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

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇒, 83 ms)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇒, 36 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 AND
    ↳7 PiDP
        ↳8 UsableRulesProof (⇔, 0 ms)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇒, 2 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

(0) Obligation:

Clauses:

mult(0, Y, 0).
mult(s(X), Y, Z) :- ','(mult(X, Y, Z1), add(Z1, Y, Z)).
add(0, Y, Y).
add(s(X), Y, s(Z)) :- add(X, Y, Z).

Query: mult(g,g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

(2) Obligation:

Triples:

multA(s(X1), X2, X3) :- multA(X1, X2, X4).
multA(s(X1), X2, X3) :- ','(multcA(X1, X2, X4), addB(X4, X2, X3)).
addB(s(X1), X2, s(X3)) :- addB(X1, X2, X3).
addC(s(X1), X2, s(X3)) :- addC(X1, X2, X3).
multD(s(s(X1)), X2, X3) :- multA(X1, X2, X4).
multD(s(s(X1)), X2, X3) :- ','(multcA(X1, X2, X4), addB(X4, X2, X5)).
multD(s(s(X1)), X2, X3) :- ','(multcA(X1, X2, X4), ','(addcB(X4, X2, X5), addC(X5, X2, X3))).

Clauses:

multcA(0, X1, 0).
multcA(s(X1), X2, X3) :- ','(multcA(X1, X2, X4), addcB(X4, X2, X3)).
addcB(0, X1, X1).
addcB(s(X1), X2, s(X3)) :- addcB(X1, X2, X3).
addcC(0, X1, X1).
addcC(s(X1), X2, s(X3)) :- addcC(X1, X2, X3).

Afs:

multD(x1, x2, x3)  =  multD(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:
multD_in: (b,b,f)
multA_in: (b,b,f)
multcA_in: (b,b,f)
addcB_in: (b,b,f)
addB_in: (b,b,f)
addC_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

MULTD_IN_GGA(s(s(X1)), X2, X3) → U6_GGA(X1, X2, X3, multA_in_gga(X1, X2, X4))
MULTD_IN_GGA(s(s(X1)), X2, X3) → MULTA_IN_GGA(X1, X2, X4)
MULTA_IN_GGA(s(X1), X2, X3) → U1_GGA(X1, X2, X3, multA_in_gga(X1, X2, X4))
MULTA_IN_GGA(s(X1), X2, X3) → MULTA_IN_GGA(X1, X2, X4)
MULTA_IN_GGA(s(X1), X2, X3) → U2_GGA(X1, X2, X3, multcA_in_gga(X1, X2, X4))
U2_GGA(X1, X2, X3, multcA_out_gga(X1, X2, X4)) → U3_GGA(X1, X2, X3, addB_in_gga(X4, X2, X3))
U2_GGA(X1, X2, X3, multcA_out_gga(X1, X2, X4)) → ADDB_IN_GGA(X4, X2, X3)
ADDB_IN_GGA(s(X1), X2, s(X3)) → U4_GGA(X1, X2, X3, addB_in_gga(X1, X2, X3))
ADDB_IN_GGA(s(X1), X2, s(X3)) → ADDB_IN_GGA(X1, X2, X3)
MULTD_IN_GGA(s(s(X1)), X2, X3) → U7_GGA(X1, X2, X3, multcA_in_gga(X1, X2, X4))
U7_GGA(X1, X2, X3, multcA_out_gga(X1, X2, X4)) → U8_GGA(X1, X2, X3, addB_in_gga(X4, X2, X5))
U7_GGA(X1, X2, X3, multcA_out_gga(X1, X2, X4)) → ADDB_IN_GGA(X4, X2, X5)
U7_GGA(X1, X2, X3, multcA_out_gga(X1, X2, X4)) → U9_GGA(X1, X2, X3, addcB_in_gga(X4, X2, X5))
U9_GGA(X1, X2, X3, addcB_out_gga(X4, X2, X5)) → U10_GGA(X1, X2, X3, addC_in_gga(X5, X2, X3))
U9_GGA(X1, X2, X3, addcB_out_gga(X4, X2, X5)) → ADDC_IN_GGA(X5, X2, X3)
ADDC_IN_GGA(s(X1), X2, s(X3)) → U5_GGA(X1, X2, X3, addC_in_gga(X1, X2, X3))
ADDC_IN_GGA(s(X1), X2, s(X3)) → ADDC_IN_GGA(X1, X2, X3)

The TRS R consists of the following rules:

multcA_in_gga(0, X1, 0) → multcA_out_gga(0, X1, 0)
multcA_in_gga(s(X1), X2, X3) → U12_gga(X1, X2, X3, multcA_in_gga(X1, X2, X4))
U12_gga(X1, X2, X3, multcA_out_gga(X1, X2, X4)) → U13_gga(X1, X2, X3, addcB_in_gga(X4, X2, X3))
addcB_in_gga(0, X1, X1) → addcB_out_gga(0, X1, X1)
addcB_in_gga(s(X1), X2, s(X3)) → U14_gga(X1, X2, X3, addcB_in_gga(X1, X2, X3))
U14_gga(X1, X2, X3, addcB_out_gga(X1, X2, X3)) → addcB_out_gga(s(X1), X2, s(X3))
U13_gga(X1, X2, X3, addcB_out_gga(X4, X2, X3)) → multcA_out_gga(s(X1), X2, X3)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
multA_in_gga(x1, x2, x3)  =  multA_in_gga(x1, x2)
multcA_in_gga(x1, x2, x3)  =  multcA_in_gga(x1, x2)
0  =  0
multcA_out_gga(x1, x2, x3)  =  multcA_out_gga(x1, x2, x3)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
addcB_in_gga(x1, x2, x3)  =  addcB_in_gga(x1, x2)
addcB_out_gga(x1, x2, x3)  =  addcB_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
addB_in_gga(x1, x2, x3)  =  addB_in_gga(x1, x2)
addC_in_gga(x1, x2, x3)  =  addC_in_gga(x1, x2)
MULTD_IN_GGA(x1, x2, x3)  =  MULTD_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
MULTA_IN_GGA(x1, x2, x3)  =  MULTA_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)
ADDB_IN_GGA(x1, x2, x3)  =  ADDB_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x1, x2, x4)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x2, x4)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x1, x2, x4)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x1, x2, x4)
ADDC_IN_GGA(x1, x2, x3)  =  ADDC_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_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:

MULTD_IN_GGA(s(s(X1)), X2, X3) → U6_GGA(X1, X2, X3, multA_in_gga(X1, X2, X4))
MULTD_IN_GGA(s(s(X1)), X2, X3) → MULTA_IN_GGA(X1, X2, X4)
MULTA_IN_GGA(s(X1), X2, X3) → U1_GGA(X1, X2, X3, multA_in_gga(X1, X2, X4))
MULTA_IN_GGA(s(X1), X2, X3) → MULTA_IN_GGA(X1, X2, X4)
MULTA_IN_GGA(s(X1), X2, X3) → U2_GGA(X1, X2, X3, multcA_in_gga(X1, X2, X4))
U2_GGA(X1, X2, X3, multcA_out_gga(X1, X2, X4)) → U3_GGA(X1, X2, X3, addB_in_gga(X4, X2, X3))
U2_GGA(X1, X2, X3, multcA_out_gga(X1, X2, X4)) → ADDB_IN_GGA(X4, X2, X3)
ADDB_IN_GGA(s(X1), X2, s(X3)) → U4_GGA(X1, X2, X3, addB_in_gga(X1, X2, X3))
ADDB_IN_GGA(s(X1), X2, s(X3)) → ADDB_IN_GGA(X1, X2, X3)
MULTD_IN_GGA(s(s(X1)), X2, X3) → U7_GGA(X1, X2, X3, multcA_in_gga(X1, X2, X4))
U7_GGA(X1, X2, X3, multcA_out_gga(X1, X2, X4)) → U8_GGA(X1, X2, X3, addB_in_gga(X4, X2, X5))
U7_GGA(X1, X2, X3, multcA_out_gga(X1, X2, X4)) → ADDB_IN_GGA(X4, X2, X5)
U7_GGA(X1, X2, X3, multcA_out_gga(X1, X2, X4)) → U9_GGA(X1, X2, X3, addcB_in_gga(X4, X2, X5))
U9_GGA(X1, X2, X3, addcB_out_gga(X4, X2, X5)) → U10_GGA(X1, X2, X3, addC_in_gga(X5, X2, X3))
U9_GGA(X1, X2, X3, addcB_out_gga(X4, X2, X5)) → ADDC_IN_GGA(X5, X2, X3)
ADDC_IN_GGA(s(X1), X2, s(X3)) → U5_GGA(X1, X2, X3, addC_in_gga(X1, X2, X3))
ADDC_IN_GGA(s(X1), X2, s(X3)) → ADDC_IN_GGA(X1, X2, X3)

The TRS R consists of the following rules:

multcA_in_gga(0, X1, 0) → multcA_out_gga(0, X1, 0)
multcA_in_gga(s(X1), X2, X3) → U12_gga(X1, X2, X3, multcA_in_gga(X1, X2, X4))
U12_gga(X1, X2, X3, multcA_out_gga(X1, X2, X4)) → U13_gga(X1, X2, X3, addcB_in_gga(X4, X2, X3))
addcB_in_gga(0, X1, X1) → addcB_out_gga(0, X1, X1)
addcB_in_gga(s(X1), X2, s(X3)) → U14_gga(X1, X2, X3, addcB_in_gga(X1, X2, X3))
U14_gga(X1, X2, X3, addcB_out_gga(X1, X2, X3)) → addcB_out_gga(s(X1), X2, s(X3))
U13_gga(X1, X2, X3, addcB_out_gga(X4, X2, X3)) → multcA_out_gga(s(X1), X2, X3)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
multA_in_gga(x1, x2, x3)  =  multA_in_gga(x1, x2)
multcA_in_gga(x1, x2, x3)  =  multcA_in_gga(x1, x2)
0  =  0
multcA_out_gga(x1, x2, x3)  =  multcA_out_gga(x1, x2, x3)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
addcB_in_gga(x1, x2, x3)  =  addcB_in_gga(x1, x2)
addcB_out_gga(x1, x2, x3)  =  addcB_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
addB_in_gga(x1, x2, x3)  =  addB_in_gga(x1, x2)
addC_in_gga(x1, x2, x3)  =  addC_in_gga(x1, x2)
MULTD_IN_GGA(x1, x2, x3)  =  MULTD_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
MULTA_IN_GGA(x1, x2, x3)  =  MULTA_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)
ADDB_IN_GGA(x1, x2, x3)  =  ADDB_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x1, x2, x4)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x2, x4)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x1, x2, x4)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x1, x2, x4)
ADDC_IN_GGA(x1, x2, x3)  =  ADDC_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_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 3 SCCs with 14 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

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

The TRS R consists of the following rules:

multcA_in_gga(0, X1, 0) → multcA_out_gga(0, X1, 0)
multcA_in_gga(s(X1), X2, X3) → U12_gga(X1, X2, X3, multcA_in_gga(X1, X2, X4))
U12_gga(X1, X2, X3, multcA_out_gga(X1, X2, X4)) → U13_gga(X1, X2, X3, addcB_in_gga(X4, X2, X3))
addcB_in_gga(0, X1, X1) → addcB_out_gga(0, X1, X1)
addcB_in_gga(s(X1), X2, s(X3)) → U14_gga(X1, X2, X3, addcB_in_gga(X1, X2, X3))
U14_gga(X1, X2, X3, addcB_out_gga(X1, X2, X3)) → addcB_out_gga(s(X1), X2, s(X3))
U13_gga(X1, X2, X3, addcB_out_gga(X4, X2, X3)) → multcA_out_gga(s(X1), X2, X3)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
multcA_in_gga(x1, x2, x3)  =  multcA_in_gga(x1, x2)
0  =  0
multcA_out_gga(x1, x2, x3)  =  multcA_out_gga(x1, x2, x3)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
addcB_in_gga(x1, x2, x3)  =  addcB_in_gga(x1, x2)
addcB_out_gga(x1, x2, x3)  =  addcB_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
ADDC_IN_GGA(x1, x2, x3)  =  ADDC_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:

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

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

ADDC_IN_GGA(s(X1), X2) → ADDC_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:

  • ADDC_IN_GGA(s(X1), X2) → ADDC_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:

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

The TRS R consists of the following rules:

multcA_in_gga(0, X1, 0) → multcA_out_gga(0, X1, 0)
multcA_in_gga(s(X1), X2, X3) → U12_gga(X1, X2, X3, multcA_in_gga(X1, X2, X4))
U12_gga(X1, X2, X3, multcA_out_gga(X1, X2, X4)) → U13_gga(X1, X2, X3, addcB_in_gga(X4, X2, X3))
addcB_in_gga(0, X1, X1) → addcB_out_gga(0, X1, X1)
addcB_in_gga(s(X1), X2, s(X3)) → U14_gga(X1, X2, X3, addcB_in_gga(X1, X2, X3))
U14_gga(X1, X2, X3, addcB_out_gga(X1, X2, X3)) → addcB_out_gga(s(X1), X2, s(X3))
U13_gga(X1, X2, X3, addcB_out_gga(X4, X2, X3)) → multcA_out_gga(s(X1), X2, X3)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
multcA_in_gga(x1, x2, x3)  =  multcA_in_gga(x1, x2)
0  =  0
multcA_out_gga(x1, x2, x3)  =  multcA_out_gga(x1, x2, x3)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
addcB_in_gga(x1, x2, x3)  =  addcB_in_gga(x1, x2)
addcB_out_gga(x1, x2, x3)  =  addcB_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
ADDB_IN_GGA(x1, x2, x3)  =  ADDB_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:

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

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

ADDB_IN_GGA(s(X1), X2) → ADDB_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:

  • ADDB_IN_GGA(s(X1), X2) → ADDB_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:

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

The TRS R consists of the following rules:

multcA_in_gga(0, X1, 0) → multcA_out_gga(0, X1, 0)
multcA_in_gga(s(X1), X2, X3) → U12_gga(X1, X2, X3, multcA_in_gga(X1, X2, X4))
U12_gga(X1, X2, X3, multcA_out_gga(X1, X2, X4)) → U13_gga(X1, X2, X3, addcB_in_gga(X4, X2, X3))
addcB_in_gga(0, X1, X1) → addcB_out_gga(0, X1, X1)
addcB_in_gga(s(X1), X2, s(X3)) → U14_gga(X1, X2, X3, addcB_in_gga(X1, X2, X3))
U14_gga(X1, X2, X3, addcB_out_gga(X1, X2, X3)) → addcB_out_gga(s(X1), X2, s(X3))
U13_gga(X1, X2, X3, addcB_out_gga(X4, X2, X3)) → multcA_out_gga(s(X1), X2, X3)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
multcA_in_gga(x1, x2, x3)  =  multcA_in_gga(x1, x2)
0  =  0
multcA_out_gga(x1, x2, x3)  =  multcA_out_gga(x1, x2, x3)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
addcB_in_gga(x1, x2, x3)  =  addcB_in_gga(x1, x2)
addcB_out_gga(x1, x2, x3)  =  addcB_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
MULTA_IN_GGA(x1, x2, x3)  =  MULTA_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:

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

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

MULTA_IN_GGA(s(X1), X2) → MULTA_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:

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

(27) YES