Left Termination of the query pattern log2_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:



↳ Prolog
  ↳ PrologToPiTRSProof

Clauses:

log2(X, Y) :- log2(X, 0, Y).
log2(0, I, I).
log2(s(0), I, I).
log2(s(s(X)), I, Y) :- ','(half(s(s(X)), X1), log2(X1, s(I), Y)).
half(0, 0).
half(s(0), 0).
half(s(s(X)), s(Y)) :- half(X, Y).

Queries:

log2(g,a).

We use the technique of [30].Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

log2_in(X, Y) → U1(X, Y, log2_in(X, 0, Y))
log2_in(s(s(X)), I, Y) → U2(X, I, Y, half_in(s(s(X)), X1))
half_in(s(s(X)), s(Y)) → U4(X, Y, half_in(X, Y))
half_in(s(0), 0) → half_out(s(0), 0)
half_in(0, 0) → half_out(0, 0)
U4(X, Y, half_out(X, Y)) → half_out(s(s(X)), s(Y))
U2(X, I, Y, half_out(s(s(X)), X1)) → U3(X, I, Y, log2_in(X1, s(I), Y))
log2_in(s(0), I, I) → log2_out(s(0), I, I)
log2_in(0, I, I) → log2_out(0, I, I)
U3(X, I, Y, log2_out(X1, s(I), Y)) → log2_out(s(s(X)), I, Y)
U1(X, Y, log2_out(X, 0, Y)) → log2_out(X, Y)

The argument filtering Pi contains the following mapping:
log2_in(x1, x2)  =  log2_in(x1)
U1(x1, x2, x3)  =  U1(x3)
log2_in(x1, x2, x3)  =  log2_in(x1, x2)
0  =  0
s(x1)  =  s(x1)
U2(x1, x2, x3, x4)  =  U2(x2, x4)
half_in(x1, x2)  =  half_in(x1)
U4(x1, x2, x3)  =  U4(x3)
half_out(x1, x2)  =  half_out(x2)
U3(x1, x2, x3, x4)  =  U3(x4)
log2_out(x1, x2, x3)  =  log2_out(x3)
log2_out(x1, x2)  =  log2_out(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

log2_in(X, Y) → U1(X, Y, log2_in(X, 0, Y))
log2_in(s(s(X)), I, Y) → U2(X, I, Y, half_in(s(s(X)), X1))
half_in(s(s(X)), s(Y)) → U4(X, Y, half_in(X, Y))
half_in(s(0), 0) → half_out(s(0), 0)
half_in(0, 0) → half_out(0, 0)
U4(X, Y, half_out(X, Y)) → half_out(s(s(X)), s(Y))
U2(X, I, Y, half_out(s(s(X)), X1)) → U3(X, I, Y, log2_in(X1, s(I), Y))
log2_in(s(0), I, I) → log2_out(s(0), I, I)
log2_in(0, I, I) → log2_out(0, I, I)
U3(X, I, Y, log2_out(X1, s(I), Y)) → log2_out(s(s(X)), I, Y)
U1(X, Y, log2_out(X, 0, Y)) → log2_out(X, Y)

The argument filtering Pi contains the following mapping:
log2_in(x1, x2)  =  log2_in(x1)
U1(x1, x2, x3)  =  U1(x3)
log2_in(x1, x2, x3)  =  log2_in(x1, x2)
0  =  0
s(x1)  =  s(x1)
U2(x1, x2, x3, x4)  =  U2(x2, x4)
half_in(x1, x2)  =  half_in(x1)
U4(x1, x2, x3)  =  U4(x3)
half_out(x1, x2)  =  half_out(x2)
U3(x1, x2, x3, x4)  =  U3(x4)
log2_out(x1, x2, x3)  =  log2_out(x3)
log2_out(x1, x2)  =  log2_out(x2)


Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

LOG2_IN(X, Y) → U11(X, Y, log2_in(X, 0, Y))
LOG2_IN(X, Y) → LOG2_IN(X, 0, Y)
LOG2_IN(s(s(X)), I, Y) → U21(X, I, Y, half_in(s(s(X)), X1))
LOG2_IN(s(s(X)), I, Y) → HALF_IN(s(s(X)), X1)
HALF_IN(s(s(X)), s(Y)) → U41(X, Y, half_in(X, Y))
HALF_IN(s(s(X)), s(Y)) → HALF_IN(X, Y)
U21(X, I, Y, half_out(s(s(X)), X1)) → U31(X, I, Y, log2_in(X1, s(I), Y))
U21(X, I, Y, half_out(s(s(X)), X1)) → LOG2_IN(X1, s(I), Y)

The TRS R consists of the following rules:

log2_in(X, Y) → U1(X, Y, log2_in(X, 0, Y))
log2_in(s(s(X)), I, Y) → U2(X, I, Y, half_in(s(s(X)), X1))
half_in(s(s(X)), s(Y)) → U4(X, Y, half_in(X, Y))
half_in(s(0), 0) → half_out(s(0), 0)
half_in(0, 0) → half_out(0, 0)
U4(X, Y, half_out(X, Y)) → half_out(s(s(X)), s(Y))
U2(X, I, Y, half_out(s(s(X)), X1)) → U3(X, I, Y, log2_in(X1, s(I), Y))
log2_in(s(0), I, I) → log2_out(s(0), I, I)
log2_in(0, I, I) → log2_out(0, I, I)
U3(X, I, Y, log2_out(X1, s(I), Y)) → log2_out(s(s(X)), I, Y)
U1(X, Y, log2_out(X, 0, Y)) → log2_out(X, Y)

The argument filtering Pi contains the following mapping:
log2_in(x1, x2)  =  log2_in(x1)
U1(x1, x2, x3)  =  U1(x3)
log2_in(x1, x2, x3)  =  log2_in(x1, x2)
0  =  0
s(x1)  =  s(x1)
U2(x1, x2, x3, x4)  =  U2(x2, x4)
half_in(x1, x2)  =  half_in(x1)
U4(x1, x2, x3)  =  U4(x3)
half_out(x1, x2)  =  half_out(x2)
U3(x1, x2, x3, x4)  =  U3(x4)
log2_out(x1, x2, x3)  =  log2_out(x3)
log2_out(x1, x2)  =  log2_out(x2)
U31(x1, x2, x3, x4)  =  U31(x4)
U41(x1, x2, x3)  =  U41(x3)
HALF_IN(x1, x2)  =  HALF_IN(x1)
LOG2_IN(x1, x2)  =  LOG2_IN(x1)
U21(x1, x2, x3, x4)  =  U21(x2, x4)
LOG2_IN(x1, x2, x3)  =  LOG2_IN(x1, x2)
U11(x1, x2, x3)  =  U11(x3)

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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof

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

LOG2_IN(X, Y) → U11(X, Y, log2_in(X, 0, Y))
LOG2_IN(X, Y) → LOG2_IN(X, 0, Y)
LOG2_IN(s(s(X)), I, Y) → U21(X, I, Y, half_in(s(s(X)), X1))
LOG2_IN(s(s(X)), I, Y) → HALF_IN(s(s(X)), X1)
HALF_IN(s(s(X)), s(Y)) → U41(X, Y, half_in(X, Y))
HALF_IN(s(s(X)), s(Y)) → HALF_IN(X, Y)
U21(X, I, Y, half_out(s(s(X)), X1)) → U31(X, I, Y, log2_in(X1, s(I), Y))
U21(X, I, Y, half_out(s(s(X)), X1)) → LOG2_IN(X1, s(I), Y)

The TRS R consists of the following rules:

log2_in(X, Y) → U1(X, Y, log2_in(X, 0, Y))
log2_in(s(s(X)), I, Y) → U2(X, I, Y, half_in(s(s(X)), X1))
half_in(s(s(X)), s(Y)) → U4(X, Y, half_in(X, Y))
half_in(s(0), 0) → half_out(s(0), 0)
half_in(0, 0) → half_out(0, 0)
U4(X, Y, half_out(X, Y)) → half_out(s(s(X)), s(Y))
U2(X, I, Y, half_out(s(s(X)), X1)) → U3(X, I, Y, log2_in(X1, s(I), Y))
log2_in(s(0), I, I) → log2_out(s(0), I, I)
log2_in(0, I, I) → log2_out(0, I, I)
U3(X, I, Y, log2_out(X1, s(I), Y)) → log2_out(s(s(X)), I, Y)
U1(X, Y, log2_out(X, 0, Y)) → log2_out(X, Y)

The argument filtering Pi contains the following mapping:
log2_in(x1, x2)  =  log2_in(x1)
U1(x1, x2, x3)  =  U1(x3)
log2_in(x1, x2, x3)  =  log2_in(x1, x2)
0  =  0
s(x1)  =  s(x1)
U2(x1, x2, x3, x4)  =  U2(x2, x4)
half_in(x1, x2)  =  half_in(x1)
U4(x1, x2, x3)  =  U4(x3)
half_out(x1, x2)  =  half_out(x2)
U3(x1, x2, x3, x4)  =  U3(x4)
log2_out(x1, x2, x3)  =  log2_out(x3)
log2_out(x1, x2)  =  log2_out(x2)
U31(x1, x2, x3, x4)  =  U31(x4)
U41(x1, x2, x3)  =  U41(x3)
HALF_IN(x1, x2)  =  HALF_IN(x1)
LOG2_IN(x1, x2)  =  LOG2_IN(x1)
U21(x1, x2, x3, x4)  =  U21(x2, x4)
LOG2_IN(x1, x2, x3)  =  LOG2_IN(x1, x2)
U11(x1, x2, x3)  =  U11(x3)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 5 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP

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

HALF_IN(s(s(X)), s(Y)) → HALF_IN(X, Y)

The TRS R consists of the following rules:

log2_in(X, Y) → U1(X, Y, log2_in(X, 0, Y))
log2_in(s(s(X)), I, Y) → U2(X, I, Y, half_in(s(s(X)), X1))
half_in(s(s(X)), s(Y)) → U4(X, Y, half_in(X, Y))
half_in(s(0), 0) → half_out(s(0), 0)
half_in(0, 0) → half_out(0, 0)
U4(X, Y, half_out(X, Y)) → half_out(s(s(X)), s(Y))
U2(X, I, Y, half_out(s(s(X)), X1)) → U3(X, I, Y, log2_in(X1, s(I), Y))
log2_in(s(0), I, I) → log2_out(s(0), I, I)
log2_in(0, I, I) → log2_out(0, I, I)
U3(X, I, Y, log2_out(X1, s(I), Y)) → log2_out(s(s(X)), I, Y)
U1(X, Y, log2_out(X, 0, Y)) → log2_out(X, Y)

The argument filtering Pi contains the following mapping:
log2_in(x1, x2)  =  log2_in(x1)
U1(x1, x2, x3)  =  U1(x3)
log2_in(x1, x2, x3)  =  log2_in(x1, x2)
0  =  0
s(x1)  =  s(x1)
U2(x1, x2, x3, x4)  =  U2(x2, x4)
half_in(x1, x2)  =  half_in(x1)
U4(x1, x2, x3)  =  U4(x3)
half_out(x1, x2)  =  half_out(x2)
U3(x1, x2, x3, x4)  =  U3(x4)
log2_out(x1, x2, x3)  =  log2_out(x3)
log2_out(x1, x2)  =  log2_out(x2)
HALF_IN(x1, x2)  =  HALF_IN(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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

HALF_IN(s(s(X)), s(Y)) → HALF_IN(X, Y)

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

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP

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

HALF_IN(s(s(X))) → HALF_IN(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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: 



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof

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

LOG2_IN(s(s(X)), I, Y) → U21(X, I, Y, half_in(s(s(X)), X1))
U21(X, I, Y, half_out(s(s(X)), X1)) → LOG2_IN(X1, s(I), Y)

The TRS R consists of the following rules:

log2_in(X, Y) → U1(X, Y, log2_in(X, 0, Y))
log2_in(s(s(X)), I, Y) → U2(X, I, Y, half_in(s(s(X)), X1))
half_in(s(s(X)), s(Y)) → U4(X, Y, half_in(X, Y))
half_in(s(0), 0) → half_out(s(0), 0)
half_in(0, 0) → half_out(0, 0)
U4(X, Y, half_out(X, Y)) → half_out(s(s(X)), s(Y))
U2(X, I, Y, half_out(s(s(X)), X1)) → U3(X, I, Y, log2_in(X1, s(I), Y))
log2_in(s(0), I, I) → log2_out(s(0), I, I)
log2_in(0, I, I) → log2_out(0, I, I)
U3(X, I, Y, log2_out(X1, s(I), Y)) → log2_out(s(s(X)), I, Y)
U1(X, Y, log2_out(X, 0, Y)) → log2_out(X, Y)

The argument filtering Pi contains the following mapping:
log2_in(x1, x2)  =  log2_in(x1)
U1(x1, x2, x3)  =  U1(x3)
log2_in(x1, x2, x3)  =  log2_in(x1, x2)
0  =  0
s(x1)  =  s(x1)
U2(x1, x2, x3, x4)  =  U2(x2, x4)
half_in(x1, x2)  =  half_in(x1)
U4(x1, x2, x3)  =  U4(x3)
half_out(x1, x2)  =  half_out(x2)
U3(x1, x2, x3, x4)  =  U3(x4)
log2_out(x1, x2, x3)  =  log2_out(x3)
log2_out(x1, x2)  =  log2_out(x2)
U21(x1, x2, x3, x4)  =  U21(x2, x4)
LOG2_IN(x1, x2, x3)  =  LOG2_IN(x1, x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof

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

LOG2_IN(s(s(X)), I, Y) → U21(X, I, Y, half_in(s(s(X)), X1))
U21(X, I, Y, half_out(s(s(X)), X1)) → LOG2_IN(X1, s(I), Y)

The TRS R consists of the following rules:

half_in(s(s(X)), s(Y)) → U4(X, Y, half_in(X, Y))
U4(X, Y, half_out(X, Y)) → half_out(s(s(X)), s(Y))
half_in(s(0), 0) → half_out(s(0), 0)
half_in(0, 0) → half_out(0, 0)

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
half_in(x1, x2)  =  half_in(x1)
U4(x1, x2, x3)  =  U4(x3)
half_out(x1, x2)  =  half_out(x2)
U21(x1, x2, x3, x4)  =  U21(x2, x4)
LOG2_IN(x1, x2, x3)  =  LOG2_IN(x1, x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ RuleRemovalProof

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

U21(I, half_out(X1)) → LOG2_IN(X1, s(I))
LOG2_IN(s(s(X)), I) → U21(I, half_in(s(s(X))))

The TRS R consists of the following rules:

half_in(s(s(X))) → U4(half_in(X))
U4(half_out(Y)) → half_out(s(Y))
half_in(s(0)) → half_out(0)
half_in(0) → half_out(0)

The set Q consists of the following terms:

half_in(x0)
U4(x0)

We have to consider all (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

U21(I, half_out(X1)) → LOG2_IN(X1, s(I))

Strictly oriented rules of the TRS R:

half_in(s(0)) → half_out(0)

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0   
POL(LOG2_IN(x1, x2)) = 2·x1 + x2   
POL(U21(x1, x2)) = 2 + x1 + x2   
POL(U4(x1)) = 2 + x1   
POL(half_in(x1)) = x1   
POL(half_out(x1)) = 2·x1   
POL(s(x1)) = 1 + x1   



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ RuleRemovalProof
                          ↳ QDP
                            ↳ DependencyGraphProof

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

LOG2_IN(s(s(X)), I) → U21(I, half_in(s(s(X))))

The TRS R consists of the following rules:

half_in(s(s(X))) → U4(half_in(X))
U4(half_out(Y)) → half_out(s(Y))
half_in(0) → half_out(0)

The set Q consists of the following terms:

half_in(x0)
U4(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.