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, s(0), Y).
log2(s(s(X)), Half, Acc, Y) :- log2(X, s(Half), Acc, Y).
log2(X, s(s(Half)), Acc, Y) :- ','(small(X), log2(Half, s(0), s(Acc), Y)).
log2(X, Half, Y, Y) :- ','(small(X), small(Half)).
small(0).
small(s(0)).

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, s(0), Y))
log2_in(X, Half, Y, Y) → U5(X, Half, Y, small_in(X))
small_in(s(0)) → small_out(s(0))
small_in(0) → small_out(0)
U5(X, Half, Y, small_out(X)) → U6(X, Half, Y, small_in(Half))
U6(X, Half, Y, small_out(Half)) → log2_out(X, Half, Y, Y)
log2_in(X, s(s(Half)), Acc, Y) → U3(X, Half, Acc, Y, small_in(X))
U3(X, Half, Acc, Y, small_out(X)) → U4(X, Half, Acc, Y, log2_in(Half, s(0), s(Acc), Y))
log2_in(s(s(X)), Half, Acc, Y) → U2(X, Half, Acc, Y, log2_in(X, s(Half), Acc, Y))
U2(X, Half, Acc, Y, log2_out(X, s(Half), Acc, Y)) → log2_out(s(s(X)), Half, Acc, Y)
U4(X, Half, Acc, Y, log2_out(Half, s(0), s(Acc), Y)) → log2_out(X, s(s(Half)), Acc, Y)
U1(X, Y, log2_out(X, 0, s(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, x4)  =  log2_in(x1, x2, x3)
0  =  0
s(x1)  =  s(x1)
U5(x1, x2, x3, x4)  =  U5(x2, x3, x4)
small_in(x1)  =  small_in(x1)
small_out(x1)  =  small_out
U6(x1, x2, x3, x4)  =  U6(x3, x4)
log2_out(x1, x2, x3, x4)  =  log2_out(x4)
U3(x1, x2, x3, x4, x5)  =  U3(x2, x3, x5)
U4(x1, x2, x3, x4, x5)  =  U4(x5)
U2(x1, x2, x3, x4, x5)  =  U2(x5)
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, s(0), Y))
log2_in(X, Half, Y, Y) → U5(X, Half, Y, small_in(X))
small_in(s(0)) → small_out(s(0))
small_in(0) → small_out(0)
U5(X, Half, Y, small_out(X)) → U6(X, Half, Y, small_in(Half))
U6(X, Half, Y, small_out(Half)) → log2_out(X, Half, Y, Y)
log2_in(X, s(s(Half)), Acc, Y) → U3(X, Half, Acc, Y, small_in(X))
U3(X, Half, Acc, Y, small_out(X)) → U4(X, Half, Acc, Y, log2_in(Half, s(0), s(Acc), Y))
log2_in(s(s(X)), Half, Acc, Y) → U2(X, Half, Acc, Y, log2_in(X, s(Half), Acc, Y))
U2(X, Half, Acc, Y, log2_out(X, s(Half), Acc, Y)) → log2_out(s(s(X)), Half, Acc, Y)
U4(X, Half, Acc, Y, log2_out(Half, s(0), s(Acc), Y)) → log2_out(X, s(s(Half)), Acc, Y)
U1(X, Y, log2_out(X, 0, s(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, x4)  =  log2_in(x1, x2, x3)
0  =  0
s(x1)  =  s(x1)
U5(x1, x2, x3, x4)  =  U5(x2, x3, x4)
small_in(x1)  =  small_in(x1)
small_out(x1)  =  small_out
U6(x1, x2, x3, x4)  =  U6(x3, x4)
log2_out(x1, x2, x3, x4)  =  log2_out(x4)
U3(x1, x2, x3, x4, x5)  =  U3(x2, x3, x5)
U4(x1, x2, x3, x4, x5)  =  U4(x5)
U2(x1, x2, x3, x4, x5)  =  U2(x5)
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, s(0), Y))
LOG2_IN(X, Y) → LOG2_IN(X, 0, s(0), Y)
LOG2_IN(X, Half, Y, Y) → U51(X, Half, Y, small_in(X))
LOG2_IN(X, Half, Y, Y) → SMALL_IN(X)
U51(X, Half, Y, small_out(X)) → U61(X, Half, Y, small_in(Half))
U51(X, Half, Y, small_out(X)) → SMALL_IN(Half)
LOG2_IN(X, s(s(Half)), Acc, Y) → U31(X, Half, Acc, Y, small_in(X))
LOG2_IN(X, s(s(Half)), Acc, Y) → SMALL_IN(X)
U31(X, Half, Acc, Y, small_out(X)) → U41(X, Half, Acc, Y, log2_in(Half, s(0), s(Acc), Y))
U31(X, Half, Acc, Y, small_out(X)) → LOG2_IN(Half, s(0), s(Acc), Y)
LOG2_IN(s(s(X)), Half, Acc, Y) → U21(X, Half, Acc, Y, log2_in(X, s(Half), Acc, Y))
LOG2_IN(s(s(X)), Half, Acc, Y) → LOG2_IN(X, s(Half), Acc, Y)

The TRS R consists of the following rules:

log2_in(X, Y) → U1(X, Y, log2_in(X, 0, s(0), Y))
log2_in(X, Half, Y, Y) → U5(X, Half, Y, small_in(X))
small_in(s(0)) → small_out(s(0))
small_in(0) → small_out(0)
U5(X, Half, Y, small_out(X)) → U6(X, Half, Y, small_in(Half))
U6(X, Half, Y, small_out(Half)) → log2_out(X, Half, Y, Y)
log2_in(X, s(s(Half)), Acc, Y) → U3(X, Half, Acc, Y, small_in(X))
U3(X, Half, Acc, Y, small_out(X)) → U4(X, Half, Acc, Y, log2_in(Half, s(0), s(Acc), Y))
log2_in(s(s(X)), Half, Acc, Y) → U2(X, Half, Acc, Y, log2_in(X, s(Half), Acc, Y))
U2(X, Half, Acc, Y, log2_out(X, s(Half), Acc, Y)) → log2_out(s(s(X)), Half, Acc, Y)
U4(X, Half, Acc, Y, log2_out(Half, s(0), s(Acc), Y)) → log2_out(X, s(s(Half)), Acc, Y)
U1(X, Y, log2_out(X, 0, s(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, x4)  =  log2_in(x1, x2, x3)
0  =  0
s(x1)  =  s(x1)
U5(x1, x2, x3, x4)  =  U5(x2, x3, x4)
small_in(x1)  =  small_in(x1)
small_out(x1)  =  small_out
U6(x1, x2, x3, x4)  =  U6(x3, x4)
log2_out(x1, x2, x3, x4)  =  log2_out(x4)
U3(x1, x2, x3, x4, x5)  =  U3(x2, x3, x5)
U4(x1, x2, x3, x4, x5)  =  U4(x5)
U2(x1, x2, x3, x4, x5)  =  U2(x5)
log2_out(x1, x2)  =  log2_out(x2)
SMALL_IN(x1)  =  SMALL_IN(x1)
U51(x1, x2, x3, x4)  =  U51(x2, x3, x4)
LOG2_IN(x1, x2, x3, x4)  =  LOG2_IN(x1, x2, x3)
U41(x1, x2, x3, x4, x5)  =  U41(x5)
U31(x1, x2, x3, x4, x5)  =  U31(x2, x3, x5)
LOG2_IN(x1, x2)  =  LOG2_IN(x1)
U21(x1, x2, x3, x4, x5)  =  U21(x5)
U61(x1, x2, x3, x4)  =  U61(x3, x4)
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, s(0), Y))
LOG2_IN(X, Y) → LOG2_IN(X, 0, s(0), Y)
LOG2_IN(X, Half, Y, Y) → U51(X, Half, Y, small_in(X))
LOG2_IN(X, Half, Y, Y) → SMALL_IN(X)
U51(X, Half, Y, small_out(X)) → U61(X, Half, Y, small_in(Half))
U51(X, Half, Y, small_out(X)) → SMALL_IN(Half)
LOG2_IN(X, s(s(Half)), Acc, Y) → U31(X, Half, Acc, Y, small_in(X))
LOG2_IN(X, s(s(Half)), Acc, Y) → SMALL_IN(X)
U31(X, Half, Acc, Y, small_out(X)) → U41(X, Half, Acc, Y, log2_in(Half, s(0), s(Acc), Y))
U31(X, Half, Acc, Y, small_out(X)) → LOG2_IN(Half, s(0), s(Acc), Y)
LOG2_IN(s(s(X)), Half, Acc, Y) → U21(X, Half, Acc, Y, log2_in(X, s(Half), Acc, Y))
LOG2_IN(s(s(X)), Half, Acc, Y) → LOG2_IN(X, s(Half), Acc, Y)

The TRS R consists of the following rules:

log2_in(X, Y) → U1(X, Y, log2_in(X, 0, s(0), Y))
log2_in(X, Half, Y, Y) → U5(X, Half, Y, small_in(X))
small_in(s(0)) → small_out(s(0))
small_in(0) → small_out(0)
U5(X, Half, Y, small_out(X)) → U6(X, Half, Y, small_in(Half))
U6(X, Half, Y, small_out(Half)) → log2_out(X, Half, Y, Y)
log2_in(X, s(s(Half)), Acc, Y) → U3(X, Half, Acc, Y, small_in(X))
U3(X, Half, Acc, Y, small_out(X)) → U4(X, Half, Acc, Y, log2_in(Half, s(0), s(Acc), Y))
log2_in(s(s(X)), Half, Acc, Y) → U2(X, Half, Acc, Y, log2_in(X, s(Half), Acc, Y))
U2(X, Half, Acc, Y, log2_out(X, s(Half), Acc, Y)) → log2_out(s(s(X)), Half, Acc, Y)
U4(X, Half, Acc, Y, log2_out(Half, s(0), s(Acc), Y)) → log2_out(X, s(s(Half)), Acc, Y)
U1(X, Y, log2_out(X, 0, s(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, x4)  =  log2_in(x1, x2, x3)
0  =  0
s(x1)  =  s(x1)
U5(x1, x2, x3, x4)  =  U5(x2, x3, x4)
small_in(x1)  =  small_in(x1)
small_out(x1)  =  small_out
U6(x1, x2, x3, x4)  =  U6(x3, x4)
log2_out(x1, x2, x3, x4)  =  log2_out(x4)
U3(x1, x2, x3, x4, x5)  =  U3(x2, x3, x5)
U4(x1, x2, x3, x4, x5)  =  U4(x5)
U2(x1, x2, x3, x4, x5)  =  U2(x5)
log2_out(x1, x2)  =  log2_out(x2)
SMALL_IN(x1)  =  SMALL_IN(x1)
U51(x1, x2, x3, x4)  =  U51(x2, x3, x4)
LOG2_IN(x1, x2, x3, x4)  =  LOG2_IN(x1, x2, x3)
U41(x1, x2, x3, x4, x5)  =  U41(x5)
U31(x1, x2, x3, x4, x5)  =  U31(x2, x3, x5)
LOG2_IN(x1, x2)  =  LOG2_IN(x1)
U21(x1, x2, x3, x4, x5)  =  U21(x5)
U61(x1, x2, x3, x4)  =  U61(x3, x4)
U11(x1, x2, x3)  =  U11(x3)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 9 less nodes.

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

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

U31(X, Half, Acc, Y, small_out(X)) → LOG2_IN(Half, s(0), s(Acc), Y)
LOG2_IN(X, s(s(Half)), Acc, Y) → U31(X, Half, Acc, Y, small_in(X))
LOG2_IN(s(s(X)), Half, Acc, Y) → LOG2_IN(X, s(Half), Acc, Y)

The TRS R consists of the following rules:

log2_in(X, Y) → U1(X, Y, log2_in(X, 0, s(0), Y))
log2_in(X, Half, Y, Y) → U5(X, Half, Y, small_in(X))
small_in(s(0)) → small_out(s(0))
small_in(0) → small_out(0)
U5(X, Half, Y, small_out(X)) → U6(X, Half, Y, small_in(Half))
U6(X, Half, Y, small_out(Half)) → log2_out(X, Half, Y, Y)
log2_in(X, s(s(Half)), Acc, Y) → U3(X, Half, Acc, Y, small_in(X))
U3(X, Half, Acc, Y, small_out(X)) → U4(X, Half, Acc, Y, log2_in(Half, s(0), s(Acc), Y))
log2_in(s(s(X)), Half, Acc, Y) → U2(X, Half, Acc, Y, log2_in(X, s(Half), Acc, Y))
U2(X, Half, Acc, Y, log2_out(X, s(Half), Acc, Y)) → log2_out(s(s(X)), Half, Acc, Y)
U4(X, Half, Acc, Y, log2_out(Half, s(0), s(Acc), Y)) → log2_out(X, s(s(Half)), Acc, Y)
U1(X, Y, log2_out(X, 0, s(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, x4)  =  log2_in(x1, x2, x3)
0  =  0
s(x1)  =  s(x1)
U5(x1, x2, x3, x4)  =  U5(x2, x3, x4)
small_in(x1)  =  small_in(x1)
small_out(x1)  =  small_out
U6(x1, x2, x3, x4)  =  U6(x3, x4)
log2_out(x1, x2, x3, x4)  =  log2_out(x4)
U3(x1, x2, x3, x4, x5)  =  U3(x2, x3, x5)
U4(x1, x2, x3, x4, x5)  =  U4(x5)
U2(x1, x2, x3, x4, x5)  =  U2(x5)
log2_out(x1, x2)  =  log2_out(x2)
LOG2_IN(x1, x2, x3, x4)  =  LOG2_IN(x1, x2, x3)
U31(x1, x2, x3, x4, x5)  =  U31(x2, x3, x5)

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
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof

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

U31(X, Half, Acc, Y, small_out(X)) → LOG2_IN(Half, s(0), s(Acc), Y)
LOG2_IN(X, s(s(Half)), Acc, Y) → U31(X, Half, Acc, Y, small_in(X))
LOG2_IN(s(s(X)), Half, Acc, Y) → LOG2_IN(X, s(Half), Acc, Y)

The TRS R consists of the following rules:

small_in(s(0)) → small_out(s(0))
small_in(0) → small_out(0)

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
small_in(x1)  =  small_in(x1)
small_out(x1)  =  small_out
LOG2_IN(x1, x2, x3, x4)  =  LOG2_IN(x1, x2, x3)
U31(x1, x2, x3, x4, x5)  =  U31(x2, x3, x5)

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
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
                    ↳ QDP
                      ↳ RuleRemovalProof

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

LOG2_IN(X, s(s(Half)), Acc) → U31(Half, Acc, small_in(X))
LOG2_IN(s(s(X)), Half, Acc) → LOG2_IN(X, s(Half), Acc)
U31(Half, Acc, small_out) → LOG2_IN(Half, s(0), s(Acc))

The TRS R consists of the following rules:

small_in(s(0)) → small_out
small_in(0) → small_out

The set Q consists of the following terms:

small_in(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:

LOG2_IN(X, s(s(Half)), Acc) → U31(Half, Acc, small_in(X))
LOG2_IN(s(s(X)), Half, Acc) → LOG2_IN(X, s(Half), Acc)

Strictly oriented rules of the TRS R:

small_in(s(0)) → small_out

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 1   
POL(LOG2_IN(x1, x2, x3)) = 2·x1 + 2·x2 + x3   
POL(U31(x1, x2, x3)) = 1 + 2·x1 + x2 + 2·x3   
POL(s(x1)) = 1 + x1   
POL(small_in(x1)) = 1 + x1   
POL(small_out) = 2   



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

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

U31(Half, Acc, small_out) → LOG2_IN(Half, s(0), s(Acc))

The TRS R consists of the following rules:

small_in(0) → small_out

The set Q consists of the following terms:

small_in(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.