:: GENEALG1 semantic presentation
theorem Th1: :: GENEALG1:1
theorem Th2: :: GENEALG1:2
:: deftheorem Def1 defines Individual GENEALG1:def 1 :
:: deftheorem Def2 defines crossover GENEALG1:def 2 :
definition
let c
1 be
Gene-Set;
let c
2, c
3 be
FinSequence of
GA-Space c
1;
let c
4, c
5 be
Nat;
func crossover c
2,c
3,c
4,c
5 -> FinSequence of
GA-Space a
1 equals :: GENEALG1:def 3
crossover (crossover a2,a3,a4),
(crossover a3,a2,a4),a
5;
correctness
coherence
crossover (crossover c2,c3,c4),(crossover c3,c2,c4),c5 is FinSequence of GA-Space c1;
;
end;
:: deftheorem Def3 defines crossover GENEALG1:def 3 :
for b
1 being
Gene-Setfor b
2, b
3 being
FinSequence of
GA-Space b
1for b
4, b
5 being
Nat holds
crossover b
2,b
3,b
4,b
5 = crossover (crossover b2,b3,b4),
(crossover b3,b2,b4),b
5;
definition
let c
1 be
Gene-Set;
let c
2, c
3 be
FinSequence of
GA-Space c
1;
let c
4, c
5, c
6 be
Nat;
func crossover c
2,c
3,c
4,c
5,c
6 -> FinSequence of
GA-Space a
1 equals :: GENEALG1:def 4
crossover (crossover a2,a3,a4,a5),
(crossover a3,a2,a4,a5),a
6;
correctness
coherence
crossover (crossover c2,c3,c4,c5),(crossover c3,c2,c4,c5),c6 is FinSequence of GA-Space c1;
;
end;
:: deftheorem Def4 defines crossover GENEALG1:def 4 :
for b
1 being
Gene-Setfor b
2, b
3 being
FinSequence of
GA-Space b
1for b
4, b
5, b
6 being
Nat holds
crossover b
2,b
3,b
4,b
5,b
6 = crossover (crossover b2,b3,b4,b5),
(crossover b3,b2,b4,b5),b
6;
definition
let c
1 be
Gene-Set;
let c
2, c
3 be
FinSequence of
GA-Space c
1;
let c
4, c
5, c
6, c
7 be
Nat;
func crossover c
2,c
3,c
4,c
5,c
6,c
7 -> FinSequence of
GA-Space a
1 equals :: GENEALG1:def 5
crossover (crossover a2,a3,a4,a5,a6),
(crossover a3,a2,a4,a5,a6),a
7;
correctness
coherence
crossover (crossover c2,c3,c4,c5,c6),(crossover c3,c2,c4,c5,c6),c7 is FinSequence of GA-Space c1;
;
end;
:: deftheorem Def5 defines crossover GENEALG1:def 5 :
for b
1 being
Gene-Setfor b
2, b
3 being
FinSequence of
GA-Space b
1for b
4, b
5, b
6, b
7 being
Nat holds
crossover b
2,b
3,b
4,b
5,b
6,b
7 = crossover (crossover b2,b3,b4,b5,b6),
(crossover b3,b2,b4,b5,b6),b
7;
definition
let c
1 be
Gene-Set;
let c
2, c
3 be
FinSequence of
GA-Space c
1;
let c
4, c
5, c
6, c
7, c
8 be
Nat;
func crossover c
2,c
3,c
4,c
5,c
6,c
7,c
8 -> FinSequence of
GA-Space a
1 equals :: GENEALG1:def 6
crossover (crossover a2,a3,a4,a5,a6,a7),
(crossover a3,a2,a4,a5,a6,a7),a
8;
correctness
coherence
crossover (crossover c2,c3,c4,c5,c6,c7),(crossover c3,c2,c4,c5,c6,c7),c8 is FinSequence of GA-Space c1;
;
end;
:: deftheorem Def6 defines crossover GENEALG1:def 6 :
for b
1 being
Gene-Setfor b
2, b
3 being
FinSequence of
GA-Space b
1for b
4, b
5, b
6, b
7, b
8 being
Nat holds
crossover b
2,b
3,b
4,b
5,b
6,b
7,b
8 = crossover (crossover b2,b3,b4,b5,b6,b7),
(crossover b3,b2,b4,b5,b6,b7),b
8;
definition
let c
1 be
Gene-Set;
let c
2, c
3 be
FinSequence of
GA-Space c
1;
let c
4, c
5, c
6, c
7, c
8, c
9 be
Nat;
func crossover c
2,c
3,c
4,c
5,c
6,c
7,c
8,c
9 -> FinSequence of
GA-Space a
1 equals :: GENEALG1:def 7
crossover (crossover a2,a3,a4,a5,a6,a7,a8),
(crossover a3,a2,a4,a5,a6,a7,a8),a
9;
correctness
coherence
crossover (crossover c2,c3,c4,c5,c6,c7,c8),(crossover c3,c2,c4,c5,c6,c7,c8),c9 is FinSequence of GA-Space c1;
;
end;
:: deftheorem Def7 defines crossover GENEALG1:def 7 :
for b
1 being
Gene-Setfor b
2, b
3 being
FinSequence of
GA-Space b
1for b
4, b
5, b
6, b
7, b
8, b
9 being
Nat holds
crossover b
2,b
3,b
4,b
5,b
6,b
7,b
8,b
9 = crossover (crossover b2,b3,b4,b5,b6,b7,b8),
(crossover b3,b2,b4,b5,b6,b7,b8),b
9;
theorem Th3: :: GENEALG1:3
theorem Th4: :: GENEALG1:4
theorem Th5: :: GENEALG1:5
theorem Th6: :: GENEALG1:6
theorem Th7: :: GENEALG1:7
theorem Th8: :: GENEALG1:8
theorem Th9: :: GENEALG1:9
theorem Th10: :: GENEALG1:10
theorem Th11: :: GENEALG1:11
theorem Th12: :: GENEALG1:12
theorem Th13: :: GENEALG1:13
theorem Th14: :: GENEALG1:14
definition
let c
1 be
Gene-Set;
let c
2, c
3 be
Individual of c
1;
let c
4, c
5, c
6 be
Nat;
redefine func crossover as
crossover c
2,c
3,c
4,c
5,c
6 -> Individual of a
1;
correctness
coherence
crossover c2,c3,c4,c5,c6 is Individual of c1;
by Th14;
end;
theorem Th15: :: GENEALG1:15
for b
1, b
2, b
3 being
Natfor b
4 being
Gene-Setfor b
5, b
6 being
Individual of b
4 holds
(
crossover b
5,b
6,0,b
1,b
2 = crossover b
6,b
5,b
1,b
2 &
crossover b
5,b
6,b
3,0,b
2 = crossover b
6,b
5,b
3,b
2 &
crossover b
5,b
6,b
3,b
1,0
= crossover b
6,b
5,b
3,b
1 )
theorem Th16: :: GENEALG1:16
for b
1, b
2, b
3 being
Natfor b
4 being
Gene-Setfor b
5, b
6 being
Individual of b
4 holds
(
crossover b
5,b
6,0,0,b
1 = crossover b
5,b
6,b
1 &
crossover b
5,b
6,b
2,0,0
= crossover b
5,b
6,b
2 &
crossover b
5,b
6,0,b
3,0
= crossover b
5,b
6,b
3 )
theorem Th17: :: GENEALG1:17
theorem Th18: :: GENEALG1:18
for b
1, b
2, b
3 being
Natfor b
4 being
Gene-Setfor b
5, b
6 being
Individual of b
4 holds
( b
1 >= len b
5 implies
crossover b
5,b
6,b
1,b
2,b
3 = crossover b
5,b
6,b
2,b
3 )
theorem Th19: :: GENEALG1:19
for b
1, b
2, b
3 being
Natfor b
4 being
Gene-Setfor b
5, b
6 being
Individual of b
4 holds
( b
1 >= len b
5 implies
crossover b
5,b
6,b
2,b
1,b
3 = crossover b
5,b
6,b
2,b
3 )
theorem Th20: :: GENEALG1:20
for b
1, b
2, b
3 being
Natfor b
4 being
Gene-Setfor b
5, b
6 being
Individual of b
4 holds
( b
1 >= len b
5 implies
crossover b
5,b
6,b
2,b
3,b
1 = crossover b
5,b
6,b
2,b
3 )
theorem Th21: :: GENEALG1:21
theorem Th22: :: GENEALG1:22
theorem Th23: :: GENEALG1:23
theorem Th24: :: GENEALG1:24
theorem Th25: :: GENEALG1:25
for b
1, b
2, b
3 being
Natfor b
4 being
Gene-Setfor b
5, b
6 being
Individual of b
4 holds
(
crossover b
5,b
6,b
1,b
2,b
3 = crossover b
5,b
6,b
2,b
1,b
3 &
crossover b
5,b
6,b
1,b
2,b
3 = crossover b
5,b
6,b
1,b
3,b
2 )
theorem Th26: :: GENEALG1:26
for b
1, b
2, b
3 being
Natfor b
4 being
Gene-Setfor b
5, b
6 being
Individual of b
4 holds
crossover b
5,b
6,b
1,b
2,b
3 = crossover b
5,b
6,b
3,b
1,b
2
theorem Th27: :: GENEALG1:27
for b
1, b
2, b
3 being
Natfor b
4 being
Gene-Setfor b
5, b
6 being
Individual of b
4 holds
(
crossover b
5,b
6,b
1,b
1,b
2 = crossover b
5,b
6,b
2 &
crossover b
5,b
6,b
1,b
3,b
1 = crossover b
5,b
6,b
3 &
crossover b
5,b
6,b
1,b
3,b
3 = crossover b
5,b
6,b
1 )
theorem Th28: :: GENEALG1:28
definition
let c
1 be
Gene-Set;
let c
2, c
3 be
Individual of c
1;
let c
4, c
5, c
6, c
7 be
Nat;
redefine func crossover as
crossover c
2,c
3,c
4,c
5,c
6,c
7 -> Individual of a
1;
correctness
coherence
crossover c2,c3,c4,c5,c6,c7 is Individual of c1;
by Th28;
end;
theorem Th29: :: GENEALG1:29
for b
1, b
2, b
3, b
4 being
Natfor b
5 being
Gene-Setfor b
6, b
7 being
Individual of b
5 holds
(
crossover b
6,b
7,0,b
1,b
2,b
3 = crossover b
7,b
6,b
1,b
2,b
3 &
crossover b
6,b
7,b
4,0,b
2,b
3 = crossover b
7,b
6,b
4,b
2,b
3 &
crossover b
6,b
7,b
4,b
1,0,b
3 = crossover b
7,b
6,b
4,b
1,b
3 &
crossover b
6,b
7,b
4,b
1,b
2,0
= crossover b
7,b
6,b
4,b
1,b
2 )
theorem Th30: :: GENEALG1:30
for b
1, b
2, b
3, b
4 being
Natfor b
5 being
Gene-Setfor b
6, b
7 being
Individual of b
5 holds
(
crossover b
6,b
7,0,0,b
1,b
2 = crossover b
6,b
7,b
1,b
2 &
crossover b
6,b
7,0,b
3,0,b
2 = crossover b
6,b
7,b
3,b
2 &
crossover b
6,b
7,0,b
3,b
1,0
= crossover b
6,b
7,b
3,b
1 &
crossover b
6,b
7,b
4,0,b
1,0
= crossover b
6,b
7,b
4,b
1 &
crossover b
6,b
7,b
4,0,0,b
2 = crossover b
6,b
7,b
4,b
2 &
crossover b
6,b
7,b
4,b
3,0,0
= crossover b
6,b
7,b
4,b
3 )
theorem Th31: :: GENEALG1:31
for b
1, b
2, b
3, b
4 being
Natfor b
5 being
Gene-Setfor b
6, b
7 being
Individual of b
5 holds
(
crossover b
6,b
7,b
1,0,0,0
= crossover b
7,b
6,b
1 &
crossover b
6,b
7,0,b
2,0,0
= crossover b
7,b
6,b
2 &
crossover b
6,b
7,0,0,b
3,0
= crossover b
7,b
6,b
3 &
crossover b
6,b
7,0,0,0,b
4 = crossover b
7,b
6,b
4 )
theorem Th32: :: GENEALG1:32
theorem Th33: :: GENEALG1:33
for b
1, b
2, b
3, b
4 being
Natfor b
5 being
Gene-Setfor b
6, b
7 being
Individual of b
5 holds
( ( b
1 >= len b
6 implies
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
2,b
3,b
4 ) & ( b
2 >= len b
6 implies
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
1,b
3,b
4 ) & ( b
3 >= len b
6 implies
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
1,b
2,b
4 ) & ( b
4 >= len b
6 implies
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
1,b
2,b
3 ) )
theorem Th34: :: GENEALG1:34
for b
1, b
2, b
3, b
4 being
Natfor b
5 being
Gene-Setfor b
6, b
7 being
Individual of b
5 holds
( ( b
1 >= len b
6 & b
2 >= len b
6 implies
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
3,b
4 ) & ( b
1 >= len b
6 & b
3 >= len b
6 implies
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
2,b
4 ) & ( b
1 >= len b
6 & b
4 >= len b
6 implies
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
2,b
3 ) & ( b
2 >= len b
6 & b
3 >= len b
6 implies
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
1,b
4 ) & ( b
2 >= len b
6 & b
4 >= len b
6 implies
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
1,b
3 ) & ( b
3 >= len b
6 & b
4 >= len b
6 implies
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
1,b
2 ) )
theorem Th35: :: GENEALG1:35
for b
1, b
2, b
3, b
4 being
Natfor b
5 being
Gene-Setfor b
6, b
7 being
Individual of b
5 holds
( ( b
1 >= len b
6 & b
2 >= len b
6 & b
3 >= len b
6 implies
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
4 ) & ( b
1 >= len b
6 & b
2 >= len b
6 & b
4 >= len b
6 implies
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
3 ) & ( b
1 >= len b
6 & b
3 >= len b
6 & b
4 >= len b
6 implies
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
2 ) & ( b
2 >= len b
6 & b
3 >= len b
6 & b
4 >= len b
6 implies
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
1 ) )
theorem Th36: :: GENEALG1:36
theorem Th37: :: GENEALG1:37
for b
1, b
2, b
3, b
4 being
Natfor b
5 being
Gene-Setfor b
6, b
7 being
Individual of b
5 holds
(
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
1,b
2,b
4,b
3 &
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
1,b
3,b
2,b
4 &
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
1,b
3,b
4,b
2 &
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
1,b
4,b
3,b
2 &
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
2,b
1,b
3,b
4 &
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
2,b
1,b
4,b
3 &
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
2,b
3,b
1,b
4 &
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
2,b
3,b
4,b
1 &
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
2,b
4,b
1,b
3 &
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
2,b
4,b
3,b
1 &
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
3,b
2,b
1,b
4 &
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
3,b
2,b
4,b
1 &
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
3,b
4,b
1,b
2 &
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
3,b
4,b
2,b
1 &
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
4,b
2,b
3,b
1 &
crossover b
6,b
7,b
1,b
2,b
3,b
4 = crossover b
6,b
7,b
4,b
3,b
2,b
1 )
theorem Th38: :: GENEALG1:38
for b
1, b
2, b
3, b
4 being
Natfor b
5 being
Gene-Setfor b
6, b
7 being
Individual of b
5 holds
(
crossover b
6,b
7,b
1,b
1,b
2,b
3 = crossover b
6,b
7,b
2,b
3 &
crossover b
6,b
7,b
1,b
4,b
1,b
3 = crossover b
6,b
7,b
4,b
3 &
crossover b
6,b
7,b
1,b
4,b
2,b
1 = crossover b
6,b
7,b
4,b
2 &
crossover b
6,b
7,b
1,b
4,b
4,b
3 = crossover b
6,b
7,b
1,b
3 &
crossover b
6,b
7,b
1,b
4,b
2,b
4 = crossover b
6,b
7,b
1,b
2 &
crossover b
6,b
7,b
1,b
4,b
2,b
2 = crossover b
6,b
7,b
1,b
4 )
theorem Th39: :: GENEALG1:39
for b
1, b
2, b
3 being
Natfor b
4 being
Gene-Setfor b
5, b
6 being
Individual of b
4 holds
(
crossover b
5,b
6,b
1,b
1,b
2,b
2 = b
5 &
crossover b
5,b
6,b
1,b
3,b
1,b
3 = b
5 &
crossover b
5,b
6,b
1,b
3,b
3,b
1 = b
5 )
theorem Th40: :: GENEALG1:40
for b
1, b
2, b
3, b
4, b
5 being
Natfor b
6 being
Gene-Setfor b
7, b
8 being
Individual of b
6 holds
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 is
Individual of b
6
definition
let c
1 be
Gene-Set;
let c
2, c
3 be
Individual of c
1;
let c
4, c
5, c
6, c
7, c
8 be
Nat;
redefine func crossover as
crossover c
2,c
3,c
4,c
5,c
6,c
7,c
8 -> Individual of a
1;
correctness
coherence
crossover c2,c3,c4,c5,c6,c7,c8 is Individual of c1;
by Th40;
end;
theorem Th41: :: GENEALG1:41
for b
1, b
2, b
3, b
4, b
5 being
Natfor b
6 being
Gene-Setfor b
7, b
8 being
Individual of b
6 holds
(
crossover b
7,b
8,0,b
1,b
2,b
3,b
4 = crossover b
8,b
7,b
1,b
2,b
3,b
4 &
crossover b
7,b
8,b
5,0,b
2,b
3,b
4 = crossover b
8,b
7,b
5,b
2,b
3,b
4 &
crossover b
7,b
8,b
5,b
1,0,b
3,b
4 = crossover b
8,b
7,b
5,b
1,b
3,b
4 &
crossover b
7,b
8,b
5,b
1,b
2,0,b
4 = crossover b
8,b
7,b
5,b
1,b
2,b
4 &
crossover b
7,b
8,b
5,b
1,b
2,b
3,0
= crossover b
8,b
7,b
5,b
1,b
2,b
3 )
theorem Th42: :: GENEALG1:42
for b
1, b
2, b
3, b
4, b
5 being
Natfor b
6 being
Gene-Setfor b
7, b
8 being
Individual of b
6 holds
(
crossover b
7,b
8,0,0,b
1,b
2,b
3 = crossover b
7,b
8,b
1,b
2,b
3 &
crossover b
7,b
8,0,b
4,0,b
2,b
3 = crossover b
7,b
8,b
4,b
2,b
3 &
crossover b
7,b
8,0,b
4,b
1,0,b
3 = crossover b
7,b
8,b
4,b
1,b
3 &
crossover b
7,b
8,0,b
4,b
1,b
2,0
= crossover b
7,b
8,b
4,b
1,b
2 &
crossover b
7,b
8,b
5,0,0,b
2,b
3 = crossover b
7,b
8,b
5,b
2,b
3 &
crossover b
7,b
8,b
5,0,b
1,0,b
3 = crossover b
7,b
8,b
5,b
1,b
3 &
crossover b
7,b
8,b
5,0,b
1,b
2,0
= crossover b
7,b
8,b
5,b
1,b
2 &
crossover b
7,b
8,b
5,b
4,0,0,b
3 = crossover b
7,b
8,b
5,b
4,b
3 &
crossover b
7,b
8,b
5,b
4,0,b
2,0
= crossover b
7,b
8,b
5,b
4,b
2 &
crossover b
7,b
8,b
5,b
4,b
1,0,0
= crossover b
7,b
8,b
5,b
4,b
1 )
theorem Th43: :: GENEALG1:43
for b
1, b
2, b
3, b
4, b
5 being
Natfor b
6 being
Gene-Setfor b
7, b
8 being
Individual of b
6 holds
(
crossover b
7,b
8,0,0,0,b
1,b
2 = crossover b
8,b
7,b
1,b
2 &
crossover b
7,b
8,0,0,b
3,0,b
2 = crossover b
8,b
7,b
3,b
2 &
crossover b
7,b
8,0,0,b
3,b
1,0
= crossover b
8,b
7,b
3,b
1 &
crossover b
7,b
8,0,b
4,0,0,b
2 = crossover b
8,b
7,b
4,b
2 &
crossover b
7,b
8,0,b
4,0,b
1,0
= crossover b
8,b
7,b
4,b
1 &
crossover b
7,b
8,0,b
4,b
3,0,0
= crossover b
8,b
7,b
4,b
3 &
crossover b
7,b
8,b
5,0,0,0,b
2 = crossover b
8,b
7,b
5,b
2 &
crossover b
7,b
8,b
5,0,0,b
1,0
= crossover b
8,b
7,b
5,b
1 &
crossover b
7,b
8,b
5,0,b
3,0,0
= crossover b
8,b
7,b
5,b
3 &
crossover b
7,b
8,b
5,b
4,0,0,0
= crossover b
8,b
7,b
5,b
4 )
theorem Th44: :: GENEALG1:44
for b
1, b
2, b
3, b
4, b
5 being
Natfor b
6 being
Gene-Setfor b
7, b
8 being
Individual of b
6 holds
(
crossover b
7,b
8,0,0,0,0,b
1 = crossover b
7,b
8,b
1 &
crossover b
7,b
8,0,0,0,b
2,0
= crossover b
7,b
8,b
2 &
crossover b
7,b
8,0,0,b
3,0,0
= crossover b
7,b
8,b
3 &
crossover b
7,b
8,0,b
4,0,0,0
= crossover b
7,b
8,b
4 &
crossover b
7,b
8,b
5,0,0,0,0
= crossover b
7,b
8,b
5 )
theorem Th45: :: GENEALG1:45
theorem Th46: :: GENEALG1:46
for b
1, b
2, b
3, b
4, b
5 being
Natfor b
6 being
Gene-Setfor b
7, b
8 being
Individual of b
6 holds
( ( b
1 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
2,b
3,b
4,b
5 ) & ( b
2 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
1,b
3,b
4,b
5 ) & ( b
3 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
1,b
2,b
4,b
5 ) & ( b
4 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
1,b
2,b
3,b
5 ) & ( b
5 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
1,b
2,b
3,b
4 ) )
theorem Th47: :: GENEALG1:47
for b
1, b
2, b
3, b
4, b
5 being
Natfor b
6 being
Gene-Setfor b
7, b
8 being
Individual of b
6 holds
( ( b
1 >= len b
7 & b
2 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
3,b
4,b
5 ) & ( b
1 >= len b
7 & b
3 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
2,b
4,b
5 ) & ( b
1 >= len b
7 & b
4 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
2,b
3,b
5 ) & ( b
1 >= len b
7 & b
5 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
2,b
3,b
4 ) & ( b
2 >= len b
7 & b
3 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
1,b
4,b
5 ) & ( b
2 >= len b
7 & b
4 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
1,b
3,b
5 ) & ( b
2 >= len b
7 & b
5 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
1,b
3,b
4 ) & ( b
3 >= len b
7 & b
4 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
1,b
2,b
5 ) & ( b
3 >= len b
7 & b
5 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
1,b
2,b
4 ) & ( b
4 >= len b
7 & b
5 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
1,b
2,b
3 ) )
theorem Th48: :: GENEALG1:48
for b
1, b
2, b
3, b
4, b
5 being
Natfor b
6 being
Gene-Setfor b
7, b
8 being
Individual of b
6 holds
( ( b
1 >= len b
7 & b
2 >= len b
7 & b
3 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
4,b
5 ) & ( b
1 >= len b
7 & b
2 >= len b
7 & b
4 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
3,b
5 ) & ( b
1 >= len b
7 & b
2 >= len b
7 & b
5 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
3,b
4 ) & ( b
1 >= len b
7 & b
3 >= len b
7 & b
4 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
2,b
5 ) & ( b
1 >= len b
7 & b
3 >= len b
7 & b
5 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
2,b
4 ) & ( b
1 >= len b
7 & b
4 >= len b
7 & b
5 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
2,b
3 ) & ( b
2 >= len b
7 & b
3 >= len b
7 & b
4 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
1,b
5 ) & ( b
2 >= len b
7 & b
3 >= len b
7 & b
5 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
1,b
4 ) & ( b
2 >= len b
7 & b
4 >= len b
7 & b
5 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
1,b
3 ) & ( b
3 >= len b
7 & b
4 >= len b
7 & b
5 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
1,b
2 ) )
theorem Th49: :: GENEALG1:49
for b
1, b
2, b
3, b
4, b
5 being
Natfor b
6 being
Gene-Setfor b
7, b
8 being
Individual of b
6 holds
( ( b
1 >= len b
7 & b
2 >= len b
7 & b
3 >= len b
7 & b
4 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
5 ) & ( b
1 >= len b
7 & b
2 >= len b
7 & b
3 >= len b
7 & b
5 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
4 ) & ( b
1 >= len b
7 & b
2 >= len b
7 & b
4 >= len b
7 & b
5 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
3 ) & ( b
1 >= len b
7 & b
3 >= len b
7 & b
4 >= len b
7 & b
5 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
2 ) & ( b
2 >= len b
7 & b
3 >= len b
7 & b
4 >= len b
7 & b
5 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
1 ) )
theorem Th50: :: GENEALG1:50
for b
1, b
2, b
3, b
4, b
5 being
Natfor b
6 being
Gene-Setfor b
7, b
8 being
Individual of b
6 holds
( b
1 >= len b
7 & b
2 >= len b
7 & b
3 >= len b
7 & b
4 >= len b
7 & b
5 >= len b
7 implies
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = b
7 )
theorem Th51: :: GENEALG1:51
for b
1, b
2, b
3, b
4, b
5 being
Natfor b
6 being
Gene-Setfor b
7, b
8 being
Individual of b
6 holds
(
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
2,b
1,b
3,b
4,b
5 &
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
3,b
2,b
1,b
4,b
5 &
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
4,b
2,b
3,b
1,b
5 &
crossover b
7,b
8,b
1,b
2,b
3,b
4,b
5 = crossover b
7,b
8,b
5,b
2,b
3,b
4,b
1 )
theorem Th52: :: GENEALG1:52
for b
1, b
2, b
3, b
4, b
5 being
Natfor b
6 being
Gene-Setfor b
7, b
8 being
Individual of b
6 holds
(
crossover b
7,b
8,b
1,b
1,b
2,b
3,b
4 = crossover b
7,b
8,b
2,b
3,b
4 &
crossover b
7,b
8,b
1,b
5,b
1,b
3,b
4 = crossover b
7,b
8,b
5,b
3,b
4 &
crossover b
7,b
8,b
1,b
5,b
2,b
1,b
4 = crossover b
7,b
8,b
5,b
2,b
4 &
crossover b
7,b
8,b
1,b
5,b
2,b
3,b
1 = crossover b
7,b
8,b
5,b
2,b
3 )
theorem Th53: :: GENEALG1:53
for b
1, b
2, b
3, b
4, b
5, b
6 being
Natfor b
7 being
Gene-Setfor b
8, b
9 being
Individual of b
7 holds
crossover b
8,b
9,b
1,b
2,b
3,b
4,b
5,b
6 is
Individual of b
7
definition
let c
1 be
Gene-Set;
let c
2, c
3 be
Individual of c
1;
let c
4, c
5, c
6, c
7, c
8, c
9 be
Nat;
redefine func crossover as
crossover c
2,c
3,c
4,c
5,c
6,c
7,c
8,c
9 -> Individual of a
1;
correctness
coherence
crossover c2,c3,c4,c5,c6,c7,c8,c9 is Individual of c1;
by Th53;
end;
theorem Th54: :: GENEALG1:54
for b
1, b
2, b
3, b
4, b
5, b
6 being
Natfor b
7 being
Gene-Setfor b
8, b
9 being
Individual of b
7 holds
(
crossover b
8,b
9,0,b
1,b
2,b
3,b
4,b
5 = crossover b
9,b
8,b
1,b
2,b
3,b
4,b
5 &
crossover b
8,b
9,b
6,0,b
2,b
3,b
4,b
5 = crossover b
9,b
8,b
6,b
2,b
3,b
4,b
5 &
crossover b
8,b
9,b
6,b
1,0,b
3,b
4,b
5 = crossover b
9,b
8,b
6,b
1,b
3,b
4,b
5 &
crossover b
8,b
9,b
6,b
1,b
2,0,b
4,b
5 = crossover b
9,b
8,b
6,b
1,b
2,b
4,b
5 &
crossover b
8,b
9,b
6,b
1,b
2,b
3,0,b
5 = crossover b
9,b
8,b
6,b
1,b
2,b
3,b
5 &
crossover b
8,b
9,b
6,b
1,b
2,b
3,b
4,0
= crossover b
9,b
8,b
6,b
1,b
2,b
3,b
4 )
theorem Th55: :: GENEALG1:55
for b
1, b
2, b
3, b
4, b
5, b
6 being
Natfor b
7 being
Gene-Setfor b
8, b
9 being
Individual of b
7 holds
( ( b
1 >= len b
8 implies
crossover b
8,b
9,b
1,b
2,b
3,b
4,b
5,b
6 = crossover b
8,b
9,b
2,b
3,b
4,b
5,b
6 ) & ( b
2 >= len b
8 implies
crossover b
8,b
9,b
1,b
2,b
3,b
4,b
5,b
6 = crossover b
8,b
9,b
1,b
3,b
4,b
5,b
6 ) & ( b
3 >= len b
8 implies
crossover b
8,b
9,b
1,b
2,b
3,b
4,b
5,b
6 = crossover b
8,b
9,b
1,b
2,b
4,b
5,b
6 ) & ( b
4 >= len b
8 implies
crossover b
8,b
9,b
1,b
2,b
3,b
4,b
5,b
6 = crossover b
8,b
9,b
1,b
2,b
3,b
5,b
6 ) & ( b
5 >= len b
8 implies
crossover b
8,b
9,b
1,b
2,b
3,b
4,b
5,b
6 = crossover b
8,b
9,b
1,b
2,b
3,b
4,b
6 ) & ( b
6 >= len b
8 implies
crossover b
8,b
9,b
1,b
2,b
3,b
4,b
5,b
6 = crossover b
8,b
9,b
1,b
2,b
3,b
4,b
5 ) )
theorem Th56: :: GENEALG1:56
for b
1, b
2, b
3, b
4, b
5, b
6 being
Natfor b
7 being
Gene-Setfor b
8, b
9 being
Individual of b
7 holds
(
crossover b
8,b
9,b
1,b
2,b
3,b
4,b
5,b
6 = crossover b
8,b
9,b
2,b
1,b
3,b
4,b
5,b
6 &
crossover b
8,b
9,b
1,b
2,b
3,b
4,b
5,b
6 = crossover b
8,b
9,b
3,b
2,b
1,b
4,b
5,b
6 &
crossover b
8,b
9,b
1,b
2,b
3,b
4,b
5,b
6 = crossover b
8,b
9,b
4,b
2,b
3,b
1,b
5,b
6 &
crossover b
8,b
9,b
1,b
2,b
3,b
4,b
5,b
6 = crossover b
8,b
9,b
5,b
2,b
3,b
4,b
1,b
6 &
crossover b
8,b
9,b
1,b
2,b
3,b
4,b
5,b
6 = crossover b
8,b
9,b
6,b
2,b
3,b
4,b
5,b
1 )
theorem Th57: :: GENEALG1:57
for b
1, b
2, b
3, b
4, b
5, b
6 being
Natfor b
7 being
Gene-Setfor b
8, b
9 being
Individual of b
7 holds
(
crossover b
8,b
9,b
1,b
1,b
2,b
3,b
4,b
5 = crossover b
8,b
9,b
2,b
3,b
4,b
5 &
crossover b
8,b
9,b
1,b
6,b
1,b
3,b
4,b
5 = crossover b
8,b
9,b
6,b
3,b
4,b
5 &
crossover b
8,b
9,b
1,b
6,b
2,b
1,b
4,b
5 = crossover b
8,b
9,b
6,b
2,b
4,b
5 &
crossover b
8,b
9,b
1,b
6,b
2,b
3,b
1,b
5 = crossover b
8,b
9,b
6,b
2,b
3,b
5 &
crossover b
8,b
9,b
1,b
6,b
2,b
3,b
4,b
1 = crossover b
8,b
9,b
6,b
2,b
3,b
4 )