:: Basic Properties of Genetic Algorithm
:: by Akihiko Uchibori and Noboru Endou
::
:: Received April 24, 1999
:: Copyright (c) 1999-2021 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;

:: Definitions of Gene-Set,GA-Space and Individual

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