Left Termination of the query pattern
convert(g,g,a)
w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇒, 187 ms)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇒, 329 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 21 ms)
↳6 AND
    ↳7 PiDP
        ↳8 UsableRulesProof (⇔, 0 ms)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇒, 0 ms)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔, 0 ms)
        ↳13 YES
    ↳14 PiDP
        ↳15 UsableRulesProof (⇔, 0 ms)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇒, 0 ms)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔, 0 ms)
        ↳20 YES
    ↳21 PiDP
        ↳22 UsableRulesProof (⇔, 0 ms)
        ↳23 PiDP
        ↳24 PiDPToQDPProof (⇒, 0 ms)
        ↳25 QDP
        ↳26 QDPSizeChangeProof (⇔, 0 ms)
        ↳27 YES
    ↳28 PiDP
        ↳29 UsableRulesProof (⇔, 0 ms)
        ↳30 PiDP
        ↳31 PiDPToQDPProof (⇒, 0 ms)
        ↳32 QDP
        ↳33 UsableRulesReductionPairsProof (⇔, 119 ms)
        ↳34 QDP
        ↳35 PisEmptyProof (⇔, 0 ms)
        ↳36 YES

(0) Obligation:

Clauses:

convert([], B, 0).
convert(.(0, XS), B, X) :- ','(convert(XS, B, Y), times(Y, B, X)).
convert(.(s(Y), XS), B, s(X)) :- convert(.(Y, XS), B, X).
plus(0, Y, Y).
plus(s(X), Y, s(Z)) :- plus(X, Y, Z).
times(0, Y, 0).
times(s(X), Y, Z) :- ','(times(X, Y, U), plus(Y, U, Z)).

Query: convert(g,g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

(2) Obligation:

Triples:

convertA(.(0, X1), X2, X3) :- convertA(X1, X2, X4).
convertA(.(0, X1), X2, X3) :- ','(convertcA(X1, X2, X4), timesB(X4, X2, X3)).
convertA(.(s(X1), X2), X3, s(X4)) :- convertC(.(X1, X2), X3, X4).
timesB(s(X1), X2, X3) :- timesB(X1, X2, X4).
timesB(s(X1), X2, X3) :- ','(timescB(X1, X2, X4), plusD(X2, X4, X3)).
plusD(s(X1), X2, s(X3)) :- plusD(X1, X2, X3).
plusF(s(X1), X2, s(X3)) :- plusF(X1, X2, X3).
timesE(s(X1), X2, X3) :- timesB(X1, X2, X4).
timesE(s(X1), X2, X3) :- ','(timescB(X1, X2, X4), plusF(X2, X4, X3)).
convertC(.(0, .(0, X1)), X2, X3) :- convertA(X1, X2, X4).
convertC(.(0, .(0, X1)), X2, X3) :- ','(convertcA(X1, X2, X4), timesB(X4, X2, X5)).
convertC(.(0, .(0, X1)), X2, X3) :- ','(convertcA(X1, X2, X4), ','(timescB(X4, X2, X5), timesE(X5, X2, X3))).
convertC(.(0, .(s(X1), X2)), X3, X4) :- convertC(.(X1, X2), X3, X5).
convertC(.(0, .(s(X1), X2)), X3, X4) :- ','(convertcC(.(X1, X2), X3, X5), timesE(s(X5), X3, X4)).
convertC(.(s(0), X1), X2, s(X3)) :- convertA(X1, X2, X4).
convertC(.(s(0), X1), X2, s(X3)) :- ','(convertcA(X1, X2, X4), timesE(X4, X2, X3)).
convertC(.(s(s(X1)), X2), X3, s(s(X4))) :- convertC(.(X1, X2), X3, X4).

Clauses:

convertcA([], X1, 0).
convertcA(.(0, X1), X2, X3) :- ','(convertcA(X1, X2, X4), timescB(X4, X2, X3)).
convertcA(.(s(X1), X2), X3, s(X4)) :- convertcC(.(X1, X2), X3, X4).
timescB(0, X1, 0).
timescB(s(X1), X2, X3) :- ','(timescB(X1, X2, X4), pluscD(X2, X4, X3)).
pluscD(0, X1, X1).
pluscD(s(X1), X2, s(X3)) :- pluscD(X1, X2, X3).
convertcC([], X1, 0).
convertcC(.(0, []), X1, 0).
convertcC(.(0, .(0, X1)), X2, X3) :- ','(convertcA(X1, X2, X4), ','(timescB(X4, X2, X5), timescE(X5, X2, X3))).
convertcC(.(0, .(s(X1), X2)), X3, X4) :- ','(convertcC(.(X1, X2), X3, X5), timescE(s(X5), X3, X4)).
convertcC(.(s(0), X1), X2, s(X3)) :- ','(convertcA(X1, X2, X4), timescE(X4, X2, X3)).
convertcC(.(s(s(X1)), X2), X3, s(s(X4))) :- convertcC(.(X1, X2), X3, X4).
pluscF(0, X1, X1).
pluscF(s(X1), X2, s(X3)) :- pluscF(X1, X2, X3).
timescE(0, X1, 0).
timescE(s(X1), X2, X3) :- ','(timescB(X1, X2, X4), pluscF(X2, X4, X3)).

Afs:

convertC(x1, x2, x3)  =  convertC(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:
convertC_in: (b,b,f)
convertA_in: (b,b,f)
convertcA_in: (b,b,f)
convertcC_in: (b,b,f)
timescB_in: (b,b,f)
pluscD_in: (b,b,f)
timescE_in: (b,b,f)
pluscF_in: (b,b,f)
timesB_in: (b,b,f)
plusD_in: (b,b,f)
timesE_in: (b,b,f)
plusF_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

CONVERTC_IN_GGA(.(0, .(0, X1)), X2, X3) → U13_GGA(X1, X2, X3, convertA_in_gga(X1, X2, X4))
CONVERTC_IN_GGA(.(0, .(0, X1)), X2, X3) → CONVERTA_IN_GGA(X1, X2, X4)
CONVERTA_IN_GGA(.(0, X1), X2, X3) → U1_GGA(X1, X2, X3, convertA_in_gga(X1, X2, X4))
CONVERTA_IN_GGA(.(0, X1), X2, X3) → CONVERTA_IN_GGA(X1, X2, X4)
CONVERTA_IN_GGA(.(0, X1), X2, X3) → U2_GGA(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
U2_GGA(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U3_GGA(X1, X2, X3, timesB_in_gga(X4, X2, X3))
U2_GGA(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → TIMESB_IN_GGA(X4, X2, X3)
TIMESB_IN_GGA(s(X1), X2, X3) → U5_GGA(X1, X2, X3, timesB_in_gga(X1, X2, X4))
TIMESB_IN_GGA(s(X1), X2, X3) → TIMESB_IN_GGA(X1, X2, X4)
TIMESB_IN_GGA(s(X1), X2, X3) → U6_GGA(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U6_GGA(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U7_GGA(X1, X2, X3, plusD_in_gga(X2, X4, X3))
U6_GGA(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → PLUSD_IN_GGA(X2, X4, X3)
PLUSD_IN_GGA(s(X1), X2, s(X3)) → U8_GGA(X1, X2, X3, plusD_in_gga(X1, X2, X3))
PLUSD_IN_GGA(s(X1), X2, s(X3)) → PLUSD_IN_GGA(X1, X2, X3)
CONVERTA_IN_GGA(.(s(X1), X2), X3, s(X4)) → U4_GGA(X1, X2, X3, X4, convertC_in_gga(.(X1, X2), X3, X4))
CONVERTA_IN_GGA(.(s(X1), X2), X3, s(X4)) → CONVERTC_IN_GGA(.(X1, X2), X3, X4)
CONVERTC_IN_GGA(.(0, .(0, X1)), X2, X3) → U14_GGA(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
U14_GGA(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U15_GGA(X1, X2, X3, timesB_in_gga(X4, X2, X5))
U14_GGA(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → TIMESB_IN_GGA(X4, X2, X5)
U14_GGA(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U16_GGA(X1, X2, X3, timescB_in_gga(X4, X2, X5))
U16_GGA(X1, X2, X3, timescB_out_gga(X4, X2, X5)) → U17_GGA(X1, X2, X3, timesE_in_gga(X5, X2, X3))
U16_GGA(X1, X2, X3, timescB_out_gga(X4, X2, X5)) → TIMESE_IN_GGA(X5, X2, X3)
TIMESE_IN_GGA(s(X1), X2, X3) → U10_GGA(X1, X2, X3, timesB_in_gga(X1, X2, X4))
TIMESE_IN_GGA(s(X1), X2, X3) → TIMESB_IN_GGA(X1, X2, X4)
TIMESE_IN_GGA(s(X1), X2, X3) → U11_GGA(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U11_GGA(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U12_GGA(X1, X2, X3, plusF_in_gga(X2, X4, X3))
U11_GGA(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → PLUSF_IN_GGA(X2, X4, X3)
PLUSF_IN_GGA(s(X1), X2, s(X3)) → U9_GGA(X1, X2, X3, plusF_in_gga(X1, X2, X3))
PLUSF_IN_GGA(s(X1), X2, s(X3)) → PLUSF_IN_GGA(X1, X2, X3)
CONVERTC_IN_GGA(.(0, .(s(X1), X2)), X3, X4) → U18_GGA(X1, X2, X3, X4, convertC_in_gga(.(X1, X2), X3, X5))
CONVERTC_IN_GGA(.(0, .(s(X1), X2)), X3, X4) → CONVERTC_IN_GGA(.(X1, X2), X3, X5)
CONVERTC_IN_GGA(.(0, .(s(X1), X2)), X3, X4) → U19_GGA(X1, X2, X3, X4, convertcC_in_gga(.(X1, X2), X3, X5))
U19_GGA(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X5)) → U20_GGA(X1, X2, X3, X4, timesE_in_gga(s(X5), X3, X4))
U19_GGA(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X5)) → TIMESE_IN_GGA(s(X5), X3, X4)
CONVERTC_IN_GGA(.(s(0), X1), X2, s(X3)) → U21_GGA(X1, X2, X3, convertA_in_gga(X1, X2, X4))
CONVERTC_IN_GGA(.(s(0), X1), X2, s(X3)) → CONVERTA_IN_GGA(X1, X2, X4)
CONVERTC_IN_GGA(.(s(0), X1), X2, s(X3)) → U22_GGA(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
U22_GGA(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U23_GGA(X1, X2, X3, timesE_in_gga(X4, X2, X3))
U22_GGA(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → TIMESE_IN_GGA(X4, X2, X3)
CONVERTC_IN_GGA(.(s(s(X1)), X2), X3, s(s(X4))) → U24_GGA(X1, X2, X3, X4, convertC_in_gga(.(X1, X2), X3, X4))
CONVERTC_IN_GGA(.(s(s(X1)), X2), X3, s(s(X4))) → CONVERTC_IN_GGA(.(X1, X2), X3, X4)

The TRS R consists of the following rules:

convertcA_in_gga([], X1, 0) → convertcA_out_gga([], X1, 0)
convertcA_in_gga(.(0, X1), X2, X3) → U26_gga(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
convertcA_in_gga(.(s(X1), X2), X3, s(X4)) → U28_gga(X1, X2, X3, X4, convertcC_in_gga(.(X1, X2), X3, X4))
convertcC_in_gga([], X1, 0) → convertcC_out_gga([], X1, 0)
convertcC_in_gga(.(0, []), X1, 0) → convertcC_out_gga(.(0, []), X1, 0)
convertcC_in_gga(.(0, .(0, X1)), X2, X3) → U32_gga(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
U32_gga(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U33_gga(X1, X2, X3, timescB_in_gga(X4, X2, X5))
timescB_in_gga(0, X1, 0) → timescB_out_gga(0, X1, 0)
timescB_in_gga(s(X1), X2, X3) → U29_gga(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U29_gga(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U30_gga(X1, X2, X3, pluscD_in_gga(X2, X4, X3))
pluscD_in_gga(0, X1, X1) → pluscD_out_gga(0, X1, X1)
pluscD_in_gga(s(X1), X2, s(X3)) → U31_gga(X1, X2, X3, pluscD_in_gga(X1, X2, X3))
U31_gga(X1, X2, X3, pluscD_out_gga(X1, X2, X3)) → pluscD_out_gga(s(X1), X2, s(X3))
U30_gga(X1, X2, X3, pluscD_out_gga(X2, X4, X3)) → timescB_out_gga(s(X1), X2, X3)
U33_gga(X1, X2, X3, timescB_out_gga(X4, X2, X5)) → U34_gga(X1, X2, X3, timescE_in_gga(X5, X2, X3))
timescE_in_gga(0, X1, 0) → timescE_out_gga(0, X1, 0)
timescE_in_gga(s(X1), X2, X3) → U41_gga(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U41_gga(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U42_gga(X1, X2, X3, pluscF_in_gga(X2, X4, X3))
pluscF_in_gga(0, X1, X1) → pluscF_out_gga(0, X1, X1)
pluscF_in_gga(s(X1), X2, s(X3)) → U40_gga(X1, X2, X3, pluscF_in_gga(X1, X2, X3))
U40_gga(X1, X2, X3, pluscF_out_gga(X1, X2, X3)) → pluscF_out_gga(s(X1), X2, s(X3))
U42_gga(X1, X2, X3, pluscF_out_gga(X2, X4, X3)) → timescE_out_gga(s(X1), X2, X3)
U34_gga(X1, X2, X3, timescE_out_gga(X5, X2, X3)) → convertcC_out_gga(.(0, .(0, X1)), X2, X3)
convertcC_in_gga(.(0, .(s(X1), X2)), X3, X4) → U35_gga(X1, X2, X3, X4, convertcC_in_gga(.(X1, X2), X3, X5))
convertcC_in_gga(.(s(0), X1), X2, s(X3)) → U37_gga(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
U37_gga(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U38_gga(X1, X2, X3, timescE_in_gga(X4, X2, X3))
U38_gga(X1, X2, X3, timescE_out_gga(X4, X2, X3)) → convertcC_out_gga(.(s(0), X1), X2, s(X3))
convertcC_in_gga(.(s(s(X1)), X2), X3, s(s(X4))) → U39_gga(X1, X2, X3, X4, convertcC_in_gga(.(X1, X2), X3, X4))
U39_gga(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X4)) → convertcC_out_gga(.(s(s(X1)), X2), X3, s(s(X4)))
U35_gga(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X5)) → U36_gga(X1, X2, X3, X4, timescE_in_gga(s(X5), X3, X4))
U36_gga(X1, X2, X3, X4, timescE_out_gga(s(X5), X3, X4)) → convertcC_out_gga(.(0, .(s(X1), X2)), X3, X4)
U28_gga(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X4)) → convertcA_out_gga(.(s(X1), X2), X3, s(X4))
U26_gga(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U27_gga(X1, X2, X3, timescB_in_gga(X4, X2, X3))
U27_gga(X1, X2, X3, timescB_out_gga(X4, X2, X3)) → convertcA_out_gga(.(0, X1), X2, X3)

The argument filtering Pi contains the following mapping:
convertC_in_gga(x1, x2, x3)  =  convertC_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
0  =  0
convertA_in_gga(x1, x2, x3)  =  convertA_in_gga(x1, x2)
convertcA_in_gga(x1, x2, x3)  =  convertcA_in_gga(x1, x2)
[]  =  []
convertcA_out_gga(x1, x2, x3)  =  convertcA_out_gga(x1, x2, x3)
U26_gga(x1, x2, x3, x4)  =  U26_gga(x1, x2, x4)
s(x1)  =  s(x1)
U28_gga(x1, x2, x3, x4, x5)  =  U28_gga(x1, x2, x3, x5)
convertcC_in_gga(x1, x2, x3)  =  convertcC_in_gga(x1, x2)
convertcC_out_gga(x1, x2, x3)  =  convertcC_out_gga(x1, x2, x3)
U32_gga(x1, x2, x3, x4)  =  U32_gga(x1, x2, x4)
U33_gga(x1, x2, x3, x4)  =  U33_gga(x1, x2, x4)
timescB_in_gga(x1, x2, x3)  =  timescB_in_gga(x1, x2)
timescB_out_gga(x1, x2, x3)  =  timescB_out_gga(x1, x2, x3)
U29_gga(x1, x2, x3, x4)  =  U29_gga(x1, x2, x4)
U30_gga(x1, x2, x3, x4)  =  U30_gga(x1, x2, x4)
pluscD_in_gga(x1, x2, x3)  =  pluscD_in_gga(x1, x2)
pluscD_out_gga(x1, x2, x3)  =  pluscD_out_gga(x1, x2, x3)
U31_gga(x1, x2, x3, x4)  =  U31_gga(x1, x2, x4)
U34_gga(x1, x2, x3, x4)  =  U34_gga(x1, x2, x4)
timescE_in_gga(x1, x2, x3)  =  timescE_in_gga(x1, x2)
timescE_out_gga(x1, x2, x3)  =  timescE_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4)  =  U41_gga(x1, x2, x4)
U42_gga(x1, x2, x3, x4)  =  U42_gga(x1, x2, x4)
pluscF_in_gga(x1, x2, x3)  =  pluscF_in_gga(x1, x2)
pluscF_out_gga(x1, x2, x3)  =  pluscF_out_gga(x1, x2, x3)
U40_gga(x1, x2, x3, x4)  =  U40_gga(x1, x2, x4)
U35_gga(x1, x2, x3, x4, x5)  =  U35_gga(x1, x2, x3, x5)
U37_gga(x1, x2, x3, x4)  =  U37_gga(x1, x2, x4)
U38_gga(x1, x2, x3, x4)  =  U38_gga(x1, x2, x4)
U39_gga(x1, x2, x3, x4, x5)  =  U39_gga(x1, x2, x3, x5)
U36_gga(x1, x2, x3, x4, x5)  =  U36_gga(x1, x2, x3, x5)
U27_gga(x1, x2, x3, x4)  =  U27_gga(x1, x2, x4)
timesB_in_gga(x1, x2, x3)  =  timesB_in_gga(x1, x2)
plusD_in_gga(x1, x2, x3)  =  plusD_in_gga(x1, x2)
timesE_in_gga(x1, x2, x3)  =  timesE_in_gga(x1, x2)
plusF_in_gga(x1, x2, x3)  =  plusF_in_gga(x1, x2)
CONVERTC_IN_GGA(x1, x2, x3)  =  CONVERTC_IN_GGA(x1, x2)
U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x1, x2, x4)
CONVERTA_IN_GGA(x1, x2, x3)  =  CONVERTA_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)
TIMESB_IN_GGA(x1, x2, x3)  =  TIMESB_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x1, x2, x4)
PLUSD_IN_GGA(x1, x2, x3)  =  PLUSD_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
U14_GGA(x1, x2, x3, x4)  =  U14_GGA(x1, x2, x4)
U15_GGA(x1, x2, x3, x4)  =  U15_GGA(x1, x2, x4)
U16_GGA(x1, x2, x3, x4)  =  U16_GGA(x1, x2, x4)
U17_GGA(x1, x2, x3, x4)  =  U17_GGA(x1, x2, x4)
TIMESE_IN_GGA(x1, x2, x3)  =  TIMESE_IN_GGA(x1, x2)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x1, x2, x4)
U11_GGA(x1, x2, x3, x4)  =  U11_GGA(x1, x2, x4)
U12_GGA(x1, x2, x3, x4)  =  U12_GGA(x1, x2, x4)
PLUSF_IN_GGA(x1, x2, x3)  =  PLUSF_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x1, x2, x4)
U18_GGA(x1, x2, x3, x4, x5)  =  U18_GGA(x1, x2, x3, x5)
U19_GGA(x1, x2, x3, x4, x5)  =  U19_GGA(x1, x2, x3, x5)
U20_GGA(x1, x2, x3, x4, x5)  =  U20_GGA(x1, x2, x3, x5)
U21_GGA(x1, x2, x3, x4)  =  U21_GGA(x1, x2, x4)
U22_GGA(x1, x2, x3, x4)  =  U22_GGA(x1, x2, x4)
U23_GGA(x1, x2, x3, x4)  =  U23_GGA(x1, x2, x4)
U24_GGA(x1, x2, x3, x4, x5)  =  U24_GGA(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:

CONVERTC_IN_GGA(.(0, .(0, X1)), X2, X3) → U13_GGA(X1, X2, X3, convertA_in_gga(X1, X2, X4))
CONVERTC_IN_GGA(.(0, .(0, X1)), X2, X3) → CONVERTA_IN_GGA(X1, X2, X4)
CONVERTA_IN_GGA(.(0, X1), X2, X3) → U1_GGA(X1, X2, X3, convertA_in_gga(X1, X2, X4))
CONVERTA_IN_GGA(.(0, X1), X2, X3) → CONVERTA_IN_GGA(X1, X2, X4)
CONVERTA_IN_GGA(.(0, X1), X2, X3) → U2_GGA(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
U2_GGA(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U3_GGA(X1, X2, X3, timesB_in_gga(X4, X2, X3))
U2_GGA(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → TIMESB_IN_GGA(X4, X2, X3)
TIMESB_IN_GGA(s(X1), X2, X3) → U5_GGA(X1, X2, X3, timesB_in_gga(X1, X2, X4))
TIMESB_IN_GGA(s(X1), X2, X3) → TIMESB_IN_GGA(X1, X2, X4)
TIMESB_IN_GGA(s(X1), X2, X3) → U6_GGA(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U6_GGA(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U7_GGA(X1, X2, X3, plusD_in_gga(X2, X4, X3))
U6_GGA(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → PLUSD_IN_GGA(X2, X4, X3)
PLUSD_IN_GGA(s(X1), X2, s(X3)) → U8_GGA(X1, X2, X3, plusD_in_gga(X1, X2, X3))
PLUSD_IN_GGA(s(X1), X2, s(X3)) → PLUSD_IN_GGA(X1, X2, X3)
CONVERTA_IN_GGA(.(s(X1), X2), X3, s(X4)) → U4_GGA(X1, X2, X3, X4, convertC_in_gga(.(X1, X2), X3, X4))
CONVERTA_IN_GGA(.(s(X1), X2), X3, s(X4)) → CONVERTC_IN_GGA(.(X1, X2), X3, X4)
CONVERTC_IN_GGA(.(0, .(0, X1)), X2, X3) → U14_GGA(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
U14_GGA(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U15_GGA(X1, X2, X3, timesB_in_gga(X4, X2, X5))
U14_GGA(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → TIMESB_IN_GGA(X4, X2, X5)
U14_GGA(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U16_GGA(X1, X2, X3, timescB_in_gga(X4, X2, X5))
U16_GGA(X1, X2, X3, timescB_out_gga(X4, X2, X5)) → U17_GGA(X1, X2, X3, timesE_in_gga(X5, X2, X3))
U16_GGA(X1, X2, X3, timescB_out_gga(X4, X2, X5)) → TIMESE_IN_GGA(X5, X2, X3)
TIMESE_IN_GGA(s(X1), X2, X3) → U10_GGA(X1, X2, X3, timesB_in_gga(X1, X2, X4))
TIMESE_IN_GGA(s(X1), X2, X3) → TIMESB_IN_GGA(X1, X2, X4)
TIMESE_IN_GGA(s(X1), X2, X3) → U11_GGA(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U11_GGA(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U12_GGA(X1, X2, X3, plusF_in_gga(X2, X4, X3))
U11_GGA(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → PLUSF_IN_GGA(X2, X4, X3)
PLUSF_IN_GGA(s(X1), X2, s(X3)) → U9_GGA(X1, X2, X3, plusF_in_gga(X1, X2, X3))
PLUSF_IN_GGA(s(X1), X2, s(X3)) → PLUSF_IN_GGA(X1, X2, X3)
CONVERTC_IN_GGA(.(0, .(s(X1), X2)), X3, X4) → U18_GGA(X1, X2, X3, X4, convertC_in_gga(.(X1, X2), X3, X5))
CONVERTC_IN_GGA(.(0, .(s(X1), X2)), X3, X4) → CONVERTC_IN_GGA(.(X1, X2), X3, X5)
CONVERTC_IN_GGA(.(0, .(s(X1), X2)), X3, X4) → U19_GGA(X1, X2, X3, X4, convertcC_in_gga(.(X1, X2), X3, X5))
U19_GGA(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X5)) → U20_GGA(X1, X2, X3, X4, timesE_in_gga(s(X5), X3, X4))
U19_GGA(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X5)) → TIMESE_IN_GGA(s(X5), X3, X4)
CONVERTC_IN_GGA(.(s(0), X1), X2, s(X3)) → U21_GGA(X1, X2, X3, convertA_in_gga(X1, X2, X4))
CONVERTC_IN_GGA(.(s(0), X1), X2, s(X3)) → CONVERTA_IN_GGA(X1, X2, X4)
CONVERTC_IN_GGA(.(s(0), X1), X2, s(X3)) → U22_GGA(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
U22_GGA(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U23_GGA(X1, X2, X3, timesE_in_gga(X4, X2, X3))
U22_GGA(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → TIMESE_IN_GGA(X4, X2, X3)
CONVERTC_IN_GGA(.(s(s(X1)), X2), X3, s(s(X4))) → U24_GGA(X1, X2, X3, X4, convertC_in_gga(.(X1, X2), X3, X4))
CONVERTC_IN_GGA(.(s(s(X1)), X2), X3, s(s(X4))) → CONVERTC_IN_GGA(.(X1, X2), X3, X4)

The TRS R consists of the following rules:

convertcA_in_gga([], X1, 0) → convertcA_out_gga([], X1, 0)
convertcA_in_gga(.(0, X1), X2, X3) → U26_gga(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
convertcA_in_gga(.(s(X1), X2), X3, s(X4)) → U28_gga(X1, X2, X3, X4, convertcC_in_gga(.(X1, X2), X3, X4))
convertcC_in_gga([], X1, 0) → convertcC_out_gga([], X1, 0)
convertcC_in_gga(.(0, []), X1, 0) → convertcC_out_gga(.(0, []), X1, 0)
convertcC_in_gga(.(0, .(0, X1)), X2, X3) → U32_gga(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
U32_gga(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U33_gga(X1, X2, X3, timescB_in_gga(X4, X2, X5))
timescB_in_gga(0, X1, 0) → timescB_out_gga(0, X1, 0)
timescB_in_gga(s(X1), X2, X3) → U29_gga(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U29_gga(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U30_gga(X1, X2, X3, pluscD_in_gga(X2, X4, X3))
pluscD_in_gga(0, X1, X1) → pluscD_out_gga(0, X1, X1)
pluscD_in_gga(s(X1), X2, s(X3)) → U31_gga(X1, X2, X3, pluscD_in_gga(X1, X2, X3))
U31_gga(X1, X2, X3, pluscD_out_gga(X1, X2, X3)) → pluscD_out_gga(s(X1), X2, s(X3))
U30_gga(X1, X2, X3, pluscD_out_gga(X2, X4, X3)) → timescB_out_gga(s(X1), X2, X3)
U33_gga(X1, X2, X3, timescB_out_gga(X4, X2, X5)) → U34_gga(X1, X2, X3, timescE_in_gga(X5, X2, X3))
timescE_in_gga(0, X1, 0) → timescE_out_gga(0, X1, 0)
timescE_in_gga(s(X1), X2, X3) → U41_gga(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U41_gga(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U42_gga(X1, X2, X3, pluscF_in_gga(X2, X4, X3))
pluscF_in_gga(0, X1, X1) → pluscF_out_gga(0, X1, X1)
pluscF_in_gga(s(X1), X2, s(X3)) → U40_gga(X1, X2, X3, pluscF_in_gga(X1, X2, X3))
U40_gga(X1, X2, X3, pluscF_out_gga(X1, X2, X3)) → pluscF_out_gga(s(X1), X2, s(X3))
U42_gga(X1, X2, X3, pluscF_out_gga(X2, X4, X3)) → timescE_out_gga(s(X1), X2, X3)
U34_gga(X1, X2, X3, timescE_out_gga(X5, X2, X3)) → convertcC_out_gga(.(0, .(0, X1)), X2, X3)
convertcC_in_gga(.(0, .(s(X1), X2)), X3, X4) → U35_gga(X1, X2, X3, X4, convertcC_in_gga(.(X1, X2), X3, X5))
convertcC_in_gga(.(s(0), X1), X2, s(X3)) → U37_gga(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
U37_gga(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U38_gga(X1, X2, X3, timescE_in_gga(X4, X2, X3))
U38_gga(X1, X2, X3, timescE_out_gga(X4, X2, X3)) → convertcC_out_gga(.(s(0), X1), X2, s(X3))
convertcC_in_gga(.(s(s(X1)), X2), X3, s(s(X4))) → U39_gga(X1, X2, X3, X4, convertcC_in_gga(.(X1, X2), X3, X4))
U39_gga(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X4)) → convertcC_out_gga(.(s(s(X1)), X2), X3, s(s(X4)))
U35_gga(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X5)) → U36_gga(X1, X2, X3, X4, timescE_in_gga(s(X5), X3, X4))
U36_gga(X1, X2, X3, X4, timescE_out_gga(s(X5), X3, X4)) → convertcC_out_gga(.(0, .(s(X1), X2)), X3, X4)
U28_gga(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X4)) → convertcA_out_gga(.(s(X1), X2), X3, s(X4))
U26_gga(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U27_gga(X1, X2, X3, timescB_in_gga(X4, X2, X3))
U27_gga(X1, X2, X3, timescB_out_gga(X4, X2, X3)) → convertcA_out_gga(.(0, X1), X2, X3)

The argument filtering Pi contains the following mapping:
convertC_in_gga(x1, x2, x3)  =  convertC_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
0  =  0
convertA_in_gga(x1, x2, x3)  =  convertA_in_gga(x1, x2)
convertcA_in_gga(x1, x2, x3)  =  convertcA_in_gga(x1, x2)
[]  =  []
convertcA_out_gga(x1, x2, x3)  =  convertcA_out_gga(x1, x2, x3)
U26_gga(x1, x2, x3, x4)  =  U26_gga(x1, x2, x4)
s(x1)  =  s(x1)
U28_gga(x1, x2, x3, x4, x5)  =  U28_gga(x1, x2, x3, x5)
convertcC_in_gga(x1, x2, x3)  =  convertcC_in_gga(x1, x2)
convertcC_out_gga(x1, x2, x3)  =  convertcC_out_gga(x1, x2, x3)
U32_gga(x1, x2, x3, x4)  =  U32_gga(x1, x2, x4)
U33_gga(x1, x2, x3, x4)  =  U33_gga(x1, x2, x4)
timescB_in_gga(x1, x2, x3)  =  timescB_in_gga(x1, x2)
timescB_out_gga(x1, x2, x3)  =  timescB_out_gga(x1, x2, x3)
U29_gga(x1, x2, x3, x4)  =  U29_gga(x1, x2, x4)
U30_gga(x1, x2, x3, x4)  =  U30_gga(x1, x2, x4)
pluscD_in_gga(x1, x2, x3)  =  pluscD_in_gga(x1, x2)
pluscD_out_gga(x1, x2, x3)  =  pluscD_out_gga(x1, x2, x3)
U31_gga(x1, x2, x3, x4)  =  U31_gga(x1, x2, x4)
U34_gga(x1, x2, x3, x4)  =  U34_gga(x1, x2, x4)
timescE_in_gga(x1, x2, x3)  =  timescE_in_gga(x1, x2)
timescE_out_gga(x1, x2, x3)  =  timescE_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4)  =  U41_gga(x1, x2, x4)
U42_gga(x1, x2, x3, x4)  =  U42_gga(x1, x2, x4)
pluscF_in_gga(x1, x2, x3)  =  pluscF_in_gga(x1, x2)
pluscF_out_gga(x1, x2, x3)  =  pluscF_out_gga(x1, x2, x3)
U40_gga(x1, x2, x3, x4)  =  U40_gga(x1, x2, x4)
U35_gga(x1, x2, x3, x4, x5)  =  U35_gga(x1, x2, x3, x5)
U37_gga(x1, x2, x3, x4)  =  U37_gga(x1, x2, x4)
U38_gga(x1, x2, x3, x4)  =  U38_gga(x1, x2, x4)
U39_gga(x1, x2, x3, x4, x5)  =  U39_gga(x1, x2, x3, x5)
U36_gga(x1, x2, x3, x4, x5)  =  U36_gga(x1, x2, x3, x5)
U27_gga(x1, x2, x3, x4)  =  U27_gga(x1, x2, x4)
timesB_in_gga(x1, x2, x3)  =  timesB_in_gga(x1, x2)
plusD_in_gga(x1, x2, x3)  =  plusD_in_gga(x1, x2)
timesE_in_gga(x1, x2, x3)  =  timesE_in_gga(x1, x2)
plusF_in_gga(x1, x2, x3)  =  plusF_in_gga(x1, x2)
CONVERTC_IN_GGA(x1, x2, x3)  =  CONVERTC_IN_GGA(x1, x2)
U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x1, x2, x4)
CONVERTA_IN_GGA(x1, x2, x3)  =  CONVERTA_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)
TIMESB_IN_GGA(x1, x2, x3)  =  TIMESB_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x1, x2, x4)
PLUSD_IN_GGA(x1, x2, x3)  =  PLUSD_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
U14_GGA(x1, x2, x3, x4)  =  U14_GGA(x1, x2, x4)
U15_GGA(x1, x2, x3, x4)  =  U15_GGA(x1, x2, x4)
U16_GGA(x1, x2, x3, x4)  =  U16_GGA(x1, x2, x4)
U17_GGA(x1, x2, x3, x4)  =  U17_GGA(x1, x2, x4)
TIMESE_IN_GGA(x1, x2, x3)  =  TIMESE_IN_GGA(x1, x2)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x1, x2, x4)
U11_GGA(x1, x2, x3, x4)  =  U11_GGA(x1, x2, x4)
U12_GGA(x1, x2, x3, x4)  =  U12_GGA(x1, x2, x4)
PLUSF_IN_GGA(x1, x2, x3)  =  PLUSF_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x1, x2, x4)
U18_GGA(x1, x2, x3, x4, x5)  =  U18_GGA(x1, x2, x3, x5)
U19_GGA(x1, x2, x3, x4, x5)  =  U19_GGA(x1, x2, x3, x5)
U20_GGA(x1, x2, x3, x4, x5)  =  U20_GGA(x1, x2, x3, x5)
U21_GGA(x1, x2, x3, x4)  =  U21_GGA(x1, x2, x4)
U22_GGA(x1, x2, x3, x4)  =  U22_GGA(x1, x2, x4)
U23_GGA(x1, x2, x3, x4)  =  U23_GGA(x1, x2, x4)
U24_GGA(x1, x2, x3, x4, x5)  =  U24_GGA(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 4 SCCs with 32 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

PLUSF_IN_GGA(s(X1), X2, s(X3)) → PLUSF_IN_GGA(X1, X2, X3)

The TRS R consists of the following rules:

convertcA_in_gga([], X1, 0) → convertcA_out_gga([], X1, 0)
convertcA_in_gga(.(0, X1), X2, X3) → U26_gga(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
convertcA_in_gga(.(s(X1), X2), X3, s(X4)) → U28_gga(X1, X2, X3, X4, convertcC_in_gga(.(X1, X2), X3, X4))
convertcC_in_gga([], X1, 0) → convertcC_out_gga([], X1, 0)
convertcC_in_gga(.(0, []), X1, 0) → convertcC_out_gga(.(0, []), X1, 0)
convertcC_in_gga(.(0, .(0, X1)), X2, X3) → U32_gga(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
U32_gga(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U33_gga(X1, X2, X3, timescB_in_gga(X4, X2, X5))
timescB_in_gga(0, X1, 0) → timescB_out_gga(0, X1, 0)
timescB_in_gga(s(X1), X2, X3) → U29_gga(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U29_gga(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U30_gga(X1, X2, X3, pluscD_in_gga(X2, X4, X3))
pluscD_in_gga(0, X1, X1) → pluscD_out_gga(0, X1, X1)
pluscD_in_gga(s(X1), X2, s(X3)) → U31_gga(X1, X2, X3, pluscD_in_gga(X1, X2, X3))
U31_gga(X1, X2, X3, pluscD_out_gga(X1, X2, X3)) → pluscD_out_gga(s(X1), X2, s(X3))
U30_gga(X1, X2, X3, pluscD_out_gga(X2, X4, X3)) → timescB_out_gga(s(X1), X2, X3)
U33_gga(X1, X2, X3, timescB_out_gga(X4, X2, X5)) → U34_gga(X1, X2, X3, timescE_in_gga(X5, X2, X3))
timescE_in_gga(0, X1, 0) → timescE_out_gga(0, X1, 0)
timescE_in_gga(s(X1), X2, X3) → U41_gga(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U41_gga(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U42_gga(X1, X2, X3, pluscF_in_gga(X2, X4, X3))
pluscF_in_gga(0, X1, X1) → pluscF_out_gga(0, X1, X1)
pluscF_in_gga(s(X1), X2, s(X3)) → U40_gga(X1, X2, X3, pluscF_in_gga(X1, X2, X3))
U40_gga(X1, X2, X3, pluscF_out_gga(X1, X2, X3)) → pluscF_out_gga(s(X1), X2, s(X3))
U42_gga(X1, X2, X3, pluscF_out_gga(X2, X4, X3)) → timescE_out_gga(s(X1), X2, X3)
U34_gga(X1, X2, X3, timescE_out_gga(X5, X2, X3)) → convertcC_out_gga(.(0, .(0, X1)), X2, X3)
convertcC_in_gga(.(0, .(s(X1), X2)), X3, X4) → U35_gga(X1, X2, X3, X4, convertcC_in_gga(.(X1, X2), X3, X5))
convertcC_in_gga(.(s(0), X1), X2, s(X3)) → U37_gga(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
U37_gga(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U38_gga(X1, X2, X3, timescE_in_gga(X4, X2, X3))
U38_gga(X1, X2, X3, timescE_out_gga(X4, X2, X3)) → convertcC_out_gga(.(s(0), X1), X2, s(X3))
convertcC_in_gga(.(s(s(X1)), X2), X3, s(s(X4))) → U39_gga(X1, X2, X3, X4, convertcC_in_gga(.(X1, X2), X3, X4))
U39_gga(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X4)) → convertcC_out_gga(.(s(s(X1)), X2), X3, s(s(X4)))
U35_gga(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X5)) → U36_gga(X1, X2, X3, X4, timescE_in_gga(s(X5), X3, X4))
U36_gga(X1, X2, X3, X4, timescE_out_gga(s(X5), X3, X4)) → convertcC_out_gga(.(0, .(s(X1), X2)), X3, X4)
U28_gga(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X4)) → convertcA_out_gga(.(s(X1), X2), X3, s(X4))
U26_gga(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U27_gga(X1, X2, X3, timescB_in_gga(X4, X2, X3))
U27_gga(X1, X2, X3, timescB_out_gga(X4, X2, X3)) → convertcA_out_gga(.(0, X1), X2, X3)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
0  =  0
convertcA_in_gga(x1, x2, x3)  =  convertcA_in_gga(x1, x2)
[]  =  []
convertcA_out_gga(x1, x2, x3)  =  convertcA_out_gga(x1, x2, x3)
U26_gga(x1, x2, x3, x4)  =  U26_gga(x1, x2, x4)
s(x1)  =  s(x1)
U28_gga(x1, x2, x3, x4, x5)  =  U28_gga(x1, x2, x3, x5)
convertcC_in_gga(x1, x2, x3)  =  convertcC_in_gga(x1, x2)
convertcC_out_gga(x1, x2, x3)  =  convertcC_out_gga(x1, x2, x3)
U32_gga(x1, x2, x3, x4)  =  U32_gga(x1, x2, x4)
U33_gga(x1, x2, x3, x4)  =  U33_gga(x1, x2, x4)
timescB_in_gga(x1, x2, x3)  =  timescB_in_gga(x1, x2)
timescB_out_gga(x1, x2, x3)  =  timescB_out_gga(x1, x2, x3)
U29_gga(x1, x2, x3, x4)  =  U29_gga(x1, x2, x4)
U30_gga(x1, x2, x3, x4)  =  U30_gga(x1, x2, x4)
pluscD_in_gga(x1, x2, x3)  =  pluscD_in_gga(x1, x2)
pluscD_out_gga(x1, x2, x3)  =  pluscD_out_gga(x1, x2, x3)
U31_gga(x1, x2, x3, x4)  =  U31_gga(x1, x2, x4)
U34_gga(x1, x2, x3, x4)  =  U34_gga(x1, x2, x4)
timescE_in_gga(x1, x2, x3)  =  timescE_in_gga(x1, x2)
timescE_out_gga(x1, x2, x3)  =  timescE_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4)  =  U41_gga(x1, x2, x4)
U42_gga(x1, x2, x3, x4)  =  U42_gga(x1, x2, x4)
pluscF_in_gga(x1, x2, x3)  =  pluscF_in_gga(x1, x2)
pluscF_out_gga(x1, x2, x3)  =  pluscF_out_gga(x1, x2, x3)
U40_gga(x1, x2, x3, x4)  =  U40_gga(x1, x2, x4)
U35_gga(x1, x2, x3, x4, x5)  =  U35_gga(x1, x2, x3, x5)
U37_gga(x1, x2, x3, x4)  =  U37_gga(x1, x2, x4)
U38_gga(x1, x2, x3, x4)  =  U38_gga(x1, x2, x4)
U39_gga(x1, x2, x3, x4, x5)  =  U39_gga(x1, x2, x3, x5)
U36_gga(x1, x2, x3, x4, x5)  =  U36_gga(x1, x2, x3, x5)
U27_gga(x1, x2, x3, x4)  =  U27_gga(x1, x2, x4)
PLUSF_IN_GGA(x1, x2, x3)  =  PLUSF_IN_GGA(x1, x2)

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:

PLUSF_IN_GGA(s(X1), X2, s(X3)) → PLUSF_IN_GGA(X1, X2, X3)

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

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:

PLUSF_IN_GGA(s(X1), X2) → PLUSF_IN_GGA(X1, X2)

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:

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

(13) YES

(14) Obligation:

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

PLUSD_IN_GGA(s(X1), X2, s(X3)) → PLUSD_IN_GGA(X1, X2, X3)

The TRS R consists of the following rules:

convertcA_in_gga([], X1, 0) → convertcA_out_gga([], X1, 0)
convertcA_in_gga(.(0, X1), X2, X3) → U26_gga(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
convertcA_in_gga(.(s(X1), X2), X3, s(X4)) → U28_gga(X1, X2, X3, X4, convertcC_in_gga(.(X1, X2), X3, X4))
convertcC_in_gga([], X1, 0) → convertcC_out_gga([], X1, 0)
convertcC_in_gga(.(0, []), X1, 0) → convertcC_out_gga(.(0, []), X1, 0)
convertcC_in_gga(.(0, .(0, X1)), X2, X3) → U32_gga(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
U32_gga(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U33_gga(X1, X2, X3, timescB_in_gga(X4, X2, X5))
timescB_in_gga(0, X1, 0) → timescB_out_gga(0, X1, 0)
timescB_in_gga(s(X1), X2, X3) → U29_gga(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U29_gga(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U30_gga(X1, X2, X3, pluscD_in_gga(X2, X4, X3))
pluscD_in_gga(0, X1, X1) → pluscD_out_gga(0, X1, X1)
pluscD_in_gga(s(X1), X2, s(X3)) → U31_gga(X1, X2, X3, pluscD_in_gga(X1, X2, X3))
U31_gga(X1, X2, X3, pluscD_out_gga(X1, X2, X3)) → pluscD_out_gga(s(X1), X2, s(X3))
U30_gga(X1, X2, X3, pluscD_out_gga(X2, X4, X3)) → timescB_out_gga(s(X1), X2, X3)
U33_gga(X1, X2, X3, timescB_out_gga(X4, X2, X5)) → U34_gga(X1, X2, X3, timescE_in_gga(X5, X2, X3))
timescE_in_gga(0, X1, 0) → timescE_out_gga(0, X1, 0)
timescE_in_gga(s(X1), X2, X3) → U41_gga(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U41_gga(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U42_gga(X1, X2, X3, pluscF_in_gga(X2, X4, X3))
pluscF_in_gga(0, X1, X1) → pluscF_out_gga(0, X1, X1)
pluscF_in_gga(s(X1), X2, s(X3)) → U40_gga(X1, X2, X3, pluscF_in_gga(X1, X2, X3))
U40_gga(X1, X2, X3, pluscF_out_gga(X1, X2, X3)) → pluscF_out_gga(s(X1), X2, s(X3))
U42_gga(X1, X2, X3, pluscF_out_gga(X2, X4, X3)) → timescE_out_gga(s(X1), X2, X3)
U34_gga(X1, X2, X3, timescE_out_gga(X5, X2, X3)) → convertcC_out_gga(.(0, .(0, X1)), X2, X3)
convertcC_in_gga(.(0, .(s(X1), X2)), X3, X4) → U35_gga(X1, X2, X3, X4, convertcC_in_gga(.(X1, X2), X3, X5))
convertcC_in_gga(.(s(0), X1), X2, s(X3)) → U37_gga(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
U37_gga(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U38_gga(X1, X2, X3, timescE_in_gga(X4, X2, X3))
U38_gga(X1, X2, X3, timescE_out_gga(X4, X2, X3)) → convertcC_out_gga(.(s(0), X1), X2, s(X3))
convertcC_in_gga(.(s(s(X1)), X2), X3, s(s(X4))) → U39_gga(X1, X2, X3, X4, convertcC_in_gga(.(X1, X2), X3, X4))
U39_gga(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X4)) → convertcC_out_gga(.(s(s(X1)), X2), X3, s(s(X4)))
U35_gga(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X5)) → U36_gga(X1, X2, X3, X4, timescE_in_gga(s(X5), X3, X4))
U36_gga(X1, X2, X3, X4, timescE_out_gga(s(X5), X3, X4)) → convertcC_out_gga(.(0, .(s(X1), X2)), X3, X4)
U28_gga(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X4)) → convertcA_out_gga(.(s(X1), X2), X3, s(X4))
U26_gga(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U27_gga(X1, X2, X3, timescB_in_gga(X4, X2, X3))
U27_gga(X1, X2, X3, timescB_out_gga(X4, X2, X3)) → convertcA_out_gga(.(0, X1), X2, X3)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
0  =  0
convertcA_in_gga(x1, x2, x3)  =  convertcA_in_gga(x1, x2)
[]  =  []
convertcA_out_gga(x1, x2, x3)  =  convertcA_out_gga(x1, x2, x3)
U26_gga(x1, x2, x3, x4)  =  U26_gga(x1, x2, x4)
s(x1)  =  s(x1)
U28_gga(x1, x2, x3, x4, x5)  =  U28_gga(x1, x2, x3, x5)
convertcC_in_gga(x1, x2, x3)  =  convertcC_in_gga(x1, x2)
convertcC_out_gga(x1, x2, x3)  =  convertcC_out_gga(x1, x2, x3)
U32_gga(x1, x2, x3, x4)  =  U32_gga(x1, x2, x4)
U33_gga(x1, x2, x3, x4)  =  U33_gga(x1, x2, x4)
timescB_in_gga(x1, x2, x3)  =  timescB_in_gga(x1, x2)
timescB_out_gga(x1, x2, x3)  =  timescB_out_gga(x1, x2, x3)
U29_gga(x1, x2, x3, x4)  =  U29_gga(x1, x2, x4)
U30_gga(x1, x2, x3, x4)  =  U30_gga(x1, x2, x4)
pluscD_in_gga(x1, x2, x3)  =  pluscD_in_gga(x1, x2)
pluscD_out_gga(x1, x2, x3)  =  pluscD_out_gga(x1, x2, x3)
U31_gga(x1, x2, x3, x4)  =  U31_gga(x1, x2, x4)
U34_gga(x1, x2, x3, x4)  =  U34_gga(x1, x2, x4)
timescE_in_gga(x1, x2, x3)  =  timescE_in_gga(x1, x2)
timescE_out_gga(x1, x2, x3)  =  timescE_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4)  =  U41_gga(x1, x2, x4)
U42_gga(x1, x2, x3, x4)  =  U42_gga(x1, x2, x4)
pluscF_in_gga(x1, x2, x3)  =  pluscF_in_gga(x1, x2)
pluscF_out_gga(x1, x2, x3)  =  pluscF_out_gga(x1, x2, x3)
U40_gga(x1, x2, x3, x4)  =  U40_gga(x1, x2, x4)
U35_gga(x1, x2, x3, x4, x5)  =  U35_gga(x1, x2, x3, x5)
U37_gga(x1, x2, x3, x4)  =  U37_gga(x1, x2, x4)
U38_gga(x1, x2, x3, x4)  =  U38_gga(x1, x2, x4)
U39_gga(x1, x2, x3, x4, x5)  =  U39_gga(x1, x2, x3, x5)
U36_gga(x1, x2, x3, x4, x5)  =  U36_gga(x1, x2, x3, x5)
U27_gga(x1, x2, x3, x4)  =  U27_gga(x1, x2, x4)
PLUSD_IN_GGA(x1, x2, x3)  =  PLUSD_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:

PLUSD_IN_GGA(s(X1), X2, s(X3)) → PLUSD_IN_GGA(X1, X2, X3)

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

PLUSD_IN_GGA(s(X1), X2) → PLUSD_IN_GGA(X1, X2)

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:

  • PLUSD_IN_GGA(s(X1), X2) → PLUSD_IN_GGA(X1, X2)
    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:

TIMESB_IN_GGA(s(X1), X2, X3) → TIMESB_IN_GGA(X1, X2, X4)

The TRS R consists of the following rules:

convertcA_in_gga([], X1, 0) → convertcA_out_gga([], X1, 0)
convertcA_in_gga(.(0, X1), X2, X3) → U26_gga(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
convertcA_in_gga(.(s(X1), X2), X3, s(X4)) → U28_gga(X1, X2, X3, X4, convertcC_in_gga(.(X1, X2), X3, X4))
convertcC_in_gga([], X1, 0) → convertcC_out_gga([], X1, 0)
convertcC_in_gga(.(0, []), X1, 0) → convertcC_out_gga(.(0, []), X1, 0)
convertcC_in_gga(.(0, .(0, X1)), X2, X3) → U32_gga(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
U32_gga(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U33_gga(X1, X2, X3, timescB_in_gga(X4, X2, X5))
timescB_in_gga(0, X1, 0) → timescB_out_gga(0, X1, 0)
timescB_in_gga(s(X1), X2, X3) → U29_gga(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U29_gga(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U30_gga(X1, X2, X3, pluscD_in_gga(X2, X4, X3))
pluscD_in_gga(0, X1, X1) → pluscD_out_gga(0, X1, X1)
pluscD_in_gga(s(X1), X2, s(X3)) → U31_gga(X1, X2, X3, pluscD_in_gga(X1, X2, X3))
U31_gga(X1, X2, X3, pluscD_out_gga(X1, X2, X3)) → pluscD_out_gga(s(X1), X2, s(X3))
U30_gga(X1, X2, X3, pluscD_out_gga(X2, X4, X3)) → timescB_out_gga(s(X1), X2, X3)
U33_gga(X1, X2, X3, timescB_out_gga(X4, X2, X5)) → U34_gga(X1, X2, X3, timescE_in_gga(X5, X2, X3))
timescE_in_gga(0, X1, 0) → timescE_out_gga(0, X1, 0)
timescE_in_gga(s(X1), X2, X3) → U41_gga(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U41_gga(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U42_gga(X1, X2, X3, pluscF_in_gga(X2, X4, X3))
pluscF_in_gga(0, X1, X1) → pluscF_out_gga(0, X1, X1)
pluscF_in_gga(s(X1), X2, s(X3)) → U40_gga(X1, X2, X3, pluscF_in_gga(X1, X2, X3))
U40_gga(X1, X2, X3, pluscF_out_gga(X1, X2, X3)) → pluscF_out_gga(s(X1), X2, s(X3))
U42_gga(X1, X2, X3, pluscF_out_gga(X2, X4, X3)) → timescE_out_gga(s(X1), X2, X3)
U34_gga(X1, X2, X3, timescE_out_gga(X5, X2, X3)) → convertcC_out_gga(.(0, .(0, X1)), X2, X3)
convertcC_in_gga(.(0, .(s(X1), X2)), X3, X4) → U35_gga(X1, X2, X3, X4, convertcC_in_gga(.(X1, X2), X3, X5))
convertcC_in_gga(.(s(0), X1), X2, s(X3)) → U37_gga(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
U37_gga(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U38_gga(X1, X2, X3, timescE_in_gga(X4, X2, X3))
U38_gga(X1, X2, X3, timescE_out_gga(X4, X2, X3)) → convertcC_out_gga(.(s(0), X1), X2, s(X3))
convertcC_in_gga(.(s(s(X1)), X2), X3, s(s(X4))) → U39_gga(X1, X2, X3, X4, convertcC_in_gga(.(X1, X2), X3, X4))
U39_gga(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X4)) → convertcC_out_gga(.(s(s(X1)), X2), X3, s(s(X4)))
U35_gga(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X5)) → U36_gga(X1, X2, X3, X4, timescE_in_gga(s(X5), X3, X4))
U36_gga(X1, X2, X3, X4, timescE_out_gga(s(X5), X3, X4)) → convertcC_out_gga(.(0, .(s(X1), X2)), X3, X4)
U28_gga(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X4)) → convertcA_out_gga(.(s(X1), X2), X3, s(X4))
U26_gga(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U27_gga(X1, X2, X3, timescB_in_gga(X4, X2, X3))
U27_gga(X1, X2, X3, timescB_out_gga(X4, X2, X3)) → convertcA_out_gga(.(0, X1), X2, X3)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
0  =  0
convertcA_in_gga(x1, x2, x3)  =  convertcA_in_gga(x1, x2)
[]  =  []
convertcA_out_gga(x1, x2, x3)  =  convertcA_out_gga(x1, x2, x3)
U26_gga(x1, x2, x3, x4)  =  U26_gga(x1, x2, x4)
s(x1)  =  s(x1)
U28_gga(x1, x2, x3, x4, x5)  =  U28_gga(x1, x2, x3, x5)
convertcC_in_gga(x1, x2, x3)  =  convertcC_in_gga(x1, x2)
convertcC_out_gga(x1, x2, x3)  =  convertcC_out_gga(x1, x2, x3)
U32_gga(x1, x2, x3, x4)  =  U32_gga(x1, x2, x4)
U33_gga(x1, x2, x3, x4)  =  U33_gga(x1, x2, x4)
timescB_in_gga(x1, x2, x3)  =  timescB_in_gga(x1, x2)
timescB_out_gga(x1, x2, x3)  =  timescB_out_gga(x1, x2, x3)
U29_gga(x1, x2, x3, x4)  =  U29_gga(x1, x2, x4)
U30_gga(x1, x2, x3, x4)  =  U30_gga(x1, x2, x4)
pluscD_in_gga(x1, x2, x3)  =  pluscD_in_gga(x1, x2)
pluscD_out_gga(x1, x2, x3)  =  pluscD_out_gga(x1, x2, x3)
U31_gga(x1, x2, x3, x4)  =  U31_gga(x1, x2, x4)
U34_gga(x1, x2, x3, x4)  =  U34_gga(x1, x2, x4)
timescE_in_gga(x1, x2, x3)  =  timescE_in_gga(x1, x2)
timescE_out_gga(x1, x2, x3)  =  timescE_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4)  =  U41_gga(x1, x2, x4)
U42_gga(x1, x2, x3, x4)  =  U42_gga(x1, x2, x4)
pluscF_in_gga(x1, x2, x3)  =  pluscF_in_gga(x1, x2)
pluscF_out_gga(x1, x2, x3)  =  pluscF_out_gga(x1, x2, x3)
U40_gga(x1, x2, x3, x4)  =  U40_gga(x1, x2, x4)
U35_gga(x1, x2, x3, x4, x5)  =  U35_gga(x1, x2, x3, x5)
U37_gga(x1, x2, x3, x4)  =  U37_gga(x1, x2, x4)
U38_gga(x1, x2, x3, x4)  =  U38_gga(x1, x2, x4)
U39_gga(x1, x2, x3, x4, x5)  =  U39_gga(x1, x2, x3, x5)
U36_gga(x1, x2, x3, x4, x5)  =  U36_gga(x1, x2, x3, x5)
U27_gga(x1, x2, x3, x4)  =  U27_gga(x1, x2, x4)
TIMESB_IN_GGA(x1, x2, x3)  =  TIMESB_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:

TIMESB_IN_GGA(s(X1), X2, X3) → TIMESB_IN_GGA(X1, X2, X4)

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

TIMESB_IN_GGA(s(X1), X2) → TIMESB_IN_GGA(X1, X2)

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:

  • TIMESB_IN_GGA(s(X1), X2) → TIMESB_IN_GGA(X1, X2)
    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:

CONVERTC_IN_GGA(.(0, .(0, X1)), X2, X3) → CONVERTA_IN_GGA(X1, X2, X4)
CONVERTA_IN_GGA(.(0, X1), X2, X3) → CONVERTA_IN_GGA(X1, X2, X4)
CONVERTA_IN_GGA(.(s(X1), X2), X3, s(X4)) → CONVERTC_IN_GGA(.(X1, X2), X3, X4)
CONVERTC_IN_GGA(.(0, .(s(X1), X2)), X3, X4) → CONVERTC_IN_GGA(.(X1, X2), X3, X5)
CONVERTC_IN_GGA(.(s(0), X1), X2, s(X3)) → CONVERTA_IN_GGA(X1, X2, X4)
CONVERTC_IN_GGA(.(s(s(X1)), X2), X3, s(s(X4))) → CONVERTC_IN_GGA(.(X1, X2), X3, X4)

The TRS R consists of the following rules:

convertcA_in_gga([], X1, 0) → convertcA_out_gga([], X1, 0)
convertcA_in_gga(.(0, X1), X2, X3) → U26_gga(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
convertcA_in_gga(.(s(X1), X2), X3, s(X4)) → U28_gga(X1, X2, X3, X4, convertcC_in_gga(.(X1, X2), X3, X4))
convertcC_in_gga([], X1, 0) → convertcC_out_gga([], X1, 0)
convertcC_in_gga(.(0, []), X1, 0) → convertcC_out_gga(.(0, []), X1, 0)
convertcC_in_gga(.(0, .(0, X1)), X2, X3) → U32_gga(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
U32_gga(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U33_gga(X1, X2, X3, timescB_in_gga(X4, X2, X5))
timescB_in_gga(0, X1, 0) → timescB_out_gga(0, X1, 0)
timescB_in_gga(s(X1), X2, X3) → U29_gga(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U29_gga(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U30_gga(X1, X2, X3, pluscD_in_gga(X2, X4, X3))
pluscD_in_gga(0, X1, X1) → pluscD_out_gga(0, X1, X1)
pluscD_in_gga(s(X1), X2, s(X3)) → U31_gga(X1, X2, X3, pluscD_in_gga(X1, X2, X3))
U31_gga(X1, X2, X3, pluscD_out_gga(X1, X2, X3)) → pluscD_out_gga(s(X1), X2, s(X3))
U30_gga(X1, X2, X3, pluscD_out_gga(X2, X4, X3)) → timescB_out_gga(s(X1), X2, X3)
U33_gga(X1, X2, X3, timescB_out_gga(X4, X2, X5)) → U34_gga(X1, X2, X3, timescE_in_gga(X5, X2, X3))
timescE_in_gga(0, X1, 0) → timescE_out_gga(0, X1, 0)
timescE_in_gga(s(X1), X2, X3) → U41_gga(X1, X2, X3, timescB_in_gga(X1, X2, X4))
U41_gga(X1, X2, X3, timescB_out_gga(X1, X2, X4)) → U42_gga(X1, X2, X3, pluscF_in_gga(X2, X4, X3))
pluscF_in_gga(0, X1, X1) → pluscF_out_gga(0, X1, X1)
pluscF_in_gga(s(X1), X2, s(X3)) → U40_gga(X1, X2, X3, pluscF_in_gga(X1, X2, X3))
U40_gga(X1, X2, X3, pluscF_out_gga(X1, X2, X3)) → pluscF_out_gga(s(X1), X2, s(X3))
U42_gga(X1, X2, X3, pluscF_out_gga(X2, X4, X3)) → timescE_out_gga(s(X1), X2, X3)
U34_gga(X1, X2, X3, timescE_out_gga(X5, X2, X3)) → convertcC_out_gga(.(0, .(0, X1)), X2, X3)
convertcC_in_gga(.(0, .(s(X1), X2)), X3, X4) → U35_gga(X1, X2, X3, X4, convertcC_in_gga(.(X1, X2), X3, X5))
convertcC_in_gga(.(s(0), X1), X2, s(X3)) → U37_gga(X1, X2, X3, convertcA_in_gga(X1, X2, X4))
U37_gga(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U38_gga(X1, X2, X3, timescE_in_gga(X4, X2, X3))
U38_gga(X1, X2, X3, timescE_out_gga(X4, X2, X3)) → convertcC_out_gga(.(s(0), X1), X2, s(X3))
convertcC_in_gga(.(s(s(X1)), X2), X3, s(s(X4))) → U39_gga(X1, X2, X3, X4, convertcC_in_gga(.(X1, X2), X3, X4))
U39_gga(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X4)) → convertcC_out_gga(.(s(s(X1)), X2), X3, s(s(X4)))
U35_gga(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X5)) → U36_gga(X1, X2, X3, X4, timescE_in_gga(s(X5), X3, X4))
U36_gga(X1, X2, X3, X4, timescE_out_gga(s(X5), X3, X4)) → convertcC_out_gga(.(0, .(s(X1), X2)), X3, X4)
U28_gga(X1, X2, X3, X4, convertcC_out_gga(.(X1, X2), X3, X4)) → convertcA_out_gga(.(s(X1), X2), X3, s(X4))
U26_gga(X1, X2, X3, convertcA_out_gga(X1, X2, X4)) → U27_gga(X1, X2, X3, timescB_in_gga(X4, X2, X3))
U27_gga(X1, X2, X3, timescB_out_gga(X4, X2, X3)) → convertcA_out_gga(.(0, X1), X2, X3)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
0  =  0
convertcA_in_gga(x1, x2, x3)  =  convertcA_in_gga(x1, x2)
[]  =  []
convertcA_out_gga(x1, x2, x3)  =  convertcA_out_gga(x1, x2, x3)
U26_gga(x1, x2, x3, x4)  =  U26_gga(x1, x2, x4)
s(x1)  =  s(x1)
U28_gga(x1, x2, x3, x4, x5)  =  U28_gga(x1, x2, x3, x5)
convertcC_in_gga(x1, x2, x3)  =  convertcC_in_gga(x1, x2)
convertcC_out_gga(x1, x2, x3)  =  convertcC_out_gga(x1, x2, x3)
U32_gga(x1, x2, x3, x4)  =  U32_gga(x1, x2, x4)
U33_gga(x1, x2, x3, x4)  =  U33_gga(x1, x2, x4)
timescB_in_gga(x1, x2, x3)  =  timescB_in_gga(x1, x2)
timescB_out_gga(x1, x2, x3)  =  timescB_out_gga(x1, x2, x3)
U29_gga(x1, x2, x3, x4)  =  U29_gga(x1, x2, x4)
U30_gga(x1, x2, x3, x4)  =  U30_gga(x1, x2, x4)
pluscD_in_gga(x1, x2, x3)  =  pluscD_in_gga(x1, x2)
pluscD_out_gga(x1, x2, x3)  =  pluscD_out_gga(x1, x2, x3)
U31_gga(x1, x2, x3, x4)  =  U31_gga(x1, x2, x4)
U34_gga(x1, x2, x3, x4)  =  U34_gga(x1, x2, x4)
timescE_in_gga(x1, x2, x3)  =  timescE_in_gga(x1, x2)
timescE_out_gga(x1, x2, x3)  =  timescE_out_gga(x1, x2, x3)
U41_gga(x1, x2, x3, x4)  =  U41_gga(x1, x2, x4)
U42_gga(x1, x2, x3, x4)  =  U42_gga(x1, x2, x4)
pluscF_in_gga(x1, x2, x3)  =  pluscF_in_gga(x1, x2)
pluscF_out_gga(x1, x2, x3)  =  pluscF_out_gga(x1, x2, x3)
U40_gga(x1, x2, x3, x4)  =  U40_gga(x1, x2, x4)
U35_gga(x1, x2, x3, x4, x5)  =  U35_gga(x1, x2, x3, x5)
U37_gga(x1, x2, x3, x4)  =  U37_gga(x1, x2, x4)
U38_gga(x1, x2, x3, x4)  =  U38_gga(x1, x2, x4)
U39_gga(x1, x2, x3, x4, x5)  =  U39_gga(x1, x2, x3, x5)
U36_gga(x1, x2, x3, x4, x5)  =  U36_gga(x1, x2, x3, x5)
U27_gga(x1, x2, x3, x4)  =  U27_gga(x1, x2, x4)
CONVERTC_IN_GGA(x1, x2, x3)  =  CONVERTC_IN_GGA(x1, x2)
CONVERTA_IN_GGA(x1, x2, x3)  =  CONVERTA_IN_GGA(x1, x2)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) Obligation:

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

CONVERTC_IN_GGA(.(0, .(0, X1)), X2, X3) → CONVERTA_IN_GGA(X1, X2, X4)
CONVERTA_IN_GGA(.(0, X1), X2, X3) → CONVERTA_IN_GGA(X1, X2, X4)
CONVERTA_IN_GGA(.(s(X1), X2), X3, s(X4)) → CONVERTC_IN_GGA(.(X1, X2), X3, X4)
CONVERTC_IN_GGA(.(0, .(s(X1), X2)), X3, X4) → CONVERTC_IN_GGA(.(X1, X2), X3, X5)
CONVERTC_IN_GGA(.(s(0), X1), X2, s(X3)) → CONVERTA_IN_GGA(X1, X2, X4)
CONVERTC_IN_GGA(.(s(s(X1)), X2), X3, s(s(X4))) → CONVERTC_IN_GGA(.(X1, X2), X3, X4)

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

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

(31) PiDPToQDPProof (SOUND transformation)

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

(32) Obligation:

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

CONVERTC_IN_GGA(.(0, .(0, X1)), X2) → CONVERTA_IN_GGA(X1, X2)
CONVERTA_IN_GGA(.(0, X1), X2) → CONVERTA_IN_GGA(X1, X2)
CONVERTA_IN_GGA(.(s(X1), X2), X3) → CONVERTC_IN_GGA(.(X1, X2), X3)
CONVERTC_IN_GGA(.(0, .(s(X1), X2)), X3) → CONVERTC_IN_GGA(.(X1, X2), X3)
CONVERTC_IN_GGA(.(s(0), X1), X2) → CONVERTA_IN_GGA(X1, X2)
CONVERTC_IN_GGA(.(s(s(X1)), X2), X3) → CONVERTC_IN_GGA(.(X1, X2), X3)

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

(33) 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:

CONVERTC_IN_GGA(.(0, .(0, X1)), X2) → CONVERTA_IN_GGA(X1, X2)
CONVERTA_IN_GGA(.(0, X1), X2) → CONVERTA_IN_GGA(X1, X2)
CONVERTA_IN_GGA(.(s(X1), X2), X3) → CONVERTC_IN_GGA(.(X1, X2), X3)
CONVERTC_IN_GGA(.(0, .(s(X1), X2)), X3) → CONVERTC_IN_GGA(.(X1, X2), X3)
CONVERTC_IN_GGA(.(s(0), X1), X2) → CONVERTA_IN_GGA(X1, X2)
CONVERTC_IN_GGA(.(s(s(X1)), X2), X3) → CONVERTC_IN_GGA(.(X1, X2), X3)
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x1, x2)) = x1 + 2·x2   
POL(0) = 0   
POL(CONVERTA_IN_GGA(x1, x2)) = 2·x1 + x2   
POL(CONVERTC_IN_GGA(x1, x2)) = x1 + x2   
POL(s(x1)) = 2·x1   

(34) Obligation:

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

(35) PisEmptyProof (EQUIVALENT transformation)

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

(36) YES