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

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇒, 187 ms)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇒, 154 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 AND
    ↳7 PiDP
        ↳8 UsableRulesProof (⇔, 0 ms)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇒, 33 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 QDPSizeChangeProof (⇔, 0 ms)
        ↳34 YES

(0) Obligation:

Clauses:

isNat(s(X)) :- isNat(X).
isNat(0).
notEq(s(X), s(Y)) :- notEq(X, Y).
notEq(s(X), 0).
notEq(0, s(X)).
lt(s(X), s(Y)) :- lt(X, Y).
lt(0, s(Y)).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(X), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(Y)).
le(0, 0).
even(s(X)) :- odd(X).
even(0).
odd(s(X)) :- even(X).
odd(s(0)).
add(s(X), Y, s(Z)) :- add(X, Y, Z).
add(0, X, X).
mult(s(X), Y, R) :- ','(mult(X, Y, Z), add(Y, Z, R)).
mult(0, Y, 0).
factorial(s(X), R) :- ','(factorial(X, Y), mult(s(X), Y, R)).
factorial(0, s(0)).

Query: factorial(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

(2) Obligation:

Triples:

factorialA(s(X1), X2) :- factorialA(X1, X3).
factorialA(s(X1), X2) :- ','(factorialcA(X1, X3), multB(X1, X3, X2)).
pC(s(X1), X2, X3, X4) :- pC(X1, X2, X5, X3).
pC(X1, X2, X3, X4) :- ','(multcE(X1, X2, X3), addD(X2, X3, X4)).
addD(s(X1), X2, s(X3)) :- addD(X1, X2, X3).
multB(X1, X2, X3) :- pC(X1, X2, X4, X3).
addF(s(X1), X2, s(X3)) :- addF(X1, X2, X3).
factorialH(s(s(X1)), X2) :- factorialA(X1, X3).
factorialH(s(s(X1)), X2) :- ','(factorialcA(X1, X3), multB(X1, X3, X4)).
factorialH(s(s(X1)), X2) :- ','(factorialcA(X1, X3), ','(multcB(X1, X3, X4), multB(X1, X4, X5))).
factorialH(s(s(X1)), X2) :- ','(factorialcA(X1, X3), ','(multcB(X1, X3, X4), ','(multcB(X1, X4, X5), addF(X4, X5, X2)))).

Clauses:

factorialcA(s(X1), X2) :- ','(factorialcA(X1, X3), multcB(X1, X3, X2)).
factorialcA(0, s(0)).
qcC(X1, X2, X3, X4) :- ','(multcE(X1, X2, X3), addcD(X2, X3, X4)).
addcD(s(X1), X2, s(X3)) :- addcD(X1, X2, X3).
addcD(0, X1, X1).
multcB(X1, X2, X3) :- qcC(X1, X2, X4, X3).
addcF(s(X1), X2, s(X3)) :- addcF(X1, X2, X3).
addcF(0, X1, X1).
multcE(s(X1), X2, X3) :- qcC(X1, X2, X4, X3).
multcE(0, X1, 0).
multcG(0).

Afs:

factorialH(x1, x2)  =  factorialH(x1)

(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:
factorialH_in: (b,f)
factorialA_in: (b,f)
factorialcA_in: (b,f)
multcB_in: (b,b,f)
qcC_in: (b,b,f,f)
multcE_in: (b,b,f)
addcD_in: (b,b,f)
multB_in: (b,b,f)
pC_in: (b,b,f,f)
addD_in: (b,b,f)
addF_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

FACTORIALH_IN_GA(s(s(X1)), X2) → U10_GA(X1, X2, factorialA_in_ga(X1, X3))
FACTORIALH_IN_GA(s(s(X1)), X2) → FACTORIALA_IN_GA(X1, X3)
FACTORIALA_IN_GA(s(X1), X2) → U1_GA(X1, X2, factorialA_in_ga(X1, X3))
FACTORIALA_IN_GA(s(X1), X2) → FACTORIALA_IN_GA(X1, X3)
FACTORIALA_IN_GA(s(X1), X2) → U2_GA(X1, X2, factorialcA_in_ga(X1, X3))
U2_GA(X1, X2, factorialcA_out_ga(X1, X3)) → U3_GA(X1, X2, multB_in_gga(X1, X3, X2))
U2_GA(X1, X2, factorialcA_out_ga(X1, X3)) → MULTB_IN_GGA(X1, X3, X2)
MULTB_IN_GGA(X1, X2, X3) → U8_GGA(X1, X2, X3, pC_in_ggaa(X1, X2, X4, X3))
MULTB_IN_GGA(X1, X2, X3) → PC_IN_GGAA(X1, X2, X4, X3)
PC_IN_GGAA(s(X1), X2, X3, X4) → U4_GGAA(X1, X2, X3, X4, pC_in_ggaa(X1, X2, X5, X3))
PC_IN_GGAA(s(X1), X2, X3, X4) → PC_IN_GGAA(X1, X2, X5, X3)
PC_IN_GGAA(X1, X2, X3, X4) → U5_GGAA(X1, X2, X3, X4, multcE_in_gga(X1, X2, X3))
U5_GGAA(X1, X2, X3, X4, multcE_out_gga(X1, X2, X3)) → U6_GGAA(X1, X2, X3, X4, addD_in_gga(X2, X3, X4))
U5_GGAA(X1, X2, X3, X4, multcE_out_gga(X1, X2, X3)) → ADDD_IN_GGA(X2, X3, X4)
ADDD_IN_GGA(s(X1), X2, s(X3)) → U7_GGA(X1, X2, X3, addD_in_gga(X1, X2, X3))
ADDD_IN_GGA(s(X1), X2, s(X3)) → ADDD_IN_GGA(X1, X2, X3)
FACTORIALH_IN_GA(s(s(X1)), X2) → U11_GA(X1, X2, factorialcA_in_ga(X1, X3))
U11_GA(X1, X2, factorialcA_out_ga(X1, X3)) → U12_GA(X1, X2, multB_in_gga(X1, X3, X4))
U11_GA(X1, X2, factorialcA_out_ga(X1, X3)) → MULTB_IN_GGA(X1, X3, X4)
U11_GA(X1, X2, factorialcA_out_ga(X1, X3)) → U13_GA(X1, X2, multcB_in_gga(X1, X3, X4))
U13_GA(X1, X2, multcB_out_gga(X1, X3, X4)) → U14_GA(X1, X2, multB_in_gga(X1, X4, X5))
U13_GA(X1, X2, multcB_out_gga(X1, X3, X4)) → MULTB_IN_GGA(X1, X4, X5)
U13_GA(X1, X2, multcB_out_gga(X1, X3, X4)) → U15_GA(X1, X2, X4, multcB_in_gga(X1, X4, X5))
U15_GA(X1, X2, X4, multcB_out_gga(X1, X4, X5)) → U16_GA(X1, X2, addF_in_gga(X4, X5, X2))
U15_GA(X1, X2, X4, multcB_out_gga(X1, X4, X5)) → ADDF_IN_GGA(X4, X5, X2)
ADDF_IN_GGA(s(X1), X2, s(X3)) → U9_GGA(X1, X2, X3, addF_in_gga(X1, X2, X3))
ADDF_IN_GGA(s(X1), X2, s(X3)) → ADDF_IN_GGA(X1, X2, X3)

The TRS R consists of the following rules:

factorialcA_in_ga(s(X1), X2) → U18_ga(X1, X2, factorialcA_in_ga(X1, X3))
factorialcA_in_ga(0, s(0)) → factorialcA_out_ga(0, s(0))
U18_ga(X1, X2, factorialcA_out_ga(X1, X3)) → U19_ga(X1, X2, multcB_in_gga(X1, X3, X2))
multcB_in_gga(X1, X2, X3) → U23_gga(X1, X2, X3, qcC_in_ggaa(X1, X2, X4, X3))
qcC_in_ggaa(X1, X2, X3, X4) → U20_ggaa(X1, X2, X3, X4, multcE_in_gga(X1, X2, X3))
multcE_in_gga(s(X1), X2, X3) → U25_gga(X1, X2, X3, qcC_in_ggaa(X1, X2, X4, X3))
U25_gga(X1, X2, X3, qcC_out_ggaa(X1, X2, X4, X3)) → multcE_out_gga(s(X1), X2, X3)
multcE_in_gga(0, X1, 0) → multcE_out_gga(0, X1, 0)
U20_ggaa(X1, X2, X3, X4, multcE_out_gga(X1, X2, X3)) → U21_ggaa(X1, X2, X3, X4, addcD_in_gga(X2, X3, X4))
addcD_in_gga(s(X1), X2, s(X3)) → U22_gga(X1, X2, X3, addcD_in_gga(X1, X2, X3))
addcD_in_gga(0, X1, X1) → addcD_out_gga(0, X1, X1)
U22_gga(X1, X2, X3, addcD_out_gga(X1, X2, X3)) → addcD_out_gga(s(X1), X2, s(X3))
U21_ggaa(X1, X2, X3, X4, addcD_out_gga(X2, X3, X4)) → qcC_out_ggaa(X1, X2, X3, X4)
U23_gga(X1, X2, X3, qcC_out_ggaa(X1, X2, X4, X3)) → multcB_out_gga(X1, X2, X3)
U19_ga(X1, X2, multcB_out_gga(X1, X3, X2)) → factorialcA_out_ga(s(X1), X2)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
factorialA_in_ga(x1, x2)  =  factorialA_in_ga(x1)
factorialcA_in_ga(x1, x2)  =  factorialcA_in_ga(x1)
U18_ga(x1, x2, x3)  =  U18_ga(x1, x3)
0  =  0
factorialcA_out_ga(x1, x2)  =  factorialcA_out_ga(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
multcB_in_gga(x1, x2, x3)  =  multcB_in_gga(x1, x2)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x1, x2, x4)
qcC_in_ggaa(x1, x2, x3, x4)  =  qcC_in_ggaa(x1, x2)
U20_ggaa(x1, x2, x3, x4, x5)  =  U20_ggaa(x1, x2, x5)
multcE_in_gga(x1, x2, x3)  =  multcE_in_gga(x1, x2)
U25_gga(x1, x2, x3, x4)  =  U25_gga(x1, x2, x4)
qcC_out_ggaa(x1, x2, x3, x4)  =  qcC_out_ggaa(x1, x2, x3, x4)
multcE_out_gga(x1, x2, x3)  =  multcE_out_gga(x1, x2, x3)
U21_ggaa(x1, x2, x3, x4, x5)  =  U21_ggaa(x1, x2, x3, x5)
addcD_in_gga(x1, x2, x3)  =  addcD_in_gga(x1, x2)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x1, x2, x4)
addcD_out_gga(x1, x2, x3)  =  addcD_out_gga(x1, x2, x3)
multcB_out_gga(x1, x2, x3)  =  multcB_out_gga(x1, x2, x3)
multB_in_gga(x1, x2, x3)  =  multB_in_gga(x1, x2)
pC_in_ggaa(x1, x2, x3, x4)  =  pC_in_ggaa(x1, x2)
addD_in_gga(x1, x2, x3)  =  addD_in_gga(x1, x2)
addF_in_gga(x1, x2, x3)  =  addF_in_gga(x1, x2)
FACTORIALH_IN_GA(x1, x2)  =  FACTORIALH_IN_GA(x1)
U10_GA(x1, x2, x3)  =  U10_GA(x1, x3)
FACTORIALA_IN_GA(x1, x2)  =  FACTORIALA_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
MULTB_IN_GGA(x1, x2, x3)  =  MULTB_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x2, x4)
PC_IN_GGAA(x1, x2, x3, x4)  =  PC_IN_GGAA(x1, x2)
U4_GGAA(x1, x2, x3, x4, x5)  =  U4_GGAA(x1, x2, x5)
U5_GGAA(x1, x2, x3, x4, x5)  =  U5_GGAA(x1, x2, x5)
U6_GGAA(x1, x2, x3, x4, x5)  =  U6_GGAA(x1, x2, x5)
ADDD_IN_GGA(x1, x2, x3)  =  ADDD_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x1, x2, x4)
U11_GA(x1, x2, x3)  =  U11_GA(x1, x3)
U12_GA(x1, x2, x3)  =  U12_GA(x1, x3)
U13_GA(x1, x2, x3)  =  U13_GA(x1, x3)
U14_GA(x1, x2, x3)  =  U14_GA(x1, x3)
U15_GA(x1, x2, x3, x4)  =  U15_GA(x1, x3, x4)
U16_GA(x1, x2, x3)  =  U16_GA(x1, x3)
ADDF_IN_GGA(x1, x2, x3)  =  ADDF_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x1, x2, x4)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

FACTORIALH_IN_GA(s(s(X1)), X2) → U10_GA(X1, X2, factorialA_in_ga(X1, X3))
FACTORIALH_IN_GA(s(s(X1)), X2) → FACTORIALA_IN_GA(X1, X3)
FACTORIALA_IN_GA(s(X1), X2) → U1_GA(X1, X2, factorialA_in_ga(X1, X3))
FACTORIALA_IN_GA(s(X1), X2) → FACTORIALA_IN_GA(X1, X3)
FACTORIALA_IN_GA(s(X1), X2) → U2_GA(X1, X2, factorialcA_in_ga(X1, X3))
U2_GA(X1, X2, factorialcA_out_ga(X1, X3)) → U3_GA(X1, X2, multB_in_gga(X1, X3, X2))
U2_GA(X1, X2, factorialcA_out_ga(X1, X3)) → MULTB_IN_GGA(X1, X3, X2)
MULTB_IN_GGA(X1, X2, X3) → U8_GGA(X1, X2, X3, pC_in_ggaa(X1, X2, X4, X3))
MULTB_IN_GGA(X1, X2, X3) → PC_IN_GGAA(X1, X2, X4, X3)
PC_IN_GGAA(s(X1), X2, X3, X4) → U4_GGAA(X1, X2, X3, X4, pC_in_ggaa(X1, X2, X5, X3))
PC_IN_GGAA(s(X1), X2, X3, X4) → PC_IN_GGAA(X1, X2, X5, X3)
PC_IN_GGAA(X1, X2, X3, X4) → U5_GGAA(X1, X2, X3, X4, multcE_in_gga(X1, X2, X3))
U5_GGAA(X1, X2, X3, X4, multcE_out_gga(X1, X2, X3)) → U6_GGAA(X1, X2, X3, X4, addD_in_gga(X2, X3, X4))
U5_GGAA(X1, X2, X3, X4, multcE_out_gga(X1, X2, X3)) → ADDD_IN_GGA(X2, X3, X4)
ADDD_IN_GGA(s(X1), X2, s(X3)) → U7_GGA(X1, X2, X3, addD_in_gga(X1, X2, X3))
ADDD_IN_GGA(s(X1), X2, s(X3)) → ADDD_IN_GGA(X1, X2, X3)
FACTORIALH_IN_GA(s(s(X1)), X2) → U11_GA(X1, X2, factorialcA_in_ga(X1, X3))
U11_GA(X1, X2, factorialcA_out_ga(X1, X3)) → U12_GA(X1, X2, multB_in_gga(X1, X3, X4))
U11_GA(X1, X2, factorialcA_out_ga(X1, X3)) → MULTB_IN_GGA(X1, X3, X4)
U11_GA(X1, X2, factorialcA_out_ga(X1, X3)) → U13_GA(X1, X2, multcB_in_gga(X1, X3, X4))
U13_GA(X1, X2, multcB_out_gga(X1, X3, X4)) → U14_GA(X1, X2, multB_in_gga(X1, X4, X5))
U13_GA(X1, X2, multcB_out_gga(X1, X3, X4)) → MULTB_IN_GGA(X1, X4, X5)
U13_GA(X1, X2, multcB_out_gga(X1, X3, X4)) → U15_GA(X1, X2, X4, multcB_in_gga(X1, X4, X5))
U15_GA(X1, X2, X4, multcB_out_gga(X1, X4, X5)) → U16_GA(X1, X2, addF_in_gga(X4, X5, X2))
U15_GA(X1, X2, X4, multcB_out_gga(X1, X4, X5)) → ADDF_IN_GGA(X4, X5, X2)
ADDF_IN_GGA(s(X1), X2, s(X3)) → U9_GGA(X1, X2, X3, addF_in_gga(X1, X2, X3))
ADDF_IN_GGA(s(X1), X2, s(X3)) → ADDF_IN_GGA(X1, X2, X3)

The TRS R consists of the following rules:

factorialcA_in_ga(s(X1), X2) → U18_ga(X1, X2, factorialcA_in_ga(X1, X3))
factorialcA_in_ga(0, s(0)) → factorialcA_out_ga(0, s(0))
U18_ga(X1, X2, factorialcA_out_ga(X1, X3)) → U19_ga(X1, X2, multcB_in_gga(X1, X3, X2))
multcB_in_gga(X1, X2, X3) → U23_gga(X1, X2, X3, qcC_in_ggaa(X1, X2, X4, X3))
qcC_in_ggaa(X1, X2, X3, X4) → U20_ggaa(X1, X2, X3, X4, multcE_in_gga(X1, X2, X3))
multcE_in_gga(s(X1), X2, X3) → U25_gga(X1, X2, X3, qcC_in_ggaa(X1, X2, X4, X3))
U25_gga(X1, X2, X3, qcC_out_ggaa(X1, X2, X4, X3)) → multcE_out_gga(s(X1), X2, X3)
multcE_in_gga(0, X1, 0) → multcE_out_gga(0, X1, 0)
U20_ggaa(X1, X2, X3, X4, multcE_out_gga(X1, X2, X3)) → U21_ggaa(X1, X2, X3, X4, addcD_in_gga(X2, X3, X4))
addcD_in_gga(s(X1), X2, s(X3)) → U22_gga(X1, X2, X3, addcD_in_gga(X1, X2, X3))
addcD_in_gga(0, X1, X1) → addcD_out_gga(0, X1, X1)
U22_gga(X1, X2, X3, addcD_out_gga(X1, X2, X3)) → addcD_out_gga(s(X1), X2, s(X3))
U21_ggaa(X1, X2, X3, X4, addcD_out_gga(X2, X3, X4)) → qcC_out_ggaa(X1, X2, X3, X4)
U23_gga(X1, X2, X3, qcC_out_ggaa(X1, X2, X4, X3)) → multcB_out_gga(X1, X2, X3)
U19_ga(X1, X2, multcB_out_gga(X1, X3, X2)) → factorialcA_out_ga(s(X1), X2)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
factorialA_in_ga(x1, x2)  =  factorialA_in_ga(x1)
factorialcA_in_ga(x1, x2)  =  factorialcA_in_ga(x1)
U18_ga(x1, x2, x3)  =  U18_ga(x1, x3)
0  =  0
factorialcA_out_ga(x1, x2)  =  factorialcA_out_ga(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
multcB_in_gga(x1, x2, x3)  =  multcB_in_gga(x1, x2)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x1, x2, x4)
qcC_in_ggaa(x1, x2, x3, x4)  =  qcC_in_ggaa(x1, x2)
U20_ggaa(x1, x2, x3, x4, x5)  =  U20_ggaa(x1, x2, x5)
multcE_in_gga(x1, x2, x3)  =  multcE_in_gga(x1, x2)
U25_gga(x1, x2, x3, x4)  =  U25_gga(x1, x2, x4)
qcC_out_ggaa(x1, x2, x3, x4)  =  qcC_out_ggaa(x1, x2, x3, x4)
multcE_out_gga(x1, x2, x3)  =  multcE_out_gga(x1, x2, x3)
U21_ggaa(x1, x2, x3, x4, x5)  =  U21_ggaa(x1, x2, x3, x5)
addcD_in_gga(x1, x2, x3)  =  addcD_in_gga(x1, x2)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x1, x2, x4)
addcD_out_gga(x1, x2, x3)  =  addcD_out_gga(x1, x2, x3)
multcB_out_gga(x1, x2, x3)  =  multcB_out_gga(x1, x2, x3)
multB_in_gga(x1, x2, x3)  =  multB_in_gga(x1, x2)
pC_in_ggaa(x1, x2, x3, x4)  =  pC_in_ggaa(x1, x2)
addD_in_gga(x1, x2, x3)  =  addD_in_gga(x1, x2)
addF_in_gga(x1, x2, x3)  =  addF_in_gga(x1, x2)
FACTORIALH_IN_GA(x1, x2)  =  FACTORIALH_IN_GA(x1)
U10_GA(x1, x2, x3)  =  U10_GA(x1, x3)
FACTORIALA_IN_GA(x1, x2)  =  FACTORIALA_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
MULTB_IN_GGA(x1, x2, x3)  =  MULTB_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x2, x4)
PC_IN_GGAA(x1, x2, x3, x4)  =  PC_IN_GGAA(x1, x2)
U4_GGAA(x1, x2, x3, x4, x5)  =  U4_GGAA(x1, x2, x5)
U5_GGAA(x1, x2, x3, x4, x5)  =  U5_GGAA(x1, x2, x5)
U6_GGAA(x1, x2, x3, x4, x5)  =  U6_GGAA(x1, x2, x5)
ADDD_IN_GGA(x1, x2, x3)  =  ADDD_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x1, x2, x4)
U11_GA(x1, x2, x3)  =  U11_GA(x1, x3)
U12_GA(x1, x2, x3)  =  U12_GA(x1, x3)
U13_GA(x1, x2, x3)  =  U13_GA(x1, x3)
U14_GA(x1, x2, x3)  =  U14_GA(x1, x3)
U15_GA(x1, x2, x3, x4)  =  U15_GA(x1, x3, x4)
U16_GA(x1, x2, x3)  =  U16_GA(x1, x3)
ADDF_IN_GGA(x1, x2, x3)  =  ADDF_IN_GGA(x1, x2)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x1, x2, x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 23 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

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

The TRS R consists of the following rules:

factorialcA_in_ga(s(X1), X2) → U18_ga(X1, X2, factorialcA_in_ga(X1, X3))
factorialcA_in_ga(0, s(0)) → factorialcA_out_ga(0, s(0))
U18_ga(X1, X2, factorialcA_out_ga(X1, X3)) → U19_ga(X1, X2, multcB_in_gga(X1, X3, X2))
multcB_in_gga(X1, X2, X3) → U23_gga(X1, X2, X3, qcC_in_ggaa(X1, X2, X4, X3))
qcC_in_ggaa(X1, X2, X3, X4) → U20_ggaa(X1, X2, X3, X4, multcE_in_gga(X1, X2, X3))
multcE_in_gga(s(X1), X2, X3) → U25_gga(X1, X2, X3, qcC_in_ggaa(X1, X2, X4, X3))
U25_gga(X1, X2, X3, qcC_out_ggaa(X1, X2, X4, X3)) → multcE_out_gga(s(X1), X2, X3)
multcE_in_gga(0, X1, 0) → multcE_out_gga(0, X1, 0)
U20_ggaa(X1, X2, X3, X4, multcE_out_gga(X1, X2, X3)) → U21_ggaa(X1, X2, X3, X4, addcD_in_gga(X2, X3, X4))
addcD_in_gga(s(X1), X2, s(X3)) → U22_gga(X1, X2, X3, addcD_in_gga(X1, X2, X3))
addcD_in_gga(0, X1, X1) → addcD_out_gga(0, X1, X1)
U22_gga(X1, X2, X3, addcD_out_gga(X1, X2, X3)) → addcD_out_gga(s(X1), X2, s(X3))
U21_ggaa(X1, X2, X3, X4, addcD_out_gga(X2, X3, X4)) → qcC_out_ggaa(X1, X2, X3, X4)
U23_gga(X1, X2, X3, qcC_out_ggaa(X1, X2, X4, X3)) → multcB_out_gga(X1, X2, X3)
U19_ga(X1, X2, multcB_out_gga(X1, X3, X2)) → factorialcA_out_ga(s(X1), X2)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
factorialcA_in_ga(x1, x2)  =  factorialcA_in_ga(x1)
U18_ga(x1, x2, x3)  =  U18_ga(x1, x3)
0  =  0
factorialcA_out_ga(x1, x2)  =  factorialcA_out_ga(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
multcB_in_gga(x1, x2, x3)  =  multcB_in_gga(x1, x2)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x1, x2, x4)
qcC_in_ggaa(x1, x2, x3, x4)  =  qcC_in_ggaa(x1, x2)
U20_ggaa(x1, x2, x3, x4, x5)  =  U20_ggaa(x1, x2, x5)
multcE_in_gga(x1, x2, x3)  =  multcE_in_gga(x1, x2)
U25_gga(x1, x2, x3, x4)  =  U25_gga(x1, x2, x4)
qcC_out_ggaa(x1, x2, x3, x4)  =  qcC_out_ggaa(x1, x2, x3, x4)
multcE_out_gga(x1, x2, x3)  =  multcE_out_gga(x1, x2, x3)
U21_ggaa(x1, x2, x3, x4, x5)  =  U21_ggaa(x1, x2, x3, x5)
addcD_in_gga(x1, x2, x3)  =  addcD_in_gga(x1, x2)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x1, x2, x4)
addcD_out_gga(x1, x2, x3)  =  addcD_out_gga(x1, x2, x3)
multcB_out_gga(x1, x2, x3)  =  multcB_out_gga(x1, x2, x3)
ADDF_IN_GGA(x1, x2, x3)  =  ADDF_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:

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

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

ADDF_IN_GGA(s(X1), X2) → ADDF_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:

  • ADDF_IN_GGA(s(X1), X2) → ADDF_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:

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

The TRS R consists of the following rules:

factorialcA_in_ga(s(X1), X2) → U18_ga(X1, X2, factorialcA_in_ga(X1, X3))
factorialcA_in_ga(0, s(0)) → factorialcA_out_ga(0, s(0))
U18_ga(X1, X2, factorialcA_out_ga(X1, X3)) → U19_ga(X1, X2, multcB_in_gga(X1, X3, X2))
multcB_in_gga(X1, X2, X3) → U23_gga(X1, X2, X3, qcC_in_ggaa(X1, X2, X4, X3))
qcC_in_ggaa(X1, X2, X3, X4) → U20_ggaa(X1, X2, X3, X4, multcE_in_gga(X1, X2, X3))
multcE_in_gga(s(X1), X2, X3) → U25_gga(X1, X2, X3, qcC_in_ggaa(X1, X2, X4, X3))
U25_gga(X1, X2, X3, qcC_out_ggaa(X1, X2, X4, X3)) → multcE_out_gga(s(X1), X2, X3)
multcE_in_gga(0, X1, 0) → multcE_out_gga(0, X1, 0)
U20_ggaa(X1, X2, X3, X4, multcE_out_gga(X1, X2, X3)) → U21_ggaa(X1, X2, X3, X4, addcD_in_gga(X2, X3, X4))
addcD_in_gga(s(X1), X2, s(X3)) → U22_gga(X1, X2, X3, addcD_in_gga(X1, X2, X3))
addcD_in_gga(0, X1, X1) → addcD_out_gga(0, X1, X1)
U22_gga(X1, X2, X3, addcD_out_gga(X1, X2, X3)) → addcD_out_gga(s(X1), X2, s(X3))
U21_ggaa(X1, X2, X3, X4, addcD_out_gga(X2, X3, X4)) → qcC_out_ggaa(X1, X2, X3, X4)
U23_gga(X1, X2, X3, qcC_out_ggaa(X1, X2, X4, X3)) → multcB_out_gga(X1, X2, X3)
U19_ga(X1, X2, multcB_out_gga(X1, X3, X2)) → factorialcA_out_ga(s(X1), X2)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
factorialcA_in_ga(x1, x2)  =  factorialcA_in_ga(x1)
U18_ga(x1, x2, x3)  =  U18_ga(x1, x3)
0  =  0
factorialcA_out_ga(x1, x2)  =  factorialcA_out_ga(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
multcB_in_gga(x1, x2, x3)  =  multcB_in_gga(x1, x2)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x1, x2, x4)
qcC_in_ggaa(x1, x2, x3, x4)  =  qcC_in_ggaa(x1, x2)
U20_ggaa(x1, x2, x3, x4, x5)  =  U20_ggaa(x1, x2, x5)
multcE_in_gga(x1, x2, x3)  =  multcE_in_gga(x1, x2)
U25_gga(x1, x2, x3, x4)  =  U25_gga(x1, x2, x4)
qcC_out_ggaa(x1, x2, x3, x4)  =  qcC_out_ggaa(x1, x2, x3, x4)
multcE_out_gga(x1, x2, x3)  =  multcE_out_gga(x1, x2, x3)
U21_ggaa(x1, x2, x3, x4, x5)  =  U21_ggaa(x1, x2, x3, x5)
addcD_in_gga(x1, x2, x3)  =  addcD_in_gga(x1, x2)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x1, x2, x4)
addcD_out_gga(x1, x2, x3)  =  addcD_out_gga(x1, x2, x3)
multcB_out_gga(x1, x2, x3)  =  multcB_out_gga(x1, x2, x3)
ADDD_IN_GGA(x1, x2, x3)  =  ADDD_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:

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

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

ADDD_IN_GGA(s(X1), X2) → ADDD_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:

  • ADDD_IN_GGA(s(X1), X2) → ADDD_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:

PC_IN_GGAA(s(X1), X2, X3, X4) → PC_IN_GGAA(X1, X2, X5, X3)

The TRS R consists of the following rules:

factorialcA_in_ga(s(X1), X2) → U18_ga(X1, X2, factorialcA_in_ga(X1, X3))
factorialcA_in_ga(0, s(0)) → factorialcA_out_ga(0, s(0))
U18_ga(X1, X2, factorialcA_out_ga(X1, X3)) → U19_ga(X1, X2, multcB_in_gga(X1, X3, X2))
multcB_in_gga(X1, X2, X3) → U23_gga(X1, X2, X3, qcC_in_ggaa(X1, X2, X4, X3))
qcC_in_ggaa(X1, X2, X3, X4) → U20_ggaa(X1, X2, X3, X4, multcE_in_gga(X1, X2, X3))
multcE_in_gga(s(X1), X2, X3) → U25_gga(X1, X2, X3, qcC_in_ggaa(X1, X2, X4, X3))
U25_gga(X1, X2, X3, qcC_out_ggaa(X1, X2, X4, X3)) → multcE_out_gga(s(X1), X2, X3)
multcE_in_gga(0, X1, 0) → multcE_out_gga(0, X1, 0)
U20_ggaa(X1, X2, X3, X4, multcE_out_gga(X1, X2, X3)) → U21_ggaa(X1, X2, X3, X4, addcD_in_gga(X2, X3, X4))
addcD_in_gga(s(X1), X2, s(X3)) → U22_gga(X1, X2, X3, addcD_in_gga(X1, X2, X3))
addcD_in_gga(0, X1, X1) → addcD_out_gga(0, X1, X1)
U22_gga(X1, X2, X3, addcD_out_gga(X1, X2, X3)) → addcD_out_gga(s(X1), X2, s(X3))
U21_ggaa(X1, X2, X3, X4, addcD_out_gga(X2, X3, X4)) → qcC_out_ggaa(X1, X2, X3, X4)
U23_gga(X1, X2, X3, qcC_out_ggaa(X1, X2, X4, X3)) → multcB_out_gga(X1, X2, X3)
U19_ga(X1, X2, multcB_out_gga(X1, X3, X2)) → factorialcA_out_ga(s(X1), X2)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
factorialcA_in_ga(x1, x2)  =  factorialcA_in_ga(x1)
U18_ga(x1, x2, x3)  =  U18_ga(x1, x3)
0  =  0
factorialcA_out_ga(x1, x2)  =  factorialcA_out_ga(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
multcB_in_gga(x1, x2, x3)  =  multcB_in_gga(x1, x2)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x1, x2, x4)
qcC_in_ggaa(x1, x2, x3, x4)  =  qcC_in_ggaa(x1, x2)
U20_ggaa(x1, x2, x3, x4, x5)  =  U20_ggaa(x1, x2, x5)
multcE_in_gga(x1, x2, x3)  =  multcE_in_gga(x1, x2)
U25_gga(x1, x2, x3, x4)  =  U25_gga(x1, x2, x4)
qcC_out_ggaa(x1, x2, x3, x4)  =  qcC_out_ggaa(x1, x2, x3, x4)
multcE_out_gga(x1, x2, x3)  =  multcE_out_gga(x1, x2, x3)
U21_ggaa(x1, x2, x3, x4, x5)  =  U21_ggaa(x1, x2, x3, x5)
addcD_in_gga(x1, x2, x3)  =  addcD_in_gga(x1, x2)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x1, x2, x4)
addcD_out_gga(x1, x2, x3)  =  addcD_out_gga(x1, x2, x3)
multcB_out_gga(x1, x2, x3)  =  multcB_out_gga(x1, x2, x3)
PC_IN_GGAA(x1, x2, x3, x4)  =  PC_IN_GGAA(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:

PC_IN_GGAA(s(X1), X2, X3, X4) → PC_IN_GGAA(X1, X2, X5, X3)

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

PC_IN_GGAA(s(X1), X2) → PC_IN_GGAA(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:

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

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

The TRS R consists of the following rules:

factorialcA_in_ga(s(X1), X2) → U18_ga(X1, X2, factorialcA_in_ga(X1, X3))
factorialcA_in_ga(0, s(0)) → factorialcA_out_ga(0, s(0))
U18_ga(X1, X2, factorialcA_out_ga(X1, X3)) → U19_ga(X1, X2, multcB_in_gga(X1, X3, X2))
multcB_in_gga(X1, X2, X3) → U23_gga(X1, X2, X3, qcC_in_ggaa(X1, X2, X4, X3))
qcC_in_ggaa(X1, X2, X3, X4) → U20_ggaa(X1, X2, X3, X4, multcE_in_gga(X1, X2, X3))
multcE_in_gga(s(X1), X2, X3) → U25_gga(X1, X2, X3, qcC_in_ggaa(X1, X2, X4, X3))
U25_gga(X1, X2, X3, qcC_out_ggaa(X1, X2, X4, X3)) → multcE_out_gga(s(X1), X2, X3)
multcE_in_gga(0, X1, 0) → multcE_out_gga(0, X1, 0)
U20_ggaa(X1, X2, X3, X4, multcE_out_gga(X1, X2, X3)) → U21_ggaa(X1, X2, X3, X4, addcD_in_gga(X2, X3, X4))
addcD_in_gga(s(X1), X2, s(X3)) → U22_gga(X1, X2, X3, addcD_in_gga(X1, X2, X3))
addcD_in_gga(0, X1, X1) → addcD_out_gga(0, X1, X1)
U22_gga(X1, X2, X3, addcD_out_gga(X1, X2, X3)) → addcD_out_gga(s(X1), X2, s(X3))
U21_ggaa(X1, X2, X3, X4, addcD_out_gga(X2, X3, X4)) → qcC_out_ggaa(X1, X2, X3, X4)
U23_gga(X1, X2, X3, qcC_out_ggaa(X1, X2, X4, X3)) → multcB_out_gga(X1, X2, X3)
U19_ga(X1, X2, multcB_out_gga(X1, X3, X2)) → factorialcA_out_ga(s(X1), X2)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
factorialcA_in_ga(x1, x2)  =  factorialcA_in_ga(x1)
U18_ga(x1, x2, x3)  =  U18_ga(x1, x3)
0  =  0
factorialcA_out_ga(x1, x2)  =  factorialcA_out_ga(x1, x2)
U19_ga(x1, x2, x3)  =  U19_ga(x1, x3)
multcB_in_gga(x1, x2, x3)  =  multcB_in_gga(x1, x2)
U23_gga(x1, x2, x3, x4)  =  U23_gga(x1, x2, x4)
qcC_in_ggaa(x1, x2, x3, x4)  =  qcC_in_ggaa(x1, x2)
U20_ggaa(x1, x2, x3, x4, x5)  =  U20_ggaa(x1, x2, x5)
multcE_in_gga(x1, x2, x3)  =  multcE_in_gga(x1, x2)
U25_gga(x1, x2, x3, x4)  =  U25_gga(x1, x2, x4)
qcC_out_ggaa(x1, x2, x3, x4)  =  qcC_out_ggaa(x1, x2, x3, x4)
multcE_out_gga(x1, x2, x3)  =  multcE_out_gga(x1, x2, x3)
U21_ggaa(x1, x2, x3, x4, x5)  =  U21_ggaa(x1, x2, x3, x5)
addcD_in_gga(x1, x2, x3)  =  addcD_in_gga(x1, x2)
U22_gga(x1, x2, x3, x4)  =  U22_gga(x1, x2, x4)
addcD_out_gga(x1, x2, x3)  =  addcD_out_gga(x1, x2, x3)
multcB_out_gga(x1, x2, x3)  =  multcB_out_gga(x1, x2, x3)
FACTORIALA_IN_GA(x1, x2)  =  FACTORIALA_IN_GA(x1)

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:

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

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

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:

FACTORIALA_IN_GA(s(X1)) → FACTORIALA_IN_GA(X1)

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

(33) 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:

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

(34) YES