(0) Obligation:

Clauses:

div(X, Y, Z) :- quot(X, Y, Y, Z).
quot(0, s(Y), s(Z), 0).
quot(X, 0, s(Z), s(U)) :- quot(X, s(Z), s(Z), U).
quot(X, Y, Z, U) :- ','(no(zero(X)), ','(no(zero(Y)), ','(p(X, Px), ','(p(Y, Py), quot(Px, Py, Z, U))))).
p(0, 0).
p(s(X), X).
zero(0).
no(X) :- ','(X, ','(!, failure(a))).
no(X1).
failure(b).

Query: div(g,g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

(2) Obligation:

Triples:

quotA(s(X1), s(X2)) :- quotA(X1, X2).
quotB(s(s(X1)), s(X2)) :- quotB(X1, X2).
quotC(s(s(s(X1))), s(X2)) :- quotC(X1, X2).
quotD(s(s(s(s(X1)))), s(X2)) :- quotD(X1, X2).
quotE(s(s(s(s(s(X1))))), s(X2)) :- quotE(X1, X2).
quotF(s(s(s(s(s(s(X1)))))), s(X2)) :- quotF(X1, X2).
quotG(s(s(s(s(s(s(s(X1))))))), s(X2)) :- quotG(X1, X2).
quotH(s(X1), s(0), s(X2)) :- quotA(X1, X2).
quotH(s(s(X1)), s(s(0)), s(X2)) :- quotB(X1, X2).
quotH(s(s(s(X1))), s(s(s(0))), s(X2)) :- quotC(X1, X2).
quotH(s(s(s(s(X1)))), s(s(s(s(0)))), s(X2)) :- quotD(X1, X2).
quotH(s(s(s(s(s(X1))))), s(s(s(s(s(0))))), s(X2)) :- quotE(X1, X2).
quotH(s(s(s(s(s(s(X1)))))), s(s(s(s(s(s(0)))))), s(X2)) :- quotF(X1, X2).
quotH(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(0))))))), s(X2)) :- quotG(X1, X2).
quotH(s(s(s(s(s(s(s(s(X1)))))))), s(s(s(s(s(s(s(s(X2)))))))), X3) :- quotI(X1, X2, s(s(s(s(s(s(s(X2))))))), X3).
quotI(X1, 0, X2, s(X3)) :- quotH(X1, s(X2), X3).
quotI(s(X1), s(X2), X3, X4) :- quotI(X1, X2, X3, X4).
divJ(X1, X2, X3) :- quotH(X1, X2, X3).

Clauses:

quotcA(0, 0).
quotcA(s(X1), s(X2)) :- quotcA(X1, X2).
quotcB(0, 0).
quotcB(s(0), 0).
quotcB(s(s(X1)), s(X2)) :- quotcB(X1, X2).
quotcC(0, 0).
quotcC(s(0), 0).
quotcC(s(s(0)), 0).
quotcC(s(s(s(X1))), s(X2)) :- quotcC(X1, X2).
quotcD(0, 0).
quotcD(s(0), 0).
quotcD(s(s(0)), 0).
quotcD(s(s(s(0))), 0).
quotcD(s(s(s(s(X1)))), s(X2)) :- quotcD(X1, X2).
quotcE(0, 0).
quotcE(s(0), 0).
quotcE(s(s(0)), 0).
quotcE(s(s(s(0))), 0).
quotcE(s(s(s(s(0)))), 0).
quotcE(s(s(s(s(s(X1))))), s(X2)) :- quotcE(X1, X2).
quotcF(0, 0).
quotcF(s(0), 0).
quotcF(s(s(0)), 0).
quotcF(s(s(s(0))), 0).
quotcF(s(s(s(s(0)))), 0).
quotcF(s(s(s(s(s(0))))), 0).
quotcF(s(s(s(s(s(s(X1)))))), s(X2)) :- quotcF(X1, X2).
quotcG(0, 0).
quotcG(s(0), 0).
quotcG(s(s(0)), 0).
quotcG(s(s(s(0))), 0).
quotcG(s(s(s(s(0)))), 0).
quotcG(s(s(s(s(s(0))))), 0).
quotcG(s(s(s(s(s(s(0)))))), 0).
quotcG(s(s(s(s(s(s(s(X1))))))), s(X2)) :- quotcG(X1, X2).
quotcH(0, s(X1), 0).
quotcH(s(0), s(s(X1)), 0).
quotcH(s(X1), s(0), s(X2)) :- quotcA(X1, X2).
quotcH(s(s(0)), s(s(s(X1))), 0).
quotcH(s(s(X1)), s(s(0)), s(X2)) :- quotcB(X1, X2).
quotcH(s(s(s(0))), s(s(s(s(X1)))), 0).
quotcH(s(s(s(X1))), s(s(s(0))), s(X2)) :- quotcC(X1, X2).
quotcH(s(s(s(s(0)))), s(s(s(s(s(X1))))), 0).
quotcH(s(s(s(s(X1)))), s(s(s(s(0)))), s(X2)) :- quotcD(X1, X2).
quotcH(s(s(s(s(s(0))))), s(s(s(s(s(s(X1)))))), 0).
quotcH(s(s(s(s(s(X1))))), s(s(s(s(s(0))))), s(X2)) :- quotcE(X1, X2).
quotcH(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X1))))))), 0).
quotcH(s(s(s(s(s(s(X1)))))), s(s(s(s(s(s(0)))))), s(X2)) :- quotcF(X1, X2).
quotcH(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(X1)))))))), 0).
quotcH(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(0))))))), s(X2)) :- quotcG(X1, X2).
quotcH(s(s(s(s(s(s(s(s(X1)))))))), s(s(s(s(s(s(s(s(X2)))))))), X3) :- quotcI(X1, X2, s(s(s(s(s(s(s(X2))))))), X3).
quotcI(0, s(X1), X2, 0).
quotcI(X1, 0, X2, s(X3)) :- quotcH(X1, s(X2), X3).
quotcI(s(X1), s(X2), X3, X4) :- quotcI(X1, X2, X3, X4).

Afs:

divJ(x1, x2, x3)  =  divJ(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
divJ_in: (b,b,f)
quotH_in: (b,b,f)
quotA_in: (b,f)
quotB_in: (b,f)
quotC_in: (b,f)
quotD_in: (b,f)
quotE_in: (b,f)
quotF_in: (b,f)
quotG_in: (b,f)
quotI_in: (b,b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

DIVJ_IN_GGA(X1, X2, X3) → U18_GGA(X1, X2, X3, quotH_in_gga(X1, X2, X3))
DIVJ_IN_GGA(X1, X2, X3) → QUOTH_IN_GGA(X1, X2, X3)
QUOTH_IN_GGA(s(X1), s(0), s(X2)) → U8_GGA(X1, X2, quotA_in_ga(X1, X2))
QUOTH_IN_GGA(s(X1), s(0), s(X2)) → QUOTA_IN_GA(X1, X2)
QUOTA_IN_GA(s(X1), s(X2)) → U1_GA(X1, X2, quotA_in_ga(X1, X2))
QUOTA_IN_GA(s(X1), s(X2)) → QUOTA_IN_GA(X1, X2)
QUOTH_IN_GGA(s(s(X1)), s(s(0)), s(X2)) → U9_GGA(X1, X2, quotB_in_ga(X1, X2))
QUOTH_IN_GGA(s(s(X1)), s(s(0)), s(X2)) → QUOTB_IN_GA(X1, X2)
QUOTB_IN_GA(s(s(X1)), s(X2)) → U2_GA(X1, X2, quotB_in_ga(X1, X2))
QUOTB_IN_GA(s(s(X1)), s(X2)) → QUOTB_IN_GA(X1, X2)
QUOTH_IN_GGA(s(s(s(X1))), s(s(s(0))), s(X2)) → U10_GGA(X1, X2, quotC_in_ga(X1, X2))
QUOTH_IN_GGA(s(s(s(X1))), s(s(s(0))), s(X2)) → QUOTC_IN_GA(X1, X2)
QUOTC_IN_GA(s(s(s(X1))), s(X2)) → U3_GA(X1, X2, quotC_in_ga(X1, X2))
QUOTC_IN_GA(s(s(s(X1))), s(X2)) → QUOTC_IN_GA(X1, X2)
QUOTH_IN_GGA(s(s(s(s(X1)))), s(s(s(s(0)))), s(X2)) → U11_GGA(X1, X2, quotD_in_ga(X1, X2))
QUOTH_IN_GGA(s(s(s(s(X1)))), s(s(s(s(0)))), s(X2)) → QUOTD_IN_GA(X1, X2)
QUOTD_IN_GA(s(s(s(s(X1)))), s(X2)) → U4_GA(X1, X2, quotD_in_ga(X1, X2))
QUOTD_IN_GA(s(s(s(s(X1)))), s(X2)) → QUOTD_IN_GA(X1, X2)
QUOTH_IN_GGA(s(s(s(s(s(X1))))), s(s(s(s(s(0))))), s(X2)) → U12_GGA(X1, X2, quotE_in_ga(X1, X2))
QUOTH_IN_GGA(s(s(s(s(s(X1))))), s(s(s(s(s(0))))), s(X2)) → QUOTE_IN_GA(X1, X2)
QUOTE_IN_GA(s(s(s(s(s(X1))))), s(X2)) → U5_GA(X1, X2, quotE_in_ga(X1, X2))
QUOTE_IN_GA(s(s(s(s(s(X1))))), s(X2)) → QUOTE_IN_GA(X1, X2)
QUOTH_IN_GGA(s(s(s(s(s(s(X1)))))), s(s(s(s(s(s(0)))))), s(X2)) → U13_GGA(X1, X2, quotF_in_ga(X1, X2))
QUOTH_IN_GGA(s(s(s(s(s(s(X1)))))), s(s(s(s(s(s(0)))))), s(X2)) → QUOTF_IN_GA(X1, X2)
QUOTF_IN_GA(s(s(s(s(s(s(X1)))))), s(X2)) → U6_GA(X1, X2, quotF_in_ga(X1, X2))
QUOTF_IN_GA(s(s(s(s(s(s(X1)))))), s(X2)) → QUOTF_IN_GA(X1, X2)
QUOTH_IN_GGA(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(0))))))), s(X2)) → U14_GGA(X1, X2, quotG_in_ga(X1, X2))
QUOTH_IN_GGA(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(0))))))), s(X2)) → QUOTG_IN_GA(X1, X2)
QUOTG_IN_GA(s(s(s(s(s(s(s(X1))))))), s(X2)) → U7_GA(X1, X2, quotG_in_ga(X1, X2))
QUOTG_IN_GA(s(s(s(s(s(s(s(X1))))))), s(X2)) → QUOTG_IN_GA(X1, X2)
QUOTH_IN_GGA(s(s(s(s(s(s(s(s(X1)))))))), s(s(s(s(s(s(s(s(X2)))))))), X3) → U15_GGA(X1, X2, X3, quotI_in_ggga(X1, X2, s(s(s(s(s(s(s(X2))))))), X3))
QUOTH_IN_GGA(s(s(s(s(s(s(s(s(X1)))))))), s(s(s(s(s(s(s(s(X2)))))))), X3) → QUOTI_IN_GGGA(X1, X2, s(s(s(s(s(s(s(X2))))))), X3)
QUOTI_IN_GGGA(X1, 0, X2, s(X3)) → U16_GGGA(X1, X2, X3, quotH_in_gga(X1, s(X2), X3))
QUOTI_IN_GGGA(X1, 0, X2, s(X3)) → QUOTH_IN_GGA(X1, s(X2), X3)
QUOTI_IN_GGGA(s(X1), s(X2), X3, X4) → U17_GGGA(X1, X2, X3, X4, quotI_in_ggga(X1, X2, X3, X4))
QUOTI_IN_GGGA(s(X1), s(X2), X3, X4) → QUOTI_IN_GGGA(X1, X2, X3, X4)

R is empty.
The argument filtering Pi contains the following mapping:
quotH_in_gga(x1, x2, x3)  =  quotH_in_gga(x1, x2)
s(x1)  =  s(x1)
0  =  0
quotA_in_ga(x1, x2)  =  quotA_in_ga(x1)
quotB_in_ga(x1, x2)  =  quotB_in_ga(x1)
quotC_in_ga(x1, x2)  =  quotC_in_ga(x1)
quotD_in_ga(x1, x2)  =  quotD_in_ga(x1)
quotE_in_ga(x1, x2)  =  quotE_in_ga(x1)
quotF_in_ga(x1, x2)  =  quotF_in_ga(x1)
quotG_in_ga(x1, x2)  =  quotG_in_ga(x1)
quotI_in_ggga(x1, x2, x3, x4)  =  quotI_in_ggga(x1, x2, x3)
DIVJ_IN_GGA(x1, x2, x3)  =  DIVJ_IN_GGA(x1, x2)
U18_GGA(x1, x2, x3, x4)  =  U18_GGA(x1, x2, x4)
QUOTH_IN_GGA(x1, x2, x3)  =  QUOTH_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3)  =  U8_GGA(x1, x3)
QUOTA_IN_GA(x1, x2)  =  QUOTA_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U9_GGA(x1, x2, x3)  =  U9_GGA(x1, x3)
QUOTB_IN_GA(x1, x2)  =  QUOTB_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U10_GGA(x1, x2, x3)  =  U10_GGA(x1, x3)
QUOTC_IN_GA(x1, x2)  =  QUOTC_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U11_GGA(x1, x2, x3)  =  U11_GGA(x1, x3)
QUOTD_IN_GA(x1, x2)  =  QUOTD_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U12_GGA(x1, x2, x3)  =  U12_GGA(x1, x3)
QUOTE_IN_GA(x1, x2)  =  QUOTE_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U13_GGA(x1, x2, x3)  =  U13_GGA(x1, x3)
QUOTF_IN_GA(x1, x2)  =  QUOTF_IN_GA(x1)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
U14_GGA(x1, x2, x3)  =  U14_GGA(x1, x3)
QUOTG_IN_GA(x1, x2)  =  QUOTG_IN_GA(x1)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U15_GGA(x1, x2, x3, x4)  =  U15_GGA(x1, x2, x4)
QUOTI_IN_GGGA(x1, x2, x3, x4)  =  QUOTI_IN_GGGA(x1, x2, x3)
U16_GGGA(x1, x2, x3, x4)  =  U16_GGGA(x1, x2, x4)
U17_GGGA(x1, x2, x3, x4, x5)  =  U17_GGGA(x1, x2, x3, x5)

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:

DIVJ_IN_GGA(X1, X2, X3) → U18_GGA(X1, X2, X3, quotH_in_gga(X1, X2, X3))
DIVJ_IN_GGA(X1, X2, X3) → QUOTH_IN_GGA(X1, X2, X3)
QUOTH_IN_GGA(s(X1), s(0), s(X2)) → U8_GGA(X1, X2, quotA_in_ga(X1, X2))
QUOTH_IN_GGA(s(X1), s(0), s(X2)) → QUOTA_IN_GA(X1, X2)
QUOTA_IN_GA(s(X1), s(X2)) → U1_GA(X1, X2, quotA_in_ga(X1, X2))
QUOTA_IN_GA(s(X1), s(X2)) → QUOTA_IN_GA(X1, X2)
QUOTH_IN_GGA(s(s(X1)), s(s(0)), s(X2)) → U9_GGA(X1, X2, quotB_in_ga(X1, X2))
QUOTH_IN_GGA(s(s(X1)), s(s(0)), s(X2)) → QUOTB_IN_GA(X1, X2)
QUOTB_IN_GA(s(s(X1)), s(X2)) → U2_GA(X1, X2, quotB_in_ga(X1, X2))
QUOTB_IN_GA(s(s(X1)), s(X2)) → QUOTB_IN_GA(X1, X2)
QUOTH_IN_GGA(s(s(s(X1))), s(s(s(0))), s(X2)) → U10_GGA(X1, X2, quotC_in_ga(X1, X2))
QUOTH_IN_GGA(s(s(s(X1))), s(s(s(0))), s(X2)) → QUOTC_IN_GA(X1, X2)
QUOTC_IN_GA(s(s(s(X1))), s(X2)) → U3_GA(X1, X2, quotC_in_ga(X1, X2))
QUOTC_IN_GA(s(s(s(X1))), s(X2)) → QUOTC_IN_GA(X1, X2)
QUOTH_IN_GGA(s(s(s(s(X1)))), s(s(s(s(0)))), s(X2)) → U11_GGA(X1, X2, quotD_in_ga(X1, X2))
QUOTH_IN_GGA(s(s(s(s(X1)))), s(s(s(s(0)))), s(X2)) → QUOTD_IN_GA(X1, X2)
QUOTD_IN_GA(s(s(s(s(X1)))), s(X2)) → U4_GA(X1, X2, quotD_in_ga(X1, X2))
QUOTD_IN_GA(s(s(s(s(X1)))), s(X2)) → QUOTD_IN_GA(X1, X2)
QUOTH_IN_GGA(s(s(s(s(s(X1))))), s(s(s(s(s(0))))), s(X2)) → U12_GGA(X1, X2, quotE_in_ga(X1, X2))
QUOTH_IN_GGA(s(s(s(s(s(X1))))), s(s(s(s(s(0))))), s(X2)) → QUOTE_IN_GA(X1, X2)
QUOTE_IN_GA(s(s(s(s(s(X1))))), s(X2)) → U5_GA(X1, X2, quotE_in_ga(X1, X2))
QUOTE_IN_GA(s(s(s(s(s(X1))))), s(X2)) → QUOTE_IN_GA(X1, X2)
QUOTH_IN_GGA(s(s(s(s(s(s(X1)))))), s(s(s(s(s(s(0)))))), s(X2)) → U13_GGA(X1, X2, quotF_in_ga(X1, X2))
QUOTH_IN_GGA(s(s(s(s(s(s(X1)))))), s(s(s(s(s(s(0)))))), s(X2)) → QUOTF_IN_GA(X1, X2)
QUOTF_IN_GA(s(s(s(s(s(s(X1)))))), s(X2)) → U6_GA(X1, X2, quotF_in_ga(X1, X2))
QUOTF_IN_GA(s(s(s(s(s(s(X1)))))), s(X2)) → QUOTF_IN_GA(X1, X2)
QUOTH_IN_GGA(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(0))))))), s(X2)) → U14_GGA(X1, X2, quotG_in_ga(X1, X2))
QUOTH_IN_GGA(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(0))))))), s(X2)) → QUOTG_IN_GA(X1, X2)
QUOTG_IN_GA(s(s(s(s(s(s(s(X1))))))), s(X2)) → U7_GA(X1, X2, quotG_in_ga(X1, X2))
QUOTG_IN_GA(s(s(s(s(s(s(s(X1))))))), s(X2)) → QUOTG_IN_GA(X1, X2)
QUOTH_IN_GGA(s(s(s(s(s(s(s(s(X1)))))))), s(s(s(s(s(s(s(s(X2)))))))), X3) → U15_GGA(X1, X2, X3, quotI_in_ggga(X1, X2, s(s(s(s(s(s(s(X2))))))), X3))
QUOTH_IN_GGA(s(s(s(s(s(s(s(s(X1)))))))), s(s(s(s(s(s(s(s(X2)))))))), X3) → QUOTI_IN_GGGA(X1, X2, s(s(s(s(s(s(s(X2))))))), X3)
QUOTI_IN_GGGA(X1, 0, X2, s(X3)) → U16_GGGA(X1, X2, X3, quotH_in_gga(X1, s(X2), X3))
QUOTI_IN_GGGA(X1, 0, X2, s(X3)) → QUOTH_IN_GGA(X1, s(X2), X3)
QUOTI_IN_GGGA(s(X1), s(X2), X3, X4) → U17_GGGA(X1, X2, X3, X4, quotI_in_ggga(X1, X2, X3, X4))
QUOTI_IN_GGGA(s(X1), s(X2), X3, X4) → QUOTI_IN_GGGA(X1, X2, X3, X4)

R is empty.
The argument filtering Pi contains the following mapping:
quotH_in_gga(x1, x2, x3)  =  quotH_in_gga(x1, x2)
s(x1)  =  s(x1)
0  =  0
quotA_in_ga(x1, x2)  =  quotA_in_ga(x1)
quotB_in_ga(x1, x2)  =  quotB_in_ga(x1)
quotC_in_ga(x1, x2)  =  quotC_in_ga(x1)
quotD_in_ga(x1, x2)  =  quotD_in_ga(x1)
quotE_in_ga(x1, x2)  =  quotE_in_ga(x1)
quotF_in_ga(x1, x2)  =  quotF_in_ga(x1)
quotG_in_ga(x1, x2)  =  quotG_in_ga(x1)
quotI_in_ggga(x1, x2, x3, x4)  =  quotI_in_ggga(x1, x2, x3)
DIVJ_IN_GGA(x1, x2, x3)  =  DIVJ_IN_GGA(x1, x2)
U18_GGA(x1, x2, x3, x4)  =  U18_GGA(x1, x2, x4)
QUOTH_IN_GGA(x1, x2, x3)  =  QUOTH_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3)  =  U8_GGA(x1, x3)
QUOTA_IN_GA(x1, x2)  =  QUOTA_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U9_GGA(x1, x2, x3)  =  U9_GGA(x1, x3)
QUOTB_IN_GA(x1, x2)  =  QUOTB_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U10_GGA(x1, x2, x3)  =  U10_GGA(x1, x3)
QUOTC_IN_GA(x1, x2)  =  QUOTC_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U11_GGA(x1, x2, x3)  =  U11_GGA(x1, x3)
QUOTD_IN_GA(x1, x2)  =  QUOTD_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U12_GGA(x1, x2, x3)  =  U12_GGA(x1, x3)
QUOTE_IN_GA(x1, x2)  =  QUOTE_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U13_GGA(x1, x2, x3)  =  U13_GGA(x1, x3)
QUOTF_IN_GA(x1, x2)  =  QUOTF_IN_GA(x1)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
U14_GGA(x1, x2, x3)  =  U14_GGA(x1, x3)
QUOTG_IN_GA(x1, x2)  =  QUOTG_IN_GA(x1)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U15_GGA(x1, x2, x3, x4)  =  U15_GGA(x1, x2, x4)
QUOTI_IN_GGGA(x1, x2, x3, x4)  =  QUOTI_IN_GGGA(x1, x2, x3)
U16_GGGA(x1, x2, x3, x4)  =  U16_GGGA(x1, x2, x4)
U17_GGGA(x1, x2, x3, x4, x5)  =  U17_GGGA(x1, x2, x3, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 8 SCCs with 26 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

QUOTG_IN_GA(s(s(s(s(s(s(s(X1))))))), s(X2)) → QUOTG_IN_GA(X1, X2)

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

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

(8) PiDPToQDPProof (SOUND transformation)

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

(9) Obligation:

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

QUOTG_IN_GA(s(s(s(s(s(s(s(X1)))))))) → QUOTG_IN_GA(X1)

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

(10) 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:

  • QUOTG_IN_GA(s(s(s(s(s(s(s(X1)))))))) → QUOTG_IN_GA(X1)
    The graph contains the following edges 1 > 1

(11) YES

(12) Obligation:

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

QUOTF_IN_GA(s(s(s(s(s(s(X1)))))), s(X2)) → QUOTF_IN_GA(X1, X2)

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

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

(13) PiDPToQDPProof (SOUND transformation)

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

(14) Obligation:

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

QUOTF_IN_GA(s(s(s(s(s(s(X1))))))) → QUOTF_IN_GA(X1)

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

(15) 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:

  • QUOTF_IN_GA(s(s(s(s(s(s(X1))))))) → QUOTF_IN_GA(X1)
    The graph contains the following edges 1 > 1

(16) YES

(17) Obligation:

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

QUOTE_IN_GA(s(s(s(s(s(X1))))), s(X2)) → QUOTE_IN_GA(X1, X2)

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

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

(18) PiDPToQDPProof (SOUND transformation)

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

(19) Obligation:

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

QUOTE_IN_GA(s(s(s(s(s(X1)))))) → QUOTE_IN_GA(X1)

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

(20) 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:

  • QUOTE_IN_GA(s(s(s(s(s(X1)))))) → QUOTE_IN_GA(X1)
    The graph contains the following edges 1 > 1

(21) YES

(22) Obligation:

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

QUOTD_IN_GA(s(s(s(s(X1)))), s(X2)) → QUOTD_IN_GA(X1, X2)

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

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

(23) PiDPToQDPProof (SOUND transformation)

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

(24) Obligation:

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

QUOTD_IN_GA(s(s(s(s(X1))))) → QUOTD_IN_GA(X1)

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

(25) 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:

  • QUOTD_IN_GA(s(s(s(s(X1))))) → QUOTD_IN_GA(X1)
    The graph contains the following edges 1 > 1

(26) YES

(27) Obligation:

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

QUOTC_IN_GA(s(s(s(X1))), s(X2)) → QUOTC_IN_GA(X1, X2)

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

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

(28) PiDPToQDPProof (SOUND transformation)

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

(29) Obligation:

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

QUOTC_IN_GA(s(s(s(X1)))) → QUOTC_IN_GA(X1)

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

(30) 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:

  • QUOTC_IN_GA(s(s(s(X1)))) → QUOTC_IN_GA(X1)
    The graph contains the following edges 1 > 1

(31) YES

(32) Obligation:

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

QUOTB_IN_GA(s(s(X1)), s(X2)) → QUOTB_IN_GA(X1, X2)

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

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:

QUOTB_IN_GA(s(s(X1))) → QUOTB_IN_GA(X1)

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:

  • QUOTB_IN_GA(s(s(X1))) → QUOTB_IN_GA(X1)
    The graph contains the following edges 1 > 1

(36) YES

(37) Obligation:

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

QUOTA_IN_GA(s(X1), s(X2)) → QUOTA_IN_GA(X1, X2)

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

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

(38) PiDPToQDPProof (SOUND transformation)

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

(39) Obligation:

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

QUOTA_IN_GA(s(X1)) → QUOTA_IN_GA(X1)

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

(40) 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:

  • QUOTA_IN_GA(s(X1)) → QUOTA_IN_GA(X1)
    The graph contains the following edges 1 > 1

(41) YES

(42) Obligation:

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

QUOTH_IN_GGA(s(s(s(s(s(s(s(s(X1)))))))), s(s(s(s(s(s(s(s(X2)))))))), X3) → QUOTI_IN_GGGA(X1, X2, s(s(s(s(s(s(s(X2))))))), X3)
QUOTI_IN_GGGA(X1, 0, X2, s(X3)) → QUOTH_IN_GGA(X1, s(X2), X3)
QUOTI_IN_GGGA(s(X1), s(X2), X3, X4) → QUOTI_IN_GGGA(X1, X2, X3, X4)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
QUOTH_IN_GGA(x1, x2, x3)  =  QUOTH_IN_GGA(x1, x2)
QUOTI_IN_GGGA(x1, x2, x3, x4)  =  QUOTI_IN_GGGA(x1, x2, x3)

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

(43) PiDPToQDPProof (SOUND transformation)

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

(44) Obligation:

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

QUOTH_IN_GGA(s(s(s(s(s(s(s(s(X1)))))))), s(s(s(s(s(s(s(s(X2))))))))) → QUOTI_IN_GGGA(X1, X2, s(s(s(s(s(s(s(X2))))))))
QUOTI_IN_GGGA(X1, 0, X2) → QUOTH_IN_GGA(X1, s(X2))
QUOTI_IN_GGGA(s(X1), s(X2), X3) → QUOTI_IN_GGGA(X1, X2, X3)

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

(45) 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:

  • QUOTI_IN_GGGA(X1, 0, X2) → QUOTH_IN_GGA(X1, s(X2))
    The graph contains the following edges 1 >= 1

  • QUOTI_IN_GGGA(s(X1), s(X2), X3) → QUOTI_IN_GGGA(X1, X2, X3)
    The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3

  • QUOTH_IN_GGA(s(s(s(s(s(s(s(s(X1)))))))), s(s(s(s(s(s(s(s(X2))))))))) → QUOTI_IN_GGGA(X1, X2, s(s(s(s(s(s(s(X2))))))))
    The graph contains the following edges 1 > 1, 2 > 2, 2 > 3

(46) YES