(0) Obligation:

Clauses:

p(cons(X, nil)).
p(cons(s(s(X)), cons(Y, Xs))) :- ','(p(cons(X, cons(Y, Xs))), ','(mult(X, Y, Z), p(cons(Z, Xs)))).
p(cons(0, Xs)) :- p(Xs).
sum(X, 0, X).
sum(X, s(Y), s(Z)) :- sum(X, Y, Z).
mult(X1, 0, 0).
mult(X, s(Y), Z) :- ','(mult(X, Y, W), sum(W, X, Z)).

Queries:

p(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

mult23(T46, 0, 0).
mult23(T51, s(T52), X83) :- mult23(T51, T52, X82).
mult23(T51, s(T52), X83) :- ','(mult23(T51, T52, T55), sum34(T55, T51, X83)).
sum34(T64, 0, T64).
sum34(T69, s(T70), s(X110)) :- sum34(T69, T70, X110).
mult53(0, 0).
mult53(s(T101), X159) :- mult53(T101, X158).
mult53(s(T101), T108) :- mult53(T101, T108).
p1(cons(T3, nil)).
p1(cons(s(s(s(s(T24)))), cons(T25, T26))) :- p1(cons(T24, cons(T25, T26))).
p1(cons(s(s(s(s(T24)))), cons(T25, T26))) :- ','(p1(cons(T24, cons(T25, T26))), mult23(T24, T25, X44)).
p1(cons(s(s(s(s(T24)))), cons(T25, T26))) :- ','(p1(cons(T24, cons(T25, T26))), ','(mult23(T24, T25, T39), p1(cons(T39, T26)))).
p1(cons(s(s(s(s(T24)))), cons(T25, T26))) :- ','(p1(cons(T24, cons(T25, T26))), ','(mult23(T24, T25, T39), ','(p1(cons(T39, T26)), mult23(s(s(T24)), T25, X22)))).
p1(cons(s(s(s(s(T24)))), cons(T25, T26))) :- ','(p1(cons(T24, cons(T25, T26))), ','(mult23(T24, T25, T39), ','(p1(cons(T39, T26)), ','(mult23(s(s(T24)), T25, T79), p1(cons(T79, T26)))))).
p1(cons(s(s(0)), cons(T92, T93))) :- p1(cons(T92, T93)).
p1(cons(s(s(0)), cons(T92, T93))) :- ','(p1(cons(T92, T93)), mult53(T92, X22)).
p1(cons(s(s(0)), cons(T92, T93))) :- ','(p1(cons(T92, T93)), ','(mult53(T92, T98), p1(cons(T98, T93)))).
p1(cons(0, cons(T118, nil))).
p1(cons(0, cons(s(s(T131)), cons(T132, T133)))) :- p1(cons(T131, cons(T132, T133))).
p1(cons(0, cons(s(s(T131)), cons(T132, T133)))) :- ','(p1(cons(T131, cons(T132, T133))), mult23(T131, T132, X205)).
p1(cons(0, cons(s(s(T131)), cons(T132, T133)))) :- ','(p1(cons(T131, cons(T132, T133))), ','(mult23(T131, T132, T146), p1(cons(T146, T133)))).
p1(cons(0, cons(0, T153))) :- p1(T153).

Queries:

p1(g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
p1_in: (b) (f)
mult23_in: (b,b,f) (f,f,f) (f,b,f)
sum34_in: (f,b,f) (f,f,f)
mult53_in: (f,f) (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x4)
U22_g(x1, x2)  =  U22_g(x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga
U4_aga(x1, x2, x3, x4)  =  U4_aga(x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U16_g(x1, x2, x3)  =  U16_g(x3)
U17_g(x1, x2, x3)  =  U17_g(x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x4)
U22_g(x1, x2)  =  U22_g(x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga
U4_aga(x1, x2, x3, x4)  =  U4_aga(x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U16_g(x1, x2, x3)  =  U16_g(x3)
U17_g(x1, x2, x3)  =  U17_g(x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x4)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_G(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_G(cons(T24, cons(T25, T26)))
P1_IN_G(cons(s(s(0)), cons(T92, T93))) → U14_G(T92, T93, p1_in_g(cons(T92, T93)))
P1_IN_G(cons(s(s(0)), cons(T92, T93))) → P1_IN_G(cons(T92, T93))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_G(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_G(cons(T131, cons(T132, T133)))
P1_IN_G(cons(0, cons(0, T153))) → U22_G(T153, p1_in_g(T153))
P1_IN_G(cons(0, cons(0, T153))) → P1_IN_G(T153)
U18_G(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_G(T131, T132, T133, mult23_in_gga(T131, T132, X205))
U18_G(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → MULT23_IN_GGA(T131, T132, X205)
MULT23_IN_GGA(T51, s(T52), X83) → U1_GGA(T51, T52, X83, mult23_in_gga(T51, T52, X82))
MULT23_IN_GGA(T51, s(T52), X83) → MULT23_IN_GGA(T51, T52, X82)
MULT23_IN_GGA(T51, s(T52), X83) → U2_GGA(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_GGA(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_GGA(T51, T52, X83, sum34_in_aga(T55, T51, X83))
U2_GGA(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → SUM34_IN_AGA(T55, T51, X83)
SUM34_IN_AGA(T69, s(T70), s(X110)) → U4_AGA(T69, T70, X110, sum34_in_aga(T69, T70, X110))
SUM34_IN_AGA(T69, s(T70), s(X110)) → SUM34_IN_AGA(T69, T70, X110)
U18_G(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_G(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_G(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_G(T131, T132, T133, p1_in_a(cons(T146, T133)))
U20_G(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → P1_IN_A(cons(T146, T133))
P1_IN_A(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_A(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
P1_IN_A(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_A(cons(T24, cons(T25, T26)))
P1_IN_A(cons(s(s(0)), cons(T92, T93))) → U14_A(T92, T93, p1_in_a(cons(T92, T93)))
P1_IN_A(cons(s(s(0)), cons(T92, T93))) → P1_IN_A(cons(T92, T93))
P1_IN_A(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_A(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
P1_IN_A(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_A(cons(T131, cons(T132, T133)))
P1_IN_A(cons(0, cons(0, T153))) → U22_A(T153, p1_in_a(T153))
P1_IN_A(cons(0, cons(0, T153))) → P1_IN_A(T153)
U18_A(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_A(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
U18_A(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → MULT23_IN_AAA(T131, T132, X205)
MULT23_IN_AAA(T51, s(T52), X83) → U1_AAA(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
MULT23_IN_AAA(T51, s(T52), X83) → MULT23_IN_AAA(T51, T52, X82)
MULT23_IN_AAA(T51, s(T52), X83) → U2_AAA(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_AAA(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_AAA(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U2_AAA(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → SUM34_IN_AAA(T55, T51, X83)
SUM34_IN_AAA(T69, s(T70), s(X110)) → U4_AAA(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
SUM34_IN_AAA(T69, s(T70), s(X110)) → SUM34_IN_AAA(T69, T70, X110)
U18_A(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_A(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_A(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_A(T131, T132, T133, p1_in_a(cons(T146, T133)))
U20_A(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → P1_IN_A(cons(T146, T133))
U14_A(T92, T93, p1_out_a(cons(T92, T93))) → U15_A(T92, T93, mult53_in_aa(T92, X22))
U14_A(T92, T93, p1_out_a(cons(T92, T93))) → MULT53_IN_AA(T92, X22)
MULT53_IN_AA(s(T101), X159) → U5_AA(T101, X159, mult53_in_aa(T101, X158))
MULT53_IN_AA(s(T101), X159) → MULT53_IN_AA(T101, X158)
MULT53_IN_AA(s(T101), T108) → U6_AA(T101, T108, mult53_in_aa(T101, T108))
MULT53_IN_AA(s(T101), T108) → MULT53_IN_AA(T101, T108)
U14_A(T92, T93, p1_out_a(cons(T92, T93))) → U16_A(T92, T93, mult53_in_aa(T92, T98))
U16_A(T92, T93, mult53_out_aa(T92, T98)) → U17_A(T92, T93, p1_in_a(cons(T98, T93)))
U16_A(T92, T93, mult53_out_aa(T92, T98)) → P1_IN_A(cons(T98, T93))
U7_A(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_A(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U7_A(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → MULT23_IN_AAA(T24, T25, X44)
U7_A(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_A(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_A(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_A(T24, T25, T26, p1_in_a(cons(T39, T26)))
U9_A(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → P1_IN_A(cons(T39, T26))
U10_A(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_A(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
U10_A(T24, T25, T26, p1_out_a(cons(T39, T26))) → MULT23_IN_AGA(s(s(T24)), T25, X22)
MULT23_IN_AGA(T51, s(T52), X83) → U1_AGA(T51, T52, X83, mult23_in_aga(T51, T52, X82))
MULT23_IN_AGA(T51, s(T52), X83) → MULT23_IN_AGA(T51, T52, X82)
MULT23_IN_AGA(T51, s(T52), X83) → U2_AGA(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_AGA(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_AGA(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U2_AGA(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → SUM34_IN_AAA(T55, T51, X83)
U10_A(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_A(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_A(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_A(T24, T25, T26, p1_in_a(cons(T79, T26)))
U12_A(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → P1_IN_A(cons(T79, T26))
U14_G(T92, T93, p1_out_g(cons(T92, T93))) → U15_G(T92, T93, mult53_in_ga(T92, X22))
U14_G(T92, T93, p1_out_g(cons(T92, T93))) → MULT53_IN_GA(T92, X22)
MULT53_IN_GA(s(T101), X159) → U5_GA(T101, X159, mult53_in_ga(T101, X158))
MULT53_IN_GA(s(T101), X159) → MULT53_IN_GA(T101, X158)
MULT53_IN_GA(s(T101), T108) → U6_GA(T101, T108, mult53_in_ga(T101, T108))
MULT53_IN_GA(s(T101), T108) → MULT53_IN_GA(T101, T108)
U14_G(T92, T93, p1_out_g(cons(T92, T93))) → U16_G(T92, T93, mult53_in_ga(T92, T98))
U16_G(T92, T93, mult53_out_ga(T92, T98)) → U17_G(T92, T93, p1_in_a(cons(T98, T93)))
U16_G(T92, T93, mult53_out_ga(T92, T98)) → P1_IN_A(cons(T98, T93))
U7_G(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_G(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U7_G(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → MULT23_IN_GGA(T24, T25, X44)
U7_G(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_G(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_G(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_G(T24, T25, T26, p1_in_a(cons(T39, T26)))
U9_G(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → P1_IN_A(cons(T39, T26))
U10_G(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_G(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U10_G(T24, T25, T26, p1_out_a(cons(T39, T26))) → MULT23_IN_GGA(s(s(T24)), T25, X22)
U10_G(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_G(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_G(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_G(T24, T25, T26, p1_in_a(cons(T79, T26)))
U12_G(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → P1_IN_A(cons(T79, T26))

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x4)
U22_g(x1, x2)  =  U22_g(x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga
U4_aga(x1, x2, x3, x4)  =  U4_aga(x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U16_g(x1, x2, x3)  =  U16_g(x3)
U17_g(x1, x2, x3)  =  U17_g(x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x4)
P1_IN_G(x1)  =  P1_IN_G(x1)
U7_G(x1, x2, x3, x4)  =  U7_G(x1, x2, x4)
U14_G(x1, x2, x3)  =  U14_G(x1, x3)
U18_G(x1, x2, x3, x4)  =  U18_G(x1, x2, x4)
U22_G(x1, x2)  =  U22_G(x2)
U19_G(x1, x2, x3, x4)  =  U19_G(x4)
MULT23_IN_GGA(x1, x2, x3)  =  MULT23_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
SUM34_IN_AGA(x1, x2, x3)  =  SUM34_IN_AGA(x2)
U4_AGA(x1, x2, x3, x4)  =  U4_AGA(x4)
U20_G(x1, x2, x3, x4)  =  U20_G(x4)
U21_G(x1, x2, x3, x4)  =  U21_G(x4)
P1_IN_A(x1)  =  P1_IN_A
U7_A(x1, x2, x3, x4)  =  U7_A(x4)
U14_A(x1, x2, x3)  =  U14_A(x3)
U18_A(x1, x2, x3, x4)  =  U18_A(x4)
U22_A(x1, x2)  =  U22_A(x2)
U19_A(x1, x2, x3, x4)  =  U19_A(x4)
MULT23_IN_AAA(x1, x2, x3)  =  MULT23_IN_AAA
U1_AAA(x1, x2, x3, x4)  =  U1_AAA(x4)
U2_AAA(x1, x2, x3, x4)  =  U2_AAA(x4)
U3_AAA(x1, x2, x3, x4)  =  U3_AAA(x2, x4)
SUM34_IN_AAA(x1, x2, x3)  =  SUM34_IN_AAA
U4_AAA(x1, x2, x3, x4)  =  U4_AAA(x4)
U20_A(x1, x2, x3, x4)  =  U20_A(x4)
U21_A(x1, x2, x3, x4)  =  U21_A(x4)
U15_A(x1, x2, x3)  =  U15_A(x3)
MULT53_IN_AA(x1, x2)  =  MULT53_IN_AA
U5_AA(x1, x2, x3)  =  U5_AA(x3)
U6_AA(x1, x2, x3)  =  U6_AA(x3)
U16_A(x1, x2, x3)  =  U16_A(x3)
U17_A(x1, x2, x3)  =  U17_A(x3)
U8_A(x1, x2, x3, x4)  =  U8_A(x4)
U9_A(x1, x2, x3, x4)  =  U9_A(x4)
U10_A(x1, x2, x3, x4)  =  U10_A(x2, x4)
U11_A(x1, x2, x3, x4)  =  U11_A(x4)
MULT23_IN_AGA(x1, x2, x3)  =  MULT23_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4)  =  U1_AGA(x4)
U2_AGA(x1, x2, x3, x4)  =  U2_AGA(x4)
U3_AGA(x1, x2, x3, x4)  =  U3_AGA(x4)
U12_A(x1, x2, x3, x4)  =  U12_A(x4)
U13_A(x1, x2, x3, x4)  =  U13_A(x4)
U15_G(x1, x2, x3)  =  U15_G(x3)
MULT53_IN_GA(x1, x2)  =  MULT53_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U6_GA(x1, x2, x3)  =  U6_GA(x3)
U16_G(x1, x2, x3)  =  U16_G(x3)
U17_G(x1, x2, x3)  =  U17_G(x3)
U8_G(x1, x2, x3, x4)  =  U8_G(x4)
U9_G(x1, x2, x3, x4)  =  U9_G(x1, x2, x4)
U10_G(x1, x2, x3, x4)  =  U10_G(x1, x2, x4)
U11_G(x1, x2, x3, x4)  =  U11_G(x4)
U12_G(x1, x2, x3, x4)  =  U12_G(x4)
U13_G(x1, x2, x3, x4)  =  U13_G(x4)

We have to consider all (P,R,Pi)-chains

(6) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_G(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_G(cons(T24, cons(T25, T26)))
P1_IN_G(cons(s(s(0)), cons(T92, T93))) → U14_G(T92, T93, p1_in_g(cons(T92, T93)))
P1_IN_G(cons(s(s(0)), cons(T92, T93))) → P1_IN_G(cons(T92, T93))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_G(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_G(cons(T131, cons(T132, T133)))
P1_IN_G(cons(0, cons(0, T153))) → U22_G(T153, p1_in_g(T153))
P1_IN_G(cons(0, cons(0, T153))) → P1_IN_G(T153)
U18_G(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_G(T131, T132, T133, mult23_in_gga(T131, T132, X205))
U18_G(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → MULT23_IN_GGA(T131, T132, X205)
MULT23_IN_GGA(T51, s(T52), X83) → U1_GGA(T51, T52, X83, mult23_in_gga(T51, T52, X82))
MULT23_IN_GGA(T51, s(T52), X83) → MULT23_IN_GGA(T51, T52, X82)
MULT23_IN_GGA(T51, s(T52), X83) → U2_GGA(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_GGA(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_GGA(T51, T52, X83, sum34_in_aga(T55, T51, X83))
U2_GGA(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → SUM34_IN_AGA(T55, T51, X83)
SUM34_IN_AGA(T69, s(T70), s(X110)) → U4_AGA(T69, T70, X110, sum34_in_aga(T69, T70, X110))
SUM34_IN_AGA(T69, s(T70), s(X110)) → SUM34_IN_AGA(T69, T70, X110)
U18_G(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_G(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_G(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_G(T131, T132, T133, p1_in_a(cons(T146, T133)))
U20_G(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → P1_IN_A(cons(T146, T133))
P1_IN_A(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_A(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
P1_IN_A(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_A(cons(T24, cons(T25, T26)))
P1_IN_A(cons(s(s(0)), cons(T92, T93))) → U14_A(T92, T93, p1_in_a(cons(T92, T93)))
P1_IN_A(cons(s(s(0)), cons(T92, T93))) → P1_IN_A(cons(T92, T93))
P1_IN_A(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_A(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
P1_IN_A(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_A(cons(T131, cons(T132, T133)))
P1_IN_A(cons(0, cons(0, T153))) → U22_A(T153, p1_in_a(T153))
P1_IN_A(cons(0, cons(0, T153))) → P1_IN_A(T153)
U18_A(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_A(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
U18_A(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → MULT23_IN_AAA(T131, T132, X205)
MULT23_IN_AAA(T51, s(T52), X83) → U1_AAA(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
MULT23_IN_AAA(T51, s(T52), X83) → MULT23_IN_AAA(T51, T52, X82)
MULT23_IN_AAA(T51, s(T52), X83) → U2_AAA(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_AAA(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_AAA(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U2_AAA(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → SUM34_IN_AAA(T55, T51, X83)
SUM34_IN_AAA(T69, s(T70), s(X110)) → U4_AAA(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
SUM34_IN_AAA(T69, s(T70), s(X110)) → SUM34_IN_AAA(T69, T70, X110)
U18_A(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_A(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_A(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_A(T131, T132, T133, p1_in_a(cons(T146, T133)))
U20_A(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → P1_IN_A(cons(T146, T133))
U14_A(T92, T93, p1_out_a(cons(T92, T93))) → U15_A(T92, T93, mult53_in_aa(T92, X22))
U14_A(T92, T93, p1_out_a(cons(T92, T93))) → MULT53_IN_AA(T92, X22)
MULT53_IN_AA(s(T101), X159) → U5_AA(T101, X159, mult53_in_aa(T101, X158))
MULT53_IN_AA(s(T101), X159) → MULT53_IN_AA(T101, X158)
MULT53_IN_AA(s(T101), T108) → U6_AA(T101, T108, mult53_in_aa(T101, T108))
MULT53_IN_AA(s(T101), T108) → MULT53_IN_AA(T101, T108)
U14_A(T92, T93, p1_out_a(cons(T92, T93))) → U16_A(T92, T93, mult53_in_aa(T92, T98))
U16_A(T92, T93, mult53_out_aa(T92, T98)) → U17_A(T92, T93, p1_in_a(cons(T98, T93)))
U16_A(T92, T93, mult53_out_aa(T92, T98)) → P1_IN_A(cons(T98, T93))
U7_A(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_A(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U7_A(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → MULT23_IN_AAA(T24, T25, X44)
U7_A(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_A(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_A(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_A(T24, T25, T26, p1_in_a(cons(T39, T26)))
U9_A(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → P1_IN_A(cons(T39, T26))
U10_A(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_A(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
U10_A(T24, T25, T26, p1_out_a(cons(T39, T26))) → MULT23_IN_AGA(s(s(T24)), T25, X22)
MULT23_IN_AGA(T51, s(T52), X83) → U1_AGA(T51, T52, X83, mult23_in_aga(T51, T52, X82))
MULT23_IN_AGA(T51, s(T52), X83) → MULT23_IN_AGA(T51, T52, X82)
MULT23_IN_AGA(T51, s(T52), X83) → U2_AGA(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_AGA(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_AGA(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U2_AGA(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → SUM34_IN_AAA(T55, T51, X83)
U10_A(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_A(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_A(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_A(T24, T25, T26, p1_in_a(cons(T79, T26)))
U12_A(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → P1_IN_A(cons(T79, T26))
U14_G(T92, T93, p1_out_g(cons(T92, T93))) → U15_G(T92, T93, mult53_in_ga(T92, X22))
U14_G(T92, T93, p1_out_g(cons(T92, T93))) → MULT53_IN_GA(T92, X22)
MULT53_IN_GA(s(T101), X159) → U5_GA(T101, X159, mult53_in_ga(T101, X158))
MULT53_IN_GA(s(T101), X159) → MULT53_IN_GA(T101, X158)
MULT53_IN_GA(s(T101), T108) → U6_GA(T101, T108, mult53_in_ga(T101, T108))
MULT53_IN_GA(s(T101), T108) → MULT53_IN_GA(T101, T108)
U14_G(T92, T93, p1_out_g(cons(T92, T93))) → U16_G(T92, T93, mult53_in_ga(T92, T98))
U16_G(T92, T93, mult53_out_ga(T92, T98)) → U17_G(T92, T93, p1_in_a(cons(T98, T93)))
U16_G(T92, T93, mult53_out_ga(T92, T98)) → P1_IN_A(cons(T98, T93))
U7_G(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_G(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U7_G(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → MULT23_IN_GGA(T24, T25, X44)
U7_G(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_G(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_G(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_G(T24, T25, T26, p1_in_a(cons(T39, T26)))
U9_G(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → P1_IN_A(cons(T39, T26))
U10_G(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_G(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U10_G(T24, T25, T26, p1_out_a(cons(T39, T26))) → MULT23_IN_GGA(s(s(T24)), T25, X22)
U10_G(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_G(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_G(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_G(T24, T25, T26, p1_in_a(cons(T79, T26)))
U12_G(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → P1_IN_A(cons(T79, T26))

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x4)
U22_g(x1, x2)  =  U22_g(x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga
U4_aga(x1, x2, x3, x4)  =  U4_aga(x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U16_g(x1, x2, x3)  =  U16_g(x3)
U17_g(x1, x2, x3)  =  U17_g(x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x4)
P1_IN_G(x1)  =  P1_IN_G(x1)
U7_G(x1, x2, x3, x4)  =  U7_G(x1, x2, x4)
U14_G(x1, x2, x3)  =  U14_G(x1, x3)
U18_G(x1, x2, x3, x4)  =  U18_G(x1, x2, x4)
U22_G(x1, x2)  =  U22_G(x2)
U19_G(x1, x2, x3, x4)  =  U19_G(x4)
MULT23_IN_GGA(x1, x2, x3)  =  MULT23_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
SUM34_IN_AGA(x1, x2, x3)  =  SUM34_IN_AGA(x2)
U4_AGA(x1, x2, x3, x4)  =  U4_AGA(x4)
U20_G(x1, x2, x3, x4)  =  U20_G(x4)
U21_G(x1, x2, x3, x4)  =  U21_G(x4)
P1_IN_A(x1)  =  P1_IN_A
U7_A(x1, x2, x3, x4)  =  U7_A(x4)
U14_A(x1, x2, x3)  =  U14_A(x3)
U18_A(x1, x2, x3, x4)  =  U18_A(x4)
U22_A(x1, x2)  =  U22_A(x2)
U19_A(x1, x2, x3, x4)  =  U19_A(x4)
MULT23_IN_AAA(x1, x2, x3)  =  MULT23_IN_AAA
U1_AAA(x1, x2, x3, x4)  =  U1_AAA(x4)
U2_AAA(x1, x2, x3, x4)  =  U2_AAA(x4)
U3_AAA(x1, x2, x3, x4)  =  U3_AAA(x2, x4)
SUM34_IN_AAA(x1, x2, x3)  =  SUM34_IN_AAA
U4_AAA(x1, x2, x3, x4)  =  U4_AAA(x4)
U20_A(x1, x2, x3, x4)  =  U20_A(x4)
U21_A(x1, x2, x3, x4)  =  U21_A(x4)
U15_A(x1, x2, x3)  =  U15_A(x3)
MULT53_IN_AA(x1, x2)  =  MULT53_IN_AA
U5_AA(x1, x2, x3)  =  U5_AA(x3)
U6_AA(x1, x2, x3)  =  U6_AA(x3)
U16_A(x1, x2, x3)  =  U16_A(x3)
U17_A(x1, x2, x3)  =  U17_A(x3)
U8_A(x1, x2, x3, x4)  =  U8_A(x4)
U9_A(x1, x2, x3, x4)  =  U9_A(x4)
U10_A(x1, x2, x3, x4)  =  U10_A(x2, x4)
U11_A(x1, x2, x3, x4)  =  U11_A(x4)
MULT23_IN_AGA(x1, x2, x3)  =  MULT23_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4)  =  U1_AGA(x4)
U2_AGA(x1, x2, x3, x4)  =  U2_AGA(x4)
U3_AGA(x1, x2, x3, x4)  =  U3_AGA(x4)
U12_A(x1, x2, x3, x4)  =  U12_A(x4)
U13_A(x1, x2, x3, x4)  =  U13_A(x4)
U15_G(x1, x2, x3)  =  U15_G(x3)
MULT53_IN_GA(x1, x2)  =  MULT53_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U6_GA(x1, x2, x3)  =  U6_GA(x3)
U16_G(x1, x2, x3)  =  U16_G(x3)
U17_G(x1, x2, x3)  =  U17_G(x3)
U8_G(x1, x2, x3, x4)  =  U8_G(x4)
U9_G(x1, x2, x3, x4)  =  U9_G(x1, x2, x4)
U10_G(x1, x2, x3, x4)  =  U10_G(x1, x2, x4)
U11_G(x1, x2, x3, x4)  =  U11_G(x4)
U12_G(x1, x2, x3, x4)  =  U12_G(x4)
U13_G(x1, x2, x3, x4)  =  U13_G(x4)

We have to consider all (P,R,Pi)-chains

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 9 SCCs with 54 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT53_IN_GA(s(T101), T108) → MULT53_IN_GA(T101, T108)
MULT53_IN_GA(s(T101), X159) → MULT53_IN_GA(T101, X158)

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x4)
U22_g(x1, x2)  =  U22_g(x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga
U4_aga(x1, x2, x3, x4)  =  U4_aga(x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U16_g(x1, x2, x3)  =  U16_g(x3)
U17_g(x1, x2, x3)  =  U17_g(x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x4)
MULT53_IN_GA(x1, x2)  =  MULT53_IN_GA(x1)

We have to consider all (P,R,Pi)-chains

(10) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(11) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT53_IN_GA(s(T101), T108) → MULT53_IN_GA(T101, T108)
MULT53_IN_GA(s(T101), X159) → MULT53_IN_GA(T101, X158)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
MULT53_IN_GA(x1, x2)  =  MULT53_IN_GA(x1)

We have to consider all (P,R,Pi)-chains

(12) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(13) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MULT53_IN_GA(s(T101)) → MULT53_IN_GA(T101)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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.

From the DPs we obtained the following set of size-change graphs:

  • MULT53_IN_GA(s(T101)) → MULT53_IN_GA(T101)
    The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT53_IN_AA(s(T101), T108) → MULT53_IN_AA(T101, T108)
MULT53_IN_AA(s(T101), X159) → MULT53_IN_AA(T101, X158)

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x4)
U22_g(x1, x2)  =  U22_g(x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga
U4_aga(x1, x2, x3, x4)  =  U4_aga(x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U16_g(x1, x2, x3)  =  U16_g(x3)
U17_g(x1, x2, x3)  =  U17_g(x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x4)
MULT53_IN_AA(x1, x2)  =  MULT53_IN_AA

We have to consider all (P,R,Pi)-chains

(17) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(18) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT53_IN_AA(s(T101), T108) → MULT53_IN_AA(T101, T108)
MULT53_IN_AA(s(T101), X159) → MULT53_IN_AA(T101, X158)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
MULT53_IN_AA(x1, x2)  =  MULT53_IN_AA

We have to consider all (P,R,Pi)-chains

(19) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MULT53_IN_AAMULT53_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(21) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = MULT53_IN_AA evaluates to t =MULT53_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Matcher: [ ]
  • Semiunifier: [ ]




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from MULT53_IN_AA to MULT53_IN_AA.



(22) NO

(23) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SUM34_IN_AAA(T69, s(T70), s(X110)) → SUM34_IN_AAA(T69, T70, X110)

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x4)
U22_g(x1, x2)  =  U22_g(x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga
U4_aga(x1, x2, x3, x4)  =  U4_aga(x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U16_g(x1, x2, x3)  =  U16_g(x3)
U17_g(x1, x2, x3)  =  U17_g(x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x4)
SUM34_IN_AAA(x1, x2, x3)  =  SUM34_IN_AAA

We have to consider all (P,R,Pi)-chains

(24) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(25) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SUM34_IN_AAA(T69, s(T70), s(X110)) → SUM34_IN_AAA(T69, T70, X110)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
SUM34_IN_AAA(x1, x2, x3)  =  SUM34_IN_AAA

We have to consider all (P,R,Pi)-chains

(26) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(27) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SUM34_IN_AAASUM34_IN_AAA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(28) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = SUM34_IN_AAA evaluates to t =SUM34_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from SUM34_IN_AAA to SUM34_IN_AAA.



(29) NO

(30) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT23_IN_AGA(T51, s(T52), X83) → MULT23_IN_AGA(T51, T52, X82)

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x4)
U22_g(x1, x2)  =  U22_g(x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga
U4_aga(x1, x2, x3, x4)  =  U4_aga(x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U16_g(x1, x2, x3)  =  U16_g(x3)
U17_g(x1, x2, x3)  =  U17_g(x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x4)
MULT23_IN_AGA(x1, x2, x3)  =  MULT23_IN_AGA(x2)

We have to consider all (P,R,Pi)-chains

(31) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(32) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT23_IN_AGA(T51, s(T52), X83) → MULT23_IN_AGA(T51, T52, X82)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
MULT23_IN_AGA(x1, x2, x3)  =  MULT23_IN_AGA(x2)

We have to consider all (P,R,Pi)-chains

(33) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(34) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MULT23_IN_AGA(s(T52)) → MULT23_IN_AGA(T52)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(35) QDPSizeChangeProof (EQUIVALENT transformation)

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.

From the DPs we obtained the following set of size-change graphs:

  • MULT23_IN_AGA(s(T52)) → MULT23_IN_AGA(T52)
    The graph contains the following edges 1 > 1

(36) YES

(37) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT23_IN_AAA(T51, s(T52), X83) → MULT23_IN_AAA(T51, T52, X82)

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x4)
U22_g(x1, x2)  =  U22_g(x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga
U4_aga(x1, x2, x3, x4)  =  U4_aga(x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U16_g(x1, x2, x3)  =  U16_g(x3)
U17_g(x1, x2, x3)  =  U17_g(x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x4)
MULT23_IN_AAA(x1, x2, x3)  =  MULT23_IN_AAA

We have to consider all (P,R,Pi)-chains

(38) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(39) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT23_IN_AAA(T51, s(T52), X83) → MULT23_IN_AAA(T51, T52, X82)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
MULT23_IN_AAA(x1, x2, x3)  =  MULT23_IN_AAA

We have to consider all (P,R,Pi)-chains

(40) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(41) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MULT23_IN_AAAMULT23_IN_AAA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(42) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = MULT23_IN_AAA evaluates to t =MULT23_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Matcher: [ ]
  • Semiunifier: [ ]




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from MULT23_IN_AAA to MULT23_IN_AAA.



(43) NO

(44) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

U7_A(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_A(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_A(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_A(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_A(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_A(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_A(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → P1_IN_A(cons(T79, T26))
P1_IN_A(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_A(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
P1_IN_A(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_A(cons(T24, cons(T25, T26)))
P1_IN_A(cons(s(s(0)), cons(T92, T93))) → U14_A(T92, T93, p1_in_a(cons(T92, T93)))
U14_A(T92, T93, p1_out_a(cons(T92, T93))) → U16_A(T92, T93, mult53_in_aa(T92, T98))
U16_A(T92, T93, mult53_out_aa(T92, T98)) → P1_IN_A(cons(T98, T93))
P1_IN_A(cons(s(s(0)), cons(T92, T93))) → P1_IN_A(cons(T92, T93))
P1_IN_A(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_A(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
U18_A(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_A(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_A(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → P1_IN_A(cons(T146, T133))
P1_IN_A(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_A(cons(T131, cons(T132, T133)))
P1_IN_A(cons(0, cons(0, T153))) → P1_IN_A(T153)
U9_A(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → P1_IN_A(cons(T39, T26))

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x4)
U22_g(x1, x2)  =  U22_g(x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga
U4_aga(x1, x2, x3, x4)  =  U4_aga(x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U16_g(x1, x2, x3)  =  U16_g(x3)
U17_g(x1, x2, x3)  =  U17_g(x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x4)
P1_IN_A(x1)  =  P1_IN_A
U7_A(x1, x2, x3, x4)  =  U7_A(x4)
U14_A(x1, x2, x3)  =  U14_A(x3)
U18_A(x1, x2, x3, x4)  =  U18_A(x4)
U20_A(x1, x2, x3, x4)  =  U20_A(x4)
U16_A(x1, x2, x3)  =  U16_A(x3)
U9_A(x1, x2, x3, x4)  =  U9_A(x4)
U10_A(x1, x2, x3, x4)  =  U10_A(x2, x4)
U12_A(x1, x2, x3, x4)  =  U12_A(x4)

We have to consider all (P,R,Pi)-chains

(45) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(46) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

U7_A(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_A(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_A(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_A(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_A(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_A(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_A(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → P1_IN_A(cons(T79, T26))
P1_IN_A(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_A(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
P1_IN_A(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_A(cons(T24, cons(T25, T26)))
P1_IN_A(cons(s(s(0)), cons(T92, T93))) → U14_A(T92, T93, p1_in_a(cons(T92, T93)))
U14_A(T92, T93, p1_out_a(cons(T92, T93))) → U16_A(T92, T93, mult53_in_aa(T92, T98))
U16_A(T92, T93, mult53_out_aa(T92, T98)) → P1_IN_A(cons(T98, T93))
P1_IN_A(cons(s(s(0)), cons(T92, T93))) → P1_IN_A(cons(T92, T93))
P1_IN_A(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_A(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
U18_A(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_A(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_A(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → P1_IN_A(cons(T146, T133))
P1_IN_A(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_A(cons(T131, cons(T132, T133)))
P1_IN_A(cons(0, cons(0, T153))) → P1_IN_A(T153)
U9_A(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → P1_IN_A(cons(T39, T26))

The TRS R consists of the following rules:

mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
s(x1)  =  s(x1)
0  =  0
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
P1_IN_A(x1)  =  P1_IN_A
U7_A(x1, x2, x3, x4)  =  U7_A(x4)
U14_A(x1, x2, x3)  =  U14_A(x3)
U18_A(x1, x2, x3, x4)  =  U18_A(x4)
U20_A(x1, x2, x3, x4)  =  U20_A(x4)
U16_A(x1, x2, x3)  =  U16_A(x3)
U9_A(x1, x2, x3, x4)  =  U9_A(x4)
U10_A(x1, x2, x3, x4)  =  U10_A(x2, x4)
U12_A(x1, x2, x3, x4)  =  U12_A(x4)

We have to consider all (P,R,Pi)-chains

(47) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(48) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U7_A(p1_out_a) → U9_A(mult23_in_aaa)
U9_A(mult23_out_aaa(T25)) → U10_A(T25, p1_in_a)
U10_A(T25, p1_out_a) → U12_A(mult23_in_aga(T25))
U12_A(mult23_out_aga) → P1_IN_A
P1_IN_AU7_A(p1_in_a)
P1_IN_AP1_IN_A
P1_IN_AU14_A(p1_in_a)
U14_A(p1_out_a) → U16_A(mult53_in_aa)
U16_A(mult53_out_aa(T92)) → P1_IN_A
P1_IN_AU18_A(p1_in_a)
U18_A(p1_out_a) → U20_A(mult23_in_aaa)
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga
mult23_in_aga(s(T52)) → U1_aga(mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(mult23_out_aga) → mult23_out_aga
U2_aga(mult23_out_aga) → U3_aga(sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(sum34_out_aaa(T51)) → mult23_out_aga
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga) → p1_out_a
U12_a(mult23_out_aga) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0)
U2_aga(x0)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(49) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U7_A(p1_out_a) → U9_A(mult23_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))

(50) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U9_A(mult23_out_aaa(T25)) → U10_A(T25, p1_in_a)
U10_A(T25, p1_out_a) → U12_A(mult23_in_aga(T25))
U12_A(mult23_out_aga) → P1_IN_A
P1_IN_AU7_A(p1_in_a)
P1_IN_AP1_IN_A
P1_IN_AU14_A(p1_in_a)
U14_A(p1_out_a) → U16_A(mult53_in_aa)
U16_A(mult53_out_aa(T92)) → P1_IN_A
P1_IN_AU18_A(p1_in_a)
U18_A(p1_out_a) → U20_A(mult23_in_aaa)
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga
mult23_in_aga(s(T52)) → U1_aga(mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(mult23_out_aga) → mult23_out_aga
U2_aga(mult23_out_aga) → U3_aga(sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(sum34_out_aaa(T51)) → mult23_out_aga
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga) → p1_out_a
U12_a(mult23_out_aga) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0)
U2_aga(x0)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(51) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U9_A(mult23_out_aaa(T25)) → U10_A(T25, p1_in_a) at position [1] we obtained the following new rules [LPAR04]:

U9_A(mult23_out_aaa(y0)) → U10_A(y0, p1_out_a)
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))

(52) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U10_A(T25, p1_out_a) → U12_A(mult23_in_aga(T25))
U12_A(mult23_out_aga) → P1_IN_A
P1_IN_AU7_A(p1_in_a)
P1_IN_AP1_IN_A
P1_IN_AU14_A(p1_in_a)
U14_A(p1_out_a) → U16_A(mult53_in_aa)
U16_A(mult53_out_aa(T92)) → P1_IN_A
P1_IN_AU18_A(p1_in_a)
U18_A(p1_out_a) → U20_A(mult23_in_aaa)
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, p1_out_a)
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga
mult23_in_aga(s(T52)) → U1_aga(mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(mult23_out_aga) → mult23_out_aga
U2_aga(mult23_out_aga) → U3_aga(sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(sum34_out_aaa(T51)) → mult23_out_aga
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga) → p1_out_a
U12_a(mult23_out_aga) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0)
U2_aga(x0)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(53) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U10_A(T25, p1_out_a) → U12_A(mult23_in_aga(T25)) at position [0] we obtained the following new rules [LPAR04]:

U10_A(0, p1_out_a) → U12_A(mult23_out_aga)
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(mult23_in_aga(x0)))

(54) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga) → P1_IN_A
P1_IN_AU7_A(p1_in_a)
P1_IN_AP1_IN_A
P1_IN_AU14_A(p1_in_a)
U14_A(p1_out_a) → U16_A(mult53_in_aa)
U16_A(mult53_out_aa(T92)) → P1_IN_A
P1_IN_AU18_A(p1_in_a)
U18_A(p1_out_a) → U20_A(mult23_in_aaa)
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, p1_out_a)
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga)
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(mult23_in_aga(x0)))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga
mult23_in_aga(s(T52)) → U1_aga(mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(mult23_out_aga) → mult23_out_aga
U2_aga(mult23_out_aga) → U3_aga(sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(sum34_out_aaa(T51)) → mult23_out_aga
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga) → p1_out_a
U12_a(mult23_out_aga) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0)
U2_aga(x0)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(55) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P1_IN_AU7_A(p1_in_a) at position [0] we obtained the following new rules [LPAR04]:

P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))

(56) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga) → P1_IN_A
P1_IN_AP1_IN_A
P1_IN_AU14_A(p1_in_a)
U14_A(p1_out_a) → U16_A(mult53_in_aa)
U16_A(mult53_out_aa(T92)) → P1_IN_A
P1_IN_AU18_A(p1_in_a)
U18_A(p1_out_a) → U20_A(mult23_in_aaa)
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, p1_out_a)
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga)
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(mult23_in_aga(x0)))
P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga
mult23_in_aga(s(T52)) → U1_aga(mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(mult23_out_aga) → mult23_out_aga
U2_aga(mult23_out_aga) → U3_aga(sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(sum34_out_aaa(T51)) → mult23_out_aga
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga) → p1_out_a
U12_a(mult23_out_aga) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0)
U2_aga(x0)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(57) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P1_IN_AU14_A(p1_in_a) at position [0] we obtained the following new rules [LPAR04]:

P1_IN_AU14_A(p1_out_a)
P1_IN_AU14_A(U7_a(p1_in_a))
P1_IN_AU14_A(U14_a(p1_in_a))
P1_IN_AU14_A(U18_a(p1_in_a))
P1_IN_AU14_A(U22_a(p1_in_a))

(58) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga) → P1_IN_A
P1_IN_AP1_IN_A
U14_A(p1_out_a) → U16_A(mult53_in_aa)
U16_A(mult53_out_aa(T92)) → P1_IN_A
P1_IN_AU18_A(p1_in_a)
U18_A(p1_out_a) → U20_A(mult23_in_aaa)
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, p1_out_a)
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga)
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(mult23_in_aga(x0)))
P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))
P1_IN_AU14_A(p1_out_a)
P1_IN_AU14_A(U7_a(p1_in_a))
P1_IN_AU14_A(U14_a(p1_in_a))
P1_IN_AU14_A(U18_a(p1_in_a))
P1_IN_AU14_A(U22_a(p1_in_a))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga
mult23_in_aga(s(T52)) → U1_aga(mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(mult23_out_aga) → mult23_out_aga
U2_aga(mult23_out_aga) → U3_aga(sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(sum34_out_aaa(T51)) → mult23_out_aga
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga) → p1_out_a
U12_a(mult23_out_aga) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0)
U2_aga(x0)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(59) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U14_A(p1_out_a) → U16_A(mult53_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U14_A(p1_out_a) → U16_A(mult53_out_aa(0))
U14_A(p1_out_a) → U16_A(U5_aa(mult53_in_aa))
U14_A(p1_out_a) → U16_A(U6_aa(mult53_in_aa))

(60) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga) → P1_IN_A
P1_IN_AP1_IN_A
U16_A(mult53_out_aa(T92)) → P1_IN_A
P1_IN_AU18_A(p1_in_a)
U18_A(p1_out_a) → U20_A(mult23_in_aaa)
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, p1_out_a)
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga)
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(mult23_in_aga(x0)))
P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))
P1_IN_AU14_A(p1_out_a)
P1_IN_AU14_A(U7_a(p1_in_a))
P1_IN_AU14_A(U14_a(p1_in_a))
P1_IN_AU14_A(U18_a(p1_in_a))
P1_IN_AU14_A(U22_a(p1_in_a))
U14_A(p1_out_a) → U16_A(mult53_out_aa(0))
U14_A(p1_out_a) → U16_A(U5_aa(mult53_in_aa))
U14_A(p1_out_a) → U16_A(U6_aa(mult53_in_aa))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga
mult23_in_aga(s(T52)) → U1_aga(mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(mult23_out_aga) → mult23_out_aga
U2_aga(mult23_out_aga) → U3_aga(sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(sum34_out_aaa(T51)) → mult23_out_aga
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga) → p1_out_a
U12_a(mult23_out_aga) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0)
U2_aga(x0)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(61) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P1_IN_AU18_A(p1_in_a) at position [0] we obtained the following new rules [LPAR04]:

P1_IN_AU18_A(p1_out_a)
P1_IN_AU18_A(U7_a(p1_in_a))
P1_IN_AU18_A(U14_a(p1_in_a))
P1_IN_AU18_A(U18_a(p1_in_a))
P1_IN_AU18_A(U22_a(p1_in_a))

(62) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga) → P1_IN_A
P1_IN_AP1_IN_A
U16_A(mult53_out_aa(T92)) → P1_IN_A
U18_A(p1_out_a) → U20_A(mult23_in_aaa)
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, p1_out_a)
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga)
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(mult23_in_aga(x0)))
P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))
P1_IN_AU14_A(p1_out_a)
P1_IN_AU14_A(U7_a(p1_in_a))
P1_IN_AU14_A(U14_a(p1_in_a))
P1_IN_AU14_A(U18_a(p1_in_a))
P1_IN_AU14_A(U22_a(p1_in_a))
U14_A(p1_out_a) → U16_A(mult53_out_aa(0))
U14_A(p1_out_a) → U16_A(U5_aa(mult53_in_aa))
U14_A(p1_out_a) → U16_A(U6_aa(mult53_in_aa))
P1_IN_AU18_A(p1_out_a)
P1_IN_AU18_A(U7_a(p1_in_a))
P1_IN_AU18_A(U14_a(p1_in_a))
P1_IN_AU18_A(U18_a(p1_in_a))
P1_IN_AU18_A(U22_a(p1_in_a))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga
mult23_in_aga(s(T52)) → U1_aga(mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(mult23_out_aga) → mult23_out_aga
U2_aga(mult23_out_aga) → U3_aga(sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(sum34_out_aaa(T51)) → mult23_out_aga
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga) → p1_out_a
U12_a(mult23_out_aga) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0)
U2_aga(x0)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(63) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U18_A(p1_out_a) → U20_A(mult23_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

U18_A(p1_out_a) → U20_A(mult23_out_aaa(0))
U18_A(p1_out_a) → U20_A(U1_aaa(mult23_in_aaa))
U18_A(p1_out_a) → U20_A(U2_aaa(mult23_in_aaa))

(64) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga) → P1_IN_A
P1_IN_AP1_IN_A
U16_A(mult53_out_aa(T92)) → P1_IN_A
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, p1_out_a)
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga)
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(mult23_in_aga(x0)))
P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))
P1_IN_AU14_A(p1_out_a)
P1_IN_AU14_A(U7_a(p1_in_a))
P1_IN_AU14_A(U14_a(p1_in_a))
P1_IN_AU14_A(U18_a(p1_in_a))
P1_IN_AU14_A(U22_a(p1_in_a))
U14_A(p1_out_a) → U16_A(mult53_out_aa(0))
U14_A(p1_out_a) → U16_A(U5_aa(mult53_in_aa))
U14_A(p1_out_a) → U16_A(U6_aa(mult53_in_aa))
P1_IN_AU18_A(p1_out_a)
P1_IN_AU18_A(U7_a(p1_in_a))
P1_IN_AU18_A(U14_a(p1_in_a))
P1_IN_AU18_A(U18_a(p1_in_a))
P1_IN_AU18_A(U22_a(p1_in_a))
U18_A(p1_out_a) → U20_A(mult23_out_aaa(0))
U18_A(p1_out_a) → U20_A(U1_aaa(mult23_in_aaa))
U18_A(p1_out_a) → U20_A(U2_aaa(mult23_in_aaa))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga
mult23_in_aga(s(T52)) → U1_aga(mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(mult23_out_aga) → mult23_out_aga
U2_aga(mult23_out_aga) → U3_aga(sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(sum34_out_aaa(T51)) → mult23_out_aga
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga) → p1_out_a
U12_a(mult23_out_aga) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0)
U2_aga(x0)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(65) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U9_A(mult23_out_aaa(y0)) → U10_A(y0, p1_out_a) we obtained the following new rules [LPAR04]:

U9_A(mult23_out_aaa(0)) → U10_A(0, p1_out_a)
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), p1_out_a)

(66) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga) → P1_IN_A
P1_IN_AP1_IN_A
U16_A(mult53_out_aa(T92)) → P1_IN_A
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga)
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(mult23_in_aga(x0)))
P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))
P1_IN_AU14_A(p1_out_a)
P1_IN_AU14_A(U7_a(p1_in_a))
P1_IN_AU14_A(U14_a(p1_in_a))
P1_IN_AU14_A(U18_a(p1_in_a))
P1_IN_AU14_A(U22_a(p1_in_a))
U14_A(p1_out_a) → U16_A(mult53_out_aa(0))
U14_A(p1_out_a) → U16_A(U5_aa(mult53_in_aa))
U14_A(p1_out_a) → U16_A(U6_aa(mult53_in_aa))
P1_IN_AU18_A(p1_out_a)
P1_IN_AU18_A(U7_a(p1_in_a))
P1_IN_AU18_A(U14_a(p1_in_a))
P1_IN_AU18_A(U18_a(p1_in_a))
P1_IN_AU18_A(U22_a(p1_in_a))
U18_A(p1_out_a) → U20_A(mult23_out_aaa(0))
U18_A(p1_out_a) → U20_A(U1_aaa(mult23_in_aaa))
U18_A(p1_out_a) → U20_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(0)) → U10_A(0, p1_out_a)
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), p1_out_a)

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga
mult23_in_aga(s(T52)) → U1_aga(mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(mult23_out_aga) → mult23_out_aga
U2_aga(mult23_out_aga) → U3_aga(sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(sum34_out_aaa(T51)) → mult23_out_aga
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga) → p1_out_a
U12_a(mult23_out_aga) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0)
U2_aga(x0)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(67) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U9_A(mult23_out_aaa(y0)) → U10_A(y0, U7_a(p1_in_a)) we obtained the following new rules [LPAR04]:

U9_A(mult23_out_aaa(0)) → U10_A(0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U7_a(p1_in_a))

(68) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga) → P1_IN_A
P1_IN_AP1_IN_A
U16_A(mult53_out_aa(T92)) → P1_IN_A
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga)
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(mult23_in_aga(x0)))
P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))
P1_IN_AU14_A(p1_out_a)
P1_IN_AU14_A(U7_a(p1_in_a))
P1_IN_AU14_A(U14_a(p1_in_a))
P1_IN_AU14_A(U18_a(p1_in_a))
P1_IN_AU14_A(U22_a(p1_in_a))
U14_A(p1_out_a) → U16_A(mult53_out_aa(0))
U14_A(p1_out_a) → U16_A(U5_aa(mult53_in_aa))
U14_A(p1_out_a) → U16_A(U6_aa(mult53_in_aa))
P1_IN_AU18_A(p1_out_a)
P1_IN_AU18_A(U7_a(p1_in_a))
P1_IN_AU18_A(U14_a(p1_in_a))
P1_IN_AU18_A(U18_a(p1_in_a))
P1_IN_AU18_A(U22_a(p1_in_a))
U18_A(p1_out_a) → U20_A(mult23_out_aaa(0))
U18_A(p1_out_a) → U20_A(U1_aaa(mult23_in_aaa))
U18_A(p1_out_a) → U20_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(0)) → U10_A(0, p1_out_a)
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), p1_out_a)
U9_A(mult23_out_aaa(0)) → U10_A(0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U7_a(p1_in_a))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga
mult23_in_aga(s(T52)) → U1_aga(mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(mult23_out_aga) → mult23_out_aga
U2_aga(mult23_out_aga) → U3_aga(sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(sum34_out_aaa(T51)) → mult23_out_aga
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga) → p1_out_a
U12_a(mult23_out_aga) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0)
U2_aga(x0)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(69) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a)) we obtained the following new rules [LPAR04]:

U9_A(mult23_out_aaa(0)) → U10_A(0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U14_a(p1_in_a))

(70) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga) → P1_IN_A
P1_IN_AP1_IN_A
U16_A(mult53_out_aa(T92)) → P1_IN_A
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga)
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(mult23_in_aga(x0)))
P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))
P1_IN_AU14_A(p1_out_a)
P1_IN_AU14_A(U7_a(p1_in_a))
P1_IN_AU14_A(U14_a(p1_in_a))
P1_IN_AU14_A(U18_a(p1_in_a))
P1_IN_AU14_A(U22_a(p1_in_a))
U14_A(p1_out_a) → U16_A(mult53_out_aa(0))
U14_A(p1_out_a) → U16_A(U5_aa(mult53_in_aa))
U14_A(p1_out_a) → U16_A(U6_aa(mult53_in_aa))
P1_IN_AU18_A(p1_out_a)
P1_IN_AU18_A(U7_a(p1_in_a))
P1_IN_AU18_A(U14_a(p1_in_a))
P1_IN_AU18_A(U18_a(p1_in_a))
P1_IN_AU18_A(U22_a(p1_in_a))
U18_A(p1_out_a) → U20_A(mult23_out_aaa(0))
U18_A(p1_out_a) → U20_A(U1_aaa(mult23_in_aaa))
U18_A(p1_out_a) → U20_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(0)) → U10_A(0, p1_out_a)
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), p1_out_a)
U9_A(mult23_out_aaa(0)) → U10_A(0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U7_a(p1_in_a))
U9_A(mult23_out_aaa(0)) → U10_A(0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U14_a(p1_in_a))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga
mult23_in_aga(s(T52)) → U1_aga(mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(mult23_out_aga) → mult23_out_aga
U2_aga(mult23_out_aga) → U3_aga(sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(sum34_out_aaa(T51)) → mult23_out_aga
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga) → p1_out_a
U12_a(mult23_out_aga) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0)
U2_aga(x0)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(71) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a)) we obtained the following new rules [LPAR04]:

U9_A(mult23_out_aaa(0)) → U10_A(0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U18_a(p1_in_a))

(72) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga) → P1_IN_A
P1_IN_AP1_IN_A
U16_A(mult53_out_aa(T92)) → P1_IN_A
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga)
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(mult23_in_aga(x0)))
P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))
P1_IN_AU14_A(p1_out_a)
P1_IN_AU14_A(U7_a(p1_in_a))
P1_IN_AU14_A(U14_a(p1_in_a))
P1_IN_AU14_A(U18_a(p1_in_a))
P1_IN_AU14_A(U22_a(p1_in_a))
U14_A(p1_out_a) → U16_A(mult53_out_aa(0))
U14_A(p1_out_a) → U16_A(U5_aa(mult53_in_aa))
U14_A(p1_out_a) → U16_A(U6_aa(mult53_in_aa))
P1_IN_AU18_A(p1_out_a)
P1_IN_AU18_A(U7_a(p1_in_a))
P1_IN_AU18_A(U14_a(p1_in_a))
P1_IN_AU18_A(U18_a(p1_in_a))
P1_IN_AU18_A(U22_a(p1_in_a))
U18_A(p1_out_a) → U20_A(mult23_out_aaa(0))
U18_A(p1_out_a) → U20_A(U1_aaa(mult23_in_aaa))
U18_A(p1_out_a) → U20_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(0)) → U10_A(0, p1_out_a)
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), p1_out_a)
U9_A(mult23_out_aaa(0)) → U10_A(0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U7_a(p1_in_a))
U9_A(mult23_out_aaa(0)) → U10_A(0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U14_a(p1_in_a))
U9_A(mult23_out_aaa(0)) → U10_A(0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U18_a(p1_in_a))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga
mult23_in_aga(s(T52)) → U1_aga(mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(mult23_out_aga) → mult23_out_aga
U2_aga(mult23_out_aga) → U3_aga(sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(sum34_out_aaa(T51)) → mult23_out_aga
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga) → p1_out_a
U12_a(mult23_out_aga) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0)
U2_aga(x0)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(73) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a)) we obtained the following new rules [LPAR04]:

U9_A(mult23_out_aaa(0)) → U10_A(0, U22_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U22_a(p1_in_a))

(74) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga) → P1_IN_A
P1_IN_AP1_IN_A
U16_A(mult53_out_aa(T92)) → P1_IN_A
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga)
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(mult23_in_aga(x0)))
P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))
P1_IN_AU14_A(p1_out_a)
P1_IN_AU14_A(U7_a(p1_in_a))
P1_IN_AU14_A(U14_a(p1_in_a))
P1_IN_AU14_A(U18_a(p1_in_a))
P1_IN_AU14_A(U22_a(p1_in_a))
U14_A(p1_out_a) → U16_A(mult53_out_aa(0))
U14_A(p1_out_a) → U16_A(U5_aa(mult53_in_aa))
U14_A(p1_out_a) → U16_A(U6_aa(mult53_in_aa))
P1_IN_AU18_A(p1_out_a)
P1_IN_AU18_A(U7_a(p1_in_a))
P1_IN_AU18_A(U14_a(p1_in_a))
P1_IN_AU18_A(U18_a(p1_in_a))
P1_IN_AU18_A(U22_a(p1_in_a))
U18_A(p1_out_a) → U20_A(mult23_out_aaa(0))
U18_A(p1_out_a) → U20_A(U1_aaa(mult23_in_aaa))
U18_A(p1_out_a) → U20_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(0)) → U10_A(0, p1_out_a)
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), p1_out_a)
U9_A(mult23_out_aaa(0)) → U10_A(0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U7_a(p1_in_a))
U9_A(mult23_out_aaa(0)) → U10_A(0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U14_a(p1_in_a))
U9_A(mult23_out_aaa(0)) → U10_A(0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U18_a(p1_in_a))
U9_A(mult23_out_aaa(0)) → U10_A(0, U22_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U22_a(p1_in_a))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga
mult23_in_aga(s(T52)) → U1_aga(mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(mult23_out_aga) → mult23_out_aga
U2_aga(mult23_out_aga) → U3_aga(sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(sum34_out_aaa(T51)) → mult23_out_aga
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga) → p1_out_a
U12_a(mult23_out_aga) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0)
U2_aga(x0)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(75) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = P1_IN_A evaluates to t =P1_IN_A

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from P1_IN_A to P1_IN_A.



(76) NO

(77) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SUM34_IN_AGA(T69, s(T70), s(X110)) → SUM34_IN_AGA(T69, T70, X110)

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x4)
U22_g(x1, x2)  =  U22_g(x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga
U4_aga(x1, x2, x3, x4)  =  U4_aga(x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U16_g(x1, x2, x3)  =  U16_g(x3)
U17_g(x1, x2, x3)  =  U17_g(x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x4)
SUM34_IN_AGA(x1, x2, x3)  =  SUM34_IN_AGA(x2)

We have to consider all (P,R,Pi)-chains

(78) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(79) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SUM34_IN_AGA(T69, s(T70), s(X110)) → SUM34_IN_AGA(T69, T70, X110)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
SUM34_IN_AGA(x1, x2, x3)  =  SUM34_IN_AGA(x2)

We have to consider all (P,R,Pi)-chains

(80) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(81) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SUM34_IN_AGA(s(T70)) → SUM34_IN_AGA(T70)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(82) QDPSizeChangeProof (EQUIVALENT transformation)

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.

From the DPs we obtained the following set of size-change graphs:

  • SUM34_IN_AGA(s(T70)) → SUM34_IN_AGA(T70)
    The graph contains the following edges 1 > 1

(83) YES

(84) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT23_IN_GGA(T51, s(T52), X83) → MULT23_IN_GGA(T51, T52, X82)

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x4)
U22_g(x1, x2)  =  U22_g(x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga
U4_aga(x1, x2, x3, x4)  =  U4_aga(x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U16_g(x1, x2, x3)  =  U16_g(x3)
U17_g(x1, x2, x3)  =  U17_g(x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x4)
MULT23_IN_GGA(x1, x2, x3)  =  MULT23_IN_GGA(x1, x2)

We have to consider all (P,R,Pi)-chains

(85) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(86) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT23_IN_GGA(T51, s(T52), X83) → MULT23_IN_GGA(T51, T52, X82)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
MULT23_IN_GGA(x1, x2, x3)  =  MULT23_IN_GGA(x1, x2)

We have to consider all (P,R,Pi)-chains

(87) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(88) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MULT23_IN_GGA(T51, s(T52)) → MULT23_IN_GGA(T51, T52)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(89) QDPSizeChangeProof (EQUIVALENT transformation)

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.

From the DPs we obtained the following set of size-change graphs:

  • MULT23_IN_GGA(T51, s(T52)) → MULT23_IN_GGA(T51, T52)
    The graph contains the following edges 1 >= 1, 2 > 2

(90) YES

(91) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

P1_IN_G(cons(s(s(0)), cons(T92, T93))) → P1_IN_G(cons(T92, T93))
P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_G(cons(T24, cons(T25, T26)))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_G(cons(T131, cons(T132, T133)))
P1_IN_G(cons(0, cons(0, T153))) → P1_IN_G(T153)

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x4)
U22_g(x1, x2)  =  U22_g(x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga
U4_aga(x1, x2, x3, x4)  =  U4_aga(x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
U16_g(x1, x2, x3)  =  U16_g(x3)
U17_g(x1, x2, x3)  =  U17_g(x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x4)
P1_IN_G(x1)  =  P1_IN_G(x1)

We have to consider all (P,R,Pi)-chains

(92) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(93) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

P1_IN_G(cons(s(s(0)), cons(T92, T93))) → P1_IN_G(cons(T92, T93))
P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_G(cons(T24, cons(T25, T26)))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_G(cons(T131, cons(T132, T133)))
P1_IN_G(cons(0, cons(0, T153))) → P1_IN_G(T153)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains

(94) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(95) Obligation:

Q DP problem:
The TRS P consists of the following rules:

P1_IN_G(cons(s(s(0)), cons(T92, T93))) → P1_IN_G(cons(T92, T93))
P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_G(cons(T24, cons(T25, T26)))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_G(cons(T131, cons(T132, T133)))
P1_IN_G(cons(0, cons(0, T153))) → P1_IN_G(T153)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(96) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

P1_IN_G(cons(s(s(0)), cons(T92, T93))) → P1_IN_G(cons(T92, T93))
P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_G(cons(T24, cons(T25, T26)))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_G(cons(T131, cons(T132, T133)))
P1_IN_G(cons(0, cons(0, T153))) → P1_IN_G(T153)
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(0) = 0   
POL(P1_IN_G(x1)) = 2·x1   
POL(cons(x1, x2)) = 2·x1 + 2·x2   
POL(s(x1)) = 2·x1   

(97) Obligation:

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(98) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(99) YES

(100) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
p1_in: (b) (f)
mult23_in: (b,b,f) (f,f,f) (f,b,f)
sum34_in: (f,b,f) (f,f,f)
mult53_in: (f,f) (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g(x1)
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x2, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x3, x4)
U22_g(x1, x2)  =  U22_g(x1, x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x1, x2, x3, x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga(x2)
U4_aga(x1, x2, x3, x4)  =  U4_aga(x2, x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x1, x2, x3, x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x1, x2, x3, x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga(x2)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x2, x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x2, x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x2, x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x1, x2, x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U16_g(x1, x2, x3)  =  U16_g(x1, x2, x3)
U17_g(x1, x2, x3)  =  U17_g(x1, x2, x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x3, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x3, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x1, x2, x3, x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x1, x2, x3, x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x1, x2, x3, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(101) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g(x1)
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x2, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x3, x4)
U22_g(x1, x2)  =  U22_g(x1, x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x1, x2, x3, x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga(x2)
U4_aga(x1, x2, x3, x4)  =  U4_aga(x2, x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x1, x2, x3, x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x1, x2, x3, x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga(x2)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x2, x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x2, x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x2, x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x1, x2, x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U16_g(x1, x2, x3)  =  U16_g(x1, x2, x3)
U17_g(x1, x2, x3)  =  U17_g(x1, x2, x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x3, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x3, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x1, x2, x3, x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x1, x2, x3, x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x1, x2, x3, x4)

(102) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_G(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_G(cons(T24, cons(T25, T26)))
P1_IN_G(cons(s(s(0)), cons(T92, T93))) → U14_G(T92, T93, p1_in_g(cons(T92, T93)))
P1_IN_G(cons(s(s(0)), cons(T92, T93))) → P1_IN_G(cons(T92, T93))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_G(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_G(cons(T131, cons(T132, T133)))
P1_IN_G(cons(0, cons(0, T153))) → U22_G(T153, p1_in_g(T153))
P1_IN_G(cons(0, cons(0, T153))) → P1_IN_G(T153)
U18_G(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_G(T131, T132, T133, mult23_in_gga(T131, T132, X205))
U18_G(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → MULT23_IN_GGA(T131, T132, X205)
MULT23_IN_GGA(T51, s(T52), X83) → U1_GGA(T51, T52, X83, mult23_in_gga(T51, T52, X82))
MULT23_IN_GGA(T51, s(T52), X83) → MULT23_IN_GGA(T51, T52, X82)
MULT23_IN_GGA(T51, s(T52), X83) → U2_GGA(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_GGA(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_GGA(T51, T52, X83, sum34_in_aga(T55, T51, X83))
U2_GGA(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → SUM34_IN_AGA(T55, T51, X83)
SUM34_IN_AGA(T69, s(T70), s(X110)) → U4_AGA(T69, T70, X110, sum34_in_aga(T69, T70, X110))
SUM34_IN_AGA(T69, s(T70), s(X110)) → SUM34_IN_AGA(T69, T70, X110)
U18_G(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_G(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_G(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_G(T131, T132, T133, p1_in_a(cons(T146, T133)))
U20_G(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → P1_IN_A(cons(T146, T133))
P1_IN_A(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_A(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
P1_IN_A(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_A(cons(T24, cons(T25, T26)))
P1_IN_A(cons(s(s(0)), cons(T92, T93))) → U14_A(T92, T93, p1_in_a(cons(T92, T93)))
P1_IN_A(cons(s(s(0)), cons(T92, T93))) → P1_IN_A(cons(T92, T93))
P1_IN_A(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_A(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
P1_IN_A(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_A(cons(T131, cons(T132, T133)))
P1_IN_A(cons(0, cons(0, T153))) → U22_A(T153, p1_in_a(T153))
P1_IN_A(cons(0, cons(0, T153))) → P1_IN_A(T153)
U18_A(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_A(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
U18_A(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → MULT23_IN_AAA(T131, T132, X205)
MULT23_IN_AAA(T51, s(T52), X83) → U1_AAA(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
MULT23_IN_AAA(T51, s(T52), X83) → MULT23_IN_AAA(T51, T52, X82)
MULT23_IN_AAA(T51, s(T52), X83) → U2_AAA(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_AAA(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_AAA(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U2_AAA(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → SUM34_IN_AAA(T55, T51, X83)
SUM34_IN_AAA(T69, s(T70), s(X110)) → U4_AAA(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
SUM34_IN_AAA(T69, s(T70), s(X110)) → SUM34_IN_AAA(T69, T70, X110)
U18_A(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_A(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_A(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_A(T131, T132, T133, p1_in_a(cons(T146, T133)))
U20_A(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → P1_IN_A(cons(T146, T133))
U14_A(T92, T93, p1_out_a(cons(T92, T93))) → U15_A(T92, T93, mult53_in_aa(T92, X22))
U14_A(T92, T93, p1_out_a(cons(T92, T93))) → MULT53_IN_AA(T92, X22)
MULT53_IN_AA(s(T101), X159) → U5_AA(T101, X159, mult53_in_aa(T101, X158))
MULT53_IN_AA(s(T101), X159) → MULT53_IN_AA(T101, X158)
MULT53_IN_AA(s(T101), T108) → U6_AA(T101, T108, mult53_in_aa(T101, T108))
MULT53_IN_AA(s(T101), T108) → MULT53_IN_AA(T101, T108)
U14_A(T92, T93, p1_out_a(cons(T92, T93))) → U16_A(T92, T93, mult53_in_aa(T92, T98))
U16_A(T92, T93, mult53_out_aa(T92, T98)) → U17_A(T92, T93, p1_in_a(cons(T98, T93)))
U16_A(T92, T93, mult53_out_aa(T92, T98)) → P1_IN_A(cons(T98, T93))
U7_A(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_A(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U7_A(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → MULT23_IN_AAA(T24, T25, X44)
U7_A(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_A(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_A(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_A(T24, T25, T26, p1_in_a(cons(T39, T26)))
U9_A(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → P1_IN_A(cons(T39, T26))
U10_A(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_A(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
U10_A(T24, T25, T26, p1_out_a(cons(T39, T26))) → MULT23_IN_AGA(s(s(T24)), T25, X22)
MULT23_IN_AGA(T51, s(T52), X83) → U1_AGA(T51, T52, X83, mult23_in_aga(T51, T52, X82))
MULT23_IN_AGA(T51, s(T52), X83) → MULT23_IN_AGA(T51, T52, X82)
MULT23_IN_AGA(T51, s(T52), X83) → U2_AGA(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_AGA(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_AGA(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U2_AGA(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → SUM34_IN_AAA(T55, T51, X83)
U10_A(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_A(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_A(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_A(T24, T25, T26, p1_in_a(cons(T79, T26)))
U12_A(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → P1_IN_A(cons(T79, T26))
U14_G(T92, T93, p1_out_g(cons(T92, T93))) → U15_G(T92, T93, mult53_in_ga(T92, X22))
U14_G(T92, T93, p1_out_g(cons(T92, T93))) → MULT53_IN_GA(T92, X22)
MULT53_IN_GA(s(T101), X159) → U5_GA(T101, X159, mult53_in_ga(T101, X158))
MULT53_IN_GA(s(T101), X159) → MULT53_IN_GA(T101, X158)
MULT53_IN_GA(s(T101), T108) → U6_GA(T101, T108, mult53_in_ga(T101, T108))
MULT53_IN_GA(s(T101), T108) → MULT53_IN_GA(T101, T108)
U14_G(T92, T93, p1_out_g(cons(T92, T93))) → U16_G(T92, T93, mult53_in_ga(T92, T98))
U16_G(T92, T93, mult53_out_ga(T92, T98)) → U17_G(T92, T93, p1_in_a(cons(T98, T93)))
U16_G(T92, T93, mult53_out_ga(T92, T98)) → P1_IN_A(cons(T98, T93))
U7_G(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_G(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U7_G(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → MULT23_IN_GGA(T24, T25, X44)
U7_G(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_G(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_G(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_G(T24, T25, T26, p1_in_a(cons(T39, T26)))
U9_G(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → P1_IN_A(cons(T39, T26))
U10_G(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_G(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U10_G(T24, T25, T26, p1_out_a(cons(T39, T26))) → MULT23_IN_GGA(s(s(T24)), T25, X22)
U10_G(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_G(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_G(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_G(T24, T25, T26, p1_in_a(cons(T79, T26)))
U12_G(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → P1_IN_A(cons(T79, T26))

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g(x1)
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x2, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x3, x4)
U22_g(x1, x2)  =  U22_g(x1, x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x1, x2, x3, x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga(x2)
U4_aga(x1, x2, x3, x4)  =  U4_aga(x2, x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x1, x2, x3, x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x1, x2, x3, x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga(x2)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x2, x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x2, x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x2, x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x1, x2, x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U16_g(x1, x2, x3)  =  U16_g(x1, x2, x3)
U17_g(x1, x2, x3)  =  U17_g(x1, x2, x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x3, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x3, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x1, x2, x3, x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x1, x2, x3, x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x1, x2, x3, x4)
P1_IN_G(x1)  =  P1_IN_G(x1)
U7_G(x1, x2, x3, x4)  =  U7_G(x1, x2, x3, x4)
U14_G(x1, x2, x3)  =  U14_G(x1, x2, x3)
U18_G(x1, x2, x3, x4)  =  U18_G(x1, x2, x3, x4)
U22_G(x1, x2)  =  U22_G(x1, x2)
U19_G(x1, x2, x3, x4)  =  U19_G(x1, x2, x3, x4)
MULT23_IN_GGA(x1, x2, x3)  =  MULT23_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
SUM34_IN_AGA(x1, x2, x3)  =  SUM34_IN_AGA(x2)
U4_AGA(x1, x2, x3, x4)  =  U4_AGA(x2, x4)
U20_G(x1, x2, x3, x4)  =  U20_G(x1, x2, x3, x4)
U21_G(x1, x2, x3, x4)  =  U21_G(x1, x2, x3, x4)
P1_IN_A(x1)  =  P1_IN_A
U7_A(x1, x2, x3, x4)  =  U7_A(x4)
U14_A(x1, x2, x3)  =  U14_A(x3)
U18_A(x1, x2, x3, x4)  =  U18_A(x4)
U22_A(x1, x2)  =  U22_A(x2)
U19_A(x1, x2, x3, x4)  =  U19_A(x4)
MULT23_IN_AAA(x1, x2, x3)  =  MULT23_IN_AAA
U1_AAA(x1, x2, x3, x4)  =  U1_AAA(x4)
U2_AAA(x1, x2, x3, x4)  =  U2_AAA(x4)
U3_AAA(x1, x2, x3, x4)  =  U3_AAA(x2, x4)
SUM34_IN_AAA(x1, x2, x3)  =  SUM34_IN_AAA
U4_AAA(x1, x2, x3, x4)  =  U4_AAA(x4)
U20_A(x1, x2, x3, x4)  =  U20_A(x4)
U21_A(x1, x2, x3, x4)  =  U21_A(x4)
U15_A(x1, x2, x3)  =  U15_A(x3)
MULT53_IN_AA(x1, x2)  =  MULT53_IN_AA
U5_AA(x1, x2, x3)  =  U5_AA(x3)
U6_AA(x1, x2, x3)  =  U6_AA(x3)
U16_A(x1, x2, x3)  =  U16_A(x3)
U17_A(x1, x2, x3)  =  U17_A(x3)
U8_A(x1, x2, x3, x4)  =  U8_A(x4)
U9_A(x1, x2, x3, x4)  =  U9_A(x4)
U10_A(x1, x2, x3, x4)  =  U10_A(x2, x4)
U11_A(x1, x2, x3, x4)  =  U11_A(x4)
MULT23_IN_AGA(x1, x2, x3)  =  MULT23_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4)  =  U1_AGA(x2, x4)
U2_AGA(x1, x2, x3, x4)  =  U2_AGA(x2, x4)
U3_AGA(x1, x2, x3, x4)  =  U3_AGA(x2, x4)
U12_A(x1, x2, x3, x4)  =  U12_A(x4)
U13_A(x1, x2, x3, x4)  =  U13_A(x4)
U15_G(x1, x2, x3)  =  U15_G(x1, x2, x3)
MULT53_IN_GA(x1, x2)  =  MULT53_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
U16_G(x1, x2, x3)  =  U16_G(x1, x2, x3)
U17_G(x1, x2, x3)  =  U17_G(x1, x2, x3)
U8_G(x1, x2, x3, x4)  =  U8_G(x1, x2, x3, x4)
U9_G(x1, x2, x3, x4)  =  U9_G(x1, x2, x3, x4)
U10_G(x1, x2, x3, x4)  =  U10_G(x1, x2, x3, x4)
U11_G(x1, x2, x3, x4)  =  U11_G(x1, x2, x3, x4)
U12_G(x1, x2, x3, x4)  =  U12_G(x1, x2, x3, x4)
U13_G(x1, x2, x3, x4)  =  U13_G(x1, x2, x3, x4)

We have to consider all (P,R,Pi)-chains

(103) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_G(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_G(cons(T24, cons(T25, T26)))
P1_IN_G(cons(s(s(0)), cons(T92, T93))) → U14_G(T92, T93, p1_in_g(cons(T92, T93)))
P1_IN_G(cons(s(s(0)), cons(T92, T93))) → P1_IN_G(cons(T92, T93))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_G(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_G(cons(T131, cons(T132, T133)))
P1_IN_G(cons(0, cons(0, T153))) → U22_G(T153, p1_in_g(T153))
P1_IN_G(cons(0, cons(0, T153))) → P1_IN_G(T153)
U18_G(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_G(T131, T132, T133, mult23_in_gga(T131, T132, X205))
U18_G(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → MULT23_IN_GGA(T131, T132, X205)
MULT23_IN_GGA(T51, s(T52), X83) → U1_GGA(T51, T52, X83, mult23_in_gga(T51, T52, X82))
MULT23_IN_GGA(T51, s(T52), X83) → MULT23_IN_GGA(T51, T52, X82)
MULT23_IN_GGA(T51, s(T52), X83) → U2_GGA(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_GGA(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_GGA(T51, T52, X83, sum34_in_aga(T55, T51, X83))
U2_GGA(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → SUM34_IN_AGA(T55, T51, X83)
SUM34_IN_AGA(T69, s(T70), s(X110)) → U4_AGA(T69, T70, X110, sum34_in_aga(T69, T70, X110))
SUM34_IN_AGA(T69, s(T70), s(X110)) → SUM34_IN_AGA(T69, T70, X110)
U18_G(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_G(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_G(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_G(T131, T132, T133, p1_in_a(cons(T146, T133)))
U20_G(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → P1_IN_A(cons(T146, T133))
P1_IN_A(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_A(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
P1_IN_A(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_A(cons(T24, cons(T25, T26)))
P1_IN_A(cons(s(s(0)), cons(T92, T93))) → U14_A(T92, T93, p1_in_a(cons(T92, T93)))
P1_IN_A(cons(s(s(0)), cons(T92, T93))) → P1_IN_A(cons(T92, T93))
P1_IN_A(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_A(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
P1_IN_A(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_A(cons(T131, cons(T132, T133)))
P1_IN_A(cons(0, cons(0, T153))) → U22_A(T153, p1_in_a(T153))
P1_IN_A(cons(0, cons(0, T153))) → P1_IN_A(T153)
U18_A(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_A(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
U18_A(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → MULT23_IN_AAA(T131, T132, X205)
MULT23_IN_AAA(T51, s(T52), X83) → U1_AAA(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
MULT23_IN_AAA(T51, s(T52), X83) → MULT23_IN_AAA(T51, T52, X82)
MULT23_IN_AAA(T51, s(T52), X83) → U2_AAA(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_AAA(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_AAA(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U2_AAA(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → SUM34_IN_AAA(T55, T51, X83)
SUM34_IN_AAA(T69, s(T70), s(X110)) → U4_AAA(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
SUM34_IN_AAA(T69, s(T70), s(X110)) → SUM34_IN_AAA(T69, T70, X110)
U18_A(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_A(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_A(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_A(T131, T132, T133, p1_in_a(cons(T146, T133)))
U20_A(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → P1_IN_A(cons(T146, T133))
U14_A(T92, T93, p1_out_a(cons(T92, T93))) → U15_A(T92, T93, mult53_in_aa(T92, X22))
U14_A(T92, T93, p1_out_a(cons(T92, T93))) → MULT53_IN_AA(T92, X22)
MULT53_IN_AA(s(T101), X159) → U5_AA(T101, X159, mult53_in_aa(T101, X158))
MULT53_IN_AA(s(T101), X159) → MULT53_IN_AA(T101, X158)
MULT53_IN_AA(s(T101), T108) → U6_AA(T101, T108, mult53_in_aa(T101, T108))
MULT53_IN_AA(s(T101), T108) → MULT53_IN_AA(T101, T108)
U14_A(T92, T93, p1_out_a(cons(T92, T93))) → U16_A(T92, T93, mult53_in_aa(T92, T98))
U16_A(T92, T93, mult53_out_aa(T92, T98)) → U17_A(T92, T93, p1_in_a(cons(T98, T93)))
U16_A(T92, T93, mult53_out_aa(T92, T98)) → P1_IN_A(cons(T98, T93))
U7_A(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_A(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U7_A(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → MULT23_IN_AAA(T24, T25, X44)
U7_A(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_A(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_A(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_A(T24, T25, T26, p1_in_a(cons(T39, T26)))
U9_A(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → P1_IN_A(cons(T39, T26))
U10_A(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_A(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
U10_A(T24, T25, T26, p1_out_a(cons(T39, T26))) → MULT23_IN_AGA(s(s(T24)), T25, X22)
MULT23_IN_AGA(T51, s(T52), X83) → U1_AGA(T51, T52, X83, mult23_in_aga(T51, T52, X82))
MULT23_IN_AGA(T51, s(T52), X83) → MULT23_IN_AGA(T51, T52, X82)
MULT23_IN_AGA(T51, s(T52), X83) → U2_AGA(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_AGA(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_AGA(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U2_AGA(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → SUM34_IN_AAA(T55, T51, X83)
U10_A(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_A(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_A(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_A(T24, T25, T26, p1_in_a(cons(T79, T26)))
U12_A(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → P1_IN_A(cons(T79, T26))
U14_G(T92, T93, p1_out_g(cons(T92, T93))) → U15_G(T92, T93, mult53_in_ga(T92, X22))
U14_G(T92, T93, p1_out_g(cons(T92, T93))) → MULT53_IN_GA(T92, X22)
MULT53_IN_GA(s(T101), X159) → U5_GA(T101, X159, mult53_in_ga(T101, X158))
MULT53_IN_GA(s(T101), X159) → MULT53_IN_GA(T101, X158)
MULT53_IN_GA(s(T101), T108) → U6_GA(T101, T108, mult53_in_ga(T101, T108))
MULT53_IN_GA(s(T101), T108) → MULT53_IN_GA(T101, T108)
U14_G(T92, T93, p1_out_g(cons(T92, T93))) → U16_G(T92, T93, mult53_in_ga(T92, T98))
U16_G(T92, T93, mult53_out_ga(T92, T98)) → U17_G(T92, T93, p1_in_a(cons(T98, T93)))
U16_G(T92, T93, mult53_out_ga(T92, T98)) → P1_IN_A(cons(T98, T93))
U7_G(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_G(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U7_G(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → MULT23_IN_GGA(T24, T25, X44)
U7_G(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_G(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_G(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_G(T24, T25, T26, p1_in_a(cons(T39, T26)))
U9_G(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → P1_IN_A(cons(T39, T26))
U10_G(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_G(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U10_G(T24, T25, T26, p1_out_a(cons(T39, T26))) → MULT23_IN_GGA(s(s(T24)), T25, X22)
U10_G(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_G(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_G(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_G(T24, T25, T26, p1_in_a(cons(T79, T26)))
U12_G(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → P1_IN_A(cons(T79, T26))

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g(x1)
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x2, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x3, x4)
U22_g(x1, x2)  =  U22_g(x1, x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x1, x2, x3, x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga(x2)
U4_aga(x1, x2, x3, x4)  =  U4_aga(x2, x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x1, x2, x3, x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x1, x2, x3, x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga(x2)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x2, x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x2, x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x2, x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x1, x2, x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U16_g(x1, x2, x3)  =  U16_g(x1, x2, x3)
U17_g(x1, x2, x3)  =  U17_g(x1, x2, x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x3, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x3, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x1, x2, x3, x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x1, x2, x3, x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x1, x2, x3, x4)
P1_IN_G(x1)  =  P1_IN_G(x1)
U7_G(x1, x2, x3, x4)  =  U7_G(x1, x2, x3, x4)
U14_G(x1, x2, x3)  =  U14_G(x1, x2, x3)
U18_G(x1, x2, x3, x4)  =  U18_G(x1, x2, x3, x4)
U22_G(x1, x2)  =  U22_G(x1, x2)
U19_G(x1, x2, x3, x4)  =  U19_G(x1, x2, x3, x4)
MULT23_IN_GGA(x1, x2, x3)  =  MULT23_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
SUM34_IN_AGA(x1, x2, x3)  =  SUM34_IN_AGA(x2)
U4_AGA(x1, x2, x3, x4)  =  U4_AGA(x2, x4)
U20_G(x1, x2, x3, x4)  =  U20_G(x1, x2, x3, x4)
U21_G(x1, x2, x3, x4)  =  U21_G(x1, x2, x3, x4)
P1_IN_A(x1)  =  P1_IN_A
U7_A(x1, x2, x3, x4)  =  U7_A(x4)
U14_A(x1, x2, x3)  =  U14_A(x3)
U18_A(x1, x2, x3, x4)  =  U18_A(x4)
U22_A(x1, x2)  =  U22_A(x2)
U19_A(x1, x2, x3, x4)  =  U19_A(x4)
MULT23_IN_AAA(x1, x2, x3)  =  MULT23_IN_AAA
U1_AAA(x1, x2, x3, x4)  =  U1_AAA(x4)
U2_AAA(x1, x2, x3, x4)  =  U2_AAA(x4)
U3_AAA(x1, x2, x3, x4)  =  U3_AAA(x2, x4)
SUM34_IN_AAA(x1, x2, x3)  =  SUM34_IN_AAA
U4_AAA(x1, x2, x3, x4)  =  U4_AAA(x4)
U20_A(x1, x2, x3, x4)  =  U20_A(x4)
U21_A(x1, x2, x3, x4)  =  U21_A(x4)
U15_A(x1, x2, x3)  =  U15_A(x3)
MULT53_IN_AA(x1, x2)  =  MULT53_IN_AA
U5_AA(x1, x2, x3)  =  U5_AA(x3)
U6_AA(x1, x2, x3)  =  U6_AA(x3)
U16_A(x1, x2, x3)  =  U16_A(x3)
U17_A(x1, x2, x3)  =  U17_A(x3)
U8_A(x1, x2, x3, x4)  =  U8_A(x4)
U9_A(x1, x2, x3, x4)  =  U9_A(x4)
U10_A(x1, x2, x3, x4)  =  U10_A(x2, x4)
U11_A(x1, x2, x3, x4)  =  U11_A(x4)
MULT23_IN_AGA(x1, x2, x3)  =  MULT23_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4)  =  U1_AGA(x2, x4)
U2_AGA(x1, x2, x3, x4)  =  U2_AGA(x2, x4)
U3_AGA(x1, x2, x3, x4)  =  U3_AGA(x2, x4)
U12_A(x1, x2, x3, x4)  =  U12_A(x4)
U13_A(x1, x2, x3, x4)  =  U13_A(x4)
U15_G(x1, x2, x3)  =  U15_G(x1, x2, x3)
MULT53_IN_GA(x1, x2)  =  MULT53_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
U16_G(x1, x2, x3)  =  U16_G(x1, x2, x3)
U17_G(x1, x2, x3)  =  U17_G(x1, x2, x3)
U8_G(x1, x2, x3, x4)  =  U8_G(x1, x2, x3, x4)
U9_G(x1, x2, x3, x4)  =  U9_G(x1, x2, x3, x4)
U10_G(x1, x2, x3, x4)  =  U10_G(x1, x2, x3, x4)
U11_G(x1, x2, x3, x4)  =  U11_G(x1, x2, x3, x4)
U12_G(x1, x2, x3, x4)  =  U12_G(x1, x2, x3, x4)
U13_G(x1, x2, x3, x4)  =  U13_G(x1, x2, x3, x4)

We have to consider all (P,R,Pi)-chains

(104) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 9 SCCs with 54 less nodes.

(105) Complex Obligation (AND)

(106) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT53_IN_GA(s(T101), T108) → MULT53_IN_GA(T101, T108)
MULT53_IN_GA(s(T101), X159) → MULT53_IN_GA(T101, X158)

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g(x1)
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x2, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x3, x4)
U22_g(x1, x2)  =  U22_g(x1, x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x1, x2, x3, x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga(x2)
U4_aga(x1, x2, x3, x4)  =  U4_aga(x2, x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x1, x2, x3, x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x1, x2, x3, x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga(x2)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x2, x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x2, x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x2, x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x1, x2, x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U16_g(x1, x2, x3)  =  U16_g(x1, x2, x3)
U17_g(x1, x2, x3)  =  U17_g(x1, x2, x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x3, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x3, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x1, x2, x3, x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x1, x2, x3, x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x1, x2, x3, x4)
MULT53_IN_GA(x1, x2)  =  MULT53_IN_GA(x1)

We have to consider all (P,R,Pi)-chains

(107) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(108) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT53_IN_GA(s(T101), T108) → MULT53_IN_GA(T101, T108)
MULT53_IN_GA(s(T101), X159) → MULT53_IN_GA(T101, X158)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
MULT53_IN_GA(x1, x2)  =  MULT53_IN_GA(x1)

We have to consider all (P,R,Pi)-chains

(109) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(110) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MULT53_IN_GA(s(T101)) → MULT53_IN_GA(T101)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(111) QDPSizeChangeProof (EQUIVALENT transformation)

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.

From the DPs we obtained the following set of size-change graphs:

  • MULT53_IN_GA(s(T101)) → MULT53_IN_GA(T101)
    The graph contains the following edges 1 > 1

(112) YES

(113) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT53_IN_AA(s(T101), T108) → MULT53_IN_AA(T101, T108)
MULT53_IN_AA(s(T101), X159) → MULT53_IN_AA(T101, X158)

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g(x1)
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x2, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x3, x4)
U22_g(x1, x2)  =  U22_g(x1, x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x1, x2, x3, x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga(x2)
U4_aga(x1, x2, x3, x4)  =  U4_aga(x2, x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x1, x2, x3, x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x1, x2, x3, x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga(x2)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x2, x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x2, x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x2, x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x1, x2, x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U16_g(x1, x2, x3)  =  U16_g(x1, x2, x3)
U17_g(x1, x2, x3)  =  U17_g(x1, x2, x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x3, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x3, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x1, x2, x3, x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x1, x2, x3, x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x1, x2, x3, x4)
MULT53_IN_AA(x1, x2)  =  MULT53_IN_AA

We have to consider all (P,R,Pi)-chains

(114) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(115) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT53_IN_AA(s(T101), T108) → MULT53_IN_AA(T101, T108)
MULT53_IN_AA(s(T101), X159) → MULT53_IN_AA(T101, X158)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
MULT53_IN_AA(x1, x2)  =  MULT53_IN_AA

We have to consider all (P,R,Pi)-chains

(116) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(117) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MULT53_IN_AAMULT53_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(118) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = MULT53_IN_AA evaluates to t =MULT53_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from MULT53_IN_AA to MULT53_IN_AA.



(119) NO

(120) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SUM34_IN_AAA(T69, s(T70), s(X110)) → SUM34_IN_AAA(T69, T70, X110)

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g(x1)
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x2, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x3, x4)
U22_g(x1, x2)  =  U22_g(x1, x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x1, x2, x3, x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga(x2)
U4_aga(x1, x2, x3, x4)  =  U4_aga(x2, x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x1, x2, x3, x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x1, x2, x3, x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga(x2)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x2, x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x2, x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x2, x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x1, x2, x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U16_g(x1, x2, x3)  =  U16_g(x1, x2, x3)
U17_g(x1, x2, x3)  =  U17_g(x1, x2, x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x3, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x3, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x1, x2, x3, x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x1, x2, x3, x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x1, x2, x3, x4)
SUM34_IN_AAA(x1, x2, x3)  =  SUM34_IN_AAA

We have to consider all (P,R,Pi)-chains

(121) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(122) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SUM34_IN_AAA(T69, s(T70), s(X110)) → SUM34_IN_AAA(T69, T70, X110)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
SUM34_IN_AAA(x1, x2, x3)  =  SUM34_IN_AAA

We have to consider all (P,R,Pi)-chains

(123) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(124) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SUM34_IN_AAASUM34_IN_AAA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(125) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = SUM34_IN_AAA evaluates to t =SUM34_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from SUM34_IN_AAA to SUM34_IN_AAA.



(126) NO

(127) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT23_IN_AGA(T51, s(T52), X83) → MULT23_IN_AGA(T51, T52, X82)

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g(x1)
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x2, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x3, x4)
U22_g(x1, x2)  =  U22_g(x1, x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x1, x2, x3, x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga(x2)
U4_aga(x1, x2, x3, x4)  =  U4_aga(x2, x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x1, x2, x3, x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x1, x2, x3, x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga(x2)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x2, x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x2, x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x2, x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x1, x2, x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U16_g(x1, x2, x3)  =  U16_g(x1, x2, x3)
U17_g(x1, x2, x3)  =  U17_g(x1, x2, x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x3, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x3, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x1, x2, x3, x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x1, x2, x3, x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x1, x2, x3, x4)
MULT23_IN_AGA(x1, x2, x3)  =  MULT23_IN_AGA(x2)

We have to consider all (P,R,Pi)-chains

(128) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(129) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT23_IN_AGA(T51, s(T52), X83) → MULT23_IN_AGA(T51, T52, X82)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
MULT23_IN_AGA(x1, x2, x3)  =  MULT23_IN_AGA(x2)

We have to consider all (P,R,Pi)-chains

(130) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(131) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MULT23_IN_AGA(s(T52)) → MULT23_IN_AGA(T52)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(132) QDPSizeChangeProof (EQUIVALENT transformation)

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.

From the DPs we obtained the following set of size-change graphs:

  • MULT23_IN_AGA(s(T52)) → MULT23_IN_AGA(T52)
    The graph contains the following edges 1 > 1

(133) YES

(134) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT23_IN_AAA(T51, s(T52), X83) → MULT23_IN_AAA(T51, T52, X82)

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g(x1)
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x2, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x3, x4)
U22_g(x1, x2)  =  U22_g(x1, x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x1, x2, x3, x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga(x2)
U4_aga(x1, x2, x3, x4)  =  U4_aga(x2, x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x1, x2, x3, x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x1, x2, x3, x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga(x2)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x2, x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x2, x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x2, x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x1, x2, x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U16_g(x1, x2, x3)  =  U16_g(x1, x2, x3)
U17_g(x1, x2, x3)  =  U17_g(x1, x2, x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x3, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x3, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x1, x2, x3, x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x1, x2, x3, x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x1, x2, x3, x4)
MULT23_IN_AAA(x1, x2, x3)  =  MULT23_IN_AAA

We have to consider all (P,R,Pi)-chains

(135) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(136) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT23_IN_AAA(T51, s(T52), X83) → MULT23_IN_AAA(T51, T52, X82)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
MULT23_IN_AAA(x1, x2, x3)  =  MULT23_IN_AAA

We have to consider all (P,R,Pi)-chains

(137) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(138) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MULT23_IN_AAAMULT23_IN_AAA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(139) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = MULT23_IN_AAA evaluates to t =MULT23_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from MULT23_IN_AAA to MULT23_IN_AAA.



(140) NO

(141) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

U7_A(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_A(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_A(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_A(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_A(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_A(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_A(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → P1_IN_A(cons(T79, T26))
P1_IN_A(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_A(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
P1_IN_A(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_A(cons(T24, cons(T25, T26)))
P1_IN_A(cons(s(s(0)), cons(T92, T93))) → U14_A(T92, T93, p1_in_a(cons(T92, T93)))
U14_A(T92, T93, p1_out_a(cons(T92, T93))) → U16_A(T92, T93, mult53_in_aa(T92, T98))
U16_A(T92, T93, mult53_out_aa(T92, T98)) → P1_IN_A(cons(T98, T93))
P1_IN_A(cons(s(s(0)), cons(T92, T93))) → P1_IN_A(cons(T92, T93))
P1_IN_A(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_A(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
U18_A(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_A(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_A(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → P1_IN_A(cons(T146, T133))
P1_IN_A(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_A(cons(T131, cons(T132, T133)))
P1_IN_A(cons(0, cons(0, T153))) → P1_IN_A(T153)
U9_A(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → P1_IN_A(cons(T39, T26))

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g(x1)
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x2, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x3, x4)
U22_g(x1, x2)  =  U22_g(x1, x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x1, x2, x3, x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga(x2)
U4_aga(x1, x2, x3, x4)  =  U4_aga(x2, x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x1, x2, x3, x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x1, x2, x3, x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga(x2)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x2, x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x2, x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x2, x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x1, x2, x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U16_g(x1, x2, x3)  =  U16_g(x1, x2, x3)
U17_g(x1, x2, x3)  =  U17_g(x1, x2, x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x3, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x3, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x1, x2, x3, x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x1, x2, x3, x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x1, x2, x3, x4)
P1_IN_A(x1)  =  P1_IN_A
U7_A(x1, x2, x3, x4)  =  U7_A(x4)
U14_A(x1, x2, x3)  =  U14_A(x3)
U18_A(x1, x2, x3, x4)  =  U18_A(x4)
U20_A(x1, x2, x3, x4)  =  U20_A(x4)
U16_A(x1, x2, x3)  =  U16_A(x3)
U9_A(x1, x2, x3, x4)  =  U9_A(x4)
U10_A(x1, x2, x3, x4)  =  U10_A(x2, x4)
U12_A(x1, x2, x3, x4)  =  U12_A(x4)

We have to consider all (P,R,Pi)-chains

(142) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(143) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

U7_A(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_A(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_A(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_A(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_A(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_A(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_A(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → P1_IN_A(cons(T79, T26))
P1_IN_A(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_A(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
P1_IN_A(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_A(cons(T24, cons(T25, T26)))
P1_IN_A(cons(s(s(0)), cons(T92, T93))) → U14_A(T92, T93, p1_in_a(cons(T92, T93)))
U14_A(T92, T93, p1_out_a(cons(T92, T93))) → U16_A(T92, T93, mult53_in_aa(T92, T98))
U16_A(T92, T93, mult53_out_aa(T92, T98)) → P1_IN_A(cons(T98, T93))
P1_IN_A(cons(s(s(0)), cons(T92, T93))) → P1_IN_A(cons(T92, T93))
P1_IN_A(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_A(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
U18_A(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_A(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_A(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → P1_IN_A(cons(T146, T133))
P1_IN_A(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_A(cons(T131, cons(T132, T133)))
P1_IN_A(cons(0, cons(0, T153))) → P1_IN_A(T153)
U9_A(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → P1_IN_A(cons(T39, T26))

The TRS R consists of the following rules:

mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
s(x1)  =  s(x1)
0  =  0
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga(x2)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x2, x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x2, x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x2, x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
P1_IN_A(x1)  =  P1_IN_A
U7_A(x1, x2, x3, x4)  =  U7_A(x4)
U14_A(x1, x2, x3)  =  U14_A(x3)
U18_A(x1, x2, x3, x4)  =  U18_A(x4)
U20_A(x1, x2, x3, x4)  =  U20_A(x4)
U16_A(x1, x2, x3)  =  U16_A(x3)
U9_A(x1, x2, x3, x4)  =  U9_A(x4)
U10_A(x1, x2, x3, x4)  =  U10_A(x2, x4)
U12_A(x1, x2, x3, x4)  =  U12_A(x4)

We have to consider all (P,R,Pi)-chains

(144) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(145) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U7_A(p1_out_a) → U9_A(mult23_in_aaa)
U9_A(mult23_out_aaa(T25)) → U10_A(T25, p1_in_a)
U10_A(T25, p1_out_a) → U12_A(mult23_in_aga(T25))
U12_A(mult23_out_aga(T25)) → P1_IN_A
P1_IN_AU7_A(p1_in_a)
P1_IN_AP1_IN_A
P1_IN_AU14_A(p1_in_a)
U14_A(p1_out_a) → U16_A(mult53_in_aa)
U16_A(mult53_out_aa(T92)) → P1_IN_A
P1_IN_AU18_A(p1_in_a)
U18_A(p1_out_a) → U20_A(mult23_in_aaa)
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga(0)
mult23_in_aga(s(T52)) → U1_aga(T52, mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(T52, mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(T52, mult23_out_aga(T52)) → mult23_out_aga(s(T52))
U2_aga(T52, mult23_out_aga(T52)) → U3_aga(T52, sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(T52, sum34_out_aaa(T51)) → mult23_out_aga(s(T52))
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga(T25)) → p1_out_a
U12_a(mult23_out_aga(T25)) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0, x1)
U2_aga(x0, x1)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0, x1)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(146) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U7_A(p1_out_a) → U9_A(mult23_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))

(147) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U9_A(mult23_out_aaa(T25)) → U10_A(T25, p1_in_a)
U10_A(T25, p1_out_a) → U12_A(mult23_in_aga(T25))
U12_A(mult23_out_aga(T25)) → P1_IN_A
P1_IN_AU7_A(p1_in_a)
P1_IN_AP1_IN_A
P1_IN_AU14_A(p1_in_a)
U14_A(p1_out_a) → U16_A(mult53_in_aa)
U16_A(mult53_out_aa(T92)) → P1_IN_A
P1_IN_AU18_A(p1_in_a)
U18_A(p1_out_a) → U20_A(mult23_in_aaa)
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga(0)
mult23_in_aga(s(T52)) → U1_aga(T52, mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(T52, mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(T52, mult23_out_aga(T52)) → mult23_out_aga(s(T52))
U2_aga(T52, mult23_out_aga(T52)) → U3_aga(T52, sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(T52, sum34_out_aaa(T51)) → mult23_out_aga(s(T52))
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga(T25)) → p1_out_a
U12_a(mult23_out_aga(T25)) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0, x1)
U2_aga(x0, x1)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0, x1)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(148) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U9_A(mult23_out_aaa(T25)) → U10_A(T25, p1_in_a) at position [1] we obtained the following new rules [LPAR04]:

U9_A(mult23_out_aaa(y0)) → U10_A(y0, p1_out_a)
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))

(149) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U10_A(T25, p1_out_a) → U12_A(mult23_in_aga(T25))
U12_A(mult23_out_aga(T25)) → P1_IN_A
P1_IN_AU7_A(p1_in_a)
P1_IN_AP1_IN_A
P1_IN_AU14_A(p1_in_a)
U14_A(p1_out_a) → U16_A(mult53_in_aa)
U16_A(mult53_out_aa(T92)) → P1_IN_A
P1_IN_AU18_A(p1_in_a)
U18_A(p1_out_a) → U20_A(mult23_in_aaa)
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, p1_out_a)
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga(0)
mult23_in_aga(s(T52)) → U1_aga(T52, mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(T52, mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(T52, mult23_out_aga(T52)) → mult23_out_aga(s(T52))
U2_aga(T52, mult23_out_aga(T52)) → U3_aga(T52, sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(T52, sum34_out_aaa(T51)) → mult23_out_aga(s(T52))
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga(T25)) → p1_out_a
U12_a(mult23_out_aga(T25)) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0, x1)
U2_aga(x0, x1)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0, x1)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(150) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U10_A(T25, p1_out_a) → U12_A(mult23_in_aga(T25)) at position [0] we obtained the following new rules [LPAR04]:

U10_A(0, p1_out_a) → U12_A(mult23_out_aga(0))
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(x0, mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(x0, mult23_in_aga(x0)))

(151) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga(T25)) → P1_IN_A
P1_IN_AU7_A(p1_in_a)
P1_IN_AP1_IN_A
P1_IN_AU14_A(p1_in_a)
U14_A(p1_out_a) → U16_A(mult53_in_aa)
U16_A(mult53_out_aa(T92)) → P1_IN_A
P1_IN_AU18_A(p1_in_a)
U18_A(p1_out_a) → U20_A(mult23_in_aaa)
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, p1_out_a)
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga(0))
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(x0, mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(x0, mult23_in_aga(x0)))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga(0)
mult23_in_aga(s(T52)) → U1_aga(T52, mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(T52, mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(T52, mult23_out_aga(T52)) → mult23_out_aga(s(T52))
U2_aga(T52, mult23_out_aga(T52)) → U3_aga(T52, sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(T52, sum34_out_aaa(T51)) → mult23_out_aga(s(T52))
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga(T25)) → p1_out_a
U12_a(mult23_out_aga(T25)) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0, x1)
U2_aga(x0, x1)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0, x1)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(152) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P1_IN_AU7_A(p1_in_a) at position [0] we obtained the following new rules [LPAR04]:

P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))

(153) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga(T25)) → P1_IN_A
P1_IN_AP1_IN_A
P1_IN_AU14_A(p1_in_a)
U14_A(p1_out_a) → U16_A(mult53_in_aa)
U16_A(mult53_out_aa(T92)) → P1_IN_A
P1_IN_AU18_A(p1_in_a)
U18_A(p1_out_a) → U20_A(mult23_in_aaa)
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, p1_out_a)
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga(0))
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(x0, mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(x0, mult23_in_aga(x0)))
P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga(0)
mult23_in_aga(s(T52)) → U1_aga(T52, mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(T52, mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(T52, mult23_out_aga(T52)) → mult23_out_aga(s(T52))
U2_aga(T52, mult23_out_aga(T52)) → U3_aga(T52, sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(T52, sum34_out_aaa(T51)) → mult23_out_aga(s(T52))
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga(T25)) → p1_out_a
U12_a(mult23_out_aga(T25)) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0, x1)
U2_aga(x0, x1)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0, x1)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(154) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P1_IN_AU14_A(p1_in_a) at position [0] we obtained the following new rules [LPAR04]:

P1_IN_AU14_A(p1_out_a)
P1_IN_AU14_A(U7_a(p1_in_a))
P1_IN_AU14_A(U14_a(p1_in_a))
P1_IN_AU14_A(U18_a(p1_in_a))
P1_IN_AU14_A(U22_a(p1_in_a))

(155) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga(T25)) → P1_IN_A
P1_IN_AP1_IN_A
U14_A(p1_out_a) → U16_A(mult53_in_aa)
U16_A(mult53_out_aa(T92)) → P1_IN_A
P1_IN_AU18_A(p1_in_a)
U18_A(p1_out_a) → U20_A(mult23_in_aaa)
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, p1_out_a)
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga(0))
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(x0, mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(x0, mult23_in_aga(x0)))
P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))
P1_IN_AU14_A(p1_out_a)
P1_IN_AU14_A(U7_a(p1_in_a))
P1_IN_AU14_A(U14_a(p1_in_a))
P1_IN_AU14_A(U18_a(p1_in_a))
P1_IN_AU14_A(U22_a(p1_in_a))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga(0)
mult23_in_aga(s(T52)) → U1_aga(T52, mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(T52, mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(T52, mult23_out_aga(T52)) → mult23_out_aga(s(T52))
U2_aga(T52, mult23_out_aga(T52)) → U3_aga(T52, sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(T52, sum34_out_aaa(T51)) → mult23_out_aga(s(T52))
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga(T25)) → p1_out_a
U12_a(mult23_out_aga(T25)) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0, x1)
U2_aga(x0, x1)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0, x1)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(156) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U14_A(p1_out_a) → U16_A(mult53_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U14_A(p1_out_a) → U16_A(mult53_out_aa(0))
U14_A(p1_out_a) → U16_A(U5_aa(mult53_in_aa))
U14_A(p1_out_a) → U16_A(U6_aa(mult53_in_aa))

(157) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga(T25)) → P1_IN_A
P1_IN_AP1_IN_A
U16_A(mult53_out_aa(T92)) → P1_IN_A
P1_IN_AU18_A(p1_in_a)
U18_A(p1_out_a) → U20_A(mult23_in_aaa)
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, p1_out_a)
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga(0))
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(x0, mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(x0, mult23_in_aga(x0)))
P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))
P1_IN_AU14_A(p1_out_a)
P1_IN_AU14_A(U7_a(p1_in_a))
P1_IN_AU14_A(U14_a(p1_in_a))
P1_IN_AU14_A(U18_a(p1_in_a))
P1_IN_AU14_A(U22_a(p1_in_a))
U14_A(p1_out_a) → U16_A(mult53_out_aa(0))
U14_A(p1_out_a) → U16_A(U5_aa(mult53_in_aa))
U14_A(p1_out_a) → U16_A(U6_aa(mult53_in_aa))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga(0)
mult23_in_aga(s(T52)) → U1_aga(T52, mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(T52, mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(T52, mult23_out_aga(T52)) → mult23_out_aga(s(T52))
U2_aga(T52, mult23_out_aga(T52)) → U3_aga(T52, sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(T52, sum34_out_aaa(T51)) → mult23_out_aga(s(T52))
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga(T25)) → p1_out_a
U12_a(mult23_out_aga(T25)) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0, x1)
U2_aga(x0, x1)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0, x1)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(158) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule P1_IN_AU18_A(p1_in_a) at position [0] we obtained the following new rules [LPAR04]:

P1_IN_AU18_A(p1_out_a)
P1_IN_AU18_A(U7_a(p1_in_a))
P1_IN_AU18_A(U14_a(p1_in_a))
P1_IN_AU18_A(U18_a(p1_in_a))
P1_IN_AU18_A(U22_a(p1_in_a))

(159) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga(T25)) → P1_IN_A
P1_IN_AP1_IN_A
U16_A(mult53_out_aa(T92)) → P1_IN_A
U18_A(p1_out_a) → U20_A(mult23_in_aaa)
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, p1_out_a)
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga(0))
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(x0, mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(x0, mult23_in_aga(x0)))
P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))
P1_IN_AU14_A(p1_out_a)
P1_IN_AU14_A(U7_a(p1_in_a))
P1_IN_AU14_A(U14_a(p1_in_a))
P1_IN_AU14_A(U18_a(p1_in_a))
P1_IN_AU14_A(U22_a(p1_in_a))
U14_A(p1_out_a) → U16_A(mult53_out_aa(0))
U14_A(p1_out_a) → U16_A(U5_aa(mult53_in_aa))
U14_A(p1_out_a) → U16_A(U6_aa(mult53_in_aa))
P1_IN_AU18_A(p1_out_a)
P1_IN_AU18_A(U7_a(p1_in_a))
P1_IN_AU18_A(U14_a(p1_in_a))
P1_IN_AU18_A(U18_a(p1_in_a))
P1_IN_AU18_A(U22_a(p1_in_a))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga(0)
mult23_in_aga(s(T52)) → U1_aga(T52, mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(T52, mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(T52, mult23_out_aga(T52)) → mult23_out_aga(s(T52))
U2_aga(T52, mult23_out_aga(T52)) → U3_aga(T52, sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(T52, sum34_out_aaa(T51)) → mult23_out_aga(s(T52))
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga(T25)) → p1_out_a
U12_a(mult23_out_aga(T25)) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0, x1)
U2_aga(x0, x1)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0, x1)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(160) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U18_A(p1_out_a) → U20_A(mult23_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

U18_A(p1_out_a) → U20_A(mult23_out_aaa(0))
U18_A(p1_out_a) → U20_A(U1_aaa(mult23_in_aaa))
U18_A(p1_out_a) → U20_A(U2_aaa(mult23_in_aaa))

(161) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga(T25)) → P1_IN_A
P1_IN_AP1_IN_A
U16_A(mult53_out_aa(T92)) → P1_IN_A
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, p1_out_a)
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga(0))
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(x0, mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(x0, mult23_in_aga(x0)))
P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))
P1_IN_AU14_A(p1_out_a)
P1_IN_AU14_A(U7_a(p1_in_a))
P1_IN_AU14_A(U14_a(p1_in_a))
P1_IN_AU14_A(U18_a(p1_in_a))
P1_IN_AU14_A(U22_a(p1_in_a))
U14_A(p1_out_a) → U16_A(mult53_out_aa(0))
U14_A(p1_out_a) → U16_A(U5_aa(mult53_in_aa))
U14_A(p1_out_a) → U16_A(U6_aa(mult53_in_aa))
P1_IN_AU18_A(p1_out_a)
P1_IN_AU18_A(U7_a(p1_in_a))
P1_IN_AU18_A(U14_a(p1_in_a))
P1_IN_AU18_A(U18_a(p1_in_a))
P1_IN_AU18_A(U22_a(p1_in_a))
U18_A(p1_out_a) → U20_A(mult23_out_aaa(0))
U18_A(p1_out_a) → U20_A(U1_aaa(mult23_in_aaa))
U18_A(p1_out_a) → U20_A(U2_aaa(mult23_in_aaa))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga(0)
mult23_in_aga(s(T52)) → U1_aga(T52, mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(T52, mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(T52, mult23_out_aga(T52)) → mult23_out_aga(s(T52))
U2_aga(T52, mult23_out_aga(T52)) → U3_aga(T52, sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(T52, sum34_out_aaa(T51)) → mult23_out_aga(s(T52))
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga(T25)) → p1_out_a
U12_a(mult23_out_aga(T25)) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0, x1)
U2_aga(x0, x1)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0, x1)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(162) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U9_A(mult23_out_aaa(y0)) → U10_A(y0, p1_out_a) we obtained the following new rules [LPAR04]:

U9_A(mult23_out_aaa(0)) → U10_A(0, p1_out_a)
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), p1_out_a)

(163) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga(T25)) → P1_IN_A
P1_IN_AP1_IN_A
U16_A(mult53_out_aa(T92)) → P1_IN_A
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga(0))
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(x0, mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(x0, mult23_in_aga(x0)))
P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))
P1_IN_AU14_A(p1_out_a)
P1_IN_AU14_A(U7_a(p1_in_a))
P1_IN_AU14_A(U14_a(p1_in_a))
P1_IN_AU14_A(U18_a(p1_in_a))
P1_IN_AU14_A(U22_a(p1_in_a))
U14_A(p1_out_a) → U16_A(mult53_out_aa(0))
U14_A(p1_out_a) → U16_A(U5_aa(mult53_in_aa))
U14_A(p1_out_a) → U16_A(U6_aa(mult53_in_aa))
P1_IN_AU18_A(p1_out_a)
P1_IN_AU18_A(U7_a(p1_in_a))
P1_IN_AU18_A(U14_a(p1_in_a))
P1_IN_AU18_A(U18_a(p1_in_a))
P1_IN_AU18_A(U22_a(p1_in_a))
U18_A(p1_out_a) → U20_A(mult23_out_aaa(0))
U18_A(p1_out_a) → U20_A(U1_aaa(mult23_in_aaa))
U18_A(p1_out_a) → U20_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(0)) → U10_A(0, p1_out_a)
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), p1_out_a)

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga(0)
mult23_in_aga(s(T52)) → U1_aga(T52, mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(T52, mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(T52, mult23_out_aga(T52)) → mult23_out_aga(s(T52))
U2_aga(T52, mult23_out_aga(T52)) → U3_aga(T52, sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(T52, sum34_out_aaa(T51)) → mult23_out_aga(s(T52))
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga(T25)) → p1_out_a
U12_a(mult23_out_aga(T25)) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0, x1)
U2_aga(x0, x1)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0, x1)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(164) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U9_A(mult23_out_aaa(y0)) → U10_A(y0, U7_a(p1_in_a)) we obtained the following new rules [LPAR04]:

U9_A(mult23_out_aaa(0)) → U10_A(0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U7_a(p1_in_a))

(165) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga(T25)) → P1_IN_A
P1_IN_AP1_IN_A
U16_A(mult53_out_aa(T92)) → P1_IN_A
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga(0))
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(x0, mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(x0, mult23_in_aga(x0)))
P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))
P1_IN_AU14_A(p1_out_a)
P1_IN_AU14_A(U7_a(p1_in_a))
P1_IN_AU14_A(U14_a(p1_in_a))
P1_IN_AU14_A(U18_a(p1_in_a))
P1_IN_AU14_A(U22_a(p1_in_a))
U14_A(p1_out_a) → U16_A(mult53_out_aa(0))
U14_A(p1_out_a) → U16_A(U5_aa(mult53_in_aa))
U14_A(p1_out_a) → U16_A(U6_aa(mult53_in_aa))
P1_IN_AU18_A(p1_out_a)
P1_IN_AU18_A(U7_a(p1_in_a))
P1_IN_AU18_A(U14_a(p1_in_a))
P1_IN_AU18_A(U18_a(p1_in_a))
P1_IN_AU18_A(U22_a(p1_in_a))
U18_A(p1_out_a) → U20_A(mult23_out_aaa(0))
U18_A(p1_out_a) → U20_A(U1_aaa(mult23_in_aaa))
U18_A(p1_out_a) → U20_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(0)) → U10_A(0, p1_out_a)
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), p1_out_a)
U9_A(mult23_out_aaa(0)) → U10_A(0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U7_a(p1_in_a))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga(0)
mult23_in_aga(s(T52)) → U1_aga(T52, mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(T52, mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(T52, mult23_out_aga(T52)) → mult23_out_aga(s(T52))
U2_aga(T52, mult23_out_aga(T52)) → U3_aga(T52, sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(T52, sum34_out_aaa(T51)) → mult23_out_aga(s(T52))
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga(T25)) → p1_out_a
U12_a(mult23_out_aga(T25)) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0, x1)
U2_aga(x0, x1)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0, x1)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(166) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U9_A(mult23_out_aaa(y0)) → U10_A(y0, U14_a(p1_in_a)) we obtained the following new rules [LPAR04]:

U9_A(mult23_out_aaa(0)) → U10_A(0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U14_a(p1_in_a))

(167) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga(T25)) → P1_IN_A
P1_IN_AP1_IN_A
U16_A(mult53_out_aa(T92)) → P1_IN_A
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga(0))
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(x0, mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(x0, mult23_in_aga(x0)))
P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))
P1_IN_AU14_A(p1_out_a)
P1_IN_AU14_A(U7_a(p1_in_a))
P1_IN_AU14_A(U14_a(p1_in_a))
P1_IN_AU14_A(U18_a(p1_in_a))
P1_IN_AU14_A(U22_a(p1_in_a))
U14_A(p1_out_a) → U16_A(mult53_out_aa(0))
U14_A(p1_out_a) → U16_A(U5_aa(mult53_in_aa))
U14_A(p1_out_a) → U16_A(U6_aa(mult53_in_aa))
P1_IN_AU18_A(p1_out_a)
P1_IN_AU18_A(U7_a(p1_in_a))
P1_IN_AU18_A(U14_a(p1_in_a))
P1_IN_AU18_A(U18_a(p1_in_a))
P1_IN_AU18_A(U22_a(p1_in_a))
U18_A(p1_out_a) → U20_A(mult23_out_aaa(0))
U18_A(p1_out_a) → U20_A(U1_aaa(mult23_in_aaa))
U18_A(p1_out_a) → U20_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(0)) → U10_A(0, p1_out_a)
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), p1_out_a)
U9_A(mult23_out_aaa(0)) → U10_A(0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U7_a(p1_in_a))
U9_A(mult23_out_aaa(0)) → U10_A(0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U14_a(p1_in_a))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga(0)
mult23_in_aga(s(T52)) → U1_aga(T52, mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(T52, mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(T52, mult23_out_aga(T52)) → mult23_out_aga(s(T52))
U2_aga(T52, mult23_out_aga(T52)) → U3_aga(T52, sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(T52, sum34_out_aaa(T51)) → mult23_out_aga(s(T52))
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga(T25)) → p1_out_a
U12_a(mult23_out_aga(T25)) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0, x1)
U2_aga(x0, x1)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0, x1)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(168) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U9_A(mult23_out_aaa(y0)) → U10_A(y0, U18_a(p1_in_a)) we obtained the following new rules [LPAR04]:

U9_A(mult23_out_aaa(0)) → U10_A(0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U18_a(p1_in_a))

(169) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga(T25)) → P1_IN_A
P1_IN_AP1_IN_A
U16_A(mult53_out_aa(T92)) → P1_IN_A
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga(0))
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(x0, mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(x0, mult23_in_aga(x0)))
P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))
P1_IN_AU14_A(p1_out_a)
P1_IN_AU14_A(U7_a(p1_in_a))
P1_IN_AU14_A(U14_a(p1_in_a))
P1_IN_AU14_A(U18_a(p1_in_a))
P1_IN_AU14_A(U22_a(p1_in_a))
U14_A(p1_out_a) → U16_A(mult53_out_aa(0))
U14_A(p1_out_a) → U16_A(U5_aa(mult53_in_aa))
U14_A(p1_out_a) → U16_A(U6_aa(mult53_in_aa))
P1_IN_AU18_A(p1_out_a)
P1_IN_AU18_A(U7_a(p1_in_a))
P1_IN_AU18_A(U14_a(p1_in_a))
P1_IN_AU18_A(U18_a(p1_in_a))
P1_IN_AU18_A(U22_a(p1_in_a))
U18_A(p1_out_a) → U20_A(mult23_out_aaa(0))
U18_A(p1_out_a) → U20_A(U1_aaa(mult23_in_aaa))
U18_A(p1_out_a) → U20_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(0)) → U10_A(0, p1_out_a)
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), p1_out_a)
U9_A(mult23_out_aaa(0)) → U10_A(0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U7_a(p1_in_a))
U9_A(mult23_out_aaa(0)) → U10_A(0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U14_a(p1_in_a))
U9_A(mult23_out_aaa(0)) → U10_A(0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U18_a(p1_in_a))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga(0)
mult23_in_aga(s(T52)) → U1_aga(T52, mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(T52, mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(T52, mult23_out_aga(T52)) → mult23_out_aga(s(T52))
U2_aga(T52, mult23_out_aga(T52)) → U3_aga(T52, sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(T52, sum34_out_aaa(T51)) → mult23_out_aga(s(T52))
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga(T25)) → p1_out_a
U12_a(mult23_out_aga(T25)) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0, x1)
U2_aga(x0, x1)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0, x1)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(170) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U9_A(mult23_out_aaa(y0)) → U10_A(y0, U22_a(p1_in_a)) we obtained the following new rules [LPAR04]:

U9_A(mult23_out_aaa(0)) → U10_A(0, U22_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U22_a(p1_in_a))

(171) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_A(mult23_out_aga(T25)) → P1_IN_A
P1_IN_AP1_IN_A
U16_A(mult53_out_aa(T92)) → P1_IN_A
U20_A(mult23_out_aaa(T132)) → P1_IN_A
U9_A(mult23_out_aaa(T25)) → P1_IN_A
U7_A(p1_out_a) → U9_A(mult23_out_aaa(0))
U7_A(p1_out_a) → U9_A(U1_aaa(mult23_in_aaa))
U7_A(p1_out_a) → U9_A(U2_aaa(mult23_in_aaa))
U10_A(0, p1_out_a) → U12_A(mult23_out_aga(0))
U10_A(s(x0), p1_out_a) → U12_A(U1_aga(x0, mult23_in_aga(x0)))
U10_A(s(x0), p1_out_a) → U12_A(U2_aga(x0, mult23_in_aga(x0)))
P1_IN_AU7_A(p1_out_a)
P1_IN_AU7_A(U7_a(p1_in_a))
P1_IN_AU7_A(U14_a(p1_in_a))
P1_IN_AU7_A(U18_a(p1_in_a))
P1_IN_AU7_A(U22_a(p1_in_a))
P1_IN_AU14_A(p1_out_a)
P1_IN_AU14_A(U7_a(p1_in_a))
P1_IN_AU14_A(U14_a(p1_in_a))
P1_IN_AU14_A(U18_a(p1_in_a))
P1_IN_AU14_A(U22_a(p1_in_a))
U14_A(p1_out_a) → U16_A(mult53_out_aa(0))
U14_A(p1_out_a) → U16_A(U5_aa(mult53_in_aa))
U14_A(p1_out_a) → U16_A(U6_aa(mult53_in_aa))
P1_IN_AU18_A(p1_out_a)
P1_IN_AU18_A(U7_a(p1_in_a))
P1_IN_AU18_A(U14_a(p1_in_a))
P1_IN_AU18_A(U18_a(p1_in_a))
P1_IN_AU18_A(U22_a(p1_in_a))
U18_A(p1_out_a) → U20_A(mult23_out_aaa(0))
U18_A(p1_out_a) → U20_A(U1_aaa(mult23_in_aaa))
U18_A(p1_out_a) → U20_A(U2_aaa(mult23_in_aaa))
U9_A(mult23_out_aaa(0)) → U10_A(0, p1_out_a)
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), p1_out_a)
U9_A(mult23_out_aaa(0)) → U10_A(0, U7_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U7_a(p1_in_a))
U9_A(mult23_out_aaa(0)) → U10_A(0, U14_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U14_a(p1_in_a))
U9_A(mult23_out_aaa(0)) → U10_A(0, U18_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U18_a(p1_in_a))
U9_A(mult23_out_aaa(0)) → U10_A(0, U22_a(p1_in_a))
U9_A(mult23_out_aaa(s(y_0))) → U10_A(s(y_0), U22_a(p1_in_a))

The TRS R consists of the following rules:

mult23_in_aaamult23_out_aaa(0)
mult23_in_aaaU1_aaa(mult23_in_aaa)
mult23_in_aaaU2_aaa(mult23_in_aaa)
p1_in_ap1_out_a
p1_in_aU7_a(p1_in_a)
p1_in_aU14_a(p1_in_a)
p1_in_aU18_a(p1_in_a)
p1_in_aU22_a(p1_in_a)
mult23_in_aga(0) → mult23_out_aga(0)
mult23_in_aga(s(T52)) → U1_aga(T52, mult23_in_aga(T52))
mult23_in_aga(s(T52)) → U2_aga(T52, mult23_in_aga(T52))
mult53_in_aamult53_out_aa(0)
mult53_in_aaU5_aa(mult53_in_aa)
mult53_in_aaU6_aa(mult53_in_aa)
U1_aaa(mult23_out_aaa(T52)) → mult23_out_aaa(s(T52))
U2_aaa(mult23_out_aaa(T52)) → U3_aaa(T52, sum34_in_aaa)
U7_a(p1_out_a) → p1_out_a
U7_a(p1_out_a) → U8_a(mult23_in_aaa)
U7_a(p1_out_a) → U9_a(mult23_in_aaa)
U14_a(p1_out_a) → p1_out_a
U14_a(p1_out_a) → U15_a(mult53_in_aa)
U14_a(p1_out_a) → U16_a(mult53_in_aa)
U18_a(p1_out_a) → p1_out_a
U18_a(p1_out_a) → U19_a(mult23_in_aaa)
U18_a(p1_out_a) → U20_a(mult23_in_aaa)
U22_a(p1_out_a) → p1_out_a
U1_aga(T52, mult23_out_aga(T52)) → mult23_out_aga(s(T52))
U2_aga(T52, mult23_out_aga(T52)) → U3_aga(T52, sum34_in_aaa)
U5_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U6_aa(mult53_out_aa(T101)) → mult53_out_aa(s(T101))
U3_aaa(T52, sum34_out_aaa(T51)) → mult23_out_aaa(s(T52))
U8_a(mult23_out_aaa(T25)) → p1_out_a
U9_a(mult23_out_aaa(T25)) → U10_a(T25, p1_in_a)
U15_a(mult53_out_aa(T92)) → p1_out_a
U16_a(mult53_out_aa(T92)) → U17_a(p1_in_a)
U19_a(mult23_out_aaa(T132)) → p1_out_a
U20_a(mult23_out_aaa(T132)) → U21_a(p1_in_a)
U3_aga(T52, sum34_out_aaa(T51)) → mult23_out_aga(s(T52))
sum34_in_aaasum34_out_aaa(0)
sum34_in_aaaU4_aaa(sum34_in_aaa)
U10_a(T25, p1_out_a) → p1_out_a
U10_a(T25, p1_out_a) → U11_a(mult23_in_aga(T25))
U10_a(T25, p1_out_a) → U12_a(mult23_in_aga(T25))
U17_a(p1_out_a) → p1_out_a
U21_a(p1_out_a) → p1_out_a
U4_aaa(sum34_out_aaa(T70)) → sum34_out_aaa(s(T70))
U11_a(mult23_out_aga(T25)) → p1_out_a
U12_a(mult23_out_aga(T25)) → U13_a(p1_in_a)
U13_a(p1_out_a) → p1_out_a

The set Q consists of the following terms:

mult23_in_aaa
p1_in_a
mult23_in_aga(x0)
mult53_in_aa
U1_aaa(x0)
U2_aaa(x0)
U7_a(x0)
U14_a(x0)
U18_a(x0)
U22_a(x0)
U1_aga(x0, x1)
U2_aga(x0, x1)
U5_aa(x0)
U6_aa(x0)
U3_aaa(x0, x1)
U8_a(x0)
U9_a(x0)
U15_a(x0)
U16_a(x0)
U19_a(x0)
U20_a(x0)
U3_aga(x0, x1)
sum34_in_aaa
U10_a(x0, x1)
U17_a(x0)
U21_a(x0)
U4_aaa(x0)
U11_a(x0)
U12_a(x0)
U13_a(x0)

We have to consider all (P,Q,R)-chains.

(172) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = P1_IN_A evaluates to t =P1_IN_A

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Matcher: [ ]
  • Semiunifier: [ ]




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from P1_IN_A to P1_IN_A.



(173) NO

(174) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SUM34_IN_AGA(T69, s(T70), s(X110)) → SUM34_IN_AGA(T69, T70, X110)

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g(x1)
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x2, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x3, x4)
U22_g(x1, x2)  =  U22_g(x1, x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x1, x2, x3, x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga(x2)
U4_aga(x1, x2, x3, x4)  =  U4_aga(x2, x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x1, x2, x3, x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x1, x2, x3, x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga(x2)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x2, x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x2, x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x2, x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x1, x2, x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U16_g(x1, x2, x3)  =  U16_g(x1, x2, x3)
U17_g(x1, x2, x3)  =  U17_g(x1, x2, x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x3, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x3, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x1, x2, x3, x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x1, x2, x3, x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x1, x2, x3, x4)
SUM34_IN_AGA(x1, x2, x3)  =  SUM34_IN_AGA(x2)

We have to consider all (P,R,Pi)-chains

(175) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(176) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SUM34_IN_AGA(T69, s(T70), s(X110)) → SUM34_IN_AGA(T69, T70, X110)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
SUM34_IN_AGA(x1, x2, x3)  =  SUM34_IN_AGA(x2)

We have to consider all (P,R,Pi)-chains

(177) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(178) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SUM34_IN_AGA(s(T70)) → SUM34_IN_AGA(T70)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(179) QDPSizeChangeProof (EQUIVALENT transformation)

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.

From the DPs we obtained the following set of size-change graphs:

  • SUM34_IN_AGA(s(T70)) → SUM34_IN_AGA(T70)
    The graph contains the following edges 1 > 1

(180) YES

(181) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT23_IN_GGA(T51, s(T52), X83) → MULT23_IN_GGA(T51, T52, X82)

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g(x1)
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x2, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x3, x4)
U22_g(x1, x2)  =  U22_g(x1, x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x1, x2, x3, x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga(x2)
U4_aga(x1, x2, x3, x4)  =  U4_aga(x2, x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x1, x2, x3, x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x1, x2, x3, x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga(x2)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x2, x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x2, x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x2, x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x1, x2, x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U16_g(x1, x2, x3)  =  U16_g(x1, x2, x3)
U17_g(x1, x2, x3)  =  U17_g(x1, x2, x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x3, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x3, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x1, x2, x3, x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x1, x2, x3, x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x1, x2, x3, x4)
MULT23_IN_GGA(x1, x2, x3)  =  MULT23_IN_GGA(x1, x2)

We have to consider all (P,R,Pi)-chains

(182) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(183) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MULT23_IN_GGA(T51, s(T52), X83) → MULT23_IN_GGA(T51, T52, X82)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
MULT23_IN_GGA(x1, x2, x3)  =  MULT23_IN_GGA(x1, x2)

We have to consider all (P,R,Pi)-chains

(184) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(185) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MULT23_IN_GGA(T51, s(T52)) → MULT23_IN_GGA(T51, T52)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(186) QDPSizeChangeProof (EQUIVALENT transformation)

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.

From the DPs we obtained the following set of size-change graphs:

  • MULT23_IN_GGA(T51, s(T52)) → MULT23_IN_GGA(T51, T52)
    The graph contains the following edges 1 >= 1, 2 > 2

(187) YES

(188) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

P1_IN_G(cons(s(s(0)), cons(T92, T93))) → P1_IN_G(cons(T92, T93))
P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_G(cons(T24, cons(T25, T26)))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_G(cons(T131, cons(T132, T133)))
P1_IN_G(cons(0, cons(0, T153))) → P1_IN_G(T153)

The TRS R consists of the following rules:

p1_in_g(cons(T3, nil)) → p1_out_g(cons(T3, nil))
p1_in_g(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_g(T24, T25, T26, p1_in_g(cons(T24, cons(T25, T26))))
p1_in_g(cons(s(s(0)), cons(T92, T93))) → U14_g(T92, T93, p1_in_g(cons(T92, T93)))
p1_in_g(cons(0, cons(T118, nil))) → p1_out_g(cons(0, cons(T118, nil)))
p1_in_g(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_g(T131, T132, T133, p1_in_g(cons(T131, cons(T132, T133))))
p1_in_g(cons(0, cons(0, T153))) → U22_g(T153, p1_in_g(T153))
U22_g(T153, p1_out_g(T153)) → p1_out_g(cons(0, cons(0, T153)))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U19_g(T131, T132, T133, mult23_in_gga(T131, T132, X205))
mult23_in_gga(T46, 0, 0) → mult23_out_gga(T46, 0, 0)
mult23_in_gga(T51, s(T52), X83) → U1_gga(T51, T52, X83, mult23_in_gga(T51, T52, X82))
mult23_in_gga(T51, s(T52), X83) → U2_gga(T51, T52, X83, mult23_in_gga(T51, T52, T55))
U2_gga(T51, T52, X83, mult23_out_gga(T51, T52, T55)) → U3_gga(T51, T52, X83, sum34_in_aga(T55, T51, X83))
sum34_in_aga(T64, 0, T64) → sum34_out_aga(T64, 0, T64)
sum34_in_aga(T69, s(T70), s(X110)) → U4_aga(T69, T70, X110, sum34_in_aga(T69, T70, X110))
U4_aga(T69, T70, X110, sum34_out_aga(T69, T70, X110)) → sum34_out_aga(T69, s(T70), s(X110))
U3_gga(T51, T52, X83, sum34_out_aga(T55, T51, X83)) → mult23_out_gga(T51, s(T52), X83)
U1_gga(T51, T52, X83, mult23_out_gga(T51, T52, X82)) → mult23_out_gga(T51, s(T52), X83)
U19_g(T131, T132, T133, mult23_out_gga(T131, T132, X205)) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_g(T131, T132, T133, p1_out_g(cons(T131, cons(T132, T133)))) → U20_g(T131, T132, T133, mult23_in_gga(T131, T132, T146))
U20_g(T131, T132, T133, mult23_out_gga(T131, T132, T146)) → U21_g(T131, T132, T133, p1_in_a(cons(T146, T133)))
p1_in_a(cons(T3, nil)) → p1_out_a(cons(T3, nil))
p1_in_a(cons(s(s(s(s(T24)))), cons(T25, T26))) → U7_a(T24, T25, T26, p1_in_a(cons(T24, cons(T25, T26))))
p1_in_a(cons(s(s(0)), cons(T92, T93))) → U14_a(T92, T93, p1_in_a(cons(T92, T93)))
p1_in_a(cons(0, cons(T118, nil))) → p1_out_a(cons(0, cons(T118, nil)))
p1_in_a(cons(0, cons(s(s(T131)), cons(T132, T133)))) → U18_a(T131, T132, T133, p1_in_a(cons(T131, cons(T132, T133))))
p1_in_a(cons(0, cons(0, T153))) → U22_a(T153, p1_in_a(T153))
U22_a(T153, p1_out_a(T153)) → p1_out_a(cons(0, cons(0, T153)))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U19_a(T131, T132, T133, mult23_in_aaa(T131, T132, X205))
mult23_in_aaa(T46, 0, 0) → mult23_out_aaa(T46, 0, 0)
mult23_in_aaa(T51, s(T52), X83) → U1_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, X82))
mult23_in_aaa(T51, s(T52), X83) → U2_aaa(T51, T52, X83, mult23_in_aaa(T51, T52, T55))
U2_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, T55)) → U3_aaa(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
sum34_in_aaa(T64, 0, T64) → sum34_out_aaa(T64, 0, T64)
sum34_in_aaa(T69, s(T70), s(X110)) → U4_aaa(T69, T70, X110, sum34_in_aaa(T69, T70, X110))
U4_aaa(T69, T70, X110, sum34_out_aaa(T69, T70, X110)) → sum34_out_aaa(T69, s(T70), s(X110))
U3_aaa(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aaa(T51, s(T52), X83)
U1_aaa(T51, T52, X83, mult23_out_aaa(T51, T52, X82)) → mult23_out_aaa(T51, s(T52), X83)
U19_a(T131, T132, T133, mult23_out_aaa(T131, T132, X205)) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U18_a(T131, T132, T133, p1_out_a(cons(T131, cons(T132, T133)))) → U20_a(T131, T132, T133, mult23_in_aaa(T131, T132, T146))
U20_a(T131, T132, T133, mult23_out_aaa(T131, T132, T146)) → U21_a(T131, T132, T133, p1_in_a(cons(T146, T133)))
U21_a(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_a(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U15_a(T92, T93, mult53_in_aa(T92, X22))
mult53_in_aa(0, 0) → mult53_out_aa(0, 0)
mult53_in_aa(s(T101), X159) → U5_aa(T101, X159, mult53_in_aa(T101, X158))
mult53_in_aa(s(T101), T108) → U6_aa(T101, T108, mult53_in_aa(T101, T108))
U6_aa(T101, T108, mult53_out_aa(T101, T108)) → mult53_out_aa(s(T101), T108)
U5_aa(T101, X159, mult53_out_aa(T101, X158)) → mult53_out_aa(s(T101), X159)
U15_a(T92, T93, mult53_out_aa(T92, X22)) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U14_a(T92, T93, p1_out_a(cons(T92, T93))) → U16_a(T92, T93, mult53_in_aa(T92, T98))
U16_a(T92, T93, mult53_out_aa(T92, T98)) → U17_a(T92, T93, p1_in_a(cons(T98, T93)))
U17_a(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_a(cons(s(s(0)), cons(T92, T93)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U8_a(T24, T25, T26, mult23_in_aaa(T24, T25, X44))
U8_a(T24, T25, T26, mult23_out_aaa(T24, T25, X44)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_a(T24, T25, T26, p1_out_a(cons(T24, cons(T25, T26)))) → U9_a(T24, T25, T26, mult23_in_aaa(T24, T25, T39))
U9_a(T24, T25, T26, mult23_out_aaa(T24, T25, T39)) → U10_a(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, X22))
mult23_in_aga(T46, 0, 0) → mult23_out_aga(T46, 0, 0)
mult23_in_aga(T51, s(T52), X83) → U1_aga(T51, T52, X83, mult23_in_aga(T51, T52, X82))
mult23_in_aga(T51, s(T52), X83) → U2_aga(T51, T52, X83, mult23_in_aga(T51, T52, T55))
U2_aga(T51, T52, X83, mult23_out_aga(T51, T52, T55)) → U3_aga(T51, T52, X83, sum34_in_aaa(T55, T51, X83))
U3_aga(T51, T52, X83, sum34_out_aaa(T55, T51, X83)) → mult23_out_aga(T51, s(T52), X83)
U1_aga(T51, T52, X83, mult23_out_aga(T51, T52, X82)) → mult23_out_aga(T51, s(T52), X83)
U11_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, X22)) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_a(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_a(T24, T25, T26, mult23_in_aga(s(s(T24)), T25, T79))
U12_a(T24, T25, T26, mult23_out_aga(s(s(T24)), T25, T79)) → U13_a(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_a(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_a(cons(s(s(s(s(T24)))), cons(T25, T26)))
U21_g(T131, T132, T133, p1_out_a(cons(T146, T133))) → p1_out_g(cons(0, cons(s(s(T131)), cons(T132, T133))))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U15_g(T92, T93, mult53_in_ga(T92, X22))
mult53_in_ga(0, 0) → mult53_out_ga(0, 0)
mult53_in_ga(s(T101), X159) → U5_ga(T101, X159, mult53_in_ga(T101, X158))
mult53_in_ga(s(T101), T108) → U6_ga(T101, T108, mult53_in_ga(T101, T108))
U6_ga(T101, T108, mult53_out_ga(T101, T108)) → mult53_out_ga(s(T101), T108)
U5_ga(T101, X159, mult53_out_ga(T101, X158)) → mult53_out_ga(s(T101), X159)
U15_g(T92, T93, mult53_out_ga(T92, X22)) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U14_g(T92, T93, p1_out_g(cons(T92, T93))) → U16_g(T92, T93, mult53_in_ga(T92, T98))
U16_g(T92, T93, mult53_out_ga(T92, T98)) → U17_g(T92, T93, p1_in_a(cons(T98, T93)))
U17_g(T92, T93, p1_out_a(cons(T98, T93))) → p1_out_g(cons(s(s(0)), cons(T92, T93)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U8_g(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U8_g(T24, T25, T26, mult23_out_gga(T24, T25, X44)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U7_g(T24, T25, T26, p1_out_g(cons(T24, cons(T25, T26)))) → U9_g(T24, T25, T26, mult23_in_gga(T24, T25, T39))
U9_g(T24, T25, T26, mult23_out_gga(T24, T25, T39)) → U10_g(T24, T25, T26, p1_in_a(cons(T39, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U11_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, X22)) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))
U10_g(T24, T25, T26, p1_out_a(cons(T39, T26))) → U12_g(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, T79))
U12_g(T24, T25, T26, mult23_out_gga(s(s(T24)), T25, T79)) → U13_g(T24, T25, T26, p1_in_a(cons(T79, T26)))
U13_g(T24, T25, T26, p1_out_a(cons(T79, T26))) → p1_out_g(cons(s(s(s(s(T24)))), cons(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
p1_out_g(x1)  =  p1_out_g(x1)
s(x1)  =  s(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
0  =  0
U14_g(x1, x2, x3)  =  U14_g(x1, x2, x3)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x2, x3, x4)
U22_g(x1, x2)  =  U22_g(x1, x2)
U19_g(x1, x2, x3, x4)  =  U19_g(x1, x2, x3, x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
mult23_out_gga(x1, x2, x3)  =  mult23_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
sum34_in_aga(x1, x2, x3)  =  sum34_in_aga(x2)
sum34_out_aga(x1, x2, x3)  =  sum34_out_aga(x2)
U4_aga(x1, x2, x3, x4)  =  U4_aga(x2, x4)
U20_g(x1, x2, x3, x4)  =  U20_g(x1, x2, x3, x4)
U21_g(x1, x2, x3, x4)  =  U21_g(x1, x2, x3, x4)
p1_in_a(x1)  =  p1_in_a
p1_out_a(x1)  =  p1_out_a
U7_a(x1, x2, x3, x4)  =  U7_a(x4)
U14_a(x1, x2, x3)  =  U14_a(x3)
U18_a(x1, x2, x3, x4)  =  U18_a(x4)
U22_a(x1, x2)  =  U22_a(x2)
U19_a(x1, x2, x3, x4)  =  U19_a(x4)
mult23_in_aaa(x1, x2, x3)  =  mult23_in_aaa
mult23_out_aaa(x1, x2, x3)  =  mult23_out_aaa(x2)
U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
U2_aaa(x1, x2, x3, x4)  =  U2_aaa(x4)
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x2, x4)
sum34_in_aaa(x1, x2, x3)  =  sum34_in_aaa
sum34_out_aaa(x1, x2, x3)  =  sum34_out_aaa(x2)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
U20_a(x1, x2, x3, x4)  =  U20_a(x4)
U21_a(x1, x2, x3, x4)  =  U21_a(x4)
U15_a(x1, x2, x3)  =  U15_a(x3)
mult53_in_aa(x1, x2)  =  mult53_in_aa
mult53_out_aa(x1, x2)  =  mult53_out_aa(x1)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U16_a(x1, x2, x3)  =  U16_a(x3)
U17_a(x1, x2, x3)  =  U17_a(x3)
U8_a(x1, x2, x3, x4)  =  U8_a(x4)
U9_a(x1, x2, x3, x4)  =  U9_a(x4)
U10_a(x1, x2, x3, x4)  =  U10_a(x2, x4)
U11_a(x1, x2, x3, x4)  =  U11_a(x4)
mult23_in_aga(x1, x2, x3)  =  mult23_in_aga(x2)
mult23_out_aga(x1, x2, x3)  =  mult23_out_aga(x2)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x2, x4)
U2_aga(x1, x2, x3, x4)  =  U2_aga(x2, x4)
U3_aga(x1, x2, x3, x4)  =  U3_aga(x2, x4)
U12_a(x1, x2, x3, x4)  =  U12_a(x4)
U13_a(x1, x2, x3, x4)  =  U13_a(x4)
U15_g(x1, x2, x3)  =  U15_g(x1, x2, x3)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
mult53_out_ga(x1, x2)  =  mult53_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
U16_g(x1, x2, x3)  =  U16_g(x1, x2, x3)
U17_g(x1, x2, x3)  =  U17_g(x1, x2, x3)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
U9_g(x1, x2, x3, x4)  =  U9_g(x1, x2, x3, x4)
U10_g(x1, x2, x3, x4)  =  U10_g(x1, x2, x3, x4)
U11_g(x1, x2, x3, x4)  =  U11_g(x1, x2, x3, x4)
U12_g(x1, x2, x3, x4)  =  U12_g(x1, x2, x3, x4)
U13_g(x1, x2, x3, x4)  =  U13_g(x1, x2, x3, x4)
P1_IN_G(x1)  =  P1_IN_G(x1)

We have to consider all (P,R,Pi)-chains

(189) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(190) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

P1_IN_G(cons(s(s(0)), cons(T92, T93))) → P1_IN_G(cons(T92, T93))
P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_G(cons(T24, cons(T25, T26)))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_G(cons(T131, cons(T132, T133)))
P1_IN_G(cons(0, cons(0, T153))) → P1_IN_G(T153)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains

(191) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(192) Obligation:

Q DP problem:
The TRS P consists of the following rules:

P1_IN_G(cons(s(s(0)), cons(T92, T93))) → P1_IN_G(cons(T92, T93))
P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_G(cons(T24, cons(T25, T26)))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_G(cons(T131, cons(T132, T133)))
P1_IN_G(cons(0, cons(0, T153))) → P1_IN_G(T153)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(193) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

P1_IN_G(cons(s(s(0)), cons(T92, T93))) → P1_IN_G(cons(T92, T93))
P1_IN_G(cons(s(s(s(s(T24)))), cons(T25, T26))) → P1_IN_G(cons(T24, cons(T25, T26)))
P1_IN_G(cons(0, cons(s(s(T131)), cons(T132, T133)))) → P1_IN_G(cons(T131, cons(T132, T133)))
P1_IN_G(cons(0, cons(0, T153))) → P1_IN_G(T153)
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(0) = 0   
POL(P1_IN_G(x1)) = 2·x1   
POL(cons(x1, x2)) = 2·x1 + 2·x2   
POL(s(x1)) = 2·x1   

(194) Obligation:

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(195) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(196) YES