(0) Obligation:

Clauses:

p(.(X, [])).
p(.(s(s(X)), .(Y, Xs))) :- ','(p(.(X, .(Y, Xs))), ','(mult(X, Y, Z), p(.(Z, Xs)))).
p(.(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) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

mult23(T51, s(T52), X83) :- mult23(T51, T52, X82).
mult23(T51, s(T52), X83) :- ','(multc23(T51, T52, T55), sum34(T55, T51, X83)).
sum34(T69, s(T70), s(X110)) :- sum34(T69, T70, X110).
mult53(s(T101), X159) :- mult53(T101, X158).
p1(.(s(s(s(s(T24)))), .(T25, T26))) :- p1(.(T24, .(T25, T26))).
p1(.(s(s(s(s(T24)))), .(T25, T26))) :- ','(pc1(.(T24, .(T25, T26))), mult23(T24, T25, X44)).
p1(.(s(s(s(s(T24)))), .(T25, T26))) :- ','(pc1(.(T24, .(T25, T26))), ','(multc23(T24, T25, T39), p1(.(T39, T26)))).
p1(.(s(s(s(s(T24)))), .(T25, T26))) :- ','(pc1(.(T24, .(T25, T26))), ','(multc23(T24, T25, T39), ','(pc1(.(T39, T26)), mult23(s(s(T24)), T25, X22)))).
p1(.(s(s(s(s(T24)))), .(T25, T26))) :- ','(pc1(.(T24, .(T25, T26))), ','(multc23(T24, T25, T39), ','(pc1(.(T39, T26)), ','(multc23(s(s(T24)), T25, T79), p1(.(T79, T26)))))).
p1(.(s(s(0)), .(T92, T93))) :- p1(.(T92, T93)).
p1(.(s(s(0)), .(T92, T93))) :- ','(pc1(.(T92, T93)), mult53(T92, X22)).
p1(.(s(s(0)), .(T92, T93))) :- ','(pc1(.(T92, T93)), ','(multc53(T92, T98), p1(.(T98, T93)))).
p1(.(0, .(s(s(T131)), .(T132, T133)))) :- p1(.(T131, .(T132, T133))).
p1(.(0, .(s(s(T131)), .(T132, T133)))) :- ','(pc1(.(T131, .(T132, T133))), mult23(T131, T132, X205)).
p1(.(0, .(s(s(T131)), .(T132, T133)))) :- ','(pc1(.(T131, .(T132, T133))), ','(multc23(T131, T132, T146), p1(.(T146, T133)))).
p1(.(0, .(0, T153))) :- p1(T153).

Clauses:

pc1(.(T3, [])).
pc1(.(s(s(s(s(T24)))), .(T25, T26))) :- ','(pc1(.(T24, .(T25, T26))), ','(multc23(T24, T25, T39), ','(pc1(.(T39, T26)), ','(multc23(s(s(T24)), T25, T79), pc1(.(T79, T26)))))).
pc1(.(s(s(0)), .(T92, T93))) :- ','(pc1(.(T92, T93)), ','(multc53(T92, T98), pc1(.(T98, T93)))).
pc1(.(0, .(T118, []))).
pc1(.(0, .(s(s(T131)), .(T132, T133)))) :- ','(pc1(.(T131, .(T132, T133))), ','(multc23(T131, T132, T146), pc1(.(T146, T133)))).
pc1(.(0, .(0, T153))) :- pc1(T153).
multc23(T46, 0, 0).
multc23(T51, s(T52), X83) :- ','(multc23(T51, T52, T55), sumc34(T55, T51, X83)).
sumc34(T64, 0, T64).
sumc34(T69, s(T70), s(X110)) :- sumc34(T69, T70, X110).
multc53(0, 0).
multc53(s(T101), T108) :- multc53(T101, T108).

Afs:

p1(x1)  =  p1(x1)

(3) TriplesToPiDPProof (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)
pc1_in: (b)
multc23_in: (b,b,f)
sumc34_in: (b,b,f)
multc53_in: (b,f)
mult23_in: (b,b,f)
sum34_in: (b,b,f)
mult53_in: (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → U6_G(T24, T25, T26, p1_in_g(.(T24, .(T25, T26))))
P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → P1_IN_G(.(T24, .(T25, T26)))
P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → U7_G(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
U7_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U8_G(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U7_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → MULT23_IN_GGA(T24, T25, X44)
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, multc23_in_gga(T51, T52, T55))
U2_GGA(T51, T52, X83, multc23_out_gga(T51, T52, T55)) → U3_GGA(T51, T52, X83, sum34_in_gga(T55, T51, X83))
U2_GGA(T51, T52, X83, multc23_out_gga(T51, T52, T55)) → SUM34_IN_GGA(T55, T51, X83)
SUM34_IN_GGA(T69, s(T70), s(X110)) → U4_GGA(T69, T70, X110, sum34_in_gga(T69, T70, X110))
SUM34_IN_GGA(T69, s(T70), s(X110)) → SUM34_IN_GGA(T69, T70, X110)
U7_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U9_G(T24, T25, T26, multc23_in_gga(T24, T25, T39))
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U10_G(T24, T25, T26, p1_in_g(.(T39, T26)))
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → P1_IN_G(.(T39, T26))
P1_IN_G(.(s(s(0)), .(T92, T93))) → U15_G(T92, T93, p1_in_g(.(T92, T93)))
P1_IN_G(.(s(s(0)), .(T92, T93))) → P1_IN_G(.(T92, T93))
P1_IN_G(.(s(s(0)), .(T92, T93))) → U16_G(T92, T93, pc1_in_g(.(T92, T93)))
U16_G(T92, T93, pc1_out_g(.(T92, T93))) → U17_G(T92, T93, mult53_in_ga(T92, X22))
U16_G(T92, T93, pc1_out_g(.(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)
U16_G(T92, T93, pc1_out_g(.(T92, T93))) → U18_G(T92, T93, multc53_in_ga(T92, T98))
U18_G(T92, T93, multc53_out_ga(T92, T98)) → U19_G(T92, T93, p1_in_g(.(T98, T93)))
U18_G(T92, T93, multc53_out_ga(T92, T98)) → P1_IN_G(.(T98, T93))
P1_IN_G(.(0, .(s(s(T131)), .(T132, T133)))) → U20_G(T131, T132, T133, p1_in_g(.(T131, .(T132, T133))))
P1_IN_G(.(0, .(s(s(T131)), .(T132, T133)))) → P1_IN_G(.(T131, .(T132, T133)))
P1_IN_G(.(0, .(s(s(T131)), .(T132, T133)))) → U21_G(T131, T132, T133, pc1_in_g(.(T131, .(T132, T133))))
U21_G(T131, T132, T133, pc1_out_g(.(T131, .(T132, T133)))) → U22_G(T131, T132, T133, mult23_in_gga(T131, T132, X205))
U21_G(T131, T132, T133, pc1_out_g(.(T131, .(T132, T133)))) → MULT23_IN_GGA(T131, T132, X205)
U21_G(T131, T132, T133, pc1_out_g(.(T131, .(T132, T133)))) → U23_G(T131, T132, T133, multc23_in_gga(T131, T132, T146))
U23_G(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U24_G(T131, T132, T133, p1_in_g(.(T146, T133)))
U23_G(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → P1_IN_G(.(T146, T133))
P1_IN_G(.(0, .(0, T153))) → U25_G(T153, p1_in_g(T153))
P1_IN_G(.(0, .(0, T153))) → P1_IN_G(T153)
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U11_G(T24, T25, T26, pc1_in_g(.(T39, T26)))
U11_G(T24, T25, T26, pc1_out_g(.(T39, T26))) → U12_G(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_G(T24, T25, T26, pc1_out_g(.(T39, T26))) → MULT23_IN_GGA(s(s(T24)), T25, X22)
U11_G(T24, T25, T26, pc1_out_g(.(T39, T26))) → U13_G(T24, T25, T26, multc23_in_gga(s(s(T24)), T25, T79))
U13_G(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U14_G(T24, T25, T26, p1_in_g(.(T79, T26)))
U13_G(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → P1_IN_G(.(T79, T26))

The TRS R consists of the following rules:

pc1_in_g(.(T3, [])) → pc1_out_g(.(T3, []))
pc1_in_g(.(s(s(s(s(T24)))), .(T25, T26))) → U27_g(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
pc1_in_g(.(s(s(0)), .(T92, T93))) → U32_g(T92, T93, pc1_in_g(.(T92, T93)))
pc1_in_g(.(0, .(T118, []))) → pc1_out_g(.(0, .(T118, [])))
pc1_in_g(.(0, .(s(s(T131)), .(T132, T133)))) → U35_g(T131, T132, T133, pc1_in_g(.(T131, .(T132, T133))))
pc1_in_g(.(0, .(0, T153))) → U38_g(T153, pc1_in_g(T153))
U38_g(T153, pc1_out_g(T153)) → pc1_out_g(.(0, .(0, T153)))
U35_g(T131, T132, T133, pc1_out_g(.(T131, .(T132, T133)))) → U36_g(T131, T132, T133, multc23_in_gga(T131, T132, T146))
multc23_in_gga(T46, 0, 0) → multc23_out_gga(T46, 0, 0)
multc23_in_gga(T51, s(T52), X83) → U39_gga(T51, T52, X83, multc23_in_gga(T51, T52, T55))
U39_gga(T51, T52, X83, multc23_out_gga(T51, T52, T55)) → U40_gga(T51, T52, X83, sumc34_in_gga(T55, T51, X83))
sumc34_in_gga(T64, 0, T64) → sumc34_out_gga(T64, 0, T64)
sumc34_in_gga(T69, s(T70), s(X110)) → U41_gga(T69, T70, X110, sumc34_in_gga(T69, T70, X110))
U41_gga(T69, T70, X110, sumc34_out_gga(T69, T70, X110)) → sumc34_out_gga(T69, s(T70), s(X110))
U40_gga(T51, T52, X83, sumc34_out_gga(T55, T51, X83)) → multc23_out_gga(T51, s(T52), X83)
U36_g(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U37_g(T131, T132, T133, pc1_in_g(.(T146, T133)))
U37_g(T131, T132, T133, pc1_out_g(.(T146, T133))) → pc1_out_g(.(0, .(s(s(T131)), .(T132, T133))))
U32_g(T92, T93, pc1_out_g(.(T92, T93))) → U33_g(T92, T93, multc53_in_ga(T92, T98))
multc53_in_ga(0, 0) → multc53_out_ga(0, 0)
multc53_in_ga(s(T101), T108) → U42_ga(T101, T108, multc53_in_ga(T101, T108))
U42_ga(T101, T108, multc53_out_ga(T101, T108)) → multc53_out_ga(s(T101), T108)
U33_g(T92, T93, multc53_out_ga(T92, T98)) → U34_g(T92, T93, pc1_in_g(.(T98, T93)))
U34_g(T92, T93, pc1_out_g(.(T98, T93))) → pc1_out_g(.(s(s(0)), .(T92, T93)))
U27_g(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U28_g(T24, T25, T26, multc23_in_gga(T24, T25, T39))
U28_g(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U29_g(T24, T25, T26, pc1_in_g(.(T39, T26)))
U29_g(T24, T25, T26, pc1_out_g(.(T39, T26))) → U30_g(T24, T25, T26, multc23_in_gga(s(s(T24)), T25, T79))
U30_g(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U31_g(T24, T25, T26, pc1_in_g(.(T79, T26)))
U31_g(T24, T25, T26, pc1_out_g(.(T79, T26))) → pc1_out_g(.(s(s(s(s(T24)))), .(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
pc1_in_g(x1)  =  pc1_in_g(x1)
[]  =  []
pc1_out_g(x1)  =  pc1_out_g(x1)
U27_g(x1, x2, x3, x4)  =  U27_g(x1, x2, x3, x4)
0  =  0
U32_g(x1, x2, x3)  =  U32_g(x1, x2, x3)
U35_g(x1, x2, x3, x4)  =  U35_g(x1, x2, x3, x4)
U38_g(x1, x2)  =  U38_g(x1, x2)
U36_g(x1, x2, x3, x4)  =  U36_g(x1, x2, x3, x4)
multc23_in_gga(x1, x2, x3)  =  multc23_in_gga(x1, x2)
multc23_out_gga(x1, x2, x3)  =  multc23_out_gga(x1, x2, x3)
U39_gga(x1, x2, x3, x4)  =  U39_gga(x1, x2, x4)
U40_gga(x1, x2, x3, x4)  =  U40_gga(x1, x2, x4)
sumc34_in_gga(x1, x2, x3)  =  sumc34_in_gga(x1, x2)
sumc34_out_gga(x1, x2, x3)  =  sumc34_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4)  =  U41_gga(x1, x2, x4)
U37_g(x1, x2, x3, x4)  =  U37_g(x1, x2, x3, x4)
U33_g(x1, x2, x3)  =  U33_g(x1, x2, x3)
multc53_in_ga(x1, x2)  =  multc53_in_ga(x1)
multc53_out_ga(x1, x2)  =  multc53_out_ga(x1, x2)
U42_ga(x1, x2, x3)  =  U42_ga(x1, x3)
U34_g(x1, x2, x3)  =  U34_g(x1, x2, x3)
U28_g(x1, x2, x3, x4)  =  U28_g(x1, x2, x3, x4)
U29_g(x1, x2, x3, x4)  =  U29_g(x1, x2, x3, x4)
U30_g(x1, x2, x3, x4)  =  U30_g(x1, x2, x3, x4)
U31_g(x1, x2, x3, x4)  =  U31_g(x1, x2, x3, x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
sum34_in_gga(x1, x2, x3)  =  sum34_in_gga(x1, x2)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
P1_IN_G(x1)  =  P1_IN_G(x1)
U6_G(x1, x2, x3, x4)  =  U6_G(x1, x2, x3, x4)
U7_G(x1, x2, x3, x4)  =  U7_G(x1, x2, x3, x4)
U8_G(x1, x2, x3, x4)  =  U8_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_GGA(x1, x2, x3)  =  SUM34_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, 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)
U15_G(x1, x2, x3)  =  U15_G(x1, x2, x3)
U16_G(x1, x2, x3)  =  U16_G(x1, x2, x3)
U17_G(x1, x2, x3)  =  U17_G(x1, x2, x3)
MULT53_IN_GA(x1, x2)  =  MULT53_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U18_G(x1, x2, x3)  =  U18_G(x1, x2, x3)
U19_G(x1, x2, x3)  =  U19_G(x1, x2, x3)
U20_G(x1, x2, x3, x4)  =  U20_G(x1, x2, x3, x4)
U21_G(x1, x2, x3, x4)  =  U21_G(x1, x2, x3, x4)
U22_G(x1, x2, x3, x4)  =  U22_G(x1, x2, x3, x4)
U23_G(x1, x2, x3, x4)  =  U23_G(x1, x2, x3, x4)
U24_G(x1, x2, x3, x4)  =  U24_G(x1, x2, x3, x4)
U25_G(x1, x2)  =  U25_G(x1, x2)
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)
U14_G(x1, x2, x3, x4)  =  U14_G(x1, x2, x3, x4)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → U6_G(T24, T25, T26, p1_in_g(.(T24, .(T25, T26))))
P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → P1_IN_G(.(T24, .(T25, T26)))
P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → U7_G(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
U7_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U8_G(T24, T25, T26, mult23_in_gga(T24, T25, X44))
U7_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → MULT23_IN_GGA(T24, T25, X44)
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, multc23_in_gga(T51, T52, T55))
U2_GGA(T51, T52, X83, multc23_out_gga(T51, T52, T55)) → U3_GGA(T51, T52, X83, sum34_in_gga(T55, T51, X83))
U2_GGA(T51, T52, X83, multc23_out_gga(T51, T52, T55)) → SUM34_IN_GGA(T55, T51, X83)
SUM34_IN_GGA(T69, s(T70), s(X110)) → U4_GGA(T69, T70, X110, sum34_in_gga(T69, T70, X110))
SUM34_IN_GGA(T69, s(T70), s(X110)) → SUM34_IN_GGA(T69, T70, X110)
U7_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U9_G(T24, T25, T26, multc23_in_gga(T24, T25, T39))
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U10_G(T24, T25, T26, p1_in_g(.(T39, T26)))
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → P1_IN_G(.(T39, T26))
P1_IN_G(.(s(s(0)), .(T92, T93))) → U15_G(T92, T93, p1_in_g(.(T92, T93)))
P1_IN_G(.(s(s(0)), .(T92, T93))) → P1_IN_G(.(T92, T93))
P1_IN_G(.(s(s(0)), .(T92, T93))) → U16_G(T92, T93, pc1_in_g(.(T92, T93)))
U16_G(T92, T93, pc1_out_g(.(T92, T93))) → U17_G(T92, T93, mult53_in_ga(T92, X22))
U16_G(T92, T93, pc1_out_g(.(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)
U16_G(T92, T93, pc1_out_g(.(T92, T93))) → U18_G(T92, T93, multc53_in_ga(T92, T98))
U18_G(T92, T93, multc53_out_ga(T92, T98)) → U19_G(T92, T93, p1_in_g(.(T98, T93)))
U18_G(T92, T93, multc53_out_ga(T92, T98)) → P1_IN_G(.(T98, T93))
P1_IN_G(.(0, .(s(s(T131)), .(T132, T133)))) → U20_G(T131, T132, T133, p1_in_g(.(T131, .(T132, T133))))
P1_IN_G(.(0, .(s(s(T131)), .(T132, T133)))) → P1_IN_G(.(T131, .(T132, T133)))
P1_IN_G(.(0, .(s(s(T131)), .(T132, T133)))) → U21_G(T131, T132, T133, pc1_in_g(.(T131, .(T132, T133))))
U21_G(T131, T132, T133, pc1_out_g(.(T131, .(T132, T133)))) → U22_G(T131, T132, T133, mult23_in_gga(T131, T132, X205))
U21_G(T131, T132, T133, pc1_out_g(.(T131, .(T132, T133)))) → MULT23_IN_GGA(T131, T132, X205)
U21_G(T131, T132, T133, pc1_out_g(.(T131, .(T132, T133)))) → U23_G(T131, T132, T133, multc23_in_gga(T131, T132, T146))
U23_G(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U24_G(T131, T132, T133, p1_in_g(.(T146, T133)))
U23_G(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → P1_IN_G(.(T146, T133))
P1_IN_G(.(0, .(0, T153))) → U25_G(T153, p1_in_g(T153))
P1_IN_G(.(0, .(0, T153))) → P1_IN_G(T153)
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U11_G(T24, T25, T26, pc1_in_g(.(T39, T26)))
U11_G(T24, T25, T26, pc1_out_g(.(T39, T26))) → U12_G(T24, T25, T26, mult23_in_gga(s(s(T24)), T25, X22))
U11_G(T24, T25, T26, pc1_out_g(.(T39, T26))) → MULT23_IN_GGA(s(s(T24)), T25, X22)
U11_G(T24, T25, T26, pc1_out_g(.(T39, T26))) → U13_G(T24, T25, T26, multc23_in_gga(s(s(T24)), T25, T79))
U13_G(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U14_G(T24, T25, T26, p1_in_g(.(T79, T26)))
U13_G(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → P1_IN_G(.(T79, T26))

The TRS R consists of the following rules:

pc1_in_g(.(T3, [])) → pc1_out_g(.(T3, []))
pc1_in_g(.(s(s(s(s(T24)))), .(T25, T26))) → U27_g(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
pc1_in_g(.(s(s(0)), .(T92, T93))) → U32_g(T92, T93, pc1_in_g(.(T92, T93)))
pc1_in_g(.(0, .(T118, []))) → pc1_out_g(.(0, .(T118, [])))
pc1_in_g(.(0, .(s(s(T131)), .(T132, T133)))) → U35_g(T131, T132, T133, pc1_in_g(.(T131, .(T132, T133))))
pc1_in_g(.(0, .(0, T153))) → U38_g(T153, pc1_in_g(T153))
U38_g(T153, pc1_out_g(T153)) → pc1_out_g(.(0, .(0, T153)))
U35_g(T131, T132, T133, pc1_out_g(.(T131, .(T132, T133)))) → U36_g(T131, T132, T133, multc23_in_gga(T131, T132, T146))
multc23_in_gga(T46, 0, 0) → multc23_out_gga(T46, 0, 0)
multc23_in_gga(T51, s(T52), X83) → U39_gga(T51, T52, X83, multc23_in_gga(T51, T52, T55))
U39_gga(T51, T52, X83, multc23_out_gga(T51, T52, T55)) → U40_gga(T51, T52, X83, sumc34_in_gga(T55, T51, X83))
sumc34_in_gga(T64, 0, T64) → sumc34_out_gga(T64, 0, T64)
sumc34_in_gga(T69, s(T70), s(X110)) → U41_gga(T69, T70, X110, sumc34_in_gga(T69, T70, X110))
U41_gga(T69, T70, X110, sumc34_out_gga(T69, T70, X110)) → sumc34_out_gga(T69, s(T70), s(X110))
U40_gga(T51, T52, X83, sumc34_out_gga(T55, T51, X83)) → multc23_out_gga(T51, s(T52), X83)
U36_g(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U37_g(T131, T132, T133, pc1_in_g(.(T146, T133)))
U37_g(T131, T132, T133, pc1_out_g(.(T146, T133))) → pc1_out_g(.(0, .(s(s(T131)), .(T132, T133))))
U32_g(T92, T93, pc1_out_g(.(T92, T93))) → U33_g(T92, T93, multc53_in_ga(T92, T98))
multc53_in_ga(0, 0) → multc53_out_ga(0, 0)
multc53_in_ga(s(T101), T108) → U42_ga(T101, T108, multc53_in_ga(T101, T108))
U42_ga(T101, T108, multc53_out_ga(T101, T108)) → multc53_out_ga(s(T101), T108)
U33_g(T92, T93, multc53_out_ga(T92, T98)) → U34_g(T92, T93, pc1_in_g(.(T98, T93)))
U34_g(T92, T93, pc1_out_g(.(T98, T93))) → pc1_out_g(.(s(s(0)), .(T92, T93)))
U27_g(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U28_g(T24, T25, T26, multc23_in_gga(T24, T25, T39))
U28_g(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U29_g(T24, T25, T26, pc1_in_g(.(T39, T26)))
U29_g(T24, T25, T26, pc1_out_g(.(T39, T26))) → U30_g(T24, T25, T26, multc23_in_gga(s(s(T24)), T25, T79))
U30_g(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U31_g(T24, T25, T26, pc1_in_g(.(T79, T26)))
U31_g(T24, T25, T26, pc1_out_g(.(T79, T26))) → pc1_out_g(.(s(s(s(s(T24)))), .(T25, T26)))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
pc1_in_g(x1)  =  pc1_in_g(x1)
[]  =  []
pc1_out_g(x1)  =  pc1_out_g(x1)
U27_g(x1, x2, x3, x4)  =  U27_g(x1, x2, x3, x4)
0  =  0
U32_g(x1, x2, x3)  =  U32_g(x1, x2, x3)
U35_g(x1, x2, x3, x4)  =  U35_g(x1, x2, x3, x4)
U38_g(x1, x2)  =  U38_g(x1, x2)
U36_g(x1, x2, x3, x4)  =  U36_g(x1, x2, x3, x4)
multc23_in_gga(x1, x2, x3)  =  multc23_in_gga(x1, x2)
multc23_out_gga(x1, x2, x3)  =  multc23_out_gga(x1, x2, x3)
U39_gga(x1, x2, x3, x4)  =  U39_gga(x1, x2, x4)
U40_gga(x1, x2, x3, x4)  =  U40_gga(x1, x2, x4)
sumc34_in_gga(x1, x2, x3)  =  sumc34_in_gga(x1, x2)
sumc34_out_gga(x1, x2, x3)  =  sumc34_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4)  =  U41_gga(x1, x2, x4)
U37_g(x1, x2, x3, x4)  =  U37_g(x1, x2, x3, x4)
U33_g(x1, x2, x3)  =  U33_g(x1, x2, x3)
multc53_in_ga(x1, x2)  =  multc53_in_ga(x1)
multc53_out_ga(x1, x2)  =  multc53_out_ga(x1, x2)
U42_ga(x1, x2, x3)  =  U42_ga(x1, x3)
U34_g(x1, x2, x3)  =  U34_g(x1, x2, x3)
U28_g(x1, x2, x3, x4)  =  U28_g(x1, x2, x3, x4)
U29_g(x1, x2, x3, x4)  =  U29_g(x1, x2, x3, x4)
U30_g(x1, x2, x3, x4)  =  U30_g(x1, x2, x3, x4)
U31_g(x1, x2, x3, x4)  =  U31_g(x1, x2, x3, x4)
mult23_in_gga(x1, x2, x3)  =  mult23_in_gga(x1, x2)
sum34_in_gga(x1, x2, x3)  =  sum34_in_gga(x1, x2)
mult53_in_ga(x1, x2)  =  mult53_in_ga(x1)
P1_IN_G(x1)  =  P1_IN_G(x1)
U6_G(x1, x2, x3, x4)  =  U6_G(x1, x2, x3, x4)
U7_G(x1, x2, x3, x4)  =  U7_G(x1, x2, x3, x4)
U8_G(x1, x2, x3, x4)  =  U8_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_GGA(x1, x2, x3)  =  SUM34_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, 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)
U15_G(x1, x2, x3)  =  U15_G(x1, x2, x3)
U16_G(x1, x2, x3)  =  U16_G(x1, x2, x3)
U17_G(x1, x2, x3)  =  U17_G(x1, x2, x3)
MULT53_IN_GA(x1, x2)  =  MULT53_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U18_G(x1, x2, x3)  =  U18_G(x1, x2, x3)
U19_G(x1, x2, x3)  =  U19_G(x1, x2, x3)
U20_G(x1, x2, x3, x4)  =  U20_G(x1, x2, x3, x4)
U21_G(x1, x2, x3, x4)  =  U21_G(x1, x2, x3, x4)
U22_G(x1, x2, x3, x4)  =  U22_G(x1, x2, x3, x4)
U23_G(x1, x2, x3, x4)  =  U23_G(x1, x2, x3, x4)
U24_G(x1, x2, x3, x4)  =  U24_G(x1, x2, x3, x4)
U25_G(x1, x2)  =  U25_G(x1, x2)
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)
U14_G(x1, x2, x3, x4)  =  U14_G(x1, x2, x3, x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 22 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

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

The TRS R consists of the following rules:

pc1_in_g(.(T3, [])) → pc1_out_g(.(T3, []))
pc1_in_g(.(s(s(s(s(T24)))), .(T25, T26))) → U27_g(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
pc1_in_g(.(s(s(0)), .(T92, T93))) → U32_g(T92, T93, pc1_in_g(.(T92, T93)))
pc1_in_g(.(0, .(T118, []))) → pc1_out_g(.(0, .(T118, [])))
pc1_in_g(.(0, .(s(s(T131)), .(T132, T133)))) → U35_g(T131, T132, T133, pc1_in_g(.(T131, .(T132, T133))))
pc1_in_g(.(0, .(0, T153))) → U38_g(T153, pc1_in_g(T153))
U38_g(T153, pc1_out_g(T153)) → pc1_out_g(.(0, .(0, T153)))
U35_g(T131, T132, T133, pc1_out_g(.(T131, .(T132, T133)))) → U36_g(T131, T132, T133, multc23_in_gga(T131, T132, T146))
multc23_in_gga(T46, 0, 0) → multc23_out_gga(T46, 0, 0)
multc23_in_gga(T51, s(T52), X83) → U39_gga(T51, T52, X83, multc23_in_gga(T51, T52, T55))
U39_gga(T51, T52, X83, multc23_out_gga(T51, T52, T55)) → U40_gga(T51, T52, X83, sumc34_in_gga(T55, T51, X83))
sumc34_in_gga(T64, 0, T64) → sumc34_out_gga(T64, 0, T64)
sumc34_in_gga(T69, s(T70), s(X110)) → U41_gga(T69, T70, X110, sumc34_in_gga(T69, T70, X110))
U41_gga(T69, T70, X110, sumc34_out_gga(T69, T70, X110)) → sumc34_out_gga(T69, s(T70), s(X110))
U40_gga(T51, T52, X83, sumc34_out_gga(T55, T51, X83)) → multc23_out_gga(T51, s(T52), X83)
U36_g(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U37_g(T131, T132, T133, pc1_in_g(.(T146, T133)))
U37_g(T131, T132, T133, pc1_out_g(.(T146, T133))) → pc1_out_g(.(0, .(s(s(T131)), .(T132, T133))))
U32_g(T92, T93, pc1_out_g(.(T92, T93))) → U33_g(T92, T93, multc53_in_ga(T92, T98))
multc53_in_ga(0, 0) → multc53_out_ga(0, 0)
multc53_in_ga(s(T101), T108) → U42_ga(T101, T108, multc53_in_ga(T101, T108))
U42_ga(T101, T108, multc53_out_ga(T101, T108)) → multc53_out_ga(s(T101), T108)
U33_g(T92, T93, multc53_out_ga(T92, T98)) → U34_g(T92, T93, pc1_in_g(.(T98, T93)))
U34_g(T92, T93, pc1_out_g(.(T98, T93))) → pc1_out_g(.(s(s(0)), .(T92, T93)))
U27_g(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U28_g(T24, T25, T26, multc23_in_gga(T24, T25, T39))
U28_g(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U29_g(T24, T25, T26, pc1_in_g(.(T39, T26)))
U29_g(T24, T25, T26, pc1_out_g(.(T39, T26))) → U30_g(T24, T25, T26, multc23_in_gga(s(s(T24)), T25, T79))
U30_g(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U31_g(T24, T25, T26, pc1_in_g(.(T79, T26)))
U31_g(T24, T25, T26, pc1_out_g(.(T79, T26))) → pc1_out_g(.(s(s(s(s(T24)))), .(T25, T26)))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
pc1_in_g(x1)  =  pc1_in_g(x1)
[]  =  []
pc1_out_g(x1)  =  pc1_out_g(x1)
U27_g(x1, x2, x3, x4)  =  U27_g(x1, x2, x3, x4)
0  =  0
U32_g(x1, x2, x3)  =  U32_g(x1, x2, x3)
U35_g(x1, x2, x3, x4)  =  U35_g(x1, x2, x3, x4)
U38_g(x1, x2)  =  U38_g(x1, x2)
U36_g(x1, x2, x3, x4)  =  U36_g(x1, x2, x3, x4)
multc23_in_gga(x1, x2, x3)  =  multc23_in_gga(x1, x2)
multc23_out_gga(x1, x2, x3)  =  multc23_out_gga(x1, x2, x3)
U39_gga(x1, x2, x3, x4)  =  U39_gga(x1, x2, x4)
U40_gga(x1, x2, x3, x4)  =  U40_gga(x1, x2, x4)
sumc34_in_gga(x1, x2, x3)  =  sumc34_in_gga(x1, x2)
sumc34_out_gga(x1, x2, x3)  =  sumc34_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4)  =  U41_gga(x1, x2, x4)
U37_g(x1, x2, x3, x4)  =  U37_g(x1, x2, x3, x4)
U33_g(x1, x2, x3)  =  U33_g(x1, x2, x3)
multc53_in_ga(x1, x2)  =  multc53_in_ga(x1)
multc53_out_ga(x1, x2)  =  multc53_out_ga(x1, x2)
U42_ga(x1, x2, x3)  =  U42_ga(x1, x3)
U34_g(x1, x2, x3)  =  U34_g(x1, x2, x3)
U28_g(x1, x2, x3, x4)  =  U28_g(x1, x2, x3, x4)
U29_g(x1, x2, x3, x4)  =  U29_g(x1, x2, x3, x4)
U30_g(x1, x2, x3, x4)  =  U30_g(x1, x2, x3, x4)
U31_g(x1, x2, x3, x4)  =  U31_g(x1, x2, x3, x4)
MULT53_IN_GA(x1, x2)  =  MULT53_IN_GA(x1)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) 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.

(12) 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

(13) YES

(14) Obligation:

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

SUM34_IN_GGA(T69, s(T70), s(X110)) → SUM34_IN_GGA(T69, T70, X110)

The TRS R consists of the following rules:

pc1_in_g(.(T3, [])) → pc1_out_g(.(T3, []))
pc1_in_g(.(s(s(s(s(T24)))), .(T25, T26))) → U27_g(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
pc1_in_g(.(s(s(0)), .(T92, T93))) → U32_g(T92, T93, pc1_in_g(.(T92, T93)))
pc1_in_g(.(0, .(T118, []))) → pc1_out_g(.(0, .(T118, [])))
pc1_in_g(.(0, .(s(s(T131)), .(T132, T133)))) → U35_g(T131, T132, T133, pc1_in_g(.(T131, .(T132, T133))))
pc1_in_g(.(0, .(0, T153))) → U38_g(T153, pc1_in_g(T153))
U38_g(T153, pc1_out_g(T153)) → pc1_out_g(.(0, .(0, T153)))
U35_g(T131, T132, T133, pc1_out_g(.(T131, .(T132, T133)))) → U36_g(T131, T132, T133, multc23_in_gga(T131, T132, T146))
multc23_in_gga(T46, 0, 0) → multc23_out_gga(T46, 0, 0)
multc23_in_gga(T51, s(T52), X83) → U39_gga(T51, T52, X83, multc23_in_gga(T51, T52, T55))
U39_gga(T51, T52, X83, multc23_out_gga(T51, T52, T55)) → U40_gga(T51, T52, X83, sumc34_in_gga(T55, T51, X83))
sumc34_in_gga(T64, 0, T64) → sumc34_out_gga(T64, 0, T64)
sumc34_in_gga(T69, s(T70), s(X110)) → U41_gga(T69, T70, X110, sumc34_in_gga(T69, T70, X110))
U41_gga(T69, T70, X110, sumc34_out_gga(T69, T70, X110)) → sumc34_out_gga(T69, s(T70), s(X110))
U40_gga(T51, T52, X83, sumc34_out_gga(T55, T51, X83)) → multc23_out_gga(T51, s(T52), X83)
U36_g(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U37_g(T131, T132, T133, pc1_in_g(.(T146, T133)))
U37_g(T131, T132, T133, pc1_out_g(.(T146, T133))) → pc1_out_g(.(0, .(s(s(T131)), .(T132, T133))))
U32_g(T92, T93, pc1_out_g(.(T92, T93))) → U33_g(T92, T93, multc53_in_ga(T92, T98))
multc53_in_ga(0, 0) → multc53_out_ga(0, 0)
multc53_in_ga(s(T101), T108) → U42_ga(T101, T108, multc53_in_ga(T101, T108))
U42_ga(T101, T108, multc53_out_ga(T101, T108)) → multc53_out_ga(s(T101), T108)
U33_g(T92, T93, multc53_out_ga(T92, T98)) → U34_g(T92, T93, pc1_in_g(.(T98, T93)))
U34_g(T92, T93, pc1_out_g(.(T98, T93))) → pc1_out_g(.(s(s(0)), .(T92, T93)))
U27_g(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U28_g(T24, T25, T26, multc23_in_gga(T24, T25, T39))
U28_g(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U29_g(T24, T25, T26, pc1_in_g(.(T39, T26)))
U29_g(T24, T25, T26, pc1_out_g(.(T39, T26))) → U30_g(T24, T25, T26, multc23_in_gga(s(s(T24)), T25, T79))
U30_g(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U31_g(T24, T25, T26, pc1_in_g(.(T79, T26)))
U31_g(T24, T25, T26, pc1_out_g(.(T79, T26))) → pc1_out_g(.(s(s(s(s(T24)))), .(T25, T26)))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
pc1_in_g(x1)  =  pc1_in_g(x1)
[]  =  []
pc1_out_g(x1)  =  pc1_out_g(x1)
U27_g(x1, x2, x3, x4)  =  U27_g(x1, x2, x3, x4)
0  =  0
U32_g(x1, x2, x3)  =  U32_g(x1, x2, x3)
U35_g(x1, x2, x3, x4)  =  U35_g(x1, x2, x3, x4)
U38_g(x1, x2)  =  U38_g(x1, x2)
U36_g(x1, x2, x3, x4)  =  U36_g(x1, x2, x3, x4)
multc23_in_gga(x1, x2, x3)  =  multc23_in_gga(x1, x2)
multc23_out_gga(x1, x2, x3)  =  multc23_out_gga(x1, x2, x3)
U39_gga(x1, x2, x3, x4)  =  U39_gga(x1, x2, x4)
U40_gga(x1, x2, x3, x4)  =  U40_gga(x1, x2, x4)
sumc34_in_gga(x1, x2, x3)  =  sumc34_in_gga(x1, x2)
sumc34_out_gga(x1, x2, x3)  =  sumc34_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4)  =  U41_gga(x1, x2, x4)
U37_g(x1, x2, x3, x4)  =  U37_g(x1, x2, x3, x4)
U33_g(x1, x2, x3)  =  U33_g(x1, x2, x3)
multc53_in_ga(x1, x2)  =  multc53_in_ga(x1)
multc53_out_ga(x1, x2)  =  multc53_out_ga(x1, x2)
U42_ga(x1, x2, x3)  =  U42_ga(x1, x3)
U34_g(x1, x2, x3)  =  U34_g(x1, x2, x3)
U28_g(x1, x2, x3, x4)  =  U28_g(x1, x2, x3, x4)
U29_g(x1, x2, x3, x4)  =  U29_g(x1, x2, x3, x4)
U30_g(x1, x2, x3, x4)  =  U30_g(x1, x2, x3, x4)
U31_g(x1, x2, x3, x4)  =  U31_g(x1, x2, x3, x4)
SUM34_IN_GGA(x1, x2, x3)  =  SUM34_IN_GGA(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

SUM34_IN_GGA(T69, s(T70), s(X110)) → SUM34_IN_GGA(T69, T70, X110)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
SUM34_IN_GGA(x1, x2, x3)  =  SUM34_IN_GGA(x1, x2)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

SUM34_IN_GGA(T69, s(T70)) → SUM34_IN_GGA(T69, T70)

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

(19) 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_GGA(T69, s(T70)) → SUM34_IN_GGA(T69, T70)
    The graph contains the following edges 1 >= 1, 2 > 2

(20) YES

(21) 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:

pc1_in_g(.(T3, [])) → pc1_out_g(.(T3, []))
pc1_in_g(.(s(s(s(s(T24)))), .(T25, T26))) → U27_g(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
pc1_in_g(.(s(s(0)), .(T92, T93))) → U32_g(T92, T93, pc1_in_g(.(T92, T93)))
pc1_in_g(.(0, .(T118, []))) → pc1_out_g(.(0, .(T118, [])))
pc1_in_g(.(0, .(s(s(T131)), .(T132, T133)))) → U35_g(T131, T132, T133, pc1_in_g(.(T131, .(T132, T133))))
pc1_in_g(.(0, .(0, T153))) → U38_g(T153, pc1_in_g(T153))
U38_g(T153, pc1_out_g(T153)) → pc1_out_g(.(0, .(0, T153)))
U35_g(T131, T132, T133, pc1_out_g(.(T131, .(T132, T133)))) → U36_g(T131, T132, T133, multc23_in_gga(T131, T132, T146))
multc23_in_gga(T46, 0, 0) → multc23_out_gga(T46, 0, 0)
multc23_in_gga(T51, s(T52), X83) → U39_gga(T51, T52, X83, multc23_in_gga(T51, T52, T55))
U39_gga(T51, T52, X83, multc23_out_gga(T51, T52, T55)) → U40_gga(T51, T52, X83, sumc34_in_gga(T55, T51, X83))
sumc34_in_gga(T64, 0, T64) → sumc34_out_gga(T64, 0, T64)
sumc34_in_gga(T69, s(T70), s(X110)) → U41_gga(T69, T70, X110, sumc34_in_gga(T69, T70, X110))
U41_gga(T69, T70, X110, sumc34_out_gga(T69, T70, X110)) → sumc34_out_gga(T69, s(T70), s(X110))
U40_gga(T51, T52, X83, sumc34_out_gga(T55, T51, X83)) → multc23_out_gga(T51, s(T52), X83)
U36_g(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U37_g(T131, T132, T133, pc1_in_g(.(T146, T133)))
U37_g(T131, T132, T133, pc1_out_g(.(T146, T133))) → pc1_out_g(.(0, .(s(s(T131)), .(T132, T133))))
U32_g(T92, T93, pc1_out_g(.(T92, T93))) → U33_g(T92, T93, multc53_in_ga(T92, T98))
multc53_in_ga(0, 0) → multc53_out_ga(0, 0)
multc53_in_ga(s(T101), T108) → U42_ga(T101, T108, multc53_in_ga(T101, T108))
U42_ga(T101, T108, multc53_out_ga(T101, T108)) → multc53_out_ga(s(T101), T108)
U33_g(T92, T93, multc53_out_ga(T92, T98)) → U34_g(T92, T93, pc1_in_g(.(T98, T93)))
U34_g(T92, T93, pc1_out_g(.(T98, T93))) → pc1_out_g(.(s(s(0)), .(T92, T93)))
U27_g(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U28_g(T24, T25, T26, multc23_in_gga(T24, T25, T39))
U28_g(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U29_g(T24, T25, T26, pc1_in_g(.(T39, T26)))
U29_g(T24, T25, T26, pc1_out_g(.(T39, T26))) → U30_g(T24, T25, T26, multc23_in_gga(s(s(T24)), T25, T79))
U30_g(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U31_g(T24, T25, T26, pc1_in_g(.(T79, T26)))
U31_g(T24, T25, T26, pc1_out_g(.(T79, T26))) → pc1_out_g(.(s(s(s(s(T24)))), .(T25, T26)))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
pc1_in_g(x1)  =  pc1_in_g(x1)
[]  =  []
pc1_out_g(x1)  =  pc1_out_g(x1)
U27_g(x1, x2, x3, x4)  =  U27_g(x1, x2, x3, x4)
0  =  0
U32_g(x1, x2, x3)  =  U32_g(x1, x2, x3)
U35_g(x1, x2, x3, x4)  =  U35_g(x1, x2, x3, x4)
U38_g(x1, x2)  =  U38_g(x1, x2)
U36_g(x1, x2, x3, x4)  =  U36_g(x1, x2, x3, x4)
multc23_in_gga(x1, x2, x3)  =  multc23_in_gga(x1, x2)
multc23_out_gga(x1, x2, x3)  =  multc23_out_gga(x1, x2, x3)
U39_gga(x1, x2, x3, x4)  =  U39_gga(x1, x2, x4)
U40_gga(x1, x2, x3, x4)  =  U40_gga(x1, x2, x4)
sumc34_in_gga(x1, x2, x3)  =  sumc34_in_gga(x1, x2)
sumc34_out_gga(x1, x2, x3)  =  sumc34_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4)  =  U41_gga(x1, x2, x4)
U37_g(x1, x2, x3, x4)  =  U37_g(x1, x2, x3, x4)
U33_g(x1, x2, x3)  =  U33_g(x1, x2, x3)
multc53_in_ga(x1, x2)  =  multc53_in_ga(x1)
multc53_out_ga(x1, x2)  =  multc53_out_ga(x1, x2)
U42_ga(x1, x2, x3)  =  U42_ga(x1, x3)
U34_g(x1, x2, x3)  =  U34_g(x1, x2, x3)
U28_g(x1, x2, x3, x4)  =  U28_g(x1, x2, x3, x4)
U29_g(x1, x2, x3, x4)  =  U29_g(x1, x2, x3, x4)
U30_g(x1, x2, x3, x4)  =  U30_g(x1, x2, x3, x4)
U31_g(x1, x2, x3, x4)  =  U31_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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) 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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) 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.

(26) 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

(27) YES

(28) Obligation:

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

P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → U7_G(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
U7_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U9_G(T24, T25, T26, multc23_in_gga(T24, T25, T39))
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → P1_IN_G(.(T39, T26))
P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → P1_IN_G(.(T24, .(T25, T26)))
P1_IN_G(.(s(s(0)), .(T92, T93))) → P1_IN_G(.(T92, T93))
P1_IN_G(.(s(s(0)), .(T92, T93))) → U16_G(T92, T93, pc1_in_g(.(T92, T93)))
U16_G(T92, T93, pc1_out_g(.(T92, T93))) → U18_G(T92, T93, multc53_in_ga(T92, T98))
U18_G(T92, T93, multc53_out_ga(T92, T98)) → P1_IN_G(.(T98, T93))
P1_IN_G(.(0, .(s(s(T131)), .(T132, T133)))) → P1_IN_G(.(T131, .(T132, T133)))
P1_IN_G(.(0, .(s(s(T131)), .(T132, T133)))) → U21_G(T131, T132, T133, pc1_in_g(.(T131, .(T132, T133))))
U21_G(T131, T132, T133, pc1_out_g(.(T131, .(T132, T133)))) → U23_G(T131, T132, T133, multc23_in_gga(T131, T132, T146))
U23_G(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → P1_IN_G(.(T146, T133))
P1_IN_G(.(0, .(0, T153))) → P1_IN_G(T153)
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U11_G(T24, T25, T26, pc1_in_g(.(T39, T26)))
U11_G(T24, T25, T26, pc1_out_g(.(T39, T26))) → U13_G(T24, T25, T26, multc23_in_gga(s(s(T24)), T25, T79))
U13_G(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → P1_IN_G(.(T79, T26))

The TRS R consists of the following rules:

pc1_in_g(.(T3, [])) → pc1_out_g(.(T3, []))
pc1_in_g(.(s(s(s(s(T24)))), .(T25, T26))) → U27_g(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
pc1_in_g(.(s(s(0)), .(T92, T93))) → U32_g(T92, T93, pc1_in_g(.(T92, T93)))
pc1_in_g(.(0, .(T118, []))) → pc1_out_g(.(0, .(T118, [])))
pc1_in_g(.(0, .(s(s(T131)), .(T132, T133)))) → U35_g(T131, T132, T133, pc1_in_g(.(T131, .(T132, T133))))
pc1_in_g(.(0, .(0, T153))) → U38_g(T153, pc1_in_g(T153))
U38_g(T153, pc1_out_g(T153)) → pc1_out_g(.(0, .(0, T153)))
U35_g(T131, T132, T133, pc1_out_g(.(T131, .(T132, T133)))) → U36_g(T131, T132, T133, multc23_in_gga(T131, T132, T146))
multc23_in_gga(T46, 0, 0) → multc23_out_gga(T46, 0, 0)
multc23_in_gga(T51, s(T52), X83) → U39_gga(T51, T52, X83, multc23_in_gga(T51, T52, T55))
U39_gga(T51, T52, X83, multc23_out_gga(T51, T52, T55)) → U40_gga(T51, T52, X83, sumc34_in_gga(T55, T51, X83))
sumc34_in_gga(T64, 0, T64) → sumc34_out_gga(T64, 0, T64)
sumc34_in_gga(T69, s(T70), s(X110)) → U41_gga(T69, T70, X110, sumc34_in_gga(T69, T70, X110))
U41_gga(T69, T70, X110, sumc34_out_gga(T69, T70, X110)) → sumc34_out_gga(T69, s(T70), s(X110))
U40_gga(T51, T52, X83, sumc34_out_gga(T55, T51, X83)) → multc23_out_gga(T51, s(T52), X83)
U36_g(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U37_g(T131, T132, T133, pc1_in_g(.(T146, T133)))
U37_g(T131, T132, T133, pc1_out_g(.(T146, T133))) → pc1_out_g(.(0, .(s(s(T131)), .(T132, T133))))
U32_g(T92, T93, pc1_out_g(.(T92, T93))) → U33_g(T92, T93, multc53_in_ga(T92, T98))
multc53_in_ga(0, 0) → multc53_out_ga(0, 0)
multc53_in_ga(s(T101), T108) → U42_ga(T101, T108, multc53_in_ga(T101, T108))
U42_ga(T101, T108, multc53_out_ga(T101, T108)) → multc53_out_ga(s(T101), T108)
U33_g(T92, T93, multc53_out_ga(T92, T98)) → U34_g(T92, T93, pc1_in_g(.(T98, T93)))
U34_g(T92, T93, pc1_out_g(.(T98, T93))) → pc1_out_g(.(s(s(0)), .(T92, T93)))
U27_g(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U28_g(T24, T25, T26, multc23_in_gga(T24, T25, T39))
U28_g(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U29_g(T24, T25, T26, pc1_in_g(.(T39, T26)))
U29_g(T24, T25, T26, pc1_out_g(.(T39, T26))) → U30_g(T24, T25, T26, multc23_in_gga(s(s(T24)), T25, T79))
U30_g(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U31_g(T24, T25, T26, pc1_in_g(.(T79, T26)))
U31_g(T24, T25, T26, pc1_out_g(.(T79, T26))) → pc1_out_g(.(s(s(s(s(T24)))), .(T25, T26)))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
pc1_in_g(x1)  =  pc1_in_g(x1)
[]  =  []
pc1_out_g(x1)  =  pc1_out_g(x1)
U27_g(x1, x2, x3, x4)  =  U27_g(x1, x2, x3, x4)
0  =  0
U32_g(x1, x2, x3)  =  U32_g(x1, x2, x3)
U35_g(x1, x2, x3, x4)  =  U35_g(x1, x2, x3, x4)
U38_g(x1, x2)  =  U38_g(x1, x2)
U36_g(x1, x2, x3, x4)  =  U36_g(x1, x2, x3, x4)
multc23_in_gga(x1, x2, x3)  =  multc23_in_gga(x1, x2)
multc23_out_gga(x1, x2, x3)  =  multc23_out_gga(x1, x2, x3)
U39_gga(x1, x2, x3, x4)  =  U39_gga(x1, x2, x4)
U40_gga(x1, x2, x3, x4)  =  U40_gga(x1, x2, x4)
sumc34_in_gga(x1, x2, x3)  =  sumc34_in_gga(x1, x2)
sumc34_out_gga(x1, x2, x3)  =  sumc34_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4)  =  U41_gga(x1, x2, x4)
U37_g(x1, x2, x3, x4)  =  U37_g(x1, x2, x3, x4)
U33_g(x1, x2, x3)  =  U33_g(x1, x2, x3)
multc53_in_ga(x1, x2)  =  multc53_in_ga(x1)
multc53_out_ga(x1, x2)  =  multc53_out_ga(x1, x2)
U42_ga(x1, x2, x3)  =  U42_ga(x1, x3)
U34_g(x1, x2, x3)  =  U34_g(x1, x2, x3)
U28_g(x1, x2, x3, x4)  =  U28_g(x1, x2, x3, x4)
U29_g(x1, x2, x3, x4)  =  U29_g(x1, x2, x3, x4)
U30_g(x1, x2, x3, x4)  =  U30_g(x1, x2, x3, x4)
U31_g(x1, x2, x3, x4)  =  U31_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)
U9_G(x1, x2, x3, x4)  =  U9_G(x1, x2, x3, x4)
U16_G(x1, x2, x3)  =  U16_G(x1, x2, x3)
U18_G(x1, x2, x3)  =  U18_G(x1, x2, x3)
U21_G(x1, x2, x3, x4)  =  U21_G(x1, x2, x3, x4)
U23_G(x1, x2, x3, x4)  =  U23_G(x1, x2, x3, x4)
U11_G(x1, x2, x3, x4)  =  U11_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

(29) PiDPToQDPProof (SOUND transformation)

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

(30) Obligation:

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

P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → U7_G(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
U7_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U9_G(T24, T25, T26, multc23_in_gga(T24, T25))
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → P1_IN_G(.(T39, T26))
P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → P1_IN_G(.(T24, .(T25, T26)))
P1_IN_G(.(s(s(0)), .(T92, T93))) → P1_IN_G(.(T92, T93))
P1_IN_G(.(s(s(0)), .(T92, T93))) → U16_G(T92, T93, pc1_in_g(.(T92, T93)))
U16_G(T92, T93, pc1_out_g(.(T92, T93))) → U18_G(T92, T93, multc53_in_ga(T92))
U18_G(T92, T93, multc53_out_ga(T92, T98)) → P1_IN_G(.(T98, T93))
P1_IN_G(.(0, .(s(s(T131)), .(T132, T133)))) → P1_IN_G(.(T131, .(T132, T133)))
P1_IN_G(.(0, .(s(s(T131)), .(T132, T133)))) → U21_G(T131, T132, T133, pc1_in_g(.(T131, .(T132, T133))))
U21_G(T131, T132, T133, pc1_out_g(.(T131, .(T132, T133)))) → U23_G(T131, T132, T133, multc23_in_gga(T131, T132))
U23_G(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → P1_IN_G(.(T146, T133))
P1_IN_G(.(0, .(0, T153))) → P1_IN_G(T153)
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U11_G(T24, T25, T26, pc1_in_g(.(T39, T26)))
U11_G(T24, T25, T26, pc1_out_g(.(T39, T26))) → U13_G(T24, T25, T26, multc23_in_gga(s(s(T24)), T25))
U13_G(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → P1_IN_G(.(T79, T26))

The TRS R consists of the following rules:

pc1_in_g(.(T3, [])) → pc1_out_g(.(T3, []))
pc1_in_g(.(s(s(s(s(T24)))), .(T25, T26))) → U27_g(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
pc1_in_g(.(s(s(0)), .(T92, T93))) → U32_g(T92, T93, pc1_in_g(.(T92, T93)))
pc1_in_g(.(0, .(T118, []))) → pc1_out_g(.(0, .(T118, [])))
pc1_in_g(.(0, .(s(s(T131)), .(T132, T133)))) → U35_g(T131, T132, T133, pc1_in_g(.(T131, .(T132, T133))))
pc1_in_g(.(0, .(0, T153))) → U38_g(T153, pc1_in_g(T153))
U38_g(T153, pc1_out_g(T153)) → pc1_out_g(.(0, .(0, T153)))
U35_g(T131, T132, T133, pc1_out_g(.(T131, .(T132, T133)))) → U36_g(T131, T132, T133, multc23_in_gga(T131, T132))
multc23_in_gga(T46, 0) → multc23_out_gga(T46, 0, 0)
multc23_in_gga(T51, s(T52)) → U39_gga(T51, T52, multc23_in_gga(T51, T52))
U39_gga(T51, T52, multc23_out_gga(T51, T52, T55)) → U40_gga(T51, T52, sumc34_in_gga(T55, T51))
sumc34_in_gga(T64, 0) → sumc34_out_gga(T64, 0, T64)
sumc34_in_gga(T69, s(T70)) → U41_gga(T69, T70, sumc34_in_gga(T69, T70))
U41_gga(T69, T70, sumc34_out_gga(T69, T70, X110)) → sumc34_out_gga(T69, s(T70), s(X110))
U40_gga(T51, T52, sumc34_out_gga(T55, T51, X83)) → multc23_out_gga(T51, s(T52), X83)
U36_g(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U37_g(T131, T132, T133, pc1_in_g(.(T146, T133)))
U37_g(T131, T132, T133, pc1_out_g(.(T146, T133))) → pc1_out_g(.(0, .(s(s(T131)), .(T132, T133))))
U32_g(T92, T93, pc1_out_g(.(T92, T93))) → U33_g(T92, T93, multc53_in_ga(T92))
multc53_in_ga(0) → multc53_out_ga(0, 0)
multc53_in_ga(s(T101)) → U42_ga(T101, multc53_in_ga(T101))
U42_ga(T101, multc53_out_ga(T101, T108)) → multc53_out_ga(s(T101), T108)
U33_g(T92, T93, multc53_out_ga(T92, T98)) → U34_g(T92, T93, pc1_in_g(.(T98, T93)))
U34_g(T92, T93, pc1_out_g(.(T98, T93))) → pc1_out_g(.(s(s(0)), .(T92, T93)))
U27_g(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U28_g(T24, T25, T26, multc23_in_gga(T24, T25))
U28_g(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U29_g(T24, T25, T26, pc1_in_g(.(T39, T26)))
U29_g(T24, T25, T26, pc1_out_g(.(T39, T26))) → U30_g(T24, T25, T26, multc23_in_gga(s(s(T24)), T25))
U30_g(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U31_g(T24, T25, T26, pc1_in_g(.(T79, T26)))
U31_g(T24, T25, T26, pc1_out_g(.(T79, T26))) → pc1_out_g(.(s(s(s(s(T24)))), .(T25, T26)))

The set Q consists of the following terms:

pc1_in_g(x0)
U38_g(x0, x1)
U35_g(x0, x1, x2, x3)
multc23_in_gga(x0, x1)
U39_gga(x0, x1, x2)
sumc34_in_gga(x0, x1)
U41_gga(x0, x1, x2)
U40_gga(x0, x1, x2)
U36_g(x0, x1, x2, x3)
U37_g(x0, x1, x2, x3)
U32_g(x0, x1, x2)
multc53_in_ga(x0)
U42_ga(x0, x1)
U33_g(x0, x1, x2)
U34_g(x0, x1, x2)
U27_g(x0, x1, x2, x3)
U28_g(x0, x1, x2, x3)
U29_g(x0, x1, x2, x3)
U30_g(x0, x1, x2, x3)
U31_g(x0, x1, x2, x3)

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

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → U7_G(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
P1_IN_G(.(s(s(0)), .(T92, T93))) → P1_IN_G(.(T92, T93))
P1_IN_G(.(s(s(0)), .(T92, T93))) → U16_G(T92, T93, pc1_in_g(.(T92, T93)))
P1_IN_G(.(0, .(s(s(T131)), .(T132, T133)))) → P1_IN_G(.(T131, .(T132, T133)))
P1_IN_G(.(0, .(s(s(T131)), .(T132, T133)))) → U21_G(T131, T132, T133, pc1_in_g(.(T131, .(T132, T133))))
P1_IN_G(.(0, .(0, T153))) → P1_IN_G(T153)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + x2   
POL(0) = 0   
POL(P1_IN_G(x1)) = x1   
POL(U11_G(x1, x2, x3, x4)) = 1 + x3   
POL(U13_G(x1, x2, x3, x4)) = 1 + x3   
POL(U16_G(x1, x2, x3)) = 1 + x2   
POL(U18_G(x1, x2, x3)) = 1 + x2   
POL(U21_G(x1, x2, x3, x4)) = 1 + x3   
POL(U23_G(x1, x2, x3, x4)) = 1 + x3   
POL(U27_g(x1, x2, x3, x4)) = 0   
POL(U28_g(x1, x2, x3, x4)) = 0   
POL(U29_g(x1, x2, x3, x4)) = 0   
POL(U30_g(x1, x2, x3, x4)) = 0   
POL(U31_g(x1, x2, x3, x4)) = 0   
POL(U32_g(x1, x2, x3)) = 0   
POL(U33_g(x1, x2, x3)) = 0   
POL(U34_g(x1, x2, x3)) = 0   
POL(U35_g(x1, x2, x3, x4)) = 0   
POL(U36_g(x1, x2, x3, x4)) = 0   
POL(U37_g(x1, x2, x3, x4)) = 0   
POL(U38_g(x1, x2)) = 0   
POL(U39_gga(x1, x2, x3)) = 0   
POL(U40_gga(x1, x2, x3)) = 0   
POL(U41_gga(x1, x2, x3)) = 0   
POL(U42_ga(x1, x2)) = 0   
POL(U7_G(x1, x2, x3, x4)) = 1 + x3   
POL(U9_G(x1, x2, x3, x4)) = 1 + x3   
POL([]) = 0   
POL(multc23_in_gga(x1, x2)) = 0   
POL(multc23_out_gga(x1, x2, x3)) = 0   
POL(multc53_in_ga(x1)) = 0   
POL(multc53_out_ga(x1, x2)) = 0   
POL(pc1_in_g(x1)) = 0   
POL(pc1_out_g(x1)) = 0   
POL(s(x1)) = 0   
POL(sumc34_in_gga(x1, x2)) = 0   
POL(sumc34_out_gga(x1, x2, x3)) = 0   

The following usable rules [FROCOS05] were oriented: none

(32) Obligation:

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

U7_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U9_G(T24, T25, T26, multc23_in_gga(T24, T25))
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → P1_IN_G(.(T39, T26))
P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → P1_IN_G(.(T24, .(T25, T26)))
U16_G(T92, T93, pc1_out_g(.(T92, T93))) → U18_G(T92, T93, multc53_in_ga(T92))
U18_G(T92, T93, multc53_out_ga(T92, T98)) → P1_IN_G(.(T98, T93))
U21_G(T131, T132, T133, pc1_out_g(.(T131, .(T132, T133)))) → U23_G(T131, T132, T133, multc23_in_gga(T131, T132))
U23_G(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → P1_IN_G(.(T146, T133))
U9_G(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U11_G(T24, T25, T26, pc1_in_g(.(T39, T26)))
U11_G(T24, T25, T26, pc1_out_g(.(T39, T26))) → U13_G(T24, T25, T26, multc23_in_gga(s(s(T24)), T25))
U13_G(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → P1_IN_G(.(T79, T26))

The TRS R consists of the following rules:

pc1_in_g(.(T3, [])) → pc1_out_g(.(T3, []))
pc1_in_g(.(s(s(s(s(T24)))), .(T25, T26))) → U27_g(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
pc1_in_g(.(s(s(0)), .(T92, T93))) → U32_g(T92, T93, pc1_in_g(.(T92, T93)))
pc1_in_g(.(0, .(T118, []))) → pc1_out_g(.(0, .(T118, [])))
pc1_in_g(.(0, .(s(s(T131)), .(T132, T133)))) → U35_g(T131, T132, T133, pc1_in_g(.(T131, .(T132, T133))))
pc1_in_g(.(0, .(0, T153))) → U38_g(T153, pc1_in_g(T153))
U38_g(T153, pc1_out_g(T153)) → pc1_out_g(.(0, .(0, T153)))
U35_g(T131, T132, T133, pc1_out_g(.(T131, .(T132, T133)))) → U36_g(T131, T132, T133, multc23_in_gga(T131, T132))
multc23_in_gga(T46, 0) → multc23_out_gga(T46, 0, 0)
multc23_in_gga(T51, s(T52)) → U39_gga(T51, T52, multc23_in_gga(T51, T52))
U39_gga(T51, T52, multc23_out_gga(T51, T52, T55)) → U40_gga(T51, T52, sumc34_in_gga(T55, T51))
sumc34_in_gga(T64, 0) → sumc34_out_gga(T64, 0, T64)
sumc34_in_gga(T69, s(T70)) → U41_gga(T69, T70, sumc34_in_gga(T69, T70))
U41_gga(T69, T70, sumc34_out_gga(T69, T70, X110)) → sumc34_out_gga(T69, s(T70), s(X110))
U40_gga(T51, T52, sumc34_out_gga(T55, T51, X83)) → multc23_out_gga(T51, s(T52), X83)
U36_g(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U37_g(T131, T132, T133, pc1_in_g(.(T146, T133)))
U37_g(T131, T132, T133, pc1_out_g(.(T146, T133))) → pc1_out_g(.(0, .(s(s(T131)), .(T132, T133))))
U32_g(T92, T93, pc1_out_g(.(T92, T93))) → U33_g(T92, T93, multc53_in_ga(T92))
multc53_in_ga(0) → multc53_out_ga(0, 0)
multc53_in_ga(s(T101)) → U42_ga(T101, multc53_in_ga(T101))
U42_ga(T101, multc53_out_ga(T101, T108)) → multc53_out_ga(s(T101), T108)
U33_g(T92, T93, multc53_out_ga(T92, T98)) → U34_g(T92, T93, pc1_in_g(.(T98, T93)))
U34_g(T92, T93, pc1_out_g(.(T98, T93))) → pc1_out_g(.(s(s(0)), .(T92, T93)))
U27_g(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U28_g(T24, T25, T26, multc23_in_gga(T24, T25))
U28_g(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U29_g(T24, T25, T26, pc1_in_g(.(T39, T26)))
U29_g(T24, T25, T26, pc1_out_g(.(T39, T26))) → U30_g(T24, T25, T26, multc23_in_gga(s(s(T24)), T25))
U30_g(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U31_g(T24, T25, T26, pc1_in_g(.(T79, T26)))
U31_g(T24, T25, T26, pc1_out_g(.(T79, T26))) → pc1_out_g(.(s(s(s(s(T24)))), .(T25, T26)))

The set Q consists of the following terms:

pc1_in_g(x0)
U38_g(x0, x1)
U35_g(x0, x1, x2, x3)
multc23_in_gga(x0, x1)
U39_gga(x0, x1, x2)
sumc34_in_gga(x0, x1)
U41_gga(x0, x1, x2)
U40_gga(x0, x1, x2)
U36_g(x0, x1, x2, x3)
U37_g(x0, x1, x2, x3)
U32_g(x0, x1, x2)
multc53_in_ga(x0)
U42_ga(x0, x1)
U33_g(x0, x1, x2)
U34_g(x0, x1, x2)
U27_g(x0, x1, x2, x3)
U28_g(x0, x1, x2, x3)
U29_g(x0, x1, x2, x3)
U30_g(x0, x1, x2, x3)
U31_g(x0, x1, x2, x3)

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

(33) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 9 less nodes.

(34) Obligation:

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

P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → P1_IN_G(.(T24, .(T25, T26)))

The TRS R consists of the following rules:

pc1_in_g(.(T3, [])) → pc1_out_g(.(T3, []))
pc1_in_g(.(s(s(s(s(T24)))), .(T25, T26))) → U27_g(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
pc1_in_g(.(s(s(0)), .(T92, T93))) → U32_g(T92, T93, pc1_in_g(.(T92, T93)))
pc1_in_g(.(0, .(T118, []))) → pc1_out_g(.(0, .(T118, [])))
pc1_in_g(.(0, .(s(s(T131)), .(T132, T133)))) → U35_g(T131, T132, T133, pc1_in_g(.(T131, .(T132, T133))))
pc1_in_g(.(0, .(0, T153))) → U38_g(T153, pc1_in_g(T153))
U38_g(T153, pc1_out_g(T153)) → pc1_out_g(.(0, .(0, T153)))
U35_g(T131, T132, T133, pc1_out_g(.(T131, .(T132, T133)))) → U36_g(T131, T132, T133, multc23_in_gga(T131, T132))
multc23_in_gga(T46, 0) → multc23_out_gga(T46, 0, 0)
multc23_in_gga(T51, s(T52)) → U39_gga(T51, T52, multc23_in_gga(T51, T52))
U39_gga(T51, T52, multc23_out_gga(T51, T52, T55)) → U40_gga(T51, T52, sumc34_in_gga(T55, T51))
sumc34_in_gga(T64, 0) → sumc34_out_gga(T64, 0, T64)
sumc34_in_gga(T69, s(T70)) → U41_gga(T69, T70, sumc34_in_gga(T69, T70))
U41_gga(T69, T70, sumc34_out_gga(T69, T70, X110)) → sumc34_out_gga(T69, s(T70), s(X110))
U40_gga(T51, T52, sumc34_out_gga(T55, T51, X83)) → multc23_out_gga(T51, s(T52), X83)
U36_g(T131, T132, T133, multc23_out_gga(T131, T132, T146)) → U37_g(T131, T132, T133, pc1_in_g(.(T146, T133)))
U37_g(T131, T132, T133, pc1_out_g(.(T146, T133))) → pc1_out_g(.(0, .(s(s(T131)), .(T132, T133))))
U32_g(T92, T93, pc1_out_g(.(T92, T93))) → U33_g(T92, T93, multc53_in_ga(T92))
multc53_in_ga(0) → multc53_out_ga(0, 0)
multc53_in_ga(s(T101)) → U42_ga(T101, multc53_in_ga(T101))
U42_ga(T101, multc53_out_ga(T101, T108)) → multc53_out_ga(s(T101), T108)
U33_g(T92, T93, multc53_out_ga(T92, T98)) → U34_g(T92, T93, pc1_in_g(.(T98, T93)))
U34_g(T92, T93, pc1_out_g(.(T98, T93))) → pc1_out_g(.(s(s(0)), .(T92, T93)))
U27_g(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U28_g(T24, T25, T26, multc23_in_gga(T24, T25))
U28_g(T24, T25, T26, multc23_out_gga(T24, T25, T39)) → U29_g(T24, T25, T26, pc1_in_g(.(T39, T26)))
U29_g(T24, T25, T26, pc1_out_g(.(T39, T26))) → U30_g(T24, T25, T26, multc23_in_gga(s(s(T24)), T25))
U30_g(T24, T25, T26, multc23_out_gga(s(s(T24)), T25, T79)) → U31_g(T24, T25, T26, pc1_in_g(.(T79, T26)))
U31_g(T24, T25, T26, pc1_out_g(.(T79, T26))) → pc1_out_g(.(s(s(s(s(T24)))), .(T25, T26)))

The set Q consists of the following terms:

pc1_in_g(x0)
U38_g(x0, x1)
U35_g(x0, x1, x2, x3)
multc23_in_gga(x0, x1)
U39_gga(x0, x1, x2)
sumc34_in_gga(x0, x1)
U41_gga(x0, x1, x2)
U40_gga(x0, x1, x2)
U36_g(x0, x1, x2, x3)
U37_g(x0, x1, x2, x3)
U32_g(x0, x1, x2)
multc53_in_ga(x0)
U42_ga(x0, x1)
U33_g(x0, x1, x2)
U34_g(x0, x1, x2)
U27_g(x0, x1, x2, x3)
U28_g(x0, x1, x2, x3)
U29_g(x0, x1, x2, x3)
U30_g(x0, x1, x2, x3)
U31_g(x0, x1, x2, x3)

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

(35) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(36) Obligation:

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

P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → P1_IN_G(.(T24, .(T25, T26)))

R is empty.
The set Q consists of the following terms:

pc1_in_g(x0)
U38_g(x0, x1)
U35_g(x0, x1, x2, x3)
multc23_in_gga(x0, x1)
U39_gga(x0, x1, x2)
sumc34_in_gga(x0, x1)
U41_gga(x0, x1, x2)
U40_gga(x0, x1, x2)
U36_g(x0, x1, x2, x3)
U37_g(x0, x1, x2, x3)
U32_g(x0, x1, x2)
multc53_in_ga(x0)
U42_ga(x0, x1)
U33_g(x0, x1, x2)
U34_g(x0, x1, x2)
U27_g(x0, x1, x2, x3)
U28_g(x0, x1, x2, x3)
U29_g(x0, x1, x2, x3)
U30_g(x0, x1, x2, x3)
U31_g(x0, x1, x2, x3)

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

(37) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

pc1_in_g(x0)
U38_g(x0, x1)
U35_g(x0, x1, x2, x3)
multc23_in_gga(x0, x1)
U39_gga(x0, x1, x2)
sumc34_in_gga(x0, x1)
U41_gga(x0, x1, x2)
U40_gga(x0, x1, x2)
U36_g(x0, x1, x2, x3)
U37_g(x0, x1, x2, x3)
U32_g(x0, x1, x2)
multc53_in_ga(x0)
U42_ga(x0, x1)
U33_g(x0, x1, x2)
U34_g(x0, x1, x2)
U27_g(x0, x1, x2, x3)
U28_g(x0, x1, x2, x3)
U29_g(x0, x1, x2, x3)
U30_g(x0, x1, x2, x3)
U31_g(x0, x1, x2, x3)

(38) Obligation:

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

P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → P1_IN_G(.(T24, .(T25, T26)))

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

(39) 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(.(s(s(s(s(T24)))), .(T25, T26))) → P1_IN_G(.(T24, .(T25, T26)))
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x1, x2)) = x1 + x2   
POL(P1_IN_G(x1)) = x1   
POL(s(x1)) = x1   

(40) Obligation:

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

(41) PisEmptyProof (EQUIVALENT transformation)

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

(42) YES