5.71/2.38	YES
5.71/2.40	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
5.71/2.40	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.71/2.40	
5.71/2.40	
5.71/2.40	Left Termination of the query pattern
5.71/2.40	
5.71/2.40	zebra(a,a,a,a,a,a,a)
5.71/2.40	
5.71/2.40	w.r.t. the given Prolog program could successfully be proven:
5.71/2.40	
5.71/2.40	(0) Prolog
5.71/2.40	(1) PrologToPiTRSProof [SOUND, 37 ms]
5.71/2.40	(2) PiTRS
5.71/2.40	(3) DependencyPairsProof [EQUIVALENT, 81 ms]
5.71/2.40	(4) PiDP
5.71/2.40	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
5.71/2.40	(6) AND
5.71/2.40	    (7) PiDP
5.71/2.40	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
5.71/2.40	        (9) PiDP
5.71/2.40	        (10) PiDPToQDPProof [SOUND, 0 ms]
5.71/2.40	        (11) QDP
5.71/2.40	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
5.71/2.40	        (13) YES
5.71/2.40	    (14) PiDP
5.71/2.40	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
5.71/2.40	        (16) PiDP
5.71/2.40	        (17) PiDPToQDPProof [SOUND, 0 ms]
5.71/2.40	        (18) QDP
5.71/2.40	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
5.71/2.40	        (20) YES
5.71/2.40	
5.71/2.40	
5.71/2.40	----------------------------------------
5.71/2.40	
5.71/2.40	(0)
5.71/2.40	Obligation:
5.71/2.40	Clauses:
5.71/2.40	
5.71/2.40	zebra(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water) :- ','(eq(Englishman, Red), ','(eq(Spaniard, Dog), ','(eq(Green, Coffee), ','(eq(Ukrainian, Tea), ','(to_the_right(Green, Ivory), ','(eq(Winston, Snails), ','(eq(Kool, Yellow), ','(eq(Milk, third), ','(eq(Norwegian, first), ','(next_to(Fox, Chesterfield), ','(next_to(Horse, Kool), ','(eq(Lucky, Juice), ','(eq(Japanese, Parliament), ','(next_to(Norwegian, Blue), ','(houses(.(Blue, .(Green, .(Red, .(Yellow, .(Ivory, [])))))), ','(houses(.(Norwegian, .(Englishman, .(Spaniard, .(Japanese, .(Ukrainian, [])))))), ','(houses(.(Dog, .(Zebra, .(Fox, .(Snails, .(Horse, [])))))), ','(houses(.(Parliament, .(Kool, .(Lucky, .(Chesterfield, .(Winston, [])))))), houses(.(Milk, .(Juice, .(Water, .(Tea, .(Coffee, [])))))))))))))))))))))))).
5.71/2.40	houses(Prop) :- domain(Prop, .(first, .(second, .(third, .(fourth, .(fifth, [])))))).
5.71/2.40	domain([], X1).
5.71/2.40	domain(.(X, Rest), Domain) :- ','(select(X, Domain, NewDomain), domain(Rest, NewDomain)).
5.71/2.40	select(X, .(X, R), R).
5.71/2.40	select(X, .(Y, R), .(Y, Rest)) :- select(X, R, Rest).
5.71/2.40	next_to(fifth, fourth).
5.71/2.40	next_to(fourth, fifth).
5.71/2.40	next_to(fourth, third).
5.71/2.40	next_to(third, fourth).
5.71/2.40	next_to(third, second).
5.71/2.40	next_to(second, third).
5.71/2.40	next_to(second, first).
5.71/2.40	next_to(first, second).
5.71/2.40	to_the_right(fifth, fourth).
5.71/2.40	to_the_right(fourth, third).
5.71/2.40	to_the_right(third, second).
5.71/2.40	to_the_right(second, first).
5.71/2.40	eq(X, X).
5.71/2.40	
5.71/2.40	
5.71/2.40	Query: zebra(a,a,a,a,a,a,a)
5.71/2.40	----------------------------------------
5.71/2.40	
5.71/2.40	(1) PrologToPiTRSProof (SOUND)
5.71/2.40	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
5.71/2.40	
5.71/2.40	zebra_in_7: (f,f,f,f,f,f,f)
5.71/2.40	
5.71/2.40	houses_in_1: (b)
5.71/2.40	
5.71/2.40	domain_in_2: (b,b)
5.71/2.40	
5.71/2.40	select_in_3: (f,b,f)
5.71/2.40	
5.71/2.40	Transforming Prolog into the following Term Rewriting System:
5.71/2.40	
5.71/2.40	Pi-finite rewrite system:
5.71/2.40	The TRS R consists of the following rules:
5.71/2.40	
5.71/2.40	   zebra_in_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water) -> U1_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, eq_in_aa(Englishman, Red))
5.71/2.40	   eq_in_aa(X, X) -> eq_out_aa(X, X)
5.71/2.40	   U1_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, eq_out_aa(Englishman, Red)) -> U2_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, eq_in_aa(Spaniard, Dog))
5.71/2.40	   U2_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, eq_out_aa(Spaniard, Dog)) -> U3_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, eq_in_aa(Green, Coffee))
5.71/2.40	   U3_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, eq_out_aa(Green, Coffee)) -> U4_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, eq_in_aa(Ukrainian, Tea))
5.71/2.40	   U4_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, eq_out_aa(Ukrainian, Tea)) -> U5_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, to_the_right_in_aa(Green, Ivory))
5.71/2.40	   to_the_right_in_aa(fifth, fourth) -> to_the_right_out_aa(fifth, fourth)
5.71/2.40	   to_the_right_in_aa(fourth, third) -> to_the_right_out_aa(fourth, third)
5.71/2.40	   to_the_right_in_aa(third, second) -> to_the_right_out_aa(third, second)
5.71/2.40	   to_the_right_in_aa(second, first) -> to_the_right_out_aa(second, first)
5.71/2.40	   U5_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, to_the_right_out_aa(Green, Ivory)) -> U6_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, eq_in_aa(Winston, Snails))
5.71/2.40	   U6_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, eq_out_aa(Winston, Snails)) -> U7_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, eq_in_aa(Kool, Yellow))
5.71/2.40	   U7_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, eq_out_aa(Kool, Yellow)) -> U8_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, eq_in_ag(Milk, third))
5.71/2.40	   eq_in_ag(X, X) -> eq_out_ag(X, X)
5.71/2.40	   U8_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, eq_out_ag(Milk, third)) -> U9_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, eq_in_ag(Norwegian, first))
5.71/2.40	   U9_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, eq_out_ag(Norwegian, first)) -> U10_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, next_to_in_aa(Fox, Chesterfield))
5.71/2.40	   next_to_in_aa(fifth, fourth) -> next_to_out_aa(fifth, fourth)
5.71/2.40	   next_to_in_aa(fourth, fifth) -> next_to_out_aa(fourth, fifth)
5.71/2.40	   next_to_in_aa(fourth, third) -> next_to_out_aa(fourth, third)
5.71/2.40	   next_to_in_aa(third, fourth) -> next_to_out_aa(third, fourth)
5.71/2.40	   next_to_in_aa(third, second) -> next_to_out_aa(third, second)
5.71/2.40	   next_to_in_aa(second, third) -> next_to_out_aa(second, third)
5.71/2.40	   next_to_in_aa(second, first) -> next_to_out_aa(second, first)
5.71/2.40	   next_to_in_aa(first, second) -> next_to_out_aa(first, second)
5.71/2.40	   U10_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, next_to_out_aa(Fox, Chesterfield)) -> U11_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, next_to_in_aa(Horse, Kool))
5.71/2.40	   U11_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, next_to_out_aa(Horse, Kool)) -> U12_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, eq_in_aa(Lucky, Juice))
5.71/2.40	   U12_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, eq_out_aa(Lucky, Juice)) -> U13_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, eq_in_aa(Japanese, Parliament))
5.71/2.40	   U13_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, eq_out_aa(Japanese, Parliament)) -> U14_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, next_to_in_ga(Norwegian, Blue))
5.71/2.40	   next_to_in_ga(fifth, fourth) -> next_to_out_ga(fifth, fourth)
5.71/2.40	   next_to_in_ga(fourth, fifth) -> next_to_out_ga(fourth, fifth)
5.71/2.40	   next_to_in_ga(fourth, third) -> next_to_out_ga(fourth, third)
5.71/2.40	   next_to_in_ga(third, fourth) -> next_to_out_ga(third, fourth)
5.71/2.40	   next_to_in_ga(third, second) -> next_to_out_ga(third, second)
5.71/2.40	   next_to_in_ga(second, third) -> next_to_out_ga(second, third)
5.71/2.40	   next_to_in_ga(second, first) -> next_to_out_ga(second, first)
5.71/2.40	   next_to_in_ga(first, second) -> next_to_out_ga(first, second)
5.71/2.40	   U14_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, next_to_out_ga(Norwegian, Blue)) -> U15_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_in_g(.(Blue, .(Green, .(Red, .(Yellow, .(Ivory, [])))))))
5.71/2.40	   houses_in_g(Prop) -> U20_g(Prop, domain_in_gg(Prop, .(first, .(second, .(third, .(fourth, .(fifth, [])))))))
5.71/2.40	   domain_in_gg([], X1) -> domain_out_gg([], X1)
5.71/2.40	   domain_in_gg(.(X, Rest), Domain) -> U21_gg(X, Rest, Domain, select_in_aga(X, Domain, NewDomain))
5.71/2.40	   select_in_aga(X, .(X, R), R) -> select_out_aga(X, .(X, R), R)
5.71/2.40	   select_in_aga(X, .(Y, R), .(Y, Rest)) -> U23_aga(X, Y, R, Rest, select_in_aga(X, R, Rest))
5.71/2.40	   U23_aga(X, Y, R, Rest, select_out_aga(X, R, Rest)) -> select_out_aga(X, .(Y, R), .(Y, Rest))
5.71/2.40	   U21_gg(X, Rest, Domain, select_out_aga(X, Domain, NewDomain)) -> U22_gg(X, Rest, Domain, domain_in_gg(Rest, NewDomain))
5.71/2.40	   U22_gg(X, Rest, Domain, domain_out_gg(Rest, NewDomain)) -> domain_out_gg(.(X, Rest), Domain)
5.71/2.40	   U20_g(Prop, domain_out_gg(Prop, .(first, .(second, .(third, .(fourth, .(fifth, []))))))) -> houses_out_g(Prop)
5.71/2.40	   U15_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_out_g(.(Blue, .(Green, .(Red, .(Yellow, .(Ivory, []))))))) -> U16_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_in_g(.(Norwegian, .(Englishman, .(Spaniard, .(Japanese, .(Ukrainian, [])))))))
5.71/2.40	   U16_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_out_g(.(Norwegian, .(Englishman, .(Spaniard, .(Japanese, .(Ukrainian, []))))))) -> U17_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Winston, Kool, Milk, Chesterfield, Lucky, Juice, Parliament, houses_in_g(.(Dog, .(Zebra, .(Fox, .(Snails, .(Horse, [])))))))
5.71/2.40	   U17_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Winston, Kool, Milk, Chesterfield, Lucky, Juice, Parliament, houses_out_g(.(Dog, .(Zebra, .(Fox, .(Snails, .(Horse, []))))))) -> U18_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Milk, Juice, houses_in_g(.(Parliament, .(Kool, .(Lucky, .(Chesterfield, .(Winston, [])))))))
5.71/2.40	   U18_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Milk, Juice, houses_out_g(.(Parliament, .(Kool, .(Lucky, .(Chesterfield, .(Winston, []))))))) -> U19_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, houses_in_g(.(Milk, .(Juice, .(Water, .(Tea, .(Coffee, [])))))))
5.71/2.40	   U19_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, houses_out_g(.(Milk, .(Juice, .(Water, .(Tea, .(Coffee, []))))))) -> zebra_out_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water)
5.71/2.40	
5.71/2.40	The argument filtering Pi contains the following mapping:
5.71/2.40	zebra_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  zebra_in_aaaaaaa
5.71/2.40	
5.71/2.40	U1_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U1_aaaaaaa(x8)
5.71/2.40	
5.71/2.40	eq_in_aa(x1, x2)  =  eq_in_aa
5.71/2.40	
5.71/2.40	eq_out_aa(x1, x2)  =  eq_out_aa
5.71/2.40	
5.71/2.40	U2_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U2_aaaaaaa(x9)
5.71/2.40	
5.71/2.40	U3_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U3_aaaaaaa(x10)
5.71/2.40	
5.71/2.40	U4_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)  =  U4_aaaaaaa(x12)
5.71/2.40	
5.71/2.40	U5_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13)  =  U5_aaaaaaa(x13)
5.71/2.40	
5.71/2.40	to_the_right_in_aa(x1, x2)  =  to_the_right_in_aa
5.71/2.40	
5.71/2.40	to_the_right_out_aa(x1, x2)  =  to_the_right_out_aa(x1, x2)
5.71/2.40	
5.71/2.40	U6_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14)  =  U6_aaaaaaa(x14)
5.71/2.40	
5.71/2.40	U7_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16)  =  U7_aaaaaaa(x16)
5.71/2.40	
5.71/2.40	U8_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18)  =  U8_aaaaaaa(x18)
5.71/2.40	
5.71/2.40	eq_in_ag(x1, x2)  =  eq_in_ag(x2)
5.71/2.40	
5.71/2.40	eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
5.71/2.40	
5.71/2.40	third  =  third
5.71/2.40	
5.71/2.40	U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19)  =  U9_aaaaaaa(x19)
5.71/2.40	
5.71/2.40	first  =  first
5.71/2.40	
5.71/2.40	U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19)  =  U10_aaaaaaa(x5, x19)
5.71/2.40	
5.71/2.40	next_to_in_aa(x1, x2)  =  next_to_in_aa
5.71/2.40	
5.71/2.40	next_to_out_aa(x1, x2)  =  next_to_out_aa(x1, x2)
5.71/2.40	
5.71/2.40	U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U11_aaaaaaa(x5, x21)
5.71/2.40	
5.71/2.40	U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22)  =  U12_aaaaaaa(x5, x22)
5.71/2.40	
5.71/2.40	U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24)  =  U13_aaaaaaa(x5, x24)
5.71/2.40	
5.71/2.40	U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25)  =  U14_aaaaaaa(x5, x25)
5.71/2.40	
5.71/2.40	next_to_in_ga(x1, x2)  =  next_to_in_ga(x1)
5.71/2.40	
5.71/2.40	fifth  =  fifth
5.71/2.40	
5.71/2.40	next_to_out_ga(x1, x2)  =  next_to_out_ga(x1, x2)
5.71/2.40	
5.71/2.40	fourth  =  fourth
5.71/2.40	
5.71/2.40	second  =  second
5.71/2.40	
5.71/2.40	U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U15_aaaaaaa(x5, x21)
5.71/2.40	
5.71/2.40	houses_in_g(x1)  =  houses_in_g(x1)
5.71/2.40	
5.71/2.40	.(x1, x2)  =  .(x2)
5.71/2.40	
5.71/2.40	U20_g(x1, x2)  =  U20_g(x1, x2)
5.71/2.40	
5.71/2.40	domain_in_gg(x1, x2)  =  domain_in_gg(x1, x2)
5.71/2.40	
5.71/2.40	[]  =  []
5.71/2.40	
5.71/2.40	domain_out_gg(x1, x2)  =  domain_out_gg(x1, x2)
5.71/2.40	
5.71/2.40	U21_gg(x1, x2, x3, x4)  =  U21_gg(x2, x3, x4)
5.71/2.40	
5.71/2.40	select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
5.71/2.40	
5.71/2.40	select_out_aga(x1, x2, x3)  =  select_out_aga(x2, x3)
5.71/2.40	
5.71/2.40	U23_aga(x1, x2, x3, x4, x5)  =  U23_aga(x3, x5)
5.71/2.40	
5.71/2.40	U22_gg(x1, x2, x3, x4)  =  U22_gg(x2, x3, x4)
5.71/2.40	
5.71/2.40	houses_out_g(x1)  =  houses_out_g(x1)
5.71/2.40	
5.71/2.40	U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U16_aaaaaaa(x5, x21)
5.71/2.40	
5.71/2.40	U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17)  =  U17_aaaaaaa(x5, x17)
5.71/2.40	
5.71/2.40	U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)  =  U18_aaaaaaa(x5, x12)
5.71/2.40	
5.71/2.40	U19_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_aaaaaaa(x5, x8)
5.71/2.40	
5.71/2.40	zebra_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  zebra_out_aaaaaaa(x5)
5.71/2.40	
5.71/2.40	
5.71/2.40	
5.71/2.40	
5.71/2.40	
5.71/2.40	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
5.71/2.40	
5.71/2.40	
5.71/2.40	
5.71/2.40	----------------------------------------
5.71/2.40	
5.71/2.40	(2)
5.71/2.40	Obligation:
5.71/2.40	Pi-finite rewrite system:
5.71/2.40	The TRS R consists of the following rules:
5.71/2.40	
5.71/2.40	   zebra_in_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water) -> U1_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, eq_in_aa(Englishman, Red))
5.71/2.40	   eq_in_aa(X, X) -> eq_out_aa(X, X)
5.71/2.40	   U1_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, eq_out_aa(Englishman, Red)) -> U2_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, eq_in_aa(Spaniard, Dog))
5.71/2.40	   U2_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, eq_out_aa(Spaniard, Dog)) -> U3_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, eq_in_aa(Green, Coffee))
5.71/2.40	   U3_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, eq_out_aa(Green, Coffee)) -> U4_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, eq_in_aa(Ukrainian, Tea))
5.71/2.40	   U4_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, eq_out_aa(Ukrainian, Tea)) -> U5_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, to_the_right_in_aa(Green, Ivory))
5.71/2.40	   to_the_right_in_aa(fifth, fourth) -> to_the_right_out_aa(fifth, fourth)
5.71/2.40	   to_the_right_in_aa(fourth, third) -> to_the_right_out_aa(fourth, third)
5.71/2.40	   to_the_right_in_aa(third, second) -> to_the_right_out_aa(third, second)
5.71/2.40	   to_the_right_in_aa(second, first) -> to_the_right_out_aa(second, first)
5.71/2.40	   U5_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, to_the_right_out_aa(Green, Ivory)) -> U6_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, eq_in_aa(Winston, Snails))
5.71/2.40	   U6_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, eq_out_aa(Winston, Snails)) -> U7_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, eq_in_aa(Kool, Yellow))
5.71/2.40	   U7_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, eq_out_aa(Kool, Yellow)) -> U8_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, eq_in_ag(Milk, third))
5.71/2.40	   eq_in_ag(X, X) -> eq_out_ag(X, X)
5.71/2.40	   U8_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, eq_out_ag(Milk, third)) -> U9_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, eq_in_ag(Norwegian, first))
5.71/2.40	   U9_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, eq_out_ag(Norwegian, first)) -> U10_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, next_to_in_aa(Fox, Chesterfield))
5.71/2.40	   next_to_in_aa(fifth, fourth) -> next_to_out_aa(fifth, fourth)
5.71/2.40	   next_to_in_aa(fourth, fifth) -> next_to_out_aa(fourth, fifth)
5.71/2.40	   next_to_in_aa(fourth, third) -> next_to_out_aa(fourth, third)
5.71/2.40	   next_to_in_aa(third, fourth) -> next_to_out_aa(third, fourth)
5.71/2.40	   next_to_in_aa(third, second) -> next_to_out_aa(third, second)
5.71/2.40	   next_to_in_aa(second, third) -> next_to_out_aa(second, third)
5.71/2.40	   next_to_in_aa(second, first) -> next_to_out_aa(second, first)
5.71/2.40	   next_to_in_aa(first, second) -> next_to_out_aa(first, second)
5.71/2.40	   U10_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, next_to_out_aa(Fox, Chesterfield)) -> U11_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, next_to_in_aa(Horse, Kool))
5.71/2.40	   U11_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, next_to_out_aa(Horse, Kool)) -> U12_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, eq_in_aa(Lucky, Juice))
5.71/2.40	   U12_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, eq_out_aa(Lucky, Juice)) -> U13_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, eq_in_aa(Japanese, Parliament))
5.71/2.40	   U13_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, eq_out_aa(Japanese, Parliament)) -> U14_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, next_to_in_ga(Norwegian, Blue))
5.71/2.40	   next_to_in_ga(fifth, fourth) -> next_to_out_ga(fifth, fourth)
5.71/2.40	   next_to_in_ga(fourth, fifth) -> next_to_out_ga(fourth, fifth)
5.71/2.40	   next_to_in_ga(fourth, third) -> next_to_out_ga(fourth, third)
5.71/2.40	   next_to_in_ga(third, fourth) -> next_to_out_ga(third, fourth)
5.71/2.40	   next_to_in_ga(third, second) -> next_to_out_ga(third, second)
5.71/2.40	   next_to_in_ga(second, third) -> next_to_out_ga(second, third)
5.71/2.40	   next_to_in_ga(second, first) -> next_to_out_ga(second, first)
5.71/2.40	   next_to_in_ga(first, second) -> next_to_out_ga(first, second)
5.71/2.40	   U14_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, next_to_out_ga(Norwegian, Blue)) -> U15_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_in_g(.(Blue, .(Green, .(Red, .(Yellow, .(Ivory, [])))))))
5.71/2.40	   houses_in_g(Prop) -> U20_g(Prop, domain_in_gg(Prop, .(first, .(second, .(third, .(fourth, .(fifth, [])))))))
5.71/2.40	   domain_in_gg([], X1) -> domain_out_gg([], X1)
5.71/2.40	   domain_in_gg(.(X, Rest), Domain) -> U21_gg(X, Rest, Domain, select_in_aga(X, Domain, NewDomain))
5.71/2.40	   select_in_aga(X, .(X, R), R) -> select_out_aga(X, .(X, R), R)
5.71/2.40	   select_in_aga(X, .(Y, R), .(Y, Rest)) -> U23_aga(X, Y, R, Rest, select_in_aga(X, R, Rest))
5.71/2.40	   U23_aga(X, Y, R, Rest, select_out_aga(X, R, Rest)) -> select_out_aga(X, .(Y, R), .(Y, Rest))
5.71/2.40	   U21_gg(X, Rest, Domain, select_out_aga(X, Domain, NewDomain)) -> U22_gg(X, Rest, Domain, domain_in_gg(Rest, NewDomain))
5.71/2.40	   U22_gg(X, Rest, Domain, domain_out_gg(Rest, NewDomain)) -> domain_out_gg(.(X, Rest), Domain)
5.71/2.40	   U20_g(Prop, domain_out_gg(Prop, .(first, .(second, .(third, .(fourth, .(fifth, []))))))) -> houses_out_g(Prop)
5.71/2.40	   U15_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_out_g(.(Blue, .(Green, .(Red, .(Yellow, .(Ivory, []))))))) -> U16_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_in_g(.(Norwegian, .(Englishman, .(Spaniard, .(Japanese, .(Ukrainian, [])))))))
5.71/2.40	   U16_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_out_g(.(Norwegian, .(Englishman, .(Spaniard, .(Japanese, .(Ukrainian, []))))))) -> U17_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Winston, Kool, Milk, Chesterfield, Lucky, Juice, Parliament, houses_in_g(.(Dog, .(Zebra, .(Fox, .(Snails, .(Horse, [])))))))
5.71/2.40	   U17_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Winston, Kool, Milk, Chesterfield, Lucky, Juice, Parliament, houses_out_g(.(Dog, .(Zebra, .(Fox, .(Snails, .(Horse, []))))))) -> U18_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Milk, Juice, houses_in_g(.(Parliament, .(Kool, .(Lucky, .(Chesterfield, .(Winston, [])))))))
5.71/2.40	   U18_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Milk, Juice, houses_out_g(.(Parliament, .(Kool, .(Lucky, .(Chesterfield, .(Winston, []))))))) -> U19_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, houses_in_g(.(Milk, .(Juice, .(Water, .(Tea, .(Coffee, [])))))))
5.71/2.40	   U19_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, houses_out_g(.(Milk, .(Juice, .(Water, .(Tea, .(Coffee, []))))))) -> zebra_out_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water)
5.71/2.40	
5.71/2.40	The argument filtering Pi contains the following mapping:
5.71/2.40	zebra_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  zebra_in_aaaaaaa
5.71/2.40	
5.71/2.40	U1_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U1_aaaaaaa(x8)
5.71/2.40	
5.71/2.40	eq_in_aa(x1, x2)  =  eq_in_aa
5.71/2.40	
5.71/2.40	eq_out_aa(x1, x2)  =  eq_out_aa
5.71/2.40	
5.71/2.40	U2_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U2_aaaaaaa(x9)
5.71/2.40	
5.71/2.40	U3_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U3_aaaaaaa(x10)
5.71/2.40	
5.71/2.40	U4_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)  =  U4_aaaaaaa(x12)
5.71/2.40	
5.71/2.40	U5_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13)  =  U5_aaaaaaa(x13)
5.71/2.40	
5.71/2.40	to_the_right_in_aa(x1, x2)  =  to_the_right_in_aa
5.71/2.40	
5.71/2.40	to_the_right_out_aa(x1, x2)  =  to_the_right_out_aa(x1, x2)
5.71/2.40	
5.71/2.40	U6_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14)  =  U6_aaaaaaa(x14)
5.71/2.40	
5.71/2.40	U7_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16)  =  U7_aaaaaaa(x16)
5.71/2.40	
5.71/2.40	U8_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18)  =  U8_aaaaaaa(x18)
5.71/2.40	
5.71/2.40	eq_in_ag(x1, x2)  =  eq_in_ag(x2)
5.71/2.40	
5.71/2.40	eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
5.71/2.40	
5.71/2.40	third  =  third
5.71/2.40	
5.71/2.40	U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19)  =  U9_aaaaaaa(x19)
5.71/2.40	
5.71/2.40	first  =  first
5.71/2.40	
5.71/2.40	U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19)  =  U10_aaaaaaa(x5, x19)
5.71/2.40	
5.71/2.40	next_to_in_aa(x1, x2)  =  next_to_in_aa
5.71/2.40	
5.71/2.40	next_to_out_aa(x1, x2)  =  next_to_out_aa(x1, x2)
5.71/2.40	
5.71/2.40	U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U11_aaaaaaa(x5, x21)
5.71/2.40	
5.71/2.40	U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22)  =  U12_aaaaaaa(x5, x22)
5.71/2.40	
5.71/2.40	U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24)  =  U13_aaaaaaa(x5, x24)
5.71/2.40	
5.71/2.40	U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25)  =  U14_aaaaaaa(x5, x25)
5.71/2.40	
5.71/2.40	next_to_in_ga(x1, x2)  =  next_to_in_ga(x1)
5.71/2.40	
5.71/2.40	fifth  =  fifth
5.71/2.40	
5.71/2.40	next_to_out_ga(x1, x2)  =  next_to_out_ga(x1, x2)
5.71/2.40	
5.71/2.40	fourth  =  fourth
5.71/2.40	
5.71/2.40	second  =  second
5.71/2.40	
5.71/2.40	U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U15_aaaaaaa(x5, x21)
5.71/2.40	
5.71/2.40	houses_in_g(x1)  =  houses_in_g(x1)
5.71/2.40	
5.71/2.40	.(x1, x2)  =  .(x2)
5.71/2.40	
5.71/2.40	U20_g(x1, x2)  =  U20_g(x1, x2)
5.71/2.40	
5.71/2.40	domain_in_gg(x1, x2)  =  domain_in_gg(x1, x2)
5.71/2.40	
5.71/2.40	[]  =  []
5.71/2.40	
5.71/2.40	domain_out_gg(x1, x2)  =  domain_out_gg(x1, x2)
5.71/2.40	
5.71/2.40	U21_gg(x1, x2, x3, x4)  =  U21_gg(x2, x3, x4)
5.71/2.40	
5.71/2.40	select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
5.71/2.40	
5.71/2.40	select_out_aga(x1, x2, x3)  =  select_out_aga(x2, x3)
5.71/2.40	
5.71/2.40	U23_aga(x1, x2, x3, x4, x5)  =  U23_aga(x3, x5)
5.71/2.40	
5.71/2.40	U22_gg(x1, x2, x3, x4)  =  U22_gg(x2, x3, x4)
5.71/2.40	
5.71/2.40	houses_out_g(x1)  =  houses_out_g(x1)
5.71/2.40	
5.71/2.40	U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U16_aaaaaaa(x5, x21)
5.71/2.40	
5.71/2.40	U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17)  =  U17_aaaaaaa(x5, x17)
5.71/2.40	
5.71/2.40	U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)  =  U18_aaaaaaa(x5, x12)
5.71/2.40	
5.71/2.40	U19_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_aaaaaaa(x5, x8)
5.71/2.40	
5.71/2.40	zebra_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  zebra_out_aaaaaaa(x5)
5.71/2.40	
5.71/2.40	
5.71/2.40	
5.71/2.40	----------------------------------------
5.71/2.40	
5.71/2.40	(3) DependencyPairsProof (EQUIVALENT)
5.71/2.40	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
5.71/2.40	Pi DP problem:
5.71/2.40	The TRS P consists of the following rules:
5.71/2.40	
5.71/2.40	   ZEBRA_IN_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water) -> U1_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, eq_in_aa(Englishman, Red))
5.71/2.40	   ZEBRA_IN_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water) -> EQ_IN_AA(Englishman, Red)
5.71/2.40	   U1_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, eq_out_aa(Englishman, Red)) -> U2_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, eq_in_aa(Spaniard, Dog))
5.71/2.40	   U1_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, eq_out_aa(Englishman, Red)) -> EQ_IN_AA(Spaniard, Dog)
5.71/2.40	   U2_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, eq_out_aa(Spaniard, Dog)) -> U3_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, eq_in_aa(Green, Coffee))
5.71/2.40	   U2_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, eq_out_aa(Spaniard, Dog)) -> EQ_IN_AA(Green, Coffee)
5.71/2.40	   U3_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, eq_out_aa(Green, Coffee)) -> U4_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, eq_in_aa(Ukrainian, Tea))
5.71/2.40	   U3_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, eq_out_aa(Green, Coffee)) -> EQ_IN_AA(Ukrainian, Tea)
5.71/2.40	   U4_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, eq_out_aa(Ukrainian, Tea)) -> U5_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, to_the_right_in_aa(Green, Ivory))
5.71/2.40	   U4_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, eq_out_aa(Ukrainian, Tea)) -> TO_THE_RIGHT_IN_AA(Green, Ivory)
5.71/2.40	   U5_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, to_the_right_out_aa(Green, Ivory)) -> U6_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, eq_in_aa(Winston, Snails))
5.71/2.40	   U5_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, to_the_right_out_aa(Green, Ivory)) -> EQ_IN_AA(Winston, Snails)
5.71/2.40	   U6_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, eq_out_aa(Winston, Snails)) -> U7_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, eq_in_aa(Kool, Yellow))
5.71/2.40	   U6_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, eq_out_aa(Winston, Snails)) -> EQ_IN_AA(Kool, Yellow)
5.71/2.40	   U7_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, eq_out_aa(Kool, Yellow)) -> U8_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, eq_in_ag(Milk, third))
5.71/2.40	   U7_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, eq_out_aa(Kool, Yellow)) -> EQ_IN_AG(Milk, third)
5.71/2.40	   U8_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, eq_out_ag(Milk, third)) -> U9_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, eq_in_ag(Norwegian, first))
5.71/2.40	   U8_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, eq_out_ag(Milk, third)) -> EQ_IN_AG(Norwegian, first)
5.71/2.40	   U9_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, eq_out_ag(Norwegian, first)) -> U10_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, next_to_in_aa(Fox, Chesterfield))
5.71/2.40	   U9_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, eq_out_ag(Norwegian, first)) -> NEXT_TO_IN_AA(Fox, Chesterfield)
5.71/2.40	   U10_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, next_to_out_aa(Fox, Chesterfield)) -> U11_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, next_to_in_aa(Horse, Kool))
5.71/2.40	   U10_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, next_to_out_aa(Fox, Chesterfield)) -> NEXT_TO_IN_AA(Horse, Kool)
5.71/2.40	   U11_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, next_to_out_aa(Horse, Kool)) -> U12_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, eq_in_aa(Lucky, Juice))
5.71/2.40	   U11_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, next_to_out_aa(Horse, Kool)) -> EQ_IN_AA(Lucky, Juice)
5.71/2.40	   U12_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, eq_out_aa(Lucky, Juice)) -> U13_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, eq_in_aa(Japanese, Parliament))
5.71/2.40	   U12_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, eq_out_aa(Lucky, Juice)) -> EQ_IN_AA(Japanese, Parliament)
5.71/2.40	   U13_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, eq_out_aa(Japanese, Parliament)) -> U14_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, next_to_in_ga(Norwegian, Blue))
5.71/2.40	   U13_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, eq_out_aa(Japanese, Parliament)) -> NEXT_TO_IN_GA(Norwegian, Blue)
5.71/2.40	   U14_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, next_to_out_ga(Norwegian, Blue)) -> U15_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_in_g(.(Blue, .(Green, .(Red, .(Yellow, .(Ivory, [])))))))
5.71/2.40	   U14_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, next_to_out_ga(Norwegian, Blue)) -> HOUSES_IN_G(.(Blue, .(Green, .(Red, .(Yellow, .(Ivory, []))))))
5.71/2.40	   HOUSES_IN_G(Prop) -> U20_G(Prop, domain_in_gg(Prop, .(first, .(second, .(third, .(fourth, .(fifth, [])))))))
5.71/2.40	   HOUSES_IN_G(Prop) -> DOMAIN_IN_GG(Prop, .(first, .(second, .(third, .(fourth, .(fifth, []))))))
5.71/2.40	   DOMAIN_IN_GG(.(X, Rest), Domain) -> U21_GG(X, Rest, Domain, select_in_aga(X, Domain, NewDomain))
5.71/2.40	   DOMAIN_IN_GG(.(X, Rest), Domain) -> SELECT_IN_AGA(X, Domain, NewDomain)
5.71/2.40	   SELECT_IN_AGA(X, .(Y, R), .(Y, Rest)) -> U23_AGA(X, Y, R, Rest, select_in_aga(X, R, Rest))
5.71/2.40	   SELECT_IN_AGA(X, .(Y, R), .(Y, Rest)) -> SELECT_IN_AGA(X, R, Rest)
5.71/2.40	   U21_GG(X, Rest, Domain, select_out_aga(X, Domain, NewDomain)) -> U22_GG(X, Rest, Domain, domain_in_gg(Rest, NewDomain))
5.71/2.40	   U21_GG(X, Rest, Domain, select_out_aga(X, Domain, NewDomain)) -> DOMAIN_IN_GG(Rest, NewDomain)
5.71/2.40	   U15_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_out_g(.(Blue, .(Green, .(Red, .(Yellow, .(Ivory, []))))))) -> U16_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_in_g(.(Norwegian, .(Englishman, .(Spaniard, .(Japanese, .(Ukrainian, [])))))))
5.71/2.40	   U15_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_out_g(.(Blue, .(Green, .(Red, .(Yellow, .(Ivory, []))))))) -> HOUSES_IN_G(.(Norwegian, .(Englishman, .(Spaniard, .(Japanese, .(Ukrainian, []))))))
5.71/2.40	   U16_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_out_g(.(Norwegian, .(Englishman, .(Spaniard, .(Japanese, .(Ukrainian, []))))))) -> U17_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Winston, Kool, Milk, Chesterfield, Lucky, Juice, Parliament, houses_in_g(.(Dog, .(Zebra, .(Fox, .(Snails, .(Horse, [])))))))
5.71/2.40	   U16_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_out_g(.(Norwegian, .(Englishman, .(Spaniard, .(Japanese, .(Ukrainian, []))))))) -> HOUSES_IN_G(.(Dog, .(Zebra, .(Fox, .(Snails, .(Horse, []))))))
5.71/2.40	   U17_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Winston, Kool, Milk, Chesterfield, Lucky, Juice, Parliament, houses_out_g(.(Dog, .(Zebra, .(Fox, .(Snails, .(Horse, []))))))) -> U18_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Milk, Juice, houses_in_g(.(Parliament, .(Kool, .(Lucky, .(Chesterfield, .(Winston, [])))))))
5.71/2.40	   U17_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Winston, Kool, Milk, Chesterfield, Lucky, Juice, Parliament, houses_out_g(.(Dog, .(Zebra, .(Fox, .(Snails, .(Horse, []))))))) -> HOUSES_IN_G(.(Parliament, .(Kool, .(Lucky, .(Chesterfield, .(Winston, []))))))
5.71/2.40	   U18_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Milk, Juice, houses_out_g(.(Parliament, .(Kool, .(Lucky, .(Chesterfield, .(Winston, []))))))) -> U19_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, houses_in_g(.(Milk, .(Juice, .(Water, .(Tea, .(Coffee, [])))))))
5.71/2.40	   U18_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Milk, Juice, houses_out_g(.(Parliament, .(Kool, .(Lucky, .(Chesterfield, .(Winston, []))))))) -> HOUSES_IN_G(.(Milk, .(Juice, .(Water, .(Tea, .(Coffee, []))))))
5.71/2.40	
5.71/2.40	The TRS R consists of the following rules:
5.71/2.40	
5.71/2.40	   zebra_in_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water) -> U1_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, eq_in_aa(Englishman, Red))
5.71/2.40	   eq_in_aa(X, X) -> eq_out_aa(X, X)
5.71/2.40	   U1_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, eq_out_aa(Englishman, Red)) -> U2_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, eq_in_aa(Spaniard, Dog))
5.71/2.40	   U2_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, eq_out_aa(Spaniard, Dog)) -> U3_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, eq_in_aa(Green, Coffee))
5.71/2.40	   U3_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, eq_out_aa(Green, Coffee)) -> U4_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, eq_in_aa(Ukrainian, Tea))
5.71/2.40	   U4_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, eq_out_aa(Ukrainian, Tea)) -> U5_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, to_the_right_in_aa(Green, Ivory))
5.71/2.40	   to_the_right_in_aa(fifth, fourth) -> to_the_right_out_aa(fifth, fourth)
5.71/2.40	   to_the_right_in_aa(fourth, third) -> to_the_right_out_aa(fourth, third)
5.71/2.40	   to_the_right_in_aa(third, second) -> to_the_right_out_aa(third, second)
5.71/2.40	   to_the_right_in_aa(second, first) -> to_the_right_out_aa(second, first)
5.71/2.40	   U5_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, to_the_right_out_aa(Green, Ivory)) -> U6_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, eq_in_aa(Winston, Snails))
5.71/2.40	   U6_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, eq_out_aa(Winston, Snails)) -> U7_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, eq_in_aa(Kool, Yellow))
5.71/2.40	   U7_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, eq_out_aa(Kool, Yellow)) -> U8_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, eq_in_ag(Milk, third))
5.71/2.40	   eq_in_ag(X, X) -> eq_out_ag(X, X)
5.71/2.40	   U8_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, eq_out_ag(Milk, third)) -> U9_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, eq_in_ag(Norwegian, first))
5.71/2.40	   U9_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, eq_out_ag(Norwegian, first)) -> U10_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, next_to_in_aa(Fox, Chesterfield))
5.71/2.40	   next_to_in_aa(fifth, fourth) -> next_to_out_aa(fifth, fourth)
5.71/2.40	   next_to_in_aa(fourth, fifth) -> next_to_out_aa(fourth, fifth)
5.71/2.40	   next_to_in_aa(fourth, third) -> next_to_out_aa(fourth, third)
5.71/2.40	   next_to_in_aa(third, fourth) -> next_to_out_aa(third, fourth)
5.71/2.40	   next_to_in_aa(third, second) -> next_to_out_aa(third, second)
5.71/2.40	   next_to_in_aa(second, third) -> next_to_out_aa(second, third)
5.71/2.40	   next_to_in_aa(second, first) -> next_to_out_aa(second, first)
5.71/2.40	   next_to_in_aa(first, second) -> next_to_out_aa(first, second)
5.71/2.40	   U10_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, next_to_out_aa(Fox, Chesterfield)) -> U11_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, next_to_in_aa(Horse, Kool))
5.71/2.40	   U11_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, next_to_out_aa(Horse, Kool)) -> U12_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, eq_in_aa(Lucky, Juice))
5.71/2.40	   U12_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, eq_out_aa(Lucky, Juice)) -> U13_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, eq_in_aa(Japanese, Parliament))
5.71/2.40	   U13_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, eq_out_aa(Japanese, Parliament)) -> U14_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, next_to_in_ga(Norwegian, Blue))
5.71/2.40	   next_to_in_ga(fifth, fourth) -> next_to_out_ga(fifth, fourth)
5.71/2.40	   next_to_in_ga(fourth, fifth) -> next_to_out_ga(fourth, fifth)
5.71/2.40	   next_to_in_ga(fourth, third) -> next_to_out_ga(fourth, third)
5.71/2.40	   next_to_in_ga(third, fourth) -> next_to_out_ga(third, fourth)
5.71/2.40	   next_to_in_ga(third, second) -> next_to_out_ga(third, second)
5.71/2.40	   next_to_in_ga(second, third) -> next_to_out_ga(second, third)
5.71/2.40	   next_to_in_ga(second, first) -> next_to_out_ga(second, first)
5.71/2.40	   next_to_in_ga(first, second) -> next_to_out_ga(first, second)
5.71/2.40	   U14_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, next_to_out_ga(Norwegian, Blue)) -> U15_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_in_g(.(Blue, .(Green, .(Red, .(Yellow, .(Ivory, [])))))))
5.71/2.40	   houses_in_g(Prop) -> U20_g(Prop, domain_in_gg(Prop, .(first, .(second, .(third, .(fourth, .(fifth, [])))))))
5.71/2.40	   domain_in_gg([], X1) -> domain_out_gg([], X1)
5.71/2.40	   domain_in_gg(.(X, Rest), Domain) -> U21_gg(X, Rest, Domain, select_in_aga(X, Domain, NewDomain))
5.71/2.40	   select_in_aga(X, .(X, R), R) -> select_out_aga(X, .(X, R), R)
5.71/2.40	   select_in_aga(X, .(Y, R), .(Y, Rest)) -> U23_aga(X, Y, R, Rest, select_in_aga(X, R, Rest))
5.71/2.40	   U23_aga(X, Y, R, Rest, select_out_aga(X, R, Rest)) -> select_out_aga(X, .(Y, R), .(Y, Rest))
5.71/2.40	   U21_gg(X, Rest, Domain, select_out_aga(X, Domain, NewDomain)) -> U22_gg(X, Rest, Domain, domain_in_gg(Rest, NewDomain))
5.71/2.40	   U22_gg(X, Rest, Domain, domain_out_gg(Rest, NewDomain)) -> domain_out_gg(.(X, Rest), Domain)
5.71/2.40	   U20_g(Prop, domain_out_gg(Prop, .(first, .(second, .(third, .(fourth, .(fifth, []))))))) -> houses_out_g(Prop)
5.71/2.40	   U15_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_out_g(.(Blue, .(Green, .(Red, .(Yellow, .(Ivory, []))))))) -> U16_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_in_g(.(Norwegian, .(Englishman, .(Spaniard, .(Japanese, .(Ukrainian, [])))))))
5.71/2.40	   U16_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_out_g(.(Norwegian, .(Englishman, .(Spaniard, .(Japanese, .(Ukrainian, []))))))) -> U17_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Winston, Kool, Milk, Chesterfield, Lucky, Juice, Parliament, houses_in_g(.(Dog, .(Zebra, .(Fox, .(Snails, .(Horse, [])))))))
5.71/2.40	   U17_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Winston, Kool, Milk, Chesterfield, Lucky, Juice, Parliament, houses_out_g(.(Dog, .(Zebra, .(Fox, .(Snails, .(Horse, []))))))) -> U18_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Milk, Juice, houses_in_g(.(Parliament, .(Kool, .(Lucky, .(Chesterfield, .(Winston, [])))))))
5.71/2.40	   U18_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Milk, Juice, houses_out_g(.(Parliament, .(Kool, .(Lucky, .(Chesterfield, .(Winston, []))))))) -> U19_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, houses_in_g(.(Milk, .(Juice, .(Water, .(Tea, .(Coffee, [])))))))
5.71/2.40	   U19_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, houses_out_g(.(Milk, .(Juice, .(Water, .(Tea, .(Coffee, []))))))) -> zebra_out_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water)
5.71/2.40	
5.71/2.40	The argument filtering Pi contains the following mapping:
5.71/2.40	zebra_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  zebra_in_aaaaaaa
5.71/2.40	
5.71/2.40	U1_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U1_aaaaaaa(x8)
5.71/2.40	
5.71/2.40	eq_in_aa(x1, x2)  =  eq_in_aa
5.71/2.40	
5.71/2.40	eq_out_aa(x1, x2)  =  eq_out_aa
5.71/2.40	
5.71/2.40	U2_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U2_aaaaaaa(x9)
5.71/2.40	
5.71/2.40	U3_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U3_aaaaaaa(x10)
5.71/2.40	
5.71/2.40	U4_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)  =  U4_aaaaaaa(x12)
5.71/2.40	
5.71/2.40	U5_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13)  =  U5_aaaaaaa(x13)
5.71/2.40	
5.71/2.40	to_the_right_in_aa(x1, x2)  =  to_the_right_in_aa
5.71/2.40	
5.71/2.40	to_the_right_out_aa(x1, x2)  =  to_the_right_out_aa(x1, x2)
5.71/2.40	
5.71/2.40	U6_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14)  =  U6_aaaaaaa(x14)
5.71/2.40	
5.71/2.40	U7_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16)  =  U7_aaaaaaa(x16)
5.71/2.40	
5.71/2.40	U8_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18)  =  U8_aaaaaaa(x18)
5.71/2.40	
5.71/2.40	eq_in_ag(x1, x2)  =  eq_in_ag(x2)
5.71/2.40	
5.71/2.40	eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
5.71/2.40	
5.71/2.40	third  =  third
5.71/2.40	
5.71/2.40	U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19)  =  U9_aaaaaaa(x19)
5.71/2.40	
5.71/2.40	first  =  first
5.71/2.40	
5.71/2.40	U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19)  =  U10_aaaaaaa(x5, x19)
5.71/2.40	
5.71/2.40	next_to_in_aa(x1, x2)  =  next_to_in_aa
5.71/2.40	
5.71/2.40	next_to_out_aa(x1, x2)  =  next_to_out_aa(x1, x2)
5.71/2.40	
5.71/2.40	U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U11_aaaaaaa(x5, x21)
5.71/2.40	
5.71/2.40	U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22)  =  U12_aaaaaaa(x5, x22)
5.71/2.40	
5.71/2.40	U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24)  =  U13_aaaaaaa(x5, x24)
5.71/2.40	
5.71/2.40	U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25)  =  U14_aaaaaaa(x5, x25)
5.71/2.40	
5.71/2.40	next_to_in_ga(x1, x2)  =  next_to_in_ga(x1)
5.71/2.40	
5.71/2.40	fifth  =  fifth
5.71/2.40	
5.71/2.40	next_to_out_ga(x1, x2)  =  next_to_out_ga(x1, x2)
5.71/2.40	
5.71/2.40	fourth  =  fourth
5.71/2.40	
5.71/2.40	second  =  second
5.71/2.40	
5.71/2.40	U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U15_aaaaaaa(x5, x21)
5.71/2.40	
5.71/2.40	houses_in_g(x1)  =  houses_in_g(x1)
5.71/2.40	
5.71/2.40	.(x1, x2)  =  .(x2)
5.71/2.40	
5.71/2.40	U20_g(x1, x2)  =  U20_g(x1, x2)
5.71/2.40	
5.71/2.40	domain_in_gg(x1, x2)  =  domain_in_gg(x1, x2)
5.71/2.40	
5.71/2.40	[]  =  []
5.71/2.40	
5.71/2.40	domain_out_gg(x1, x2)  =  domain_out_gg(x1, x2)
5.71/2.40	
5.71/2.40	U21_gg(x1, x2, x3, x4)  =  U21_gg(x2, x3, x4)
5.71/2.40	
5.71/2.40	select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
5.71/2.40	
5.71/2.40	select_out_aga(x1, x2, x3)  =  select_out_aga(x2, x3)
5.71/2.40	
5.71/2.40	U23_aga(x1, x2, x3, x4, x5)  =  U23_aga(x3, x5)
5.71/2.40	
5.71/2.40	U22_gg(x1, x2, x3, x4)  =  U22_gg(x2, x3, x4)
5.71/2.40	
5.71/2.40	houses_out_g(x1)  =  houses_out_g(x1)
5.71/2.40	
5.71/2.41	U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U16_aaaaaaa(x5, x21)
5.71/2.41	
5.71/2.41	U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17)  =  U17_aaaaaaa(x5, x17)
5.71/2.41	
5.71/2.41	U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)  =  U18_aaaaaaa(x5, x12)
5.71/2.41	
5.71/2.41	U19_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_aaaaaaa(x5, x8)
5.71/2.41	
5.71/2.41	zebra_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  zebra_out_aaaaaaa(x5)
5.71/2.41	
5.71/2.41	ZEBRA_IN_AAAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  ZEBRA_IN_AAAAAAA
5.71/2.41	
5.71/2.41	U1_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U1_AAAAAAA(x8)
5.71/2.41	
5.71/2.41	EQ_IN_AA(x1, x2)  =  EQ_IN_AA
5.71/2.41	
5.71/2.41	U2_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U2_AAAAAAA(x9)
5.71/2.41	
5.71/2.41	U3_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U3_AAAAAAA(x10)
5.71/2.41	
5.71/2.41	U4_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)  =  U4_AAAAAAA(x12)
5.71/2.41	
5.71/2.41	U5_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13)  =  U5_AAAAAAA(x13)
5.71/2.41	
5.71/2.41	TO_THE_RIGHT_IN_AA(x1, x2)  =  TO_THE_RIGHT_IN_AA
5.71/2.41	
5.71/2.41	U6_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14)  =  U6_AAAAAAA(x14)
5.71/2.41	
5.71/2.41	U7_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16)  =  U7_AAAAAAA(x16)
5.71/2.41	
5.71/2.41	U8_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18)  =  U8_AAAAAAA(x18)
5.71/2.41	
5.71/2.41	EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
5.71/2.41	
5.71/2.41	U9_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19)  =  U9_AAAAAAA(x19)
5.71/2.41	
5.71/2.41	U10_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19)  =  U10_AAAAAAA(x5, x19)
5.71/2.41	
5.71/2.41	NEXT_TO_IN_AA(x1, x2)  =  NEXT_TO_IN_AA
5.71/2.41	
5.71/2.41	U11_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U11_AAAAAAA(x5, x21)
5.71/2.41	
5.71/2.41	U12_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22)  =  U12_AAAAAAA(x5, x22)
5.71/2.41	
5.71/2.41	U13_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24)  =  U13_AAAAAAA(x5, x24)
5.71/2.41	
5.71/2.41	U14_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25)  =  U14_AAAAAAA(x5, x25)
5.71/2.41	
5.71/2.41	NEXT_TO_IN_GA(x1, x2)  =  NEXT_TO_IN_GA(x1)
5.71/2.41	
5.71/2.41	U15_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U15_AAAAAAA(x5, x21)
5.71/2.41	
5.71/2.41	HOUSES_IN_G(x1)  =  HOUSES_IN_G(x1)
5.71/2.41	
5.71/2.41	U20_G(x1, x2)  =  U20_G(x1, x2)
5.71/2.41	
5.71/2.41	DOMAIN_IN_GG(x1, x2)  =  DOMAIN_IN_GG(x1, x2)
5.71/2.41	
5.71/2.41	U21_GG(x1, x2, x3, x4)  =  U21_GG(x2, x3, x4)
5.71/2.41	
5.71/2.41	SELECT_IN_AGA(x1, x2, x3)  =  SELECT_IN_AGA(x2)
5.71/2.41	
5.71/2.41	U23_AGA(x1, x2, x3, x4, x5)  =  U23_AGA(x3, x5)
5.71/2.41	
5.71/2.41	U22_GG(x1, x2, x3, x4)  =  U22_GG(x2, x3, x4)
5.71/2.41	
5.71/2.41	U16_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U16_AAAAAAA(x5, x21)
5.71/2.41	
5.71/2.41	U17_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17)  =  U17_AAAAAAA(x5, x17)
5.71/2.41	
5.71/2.41	U18_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)  =  U18_AAAAAAA(x5, x12)
5.71/2.41	
5.71/2.41	U19_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_AAAAAAA(x5, x8)
5.71/2.41	
5.71/2.41	
5.71/2.41	We have to consider all (P,R,Pi)-chains
5.71/2.41	----------------------------------------
5.71/2.41	
5.71/2.41	(4)
5.71/2.41	Obligation:
5.71/2.41	Pi DP problem:
5.71/2.41	The TRS P consists of the following rules:
5.71/2.41	
5.71/2.41	   ZEBRA_IN_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water) -> U1_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, eq_in_aa(Englishman, Red))
5.71/2.41	   ZEBRA_IN_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water) -> EQ_IN_AA(Englishman, Red)
5.71/2.41	   U1_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, eq_out_aa(Englishman, Red)) -> U2_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, eq_in_aa(Spaniard, Dog))
5.71/2.41	   U1_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, eq_out_aa(Englishman, Red)) -> EQ_IN_AA(Spaniard, Dog)
5.71/2.41	   U2_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, eq_out_aa(Spaniard, Dog)) -> U3_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, eq_in_aa(Green, Coffee))
5.71/2.41	   U2_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, eq_out_aa(Spaniard, Dog)) -> EQ_IN_AA(Green, Coffee)
5.71/2.41	   U3_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, eq_out_aa(Green, Coffee)) -> U4_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, eq_in_aa(Ukrainian, Tea))
5.71/2.41	   U3_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, eq_out_aa(Green, Coffee)) -> EQ_IN_AA(Ukrainian, Tea)
5.71/2.41	   U4_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, eq_out_aa(Ukrainian, Tea)) -> U5_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, to_the_right_in_aa(Green, Ivory))
5.71/2.41	   U4_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, eq_out_aa(Ukrainian, Tea)) -> TO_THE_RIGHT_IN_AA(Green, Ivory)
5.71/2.41	   U5_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, to_the_right_out_aa(Green, Ivory)) -> U6_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, eq_in_aa(Winston, Snails))
5.71/2.41	   U5_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, to_the_right_out_aa(Green, Ivory)) -> EQ_IN_AA(Winston, Snails)
5.71/2.41	   U6_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, eq_out_aa(Winston, Snails)) -> U7_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, eq_in_aa(Kool, Yellow))
5.71/2.41	   U6_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, eq_out_aa(Winston, Snails)) -> EQ_IN_AA(Kool, Yellow)
5.71/2.41	   U7_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, eq_out_aa(Kool, Yellow)) -> U8_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, eq_in_ag(Milk, third))
5.71/2.41	   U7_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, eq_out_aa(Kool, Yellow)) -> EQ_IN_AG(Milk, third)
5.71/2.41	   U8_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, eq_out_ag(Milk, third)) -> U9_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, eq_in_ag(Norwegian, first))
5.71/2.41	   U8_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, eq_out_ag(Milk, third)) -> EQ_IN_AG(Norwegian, first)
5.71/2.41	   U9_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, eq_out_ag(Norwegian, first)) -> U10_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, next_to_in_aa(Fox, Chesterfield))
5.71/2.41	   U9_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, eq_out_ag(Norwegian, first)) -> NEXT_TO_IN_AA(Fox, Chesterfield)
5.71/2.41	   U10_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, next_to_out_aa(Fox, Chesterfield)) -> U11_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, next_to_in_aa(Horse, Kool))
5.71/2.41	   U10_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, next_to_out_aa(Fox, Chesterfield)) -> NEXT_TO_IN_AA(Horse, Kool)
5.71/2.41	   U11_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, next_to_out_aa(Horse, Kool)) -> U12_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, eq_in_aa(Lucky, Juice))
5.71/2.41	   U11_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, next_to_out_aa(Horse, Kool)) -> EQ_IN_AA(Lucky, Juice)
5.71/2.41	   U12_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, eq_out_aa(Lucky, Juice)) -> U13_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, eq_in_aa(Japanese, Parliament))
5.71/2.41	   U12_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, eq_out_aa(Lucky, Juice)) -> EQ_IN_AA(Japanese, Parliament)
5.71/2.41	   U13_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, eq_out_aa(Japanese, Parliament)) -> U14_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, next_to_in_ga(Norwegian, Blue))
5.71/2.41	   U13_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, eq_out_aa(Japanese, Parliament)) -> NEXT_TO_IN_GA(Norwegian, Blue)
5.71/2.41	   U14_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, next_to_out_ga(Norwegian, Blue)) -> U15_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_in_g(.(Blue, .(Green, .(Red, .(Yellow, .(Ivory, [])))))))
5.71/2.41	   U14_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, next_to_out_ga(Norwegian, Blue)) -> HOUSES_IN_G(.(Blue, .(Green, .(Red, .(Yellow, .(Ivory, []))))))
5.71/2.41	   HOUSES_IN_G(Prop) -> U20_G(Prop, domain_in_gg(Prop, .(first, .(second, .(third, .(fourth, .(fifth, [])))))))
5.71/2.41	   HOUSES_IN_G(Prop) -> DOMAIN_IN_GG(Prop, .(first, .(second, .(third, .(fourth, .(fifth, []))))))
5.71/2.41	   DOMAIN_IN_GG(.(X, Rest), Domain) -> U21_GG(X, Rest, Domain, select_in_aga(X, Domain, NewDomain))
5.71/2.41	   DOMAIN_IN_GG(.(X, Rest), Domain) -> SELECT_IN_AGA(X, Domain, NewDomain)
5.71/2.41	   SELECT_IN_AGA(X, .(Y, R), .(Y, Rest)) -> U23_AGA(X, Y, R, Rest, select_in_aga(X, R, Rest))
5.71/2.41	   SELECT_IN_AGA(X, .(Y, R), .(Y, Rest)) -> SELECT_IN_AGA(X, R, Rest)
5.71/2.41	   U21_GG(X, Rest, Domain, select_out_aga(X, Domain, NewDomain)) -> U22_GG(X, Rest, Domain, domain_in_gg(Rest, NewDomain))
5.71/2.41	   U21_GG(X, Rest, Domain, select_out_aga(X, Domain, NewDomain)) -> DOMAIN_IN_GG(Rest, NewDomain)
5.71/2.41	   U15_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_out_g(.(Blue, .(Green, .(Red, .(Yellow, .(Ivory, []))))))) -> U16_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_in_g(.(Norwegian, .(Englishman, .(Spaniard, .(Japanese, .(Ukrainian, [])))))))
5.71/2.41	   U15_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_out_g(.(Blue, .(Green, .(Red, .(Yellow, .(Ivory, []))))))) -> HOUSES_IN_G(.(Norwegian, .(Englishman, .(Spaniard, .(Japanese, .(Ukrainian, []))))))
5.71/2.41	   U16_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_out_g(.(Norwegian, .(Englishman, .(Spaniard, .(Japanese, .(Ukrainian, []))))))) -> U17_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Winston, Kool, Milk, Chesterfield, Lucky, Juice, Parliament, houses_in_g(.(Dog, .(Zebra, .(Fox, .(Snails, .(Horse, [])))))))
5.71/2.41	   U16_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_out_g(.(Norwegian, .(Englishman, .(Spaniard, .(Japanese, .(Ukrainian, []))))))) -> HOUSES_IN_G(.(Dog, .(Zebra, .(Fox, .(Snails, .(Horse, []))))))
5.71/2.41	   U17_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Winston, Kool, Milk, Chesterfield, Lucky, Juice, Parliament, houses_out_g(.(Dog, .(Zebra, .(Fox, .(Snails, .(Horse, []))))))) -> U18_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Milk, Juice, houses_in_g(.(Parliament, .(Kool, .(Lucky, .(Chesterfield, .(Winston, [])))))))
5.71/2.41	   U17_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Winston, Kool, Milk, Chesterfield, Lucky, Juice, Parliament, houses_out_g(.(Dog, .(Zebra, .(Fox, .(Snails, .(Horse, []))))))) -> HOUSES_IN_G(.(Parliament, .(Kool, .(Lucky, .(Chesterfield, .(Winston, []))))))
5.71/2.41	   U18_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Milk, Juice, houses_out_g(.(Parliament, .(Kool, .(Lucky, .(Chesterfield, .(Winston, []))))))) -> U19_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, houses_in_g(.(Milk, .(Juice, .(Water, .(Tea, .(Coffee, [])))))))
5.71/2.41	   U18_AAAAAAA(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Milk, Juice, houses_out_g(.(Parliament, .(Kool, .(Lucky, .(Chesterfield, .(Winston, []))))))) -> HOUSES_IN_G(.(Milk, .(Juice, .(Water, .(Tea, .(Coffee, []))))))
5.71/2.41	
5.71/2.41	The TRS R consists of the following rules:
5.71/2.41	
5.71/2.41	   zebra_in_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water) -> U1_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, eq_in_aa(Englishman, Red))
5.71/2.41	   eq_in_aa(X, X) -> eq_out_aa(X, X)
5.71/2.41	   U1_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, eq_out_aa(Englishman, Red)) -> U2_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, eq_in_aa(Spaniard, Dog))
5.71/2.41	   U2_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, eq_out_aa(Spaniard, Dog)) -> U3_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, eq_in_aa(Green, Coffee))
5.71/2.41	   U3_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, eq_out_aa(Green, Coffee)) -> U4_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, eq_in_aa(Ukrainian, Tea))
5.71/2.41	   U4_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, eq_out_aa(Ukrainian, Tea)) -> U5_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, to_the_right_in_aa(Green, Ivory))
5.71/2.41	   to_the_right_in_aa(fifth, fourth) -> to_the_right_out_aa(fifth, fourth)
5.71/2.41	   to_the_right_in_aa(fourth, third) -> to_the_right_out_aa(fourth, third)
5.71/2.41	   to_the_right_in_aa(third, second) -> to_the_right_out_aa(third, second)
5.71/2.41	   to_the_right_in_aa(second, first) -> to_the_right_out_aa(second, first)
5.71/2.41	   U5_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, to_the_right_out_aa(Green, Ivory)) -> U6_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, eq_in_aa(Winston, Snails))
5.71/2.41	   U6_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, eq_out_aa(Winston, Snails)) -> U7_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, eq_in_aa(Kool, Yellow))
5.71/2.41	   U7_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, eq_out_aa(Kool, Yellow)) -> U8_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, eq_in_ag(Milk, third))
5.71/2.41	   eq_in_ag(X, X) -> eq_out_ag(X, X)
5.71/2.41	   U8_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, eq_out_ag(Milk, third)) -> U9_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, eq_in_ag(Norwegian, first))
5.71/2.41	   U9_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, eq_out_ag(Norwegian, first)) -> U10_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, next_to_in_aa(Fox, Chesterfield))
5.71/2.41	   next_to_in_aa(fifth, fourth) -> next_to_out_aa(fifth, fourth)
5.71/2.41	   next_to_in_aa(fourth, fifth) -> next_to_out_aa(fourth, fifth)
5.71/2.41	   next_to_in_aa(fourth, third) -> next_to_out_aa(fourth, third)
5.71/2.41	   next_to_in_aa(third, fourth) -> next_to_out_aa(third, fourth)
5.71/2.41	   next_to_in_aa(third, second) -> next_to_out_aa(third, second)
5.71/2.41	   next_to_in_aa(second, third) -> next_to_out_aa(second, third)
5.71/2.41	   next_to_in_aa(second, first) -> next_to_out_aa(second, first)
5.71/2.41	   next_to_in_aa(first, second) -> next_to_out_aa(first, second)
5.71/2.41	   U10_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, next_to_out_aa(Fox, Chesterfield)) -> U11_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, next_to_in_aa(Horse, Kool))
5.71/2.41	   U11_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, next_to_out_aa(Horse, Kool)) -> U12_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, eq_in_aa(Lucky, Juice))
5.71/2.41	   U12_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, eq_out_aa(Lucky, Juice)) -> U13_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, eq_in_aa(Japanese, Parliament))
5.71/2.41	   U13_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, eq_out_aa(Japanese, Parliament)) -> U14_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, next_to_in_ga(Norwegian, Blue))
5.71/2.41	   next_to_in_ga(fifth, fourth) -> next_to_out_ga(fifth, fourth)
5.71/2.41	   next_to_in_ga(fourth, fifth) -> next_to_out_ga(fourth, fifth)
5.71/2.41	   next_to_in_ga(fourth, third) -> next_to_out_ga(fourth, third)
5.71/2.41	   next_to_in_ga(third, fourth) -> next_to_out_ga(third, fourth)
5.71/2.41	   next_to_in_ga(third, second) -> next_to_out_ga(third, second)
5.71/2.41	   next_to_in_ga(second, third) -> next_to_out_ga(second, third)
5.71/2.41	   next_to_in_ga(second, first) -> next_to_out_ga(second, first)
5.71/2.41	   next_to_in_ga(first, second) -> next_to_out_ga(first, second)
5.71/2.41	   U14_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, next_to_out_ga(Norwegian, Blue)) -> U15_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_in_g(.(Blue, .(Green, .(Red, .(Yellow, .(Ivory, [])))))))
5.71/2.41	   houses_in_g(Prop) -> U20_g(Prop, domain_in_gg(Prop, .(first, .(second, .(third, .(fourth, .(fifth, [])))))))
5.71/2.41	   domain_in_gg([], X1) -> domain_out_gg([], X1)
5.71/2.41	   domain_in_gg(.(X, Rest), Domain) -> U21_gg(X, Rest, Domain, select_in_aga(X, Domain, NewDomain))
5.71/2.41	   select_in_aga(X, .(X, R), R) -> select_out_aga(X, .(X, R), R)
5.71/2.41	   select_in_aga(X, .(Y, R), .(Y, Rest)) -> U23_aga(X, Y, R, Rest, select_in_aga(X, R, Rest))
5.71/2.41	   U23_aga(X, Y, R, Rest, select_out_aga(X, R, Rest)) -> select_out_aga(X, .(Y, R), .(Y, Rest))
5.71/2.41	   U21_gg(X, Rest, Domain, select_out_aga(X, Domain, NewDomain)) -> U22_gg(X, Rest, Domain, domain_in_gg(Rest, NewDomain))
5.71/2.41	   U22_gg(X, Rest, Domain, domain_out_gg(Rest, NewDomain)) -> domain_out_gg(.(X, Rest), Domain)
5.71/2.41	   U20_g(Prop, domain_out_gg(Prop, .(first, .(second, .(third, .(fourth, .(fifth, []))))))) -> houses_out_g(Prop)
5.71/2.41	   U15_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_out_g(.(Blue, .(Green, .(Red, .(Yellow, .(Ivory, []))))))) -> U16_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_in_g(.(Norwegian, .(Englishman, .(Spaniard, .(Japanese, .(Ukrainian, [])))))))
5.71/2.41	   U16_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_out_g(.(Norwegian, .(Englishman, .(Spaniard, .(Japanese, .(Ukrainian, []))))))) -> U17_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Winston, Kool, Milk, Chesterfield, Lucky, Juice, Parliament, houses_in_g(.(Dog, .(Zebra, .(Fox, .(Snails, .(Horse, [])))))))
5.71/2.41	   U17_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Winston, Kool, Milk, Chesterfield, Lucky, Juice, Parliament, houses_out_g(.(Dog, .(Zebra, .(Fox, .(Snails, .(Horse, []))))))) -> U18_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Milk, Juice, houses_in_g(.(Parliament, .(Kool, .(Lucky, .(Chesterfield, .(Winston, [])))))))
5.71/2.41	   U18_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Milk, Juice, houses_out_g(.(Parliament, .(Kool, .(Lucky, .(Chesterfield, .(Winston, []))))))) -> U19_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, houses_in_g(.(Milk, .(Juice, .(Water, .(Tea, .(Coffee, [])))))))
5.71/2.41	   U19_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, houses_out_g(.(Milk, .(Juice, .(Water, .(Tea, .(Coffee, []))))))) -> zebra_out_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water)
5.71/2.41	
5.71/2.41	The argument filtering Pi contains the following mapping:
5.71/2.41	zebra_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  zebra_in_aaaaaaa
5.71/2.41	
5.71/2.41	U1_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U1_aaaaaaa(x8)
5.71/2.41	
5.71/2.41	eq_in_aa(x1, x2)  =  eq_in_aa
5.71/2.41	
5.71/2.41	eq_out_aa(x1, x2)  =  eq_out_aa
5.71/2.41	
5.71/2.41	U2_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U2_aaaaaaa(x9)
5.71/2.41	
5.71/2.41	U3_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U3_aaaaaaa(x10)
5.71/2.41	
5.71/2.41	U4_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)  =  U4_aaaaaaa(x12)
5.71/2.41	
5.71/2.41	U5_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13)  =  U5_aaaaaaa(x13)
5.71/2.41	
5.71/2.41	to_the_right_in_aa(x1, x2)  =  to_the_right_in_aa
5.71/2.41	
5.71/2.41	to_the_right_out_aa(x1, x2)  =  to_the_right_out_aa(x1, x2)
5.71/2.41	
5.71/2.41	U6_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14)  =  U6_aaaaaaa(x14)
5.71/2.41	
5.71/2.41	U7_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16)  =  U7_aaaaaaa(x16)
5.71/2.41	
5.71/2.41	U8_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18)  =  U8_aaaaaaa(x18)
5.71/2.41	
5.71/2.41	eq_in_ag(x1, x2)  =  eq_in_ag(x2)
5.71/2.41	
5.71/2.41	eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
5.71/2.41	
5.71/2.41	third  =  third
5.71/2.41	
5.71/2.41	U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19)  =  U9_aaaaaaa(x19)
5.71/2.41	
5.71/2.41	first  =  first
5.71/2.41	
5.71/2.41	U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19)  =  U10_aaaaaaa(x5, x19)
5.71/2.41	
5.71/2.41	next_to_in_aa(x1, x2)  =  next_to_in_aa
5.71/2.41	
5.71/2.41	next_to_out_aa(x1, x2)  =  next_to_out_aa(x1, x2)
5.71/2.41	
5.71/2.41	U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U11_aaaaaaa(x5, x21)
5.71/2.41	
5.71/2.41	U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22)  =  U12_aaaaaaa(x5, x22)
5.71/2.41	
5.71/2.41	U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24)  =  U13_aaaaaaa(x5, x24)
5.71/2.41	
5.71/2.41	U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25)  =  U14_aaaaaaa(x5, x25)
5.71/2.41	
5.71/2.41	next_to_in_ga(x1, x2)  =  next_to_in_ga(x1)
5.71/2.41	
5.71/2.41	fifth  =  fifth
5.71/2.41	
5.71/2.41	next_to_out_ga(x1, x2)  =  next_to_out_ga(x1, x2)
5.71/2.41	
5.71/2.41	fourth  =  fourth
5.71/2.41	
5.71/2.41	second  =  second
5.71/2.41	
5.71/2.41	U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U15_aaaaaaa(x5, x21)
5.71/2.41	
5.71/2.41	houses_in_g(x1)  =  houses_in_g(x1)
5.71/2.41	
5.71/2.41	.(x1, x2)  =  .(x2)
5.71/2.41	
5.71/2.41	U20_g(x1, x2)  =  U20_g(x1, x2)
5.71/2.41	
5.71/2.41	domain_in_gg(x1, x2)  =  domain_in_gg(x1, x2)
5.71/2.41	
5.71/2.41	[]  =  []
5.71/2.41	
5.71/2.41	domain_out_gg(x1, x2)  =  domain_out_gg(x1, x2)
5.71/2.41	
5.71/2.41	U21_gg(x1, x2, x3, x4)  =  U21_gg(x2, x3, x4)
5.71/2.41	
5.71/2.41	select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
5.71/2.41	
5.71/2.41	select_out_aga(x1, x2, x3)  =  select_out_aga(x2, x3)
5.71/2.41	
5.71/2.41	U23_aga(x1, x2, x3, x4, x5)  =  U23_aga(x3, x5)
5.71/2.41	
5.71/2.41	U22_gg(x1, x2, x3, x4)  =  U22_gg(x2, x3, x4)
5.71/2.41	
5.71/2.41	houses_out_g(x1)  =  houses_out_g(x1)
5.71/2.41	
5.71/2.41	U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U16_aaaaaaa(x5, x21)
5.71/2.41	
5.71/2.41	U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17)  =  U17_aaaaaaa(x5, x17)
5.71/2.41	
5.71/2.41	U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)  =  U18_aaaaaaa(x5, x12)
5.71/2.41	
5.71/2.41	U19_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_aaaaaaa(x5, x8)
5.71/2.41	
5.71/2.41	zebra_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  zebra_out_aaaaaaa(x5)
5.71/2.41	
5.71/2.41	ZEBRA_IN_AAAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  ZEBRA_IN_AAAAAAA
5.71/2.41	
5.71/2.41	U1_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U1_AAAAAAA(x8)
5.71/2.41	
5.71/2.41	EQ_IN_AA(x1, x2)  =  EQ_IN_AA
5.71/2.41	
5.71/2.41	U2_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U2_AAAAAAA(x9)
5.71/2.41	
5.71/2.41	U3_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U3_AAAAAAA(x10)
5.71/2.41	
5.71/2.41	U4_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)  =  U4_AAAAAAA(x12)
5.71/2.41	
5.71/2.41	U5_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13)  =  U5_AAAAAAA(x13)
5.71/2.41	
5.71/2.41	TO_THE_RIGHT_IN_AA(x1, x2)  =  TO_THE_RIGHT_IN_AA
5.71/2.41	
5.71/2.41	U6_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14)  =  U6_AAAAAAA(x14)
5.71/2.41	
5.71/2.41	U7_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16)  =  U7_AAAAAAA(x16)
5.71/2.41	
5.71/2.41	U8_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18)  =  U8_AAAAAAA(x18)
5.71/2.41	
5.71/2.41	EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
5.71/2.41	
5.71/2.41	U9_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19)  =  U9_AAAAAAA(x19)
5.71/2.41	
5.71/2.41	U10_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19)  =  U10_AAAAAAA(x5, x19)
5.71/2.41	
5.71/2.41	NEXT_TO_IN_AA(x1, x2)  =  NEXT_TO_IN_AA
5.71/2.41	
5.71/2.41	U11_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U11_AAAAAAA(x5, x21)
5.71/2.41	
5.71/2.41	U12_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22)  =  U12_AAAAAAA(x5, x22)
5.71/2.41	
5.71/2.41	U13_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24)  =  U13_AAAAAAA(x5, x24)
5.71/2.41	
5.71/2.41	U14_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25)  =  U14_AAAAAAA(x5, x25)
5.71/2.41	
5.71/2.41	NEXT_TO_IN_GA(x1, x2)  =  NEXT_TO_IN_GA(x1)
5.71/2.41	
5.71/2.41	U15_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U15_AAAAAAA(x5, x21)
5.71/2.41	
5.71/2.41	HOUSES_IN_G(x1)  =  HOUSES_IN_G(x1)
5.71/2.41	
5.71/2.41	U20_G(x1, x2)  =  U20_G(x1, x2)
5.71/2.41	
5.71/2.41	DOMAIN_IN_GG(x1, x2)  =  DOMAIN_IN_GG(x1, x2)
5.71/2.41	
5.71/2.41	U21_GG(x1, x2, x3, x4)  =  U21_GG(x2, x3, x4)
5.71/2.41	
5.71/2.41	SELECT_IN_AGA(x1, x2, x3)  =  SELECT_IN_AGA(x2)
5.71/2.41	
5.71/2.41	U23_AGA(x1, x2, x3, x4, x5)  =  U23_AGA(x3, x5)
5.71/2.41	
5.71/2.41	U22_GG(x1, x2, x3, x4)  =  U22_GG(x2, x3, x4)
5.71/2.41	
5.71/2.41	U16_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U16_AAAAAAA(x5, x21)
5.71/2.41	
5.71/2.41	U17_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17)  =  U17_AAAAAAA(x5, x17)
5.71/2.41	
5.71/2.41	U18_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)  =  U18_AAAAAAA(x5, x12)
5.71/2.41	
5.71/2.41	U19_AAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_AAAAAAA(x5, x8)
5.71/2.41	
5.71/2.41	
5.71/2.41	We have to consider all (P,R,Pi)-chains
5.71/2.41	----------------------------------------
5.71/2.41	
5.71/2.41	(5) DependencyGraphProof (EQUIVALENT)
5.71/2.41	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 43 less nodes.
5.71/2.41	----------------------------------------
5.71/2.41	
5.71/2.41	(6)
5.71/2.41	Complex Obligation (AND)
5.71/2.41	
5.71/2.41	----------------------------------------
5.71/2.41	
5.71/2.41	(7)
5.71/2.41	Obligation:
5.71/2.41	Pi DP problem:
5.71/2.41	The TRS P consists of the following rules:
5.71/2.41	
5.71/2.41	   SELECT_IN_AGA(X, .(Y, R), .(Y, Rest)) -> SELECT_IN_AGA(X, R, Rest)
5.71/2.41	
5.71/2.41	The TRS R consists of the following rules:
5.71/2.41	
5.71/2.41	   zebra_in_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water) -> U1_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, eq_in_aa(Englishman, Red))
5.71/2.41	   eq_in_aa(X, X) -> eq_out_aa(X, X)
5.71/2.41	   U1_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, eq_out_aa(Englishman, Red)) -> U2_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, eq_in_aa(Spaniard, Dog))
5.71/2.41	   U2_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, eq_out_aa(Spaniard, Dog)) -> U3_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, eq_in_aa(Green, Coffee))
5.71/2.41	   U3_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, eq_out_aa(Green, Coffee)) -> U4_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, eq_in_aa(Ukrainian, Tea))
5.71/2.41	   U4_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, eq_out_aa(Ukrainian, Tea)) -> U5_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, to_the_right_in_aa(Green, Ivory))
5.71/2.41	   to_the_right_in_aa(fifth, fourth) -> to_the_right_out_aa(fifth, fourth)
5.71/2.41	   to_the_right_in_aa(fourth, third) -> to_the_right_out_aa(fourth, third)
5.71/2.41	   to_the_right_in_aa(third, second) -> to_the_right_out_aa(third, second)
5.71/2.41	   to_the_right_in_aa(second, first) -> to_the_right_out_aa(second, first)
5.71/2.41	   U5_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, to_the_right_out_aa(Green, Ivory)) -> U6_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, eq_in_aa(Winston, Snails))
5.71/2.41	   U6_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, eq_out_aa(Winston, Snails)) -> U7_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, eq_in_aa(Kool, Yellow))
5.71/2.41	   U7_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, eq_out_aa(Kool, Yellow)) -> U8_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, eq_in_ag(Milk, third))
5.71/2.41	   eq_in_ag(X, X) -> eq_out_ag(X, X)
5.71/2.41	   U8_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, eq_out_ag(Milk, third)) -> U9_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, eq_in_ag(Norwegian, first))
5.71/2.41	   U9_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, eq_out_ag(Norwegian, first)) -> U10_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, next_to_in_aa(Fox, Chesterfield))
5.71/2.41	   next_to_in_aa(fifth, fourth) -> next_to_out_aa(fifth, fourth)
5.71/2.41	   next_to_in_aa(fourth, fifth) -> next_to_out_aa(fourth, fifth)
5.71/2.41	   next_to_in_aa(fourth, third) -> next_to_out_aa(fourth, third)
5.71/2.41	   next_to_in_aa(third, fourth) -> next_to_out_aa(third, fourth)
5.71/2.41	   next_to_in_aa(third, second) -> next_to_out_aa(third, second)
5.71/2.41	   next_to_in_aa(second, third) -> next_to_out_aa(second, third)
5.71/2.41	   next_to_in_aa(second, first) -> next_to_out_aa(second, first)
5.71/2.41	   next_to_in_aa(first, second) -> next_to_out_aa(first, second)
5.71/2.41	   U10_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, next_to_out_aa(Fox, Chesterfield)) -> U11_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, next_to_in_aa(Horse, Kool))
5.71/2.41	   U11_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, next_to_out_aa(Horse, Kool)) -> U12_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, eq_in_aa(Lucky, Juice))
5.71/2.41	   U12_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, eq_out_aa(Lucky, Juice)) -> U13_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, eq_in_aa(Japanese, Parliament))
5.71/2.41	   U13_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, eq_out_aa(Japanese, Parliament)) -> U14_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, next_to_in_ga(Norwegian, Blue))
5.71/2.41	   next_to_in_ga(fifth, fourth) -> next_to_out_ga(fifth, fourth)
5.71/2.41	   next_to_in_ga(fourth, fifth) -> next_to_out_ga(fourth, fifth)
5.71/2.41	   next_to_in_ga(fourth, third) -> next_to_out_ga(fourth, third)
5.71/2.41	   next_to_in_ga(third, fourth) -> next_to_out_ga(third, fourth)
5.71/2.41	   next_to_in_ga(third, second) -> next_to_out_ga(third, second)
5.71/2.41	   next_to_in_ga(second, third) -> next_to_out_ga(second, third)
5.71/2.41	   next_to_in_ga(second, first) -> next_to_out_ga(second, first)
5.71/2.41	   next_to_in_ga(first, second) -> next_to_out_ga(first, second)
5.71/2.41	   U14_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, next_to_out_ga(Norwegian, Blue)) -> U15_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_in_g(.(Blue, .(Green, .(Red, .(Yellow, .(Ivory, [])))))))
5.71/2.41	   houses_in_g(Prop) -> U20_g(Prop, domain_in_gg(Prop, .(first, .(second, .(third, .(fourth, .(fifth, [])))))))
5.71/2.41	   domain_in_gg([], X1) -> domain_out_gg([], X1)
5.71/2.41	   domain_in_gg(.(X, Rest), Domain) -> U21_gg(X, Rest, Domain, select_in_aga(X, Domain, NewDomain))
5.71/2.41	   select_in_aga(X, .(X, R), R) -> select_out_aga(X, .(X, R), R)
5.71/2.41	   select_in_aga(X, .(Y, R), .(Y, Rest)) -> U23_aga(X, Y, R, Rest, select_in_aga(X, R, Rest))
5.71/2.41	   U23_aga(X, Y, R, Rest, select_out_aga(X, R, Rest)) -> select_out_aga(X, .(Y, R), .(Y, Rest))
5.71/2.41	   U21_gg(X, Rest, Domain, select_out_aga(X, Domain, NewDomain)) -> U22_gg(X, Rest, Domain, domain_in_gg(Rest, NewDomain))
5.71/2.41	   U22_gg(X, Rest, Domain, domain_out_gg(Rest, NewDomain)) -> domain_out_gg(.(X, Rest), Domain)
5.71/2.41	   U20_g(Prop, domain_out_gg(Prop, .(first, .(second, .(third, .(fourth, .(fifth, []))))))) -> houses_out_g(Prop)
5.71/2.41	   U15_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_out_g(.(Blue, .(Green, .(Red, .(Yellow, .(Ivory, []))))))) -> U16_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_in_g(.(Norwegian, .(Englishman, .(Spaniard, .(Japanese, .(Ukrainian, [])))))))
5.71/2.41	   U16_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_out_g(.(Norwegian, .(Englishman, .(Spaniard, .(Japanese, .(Ukrainian, []))))))) -> U17_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Winston, Kool, Milk, Chesterfield, Lucky, Juice, Parliament, houses_in_g(.(Dog, .(Zebra, .(Fox, .(Snails, .(Horse, [])))))))
5.71/2.41	   U17_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Winston, Kool, Milk, Chesterfield, Lucky, Juice, Parliament, houses_out_g(.(Dog, .(Zebra, .(Fox, .(Snails, .(Horse, []))))))) -> U18_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Milk, Juice, houses_in_g(.(Parliament, .(Kool, .(Lucky, .(Chesterfield, .(Winston, [])))))))
5.71/2.41	   U18_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Milk, Juice, houses_out_g(.(Parliament, .(Kool, .(Lucky, .(Chesterfield, .(Winston, []))))))) -> U19_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, houses_in_g(.(Milk, .(Juice, .(Water, .(Tea, .(Coffee, [])))))))
5.71/2.41	   U19_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, houses_out_g(.(Milk, .(Juice, .(Water, .(Tea, .(Coffee, []))))))) -> zebra_out_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water)
5.71/2.41	
5.71/2.41	The argument filtering Pi contains the following mapping:
5.71/2.41	zebra_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  zebra_in_aaaaaaa
5.71/2.41	
5.71/2.41	U1_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U1_aaaaaaa(x8)
5.71/2.41	
5.71/2.41	eq_in_aa(x1, x2)  =  eq_in_aa
5.71/2.41	
5.71/2.41	eq_out_aa(x1, x2)  =  eq_out_aa
5.71/2.41	
5.71/2.41	U2_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U2_aaaaaaa(x9)
5.71/2.41	
5.71/2.41	U3_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U3_aaaaaaa(x10)
5.71/2.41	
5.71/2.41	U4_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)  =  U4_aaaaaaa(x12)
5.71/2.41	
5.71/2.41	U5_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13)  =  U5_aaaaaaa(x13)
5.71/2.41	
5.71/2.41	to_the_right_in_aa(x1, x2)  =  to_the_right_in_aa
5.71/2.41	
5.71/2.41	to_the_right_out_aa(x1, x2)  =  to_the_right_out_aa(x1, x2)
5.71/2.41	
5.71/2.41	U6_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14)  =  U6_aaaaaaa(x14)
5.71/2.41	
5.71/2.41	U7_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16)  =  U7_aaaaaaa(x16)
5.71/2.41	
5.71/2.41	U8_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18)  =  U8_aaaaaaa(x18)
5.71/2.41	
5.71/2.41	eq_in_ag(x1, x2)  =  eq_in_ag(x2)
5.71/2.41	
5.71/2.41	eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
5.71/2.41	
5.71/2.41	third  =  third
5.71/2.41	
5.71/2.41	U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19)  =  U9_aaaaaaa(x19)
5.71/2.41	
5.71/2.41	first  =  first
5.71/2.41	
5.71/2.41	U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19)  =  U10_aaaaaaa(x5, x19)
5.71/2.41	
5.71/2.41	next_to_in_aa(x1, x2)  =  next_to_in_aa
5.71/2.41	
5.71/2.41	next_to_out_aa(x1, x2)  =  next_to_out_aa(x1, x2)
5.71/2.41	
5.71/2.41	U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U11_aaaaaaa(x5, x21)
5.71/2.41	
5.71/2.41	U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22)  =  U12_aaaaaaa(x5, x22)
5.71/2.41	
5.71/2.41	U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24)  =  U13_aaaaaaa(x5, x24)
5.71/2.41	
5.71/2.41	U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25)  =  U14_aaaaaaa(x5, x25)
5.71/2.41	
5.71/2.41	next_to_in_ga(x1, x2)  =  next_to_in_ga(x1)
5.71/2.41	
5.71/2.41	fifth  =  fifth
5.71/2.41	
5.71/2.41	next_to_out_ga(x1, x2)  =  next_to_out_ga(x1, x2)
5.71/2.41	
5.71/2.41	fourth  =  fourth
5.71/2.41	
5.71/2.41	second  =  second
5.71/2.41	
5.71/2.41	U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U15_aaaaaaa(x5, x21)
5.71/2.41	
5.71/2.41	houses_in_g(x1)  =  houses_in_g(x1)
5.71/2.41	
5.71/2.41	.(x1, x2)  =  .(x2)
5.71/2.41	
5.71/2.41	U20_g(x1, x2)  =  U20_g(x1, x2)
5.71/2.41	
5.71/2.41	domain_in_gg(x1, x2)  =  domain_in_gg(x1, x2)
5.71/2.41	
5.71/2.41	[]  =  []
5.71/2.41	
5.71/2.41	domain_out_gg(x1, x2)  =  domain_out_gg(x1, x2)
5.71/2.41	
5.71/2.41	U21_gg(x1, x2, x3, x4)  =  U21_gg(x2, x3, x4)
5.71/2.41	
5.71/2.41	select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
5.71/2.41	
5.71/2.41	select_out_aga(x1, x2, x3)  =  select_out_aga(x2, x3)
5.71/2.41	
5.71/2.41	U23_aga(x1, x2, x3, x4, x5)  =  U23_aga(x3, x5)
5.71/2.41	
5.71/2.41	U22_gg(x1, x2, x3, x4)  =  U22_gg(x2, x3, x4)
5.71/2.41	
5.71/2.41	houses_out_g(x1)  =  houses_out_g(x1)
5.71/2.41	
5.71/2.41	U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U16_aaaaaaa(x5, x21)
5.71/2.41	
5.71/2.41	U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17)  =  U17_aaaaaaa(x5, x17)
5.71/2.41	
5.71/2.41	U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)  =  U18_aaaaaaa(x5, x12)
5.71/2.41	
5.71/2.41	U19_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_aaaaaaa(x5, x8)
5.71/2.41	
5.71/2.41	zebra_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  zebra_out_aaaaaaa(x5)
5.71/2.41	
5.71/2.41	SELECT_IN_AGA(x1, x2, x3)  =  SELECT_IN_AGA(x2)
5.71/2.41	
5.71/2.41	
5.71/2.41	We have to consider all (P,R,Pi)-chains
5.71/2.41	----------------------------------------
5.71/2.41	
5.71/2.41	(8) UsableRulesProof (EQUIVALENT)
5.71/2.41	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.71/2.41	----------------------------------------
5.71/2.41	
5.71/2.41	(9)
5.71/2.41	Obligation:
5.71/2.41	Pi DP problem:
5.71/2.41	The TRS P consists of the following rules:
5.71/2.41	
5.71/2.41	   SELECT_IN_AGA(X, .(Y, R), .(Y, Rest)) -> SELECT_IN_AGA(X, R, Rest)
5.71/2.41	
5.71/2.41	R is empty.
5.71/2.41	The argument filtering Pi contains the following mapping:
5.71/2.41	.(x1, x2)  =  .(x2)
5.71/2.41	
5.71/2.41	SELECT_IN_AGA(x1, x2, x3)  =  SELECT_IN_AGA(x2)
5.71/2.41	
5.71/2.41	
5.71/2.41	We have to consider all (P,R,Pi)-chains
5.71/2.41	----------------------------------------
5.71/2.41	
5.71/2.41	(10) PiDPToQDPProof (SOUND)
5.71/2.41	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.71/2.41	----------------------------------------
5.71/2.41	
5.71/2.41	(11)
5.71/2.41	Obligation:
5.71/2.41	Q DP problem:
5.71/2.41	The TRS P consists of the following rules:
5.71/2.41	
5.71/2.41	   SELECT_IN_AGA(.(R)) -> SELECT_IN_AGA(R)
5.71/2.41	
5.71/2.41	R is empty.
5.71/2.41	Q is empty.
5.71/2.41	We have to consider all (P,Q,R)-chains.
5.71/2.41	----------------------------------------
5.71/2.41	
5.71/2.41	(12) QDPSizeChangeProof (EQUIVALENT)
5.71/2.41	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
5.71/2.41	
5.71/2.41	From the DPs we obtained the following set of size-change graphs:
5.71/2.41	*SELECT_IN_AGA(.(R)) -> SELECT_IN_AGA(R)
5.71/2.41	The graph contains the following edges 1 > 1
5.71/2.41	
5.71/2.41	
5.71/2.41	----------------------------------------
5.71/2.41	
5.71/2.41	(13)
5.71/2.41	YES
5.71/2.41	
5.71/2.41	----------------------------------------
5.71/2.41	
5.71/2.41	(14)
5.71/2.41	Obligation:
5.71/2.41	Pi DP problem:
5.71/2.41	The TRS P consists of the following rules:
5.71/2.41	
5.71/2.41	   U21_GG(X, Rest, Domain, select_out_aga(X, Domain, NewDomain)) -> DOMAIN_IN_GG(Rest, NewDomain)
5.71/2.41	   DOMAIN_IN_GG(.(X, Rest), Domain) -> U21_GG(X, Rest, Domain, select_in_aga(X, Domain, NewDomain))
5.71/2.41	
5.71/2.41	The TRS R consists of the following rules:
5.71/2.41	
5.71/2.41	   zebra_in_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water) -> U1_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, eq_in_aa(Englishman, Red))
5.71/2.41	   eq_in_aa(X, X) -> eq_out_aa(X, X)
5.71/2.41	   U1_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, eq_out_aa(Englishman, Red)) -> U2_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, eq_in_aa(Spaniard, Dog))
5.71/2.41	   U2_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, eq_out_aa(Spaniard, Dog)) -> U3_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, eq_in_aa(Green, Coffee))
5.71/2.41	   U3_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, eq_out_aa(Green, Coffee)) -> U4_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, eq_in_aa(Ukrainian, Tea))
5.71/2.41	   U4_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, eq_out_aa(Ukrainian, Tea)) -> U5_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, to_the_right_in_aa(Green, Ivory))
5.71/2.41	   to_the_right_in_aa(fifth, fourth) -> to_the_right_out_aa(fifth, fourth)
5.71/2.41	   to_the_right_in_aa(fourth, third) -> to_the_right_out_aa(fourth, third)
5.71/2.41	   to_the_right_in_aa(third, second) -> to_the_right_out_aa(third, second)
5.71/2.41	   to_the_right_in_aa(second, first) -> to_the_right_out_aa(second, first)
5.71/2.41	   U5_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, to_the_right_out_aa(Green, Ivory)) -> U6_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, eq_in_aa(Winston, Snails))
5.71/2.41	   U6_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, eq_out_aa(Winston, Snails)) -> U7_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, eq_in_aa(Kool, Yellow))
5.71/2.41	   U7_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, eq_out_aa(Kool, Yellow)) -> U8_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, eq_in_ag(Milk, third))
5.71/2.41	   eq_in_ag(X, X) -> eq_out_ag(X, X)
5.71/2.41	   U8_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, eq_out_ag(Milk, third)) -> U9_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, eq_in_ag(Norwegian, first))
5.71/2.41	   U9_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, eq_out_ag(Norwegian, first)) -> U10_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, next_to_in_aa(Fox, Chesterfield))
5.71/2.41	   next_to_in_aa(fifth, fourth) -> next_to_out_aa(fifth, fourth)
5.71/2.41	   next_to_in_aa(fourth, fifth) -> next_to_out_aa(fourth, fifth)
5.71/2.41	   next_to_in_aa(fourth, third) -> next_to_out_aa(fourth, third)
5.71/2.41	   next_to_in_aa(third, fourth) -> next_to_out_aa(third, fourth)
5.71/2.41	   next_to_in_aa(third, second) -> next_to_out_aa(third, second)
5.71/2.41	   next_to_in_aa(second, third) -> next_to_out_aa(second, third)
5.71/2.41	   next_to_in_aa(second, first) -> next_to_out_aa(second, first)
5.71/2.41	   next_to_in_aa(first, second) -> next_to_out_aa(first, second)
5.71/2.41	   U10_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, next_to_out_aa(Fox, Chesterfield)) -> U11_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, next_to_in_aa(Horse, Kool))
5.71/2.41	   U11_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, next_to_out_aa(Horse, Kool)) -> U12_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, eq_in_aa(Lucky, Juice))
5.71/2.41	   U12_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, eq_out_aa(Lucky, Juice)) -> U13_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, eq_in_aa(Japanese, Parliament))
5.71/2.41	   U13_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, eq_out_aa(Japanese, Parliament)) -> U14_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, next_to_in_ga(Norwegian, Blue))
5.71/2.41	   next_to_in_ga(fifth, fourth) -> next_to_out_ga(fifth, fourth)
5.71/2.41	   next_to_in_ga(fourth, fifth) -> next_to_out_ga(fourth, fifth)
5.71/2.41	   next_to_in_ga(fourth, third) -> next_to_out_ga(fourth, third)
5.71/2.41	   next_to_in_ga(third, fourth) -> next_to_out_ga(third, fourth)
5.71/2.41	   next_to_in_ga(third, second) -> next_to_out_ga(third, second)
5.71/2.41	   next_to_in_ga(second, third) -> next_to_out_ga(second, third)
5.71/2.41	   next_to_in_ga(second, first) -> next_to_out_ga(second, first)
5.71/2.41	   next_to_in_ga(first, second) -> next_to_out_ga(first, second)
5.71/2.41	   U14_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Red, Dog, Green, Coffee, Tea, Ivory, Winston, Snails, Kool, Yellow, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, next_to_out_ga(Norwegian, Blue)) -> U15_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_in_g(.(Blue, .(Green, .(Red, .(Yellow, .(Ivory, [])))))))
5.71/2.41	   houses_in_g(Prop) -> U20_g(Prop, domain_in_gg(Prop, .(first, .(second, .(third, .(fourth, .(fifth, [])))))))
5.71/2.41	   domain_in_gg([], X1) -> domain_out_gg([], X1)
5.71/2.41	   domain_in_gg(.(X, Rest), Domain) -> U21_gg(X, Rest, Domain, select_in_aga(X, Domain, NewDomain))
5.71/2.41	   select_in_aga(X, .(X, R), R) -> select_out_aga(X, .(X, R), R)
5.71/2.41	   select_in_aga(X, .(Y, R), .(Y, Rest)) -> U23_aga(X, Y, R, Rest, select_in_aga(X, R, Rest))
5.71/2.41	   U23_aga(X, Y, R, Rest, select_out_aga(X, R, Rest)) -> select_out_aga(X, .(Y, R), .(Y, Rest))
5.71/2.41	   U21_gg(X, Rest, Domain, select_out_aga(X, Domain, NewDomain)) -> U22_gg(X, Rest, Domain, domain_in_gg(Rest, NewDomain))
5.71/2.41	   U22_gg(X, Rest, Domain, domain_out_gg(Rest, NewDomain)) -> domain_out_gg(.(X, Rest), Domain)
5.71/2.41	   U20_g(Prop, domain_out_gg(Prop, .(first, .(second, .(third, .(fourth, .(fifth, []))))))) -> houses_out_g(Prop)
5.71/2.41	   U15_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_out_g(.(Blue, .(Green, .(Red, .(Yellow, .(Ivory, []))))))) -> U16_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_in_g(.(Norwegian, .(Englishman, .(Spaniard, .(Japanese, .(Ukrainian, [])))))))
5.71/2.41	   U16_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Dog, Coffee, Tea, Winston, Snails, Kool, Milk, Fox, Chesterfield, Horse, Lucky, Juice, Parliament, houses_out_g(.(Norwegian, .(Englishman, .(Spaniard, .(Japanese, .(Ukrainian, []))))))) -> U17_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Winston, Kool, Milk, Chesterfield, Lucky, Juice, Parliament, houses_in_g(.(Dog, .(Zebra, .(Fox, .(Snails, .(Horse, [])))))))
5.71/2.41	   U17_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Winston, Kool, Milk, Chesterfield, Lucky, Juice, Parliament, houses_out_g(.(Dog, .(Zebra, .(Fox, .(Snails, .(Horse, []))))))) -> U18_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Milk, Juice, houses_in_g(.(Parliament, .(Kool, .(Lucky, .(Chesterfield, .(Winston, [])))))))
5.71/2.41	   U18_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, Coffee, Tea, Milk, Juice, houses_out_g(.(Parliament, .(Kool, .(Lucky, .(Chesterfield, .(Winston, []))))))) -> U19_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, houses_in_g(.(Milk, .(Juice, .(Water, .(Tea, .(Coffee, [])))))))
5.71/2.41	   U19_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water, houses_out_g(.(Milk, .(Juice, .(Water, .(Tea, .(Coffee, []))))))) -> zebra_out_aaaaaaa(Englishman, Spaniard, Japanese, Ukrainian, Norwegian, Zebra, Water)
5.71/2.41	
5.71/2.41	The argument filtering Pi contains the following mapping:
5.71/2.41	zebra_in_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  zebra_in_aaaaaaa
5.71/2.41	
5.71/2.41	U1_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U1_aaaaaaa(x8)
5.71/2.41	
5.71/2.41	eq_in_aa(x1, x2)  =  eq_in_aa
5.71/2.41	
5.71/2.41	eq_out_aa(x1, x2)  =  eq_out_aa
5.71/2.41	
5.71/2.41	U2_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U2_aaaaaaa(x9)
5.71/2.41	
5.71/2.41	U3_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)  =  U3_aaaaaaa(x10)
5.71/2.41	
5.71/2.41	U4_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)  =  U4_aaaaaaa(x12)
5.71/2.41	
5.71/2.41	U5_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13)  =  U5_aaaaaaa(x13)
5.71/2.41	
5.71/2.41	to_the_right_in_aa(x1, x2)  =  to_the_right_in_aa
5.71/2.41	
5.71/2.41	to_the_right_out_aa(x1, x2)  =  to_the_right_out_aa(x1, x2)
5.71/2.41	
5.71/2.41	U6_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14)  =  U6_aaaaaaa(x14)
5.71/2.41	
5.71/2.41	U7_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16)  =  U7_aaaaaaa(x16)
5.71/2.41	
5.71/2.41	U8_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18)  =  U8_aaaaaaa(x18)
5.71/2.41	
5.71/2.41	eq_in_ag(x1, x2)  =  eq_in_ag(x2)
5.71/2.41	
5.71/2.41	eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
5.71/2.41	
5.71/2.41	third  =  third
5.71/2.41	
5.71/2.41	U9_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19)  =  U9_aaaaaaa(x19)
5.71/2.41	
5.71/2.41	first  =  first
5.71/2.41	
5.71/2.41	U10_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19)  =  U10_aaaaaaa(x5, x19)
5.71/2.41	
5.71/2.41	next_to_in_aa(x1, x2)  =  next_to_in_aa
5.71/2.41	
5.71/2.41	next_to_out_aa(x1, x2)  =  next_to_out_aa(x1, x2)
5.71/2.41	
5.71/2.41	U11_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U11_aaaaaaa(x5, x21)
5.71/2.41	
5.71/2.41	U12_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22)  =  U12_aaaaaaa(x5, x22)
5.71/2.41	
5.71/2.41	U13_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24)  =  U13_aaaaaaa(x5, x24)
5.71/2.41	
5.71/2.41	U14_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25)  =  U14_aaaaaaa(x5, x25)
5.71/2.41	
5.71/2.41	next_to_in_ga(x1, x2)  =  next_to_in_ga(x1)
5.71/2.41	
5.71/2.41	fifth  =  fifth
5.71/2.41	
5.71/2.41	next_to_out_ga(x1, x2)  =  next_to_out_ga(x1, x2)
5.71/2.41	
5.71/2.41	fourth  =  fourth
5.71/2.41	
5.71/2.41	second  =  second
5.71/2.41	
5.71/2.41	U15_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U15_aaaaaaa(x5, x21)
5.71/2.41	
5.71/2.41	houses_in_g(x1)  =  houses_in_g(x1)
5.71/2.41	
5.71/2.41	.(x1, x2)  =  .(x2)
5.71/2.41	
5.71/2.41	U20_g(x1, x2)  =  U20_g(x1, x2)
5.71/2.41	
5.71/2.41	domain_in_gg(x1, x2)  =  domain_in_gg(x1, x2)
5.71/2.41	
5.71/2.41	[]  =  []
5.71/2.41	
5.71/2.41	domain_out_gg(x1, x2)  =  domain_out_gg(x1, x2)
5.71/2.41	
5.71/2.41	U21_gg(x1, x2, x3, x4)  =  U21_gg(x2, x3, x4)
5.71/2.41	
5.71/2.41	select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
5.71/2.41	
5.71/2.41	select_out_aga(x1, x2, x3)  =  select_out_aga(x2, x3)
5.71/2.41	
5.71/2.41	U23_aga(x1, x2, x3, x4, x5)  =  U23_aga(x3, x5)
5.71/2.41	
5.71/2.41	U22_gg(x1, x2, x3, x4)  =  U22_gg(x2, x3, x4)
5.71/2.41	
5.71/2.41	houses_out_g(x1)  =  houses_out_g(x1)
5.71/2.41	
5.71/2.41	U16_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U16_aaaaaaa(x5, x21)
5.71/2.41	
5.71/2.41	U17_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17)  =  U17_aaaaaaa(x5, x17)
5.71/2.41	
5.71/2.41	U18_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)  =  U18_aaaaaaa(x5, x12)
5.71/2.41	
5.71/2.41	U19_aaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U19_aaaaaaa(x5, x8)
5.71/2.41	
5.71/2.41	zebra_out_aaaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  zebra_out_aaaaaaa(x5)
5.71/2.41	
5.71/2.41	DOMAIN_IN_GG(x1, x2)  =  DOMAIN_IN_GG(x1, x2)
5.71/2.41	
5.71/2.41	U21_GG(x1, x2, x3, x4)  =  U21_GG(x2, x3, x4)
5.71/2.41	
5.71/2.41	
5.71/2.41	We have to consider all (P,R,Pi)-chains
5.71/2.41	----------------------------------------
5.71/2.41	
5.71/2.41	(15) UsableRulesProof (EQUIVALENT)
5.71/2.41	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.71/2.41	----------------------------------------
5.71/2.41	
5.71/2.41	(16)
5.71/2.41	Obligation:
5.71/2.41	Pi DP problem:
5.71/2.41	The TRS P consists of the following rules:
5.71/2.41	
5.71/2.41	   U21_GG(X, Rest, Domain, select_out_aga(X, Domain, NewDomain)) -> DOMAIN_IN_GG(Rest, NewDomain)
5.71/2.41	   DOMAIN_IN_GG(.(X, Rest), Domain) -> U21_GG(X, Rest, Domain, select_in_aga(X, Domain, NewDomain))
5.71/2.41	
5.71/2.41	The TRS R consists of the following rules:
5.71/2.41	
5.71/2.41	   select_in_aga(X, .(X, R), R) -> select_out_aga(X, .(X, R), R)
5.71/2.41	   select_in_aga(X, .(Y, R), .(Y, Rest)) -> U23_aga(X, Y, R, Rest, select_in_aga(X, R, Rest))
5.71/2.41	   U23_aga(X, Y, R, Rest, select_out_aga(X, R, Rest)) -> select_out_aga(X, .(Y, R), .(Y, Rest))
5.71/2.41	
5.71/2.41	The argument filtering Pi contains the following mapping:
5.71/2.41	.(x1, x2)  =  .(x2)
5.71/2.41	
5.71/2.41	select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
5.71/2.41	
5.71/2.41	select_out_aga(x1, x2, x3)  =  select_out_aga(x2, x3)
5.71/2.41	
5.71/2.41	U23_aga(x1, x2, x3, x4, x5)  =  U23_aga(x3, x5)
5.71/2.41	
5.71/2.41	DOMAIN_IN_GG(x1, x2)  =  DOMAIN_IN_GG(x1, x2)
5.71/2.41	
5.71/2.41	U21_GG(x1, x2, x3, x4)  =  U21_GG(x2, x3, x4)
5.71/2.41	
5.71/2.41	
5.71/2.41	We have to consider all (P,R,Pi)-chains
5.71/2.41	----------------------------------------
5.71/2.41	
5.71/2.41	(17) PiDPToQDPProof (SOUND)
5.71/2.41	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.71/2.41	----------------------------------------
5.71/2.41	
5.71/2.41	(18)
5.71/2.41	Obligation:
5.71/2.41	Q DP problem:
5.71/2.41	The TRS P consists of the following rules:
5.71/2.41	
5.71/2.41	   U21_GG(Rest, Domain, select_out_aga(Domain, NewDomain)) -> DOMAIN_IN_GG(Rest, NewDomain)
5.71/2.41	   DOMAIN_IN_GG(.(Rest), Domain) -> U21_GG(Rest, Domain, select_in_aga(Domain))
5.71/2.41	
5.71/2.41	The TRS R consists of the following rules:
5.71/2.41	
5.71/2.41	   select_in_aga(.(R)) -> select_out_aga(.(R), R)
5.71/2.41	   select_in_aga(.(R)) -> U23_aga(R, select_in_aga(R))
5.71/2.41	   U23_aga(R, select_out_aga(R, Rest)) -> select_out_aga(.(R), .(Rest))
5.71/2.41	
5.71/2.41	The set Q consists of the following terms:
5.71/2.41	
5.71/2.41	   select_in_aga(x0)
5.71/2.41	   U23_aga(x0, x1)
5.71/2.41	
5.71/2.41	We have to consider all (P,Q,R)-chains.
5.71/2.41	----------------------------------------
5.71/2.41	
5.71/2.41	(19) QDPSizeChangeProof (EQUIVALENT)
5.71/2.41	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
5.71/2.41	
5.71/2.41	From the DPs we obtained the following set of size-change graphs:
5.71/2.41	*DOMAIN_IN_GG(.(Rest), Domain) -> U21_GG(Rest, Domain, select_in_aga(Domain))
5.71/2.41	The graph contains the following edges 1 > 1, 2 >= 2
5.71/2.41	
5.71/2.41	
5.71/2.41	*U21_GG(Rest, Domain, select_out_aga(Domain, NewDomain)) -> DOMAIN_IN_GG(Rest, NewDomain)
5.71/2.41	The graph contains the following edges 1 >= 1, 3 > 2
5.71/2.41	
5.71/2.41	
5.71/2.41	----------------------------------------
5.71/2.41	
5.71/2.41	(20)
5.71/2.41	YES
6.08/2.45	EOF