(0) Obligation:
Clauses:
div(X, Y, Z) :- quot(X, Y, Y, Z).
quot(0, s(Y), s(Z), 0).
quot(s(X), s(Y), Z, U) :- quot(X, Y, Z, U).
quot(X, 0, s(Z), s(U)) :- quot(X, s(Z), s(Z), U).
Query: div(g,g,a)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
quotA(s(X1), s(X2), X3, X4) :- quotA(X1, X2, X3, X4).
quotA(X1, 0, X2, s(X3)) :- quotB(X1, s(X2), X3).
quotB(s(s(s(s(s(s(s(s(X1)))))))), s(s(s(s(s(s(s(s(X2)))))))), X3) :- quotA(X1, X2, s(s(s(s(s(s(s(X2))))))), X3).
quotB(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(0))))))), s(X2)) :- quotC(X1, X2).
quotB(s(s(s(s(s(s(X1)))))), s(s(s(s(s(s(0)))))), s(X2)) :- quotD(X1, X2).
quotB(s(s(s(s(s(X1))))), s(s(s(s(s(0))))), s(X2)) :- quotE(X1, X2).
quotB(s(s(s(s(X1)))), s(s(s(s(0)))), s(X2)) :- quotF(X1, X2).
quotB(s(s(s(X1))), s(s(s(0))), s(X2)) :- quotG(X1, X2).
quotB(s(s(X1)), s(s(0)), s(X2)) :- quotH(X1, X2).
quotB(s(X1), s(0), s(X2)) :- quotI(X1, X2).
quotC(s(s(s(s(s(s(s(X1))))))), s(X2)) :- quotC(X1, X2).
quotD(s(s(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(X1)))), s(X2)) :- quotF(X1, X2).
quotG(s(s(s(X1))), s(X2)) :- quotG(X1, X2).
quotH(s(s(X1)), s(X2)) :- quotH(X1, X2).
quotI(s(X1), s(X2)) :- quotI(X1, X2).
divJ(X1, X2, X3) :- quotB(X1, X2, X3).
Clauses:
quotcA(0, s(X1), X2, 0).
quotcA(s(X1), s(X2), X3, X4) :- quotcA(X1, X2, X3, X4).
quotcA(X1, 0, X2, s(X3)) :- quotcB(X1, s(X2), X3).
quotcB(0, s(X1), 0).
quotcB(s(0), s(s(X1)), 0).
quotcB(s(s(0)), s(s(s(X1))), 0).
quotcB(s(s(s(0))), s(s(s(s(X1)))), 0).
quotcB(s(s(s(s(0)))), s(s(s(s(s(X1))))), 0).
quotcB(s(s(s(s(s(0))))), s(s(s(s(s(s(X1)))))), 0).
quotcB(s(s(s(s(s(s(0)))))), s(s(s(s(s(s(s(X1))))))), 0).
quotcB(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(s(X1)))))))), 0).
quotcB(s(s(s(s(s(s(s(s(X1)))))))), s(s(s(s(s(s(s(s(X2)))))))), X3) :- quotcA(X1, X2, s(s(s(s(s(s(s(X2))))))), X3).
quotcB(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(0))))))), s(X2)) :- quotcC(X1, X2).
quotcB(s(s(s(s(s(s(X1)))))), s(s(s(s(s(s(0)))))), s(X2)) :- quotcD(X1, X2).
quotcB(s(s(s(s(s(X1))))), s(s(s(s(s(0))))), s(X2)) :- quotcE(X1, X2).
quotcB(s(s(s(s(X1)))), s(s(s(s(0)))), s(X2)) :- quotcF(X1, X2).
quotcB(s(s(s(X1))), s(s(s(0))), s(X2)) :- quotcG(X1, X2).
quotcB(s(s(X1)), s(s(0)), s(X2)) :- quotcH(X1, X2).
quotcB(s(X1), s(0), s(X2)) :- quotcI(X1, X2).
quotcC(0, 0).
quotcC(s(0), 0).
quotcC(s(s(0)), 0).
quotcC(s(s(s(0))), 0).
quotcC(s(s(s(s(0)))), 0).
quotcC(s(s(s(s(s(0))))), 0).
quotcC(s(s(s(s(s(s(0)))))), 0).
quotcC(s(s(s(s(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(0)))), 0).
quotcD(s(s(s(s(s(0))))), 0).
quotcD(s(s(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(X1)))), s(X2)) :- quotcF(X1, X2).
quotcG(0, 0).
quotcG(s(0), 0).
quotcG(s(s(0)), 0).
quotcG(s(s(s(X1))), s(X2)) :- quotcG(X1, X2).
quotcH(0, 0).
quotcH(s(0), 0).
quotcH(s(s(X1)), s(X2)) :- quotcH(X1, X2).
quotcI(0, 0).
quotcI(s(X1), s(X2)) :- quotcI(X1, X2).
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)
quotB_in: (b,b,f)
quotA_in: (b,b,b,f)
quotC_in: (b,f)
quotD_in: (b,f)
quotE_in: (b,f)
quotF_in: (b,f)
quotG_in: (b,f)
quotH_in: (b,f)
quotI_in: (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, quotB_in_gga(X1, X2, X3))
DIVJ_IN_GGA(X1, X2, X3) → QUOTB_IN_GGA(X1, X2, X3)
QUOTB_IN_GGA(s(s(s(s(s(s(s(s(X1)))))))), s(s(s(s(s(s(s(s(X2)))))))), X3) → U3_GGA(X1, X2, X3, quotA_in_ggga(X1, X2, s(s(s(s(s(s(s(X2))))))), X3))
QUOTB_IN_GGA(s(s(s(s(s(s(s(s(X1)))))))), s(s(s(s(s(s(s(s(X2)))))))), X3) → QUOTA_IN_GGGA(X1, X2, s(s(s(s(s(s(s(X2))))))), X3)
QUOTA_IN_GGGA(s(X1), s(X2), X3, X4) → U1_GGGA(X1, X2, X3, X4, quotA_in_ggga(X1, X2, X3, X4))
QUOTA_IN_GGGA(s(X1), s(X2), X3, X4) → QUOTA_IN_GGGA(X1, X2, X3, X4)
QUOTA_IN_GGGA(X1, 0, X2, s(X3)) → U2_GGGA(X1, X2, X3, quotB_in_gga(X1, s(X2), X3))
QUOTA_IN_GGGA(X1, 0, X2, s(X3)) → QUOTB_IN_GGA(X1, s(X2), X3)
QUOTB_IN_GGA(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(0))))))), s(X2)) → U4_GGA(X1, X2, quotC_in_ga(X1, X2))
QUOTB_IN_GGA(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(0))))))), s(X2)) → QUOTC_IN_GA(X1, X2)
QUOTC_IN_GA(s(s(s(s(s(s(s(X1))))))), s(X2)) → U11_GA(X1, X2, quotC_in_ga(X1, X2))
QUOTC_IN_GA(s(s(s(s(s(s(s(X1))))))), s(X2)) → QUOTC_IN_GA(X1, X2)
QUOTB_IN_GGA(s(s(s(s(s(s(X1)))))), s(s(s(s(s(s(0)))))), s(X2)) → U5_GGA(X1, X2, quotD_in_ga(X1, X2))
QUOTB_IN_GGA(s(s(s(s(s(s(X1)))))), s(s(s(s(s(s(0)))))), s(X2)) → QUOTD_IN_GA(X1, X2)
QUOTD_IN_GA(s(s(s(s(s(s(X1)))))), s(X2)) → U12_GA(X1, X2, quotD_in_ga(X1, X2))
QUOTD_IN_GA(s(s(s(s(s(s(X1)))))), s(X2)) → QUOTD_IN_GA(X1, X2)
QUOTB_IN_GGA(s(s(s(s(s(X1))))), s(s(s(s(s(0))))), s(X2)) → U6_GGA(X1, X2, quotE_in_ga(X1, X2))
QUOTB_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)) → U13_GA(X1, X2, quotE_in_ga(X1, X2))
QUOTE_IN_GA(s(s(s(s(s(X1))))), s(X2)) → QUOTE_IN_GA(X1, X2)
QUOTB_IN_GGA(s(s(s(s(X1)))), s(s(s(s(0)))), s(X2)) → U7_GGA(X1, X2, quotF_in_ga(X1, X2))
QUOTB_IN_GGA(s(s(s(s(X1)))), s(s(s(s(0)))), s(X2)) → QUOTF_IN_GA(X1, X2)
QUOTF_IN_GA(s(s(s(s(X1)))), s(X2)) → U14_GA(X1, X2, quotF_in_ga(X1, X2))
QUOTF_IN_GA(s(s(s(s(X1)))), s(X2)) → QUOTF_IN_GA(X1, X2)
QUOTB_IN_GGA(s(s(s(X1))), s(s(s(0))), s(X2)) → U8_GGA(X1, X2, quotG_in_ga(X1, X2))
QUOTB_IN_GGA(s(s(s(X1))), s(s(s(0))), s(X2)) → QUOTG_IN_GA(X1, X2)
QUOTG_IN_GA(s(s(s(X1))), s(X2)) → U15_GA(X1, X2, quotG_in_ga(X1, X2))
QUOTG_IN_GA(s(s(s(X1))), s(X2)) → QUOTG_IN_GA(X1, X2)
QUOTB_IN_GGA(s(s(X1)), s(s(0)), s(X2)) → U9_GGA(X1, X2, quotH_in_ga(X1, X2))
QUOTB_IN_GGA(s(s(X1)), s(s(0)), s(X2)) → QUOTH_IN_GA(X1, X2)
QUOTH_IN_GA(s(s(X1)), s(X2)) → U16_GA(X1, X2, quotH_in_ga(X1, X2))
QUOTH_IN_GA(s(s(X1)), s(X2)) → QUOTH_IN_GA(X1, X2)
QUOTB_IN_GGA(s(X1), s(0), s(X2)) → U10_GGA(X1, X2, quotI_in_ga(X1, X2))
QUOTB_IN_GGA(s(X1), s(0), s(X2)) → QUOTI_IN_GA(X1, X2)
QUOTI_IN_GA(s(X1), s(X2)) → U17_GA(X1, X2, quotI_in_ga(X1, X2))
QUOTI_IN_GA(s(X1), s(X2)) → QUOTI_IN_GA(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
quotB_in_gga(
x1,
x2,
x3) =
quotB_in_gga(
x1,
x2)
s(
x1) =
s(
x1)
quotA_in_ggga(
x1,
x2,
x3,
x4) =
quotA_in_ggga(
x1,
x2,
x3)
0 =
0
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)
quotH_in_ga(
x1,
x2) =
quotH_in_ga(
x1)
quotI_in_ga(
x1,
x2) =
quotI_in_ga(
x1)
DIVJ_IN_GGA(
x1,
x2,
x3) =
DIVJ_IN_GGA(
x1,
x2)
U18_GGA(
x1,
x2,
x3,
x4) =
U18_GGA(
x1,
x2,
x4)
QUOTB_IN_GGA(
x1,
x2,
x3) =
QUOTB_IN_GGA(
x1,
x2)
U3_GGA(
x1,
x2,
x3,
x4) =
U3_GGA(
x1,
x2,
x4)
QUOTA_IN_GGGA(
x1,
x2,
x3,
x4) =
QUOTA_IN_GGGA(
x1,
x2,
x3)
U1_GGGA(
x1,
x2,
x3,
x4,
x5) =
U1_GGGA(
x1,
x2,
x3,
x5)
U2_GGGA(
x1,
x2,
x3,
x4) =
U2_GGGA(
x1,
x2,
x4)
U4_GGA(
x1,
x2,
x3) =
U4_GGA(
x1,
x3)
QUOTC_IN_GA(
x1,
x2) =
QUOTC_IN_GA(
x1)
U11_GA(
x1,
x2,
x3) =
U11_GA(
x1,
x3)
U5_GGA(
x1,
x2,
x3) =
U5_GGA(
x1,
x3)
QUOTD_IN_GA(
x1,
x2) =
QUOTD_IN_GA(
x1)
U12_GA(
x1,
x2,
x3) =
U12_GA(
x1,
x3)
U6_GGA(
x1,
x2,
x3) =
U6_GGA(
x1,
x3)
QUOTE_IN_GA(
x1,
x2) =
QUOTE_IN_GA(
x1)
U13_GA(
x1,
x2,
x3) =
U13_GA(
x1,
x3)
U7_GGA(
x1,
x2,
x3) =
U7_GGA(
x1,
x3)
QUOTF_IN_GA(
x1,
x2) =
QUOTF_IN_GA(
x1)
U14_GA(
x1,
x2,
x3) =
U14_GA(
x1,
x3)
U8_GGA(
x1,
x2,
x3) =
U8_GGA(
x1,
x3)
QUOTG_IN_GA(
x1,
x2) =
QUOTG_IN_GA(
x1)
U15_GA(
x1,
x2,
x3) =
U15_GA(
x1,
x3)
U9_GGA(
x1,
x2,
x3) =
U9_GGA(
x1,
x3)
QUOTH_IN_GA(
x1,
x2) =
QUOTH_IN_GA(
x1)
U16_GA(
x1,
x2,
x3) =
U16_GA(
x1,
x3)
U10_GGA(
x1,
x2,
x3) =
U10_GGA(
x1,
x3)
QUOTI_IN_GA(
x1,
x2) =
QUOTI_IN_GA(
x1)
U17_GA(
x1,
x2,
x3) =
U17_GA(
x1,
x3)
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, quotB_in_gga(X1, X2, X3))
DIVJ_IN_GGA(X1, X2, X3) → QUOTB_IN_GGA(X1, X2, X3)
QUOTB_IN_GGA(s(s(s(s(s(s(s(s(X1)))))))), s(s(s(s(s(s(s(s(X2)))))))), X3) → U3_GGA(X1, X2, X3, quotA_in_ggga(X1, X2, s(s(s(s(s(s(s(X2))))))), X3))
QUOTB_IN_GGA(s(s(s(s(s(s(s(s(X1)))))))), s(s(s(s(s(s(s(s(X2)))))))), X3) → QUOTA_IN_GGGA(X1, X2, s(s(s(s(s(s(s(X2))))))), X3)
QUOTA_IN_GGGA(s(X1), s(X2), X3, X4) → U1_GGGA(X1, X2, X3, X4, quotA_in_ggga(X1, X2, X3, X4))
QUOTA_IN_GGGA(s(X1), s(X2), X3, X4) → QUOTA_IN_GGGA(X1, X2, X3, X4)
QUOTA_IN_GGGA(X1, 0, X2, s(X3)) → U2_GGGA(X1, X2, X3, quotB_in_gga(X1, s(X2), X3))
QUOTA_IN_GGGA(X1, 0, X2, s(X3)) → QUOTB_IN_GGA(X1, s(X2), X3)
QUOTB_IN_GGA(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(0))))))), s(X2)) → U4_GGA(X1, X2, quotC_in_ga(X1, X2))
QUOTB_IN_GGA(s(s(s(s(s(s(s(X1))))))), s(s(s(s(s(s(s(0))))))), s(X2)) → QUOTC_IN_GA(X1, X2)
QUOTC_IN_GA(s(s(s(s(s(s(s(X1))))))), s(X2)) → U11_GA(X1, X2, quotC_in_ga(X1, X2))
QUOTC_IN_GA(s(s(s(s(s(s(s(X1))))))), s(X2)) → QUOTC_IN_GA(X1, X2)
QUOTB_IN_GGA(s(s(s(s(s(s(X1)))))), s(s(s(s(s(s(0)))))), s(X2)) → U5_GGA(X1, X2, quotD_in_ga(X1, X2))
QUOTB_IN_GGA(s(s(s(s(s(s(X1)))))), s(s(s(s(s(s(0)))))), s(X2)) → QUOTD_IN_GA(X1, X2)
QUOTD_IN_GA(s(s(s(s(s(s(X1)))))), s(X2)) → U12_GA(X1, X2, quotD_in_ga(X1, X2))
QUOTD_IN_GA(s(s(s(s(s(s(X1)))))), s(X2)) → QUOTD_IN_GA(X1, X2)
QUOTB_IN_GGA(s(s(s(s(s(X1))))), s(s(s(s(s(0))))), s(X2)) → U6_GGA(X1, X2, quotE_in_ga(X1, X2))
QUOTB_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)) → U13_GA(X1, X2, quotE_in_ga(X1, X2))
QUOTE_IN_GA(s(s(s(s(s(X1))))), s(X2)) → QUOTE_IN_GA(X1, X2)
QUOTB_IN_GGA(s(s(s(s(X1)))), s(s(s(s(0)))), s(X2)) → U7_GGA(X1, X2, quotF_in_ga(X1, X2))
QUOTB_IN_GGA(s(s(s(s(X1)))), s(s(s(s(0)))), s(X2)) → QUOTF_IN_GA(X1, X2)
QUOTF_IN_GA(s(s(s(s(X1)))), s(X2)) → U14_GA(X1, X2, quotF_in_ga(X1, X2))
QUOTF_IN_GA(s(s(s(s(X1)))), s(X2)) → QUOTF_IN_GA(X1, X2)
QUOTB_IN_GGA(s(s(s(X1))), s(s(s(0))), s(X2)) → U8_GGA(X1, X2, quotG_in_ga(X1, X2))
QUOTB_IN_GGA(s(s(s(X1))), s(s(s(0))), s(X2)) → QUOTG_IN_GA(X1, X2)
QUOTG_IN_GA(s(s(s(X1))), s(X2)) → U15_GA(X1, X2, quotG_in_ga(X1, X2))
QUOTG_IN_GA(s(s(s(X1))), s(X2)) → QUOTG_IN_GA(X1, X2)
QUOTB_IN_GGA(s(s(X1)), s(s(0)), s(X2)) → U9_GGA(X1, X2, quotH_in_ga(X1, X2))
QUOTB_IN_GGA(s(s(X1)), s(s(0)), s(X2)) → QUOTH_IN_GA(X1, X2)
QUOTH_IN_GA(s(s(X1)), s(X2)) → U16_GA(X1, X2, quotH_in_ga(X1, X2))
QUOTH_IN_GA(s(s(X1)), s(X2)) → QUOTH_IN_GA(X1, X2)
QUOTB_IN_GGA(s(X1), s(0), s(X2)) → U10_GGA(X1, X2, quotI_in_ga(X1, X2))
QUOTB_IN_GGA(s(X1), s(0), s(X2)) → QUOTI_IN_GA(X1, X2)
QUOTI_IN_GA(s(X1), s(X2)) → U17_GA(X1, X2, quotI_in_ga(X1, X2))
QUOTI_IN_GA(s(X1), s(X2)) → QUOTI_IN_GA(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
quotB_in_gga(
x1,
x2,
x3) =
quotB_in_gga(
x1,
x2)
s(
x1) =
s(
x1)
quotA_in_ggga(
x1,
x2,
x3,
x4) =
quotA_in_ggga(
x1,
x2,
x3)
0 =
0
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)
quotH_in_ga(
x1,
x2) =
quotH_in_ga(
x1)
quotI_in_ga(
x1,
x2) =
quotI_in_ga(
x1)
DIVJ_IN_GGA(
x1,
x2,
x3) =
DIVJ_IN_GGA(
x1,
x2)
U18_GGA(
x1,
x2,
x3,
x4) =
U18_GGA(
x1,
x2,
x4)
QUOTB_IN_GGA(
x1,
x2,
x3) =
QUOTB_IN_GGA(
x1,
x2)
U3_GGA(
x1,
x2,
x3,
x4) =
U3_GGA(
x1,
x2,
x4)
QUOTA_IN_GGGA(
x1,
x2,
x3,
x4) =
QUOTA_IN_GGGA(
x1,
x2,
x3)
U1_GGGA(
x1,
x2,
x3,
x4,
x5) =
U1_GGGA(
x1,
x2,
x3,
x5)
U2_GGGA(
x1,
x2,
x3,
x4) =
U2_GGGA(
x1,
x2,
x4)
U4_GGA(
x1,
x2,
x3) =
U4_GGA(
x1,
x3)
QUOTC_IN_GA(
x1,
x2) =
QUOTC_IN_GA(
x1)
U11_GA(
x1,
x2,
x3) =
U11_GA(
x1,
x3)
U5_GGA(
x1,
x2,
x3) =
U5_GGA(
x1,
x3)
QUOTD_IN_GA(
x1,
x2) =
QUOTD_IN_GA(
x1)
U12_GA(
x1,
x2,
x3) =
U12_GA(
x1,
x3)
U6_GGA(
x1,
x2,
x3) =
U6_GGA(
x1,
x3)
QUOTE_IN_GA(
x1,
x2) =
QUOTE_IN_GA(
x1)
U13_GA(
x1,
x2,
x3) =
U13_GA(
x1,
x3)
U7_GGA(
x1,
x2,
x3) =
U7_GGA(
x1,
x3)
QUOTF_IN_GA(
x1,
x2) =
QUOTF_IN_GA(
x1)
U14_GA(
x1,
x2,
x3) =
U14_GA(
x1,
x3)
U8_GGA(
x1,
x2,
x3) =
U8_GGA(
x1,
x3)
QUOTG_IN_GA(
x1,
x2) =
QUOTG_IN_GA(
x1)
U15_GA(
x1,
x2,
x3) =
U15_GA(
x1,
x3)
U9_GGA(
x1,
x2,
x3) =
U9_GGA(
x1,
x3)
QUOTH_IN_GA(
x1,
x2) =
QUOTH_IN_GA(
x1)
U16_GA(
x1,
x2,
x3) =
U16_GA(
x1,
x3)
U10_GGA(
x1,
x2,
x3) =
U10_GGA(
x1,
x3)
QUOTI_IN_GA(
x1,
x2) =
QUOTI_IN_GA(
x1)
U17_GA(
x1,
x2,
x3) =
U17_GA(
x1,
x3)
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:
QUOTI_IN_GA(s(X1), s(X2)) → QUOTI_IN_GA(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
QUOTI_IN_GA(
x1,
x2) =
QUOTI_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:
QUOTI_IN_GA(s(X1)) → QUOTI_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:
- QUOTI_IN_GA(s(X1)) → QUOTI_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:
QUOTH_IN_GA(s(s(X1)), s(X2)) → QUOTH_IN_GA(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
QUOTH_IN_GA(
x1,
x2) =
QUOTH_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:
QUOTH_IN_GA(s(s(X1))) → QUOTH_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:
- QUOTH_IN_GA(s(s(X1))) → QUOTH_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:
QUOTG_IN_GA(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
(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:
QUOTG_IN_GA(s(s(s(X1)))) → QUOTG_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:
- QUOTG_IN_GA(s(s(s(X1)))) → QUOTG_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:
QUOTF_IN_GA(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
(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:
QUOTF_IN_GA(s(s(s(s(X1))))) → QUOTF_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:
- QUOTF_IN_GA(s(s(s(s(X1))))) → QUOTF_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:
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
(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:
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.
(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:
- QUOTE_IN_GA(s(s(s(s(s(X1)))))) → QUOTE_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:
QUOTD_IN_GA(s(s(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
(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:
QUOTD_IN_GA(s(s(s(s(s(s(X1))))))) → QUOTD_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:
- QUOTD_IN_GA(s(s(s(s(s(s(X1))))))) → QUOTD_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:
QUOTC_IN_GA(s(s(s(s(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
(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:
QUOTC_IN_GA(s(s(s(s(s(s(s(X1)))))))) → QUOTC_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:
- QUOTC_IN_GA(s(s(s(s(s(s(s(X1)))))))) → QUOTC_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:
QUOTB_IN_GGA(s(s(s(s(s(s(s(s(X1)))))))), s(s(s(s(s(s(s(s(X2)))))))), X3) → QUOTA_IN_GGGA(X1, X2, s(s(s(s(s(s(s(X2))))))), X3)
QUOTA_IN_GGGA(s(X1), s(X2), X3, X4) → QUOTA_IN_GGGA(X1, X2, X3, X4)
QUOTA_IN_GGGA(X1, 0, X2, s(X3)) → QUOTB_IN_GGA(X1, s(X2), X3)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
0 =
0
QUOTB_IN_GGA(
x1,
x2,
x3) =
QUOTB_IN_GGA(
x1,
x2)
QUOTA_IN_GGGA(
x1,
x2,
x3,
x4) =
QUOTA_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:
QUOTB_IN_GGA(s(s(s(s(s(s(s(s(X1)))))))), s(s(s(s(s(s(s(s(X2))))))))) → QUOTA_IN_GGGA(X1, X2, s(s(s(s(s(s(s(X2))))))))
QUOTA_IN_GGGA(s(X1), s(X2), X3) → QUOTA_IN_GGGA(X1, X2, X3)
QUOTA_IN_GGGA(X1, 0, X2) → QUOTB_IN_GGA(X1, s(X2))
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:
- QUOTA_IN_GGGA(X1, 0, X2) → QUOTB_IN_GGA(X1, s(X2))
The graph contains the following edges 1 >= 1
- QUOTA_IN_GGGA(s(X1), s(X2), X3) → QUOTA_IN_GGGA(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
- QUOTB_IN_GGA(s(s(s(s(s(s(s(s(X1)))))))), s(s(s(s(s(s(s(s(X2))))))))) → QUOTA_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