 >= len p1 & n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n1,n3 ) & ( n3 >= len p1 & n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n1,n2 ) )

proof end;

theorem Th35: :: GENEALG1:35

for n1, n2, n3, n4 being 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 = crossover p1,p2,n4 ) & ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n3 ) & ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n2 ) & ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n1 ) )

proof end;

theorem Th36: :: GENEALG1:36

for n1, n2, n3, n4 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 holds
crossover p1,p2,n1,n2,n3,n4 = p1

proof end;

theorem Th37: :: GENEALG1:37

for n1, n2, n3, n4 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n1,n2,n4,n3 & crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n1,n3,n2,n4 & crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n1,n3,n4,n2 & crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n1,n4,n3,n2 & crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n2,n1,n3,n4 & crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n2,n1,n4,n3 & crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n2,n3,n1,n4 & crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n2,n3,n4,n1 & crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n2,n4,n1,n3 & crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n2,n4,n3,n1 & crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n3,n2,n1,n4 & crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n3,n2,n4,n1 & crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n3,n4,n1,n2 & crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n3,n4,n2,n1 & crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n4,n2,n3,n1 & crossover p1,p2,n1,n2,n3,n4 = crossover p1,p2,n4,n3,n2,n1 )

proof end;

theorem Th38: :: GENEALG1:38

for n1, n3, n4, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,n1,n1,n3,n4 = crossover p1,p2,n3,n4 & crossover p1,p2,n1,n2,n1,n4 = crossover p1,p2,n2,n4 & crossover p1,p2,n1,n2,n3,n1 = crossover p1,p2,n2,n3 & crossover p1,p2,n1,n2,n2,n4 = crossover p1,p2,n1,n4 & crossover p1,p2,n1,n2,n3,n2 = crossover p1,p2,n1,n3 & crossover p1,p2,n1,n2,n3,n3 = crossover p1,p2,n1,n2 )

proof end;

theorem :: GENEALG1:39

for n1, n3, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,n1,n1,n3,n3 = p1 & crossover p1,p2,n1,n2,n1,n2 = p1 & crossover p1,p2,n1,n2,n2,n1 = p1 )

proof end;

theorem Th40: :: GENEALG1:40

for n1, n2, n3, n4, n5 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover p1,p2,n1,n2,n3,n4,n5 is Individual of S

proof end;

definition

let S be Gene-Set;
let p1, p2 be Individual of S;
let n1, n2, n3, n4, n5 be Element of NAT ;
:: original: crossover
redefine func crossover p1,p2,n1,n2,n3,n4,n5 -> Individual of S;
correctness
coherence
crossover p1,p2,n1,n2,n3,n4,n5 is Individual of S;
by Th40;

end;

theorem Th41: :: GENEALG1:41

for n2, n3, n4, n5, n1 being Element of NAT
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 )

proof end;

theorem :: GENEALG1:42

for n3, n4, n5, n2, n1 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,0 ,0 ,n3,n4,n5 = crossover p1,p2,n3,n4,n5 & crossover p1,p2,0 ,n2,0 ,n4,n5 = crossover p1,p2,n2,n4,n5 & crossover p1,p2,0 ,n2,n3,0 ,n5 = crossover p1,p2,n2,n3,n5 & crossover p1,p2,0 ,n2,n3,n4,0 = crossover p1,p2,n2,n3,n4 & crossover p1,p2,n1,0 ,0 ,n4,n5 = crossover p1,p2,n1,n4,n5 & crossover p1,p2,n1,0 ,n3,0 ,n5 = crossover p1,p2,n1,n3,n5 & crossover p1,p2,n1,0 ,n3,n4,0 = crossover p1,p2,n1,n3,n4 & crossover p1,p2,n1,n2,0 ,0 ,n5 = crossover p1,p2,n1,n2,n5 & crossover p1,p2,n1,n2,0 ,n4,0 = crossover p1,p2,n1,n2,n4 & crossover p1,p2,n1,n2,n3,0 ,0 = crossover p1,p2,n1,n2,n3 )

proof end;

theorem :: GENEALG1:43

for n4, n5, n3, n2, n1 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,0 ,0 ,0 ,n4,n5 = crossover p2,p1,n4,n5 & crossover p1,p2,0 ,0 ,n3,0 ,n5 = crossover p2,p1,n3,n5 & crossover p1,p2,0 ,0 ,n3,n4,0 = crossover p2,p1,n3,n4 & crossover p1,p2,0 ,n2,0 ,0 ,n5 = crossover p2,p1,n2,n5 & crossover p1,p2,0 ,n2,0 ,n4,0 = crossover p2,p1,n2,n4 & crossover p1,p2,0 ,n2,n3,0 ,0 = crossover p2,p1,n2,n3 & crossover p1,p2,n1,0 ,0 ,0 ,n5 = crossover p2,p1,n1,n5 & crossover p1,p2,n1,0 ,0 ,n4,0 = crossover p2,p1,n1,n4 & crossover p1,p2,n1,0 ,n3,0 ,0 = crossover p2,p1,n1,n3 & crossover p1,p2,n1,n2,0 ,0 ,0 = crossover p2,p1,n1,n2 )

proof end;

theorem :: GENEALG1:44

for n5, n4, n3, n2, n1 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,0 ,0 ,0 ,0 ,n5 = crossover p1,p2,n5 & crossover p1,p2,0 ,0 ,0 ,n4,0 = crossover p1,p2,n4 & crossover p1,p2,0 ,0 ,n3,0 ,0 = crossover p1,p2,n3 & crossover p1,p2,0 ,n2,0 ,0 ,0 = crossover p1,p2,n2 & crossover p1,p2,n1,0 ,0 ,0 ,0 = crossover p1,p2,n1 )

proof end;

theorem :: GENEALG1:45

for S being Gene-Set
for p1, p2 being Individual of S holds crossover p1,p2,0 ,0 ,0 ,0 ,0 = p2

proof end;

theorem Th46: :: GENEALG1:46

for n1, n2, n3, n4, n5 being 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 ) )

proof end;

theorem :: GENEALG1:47

for n1, n2, n3, n4, n5 being 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 ) )

proof end;

theorem :: GENEALG1:48

for n1, n2, n3, n4, n5 being 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 ) )

proof end;

theorem :: GENEALG1:49

for n1, n2, n3, n4, n5 being 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 ) )

proof end;

theorem :: GENEALG1:50

for n1, n2, n3, n4, n5 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 holds
crossover p1,p2,n1,n2,n3,n4,n5 = p1

proof end;

theorem Th51: :: GENEALG1:51

for n1, n2, n3, n4, n5 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n1,n3,n4,n5 & crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n3,n2,n1,n4,n5 & crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n4,n2,n3,n1,n5 & crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n5,n2,n3,n4,n1 )

proof end;

theorem Th52: :: GENEALG1:52

for n1, n3, n4, n5, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,n1,n1,n3,n4,n5 = crossover p1,p2,n3,n4,n5 & crossover p1,p2,n1,n2,n1,n4,n5 = crossover p1,p2,n2,n4,n5 & crossover p1,p2,n1,n2,n3,n1,n5 = crossover p1,p2,n2,n3,n5 & crossover p1,p2,n1,n2,n3,n4,n1 = crossover p1,p2,n2,n3,n4 )

proof end;

theorem Th53: :: GENEALG1:53

for n1, n2, n3, n4, n5, n6 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover p1,p2,n1,n2,n3,n4,n5,n6 is Individual of S

proof end;

definition

let S be Gene-Set;
let p1, p2 be Individual of S;
let n1, n2, n3, n4, n5, n6 be Element of NAT ;
:: original: crossover
redefine func crossover p1,p2,n1,n2,n3,n4,n5,n6 -> Individual of S;
correctness
coherence
crossover p1,p2,n1,n2,n3,n4,n5,n6 is Individual of S;
by Th53;

end;

theorem :: GENEALG1:54

for n2, n3, n4, n5, n6, n1 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,0 ,n2,n3,n4,n5,n6 = crossover p2,p1,n2,n3,n4,n5,n6 & crossover p1,p2,n1,0 ,n3,n4,n5,n6 = crossover p2,p1,n1,n3,n4,n5,n6 & crossover p1,p2,n1,n2,0 ,n4,n5,n6 = crossover p2,p1,n1,n2,n4,n5,n6 & crossover p1,p2,n1,n2,n3,0 ,n5,n6 = crossover p2,p1,n1,n2,n3,n5,n6 & crossover p1,p2,n1,n2,n3,n4,0 ,n6 = crossover p2,p1,n1,n2,n3,n4,n6 & crossover p1,p2,n1,n2,n3,n4,n5,0 = crossover p2,p1,n1,n2,n3,n4,n5 )

proof end;

theorem :: GENEALG1:55

for n1, n2, n3, n4, n5, n6 being 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,n6 = crossover p1,p2,n2,n3,n4,n5,n6 ) & ( n2 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n1,n3,n4,n5,n6 ) & ( n3 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n1,n2,n4,n5,n6 ) & ( n4 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n1,n2,n3,n5,n6 ) & ( n5 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n1,n2,n3,n4,n6 ) & ( n6 >= len p1 implies crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n1,n2,n3,n4,n5 ) )

proof end;

theorem Th56: :: GENEALG1:56

for n1, n2, n3, n4, n5, n6 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n2,n1,n3,n4,n5,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n3,n2,n1,n4,n5,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n4,n2,n3,n1,n5,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n5,n2,n3,n4,n1,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n6,n2,n3,n4,n5,n1 )

proof end;

theorem :: GENEALG1:57

for n1, n3, n4, n5, n6, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,n1,n1,n3,n4,n5,n6 = crossover p1,p2,n3,n4,n5,n6 & crossover p1,p2,n1,n2,n1,n4,n5,n6 = crossover p1,p2,n2,n4,n5,n6 & crossover p1,p2,n1,n2,n3,n1,n5,n6 = crossover p1,p2,n2,n3,n5,n6 & crossover p1,p2,n1,n2,n3,n4,n1,n6 = crossover p1,p2,n2,n3,n4,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n1 = crossover p1,p2,n2,n3,n4,n5 )

proof end;