:: GENEALG1 semantic presentation begin 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 let D be non empty set ; ::_thesis: for f1, f2 being FinSequence of D for n being Nat st n <= len f1 holds (f1 ^ f2) /^ n = (f1 /^ n) ^ f2 let f1, f2 be FinSequence of D; ::_thesis: for n being Nat st n <= len f1 holds (f1 ^ f2) /^ n = (f1 /^ n) ^ f2 let n be Nat; ::_thesis: ( n <= len f1 implies (f1 ^ f2) /^ n = (f1 /^ n) ^ f2 ) assume A1: n <= len f1 ; ::_thesis: (f1 ^ f2) /^ n = (f1 /^ n) ^ f2 reconsider n = n as Element of NAT by ORDINAL1:def_12; len (f1 ^ f2) = (len f1) + (len f2) by FINSEQ_1:22; then len f1 <= len (f1 ^ f2) by NAT_1:11; then A2: n <= len (f1 ^ f2) by A1, XXREAL_0:2; then A3: len ((f1 ^ f2) /^ n) = (len (f1 ^ f2)) - n by RFINSEQ:def_1; A4: len ((f1 /^ n) ^ f2) = (len (f1 /^ n)) + (len f2) by FINSEQ_1:22 .= ((len f1) - n) + (len f2) by A1, RFINSEQ:def_1 .= ((len f1) + (len f2)) - n ; A5: for i being Nat st 1 <= i & i <= len ((f1 ^ f2) /^ n) holds ((f1 ^ f2) /^ n) . i = ((f1 /^ n) ^ f2) . i proof let i be Nat; ::_thesis: ( 1 <= i & i <= len ((f1 ^ f2) /^ n) implies ((f1 ^ f2) /^ n) . i = ((f1 /^ n) ^ f2) . i ) assume that A6: 1 <= i and A7: i <= len ((f1 ^ f2) /^ n) ; ::_thesis: ((f1 ^ f2) /^ n) . i = ((f1 /^ n) ^ f2) . i i in Seg (len ((f1 ^ f2) /^ n)) by A6, A7, FINSEQ_1:1; then A8: i in dom ((f1 ^ f2) /^ n) by FINSEQ_1:def_3; reconsider i = i as Element of NAT by ORDINAL1:def_12; now__::_thesis:_((f1_^_f2)_/^_n)_._i_=_((f1_/^_n)_^_f2)_._i percases ( i <= (len f1) - n or (len f1) - n < i ) ; supposeA9: i <= (len f1) - n ; ::_thesis: ((f1 ^ f2) /^ n) . i = ((f1 /^ n) ^ f2) . i i <= i + n by NAT_1:11; then A10: 1 <= i + n by A6, XXREAL_0:2; i + n <= len f1 by A9, XREAL_1:19; then i + n in Seg (len f1) by A10, FINSEQ_1:1; then A11: i + n in dom f1 by FINSEQ_1:def_3; i <= len (f1 /^ n) by A1, A9, RFINSEQ:def_1; then i in Seg (len (f1 /^ n)) by A6, FINSEQ_1:1; then A12: i in dom (f1 /^ n) by FINSEQ_1:def_3; then A13: ((f1 /^ n) ^ f2) . i = (f1 /^ n) . i by FINSEQ_1:def_7 .= f1 . (i + n) by A1, A12, RFINSEQ:def_1 ; ((f1 ^ f2) /^ n) . i = (f1 ^ f2) . (i + n) by A2, A8, RFINSEQ:def_1 .= f1 . (i + n) by A11, FINSEQ_1:def_7 ; hence ((f1 ^ f2) /^ n) . i = ((f1 /^ n) ^ f2) . i by A13; ::_thesis: verum end; supposeA14: (len f1) - n < i ; ::_thesis: ((f1 ^ f2) /^ n) . i = ((f1 /^ n) ^ f2) . i then A15: len (f1 /^ n) < i by A1, RFINSEQ:def_1; i <= len ((f1 /^ n) ^ f2) by A3, A4, A7, FINSEQ_1:22; then A16: ((f1 /^ n) ^ f2) . i = f2 . (i - (len (f1 /^ n))) by A15, FINSEQ_1:24 .= f2 . (i - ((len f1) - n)) by A1, RFINSEQ:def_1 .= f2 . ((i + n) - (len f1)) ; A17: i + n <= len (f1 ^ f2) by A3, A7, XREAL_1:19; A18: len f1 < i + n by A14, XREAL_1:19; ((f1 ^ f2) /^ n) . i = (f1 ^ f2) . (i + n) by A2, A8, RFINSEQ:def_1 .= f2 . ((i + n) - (len f1)) by A18, A17, FINSEQ_1:24 ; hence ((f1 ^ f2) /^ n) . i = ((f1 /^ n) ^ f2) . i by A16; ::_thesis: verum end; end; end; hence ((f1 ^ f2) /^ n) . i = ((f1 /^ n) ^ f2) . i ; ::_thesis: verum end; len ((f1 ^ f2) /^ n) = len ((f1 /^ n) ^ f2) by A3, A4, FINSEQ_1:22; hence (f1 ^ f2) /^ n = (f1 /^ n) ^ f2 by A5, FINSEQ_1:14; ::_thesis: verum 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 let D be non empty set ; ::_thesis: for f1, f2 being FinSequence of D for i being Element of NAT holds (f1 ^ f2) | ((len f1) + i) = f1 ^ (f2 | i) let f1, f2 be FinSequence of D; ::_thesis: for i being Element of NAT holds (f1 ^ f2) | ((len f1) + i) = f1 ^ (f2 | i) let i be Element of NAT ; ::_thesis: (f1 ^ f2) | ((len f1) + i) = f1 ^ (f2 | i) ( (f1 ^ f2) | ((len f1) + i) = (f1 ^ f2) | (Seg ((len f1) + i)) & f2 | i = f2 | (Seg i) ) by FINSEQ_1:def_15; hence (f1 ^ f2) | ((len f1) + i) = f1 ^ (f2 | i) by FINSEQ_6:14; ::_thesis: verum 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 Union f = union (rng f) by CARD_3:def_4; hence not Union f is empty ; ::_thesis: verum 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 defpred S1[ set , set ] means \$2 in S . \$1; A1: for k being Nat st k in Seg (len S) holds ex x being Element of GA-Space S st S1[k,x] proof let k be Nat; ::_thesis: ( k in Seg (len S) implies ex x being Element of GA-Space S st S1[k,x] ) assume A2: k in Seg (len S) ; ::_thesis: ex x being Element of GA-Space S st S1[k,x] then reconsider k9 = k as Element of dom S by FINSEQ_1:def_3; S . k9 <> {} ; then consider x being Element of S . k such that A3: x in [#] (S . k) by SUBSET_1:4; k in dom S by A2, FINSEQ_1:def_3; then S . k in rng S by FUNCT_1:def_3; then [#] (S . k) c= union (rng S) by ZFMISC_1:74; then reconsider x = x as Element of GA-Space S by A3, CARD_3:def_4; take x ; ::_thesis: S1[k,x] thus S1[k,x] by A3; ::_thesis: verum end; consider IT being FinSequence of GA-Space S such that A4: ( dom IT = Seg (len S) & ( for k being Nat st k in Seg (len S) holds S1[k,IT . k] ) ) from FINSEQ_1:sch_5(A1); take IT ; ::_thesis: ( len IT = len S & ( for i being Element of NAT st i in dom IT holds IT . i in S . i ) ) thus ( len IT = len S & ( for i being Element of NAT st i in dom IT holds IT . i in S . i ) ) by A4, FINSEQ_1:def_3; ::_thesis: verum 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 ) ) ); begin 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); begin 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 let n be Element of NAT ; ::_thesis: for S being Gene-Set for p1, p2 being Individual of S holds crossover (p1,p2,n) is Individual of S let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n) is Individual of S let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,n) is Individual of S A1: len (crossover (p1,p2,n)) = len S proof A2: len (crossover (p1,p2,n)) = (len (p1 | n)) + (len (p2 /^ n)) by FINSEQ_1:22; now__::_thesis:_len_(crossover_(p1,p2,n))_=_len_S percases ( n <= len p1 or n > len p1 ) ; supposeA3: n <= len p1 ; ::_thesis: len (crossover (p1,p2,n)) = len S len (p2 /^ n) = (len p2) -' n by RFINSEQ:29 .= (len S) -' n by Def1 .= (len p1) -' n by Def1 .= (len p1) - n by A3, XREAL_1:233 ; then len (crossover (p1,p2,n)) = n + ((len p1) - n) by A2, A3, FINSEQ_1:59 .= len p1 ; hence len (crossover (p1,p2,n)) = len S by Def1; ::_thesis: verum end; supposeA4: n > len p1 ; ::_thesis: len (crossover (p1,p2,n)) = len S p1 | n = p1 | (Seg n) by FINSEQ_1:def_15; then A5: p1 | n = p1 by A4, FINSEQ_2:20; A6: (len p1) - n < 0 by A4, XREAL_1:49; len (p2 /^ n) = (len p2) -' n by RFINSEQ:29 .= (len S) -' n by Def1 .= (len p1) -' n by Def1 .= 0 by A6, XREAL_0:def_2 ; hence len (crossover (p1,p2,n)) = len S by A2, A5, Def1; ::_thesis: verum end; end; end; hence len (crossover (p1,p2,n)) = len S ; ::_thesis: verum end; for i being Element of NAT st i in dom (crossover (p1,p2,n)) holds (crossover (p1,p2,n)) . i in S . i proof let i be Element of NAT ; ::_thesis: ( i in dom (crossover (p1,p2,n)) implies (crossover (p1,p2,n)) . i in S . i ) assume A7: i in dom (crossover (p1,p2,n)) ; ::_thesis: (crossover (p1,p2,n)) . i in S . i now__::_thesis:_(crossover_(p1,p2,n))_._i_in_S_._i percases ( i in dom (p1 | n) or ex k being Nat st ( k in dom (p2 /^ n) & i = (len (p1 | n)) + k ) ) by A7, FINSEQ_1:25; supposeA8: i in dom (p1 | n) ; ::_thesis: (crossover (p1,p2,n)) . i in S . i A9: dom (p1 | n) c= dom p1 by FINSEQ_5:18; (p1 | n) . i = (p1 | n) /. i by A8, PARTFUN1:def_6 .= p1 /. i by A8, FINSEQ_4:70 .= p1 . i by A8, A9, PARTFUN1:def_6 ; then (p1 | n) . i in S . i by A8, A9, Def1; hence (crossover (p1,p2,n)) . i in S . i by A8, FINSEQ_1:def_7; ::_thesis: verum end; suppose ex k being Nat st ( k in dom (p2 /^ n) & i = (len (p1 | n)) + k ) ; ::_thesis: (crossover (p1,p2,n)) . i in S . i then consider k being Nat such that A10: k in dom (p2 /^ n) and A11: i = (len (p1 | n)) + k ; reconsider k = k as Element of NAT by ORDINAL1:def_12; A12: n <= len (p1 | n) proof n + k in dom p2 by A10, FINSEQ_5:26; then n + k in Seg (len p2) by FINSEQ_1:def_3; then n + k <= len p2 by FINSEQ_1:1; then n + k <= len S by Def1; then n + k <= len p1 by Def1; then A13: k <= (len p1) - n by XREAL_1:19; k in Seg (len (p2 /^ n)) by A10, FINSEQ_1:def_3; then 1 <= k by FINSEQ_1:1; then 1 <= (len p1) - n by A13, XXREAL_0:2; then 1 + n <= len p1 by XREAL_1:19; then A14: n <= (len p1) - 1 by XREAL_1:19; len p1 <= (len p1) + 1 by NAT_1:11; then A15: (len p1) - 1 <= len p1 by XREAL_1:20; assume n > len (p1 | n) ; ::_thesis: contradiction hence contradiction by A14, A15, FINSEQ_1:59, XXREAL_0:2; ::_thesis: verum end; len (p1 | n) <= n by FINSEQ_5:17; then A16: len (p1 | n) = n by A12, XXREAL_0:1; then n + k in Seg (len S) by A1, A7, A11, FINSEQ_1:def_3; then n + k in Seg (len p2) by Def1; then A17: n + k in dom p2 by FINSEQ_1:def_3; (crossover (p1,p2,n)) . i = (p2 /^ n) . k by A10, A11, FINSEQ_1:def_7 .= (p2 /^ n) /. k by A10, PARTFUN1:def_6 .= p2 /. (n + k) by A10, FINSEQ_5:27 .= p2 . (n + k) by A17, PARTFUN1:def_6 ; hence (crossover (p1,p2,n)) . i in S . i by A11, A16, A17, Def1; ::_thesis: verum end; end; end; hence (crossover (p1,p2,n)) . i in S . i ; ::_thesis: verum end; hence crossover (p1,p2,n) is Individual of S by A1, Def1; ::_thesis: verum 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 let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,0) = p2 let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,0) = p2 crossover (p1,p2,0) = p2 /^ 0 by FINSEQ_1:34 .= p2 by FINSEQ_5:28 ; hence crossover (p1,p2,0) = p2 ; ::_thesis: verum 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 let n be Element of NAT ; ::_thesis: for S being Gene-Set for p1, p2 being Individual of S st n >= len p1 holds crossover (p1,p2,n) = p1 let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S st n >= len p1 holds crossover (p1,p2,n) = p1 let p1, p2 be Individual of S; ::_thesis: ( n >= len p1 implies crossover (p1,p2,n) = p1 ) assume A1: n >= len p1 ; ::_thesis: crossover (p1,p2,n) = p1 then n >= len S by Def1; then A2: n >= len p2 by Def1; crossover (p1,p2,n) = p1 ^ (p2 /^ n) by A1, FINSEQ_1:58 .= p1 ^ {} by A2, FINSEQ_5:32 .= p1 by FINSEQ_1:34 ; hence crossover (p1,p2,n) = p1 ; ::_thesis: verum end; begin 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 let n1, n2 be Element of NAT ; ::_thesis: for S being Gene-Set for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2) is Individual of S let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2) is Individual of S let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,n1,n2) is Individual of S reconsider q1 = crossover (p1,p2,n1), q2 = crossover (p2,p1,n1) as Individual of S ; crossover (q1,q2,n2) is Individual of S ; hence crossover (p1,p2,n1,n2) is Individual of S ; ::_thesis: verum 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 let n be Element of NAT ; ::_thesis: for S being Gene-Set for p1, p2 being Individual of S holds crossover (p1,p2,0,n) = crossover (p2,p1,n) let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,0,n) = crossover (p2,p1,n) let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,0,n) = crossover (p2,p1,n) crossover (p1,p2,0,n) = crossover (p2,(crossover (p2,p1,0)),n) by Th4 .= crossover (p2,p1,n) by Th4 ; hence crossover (p1,p2,0,n) = crossover (p2,p1,n) ; ::_thesis: verum 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 let n be Element of NAT ; ::_thesis: for S being Gene-Set for p1, p2 being Individual of S holds crossover (p1,p2,n,0) = crossover (p2,p1,n) let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n,0) = crossover (p2,p1,n) let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,n,0) = crossover (p2,p1,n) reconsider q1 = crossover (p1,p2,n) as Individual of S ; reconsider q2 = crossover (p2,p1,n) as Individual of S ; crossover (p1,p2,n,0) = crossover (q1,q2,0) .= q2 by Th4 ; hence crossover (p1,p2,n,0) = crossover (p2,p1,n) ; ::_thesis: verum 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 let n1, n2 be Element of NAT ; ::_thesis: 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) let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S st n1 >= len p1 holds crossover (p1,p2,n1,n2) = crossover (p1,p2,n2) let p1, p2 be Individual of S; ::_thesis: ( n1 >= len p1 implies crossover (p1,p2,n1,n2) = crossover (p1,p2,n2) ) assume A1: n1 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2) = crossover (p1,p2,n2) then n1 >= len S by Def1; then A2: n1 >= len p2 by Def1; crossover (p1,p2,n1,n2) = crossover (p1,(crossover (p2,p1,n1)),n2) by A1, Th5; hence crossover (p1,p2,n1,n2) = crossover (p1,p2,n2) by A2, Th5; ::_thesis: verum 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 let n2, n1 be Element of NAT ; ::_thesis: 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) let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S st n2 >= len p1 holds crossover (p1,p2,n1,n2) = crossover (p1,p2,n1) let p1, p2 be Individual of S; ::_thesis: ( n2 >= len p1 implies crossover (p1,p2,n1,n2) = crossover (p1,p2,n1) ) assume n2 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2) = crossover (p1,p2,n1) then n2 >= len S by Def1; then A1: n2 >= len (crossover (p1,p2,n1)) by Def1; crossover (p1,p2,n1,n2) = crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n2) ; hence crossover (p1,p2,n1,n2) = crossover (p1,p2,n1) by A1, Th5; ::_thesis: verum 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 let n1, n2 be Element of NAT ; ::_thesis: 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 let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 holds crossover (p1,p2,n1,n2) = p1 let p1, p2 be Individual of S; ::_thesis: ( n1 >= len p1 & n2 >= len p1 implies crossover (p1,p2,n1,n2) = p1 ) assume that A1: n1 >= len p1 and A2: n2 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2) = p1 crossover (p1,p2,n1,n2) = crossover (p1,p2,n2) by A1, Th9; hence crossover (p1,p2,n1,n2) = p1 by A2, Th5; ::_thesis: verum 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 let n1 be Element of NAT ; ::_thesis: for S being Gene-Set for p1, p2 being Individual of S holds crossover (p1,p2,n1,n1) = p1 let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n1,n1) = p1 let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,n1,n1) = p1 A1: ((p2 | n1) ^ (p1 /^ n1)) /^ n1 = p1 /^ n1 proof now__::_thesis:_((p2_|_n1)_^_(p1_/^_n1))_/^_n1_=_p1_/^_n1 percases ( n1 <= len p2 or n1 > len p2 ) ; suppose n1 <= len p2 ; ::_thesis: ((p2 | n1) ^ (p1 /^ n1)) /^ n1 = p1 /^ n1 then len (p2 | n1) = n1 by FINSEQ_1:59; hence ((p2 | n1) ^ (p1 /^ n1)) /^ n1 = p1 /^ n1 by FINSEQ_5:37; ::_thesis: verum end; suppose n1 > len p2 ; ::_thesis: ((p2 | n1) ^ (p1 /^ n1)) /^ n1 = p1 /^ n1 then A2: n1 > len S by Def1; then n1 > len (crossover (p2,p1,n1)) by Def1; then A3: ((p2 | n1) ^ (p1 /^ n1)) /^ n1 = {} by FINSEQ_5:32; n1 > len p1 by A2, Def1; hence ((p2 | n1) ^ (p1 /^ n1)) /^ n1 = p1 /^ n1 by A3, FINSEQ_5:32; ::_thesis: verum end; end; end; hence ((p2 | n1) ^ (p1 /^ n1)) /^ n1 = p1 /^ n1 ; ::_thesis: verum end; ((p1 | n1) ^ (p2 /^ n1)) | n1 = p1 | n1 proof now__::_thesis:_((p1_|_n1)_^_(p2_/^_n1))_|_n1_=_p1_|_n1 percases ( n1 <= len p1 or n1 > len p1 ) ; suppose n1 <= len p1 ; ::_thesis: ((p1 | n1) ^ (p2 /^ n1)) | n1 = p1 | n1 then len (p1 | n1) = n1 by FINSEQ_1:59; hence ((p1 | n1) ^ (p2 /^ n1)) | n1 = p1 | n1 by FINSEQ_5:23; ::_thesis: verum end; supposeA4: n1 > len p1 ; ::_thesis: ((p1 | n1) ^ (p2 /^ n1)) | n1 = p1 | n1 then A5: n1 > len S by Def1; then A6: n1 > len p2 by Def1; n1 > len (crossover (p1,p2,n1)) by A5, Def1; then ((p1 | n1) ^ (p2 /^ n1)) | n1 = (p1 | n1) ^ (p2 /^ n1) by FINSEQ_1:58 .= p1 ^ (p2 /^ n1) by A4, FINSEQ_1:58 .= p1 ^ {} by A6, FINSEQ_5:32 .= p1 by FINSEQ_1:34 .= p1 | n1 by A4, FINSEQ_1:58 ; hence ((p1 | n1) ^ (p2 /^ n1)) | n1 = p1 | n1 ; ::_thesis: verum end; end; end; hence ((p1 | n1) ^ (p2 /^ n1)) | n1 = p1 | n1 ; ::_thesis: verum end; hence crossover (p1,p2,n1,n1) = p1 by A1, RFINSEQ:8; ::_thesis: verum 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 let n1, n2 be Element of NAT ; ::_thesis: for S being Gene-Set for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) A1: len p1 = len S by Def1; len ((p2 | n2) ^ (p1 /^ n2)) = len (crossover (p2,p1,n2)) ; then A2: len ((p2 | n2) ^ (p1 /^ n2)) = len S by Def1; A3: len p2 = len S by Def1; len ((p1 | n2) ^ (p2 /^ n2)) = len (crossover (p1,p2,n2)) ; then A4: len ((p1 | n2) ^ (p2 /^ n2)) = len S by Def1; now__::_thesis:_crossover_(p1,p2,n1,n2)_=_crossover_(p1,p2,n2,n1) percases ( ( n1 >= n2 & n2 > 0 ) or ( n1 < n2 & n2 > 0 ) or n2 = 0 ) by NAT_1:3; supposeA5: ( n1 >= n2 & n2 > 0 ) ; ::_thesis: crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) now__::_thesis:_crossover_(p1,p2,n1,n2)_=_crossover_(p1,p2,n2,n1) percases ( n1 >= len p1 or n1 < len p1 ) ; supposeA6: n1 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) then p2 | n1 = p2 by A1, A3, FINSEQ_1:58; then A7: (p2 | n1) ^ (p1 /^ n1) = p2 ^ {} by A6, FINSEQ_5:32 .= p2 by FINSEQ_1:34 ; p1 | n1 = p1 by A6, FINSEQ_1:58; then A8: (p1 | n1) ^ (p2 /^ n1) = p1 ^ {} by A1, A3, A6, FINSEQ_5:32 .= p1 by FINSEQ_1:34 ; ((p1 | n2) ^ (p2 /^ n2)) | n1 = (p1 | n2) ^ (p2 /^ n2) by A1, A4, A6, FINSEQ_1:58; then crossover (p1,p2,n2,n1) = ((p1 | n2) ^ (p2 /^ n2)) ^ {} by A1, A2, A6, FINSEQ_5:32 .= (p1 | n2) ^ (p2 /^ n2) by FINSEQ_1:34 ; hence crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) by A8, A7; ::_thesis: verum end; supposeA9: n1 < len p1 ; ::_thesis: crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) then len (p1 | n1) = n1 by FINSEQ_1:59; then A10: ((p1 | n1) ^ (p2 /^ n1)) | n2 = (p1 | n1) | n2 by A5, FINSEQ_5:22 .= p1 | n2 by A5, FINSEQ_5:77 ; n1 <= len p2 by A3, A9, Def1; then n2 <= len (p2 | n1) by A5, FINSEQ_1:59; then A11: crossover (p1,p2,n1,n2) = (p1 | n2) ^ (((p2 | n1) /^ n2) ^ (p1 /^ n1)) by A10, Th1 .= ((p1 | n2) ^ ((p2 | n1) /^ n2)) ^ (p1 /^ n1) by FINSEQ_1:32 ; len (p2 | n2) = n2 by A1, A3, A5, A9, FINSEQ_1:59, XXREAL_0:2; then consider i being Nat such that A12: (len (p2 | n2)) + i = n1 by A5, NAT_1:10; reconsider i = i as Element of NAT by ORDINAL1:def_12; A13: len (p1 | n2) = n2 by A5, A9, FINSEQ_1:59, XXREAL_0:2; then A14: (len (p1 | n2)) + i = n1 by A1, A3, A5, A9, A12, FINSEQ_1:59, XXREAL_0:2; then i = n1 - n2 by A13; then A15: i = n1 -' n2 by A5, XREAL_1:233; A16: ((p1 | n2) ^ (p2 /^ n2)) | n1 = (p1 | n2) ^ ((p2 /^ n2) | i) by A14, Th2 .= (p1 | n2) ^ ((p2 | n1) /^ n2) by A15, FINSEQ_5:80 ; ((p2 | n2) ^ (p1 /^ n2)) /^ n1 = (p1 /^ n2) /^ i by A12, FINSEQ_5:36 .= p1 /^ (n2 + i) by FINSEQ_6:81 .= p1 /^ n1 by A1, A3, A5, A9, A12, FINSEQ_1:59, XXREAL_0:2 ; hence crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) by A11, A16; ::_thesis: verum end; end; end; hence crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) ; ::_thesis: verum end; supposeA17: ( n1 < n2 & n2 > 0 ) ; ::_thesis: crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) now__::_thesis:_crossover_(p1,p2,n1,n2)_=_crossover_(p1,p2,n2,n1) percases ( n1 >= len p1 or n1 < len p1 ) ; supposeA18: n1 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) then n2 >= len p1 by A17, XXREAL_0:2; then A19: n2 >= len p2 by A3, Def1; A20: n1 >= len p2 by A3, A18, Def1; then p2 | n1 = p2 by FINSEQ_1:58; then (p2 | n1) ^ (p1 /^ n1) = p2 ^ {} by A18, FINSEQ_5:32 .= p2 by FINSEQ_1:34 ; then A21: ((p2 | n1) ^ (p1 /^ n1)) /^ n2 = {} by A19, FINSEQ_5:32; p1 | n2 = p1 by A17, A18, FINSEQ_1:58, XXREAL_0:2; then (p1 | n2) ^ (p2 /^ n2) = p1 ^ {} by A19, FINSEQ_5:32 .= p1 by FINSEQ_1:34 ; then A22: ((p1 | n2) ^ (p2 /^ n2)) | n1 = p1 by A18, FINSEQ_1:58; p2 | n2 = p2 by A19, FINSEQ_1:58; then (p2 | n2) ^ (p1 /^ n2) = p2 ^ {} by A17, A18, FINSEQ_5:32, XXREAL_0:2 .= p2 by FINSEQ_1:34 ; then A23: ((p2 | n2) ^ (p1 /^ n2)) /^ n1 = {} by A20, FINSEQ_5:32; p1 | n1 = p1 by A18, FINSEQ_1:58; then (p1 | n1) ^ (p2 /^ n1) = p1 ^ {} by A20, FINSEQ_5:32 .= p1 by FINSEQ_1:34 ; hence crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) by A17, A18, A21, A22, A23, FINSEQ_1:58, XXREAL_0:2; ::_thesis: verum end; supposeA24: n1 < len p1 ; ::_thesis: crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) n1 <= len (p1 | n2) proof now__::_thesis:_n1_<=_len_(p1_|_n2) percases ( n2 >= len p1 or n2 < len p1 ) ; suppose n2 >= len p1 ; ::_thesis: n1 <= len (p1 | n2) hence n1 <= len (p1 | n2) by A24, FINSEQ_1:58; ::_thesis: verum end; suppose n2 < len p1 ; ::_thesis: n1 <= len (p1 | n2) hence n1 <= len (p1 | n2) by A17, FINSEQ_1:59; ::_thesis: verum end; end; end; hence n1 <= len (p1 | n2) ; ::_thesis: verum end; then A25: ((p1 | n2) ^ (p2 /^ n2)) | n1 = (p1 | n2) | n1 by FINSEQ_5:22 .= p1 | n1 by A17, FINSEQ_5:77 ; A26: len (p1 | n1) = n1 by A24, FINSEQ_1:59; then consider i being Nat such that A27: (len (p1 | n1)) + i = n2 by A17, NAT_1:10; reconsider i = i as Element of NAT by ORDINAL1:def_12; A28: ((p1 | n1) ^ (p2 /^ n1)) | n2 = (p1 | n1) ^ ((p2 /^ n1) | i) by A27, Th2; len (p2 | n1) = n1 by A1, A3, A24, FINSEQ_1:59; then (len (p2 | n1)) + i = n2 by A24, A27, FINSEQ_1:59; then A29: crossover (p1,p2,n1,n2) = ((p1 | n1) ^ ((p2 /^ n1) | i)) ^ ((p1 /^ n1) /^ i) by A28, FINSEQ_5:36 .= ((p1 | n1) ^ ((p2 /^ n1) | i)) ^ (p1 /^ (n1 + i)) by FINSEQ_6:81 .= ((p1 | n1) ^ ((p2 /^ n1) | i)) ^ (p1 /^ n2) by A24, A27, FINSEQ_1:59 ; A30: i = n2 - n1 by A26, A27; n1 <= len (p2 | n2) proof now__::_thesis:_n1_<=_len_(p2_|_n2) percases ( n2 >= len p2 or n2 < len p2 ) ; suppose n2 >= len p2 ; ::_thesis: n1 <= len (p2 | n2) hence n1 <= len (p2 | n2) by A1, A3, A24, FINSEQ_1:58; ::_thesis: verum end; suppose n2 < len p2 ; ::_thesis: n1 <= len (p2 | n2) hence n1 <= len (p2 | n2) by A17, FINSEQ_1:59; ::_thesis: verum end; end; end; hence n1 <= len (p2 | n2) ; ::_thesis: verum end; then ((p2 | n2) ^ (p1 /^ n2)) /^ n1 = ((p2 | n2) /^ n1) ^ (p1 /^ n2) by Th1 .= ((p2 /^ n1) | (n2 -' n1)) ^ (p1 /^ n2) by FINSEQ_5:80 .= ((p2 /^ n1) | i) ^ (p1 /^ n2) by A17, A30, XREAL_1:233 ; hence crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) by A29, A25, FINSEQ_1:32; ::_thesis: verum end; end; end; hence crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) ; ::_thesis: verum end; supposeA31: n2 = 0 ; ::_thesis: crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) then crossover (p1,p2,n1,n2) = crossover (p2,p1,n1) by Th8 .= crossover (p1,p2,0,n1) by Th7 ; hence crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) by A31; ::_thesis: verum end; end; end; hence crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) ; ::_thesis: verum end; begin 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 let n1, n2, n3 be Element of NAT ; ::_thesis: for S being Gene-Set for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2,n3) is Individual of S let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2,n3) is Individual of S let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,n1,n2,n3) is Individual of S reconsider q1 = crossover (p1,p2,n1,n2), q2 = crossover (p2,p1,n1,n2) as Individual of S ; crossover (q1,q2,n3) is Individual of S ; hence crossover (p1,p2,n1,n2,n3) is Individual of S ; ::_thesis: verum 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 let n2, n3, n1 be Element of NAT ; ::_thesis: 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) ) let S be Gene-Set; ::_thesis: 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) ) let p1, p2 be Individual of S; ::_thesis: ( 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) ) crossover (p1,p2,0,n2,n3) = crossover ((crossover (p2,p1,n2)),(crossover (p2,p1,0,n2)),n3) by Th7 .= crossover ((crossover (p2,p1,n2)),(crossover (p1,p2,n2)),n3) by Th7 ; hence crossover (p1,p2,0,n2,n3) = crossover (p2,p1,n2,n3) ; ::_thesis: ( crossover (p1,p2,n1,0,n3) = crossover (p2,p1,n1,n3) & crossover (p1,p2,n1,n2,0) = crossover (p2,p1,n1,n2) ) crossover (p1,p2,n1,0,n3) = crossover ((crossover (p2,p1,n1)),(crossover (p2,p1,n1,0)),n3) by Th8 .= crossover ((crossover (p2,p1,n1)),(crossover (p1,p2,n1)),n3) by Th8 ; hence crossover (p1,p2,n1,0,n3) = crossover (p2,p1,n1,n3) ; ::_thesis: crossover (p1,p2,n1,n2,0) = crossover (p2,p1,n1,n2) crossover (p1,p2,n1,n2,0) = crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2)),0) ; hence crossover (p1,p2,n1,n2,0) = crossover (p2,p1,n1,n2) by Th4; ::_thesis: verum 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 let n3, n1, n2 be Element of NAT ; ::_thesis: 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) ) let S be Gene-Set; ::_thesis: 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) ) let p1, p2 be Individual of S; ::_thesis: ( 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) ) crossover (p1,p2,0,0,n3) = crossover (p1,(crossover (p2,p1,0,0)),n3) by Th12; hence crossover (p1,p2,0,0,n3) = crossover (p1,p2,n3) by Th12; ::_thesis: ( crossover (p1,p2,n1,0,0) = crossover (p1,p2,n1) & crossover (p1,p2,0,n2,0) = crossover (p1,p2,n2) ) crossover (p1,p2,n1,0,0) = crossover ((crossover (p2,p1,n1)),(crossover (p1,p2,n1)),0) by Th8; hence crossover (p1,p2,n1,0,0) = crossover (p1,p2,n1) by Th4; ::_thesis: crossover (p1,p2,0,n2,0) = crossover (p1,p2,n2) crossover (p1,p2,0,n2,0) = crossover ((crossover (p2,p1,n2)),(crossover (p1,p2,n2)),0) by Th7; hence crossover (p1,p2,0,n2,0) = crossover (p1,p2,n2) by Th4; ::_thesis: verum 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 let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,0,0,0) = p2 let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,0,0,0) = p2 crossover (p1,p2,0,0,0) = crossover ((crossover (p1,p2,0,0)),p2,0) by Th12; hence crossover (p1,p2,0,0,0) = p2 by Th4; ::_thesis: verum 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 let n1, n2, n3 be Element of NAT ; ::_thesis: 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) let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S st n1 >= len p1 holds crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n3) let p1, p2 be Individual of S; ::_thesis: ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n3) ) assume A1: n1 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n3) then n1 >= len S by Def1; then A2: n1 >= len p2 by Def1; crossover (p1,p2,n1,n2,n3) = crossover ((crossover (p1,p2,n2)),(crossover (p2,p1,n1,n2)),n3) by A1, Th9 .= crossover ((crossover (p1,p2,n2)),(crossover (p2,p1,n2)),n3) by A2, Th9 ; hence crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n3) ; ::_thesis: verum 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 let n2, n1, n3 be Element of NAT ; ::_thesis: 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) let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S st n2 >= len p1 holds crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3) let p1, p2 be Individual of S; ::_thesis: ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3) ) assume A1: n2 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3) then n2 >= len S by Def1; then A2: n2 >= len p2 by Def1; crossover (p1,p2,n1,n2,n3) = crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1,n2)),n3) by A1, Th10 .= crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n3) by A2, Th10 ; hence crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3) ; ::_thesis: verum 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 let n3, n1, n2 be Element of NAT ; ::_thesis: 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) let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S st n3 >= len p1 holds crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n2) let p1, p2 be Individual of S; ::_thesis: ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n2) ) assume n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n2) then n3 >= len S by Def1; then A1: n3 >= len (crossover (p1,p2,n1,n2)) by Def1; crossover (p1,p2,n1,n2,n3) = crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2)),n3) ; hence crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n2) by A1, Th5; ::_thesis: verum 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 let n1, n2, n3 be Element of NAT ; ::_thesis: 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) let S be Gene-Set; ::_thesis: 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) let p1, p2 be Individual of S; ::_thesis: ( n1 >= len p1 & n2 >= len p1 implies crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3) ) assume that A1: n1 >= len p1 and A2: n2 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3) crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n3) by A1, Th18; hence crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3) by A2, Th9; ::_thesis: verum 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 let n1, n3, n2 be Element of NAT ; ::_thesis: 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) let S be Gene-Set; ::_thesis: 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) let p1, p2 be Individual of S; ::_thesis: ( n1 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2) ) assume that A1: n1 >= len p1 and A2: n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2) crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n3) by A1, Th18; hence crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2) by A2, Th10; ::_thesis: verum 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 let n2, n3, n1 be Element of NAT ; ::_thesis: 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) let S be Gene-Set; ::_thesis: 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) let p1, p2 be Individual of S; ::_thesis: ( n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1) ) assume that A1: n2 >= len p1 and A2: n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1) crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3) by A1, Th19; hence crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1) by A2, Th10; ::_thesis: verum 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 let n1, n2, n3 be Element of NAT ; ::_thesis: 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 let S be Gene-Set; ::_thesis: 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 let p1, p2 be Individual of S; ::_thesis: ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3) = p1 ) assume that A1: ( n1 >= len p1 & n2 >= len p1 ) and A2: n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3) = p1 crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3) by A1, Th21; hence crossover (p1,p2,n1,n2,n3) = p1 by A2, Th5; ::_thesis: verum 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 let n1, n2, n3 be Element of NAT ; ::_thesis: 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) ) let S be Gene-Set; ::_thesis: 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) ) let p1, p2 be Individual of S; ::_thesis: ( crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n1,n3) & crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3,n2) ) set q1 = crossover (p1,p2,n1); set q2 = crossover (p2,p1,n1); crossover (p1,p2,n1,n2,n3) = crossover ((crossover (p1,p2,n2,n1)),(crossover (p2,p1,n1,n2)),n3) by Th13 .= crossover ((crossover (p1,p2,n2,n1)),(crossover (p2,p1,n2,n1)),n3) by Th13 ; hence crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n1,n3) ; ::_thesis: crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3,n2) crossover (p1,p2,n1,n2,n3) = crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n2,n3) .= crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n3,n2) by Th13 .= crossover ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,n3)),n2) ; hence crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3,n2) ; ::_thesis: verum 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 let n1, n2, n3 be Element of NAT ; ::_thesis: 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) let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3,n1,n2) let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3,n1,n2) crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3,n2) by Th25; hence crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3,n1,n2) by Th25; ::_thesis: verum 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 let n1, n3, n2 be Element of NAT ; ::_thesis: 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) ) let S be Gene-Set; ::_thesis: 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) ) let p1, p2 be Individual of S; ::_thesis: ( 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) ) crossover (p1,p2,n1,n1,n3) = crossover (p1,(crossover (p2,p1,n1,n1)),n3) by Th12; hence crossover (p1,p2,n1,n1,n3) = crossover (p1,p2,n3) by Th12; ::_thesis: ( crossover (p1,p2,n1,n2,n1) = crossover (p1,p2,n2) & crossover (p1,p2,n1,n2,n2) = crossover (p1,p2,n1) ) crossover (p1,p2,n1,n2,n1) = crossover (p1,p2,n1,n1,n2) by Th25 .= crossover (p1,(crossover (p2,p1,n1,n1)),n2) by Th12 ; hence crossover (p1,p2,n1,n2,n1) = crossover (p1,p2,n2) by Th12; ::_thesis: crossover (p1,p2,n1,n2,n2) = crossover (p1,p2,n1) thus crossover (p1,p2,n1,n2,n2) = crossover (p1,p2,n2,n2,n1) by Th26 .= crossover (p1,(crossover (p2,p1,n2,n2)),n1) by Th12 .= crossover (p1,p2,n1) by Th12 ; ::_thesis: verum end; begin 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 let n1, n2, n3, n4 be Element of NAT ; ::_thesis: for S being Gene-Set for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2,n3,n4) is Individual of S let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2,n3,n4) is Individual of S let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) is Individual of S reconsider q1 = crossover (p1,p2,n1,n2,n3), q2 = crossover (p2,p1,n1,n2,n3) as Individual of S ; crossover (q1,q2,n4) is Individual of S ; hence crossover (p1,p2,n1,n2,n3,n4) is Individual of S ; ::_thesis: verum 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 let n2, n3, n4, n1 be Element of NAT ; ::_thesis: 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) ) let S be Gene-Set; ::_thesis: 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) ) let p1, p2 be Individual of S; ::_thesis: ( 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) ) crossover (p1,p2,0,n2,n3,n4) = crossover ((crossover (p2,p1,n2,n3)),(crossover (p2,p1,0,n2,n3)),n4) by Th15 .= crossover ((crossover (p2,p1,n2,n3)),(crossover (p1,p2,n2,n3)),n4) by Th15 ; hence crossover (p1,p2,0,n2,n3,n4) = crossover (p2,p1,n2,n3,n4) ; ::_thesis: ( 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) ) crossover (p1,p2,n1,0,n3,n4) = crossover ((crossover (p2,p1,n1,n3)),(crossover (p2,p1,n1,0,n3)),n4) by Th15 .= crossover ((crossover (p2,p1,n1,n3)),(crossover (p1,p2,n1,n3)),n4) by Th15 ; hence crossover (p1,p2,n1,0,n3,n4) = crossover (p2,p1,n1,n3,n4) ; ::_thesis: ( 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) ) crossover (p1,p2,n1,n2,0,n4) = crossover ((crossover (p2,p1,n1,n2)),(crossover (p2,p1,n1,n2,0)),n4) by Th15 .= crossover ((crossover (p2,p1,n1,n2)),(crossover (p1,p2,n1,n2)),n4) by Th15 ; hence crossover (p1,p2,n1,n2,0,n4) = crossover (p2,p1,n1,n2,n4) ; ::_thesis: crossover (p1,p2,n1,n2,n3,0) = crossover (p2,p1,n1,n2,n3) crossover (p1,p2,n1,n2,n3,0) = crossover ((crossover (p1,p2,n1,n2,n3)),(crossover (p2,p1,n1,n2,n3)),0) ; hence crossover (p1,p2,n1,n2,n3,0) = crossover (p2,p1,n1,n2,n3) by Th4; ::_thesis: verum 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 let n3, n4, n2, n1 be Element of NAT ; ::_thesis: 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) ) let S be Gene-Set; ::_thesis: 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) ) let p1, p2 be Individual of S; ::_thesis: ( 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) ) crossover (p1,p2,0,0,n3,n4) = crossover (p2,p1,0,n3,n4) by Th29; hence crossover (p1,p2,0,0,n3,n4) = crossover (p1,p2,n3,n4) by Th15; ::_thesis: ( 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) ) crossover (p1,p2,0,n2,0,n4) = crossover (p2,p1,n2,0,n4) by Th29; hence crossover (p1,p2,0,n2,0,n4) = crossover (p1,p2,n2,n4) by Th15; ::_thesis: ( 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) ) crossover (p1,p2,0,n2,n3,0) = crossover (p2,p1,n2,n3,0) by Th29; hence crossover (p1,p2,0,n2,n3,0) = crossover (p1,p2,n2,n3) by Th15; ::_thesis: ( 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) ) crossover (p1,p2,n1,0,n3,0) = crossover (p2,p1,n1,n3,0) by Th29; hence crossover (p1,p2,n1,0,n3,0) = crossover (p1,p2,n1,n3) by Th15; ::_thesis: ( crossover (p1,p2,n1,0,0,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,0,0) = crossover (p1,p2,n1,n2) ) crossover (p1,p2,n1,0,0,n4) = crossover (p2,p1,n1,0,n4) by Th29; hence crossover (p1,p2,n1,0,0,n4) = crossover (p1,p2,n1,n4) by Th15; ::_thesis: crossover (p1,p2,n1,n2,0,0) = crossover (p1,p2,n1,n2) crossover (p1,p2,n1,n2,0,0) = crossover (p2,p1,n1,n2,0) by Th29; hence crossover (p1,p2,n1,n2,0,0) = crossover (p1,p2,n1,n2) by Th15; ::_thesis: verum 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 let n1, n2, n3, n4 be Element of NAT ; ::_thesis: 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) ) let S be Gene-Set; ::_thesis: 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) ) let p1, p2 be Individual of S; ::_thesis: ( 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) ) crossover (p1,p2,n1,0,0,0) = crossover (p1,p2,n1,0) by Th30; hence crossover (p1,p2,n1,0,0,0) = crossover (p2,p1,n1) by Th8; ::_thesis: ( 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) ) crossover (p1,p2,0,n2,0,0) = crossover (p1,p2,0,n2) by Th30; hence crossover (p1,p2,0,n2,0,0) = crossover (p2,p1,n2) by Th7; ::_thesis: ( crossover (p1,p2,0,0,n3,0) = crossover (p2,p1,n3) & crossover (p1,p2,0,0,0,n4) = crossover (p2,p1,n4) ) crossover (p1,p2,0,0,n3,0) = crossover (p1,p2,0,n3) by Th30; hence crossover (p1,p2,0,0,n3,0) = crossover (p2,p1,n3) by Th7; ::_thesis: crossover (p1,p2,0,0,0,n4) = crossover (p2,p1,n4) crossover (p1,p2,0,0,0,n4) = crossover (p1,p2,0,n4) by Th30; hence crossover (p1,p2,0,0,0,n4) = crossover (p2,p1,n4) by Th7; ::_thesis: verum 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 let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,0,0,0,0) = p1 let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,0,0,0,0) = p1 crossover (p1,p2,0,0,0,0) = crossover (p2,p1,0) by Th31; hence crossover (p1,p2,0,0,0,0) = p1 by Th4; ::_thesis: verum 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 let n1, n2, n3, n4 be Element of NAT ; ::_thesis: 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) ) ) let S be Gene-Set; ::_thesis: 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) ) ) let p1, p2 be Individual of S; ::_thesis: ( ( 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) ) ) A1: ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n3) ) proof assume n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n3) then n4 >= len S by Def1; then A2: n4 >= len (crossover (p1,p2,n1,n2,n3)) by Def1; crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n1,n2,n3)),(crossover (p2,p1,n1,n2,n3)),n4) ; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n3) by A2, Th5; ::_thesis: verum end; A3: ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3,n4) ) proof assume A4: n2 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3,n4) then n2 >= len S by Def1; then A5: n2 >= len p2 by Def1; crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,n2,n3)),n4) by A4, Th19 .= crossover ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,n3)),n4) by A5, Th19 ; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3,n4) ; ::_thesis: verum end; A6: ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n4) ) proof assume A7: n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n4) then n3 >= len S by Def1; then A8: n3 >= len p2 by Def1; crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2,n3)),n4) by A7, Th20 .= crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2)),n4) by A8, Th20 ; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n4) ; ::_thesis: verum end; ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3,n4) ) proof assume A9: n1 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3,n4) then n1 >= len S by Def1; then A10: n1 >= len p2 by Def1; crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n1,n2,n3)),n4) by A9, Th18 .= crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n2,n3)),n4) by A10, Th18 ; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3,n4) ; ::_thesis: verum end; hence ( ( 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) ) ) by A3, A6, A1; ::_thesis: verum 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 let n1, n2, n3, n4 be Element of NAT ; ::_thesis: 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) ) ) let S be Gene-Set; ::_thesis: 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) ) ) let p1, p2 be Individual of S; ::_thesis: ( ( 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) ) ) A1: ( n1 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n4) ) proof assume that A2: n1 >= len p1 and A3: n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n4) crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3,n4) by A2, Th33; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n4) by A3, Th19; ::_thesis: verum end; A4: ( n1 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3) ) proof assume that A5: n1 >= len p1 and A6: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3) crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3,n4) by A5, Th33; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3) by A6, Th20; ::_thesis: verum end; A7: ( n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2) ) proof assume that A8: n3 >= len p1 and A9: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2) crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n4) by A8, Th33; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2) by A9, Th20; ::_thesis: verum end; A10: ( n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3) ) proof assume that A11: n2 >= len p1 and A12: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3) crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3,n4) by A11, Th33; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3) by A12, Th20; ::_thesis: verum end; A13: ( n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n4) ) proof assume that A14: n2 >= len p1 and A15: n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n4) crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3,n4) by A14, Th33; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n4) by A15, Th19; ::_thesis: verum end; ( n1 >= len p1 & n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n4) ) proof assume that A16: n1 >= len p1 and A17: n2 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n4) crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3,n4) by A16, Th33; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n4) by A17, Th18; ::_thesis: verum end; hence ( ( 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) ) ) by A1, A4, A13, A10, A7; ::_thesis: verum 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 let n1, n2, n3, n4 be Element of NAT ; ::_thesis: 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) ) ) let S be Gene-Set; ::_thesis: 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) ) ) let p1, p2 be Individual of S; ::_thesis: ( ( 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) ) ) A1: ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n4) ) proof assume that A2: ( n1 >= len p1 & n2 >= len p1 ) and A3: n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n4) crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n4) by A2, Th34; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n4) by A3, Th9; ::_thesis: verum end; A4: ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2) ) proof assume that A5: ( n1 >= len p1 & n3 >= len p1 ) and A6: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2) crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n4) by A5, Th34 .= crossover (p1,p2,n4,n2) by Th13 ; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2) by A6, Th9; ::_thesis: verum end; A7: ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1) ) proof assume that A8: ( n2 >= len p1 & n3 >= len p1 ) and A9: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1) crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n4) by A8, Th34 .= crossover (p1,p2,n4,n1) by Th13 ; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1) by A9, Th9; ::_thesis: verum end; ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3) ) proof assume that A10: ( n1 >= len p1 & n2 >= len p1 ) and A11: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3) crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n4) by A10, Th34 .= crossover (p1,p2,n4,n3) by Th13 ; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3) by A11, Th9; ::_thesis: verum end; hence ( ( 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) ) ) by A1, A4, A7; ::_thesis: verum 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 let n1, n2, n3, n4 be Element of NAT ; ::_thesis: 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 let S be Gene-Set; ::_thesis: 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 let p1, p2 be Individual of S; ::_thesis: ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = p1 ) assume that A1: ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 ) and A2: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = p1 crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n4) by A1, Th35; hence crossover (p1,p2,n1,n2,n3,n4) = p1 by A2, Th5; ::_thesis: verum 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 let n1, n2, n3, n4 be Element of NAT ; ::_thesis: 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) ) let S be Gene-Set; ::_thesis: 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) ) let p1, p2 be Individual of S; ::_thesis: ( 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) ) A1: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n4,n3) proof set q2 = crossover (p2,p1,n1,n2); set q1 = crossover (p1,p2,n1,n2); crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2)),n3,n4) .= crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2)),n4,n3) by Th13 .= crossover ((crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2)),n4)),(crossover ((crossover (p2,p1,n1,n2)),(crossover (p1,p2,n1,n2)),n4)),n3) ; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n4,n3) ; ::_thesis: verum end; A2: crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n1,n3,n2)),(crossover (p2,p1,n1,n2,n3)),n4) by Th25 .= crossover ((crossover (p1,p2,n1,n3,n2)),(crossover (p2,p1,n1,n3,n2)),n4) by Th25 ; A3: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3,n4,n1) proof set q2 = crossover (p2,p1,n2,n3); set q1 = crossover (p1,p2,n2,n3); crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n2,n1,n3)),(crossover (p2,p1,n1,n2,n3)),n4) by Th25 .= crossover ((crossover (p1,p2,n2,n3,n1)),(crossover (p2,p1,n1,n2,n3)),n4) by Th25 .= crossover ((crossover (p1,p2,n2,n3,n1)),(crossover (p2,p1,n2,n1,n3)),n4) by Th25 .= crossover ((crossover (p1,p2,n2,n3,n1)),(crossover (p2,p1,n2,n3,n1)),n4) by Th25 .= crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n2,n3)),n1,n4) .= crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n2,n3)),n4,n1) by Th13 .= crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4)),n1) ; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3,n4,n1) ; ::_thesis: verum end; A4: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n4,n3,n2) proof set q2 = crossover (p2,p1,n1,n3); set q1 = crossover (p1,p2,n1,n3); crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n1,n3,n2)),(crossover (p2,p1,n1,n2,n3)),n4) by Th25 .= crossover ((crossover (p1,p2,n1,n3,n2)),(crossover (p2,p1,n1,n3,n2)),n4) by Th25 .= crossover ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,n3)),n2,n4) .= crossover ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,n3)),n4,n2) by Th13 .= crossover ((crossover (p1,p2,n1,n3,n4)),(crossover (p2,p1,n1,n3,n4)),n2) .= crossover ((crossover (p1,p2,n1,n4,n3)),(crossover (p2,p1,n1,n3,n4)),n2) by Th25 .= crossover ((crossover (p1,p2,n1,n4,n3)),(crossover (p2,p1,n1,n4,n3)),n2) by Th25 ; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n4,n3,n2) ; ::_thesis: verum end; A5: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n4,n3,n2,n1) proof set q2 = crossover (p2,p1,n3,n2); set q1 = crossover (p1,p2,n3,n2); crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n3,n1,n2)),(crossover (p2,p1,n1,n2,n3)),n4) by Th26 .= crossover ((crossover (p1,p2,n3,n1,n2)),(crossover (p2,p1,n3,n1,n2)),n4) by Th26 .= crossover ((crossover (p1,p2,n3,n2,n1)),(crossover (p2,p1,n3,n1,n2)),n4) by Th25 .= crossover ((crossover (p1,p2,n3,n2,n1)),(crossover (p2,p1,n3,n2,n1)),n4) by Th25 .= crossover ((crossover (p1,p2,n3,n2)),(crossover (p2,p1,n3,n2)),n1,n4) .= crossover ((crossover (p1,p2,n3,n2)),(crossover (p2,p1,n3,n2)),n4,n1) by Th13 .= crossover ((crossover (p1,p2,n3,n2,n4)),(crossover (p2,p1,n3,n2,n4)),n1) .= crossover ((crossover (p1,p2,n4,n3,n2)),(crossover (p2,p1,n3,n2,n4)),n1) by Th26 .= crossover ((crossover (p1,p2,n4,n3,n2)),(crossover (p2,p1,n4,n3,n2)),n1) by Th26 ; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n4,n3,n2,n1) ; ::_thesis: verum end; A6: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n1,n4,n3) proof set q2 = crossover (p2,p1,n2,n1); set q1 = crossover (p1,p2,n2,n1); crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n2,n1,n3)),(crossover (p2,p1,n1,n2,n3)),n4) by Th25 .= crossover ((crossover (p1,p2,n2,n1,n3)),(crossover (p2,p1,n2,n1,n3)),n4) by Th25 .= crossover ((crossover (p1,p2,n2,n1)),(crossover (p2,p1,n2,n1)),n3,n4) .= crossover ((crossover (p1,p2,n2,n1)),(crossover (p2,p1,n2,n1)),n4,n3) by Th13 .= crossover ((crossover (p1,p2,n2,n1,n4)),(crossover (p2,p1,n2,n1,n4)),n3) ; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n1,n4,n3) ; ::_thesis: verum end; A7: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n4,n2,n3,n1) proof set q2 = crossover (p2,p1,n2,n3); set q1 = crossover (p1,p2,n2,n3); crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n2,n1,n3)),(crossover (p2,p1,n1,n2,n3)),n4) by Th25 .= crossover ((crossover (p1,p2,n2,n1,n3)),(crossover (p2,p1,n2,n1,n3)),n4) by Th25 .= crossover ((crossover (p1,p2,n2,n3,n1)),(crossover (p2,p1,n2,n1,n3)),n4) by Th25 .= crossover ((crossover (p1,p2,n2,n3,n1)),(crossover (p2,p1,n2,n3,n1)),n4) by Th25 .= crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n2,n3)),n1,n4) .= crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n2,n3)),n4,n1) by Th13 .= crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4)),n1) .= crossover ((crossover (p1,p2,n4,n2,n3)),(crossover (p2,p1,n2,n3,n4)),n1) by Th26 .= crossover ((crossover (p1,p2,n4,n2,n3)),(crossover (p2,p1,n4,n2,n3)),n1) by Th26 ; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n4,n2,n3,n1) ; ::_thesis: verum end; A8: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3,n4,n2) proof set q2 = crossover (p2,p1,n1,n3); set q1 = crossover (p1,p2,n1,n3); crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n1,n3,n2)),(crossover (p2,p1,n1,n2,n3)),n4) by Th25 .= crossover ((crossover (p1,p2,n1,n3,n2)),(crossover (p2,p1,n1,n3,n2)),n4) by Th25 .= crossover ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,n3)),n2,n4) .= crossover ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,n3)),n4,n2) by Th13 .= crossover ((crossover ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,n3)),n4)),(crossover ((crossover (p2,p1,n1,n3)),(crossover (p1,p2,n1,n3)),n4)),n2) ; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3,n4,n2) ; ::_thesis: verum end; A9: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n2,n4,n1) proof set q2 = crossover (p2,p1,n3,n2); set q1 = crossover (p1,p2,n3,n2); crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n3,n1,n2)),(crossover (p2,p1,n1,n2,n3)),n4) by Th26 .= crossover ((crossover (p1,p2,n3,n1,n2)),(crossover (p2,p1,n3,n1,n2)),n4) by Th26 .= crossover ((crossover (p1,p2,n3,n2,n1)),(crossover (p2,p1,n3,n1,n2)),n4) by Th25 .= crossover ((crossover (p1,p2,n3,n2,n1)),(crossover (p2,p1,n3,n2,n1)),n4) by Th25 .= crossover ((crossover (p1,p2,n3,n2)),(crossover (p2,p1,n3,n2)),n1,n4) .= crossover ((crossover (p1,p2,n3,n2)),(crossover (p2,p1,n3,n2)),n4,n1) by Th13 .= crossover ((crossover (p1,p2,n3,n2,n4)),(crossover (p2,p1,n3,n2,n4)),n1) ; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n2,n4,n1) ; ::_thesis: verum end; A10: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n4,n1,n3) proof set q2 = crossover (p2,p1,n2,n1); set q1 = crossover (p1,p2,n2,n1); crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n2,n1,n3)),(crossover (p2,p1,n1,n2,n3)),n4) by Th25 .= crossover ((crossover (p1,p2,n2,n1,n3)),(crossover (p2,p1,n2,n1,n3)),n4) by Th25 .= crossover ((crossover (p1,p2,n2,n1)),(crossover (p2,p1,n2,n1)),n3,n4) .= crossover ((crossover (p1,p2,n2,n1)),(crossover (p2,p1,n2,n1)),n4,n3) by Th13 .= crossover ((crossover (p1,p2,n2,n1,n4)),(crossover (p2,p1,n2,n1,n4)),n3) .= crossover ((crossover (p1,p2,n2,n4,n1)),(crossover (p2,p1,n2,n1,n4)),n3) by Th25 .= crossover ((crossover (p1,p2,n2,n4,n1)),(crossover (p2,p1,n2,n4,n1)),n3) by Th25 ; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n4,n1,n3) ; ::_thesis: verum end; A11: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n4,n2,n1) proof set q2 = crossover (p2,p1,n3,n2); set q1 = crossover (p1,p2,n3,n2); crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n3,n1,n2)),(crossover (p2,p1,n1,n2,n3)),n4) by Th26 .= crossover ((crossover (p1,p2,n3,n1,n2)),(crossover (p2,p1,n3,n1,n2)),n4) by Th26 .= crossover ((crossover (p1,p2,n3,n2,n1)),(crossover (p2,p1,n3,n1,n2)),n4) by Th25 .= crossover ((crossover (p1,p2,n3,n2,n1)),(crossover (p2,p1,n3,n2,n1)),n4) by Th25 .= crossover ((crossover (p1,p2,n3,n2)),(crossover (p2,p1,n3,n2)),n1,n4) .= crossover ((crossover (p1,p2,n3,n2)),(crossover (p2,p1,n3,n2)),n4,n1) by Th13 .= crossover ((crossover (p1,p2,n3,n2,n4)),(crossover (p2,p1,n3,n2,n4)),n1) .= crossover ((crossover (p1,p2,n3,n4,n2)),(crossover (p2,p1,n3,n2,n4)),n1) by Th25 .= crossover ((crossover (p1,p2,n3,n4,n2)),(crossover (p2,p1,n3,n4,n2)),n1) by Th25 ; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n4,n2,n1) ; ::_thesis: verum end; A12: crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n3,n1,n2)),(crossover (p2,p1,n1,n2,n3)),n4) by Th26 .= crossover ((crossover (p1,p2,n3,n1,n2)),(crossover (p2,p1,n3,n1,n2)),n4) by Th26 .= crossover ((crossover (p1,p2,n3,n2,n1)),(crossover (p2,p1,n3,n1,n2)),n4) by Th25 .= crossover ((crossover (p1,p2,n3,n2,n1)),(crossover (p2,p1,n3,n2,n1)),n4) by Th25 ; A13: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n4,n1,n2) proof set q2 = crossover (p2,p1,n3,n1); set q1 = crossover (p1,p2,n3,n1); crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n3,n1,n2)),(crossover (p2,p1,n1,n2,n3)),n4) by Th26 .= crossover ((crossover (p1,p2,n3,n1,n2)),(crossover (p2,p1,n3,n1,n2)),n4) by Th26 .= crossover ((crossover (p1,p2,n3,n1)),(crossover (p2,p1,n3,n1)),n2,n4) .= crossover ((crossover (p1,p2,n3,n1)),(crossover (p2,p1,n3,n1)),n4,n2) by Th13 .= crossover ((crossover (p1,p2,n3,n1,n4)),(crossover (p2,p1,n3,n1,n4)),n2) .= crossover ((crossover (p1,p2,n3,n4,n1)),(crossover (p2,p1,n3,n1,n4)),n2) by Th25 .= crossover ((crossover (p1,p2,n3,n4,n1)),(crossover (p2,p1,n3,n4,n1)),n2) by Th25 ; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n4,n1,n2) ; ::_thesis: verum end; A14: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n4,n3,n1) proof set q2 = crossover (p2,p1,n2,n3); set q1 = crossover (p1,p2,n2,n3); crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n2,n1,n3)),(crossover (p2,p1,n1,n2,n3)),n4) by Th25 .= crossover ((crossover (p1,p2,n2,n3,n1)),(crossover (p2,p1,n1,n2,n3)),n4) by Th25 .= crossover ((crossover (p1,p2,n2,n3,n1)),(crossover (p2,p1,n2,n1,n3)),n4) by Th25 .= crossover ((crossover (p1,p2,n2,n3,n1)),(crossover (p2,p1,n2,n3,n1)),n4) by Th25 .= crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n2,n3)),n1,n4) .= crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n2,n3)),n4,n1) by Th13 .= crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4)),n1) .= crossover ((crossover (p1,p2,n2,n4,n3)),(crossover (p2,p1,n2,n3,n4)),n1) by Th25 .= crossover ((crossover (p1,p2,n2,n4,n3)),(crossover (p2,p1,n2,n4,n3)),n1) by Th25 ; hence crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n4,n3,n1) ; ::_thesis: verum end; A15: crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n2,n1,n3)),(crossover (p2,p1,n1,n2,n3)),n4) by Th25 .= crossover ((crossover (p1,p2,n2,n1,n3)),(crossover (p2,p1,n2,n1,n3)),n4) by Th25 ; crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n2,n1,n3)),(crossover (p2,p1,n1,n2,n3)),n4) by Th25 .= crossover ((crossover (p1,p2,n2,n1,n3)),(crossover (p2,p1,n2,n1,n3)),n4) by Th25 .= crossover ((crossover (p1,p2,n2,n3,n1)),(crossover (p2,p1,n2,n1,n3)),n4) by Th25 .= crossover ((crossover (p1,p2,n2,n3,n1)),(crossover (p2,p1,n2,n3,n1)),n4) by Th25 ; hence ( 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) ) by A2, A15, A1, A8, A4, A6, A3, A10, A14, A12, A9, A13, A11, A7, A5; ::_thesis: verum 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 let n1, n3, n4, n2 be Element of NAT ; ::_thesis: 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) ) let S be Gene-Set; ::_thesis: 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) ) let p1, p2 be Individual of S; ::_thesis: ( 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) ) crossover (p1,p2,n1,n1,n3,n4) = crossover ((crossover (p1,p2,n3)),(crossover (p2,p1,n1,n1,n3)),n4) by Th27 .= crossover ((crossover (p1,p2,n3)),(crossover (p2,p1,n3)),n4) by Th27 ; hence crossover (p1,p2,n1,n1,n3,n4) = crossover (p1,p2,n3,n4) ; ::_thesis: ( 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) ) crossover (p1,p2,n1,n2,n1,n4) = crossover ((crossover (p1,p2,n2)),(crossover (p2,p1,n1,n2,n1)),n4) by Th27 .= crossover ((crossover (p1,p2,n2)),(crossover (p2,p1,n2)),n4) by Th27 ; hence crossover (p1,p2,n1,n2,n1,n4) = crossover (p1,p2,n2,n4) ; ::_thesis: ( 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) ) crossover (p1,p2,n1,n2,n3,n1) = crossover (p1,p2,n1,n1,n2,n3) by Th37 .= crossover ((crossover (p1,p2,n2)),(crossover (p2,p1,n1,n1,n2)),n3) by Th27 .= crossover ((crossover (p1,p2,n2)),(crossover (p2,p1,n2)),n3) by Th27 ; hence crossover (p1,p2,n1,n2,n3,n1) = crossover (p1,p2,n2,n3) ; ::_thesis: ( 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) ) crossover (p1,p2,n1,n2,n2,n4) = crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1,n2,n2)),n4) by Th27 .= crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n4) by Th27 ; hence crossover (p1,p2,n1,n2,n2,n4) = crossover (p1,p2,n1,n4) ; ::_thesis: ( crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2) ) crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n2,n2,n3) by Th37 .= crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1,n2,n2)),n3) by Th27 .= crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n3) by Th27 ; hence crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n3) ; ::_thesis: crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2) crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n3,n3,n2) by Th37 .= crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1,n3,n3)),n2) by Th27 .= crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n2) by Th27 ; hence crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2) ; ::_thesis: verum 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 let n1, n3, n2 be Element of NAT ; ::_thesis: 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 ) let S be Gene-Set; ::_thesis: 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 ) let p1, p2 be Individual of S; ::_thesis: ( crossover (p1,p2,n1,n1,n3,n3) = p1 & crossover (p1,p2,n1,n2,n1,n2) = p1 & crossover (p1,p2,n1,n2,n2,n1) = p1 ) crossover (p1,p2,n1,n1,n3,n3) = crossover (p1,p2,n3,n3) by Th38; hence crossover (p1,p2,n1,n1,n3,n3) = p1 by Th12; ::_thesis: ( crossover (p1,p2,n1,n2,n1,n2) = p1 & crossover (p1,p2,n1,n2,n2,n1) = p1 ) crossover (p1,p2,n1,n2,n1,n2) = crossover (p1,p2,n2,n2) by Th38; hence crossover (p1,p2,n1,n2,n1,n2) = p1 by Th12; ::_thesis: crossover (p1,p2,n1,n2,n2,n1) = p1 crossover (p1,p2,n1,n2,n2,n1) = crossover (p1,p2,n2,n2) by Th38; hence crossover (p1,p2,n1,n2,n2,n1) = p1 by Th12; ::_thesis: verum end; begin 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 let n1, n2, n3, n4, n5 be Element of NAT ; ::_thesis: 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 let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2,n3,n4,n5) is Individual of S let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) is Individual of S reconsider q1 = crossover (p1,p2,n1,n2,n3,n4), q2 = crossover (p2,p1,n1,n2,n3,n4) as Individual of S ; crossover (q1,q2,n5) is Individual of S ; hence crossover (p1,p2,n1,n2,n3,n4,n5) is Individual of S ; ::_thesis: verum 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 let n2, n3, n4, n5, n1 be Element of NAT ; ::_thesis: 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) ) let S be Gene-Set; ::_thesis: 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) ) let p1, p2 be Individual of S; ::_thesis: ( 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) ) A1: crossover (p1,p2,n1,n2,n3,n4,0) = crossover ((crossover (p1,p2,n1,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4)),0) ; A2: crossover (p1,p2,n1,0,n3,n4,n5) = crossover ((crossover (p2,p1,n1,n3,n4)),(crossover (p2,p1,n1,0,n3,n4)),n5) by Th29 .= crossover ((crossover (p2,p1,n1,n3,n4)),(crossover (p1,p2,n1,n3,n4)),n5) by Th29 ; A3: crossover (p1,p2,n1,n2,n3,0,n5) = crossover ((crossover (p2,p1,n1,n2,n3)),(crossover (p2,p1,n1,n2,n3,0)),n5) by Th29 .= crossover ((crossover (p2,p1,n1,n2,n3)),(crossover (p1,p2,n1,n2,n3)),n5) by Th29 ; A4: crossover (p1,p2,n1,n2,0,n4,n5) = crossover ((crossover (p2,p1,n1,n2,n4)),(crossover (p2,p1,n1,n2,0,n4)),n5) by Th29 .= crossover ((crossover (p2,p1,n1,n2,n4)),(crossover (p1,p2,n1,n2,n4)),n5) by Th29 ; crossover (p1,p2,0,n2,n3,n4,n5) = crossover ((crossover (p2,p1,n2,n3,n4)),(crossover (p2,p1,0,n2,n3,n4)),n5) by Th29 .= crossover ((crossover (p2,p1,n2,n3,n4)),(crossover (p1,p2,n2,n3,n4)),n5) by Th29 ; hence ( 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) ) by A2, A4, A3, A1, Th4; ::_thesis: verum 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 let n3, n4, n5, n2, n1 be Element of NAT ; ::_thesis: 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) ) let S be Gene-Set; ::_thesis: 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) ) let p1, p2 be Individual of S; ::_thesis: ( 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) ) A1: ( crossover (p1,p2,0,n2,n3,n4,0) = crossover ((crossover (p1,p2,0,n2,n3,n4)),(crossover (p1,p2,n2,n3,n4)),0) & crossover (p1,p2,n1,0,n3,n4,0) = crossover ((crossover (p1,p2,n1,0,n3,n4)),(crossover (p1,p2,n1,n3,n4)),0) ) by Th29; A2: ( crossover (p1,p2,n1,n2,0,n4,0) = crossover ((crossover (p1,p2,n1,n2,0,n4)),(crossover (p1,p2,n1,n2,n4)),0) & crossover (p1,p2,n1,n2,n3,0,0) = crossover ((crossover (p1,p2,n1,n2,n3,0)),(crossover (p1,p2,n1,n2,n3)),0) ) by Th29; A3: crossover (p1,p2,n1,0,n3,0,n5) = crossover ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,0,n3,0)),n5) by Th30 .= crossover ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,n3)),n5) by Th30 ; A4: crossover (p1,p2,n1,0,0,n4,n5) = crossover ((crossover (p1,p2,n1,n4)),(crossover (p2,p1,n1,0,0,n4)),n5) by Th30 .= crossover ((crossover (p1,p2,n1,n4)),(crossover (p2,p1,n1,n4)),n5) by Th30 ; A5: crossover (p1,p2,n1,n2,0,0,n5) = crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2,0,0)),n5) by Th30 .= crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2)),n5) by Th30 ; A6: crossover (p1,p2,0,n2,n3,0,n5) = crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,0,n2,n3,0)),n5) by Th30 .= crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n2,n3)),n5) by Th30 ; A7: crossover (p1,p2,0,n2,0,n4,n5) = crossover ((crossover (p1,p2,n2,n4)),(crossover (p2,p1,0,n2,0,n4)),n5) by Th30 .= crossover ((crossover (p1,p2,n2,n4)),(crossover (p2,p1,n2,n4)),n5) by Th30 ; crossover (p1,p2,0,0,n3,n4,n5) = crossover ((crossover (p1,p2,n3,n4)),(crossover (p2,p1,0,0,n3,n4)),n5) by Th30 .= crossover ((crossover (p1,p2,n3,n4)),(crossover (p2,p1,n3,n4)),n5) by Th30 ; hence ( 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) ) by A7, A6, A4, A3, A1, A5, A2, Th4; ::_thesis: verum 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 let n4, n5, n3, n2, n1 be Element of NAT ; ::_thesis: 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) ) let S be Gene-Set; ::_thesis: 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) ) let p1, p2 be Individual of S; ::_thesis: ( 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) ) A1: ( crossover (p1,p2,0,0,n3,n4,0) = crossover ((crossover (p1,p2,0,0,n3,n4)),(crossover (p2,p1,n3,n4)),0) & crossover (p1,p2,0,n2,0,n4,0) = crossover ((crossover (p1,p2,0,n2,0,n4)),(crossover (p2,p1,n2,n4)),0) ) by Th30; A2: ( crossover (p1,p2,0,n2,n3,0,0) = crossover ((crossover (p1,p2,0,n2,n3,0)),(crossover (p2,p1,n2,n3)),0) & crossover (p1,p2,n1,0,0,n4,0) = crossover ((crossover (p1,p2,n1,0,0,n4)),(crossover (p2,p1,n1,n4)),0) ) by Th30; A3: crossover (p1,p2,0,0,n3,0,n5) = crossover ((crossover (p2,p1,n3)),(crossover (p2,p1,0,0,n3,0)),n5) by Th31 .= crossover ((crossover (p2,p1,n3)),(crossover (p1,p2,n3)),n5) by Th31 ; A4: ( crossover (p1,p2,n1,0,n3,0,0) = crossover ((crossover (p1,p2,n1,0,n3,0)),(crossover (p2,p1,n1,n3)),0) & crossover (p1,p2,n1,n2,0,0,0) = crossover ((crossover (p1,p2,n1,n2,0,0)),(crossover (p2,p1,n1,n2)),0) ) by Th30; A5: crossover (p1,p2,n1,0,0,0,n5) = crossover ((crossover (p2,p1,n1)),(crossover (p2,p1,n1,0,0,0)),n5) by Th31 .= crossover ((crossover (p2,p1,n1)),(crossover (p1,p2,n1)),n5) by Th31 ; A6: crossover (p1,p2,0,n2,0,0,n5) = crossover ((crossover (p2,p1,n2)),(crossover (p2,p1,0,n2,0,0)),n5) by Th31 .= crossover ((crossover (p2,p1,n2)),(crossover (p1,p2,n2)),n5) by Th31 ; crossover (p1,p2,0,0,0,n4,n5) = crossover ((crossover (p2,p1,n4)),(crossover (p2,p1,0,0,0,n4)),n5) by Th31 .= crossover ((crossover (p2,p1,n4)),(crossover (p1,p2,n4)),n5) by Th31 ; hence ( 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) ) by A3, A6, A1, A5, A2, A4, Th4; ::_thesis: verum 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 let n5, n4, n3, n2, n1 be Element of NAT ; ::_thesis: 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) ) let S be Gene-Set; ::_thesis: 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) ) let p1, p2 be Individual of S; ::_thesis: ( 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) ) A1: ( crossover (p1,p2,0,0,n3,0,0) = crossover ((crossover (p1,p2,0,0,n3,0)),(crossover (p1,p2,n3)),0) & crossover (p1,p2,0,n2,0,0,0) = crossover ((crossover (p1,p2,0,n2,0,0)),(crossover (p1,p2,n2)),0) ) by Th31; A2: crossover (p1,p2,n1,0,0,0,0) = crossover ((crossover (p1,p2,n1,0,0,0)),(crossover (p1,p2,n1)),0) by Th31; ( crossover (p1,p2,0,0,0,0,n5) = crossover (p1,(crossover (p2,p1,0,0,0,0)),n5) & crossover (p1,p2,0,0,0,n4,0) = crossover ((crossover (p1,p2,0,0,0,n4)),(crossover (p1,p2,n4)),0) ) by Th31, Th32; hence ( 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) ) by A1, A2, Th4, Th32; ::_thesis: verum 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 let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,0,0,0,0,0) = p2 let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,0,0,0,0,0) = p2 crossover (p1,p2,0,0,0,0,0) = crossover ((crossover (p1,p2,0,0,0,0)),p2,0) by Th32; hence crossover (p1,p2,0,0,0,0,0) = p2 by Th4; ::_thesis: verum 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 let n1, n2, n3, n4, n5 be Element of NAT ; ::_thesis: 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; ::_thesis: 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; ::_thesis: ( ( 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 ; ::_thesis: 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; ::_thesis: 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 ; ::_thesis: 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) ; ::_thesis: 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 ; ::_thesis: 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) ; ::_thesis: 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 ; ::_thesis: 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) ; ::_thesis: 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 ; ::_thesis: 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) ; ::_thesis: 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; ::_thesis: verum 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 let n1, n2, n3, n4, n5 be Element of NAT ; ::_thesis: 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) ) ) let S be Gene-Set; ::_thesis: 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) ) ) let p1, p2 be Individual of S; ::_thesis: ( ( 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) ) ) A1: ( n2 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4) ) proof assume that A2: n2 >= len p1 and A3: n5 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4) n5 >= len S by A3, Def1; then A4: n5 >= len (crossover (p1,p2,n1,n3,n4)) 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 A2, Th33; hence crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4) by A4, Th5; ::_thesis: verum end; A5: ( n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4) ) proof assume that A6: n3 >= len p1 and A7: n5 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4) n5 >= len S by A7, Def1; then A8: n5 >= len (crossover (p1,p2,n1,n2,n4)) 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 A6, Th33; hence crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4) by A8, Th5; ::_thesis: verum end; A9: ( n1 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n5) ) proof assume that A10: n1 >= len p1 and A11: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n5) n1 >= len S by A10, Def1; then A12: n1 >= len p2 by Def1; n4 >= len S by A11, Def1; then A13: n4 >= len p2 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 A10, A11, Th34 .= crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n2,n3)),n5) by A12, A13, Th34 ; hence crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n5) ; ::_thesis: verum end; A14: ( n1 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n4,n5) ) proof assume that A15: n1 >= len p1 and A16: n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n4,n5) n1 >= len S by A15, Def1; then A17: n1 >= len p2 by Def1; n3 >= len S by A16, Def1; then A18: n3 >= len p2 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 A15, A16, Th34 .= crossover ((crossover (p1,p2,n2,n4)),(crossover (p2,p1,n2,n4)),n5) by A17, A18, Th34 ; hence crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n4,n5) ; ::_thesis: verum end; A19: ( n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n5) ) proof assume that A20: n2 >= len p1 and A21: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n5) n2 >= len S by A20, Def1; then A22: n2 >= len p2 by Def1; n4 >= len S by A21, Def1; then A23: n4 >= len p2 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 A20, A21, Th34 .= crossover ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,n3)),n5) by A22, A23, Th34 ; hence crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n5) ; ::_thesis: verum end; A24: ( n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n4,n5) ) proof assume that A25: n2 >= len p1 and A26: n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n4,n5) n2 >= len S by A25, Def1; then A27: n2 >= len p2 by Def1; n3 >= len S by A26, Def1; then A28: n3 >= len p2 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 A25, A26, Th34 .= crossover ((crossover (p1,p2,n1,n4)),(crossover (p2,p1,n1,n4)),n5) by A27, A28, Th34 ; hence crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n4,n5) ; ::_thesis: verum end; A29: ( n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n5) ) proof assume that A30: n3 >= len p1 and A31: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n5) n3 >= len S by A30, Def1; then A32: n3 >= len p2 by Def1; n4 >= len S by A31, Def1; then A33: n4 >= len p2 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 A30, A31, Th34 .= crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2)),n5) by A32, A33, Th34 ; hence crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n5) ; ::_thesis: verum end; A34: ( n1 >= len p1 & n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) ) proof assume that A35: n1 >= len p1 and A36: n2 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) n1 >= len S by A35, Def1; then A37: n1 >= len p2 by Def1; n2 >= len S by A36, Def1; then A38: n2 >= len p2 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 A35, A36, Th34 .= crossover ((crossover (p1,p2,n3,n4)),(crossover (p2,p1,n3,n4)),n5) by A37, A38, Th34 ; hence crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) ; ::_thesis: verum end; A39: ( n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3) ) proof assume that A40: n4 >= len p1 and A41: n5 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3) n5 >= len S by A41, Def1; then A42: n5 >= len (crossover (p1,p2,n1,n2,n3)) 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 A40, Th33; hence crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3) by A42, Th5; ::_thesis: verum end; ( n1 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4) ) proof assume that A43: n1 >= len p1 and A44: n5 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4) n5 >= len S by A44, Def1; then A45: n5 >= len (crossover (p1,p2,n2,n3,n4)) 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 A43, Th33; hence crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4) by A45, Th5; ::_thesis: verum end; hence ( ( 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) ) ) by A34, A14, A9, A24, A19, A1, A29, A5, A39; ::_thesis: verum 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 let n1, n2, n3, n4, n5 be Element of NAT ; ::_thesis: 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; ::_thesis: 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; ::_thesis: ( ( 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 ; ::_thesis: 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; ::_thesis: 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 ; ::_thesis: 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; ::_thesis: 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 ; ::_thesis: 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) ; ::_thesis: 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 ; ::_thesis: 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; ::_thesis: 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 ; ::_thesis: 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) ; ::_thesis: 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 ; ::_thesis: 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) ; ::_thesis: 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 ; ::_thesis: 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; ::_thesis: 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 ; ::_thesis: 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) ; ::_thesis: 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 ; ::_thesis: 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; ::_thesis: 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 ; ::_thesis: 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; ::_thesis: 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; ::_thesis: verum 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 let n1, n2, n3, n4, n5 be Element of NAT ; ::_thesis: 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) ) ) let S be Gene-Set; ::_thesis: 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) ) ) let p1, p2 be Individual of S; ::_thesis: ( ( 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) ) ) A1: ( 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) ) proof assume that A2: ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 ) and A3: n5 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3) n5 >= len S by A3, Def1; then A4: n5 >= len (crossover (p1,p2,n3)) by Def1; crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n3)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by A2, Th35; hence crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3) by A4, Th5; ::_thesis: verum end; A5: ( 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) ) proof assume that A6: ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 ) and A7: n5 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2) n5 >= len S by A7, Def1; then A8: n5 >= len (crossover (p1,p2,n2)) by Def1; crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n2)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by A6, Th35; hence crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2) by A8, Th5; ::_thesis: verum end; A9: ( 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) ) proof assume that A10: n1 >= len p1 and A11: n2 >= len p1 and A12: n3 >= len p1 and A13: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5) n1 >= len S by A10, Def1; then A14: n1 >= len p2 by Def1; n3 >= len S by A12, Def1; then A15: n3 >= len p2 by Def1; n2 >= len S by A11, Def1; then A16: n2 >= len p2 by Def1; n4 >= len S by A13, Def1; then A17: n4 >= len p2 by Def1; crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,(crossover (p2,p1,n1,n2,n3,n4)),n5) by A10, A11, A12, A13, Th36; hence crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5) by A14, A16, A15, A17, Th36; ::_thesis: verum end; A18: ( 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 assume that A19: ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 ) and A20: n5 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1) n5 >= len S by A20, Def1; then A21: n5 >= len (crossover (p1,p2,n1)) by Def1; crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by A19, Th35; hence crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1) by A21, Th5; ::_thesis: verum end; ( 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) ) proof assume that A22: ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 ) and A23: n5 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4) n5 >= len S by A23, Def1; then A24: n5 >= len (crossover (p1,p2,n4)) by Def1; crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n4)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by A22, Th35; hence crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4) by A24, Th5; ::_thesis: verum end; hence ( ( 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) ) ) by A9, A1, A5, A18; ::_thesis: verum 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 let n1, n2, n3, n4, n5 be Element of NAT ; ::_thesis: 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 let S be Gene-Set; ::_thesis: 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 let p1, p2 be Individual of S; ::_thesis: ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = p1 ) assume that A1: ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 ) and A2: n5 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = p1 crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,(crossover (p2,p1,n1,n2,n3,n4)),n5) by A1, Th36; hence crossover (p1,p2,n1,n2,n3,n4,n5) = p1 by A2, Th5; ::_thesis: verum 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 let n1, n2, n3, n4, n5 be Element of NAT ; ::_thesis: 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) ) let S be Gene-Set; ::_thesis: 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) ) let p1, p2 be Individual of S; ::_thesis: ( 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) ) A1: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n3,n2,n1,n4)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by Th37 .= crossover ((crossover (p1,p2,n3,n2,n1,n4)),(crossover (p2,p1,n3,n2,n1,n4)),n5) by Th37 ; A2: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n4,n2,n3,n1)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by Th37 .= crossover ((crossover (p1,p2,n4,n2,n3,n1)),(crossover (p2,p1,n4,n2,n3,n1)),n5) by Th37 ; A3: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5,n2,n3,n4,n1) proof set q2 = crossover (p2,p1,n2,n3,n4); set q1 = crossover (p1,p2,n2,n3,n4); A4: crossover (p1,p2,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4)),n5) ; crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n2,n3,n4,n1)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by Th37 .= crossover ((crossover (p1,p2,n2,n3,n4,n1)),(crossover (p2,p1,n2,n3,n4,n1)),n5) by Th37 .= crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4)),n1,n5) .= crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4)),n5,n1) by Th13 .= crossover ((crossover (p1,p2,n5,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4,n5)),n1) by A4, Th37 .= crossover ((crossover (p1,p2,n5,n2,n3,n4)),(crossover (p2,p1,n5,n2,n3,n4)),n1) by Th37 ; hence crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5,n2,n3,n4,n1) ; ::_thesis: verum end; crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n2,n1,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by Th37 .= crossover ((crossover (p1,p2,n2,n1,n3,n4)),(crossover (p2,p1,n2,n1,n3,n4)),n5) by Th37 ; hence ( 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) ) by A1, A2, A3; ::_thesis: verum 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 let n1, n3, n4, n5, n2 be Element of NAT ; ::_thesis: 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) ) let S be Gene-Set; ::_thesis: 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) ) let p1, p2 be Individual of S; ::_thesis: ( 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) ) set q1 = crossover (p1,p2,n2,n3,n4); set q2 = crossover (p2,p1,n2,n3,n4); crossover (p1,p2,n1,n1,n3,n4,n5) = crossover ((crossover (p1,p2,n3,n4)),(crossover (p2,p1,n1,n1,n3,n4)),n5) by Th38 .= crossover ((crossover (p1,p2,n3,n4)),(crossover (p2,p1,n3,n4)),n5) by Th38 ; hence crossover (p1,p2,n1,n1,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) ; ::_thesis: ( 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) ) crossover (p1,p2,n1,n2,n1,n4,n5) = crossover ((crossover (p1,p2,n2,n4)),(crossover (p2,p1,n1,n2,n1,n4)),n5) by Th38 .= crossover ((crossover (p1,p2,n2,n4)),(crossover (p2,p1,n2,n4)),n5) by Th38 ; hence crossover (p1,p2,n1,n2,n1,n4,n5) = crossover (p1,p2,n2,n4,n5) ; ::_thesis: ( 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) ) crossover (p1,p2,n1,n2,n3,n1,n5) = crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n1,n2,n3,n1)),n5) by Th38 .= crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n2,n3)),n5) by Th38 ; hence crossover (p1,p2,n1,n2,n3,n1,n5) = crossover (p1,p2,n2,n3,n5) ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n1) = crossover (p1,p2,n2,n3,n4) crossover (p1,p2,n1,n2,n3,n4,n1) = crossover ((crossover (p1,p2,n2,n3,n4,n1)),(crossover (p2,p1,n1,n2,n3,n4)),n1) by Th37 .= crossover ((crossover (p1,p2,n2,n3,n4,n1)),(crossover (p2,p1,n2,n3,n4,n1)),n1) by Th37 .= crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4)),n1,n1) ; hence crossover (p1,p2,n1,n2,n3,n4,n1) = crossover (p1,p2,n2,n3,n4) by Th12; ::_thesis: verum end; begin 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 let n1, n2, n3, n4, n5, n6 be Element of NAT ; ::_thesis: 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 let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2,n3,n4,n5,n6) is Individual of S let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5,n6) is Individual of S reconsider q1 = crossover (p1,p2,n1,n2,n3,n4,n5), q2 = crossover (p2,p1,n1,n2,n3,n4,n5) as Individual of S ; crossover (q1,q2,n6) is Individual of S ; hence crossover (p1,p2,n1,n2,n3,n4,n5,n6) is Individual of S ; ::_thesis: verum 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 let n2, n3, n4, n5, n6, n1 be Element of NAT ; ::_thesis: 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) ) let S be Gene-Set; ::_thesis: 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) ) let p1, p2 be Individual of S; ::_thesis: ( 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) ) crossover (p1,p2,0,n2,n3,n4,n5,n6) = crossover ((crossover (p2,p1,n2,n3,n4,n5)),(crossover (p2,p1,0,n2,n3,n4,n5)),n6) by Th41 .= crossover ((crossover (p2,p1,n2,n3,n4,n5)),(crossover (p1,p2,n2,n3,n4,n5)),n6) by Th41 ; hence crossover (p1,p2,0,n2,n3,n4,n5,n6) = crossover (p2,p1,n2,n3,n4,n5,n6) ; ::_thesis: ( 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) ) crossover (p1,p2,n1,0,n3,n4,n5,n6) = crossover ((crossover (p2,p1,n1,n3,n4,n5)),(crossover (p2,p1,n1,0,n3,n4,n5)),n6) by Th41 .= crossover ((crossover (p2,p1,n1,n3,n4,n5)),(crossover (p1,p2,n1,n3,n4,n5)),n6) by Th41 ; hence crossover (p1,p2,n1,0,n3,n4,n5,n6) = crossover (p2,p1,n1,n3,n4,n5,n6) ; ::_thesis: ( 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) ) crossover (p1,p2,n1,n2,0,n4,n5,n6) = crossover ((crossover (p2,p1,n1,n2,n4,n5)),(crossover (p2,p1,n1,n2,0,n4,n5)),n6) by Th41 .= crossover ((crossover (p2,p1,n1,n2,n4,n5)),(crossover (p1,p2,n1,n2,n4,n5)),n6) by Th41 ; hence crossover (p1,p2,n1,n2,0,n4,n5,n6) = crossover (p2,p1,n1,n2,n4,n5,n6) ; ::_thesis: ( 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) ) crossover (p1,p2,n1,n2,n3,0,n5,n6) = crossover ((crossover (p2,p1,n1,n2,n3,n5)),(crossover (p2,p1,n1,n2,n3,0,n5)),n6) by Th41 .= crossover ((crossover (p2,p1,n1,n2,n3,n5)),(crossover (p1,p2,n1,n2,n3,n5)),n6) by Th41 ; hence crossover (p1,p2,n1,n2,n3,0,n5,n6) = crossover (p2,p1,n1,n2,n3,n5,n6) ; ::_thesis: ( 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) ) crossover (p1,p2,n1,n2,n3,n4,0,n6) = crossover ((crossover (p2,p1,n1,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4,0)),n6) by Th41 .= crossover ((crossover (p2,p1,n1,n2,n3,n4)),(crossover (p1,p2,n1,n2,n3,n4)),n6) by Th41 ; hence crossover (p1,p2,n1,n2,n3,n4,0,n6) = crossover (p2,p1,n1,n2,n3,n4,n6) ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5,0) = crossover (p2,p1,n1,n2,n3,n4,n5) thus crossover (p1,p2,n1,n2,n3,n4,n5,0) = crossover ((crossover (p1,p2,n1,n2,n3,n4,n5)),(crossover (p2,p1,n1,n2,n3,n4,n5)),0) .= crossover (p2,p1,n1,n2,n3,n4,n5) by Th4 ; ::_thesis: verum 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 let n1, n2, n3, n4, n5, n6 be Element of NAT ; ::_thesis: 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) ) ) let S be Gene-Set; ::_thesis: 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) ) ) let p1, p2 be Individual of S; ::_thesis: ( ( 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) ) ) A1: ( n6 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n5) ) proof assume n6 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n5) then n6 >= len S by Def1; then A2: n6 >= len (crossover (p1,p2,n1,n2,n3,n4,n5)) by Def1; 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) ; hence crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n5) by A2, Th5; ::_thesis: verum end; A3: ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n3,n4,n5,n6) ) proof assume A4: n2 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n3,n4,n5,n6) then n2 >= len S by Def1; then A5: n2 >= len p2 by Def1; crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n1,n3,n4,n5)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) by A4, Th46 .= crossover ((crossover (p1,p2,n1,n3,n4,n5)),(crossover (p2,p1,n1,n3,n4,n5)),n6) by A5, Th46 ; hence crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n3,n4,n5,n6) ; ::_thesis: verum end; A6: ( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n6) ) proof assume A7: n5 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n6) then n5 >= len S by Def1; then A8: n5 >= len p2 by Def1; crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n1,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) by A7, Th46 .= crossover ((crossover (p1,p2,n1,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4)),n6) by A8, Th46 ; hence crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n6) ; ::_thesis: verum end; A9: ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n5,n6) ) proof assume A10: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n5,n6) then n4 >= len S by Def1; then A11: n4 >= len p2 by Def1; crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n1,n2,n3,n5)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) by A10, Th46 .= crossover ((crossover (p1,p2,n1,n2,n3,n5)),(crossover (p2,p1,n1,n2,n3,n5)),n6) by A11, Th46 ; hence crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n5,n6) ; ::_thesis: verum end; A12: ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n4,n5,n6) ) proof assume A13: n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n4,n5,n6) then n3 >= len S by Def1; then A14: n3 >= len p2 by Def1; crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n1,n2,n4,n5)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) by A13, Th46 .= crossover ((crossover (p1,p2,n1,n2,n4,n5)),(crossover (p2,p1,n1,n2,n4,n5)),n6) by A14, Th46 ; hence crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n4,n5,n6) ; ::_thesis: verum end; ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n3,n4,n5,n6) ) proof assume A15: n1 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n3,n4,n5,n6) then n1 >= len S by Def1; then A16: n1 >= len p2 by Def1; crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n2,n3,n4,n5)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) by A15, Th46 .= crossover ((crossover (p1,p2,n2,n3,n4,n5)),(crossover (p2,p1,n2,n3,n4,n5)),n6) by A16, Th46 ; hence crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n3,n4,n5,n6) ; ::_thesis: verum end; hence ( ( 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) ) ) by A3, A12, A9, A6, A1; ::_thesis: verum 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 let n1, n2, n3, n4, n5, n6 be Element of NAT ; ::_thesis: 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) ) let S be Gene-Set; ::_thesis: 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) ) let p1, p2 be Individual of S; ::_thesis: ( 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) ) set q1 = crossover (p1,p2,n5,n2,n3,n4); set q2 = crossover (p2,p1,n5,n2,n3,n4); A1: crossover (p1,p2,n5,n2,n3,n4,n6) = crossover ((crossover (p1,p2,n5,n2,n3,n4)),(crossover (p2,p1,n5,n2,n3,n4)),n6) ; crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n2,n1,n3,n4,n5)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) by Th51 .= crossover ((crossover (p1,p2,n2,n1,n3,n4,n5)),(crossover (p2,p1,n2,n1,n3,n4,n5)),n6) by Th51 ; hence crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n1,n3,n4,n5,n6) ; ::_thesis: ( 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) ) crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n3,n2,n1,n4,n5)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) by Th51 .= crossover ((crossover (p1,p2,n3,n2,n1,n4,n5)),(crossover (p2,p1,n3,n2,n1,n4,n5)),n6) by Th51 ; hence crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n3,n2,n1,n4,n5,n6) ; ::_thesis: ( 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) ) crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n4,n2,n3,n1,n5)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) by Th51 .= crossover ((crossover (p1,p2,n4,n2,n3,n1,n5)),(crossover (p2,p1,n4,n2,n3,n1,n5)),n6) by Th51 ; hence crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n4,n2,n3,n1,n5,n6) ; ::_thesis: ( 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) ) crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n5,n2,n3,n4,n1)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) by Th51 .= crossover ((crossover (p1,p2,n5,n2,n3,n4,n1)),(crossover (p2,p1,n5,n2,n3,n4,n1)),n6) by Th51 ; hence crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n5,n2,n3,n4,n1,n6) ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n6,n2,n3,n4,n5,n1) crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n5,n2,n3,n4,n1)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) by Th51 .= crossover ((crossover (p1,p2,n5,n2,n3,n4,n1)),(crossover (p2,p1,n5,n2,n3,n4,n1)),n6) by Th51 .= crossover ((crossover (p1,p2,n5,n2,n3,n4)),(crossover (p2,p1,n5,n2,n3,n4)),n1,n6) .= crossover ((crossover (p1,p2,n5,n2,n3,n4)),(crossover (p2,p1,n5,n2,n3,n4)),n6,n1) by Th13 .= crossover ((crossover (p1,p2,n6,n2,n3,n4,n5)),(crossover (p2,p1,n5,n2,n3,n4,n6)),n1) by A1, Th51 .= crossover ((crossover (p1,p2,n6,n2,n3,n4,n5)),(crossover (p2,p1,n6,n2,n3,n4,n5)),n1) by Th51 ; hence crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n6,n2,n3,n4,n5,n1) ; ::_thesis: verum 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 let n1, n3, n4, n5, n6, n2 be Element of NAT ; ::_thesis: 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) ) let S be Gene-Set; ::_thesis: 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) ) let p1, p2 be Individual of S; ::_thesis: ( 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) ) crossover (p1,p2,n1,n1,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n3,n4,n5)),(crossover (p2,p1,n1,n1,n3,n4,n5)),n6) by Th52 .= crossover ((crossover (p1,p2,n3,n4,n5)),(crossover (p2,p1,n3,n4,n5)),n6) by Th52 ; hence crossover (p1,p2,n1,n1,n3,n4,n5,n6) = crossover (p1,p2,n3,n4,n5,n6) ; ::_thesis: ( 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) ) crossover (p1,p2,n1,n2,n1,n4,n5,n6) = crossover ((crossover (p1,p2,n2,n4,n5)),(crossover (p2,p1,n1,n2,n1,n4,n5)),n6) by Th52 .= crossover ((crossover (p1,p2,n2,n4,n5)),(crossover (p2,p1,n2,n4,n5)),n6) by Th52 ; hence crossover (p1,p2,n1,n2,n1,n4,n5,n6) = crossover (p1,p2,n2,n4,n5,n6) ; ::_thesis: ( 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) ) crossover (p1,p2,n1,n2,n3,n1,n5,n6) = crossover ((crossover (p1,p2,n2,n3,n5)),(crossover (p2,p1,n1,n2,n3,n1,n5)),n6) by Th52 .= crossover ((crossover (p1,p2,n2,n3,n5)),(crossover (p2,p1,n2,n3,n5)),n6) by Th52 ; hence crossover (p1,p2,n1,n2,n3,n1,n5,n6) = crossover (p1,p2,n2,n3,n5,n6) ; ::_thesis: ( 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) ) crossover (p1,p2,n1,n2,n3,n4,n1,n6) = crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4,n1)),n6) by Th52 .= crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4)),n6) by Th52 ; hence crossover (p1,p2,n1,n2,n3,n4,n1,n6) = crossover (p1,p2,n2,n3,n4,n6) ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n2,n3,n4,n5) crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n5,n2,n3,n4,n1,n1) by Th56 .= crossover (p1,p2,n1,n2,n3,n4,n1,n5) by Th56 .= crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4,n1)),n5) by Th52 .= crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4)),n5) by Th52 ; hence crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n2,n3,n4,n5) ; ::_thesis: verum end;