Left Termination of the query pattern log_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇐)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇐)
↳4 PiDP
↳5 DependencyGraphProof (⇔)
↳6 AND
    ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇐)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 YES
    ↳14 PiDP
        ↳15 PiDPToQDPProof (⇐)
        ↳16 QDP
        ↳17 QDPOrderProof (⇔)
        ↳18 QDP
        ↳19 DependencyGraphProof (⇔)
        ↳20 TRUE

(0) Obligation:

Clauses:

half(0, 0).
half(s(0), 0).
half(s(s(X)), s(Y)) :- half(X, Y).
log(0, s(0)).
log(s(X), s(Y)) :- ','(half(s(X), Z), log(Z, Y)).

Queries:

log(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

half24(s(s(T14)), s(X31)) :- half24(T14, X31).
log1(s(s(T10)), s(T7)) :- half24(T10, X20).
log1(s(s(T10)), s(T7)) :- ','(halfc24(T10, T11), log1(s(T11), T7)).

Clauses:

logc1(0, s(0)).
logc1(s(0), s(s(0))).
logc1(s(s(T10)), s(T7)) :- ','(halfc24(T10, T11), logc1(s(T11), T7)).
halfc24(0, 0).
halfc24(s(0), 0).
halfc24(s(s(T14)), s(X31)) :- halfc24(T14, X31).

Afs:

log1(x1, x2)  =  log1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
log1_in: (b,f)
half24_in: (b,f)
halfc24_in: (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

LOG1_IN_GA(s(s(T10)), s(T7)) → U2_GA(T10, T7, half24_in_ga(T10, X20))
LOG1_IN_GA(s(s(T10)), s(T7)) → HALF24_IN_GA(T10, X20)
HALF24_IN_GA(s(s(T14)), s(X31)) → U1_GA(T14, X31, half24_in_ga(T14, X31))
HALF24_IN_GA(s(s(T14)), s(X31)) → HALF24_IN_GA(T14, X31)
LOG1_IN_GA(s(s(T10)), s(T7)) → U3_GA(T10, T7, halfc24_in_ga(T10, T11))
U3_GA(T10, T7, halfc24_out_ga(T10, T11)) → U4_GA(T10, T7, log1_in_ga(s(T11), T7))
U3_GA(T10, T7, halfc24_out_ga(T10, T11)) → LOG1_IN_GA(s(T11), T7)

The TRS R consists of the following rules:

halfc24_in_ga(0, 0) → halfc24_out_ga(0, 0)
halfc24_in_ga(s(0), 0) → halfc24_out_ga(s(0), 0)
halfc24_in_ga(s(s(T14)), s(X31)) → U8_ga(T14, X31, halfc24_in_ga(T14, X31))
U8_ga(T14, X31, halfc24_out_ga(T14, X31)) → halfc24_out_ga(s(s(T14)), s(X31))

The argument filtering Pi contains the following mapping:
log1_in_ga(x1, x2)  =  log1_in_ga(x1)
s(x1)  =  s(x1)
half24_in_ga(x1, x2)  =  half24_in_ga(x1)
halfc24_in_ga(x1, x2)  =  halfc24_in_ga(x1)
0  =  0
halfc24_out_ga(x1, x2)  =  halfc24_out_ga(x1, x2)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
LOG1_IN_GA(x1, x2)  =  LOG1_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
HALF24_IN_GA(x1, x2)  =  HALF24_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

LOG1_IN_GA(s(s(T10)), s(T7)) → U2_GA(T10, T7, half24_in_ga(T10, X20))
LOG1_IN_GA(s(s(T10)), s(T7)) → HALF24_IN_GA(T10, X20)
HALF24_IN_GA(s(s(T14)), s(X31)) → U1_GA(T14, X31, half24_in_ga(T14, X31))
HALF24_IN_GA(s(s(T14)), s(X31)) → HALF24_IN_GA(T14, X31)
LOG1_IN_GA(s(s(T10)), s(T7)) → U3_GA(T10, T7, halfc24_in_ga(T10, T11))
U3_GA(T10, T7, halfc24_out_ga(T10, T11)) → U4_GA(T10, T7, log1_in_ga(s(T11), T7))
U3_GA(T10, T7, halfc24_out_ga(T10, T11)) → LOG1_IN_GA(s(T11), T7)

The TRS R consists of the following rules:

halfc24_in_ga(0, 0) → halfc24_out_ga(0, 0)
halfc24_in_ga(s(0), 0) → halfc24_out_ga(s(0), 0)
halfc24_in_ga(s(s(T14)), s(X31)) → U8_ga(T14, X31, halfc24_in_ga(T14, X31))
U8_ga(T14, X31, halfc24_out_ga(T14, X31)) → halfc24_out_ga(s(s(T14)), s(X31))

The argument filtering Pi contains the following mapping:
log1_in_ga(x1, x2)  =  log1_in_ga(x1)
s(x1)  =  s(x1)
half24_in_ga(x1, x2)  =  half24_in_ga(x1)
halfc24_in_ga(x1, x2)  =  halfc24_in_ga(x1)
0  =  0
halfc24_out_ga(x1, x2)  =  halfc24_out_ga(x1, x2)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
LOG1_IN_GA(x1, x2)  =  LOG1_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
HALF24_IN_GA(x1, x2)  =  HALF24_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

HALF24_IN_GA(s(s(T14)), s(X31)) → HALF24_IN_GA(T14, X31)

The TRS R consists of the following rules:

halfc24_in_ga(0, 0) → halfc24_out_ga(0, 0)
halfc24_in_ga(s(0), 0) → halfc24_out_ga(s(0), 0)
halfc24_in_ga(s(s(T14)), s(X31)) → U8_ga(T14, X31, halfc24_in_ga(T14, X31))
U8_ga(T14, X31, halfc24_out_ga(T14, X31)) → halfc24_out_ga(s(s(T14)), s(X31))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
halfc24_in_ga(x1, x2)  =  halfc24_in_ga(x1)
0  =  0
halfc24_out_ga(x1, x2)  =  halfc24_out_ga(x1, x2)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
HALF24_IN_GA(x1, x2)  =  HALF24_IN_GA(x1)

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:

HALF24_IN_GA(s(s(T14)), s(X31)) → HALF24_IN_GA(T14, X31)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
HALF24_IN_GA(x1, x2)  =  HALF24_IN_GA(x1)

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:

HALF24_IN_GA(s(s(T14))) → HALF24_IN_GA(T14)

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:

  • HALF24_IN_GA(s(s(T14))) → HALF24_IN_GA(T14)
    The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

LOG1_IN_GA(s(s(T10)), s(T7)) → U3_GA(T10, T7, halfc24_in_ga(T10, T11))
U3_GA(T10, T7, halfc24_out_ga(T10, T11)) → LOG1_IN_GA(s(T11), T7)

The TRS R consists of the following rules:

halfc24_in_ga(0, 0) → halfc24_out_ga(0, 0)
halfc24_in_ga(s(0), 0) → halfc24_out_ga(s(0), 0)
halfc24_in_ga(s(s(T14)), s(X31)) → U8_ga(T14, X31, halfc24_in_ga(T14, X31))
U8_ga(T14, X31, halfc24_out_ga(T14, X31)) → halfc24_out_ga(s(s(T14)), s(X31))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
halfc24_in_ga(x1, x2)  =  halfc24_in_ga(x1)
0  =  0
halfc24_out_ga(x1, x2)  =  halfc24_out_ga(x1, x2)
U8_ga(x1, x2, x3)  =  U8_ga(x1, x3)
LOG1_IN_GA(x1, x2)  =  LOG1_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)

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

(15) PiDPToQDPProof (SOUND transformation)

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

(16) Obligation:

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

LOG1_IN_GA(s(s(T10))) → U3_GA(T10, halfc24_in_ga(T10))
U3_GA(T10, halfc24_out_ga(T10, T11)) → LOG1_IN_GA(s(T11))

The TRS R consists of the following rules:

halfc24_in_ga(0) → halfc24_out_ga(0, 0)
halfc24_in_ga(s(0)) → halfc24_out_ga(s(0), 0)
halfc24_in_ga(s(s(T14))) → U8_ga(T14, halfc24_in_ga(T14))
U8_ga(T14, halfc24_out_ga(T14, X31)) → halfc24_out_ga(s(s(T14)), s(X31))

The set Q consists of the following terms:

halfc24_in_ga(x0)
U8_ga(x0, x1)

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


LOG1_IN_GA(s(s(T10))) → U3_GA(T10, halfc24_in_ga(T10))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(LOG1_IN_GA(x1)) = x1   
POL(U3_GA(x1, x2)) = 1 + x2   
POL(U8_ga(x1, x2)) = 1 + x2   
POL(halfc24_in_ga(x1)) = x1   
POL(halfc24_out_ga(x1, x2)) = x2   
POL(s(x1)) = 1 + x1   

The following usable rules [FROCOS05] were oriented:

halfc24_in_ga(0) → halfc24_out_ga(0, 0)
halfc24_in_ga(s(0)) → halfc24_out_ga(s(0), 0)
halfc24_in_ga(s(s(T14))) → U8_ga(T14, halfc24_in_ga(T14))
U8_ga(T14, halfc24_out_ga(T14, X31)) → halfc24_out_ga(s(s(T14)), s(X31))

(18) Obligation:

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

U3_GA(T10, halfc24_out_ga(T10, T11)) → LOG1_IN_GA(s(T11))

The TRS R consists of the following rules:

halfc24_in_ga(0) → halfc24_out_ga(0, 0)
halfc24_in_ga(s(0)) → halfc24_out_ga(s(0), 0)
halfc24_in_ga(s(s(T14))) → U8_ga(T14, halfc24_in_ga(T14))
U8_ga(T14, halfc24_out_ga(T14, X31)) → halfc24_out_ga(s(s(T14)), s(X31))

The set Q consists of the following terms:

halfc24_in_ga(x0)
U8_ga(x0, x1)

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

(19) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(20) TRUE