(0) Obligation:
Clauses:
times(X, Y, Z) :- mult(X, Y, 0, Z).
mult(0, Y, 0, 0).
mult(s(U), Y, 0, Z) :- mult(U, Y, Y, Z).
mult(X, Y, s(W), s(Z)) :- mult(X, Y, W, Z).
Query: times(g,g,a)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
multA(s(X1), X2) :- multA(X1, X2).
multB(s(X1), s(X2)) :- multB(X1, X2).
multC(s(X1), s(s(X2))) :- multC(X1, X2).
multD(s(X1), s(s(s(X2)))) :- multD(X1, X2).
multE(s(X1), s(s(s(s(X2))))) :- multE(X1, X2).
multF(s(X1), s(s(s(s(s(X2)))))) :- multF(X1, X2).
multG(s(X1), s(s(s(s(s(s(X2))))))) :- multG(X1, X2).
multH(s(X1), s(s(s(s(s(s(s(X2)))))))) :- multH(X1, X2).
multI(s(X1), 0, X2) :- multA(X1, X2).
multI(s(X1), s(0), s(X2)) :- multB(X1, X2).
multI(s(X1), s(s(0)), s(s(X2))) :- multC(X1, X2).
multI(s(X1), s(s(s(0))), s(s(s(X2)))) :- multD(X1, X2).
multI(s(X1), s(s(s(s(0)))), s(s(s(s(X2))))) :- multE(X1, X2).
multI(s(X1), s(s(s(s(s(0))))), s(s(s(s(s(X2)))))) :- multF(X1, X2).
multI(s(X1), s(s(s(s(s(s(0)))))), s(s(s(s(s(s(X2))))))) :- multG(X1, X2).
multI(s(X1), s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(X2)))))))) :- multH(X1, X2).
multI(X1, s(s(s(s(s(s(s(s(X2)))))))), s(s(s(s(s(s(s(s(X3))))))))) :- multJ(X1, s(s(s(s(s(s(s(X2))))))), X2, X3).
multJ(s(X1), X2, 0, X3) :- multI(X1, s(X2), X3).
multJ(X1, X2, s(X3), s(X4)) :- multJ(X1, X2, X3, X4).
timesK(s(X1), X2, X3) :- multI(X1, X2, X3).
Clauses:
multcA(0, 0).
multcA(s(X1), X2) :- multcA(X1, X2).
multcB(0, s(0)).
multcB(s(X1), s(X2)) :- multcB(X1, X2).
multcC(0, s(s(0))).
multcC(s(X1), s(s(X2))) :- multcC(X1, X2).
multcD(0, s(s(s(0)))).
multcD(s(X1), s(s(s(X2)))) :- multcD(X1, X2).
multcE(0, s(s(s(s(0))))).
multcE(s(X1), s(s(s(s(X2))))) :- multcE(X1, X2).
multcF(0, s(s(s(s(s(0)))))).
multcF(s(X1), s(s(s(s(s(X2)))))) :- multcF(X1, X2).
multcG(0, s(s(s(s(s(s(0))))))).
multcG(s(X1), s(s(s(s(s(s(X2))))))) :- multcG(X1, X2).
multcH(0, s(s(s(s(s(s(s(0)))))))).
multcH(s(X1), s(s(s(s(s(s(s(X2)))))))) :- multcH(X1, X2).
multcI(0, 0, 0).
multcI(s(X1), 0, X2) :- multcA(X1, X2).
multcI(0, s(0), s(0)).
multcI(s(X1), s(0), s(X2)) :- multcB(X1, X2).
multcI(0, s(s(0)), s(s(0))).
multcI(s(X1), s(s(0)), s(s(X2))) :- multcC(X1, X2).
multcI(0, s(s(s(0))), s(s(s(0)))).
multcI(s(X1), s(s(s(0))), s(s(s(X2)))) :- multcD(X1, X2).
multcI(0, s(s(s(s(0)))), s(s(s(s(0))))).
multcI(s(X1), s(s(s(s(0)))), s(s(s(s(X2))))) :- multcE(X1, X2).
multcI(0, s(s(s(s(s(0))))), s(s(s(s(s(0)))))).
multcI(s(X1), s(s(s(s(s(0))))), s(s(s(s(s(X2)))))) :- multcF(X1, X2).
multcI(0, s(s(s(s(s(s(0)))))), s(s(s(s(s(s(0))))))).
multcI(s(X1), s(s(s(s(s(s(0)))))), s(s(s(s(s(s(X2))))))) :- multcG(X1, X2).
multcI(0, s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(0)))))))).
multcI(s(X1), s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(X2)))))))) :- multcH(X1, X2).
multcI(X1, s(s(s(s(s(s(s(s(X2)))))))), s(s(s(s(s(s(s(s(X3))))))))) :- multcJ(X1, s(s(s(s(s(s(s(X2))))))), X2, X3).
multcJ(0, X1, 0, 0).
multcJ(s(X1), X2, 0, X3) :- multcI(X1, s(X2), X3).
multcJ(X1, X2, s(X3), s(X4)) :- multcJ(X1, X2, X3, X4).
Afs:
timesK(x1, x2, x3) = timesK(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:
timesK_in: (b,b,f)
multI_in: (b,b,f)
multA_in: (b,f)
multB_in: (b,f)
multC_in: (b,f)
multD_in: (b,f)
multE_in: (b,f)
multF_in: (b,f)
multG_in: (b,f)
multH_in: (b,f)
multJ_in: (b,b,b,f)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
TIMESK_IN_GGA(s(X1), X2, X3) → U20_GGA(X1, X2, X3, multI_in_gga(X1, X2, X3))
TIMESK_IN_GGA(s(X1), X2, X3) → MULTI_IN_GGA(X1, X2, X3)
MULTI_IN_GGA(s(X1), 0, X2) → U9_GGA(X1, X2, multA_in_ga(X1, X2))
MULTI_IN_GGA(s(X1), 0, X2) → MULTA_IN_GA(X1, X2)
MULTA_IN_GA(s(X1), X2) → U1_GA(X1, X2, multA_in_ga(X1, X2))
MULTA_IN_GA(s(X1), X2) → MULTA_IN_GA(X1, X2)
MULTI_IN_GGA(s(X1), s(0), s(X2)) → U10_GGA(X1, X2, multB_in_ga(X1, X2))
MULTI_IN_GGA(s(X1), s(0), s(X2)) → MULTB_IN_GA(X1, X2)
MULTB_IN_GA(s(X1), s(X2)) → U2_GA(X1, X2, multB_in_ga(X1, X2))
MULTB_IN_GA(s(X1), s(X2)) → MULTB_IN_GA(X1, X2)
MULTI_IN_GGA(s(X1), s(s(0)), s(s(X2))) → U11_GGA(X1, X2, multC_in_ga(X1, X2))
MULTI_IN_GGA(s(X1), s(s(0)), s(s(X2))) → MULTC_IN_GA(X1, X2)
MULTC_IN_GA(s(X1), s(s(X2))) → U3_GA(X1, X2, multC_in_ga(X1, X2))
MULTC_IN_GA(s(X1), s(s(X2))) → MULTC_IN_GA(X1, X2)
MULTI_IN_GGA(s(X1), s(s(s(0))), s(s(s(X2)))) → U12_GGA(X1, X2, multD_in_ga(X1, X2))
MULTI_IN_GGA(s(X1), s(s(s(0))), s(s(s(X2)))) → MULTD_IN_GA(X1, X2)
MULTD_IN_GA(s(X1), s(s(s(X2)))) → U4_GA(X1, X2, multD_in_ga(X1, X2))
MULTD_IN_GA(s(X1), s(s(s(X2)))) → MULTD_IN_GA(X1, X2)
MULTI_IN_GGA(s(X1), s(s(s(s(0)))), s(s(s(s(X2))))) → U13_GGA(X1, X2, multE_in_ga(X1, X2))
MULTI_IN_GGA(s(X1), s(s(s(s(0)))), s(s(s(s(X2))))) → MULTE_IN_GA(X1, X2)
MULTE_IN_GA(s(X1), s(s(s(s(X2))))) → U5_GA(X1, X2, multE_in_ga(X1, X2))
MULTE_IN_GA(s(X1), s(s(s(s(X2))))) → MULTE_IN_GA(X1, X2)
MULTI_IN_GGA(s(X1), s(s(s(s(s(0))))), s(s(s(s(s(X2)))))) → U14_GGA(X1, X2, multF_in_ga(X1, X2))
MULTI_IN_GGA(s(X1), s(s(s(s(s(0))))), s(s(s(s(s(X2)))))) → MULTF_IN_GA(X1, X2)
MULTF_IN_GA(s(X1), s(s(s(s(s(X2)))))) → U6_GA(X1, X2, multF_in_ga(X1, X2))
MULTF_IN_GA(s(X1), s(s(s(s(s(X2)))))) → MULTF_IN_GA(X1, X2)
MULTI_IN_GGA(s(X1), s(s(s(s(s(s(0)))))), s(s(s(s(s(s(X2))))))) → U15_GGA(X1, X2, multG_in_ga(X1, X2))
MULTI_IN_GGA(s(X1), s(s(s(s(s(s(0)))))), s(s(s(s(s(s(X2))))))) → MULTG_IN_GA(X1, X2)
MULTG_IN_GA(s(X1), s(s(s(s(s(s(X2))))))) → U7_GA(X1, X2, multG_in_ga(X1, X2))
MULTG_IN_GA(s(X1), s(s(s(s(s(s(X2))))))) → MULTG_IN_GA(X1, X2)
MULTI_IN_GGA(s(X1), s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(X2)))))))) → U16_GGA(X1, X2, multH_in_ga(X1, X2))
MULTI_IN_GGA(s(X1), s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(X2)))))))) → MULTH_IN_GA(X1, X2)
MULTH_IN_GA(s(X1), s(s(s(s(s(s(s(X2)))))))) → U8_GA(X1, X2, multH_in_ga(X1, X2))
MULTH_IN_GA(s(X1), s(s(s(s(s(s(s(X2)))))))) → MULTH_IN_GA(X1, X2)
MULTI_IN_GGA(X1, s(s(s(s(s(s(s(s(X2)))))))), s(s(s(s(s(s(s(s(X3))))))))) → U17_GGA(X1, X2, X3, multJ_in_ggga(X1, s(s(s(s(s(s(s(X2))))))), X2, X3))
MULTI_IN_GGA(X1, s(s(s(s(s(s(s(s(X2)))))))), s(s(s(s(s(s(s(s(X3))))))))) → MULTJ_IN_GGGA(X1, s(s(s(s(s(s(s(X2))))))), X2, X3)
MULTJ_IN_GGGA(s(X1), X2, 0, X3) → U18_GGGA(X1, X2, X3, multI_in_gga(X1, s(X2), X3))
MULTJ_IN_GGGA(s(X1), X2, 0, X3) → MULTI_IN_GGA(X1, s(X2), X3)
MULTJ_IN_GGGA(X1, X2, s(X3), s(X4)) → U19_GGGA(X1, X2, X3, X4, multJ_in_ggga(X1, X2, X3, X4))
MULTJ_IN_GGGA(X1, X2, s(X3), s(X4)) → MULTJ_IN_GGGA(X1, X2, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
multI_in_gga(
x1,
x2,
x3) =
multI_in_gga(
x1,
x2)
0 =
0
multA_in_ga(
x1,
x2) =
multA_in_ga(
x1)
multB_in_ga(
x1,
x2) =
multB_in_ga(
x1)
multC_in_ga(
x1,
x2) =
multC_in_ga(
x1)
multD_in_ga(
x1,
x2) =
multD_in_ga(
x1)
multE_in_ga(
x1,
x2) =
multE_in_ga(
x1)
multF_in_ga(
x1,
x2) =
multF_in_ga(
x1)
multG_in_ga(
x1,
x2) =
multG_in_ga(
x1)
multH_in_ga(
x1,
x2) =
multH_in_ga(
x1)
multJ_in_ggga(
x1,
x2,
x3,
x4) =
multJ_in_ggga(
x1,
x2,
x3)
TIMESK_IN_GGA(
x1,
x2,
x3) =
TIMESK_IN_GGA(
x1,
x2)
U20_GGA(
x1,
x2,
x3,
x4) =
U20_GGA(
x1,
x2,
x4)
MULTI_IN_GGA(
x1,
x2,
x3) =
MULTI_IN_GGA(
x1,
x2)
U9_GGA(
x1,
x2,
x3) =
U9_GGA(
x1,
x3)
MULTA_IN_GA(
x1,
x2) =
MULTA_IN_GA(
x1)
U1_GA(
x1,
x2,
x3) =
U1_GA(
x1,
x3)
U10_GGA(
x1,
x2,
x3) =
U10_GGA(
x1,
x3)
MULTB_IN_GA(
x1,
x2) =
MULTB_IN_GA(
x1)
U2_GA(
x1,
x2,
x3) =
U2_GA(
x1,
x3)
U11_GGA(
x1,
x2,
x3) =
U11_GGA(
x1,
x3)
MULTC_IN_GA(
x1,
x2) =
MULTC_IN_GA(
x1)
U3_GA(
x1,
x2,
x3) =
U3_GA(
x1,
x3)
U12_GGA(
x1,
x2,
x3) =
U12_GGA(
x1,
x3)
MULTD_IN_GA(
x1,
x2) =
MULTD_IN_GA(
x1)
U4_GA(
x1,
x2,
x3) =
U4_GA(
x1,
x3)
U13_GGA(
x1,
x2,
x3) =
U13_GGA(
x1,
x3)
MULTE_IN_GA(
x1,
x2) =
MULTE_IN_GA(
x1)
U5_GA(
x1,
x2,
x3) =
U5_GA(
x1,
x3)
U14_GGA(
x1,
x2,
x3) =
U14_GGA(
x1,
x3)
MULTF_IN_GA(
x1,
x2) =
MULTF_IN_GA(
x1)
U6_GA(
x1,
x2,
x3) =
U6_GA(
x1,
x3)
U15_GGA(
x1,
x2,
x3) =
U15_GGA(
x1,
x3)
MULTG_IN_GA(
x1,
x2) =
MULTG_IN_GA(
x1)
U7_GA(
x1,
x2,
x3) =
U7_GA(
x1,
x3)
U16_GGA(
x1,
x2,
x3) =
U16_GGA(
x1,
x3)
MULTH_IN_GA(
x1,
x2) =
MULTH_IN_GA(
x1)
U8_GA(
x1,
x2,
x3) =
U8_GA(
x1,
x3)
U17_GGA(
x1,
x2,
x3,
x4) =
U17_GGA(
x1,
x2,
x4)
MULTJ_IN_GGGA(
x1,
x2,
x3,
x4) =
MULTJ_IN_GGGA(
x1,
x2,
x3)
U18_GGGA(
x1,
x2,
x3,
x4) =
U18_GGGA(
x1,
x2,
x4)
U19_GGGA(
x1,
x2,
x3,
x4,
x5) =
U19_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:
TIMESK_IN_GGA(s(X1), X2, X3) → U20_GGA(X1, X2, X3, multI_in_gga(X1, X2, X3))
TIMESK_IN_GGA(s(X1), X2, X3) → MULTI_IN_GGA(X1, X2, X3)
MULTI_IN_GGA(s(X1), 0, X2) → U9_GGA(X1, X2, multA_in_ga(X1, X2))
MULTI_IN_GGA(s(X1), 0, X2) → MULTA_IN_GA(X1, X2)
MULTA_IN_GA(s(X1), X2) → U1_GA(X1, X2, multA_in_ga(X1, X2))
MULTA_IN_GA(s(X1), X2) → MULTA_IN_GA(X1, X2)
MULTI_IN_GGA(s(X1), s(0), s(X2)) → U10_GGA(X1, X2, multB_in_ga(X1, X2))
MULTI_IN_GGA(s(X1), s(0), s(X2)) → MULTB_IN_GA(X1, X2)
MULTB_IN_GA(s(X1), s(X2)) → U2_GA(X1, X2, multB_in_ga(X1, X2))
MULTB_IN_GA(s(X1), s(X2)) → MULTB_IN_GA(X1, X2)
MULTI_IN_GGA(s(X1), s(s(0)), s(s(X2))) → U11_GGA(X1, X2, multC_in_ga(X1, X2))
MULTI_IN_GGA(s(X1), s(s(0)), s(s(X2))) → MULTC_IN_GA(X1, X2)
MULTC_IN_GA(s(X1), s(s(X2))) → U3_GA(X1, X2, multC_in_ga(X1, X2))
MULTC_IN_GA(s(X1), s(s(X2))) → MULTC_IN_GA(X1, X2)
MULTI_IN_GGA(s(X1), s(s(s(0))), s(s(s(X2)))) → U12_GGA(X1, X2, multD_in_ga(X1, X2))
MULTI_IN_GGA(s(X1), s(s(s(0))), s(s(s(X2)))) → MULTD_IN_GA(X1, X2)
MULTD_IN_GA(s(X1), s(s(s(X2)))) → U4_GA(X1, X2, multD_in_ga(X1, X2))
MULTD_IN_GA(s(X1), s(s(s(X2)))) → MULTD_IN_GA(X1, X2)
MULTI_IN_GGA(s(X1), s(s(s(s(0)))), s(s(s(s(X2))))) → U13_GGA(X1, X2, multE_in_ga(X1, X2))
MULTI_IN_GGA(s(X1), s(s(s(s(0)))), s(s(s(s(X2))))) → MULTE_IN_GA(X1, X2)
MULTE_IN_GA(s(X1), s(s(s(s(X2))))) → U5_GA(X1, X2, multE_in_ga(X1, X2))
MULTE_IN_GA(s(X1), s(s(s(s(X2))))) → MULTE_IN_GA(X1, X2)
MULTI_IN_GGA(s(X1), s(s(s(s(s(0))))), s(s(s(s(s(X2)))))) → U14_GGA(X1, X2, multF_in_ga(X1, X2))
MULTI_IN_GGA(s(X1), s(s(s(s(s(0))))), s(s(s(s(s(X2)))))) → MULTF_IN_GA(X1, X2)
MULTF_IN_GA(s(X1), s(s(s(s(s(X2)))))) → U6_GA(X1, X2, multF_in_ga(X1, X2))
MULTF_IN_GA(s(X1), s(s(s(s(s(X2)))))) → MULTF_IN_GA(X1, X2)
MULTI_IN_GGA(s(X1), s(s(s(s(s(s(0)))))), s(s(s(s(s(s(X2))))))) → U15_GGA(X1, X2, multG_in_ga(X1, X2))
MULTI_IN_GGA(s(X1), s(s(s(s(s(s(0)))))), s(s(s(s(s(s(X2))))))) → MULTG_IN_GA(X1, X2)
MULTG_IN_GA(s(X1), s(s(s(s(s(s(X2))))))) → U7_GA(X1, X2, multG_in_ga(X1, X2))
MULTG_IN_GA(s(X1), s(s(s(s(s(s(X2))))))) → MULTG_IN_GA(X1, X2)
MULTI_IN_GGA(s(X1), s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(X2)))))))) → U16_GGA(X1, X2, multH_in_ga(X1, X2))
MULTI_IN_GGA(s(X1), s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(X2)))))))) → MULTH_IN_GA(X1, X2)
MULTH_IN_GA(s(X1), s(s(s(s(s(s(s(X2)))))))) → U8_GA(X1, X2, multH_in_ga(X1, X2))
MULTH_IN_GA(s(X1), s(s(s(s(s(s(s(X2)))))))) → MULTH_IN_GA(X1, X2)
MULTI_IN_GGA(X1, s(s(s(s(s(s(s(s(X2)))))))), s(s(s(s(s(s(s(s(X3))))))))) → U17_GGA(X1, X2, X3, multJ_in_ggga(X1, s(s(s(s(s(s(s(X2))))))), X2, X3))
MULTI_IN_GGA(X1, s(s(s(s(s(s(s(s(X2)))))))), s(s(s(s(s(s(s(s(X3))))))))) → MULTJ_IN_GGGA(X1, s(s(s(s(s(s(s(X2))))))), X2, X3)
MULTJ_IN_GGGA(s(X1), X2, 0, X3) → U18_GGGA(X1, X2, X3, multI_in_gga(X1, s(X2), X3))
MULTJ_IN_GGGA(s(X1), X2, 0, X3) → MULTI_IN_GGA(X1, s(X2), X3)
MULTJ_IN_GGGA(X1, X2, s(X3), s(X4)) → U19_GGGA(X1, X2, X3, X4, multJ_in_ggga(X1, X2, X3, X4))
MULTJ_IN_GGGA(X1, X2, s(X3), s(X4)) → MULTJ_IN_GGGA(X1, X2, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
multI_in_gga(
x1,
x2,
x3) =
multI_in_gga(
x1,
x2)
0 =
0
multA_in_ga(
x1,
x2) =
multA_in_ga(
x1)
multB_in_ga(
x1,
x2) =
multB_in_ga(
x1)
multC_in_ga(
x1,
x2) =
multC_in_ga(
x1)
multD_in_ga(
x1,
x2) =
multD_in_ga(
x1)
multE_in_ga(
x1,
x2) =
multE_in_ga(
x1)
multF_in_ga(
x1,
x2) =
multF_in_ga(
x1)
multG_in_ga(
x1,
x2) =
multG_in_ga(
x1)
multH_in_ga(
x1,
x2) =
multH_in_ga(
x1)
multJ_in_ggga(
x1,
x2,
x3,
x4) =
multJ_in_ggga(
x1,
x2,
x3)
TIMESK_IN_GGA(
x1,
x2,
x3) =
TIMESK_IN_GGA(
x1,
x2)
U20_GGA(
x1,
x2,
x3,
x4) =
U20_GGA(
x1,
x2,
x4)
MULTI_IN_GGA(
x1,
x2,
x3) =
MULTI_IN_GGA(
x1,
x2)
U9_GGA(
x1,
x2,
x3) =
U9_GGA(
x1,
x3)
MULTA_IN_GA(
x1,
x2) =
MULTA_IN_GA(
x1)
U1_GA(
x1,
x2,
x3) =
U1_GA(
x1,
x3)
U10_GGA(
x1,
x2,
x3) =
U10_GGA(
x1,
x3)
MULTB_IN_GA(
x1,
x2) =
MULTB_IN_GA(
x1)
U2_GA(
x1,
x2,
x3) =
U2_GA(
x1,
x3)
U11_GGA(
x1,
x2,
x3) =
U11_GGA(
x1,
x3)
MULTC_IN_GA(
x1,
x2) =
MULTC_IN_GA(
x1)
U3_GA(
x1,
x2,
x3) =
U3_GA(
x1,
x3)
U12_GGA(
x1,
x2,
x3) =
U12_GGA(
x1,
x3)
MULTD_IN_GA(
x1,
x2) =
MULTD_IN_GA(
x1)
U4_GA(
x1,
x2,
x3) =
U4_GA(
x1,
x3)
U13_GGA(
x1,
x2,
x3) =
U13_GGA(
x1,
x3)
MULTE_IN_GA(
x1,
x2) =
MULTE_IN_GA(
x1)
U5_GA(
x1,
x2,
x3) =
U5_GA(
x1,
x3)
U14_GGA(
x1,
x2,
x3) =
U14_GGA(
x1,
x3)
MULTF_IN_GA(
x1,
x2) =
MULTF_IN_GA(
x1)
U6_GA(
x1,
x2,
x3) =
U6_GA(
x1,
x3)
U15_GGA(
x1,
x2,
x3) =
U15_GGA(
x1,
x3)
MULTG_IN_GA(
x1,
x2) =
MULTG_IN_GA(
x1)
U7_GA(
x1,
x2,
x3) =
U7_GA(
x1,
x3)
U16_GGA(
x1,
x2,
x3) =
U16_GGA(
x1,
x3)
MULTH_IN_GA(
x1,
x2) =
MULTH_IN_GA(
x1)
U8_GA(
x1,
x2,
x3) =
U8_GA(
x1,
x3)
U17_GGA(
x1,
x2,
x3,
x4) =
U17_GGA(
x1,
x2,
x4)
MULTJ_IN_GGGA(
x1,
x2,
x3,
x4) =
MULTJ_IN_GGGA(
x1,
x2,
x3)
U18_GGGA(
x1,
x2,
x3,
x4) =
U18_GGGA(
x1,
x2,
x4)
U19_GGGA(
x1,
x2,
x3,
x4,
x5) =
U19_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 9 SCCs with 29 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
MULTH_IN_GA(s(X1), s(s(s(s(s(s(s(X2)))))))) → MULTH_IN_GA(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
MULTH_IN_GA(
x1,
x2) =
MULTH_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:
MULTH_IN_GA(s(X1)) → MULTH_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:
- MULTH_IN_GA(s(X1)) → MULTH_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:
MULTG_IN_GA(s(X1), s(s(s(s(s(s(X2))))))) → MULTG_IN_GA(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
MULTG_IN_GA(
x1,
x2) =
MULTG_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:
MULTG_IN_GA(s(X1)) → MULTG_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:
- MULTG_IN_GA(s(X1)) → MULTG_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:
MULTF_IN_GA(s(X1), s(s(s(s(s(X2)))))) → MULTF_IN_GA(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
MULTF_IN_GA(
x1,
x2) =
MULTF_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:
MULTF_IN_GA(s(X1)) → MULTF_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:
- MULTF_IN_GA(s(X1)) → MULTF_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:
MULTE_IN_GA(s(X1), s(s(s(s(X2))))) → MULTE_IN_GA(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
MULTE_IN_GA(
x1,
x2) =
MULTE_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:
MULTE_IN_GA(s(X1)) → MULTE_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:
- MULTE_IN_GA(s(X1)) → MULTE_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:
MULTD_IN_GA(s(X1), s(s(s(X2)))) → MULTD_IN_GA(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
MULTD_IN_GA(
x1,
x2) =
MULTD_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:
MULTD_IN_GA(s(X1)) → MULTD_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:
- MULTD_IN_GA(s(X1)) → MULTD_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:
MULTC_IN_GA(s(X1), s(s(X2))) → MULTC_IN_GA(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
MULTC_IN_GA(
x1,
x2) =
MULTC_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:
MULTC_IN_GA(s(X1)) → MULTC_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:
- MULTC_IN_GA(s(X1)) → MULTC_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:
MULTB_IN_GA(s(X1), s(X2)) → MULTB_IN_GA(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
MULTB_IN_GA(
x1,
x2) =
MULTB_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:
MULTB_IN_GA(s(X1)) → MULTB_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:
- MULTB_IN_GA(s(X1)) → MULTB_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:
MULTI_IN_GGA(X1, s(s(s(s(s(s(s(s(X2)))))))), s(s(s(s(s(s(s(s(X3))))))))) → MULTJ_IN_GGGA(X1, s(s(s(s(s(s(s(X2))))))), X2, X3)
MULTJ_IN_GGGA(s(X1), X2, 0, X3) → MULTI_IN_GGA(X1, s(X2), X3)
MULTJ_IN_GGGA(X1, X2, s(X3), s(X4)) → MULTJ_IN_GGGA(X1, X2, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
0 =
0
MULTI_IN_GGA(
x1,
x2,
x3) =
MULTI_IN_GGA(
x1,
x2)
MULTJ_IN_GGGA(
x1,
x2,
x3,
x4) =
MULTJ_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:
MULTI_IN_GGA(X1, s(s(s(s(s(s(s(s(X2))))))))) → MULTJ_IN_GGGA(X1, s(s(s(s(s(s(s(X2))))))), X2)
MULTJ_IN_GGGA(s(X1), X2, 0) → MULTI_IN_GGA(X1, s(X2))
MULTJ_IN_GGGA(X1, X2, s(X3)) → MULTJ_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:
- MULTJ_IN_GGGA(s(X1), X2, 0) → MULTI_IN_GGA(X1, s(X2))
The graph contains the following edges 1 > 1
- MULTJ_IN_GGGA(X1, X2, s(X3)) → MULTJ_IN_GGGA(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
- MULTI_IN_GGA(X1, s(s(s(s(s(s(s(s(X2))))))))) → MULTJ_IN_GGGA(X1, s(s(s(s(s(s(s(X2))))))), X2)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3
(46) YES
(47) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
MULTA_IN_GA(s(X1), X2) → MULTA_IN_GA(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
MULTA_IN_GA(
x1,
x2) =
MULTA_IN_GA(
x1)
We have to consider all (P,R,Pi)-chains
(48) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(49) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MULTA_IN_GA(s(X1)) → MULTA_IN_GA(X1)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(50) 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:
- MULTA_IN_GA(s(X1)) → MULTA_IN_GA(X1)
The graph contains the following edges 1 > 1
(51) YES