Left Termination of the query pattern log2_in_2(a, g) w.r.t. the given Prolog program could not be shown:



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ 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(a,g).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
log2_in: (f,b)
log2_in: (f,b,b) (b,b,b)
half_in: (f,f) (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

log2_in_ag(X, Y) → U1_ag(X, Y, log2_in_agg(X, 0, Y))
log2_in_agg(0, I, I) → log2_out_agg(0, I, I)
log2_in_agg(s(0), I, I) → log2_out_agg(s(0), I, I)
log2_in_agg(s(s(X)), I, Y) → U2_agg(X, I, Y, half_in_aa(s(s(X)), X1))
half_in_aa(0, 0) → half_out_aa(0, 0)
half_in_aa(s(0), 0) → half_out_aa(s(0), 0)
half_in_aa(s(s(X)), s(Y)) → U4_aa(X, Y, half_in_aa(X, Y))
U4_aa(X, Y, half_out_aa(X, Y)) → half_out_aa(s(s(X)), s(Y))
U2_agg(X, I, Y, half_out_aa(s(s(X)), X1)) → U3_agg(X, I, Y, log2_in_ggg(X1, s(I), Y))
log2_in_ggg(0, I, I) → log2_out_ggg(0, I, I)
log2_in_ggg(s(0), I, I) → log2_out_ggg(s(0), I, I)
log2_in_ggg(s(s(X)), I, Y) → U2_ggg(X, I, Y, half_in_ga(s(s(X)), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(0), 0) → half_out_ga(s(0), 0)
half_in_ga(s(s(X)), s(Y)) → U4_ga(X, Y, half_in_ga(X, Y))
U4_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U2_ggg(X, I, Y, half_out_ga(s(s(X)), X1)) → U3_ggg(X, I, Y, log2_in_ggg(X1, s(I), Y))
U3_ggg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_ggg(s(s(X)), I, Y)
U3_agg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_agg(s(s(X)), I, Y)
U1_ag(X, Y, log2_out_agg(X, 0, Y)) → log2_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
log2_in_ag(x1, x2)  =  log2_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
log2_in_agg(x1, x2, x3)  =  log2_in_agg(x2, x3)
log2_out_agg(x1, x2, x3)  =  log2_out_agg(x1)
U2_agg(x1, x2, x3, x4)  =  U2_agg(x2, x3, x4)
half_in_aa(x1, x2)  =  half_in_aa
half_out_aa(x1, x2)  =  half_out_aa(x1, x2)
U4_aa(x1, x2, x3)  =  U4_aa(x3)
s(x1)  =  s(x1)
U3_agg(x1, x2, x3, x4)  =  U3_agg(x1, x4)
log2_in_ggg(x1, x2, x3)  =  log2_in_ggg(x1, x2, x3)
0  =  0
log2_out_ggg(x1, x2, x3)  =  log2_out_ggg
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x2)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
log2_out_ag(x1, x2)  =  log2_out_ag(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
  ↳ PrologToPiTRSProof

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

log2_in_ag(X, Y) → U1_ag(X, Y, log2_in_agg(X, 0, Y))
log2_in_agg(0, I, I) → log2_out_agg(0, I, I)
log2_in_agg(s(0), I, I) → log2_out_agg(s(0), I, I)
log2_in_agg(s(s(X)), I, Y) → U2_agg(X, I, Y, half_in_aa(s(s(X)), X1))
half_in_aa(0, 0) → half_out_aa(0, 0)
half_in_aa(s(0), 0) → half_out_aa(s(0), 0)
half_in_aa(s(s(X)), s(Y)) → U4_aa(X, Y, half_in_aa(X, Y))
U4_aa(X, Y, half_out_aa(X, Y)) → half_out_aa(s(s(X)), s(Y))
U2_agg(X, I, Y, half_out_aa(s(s(X)), X1)) → U3_agg(X, I, Y, log2_in_ggg(X1, s(I), Y))
log2_in_ggg(0, I, I) → log2_out_ggg(0, I, I)
log2_in_ggg(s(0), I, I) → log2_out_ggg(s(0), I, I)
log2_in_ggg(s(s(X)), I, Y) → U2_ggg(X, I, Y, half_in_ga(s(s(X)), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(0), 0) → half_out_ga(s(0), 0)
half_in_ga(s(s(X)), s(Y)) → U4_ga(X, Y, half_in_ga(X, Y))
U4_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U2_ggg(X, I, Y, half_out_ga(s(s(X)), X1)) → U3_ggg(X, I, Y, log2_in_ggg(X1, s(I), Y))
U3_ggg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_ggg(s(s(X)), I, Y)
U3_agg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_agg(s(s(X)), I, Y)
U1_ag(X, Y, log2_out_agg(X, 0, Y)) → log2_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
log2_in_ag(x1, x2)  =  log2_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
log2_in_agg(x1, x2, x3)  =  log2_in_agg(x2, x3)
log2_out_agg(x1, x2, x3)  =  log2_out_agg(x1)
U2_agg(x1, x2, x3, x4)  =  U2_agg(x2, x3, x4)
half_in_aa(x1, x2)  =  half_in_aa
half_out_aa(x1, x2)  =  half_out_aa(x1, x2)
U4_aa(x1, x2, x3)  =  U4_aa(x3)
s(x1)  =  s(x1)
U3_agg(x1, x2, x3, x4)  =  U3_agg(x1, x4)
log2_in_ggg(x1, x2, x3)  =  log2_in_ggg(x1, x2, x3)
0  =  0
log2_out_ggg(x1, x2, x3)  =  log2_out_ggg
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x2)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
log2_out_ag(x1, x2)  =  log2_out_ag(x1)


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_AG(X, Y) → U1_AG(X, Y, log2_in_agg(X, 0, Y))
LOG2_IN_AG(X, Y) → LOG2_IN_AGG(X, 0, Y)
LOG2_IN_AGG(s(s(X)), I, Y) → U2_AGG(X, I, Y, half_in_aa(s(s(X)), X1))
LOG2_IN_AGG(s(s(X)), I, Y) → HALF_IN_AA(s(s(X)), X1)
HALF_IN_AA(s(s(X)), s(Y)) → U4_AA(X, Y, half_in_aa(X, Y))
HALF_IN_AA(s(s(X)), s(Y)) → HALF_IN_AA(X, Y)
U2_AGG(X, I, Y, half_out_aa(s(s(X)), X1)) → U3_AGG(X, I, Y, log2_in_ggg(X1, s(I), Y))
U2_AGG(X, I, Y, half_out_aa(s(s(X)), X1)) → LOG2_IN_GGG(X1, s(I), Y)
LOG2_IN_GGG(s(s(X)), I, Y) → U2_GGG(X, I, Y, half_in_ga(s(s(X)), X1))
LOG2_IN_GGG(s(s(X)), I, Y) → HALF_IN_GA(s(s(X)), X1)
HALF_IN_GA(s(s(X)), s(Y)) → U4_GA(X, Y, half_in_ga(X, Y))
HALF_IN_GA(s(s(X)), s(Y)) → HALF_IN_GA(X, Y)
U2_GGG(X, I, Y, half_out_ga(s(s(X)), X1)) → U3_GGG(X, I, Y, log2_in_ggg(X1, s(I), Y))
U2_GGG(X, I, Y, half_out_ga(s(s(X)), X1)) → LOG2_IN_GGG(X1, s(I), Y)

The TRS R consists of the following rules:

log2_in_ag(X, Y) → U1_ag(X, Y, log2_in_agg(X, 0, Y))
log2_in_agg(0, I, I) → log2_out_agg(0, I, I)
log2_in_agg(s(0), I, I) → log2_out_agg(s(0), I, I)
log2_in_agg(s(s(X)), I, Y) → U2_agg(X, I, Y, half_in_aa(s(s(X)), X1))
half_in_aa(0, 0) → half_out_aa(0, 0)
half_in_aa(s(0), 0) → half_out_aa(s(0), 0)
half_in_aa(s(s(X)), s(Y)) → U4_aa(X, Y, half_in_aa(X, Y))
U4_aa(X, Y, half_out_aa(X, Y)) → half_out_aa(s(s(X)), s(Y))
U2_agg(X, I, Y, half_out_aa(s(s(X)), X1)) → U3_agg(X, I, Y, log2_in_ggg(X1, s(I), Y))
log2_in_ggg(0, I, I) → log2_out_ggg(0, I, I)
log2_in_ggg(s(0), I, I) → log2_out_ggg(s(0), I, I)
log2_in_ggg(s(s(X)), I, Y) → U2_ggg(X, I, Y, half_in_ga(s(s(X)), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(0), 0) → half_out_ga(s(0), 0)
half_in_ga(s(s(X)), s(Y)) → U4_ga(X, Y, half_in_ga(X, Y))
U4_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U2_ggg(X, I, Y, half_out_ga(s(s(X)), X1)) → U3_ggg(X, I, Y, log2_in_ggg(X1, s(I), Y))
U3_ggg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_ggg(s(s(X)), I, Y)
U3_agg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_agg(s(s(X)), I, Y)
U1_ag(X, Y, log2_out_agg(X, 0, Y)) → log2_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
log2_in_ag(x1, x2)  =  log2_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
log2_in_agg(x1, x2, x3)  =  log2_in_agg(x2, x3)
log2_out_agg(x1, x2, x3)  =  log2_out_agg(x1)
U2_agg(x1, x2, x3, x4)  =  U2_agg(x2, x3, x4)
half_in_aa(x1, x2)  =  half_in_aa
half_out_aa(x1, x2)  =  half_out_aa(x1, x2)
U4_aa(x1, x2, x3)  =  U4_aa(x3)
s(x1)  =  s(x1)
U3_agg(x1, x2, x3, x4)  =  U3_agg(x1, x4)
log2_in_ggg(x1, x2, x3)  =  log2_in_ggg(x1, x2, x3)
0  =  0
log2_out_ggg(x1, x2, x3)  =  log2_out_ggg
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x2)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
log2_out_ag(x1, x2)  =  log2_out_ag(x1)
U3_AGG(x1, x2, x3, x4)  =  U3_AGG(x1, x4)
LOG2_IN_AGG(x1, x2, x3)  =  LOG2_IN_AGG(x2, x3)
LOG2_IN_GGG(x1, x2, x3)  =  LOG2_IN_GGG(x1, x2, x3)
U3_GGG(x1, x2, x3, x4)  =  U3_GGG(x4)
HALF_IN_GA(x1, x2)  =  HALF_IN_GA(x1)
U2_AGG(x1, x2, x3, x4)  =  U2_AGG(x2, x3, x4)
U4_GA(x1, x2, x3)  =  U4_GA(x3)
U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x2, x3, x4)
HALF_IN_AA(x1, x2)  =  HALF_IN_AA
U1_AG(x1, x2, x3)  =  U1_AG(x3)
U4_AA(x1, x2, x3)  =  U4_AA(x3)
LOG2_IN_AG(x1, x2)  =  LOG2_IN_AG(x2)

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

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

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

LOG2_IN_AG(X, Y) → U1_AG(X, Y, log2_in_agg(X, 0, Y))
LOG2_IN_AG(X, Y) → LOG2_IN_AGG(X, 0, Y)
LOG2_IN_AGG(s(s(X)), I, Y) → U2_AGG(X, I, Y, half_in_aa(s(s(X)), X1))
LOG2_IN_AGG(s(s(X)), I, Y) → HALF_IN_AA(s(s(X)), X1)
HALF_IN_AA(s(s(X)), s(Y)) → U4_AA(X, Y, half_in_aa(X, Y))
HALF_IN_AA(s(s(X)), s(Y)) → HALF_IN_AA(X, Y)
U2_AGG(X, I, Y, half_out_aa(s(s(X)), X1)) → U3_AGG(X, I, Y, log2_in_ggg(X1, s(I), Y))
U2_AGG(X, I, Y, half_out_aa(s(s(X)), X1)) → LOG2_IN_GGG(X1, s(I), Y)
LOG2_IN_GGG(s(s(X)), I, Y) → U2_GGG(X, I, Y, half_in_ga(s(s(X)), X1))
LOG2_IN_GGG(s(s(X)), I, Y) → HALF_IN_GA(s(s(X)), X1)
HALF_IN_GA(s(s(X)), s(Y)) → U4_GA(X, Y, half_in_ga(X, Y))
HALF_IN_GA(s(s(X)), s(Y)) → HALF_IN_GA(X, Y)
U2_GGG(X, I, Y, half_out_ga(s(s(X)), X1)) → U3_GGG(X, I, Y, log2_in_ggg(X1, s(I), Y))
U2_GGG(X, I, Y, half_out_ga(s(s(X)), X1)) → LOG2_IN_GGG(X1, s(I), Y)

The TRS R consists of the following rules:

log2_in_ag(X, Y) → U1_ag(X, Y, log2_in_agg(X, 0, Y))
log2_in_agg(0, I, I) → log2_out_agg(0, I, I)
log2_in_agg(s(0), I, I) → log2_out_agg(s(0), I, I)
log2_in_agg(s(s(X)), I, Y) → U2_agg(X, I, Y, half_in_aa(s(s(X)), X1))
half_in_aa(0, 0) → half_out_aa(0, 0)
half_in_aa(s(0), 0) → half_out_aa(s(0), 0)
half_in_aa(s(s(X)), s(Y)) → U4_aa(X, Y, half_in_aa(X, Y))
U4_aa(X, Y, half_out_aa(X, Y)) → half_out_aa(s(s(X)), s(Y))
U2_agg(X, I, Y, half_out_aa(s(s(X)), X1)) → U3_agg(X, I, Y, log2_in_ggg(X1, s(I), Y))
log2_in_ggg(0, I, I) → log2_out_ggg(0, I, I)
log2_in_ggg(s(0), I, I) → log2_out_ggg(s(0), I, I)
log2_in_ggg(s(s(X)), I, Y) → U2_ggg(X, I, Y, half_in_ga(s(s(X)), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(0), 0) → half_out_ga(s(0), 0)
half_in_ga(s(s(X)), s(Y)) → U4_ga(X, Y, half_in_ga(X, Y))
U4_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U2_ggg(X, I, Y, half_out_ga(s(s(X)), X1)) → U3_ggg(X, I, Y, log2_in_ggg(X1, s(I), Y))
U3_ggg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_ggg(s(s(X)), I, Y)
U3_agg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_agg(s(s(X)), I, Y)
U1_ag(X, Y, log2_out_agg(X, 0, Y)) → log2_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
log2_in_ag(x1, x2)  =  log2_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
log2_in_agg(x1, x2, x3)  =  log2_in_agg(x2, x3)
log2_out_agg(x1, x2, x3)  =  log2_out_agg(x1)
U2_agg(x1, x2, x3, x4)  =  U2_agg(x2, x3, x4)
half_in_aa(x1, x2)  =  half_in_aa
half_out_aa(x1, x2)  =  half_out_aa(x1, x2)
U4_aa(x1, x2, x3)  =  U4_aa(x3)
s(x1)  =  s(x1)
U3_agg(x1, x2, x3, x4)  =  U3_agg(x1, x4)
log2_in_ggg(x1, x2, x3)  =  log2_in_ggg(x1, x2, x3)
0  =  0
log2_out_ggg(x1, x2, x3)  =  log2_out_ggg
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x2)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
log2_out_ag(x1, x2)  =  log2_out_ag(x1)
U3_AGG(x1, x2, x3, x4)  =  U3_AGG(x1, x4)
LOG2_IN_AGG(x1, x2, x3)  =  LOG2_IN_AGG(x2, x3)
LOG2_IN_GGG(x1, x2, x3)  =  LOG2_IN_GGG(x1, x2, x3)
U3_GGG(x1, x2, x3, x4)  =  U3_GGG(x4)
HALF_IN_GA(x1, x2)  =  HALF_IN_GA(x1)
U2_AGG(x1, x2, x3, x4)  =  U2_AGG(x2, x3, x4)
U4_GA(x1, x2, x3)  =  U4_GA(x3)
U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x2, x3, x4)
HALF_IN_AA(x1, x2)  =  HALF_IN_AA
U1_AG(x1, x2, x3)  =  U1_AG(x3)
U4_AA(x1, x2, x3)  =  U4_AA(x3)
LOG2_IN_AG(x1, x2)  =  LOG2_IN_AG(x2)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 3 SCCs with 10 less nodes.

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

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

HALF_IN_GA(s(s(X)), s(Y)) → HALF_IN_GA(X, Y)

The TRS R consists of the following rules:

log2_in_ag(X, Y) → U1_ag(X, Y, log2_in_agg(X, 0, Y))
log2_in_agg(0, I, I) → log2_out_agg(0, I, I)
log2_in_agg(s(0), I, I) → log2_out_agg(s(0), I, I)
log2_in_agg(s(s(X)), I, Y) → U2_agg(X, I, Y, half_in_aa(s(s(X)), X1))
half_in_aa(0, 0) → half_out_aa(0, 0)
half_in_aa(s(0), 0) → half_out_aa(s(0), 0)
half_in_aa(s(s(X)), s(Y)) → U4_aa(X, Y, half_in_aa(X, Y))
U4_aa(X, Y, half_out_aa(X, Y)) → half_out_aa(s(s(X)), s(Y))
U2_agg(X, I, Y, half_out_aa(s(s(X)), X1)) → U3_agg(X, I, Y, log2_in_ggg(X1, s(I), Y))
log2_in_ggg(0, I, I) → log2_out_ggg(0, I, I)
log2_in_ggg(s(0), I, I) → log2_out_ggg(s(0), I, I)
log2_in_ggg(s(s(X)), I, Y) → U2_ggg(X, I, Y, half_in_ga(s(s(X)), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(0), 0) → half_out_ga(s(0), 0)
half_in_ga(s(s(X)), s(Y)) → U4_ga(X, Y, half_in_ga(X, Y))
U4_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U2_ggg(X, I, Y, half_out_ga(s(s(X)), X1)) → U3_ggg(X, I, Y, log2_in_ggg(X1, s(I), Y))
U3_ggg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_ggg(s(s(X)), I, Y)
U3_agg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_agg(s(s(X)), I, Y)
U1_ag(X, Y, log2_out_agg(X, 0, Y)) → log2_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
log2_in_ag(x1, x2)  =  log2_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
log2_in_agg(x1, x2, x3)  =  log2_in_agg(x2, x3)
log2_out_agg(x1, x2, x3)  =  log2_out_agg(x1)
U2_agg(x1, x2, x3, x4)  =  U2_agg(x2, x3, x4)
half_in_aa(x1, x2)  =  half_in_aa
half_out_aa(x1, x2)  =  half_out_aa(x1, x2)
U4_aa(x1, x2, x3)  =  U4_aa(x3)
s(x1)  =  s(x1)
U3_agg(x1, x2, x3, x4)  =  U3_agg(x1, x4)
log2_in_ggg(x1, x2, x3)  =  log2_in_ggg(x1, x2, x3)
0  =  0
log2_out_ggg(x1, x2, x3)  =  log2_out_ggg
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x2)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
log2_out_ag(x1, x2)  =  log2_out_ag(x1)
HALF_IN_GA(x1, x2)  =  HALF_IN_GA(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
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

HALF_IN_GA(s(s(X)), s(Y)) → HALF_IN_GA(X, Y)

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

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

HALF_IN_GA(s(s(X))) → HALF_IN_GA(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
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LOG2_IN_GGG(s(s(X)), I, Y) → U2_GGG(X, I, Y, half_in_ga(s(s(X)), X1))
U2_GGG(X, I, Y, half_out_ga(s(s(X)), X1)) → LOG2_IN_GGG(X1, s(I), Y)

The TRS R consists of the following rules:

log2_in_ag(X, Y) → U1_ag(X, Y, log2_in_agg(X, 0, Y))
log2_in_agg(0, I, I) → log2_out_agg(0, I, I)
log2_in_agg(s(0), I, I) → log2_out_agg(s(0), I, I)
log2_in_agg(s(s(X)), I, Y) → U2_agg(X, I, Y, half_in_aa(s(s(X)), X1))
half_in_aa(0, 0) → half_out_aa(0, 0)
half_in_aa(s(0), 0) → half_out_aa(s(0), 0)
half_in_aa(s(s(X)), s(Y)) → U4_aa(X, Y, half_in_aa(X, Y))
U4_aa(X, Y, half_out_aa(X, Y)) → half_out_aa(s(s(X)), s(Y))
U2_agg(X, I, Y, half_out_aa(s(s(X)), X1)) → U3_agg(X, I, Y, log2_in_ggg(X1, s(I), Y))
log2_in_ggg(0, I, I) → log2_out_ggg(0, I, I)
log2_in_ggg(s(0), I, I) → log2_out_ggg(s(0), I, I)
log2_in_ggg(s(s(X)), I, Y) → U2_ggg(X, I, Y, half_in_ga(s(s(X)), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(0), 0) → half_out_ga(s(0), 0)
half_in_ga(s(s(X)), s(Y)) → U4_ga(X, Y, half_in_ga(X, Y))
U4_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U2_ggg(X, I, Y, half_out_ga(s(s(X)), X1)) → U3_ggg(X, I, Y, log2_in_ggg(X1, s(I), Y))
U3_ggg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_ggg(s(s(X)), I, Y)
U3_agg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_agg(s(s(X)), I, Y)
U1_ag(X, Y, log2_out_agg(X, 0, Y)) → log2_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
log2_in_ag(x1, x2)  =  log2_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
log2_in_agg(x1, x2, x3)  =  log2_in_agg(x2, x3)
log2_out_agg(x1, x2, x3)  =  log2_out_agg(x1)
U2_agg(x1, x2, x3, x4)  =  U2_agg(x2, x3, x4)
half_in_aa(x1, x2)  =  half_in_aa
half_out_aa(x1, x2)  =  half_out_aa(x1, x2)
U4_aa(x1, x2, x3)  =  U4_aa(x3)
s(x1)  =  s(x1)
U3_agg(x1, x2, x3, x4)  =  U3_agg(x1, x4)
log2_in_ggg(x1, x2, x3)  =  log2_in_ggg(x1, x2, x3)
0  =  0
log2_out_ggg(x1, x2, x3)  =  log2_out_ggg
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x2)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
log2_out_ag(x1, x2)  =  log2_out_ag(x1)
LOG2_IN_GGG(x1, x2, x3)  =  LOG2_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x2, x3, x4)

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
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LOG2_IN_GGG(s(s(X)), I, Y) → U2_GGG(X, I, Y, half_in_ga(s(s(X)), X1))
U2_GGG(X, I, Y, half_out_ga(s(s(X)), X1)) → LOG2_IN_GGG(X1, s(I), Y)

The TRS R consists of the following rules:

half_in_ga(s(s(X)), s(Y)) → U4_ga(X, Y, half_in_ga(X, Y))
U4_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(0), 0) → half_out_ga(s(0), 0)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x2)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
LOG2_IN_GGG(x1, x2, x3)  =  LOG2_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x2, x3, x4)

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
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LOG2_IN_GGG(s(s(X)), I, Y) → U2_GGG(I, Y, half_in_ga(s(s(X))))
U2_GGG(I, Y, half_out_ga(X1)) → LOG2_IN_GGG(X1, s(I), Y)

The TRS R consists of the following rules:

half_in_ga(s(s(X))) → U4_ga(half_in_ga(X))
U4_ga(half_out_ga(Y)) → half_out_ga(s(Y))
half_in_ga(0) → half_out_ga(0)
half_in_ga(s(0)) → half_out_ga(0)

The set Q consists of the following terms:

half_in_ga(x0)
U4_ga(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:

U2_GGG(I, Y, half_out_ga(X1)) → LOG2_IN_GGG(X1, s(I), Y)

Strictly oriented rules of the TRS R:

half_in_ga(s(0)) → half_out_ga(0)

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0   
POL(LOG2_IN_GGG(x1, x2, x3)) = 2·x1 + x2 + x3   
POL(U2_GGG(x1, x2, x3)) = 2 + x1 + x2 + x3   
POL(U4_ga(x1)) = 2 + x1   
POL(half_in_ga(x1)) = x1   
POL(half_out_ga(x1)) = 2·x1   
POL(s(x1)) = 1 + x1   



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

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

LOG2_IN_GGG(s(s(X)), I, Y) → U2_GGG(I, Y, half_in_ga(s(s(X))))

The TRS R consists of the following rules:

half_in_ga(s(s(X))) → U4_ga(half_in_ga(X))
U4_ga(half_out_ga(Y)) → half_out_ga(s(Y))
half_in_ga(0) → half_out_ga(0)

The set Q consists of the following terms:

half_in_ga(x0)
U4_ga(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.

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

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

HALF_IN_AA(s(s(X)), s(Y)) → HALF_IN_AA(X, Y)

The TRS R consists of the following rules:

log2_in_ag(X, Y) → U1_ag(X, Y, log2_in_agg(X, 0, Y))
log2_in_agg(0, I, I) → log2_out_agg(0, I, I)
log2_in_agg(s(0), I, I) → log2_out_agg(s(0), I, I)
log2_in_agg(s(s(X)), I, Y) → U2_agg(X, I, Y, half_in_aa(s(s(X)), X1))
half_in_aa(0, 0) → half_out_aa(0, 0)
half_in_aa(s(0), 0) → half_out_aa(s(0), 0)
half_in_aa(s(s(X)), s(Y)) → U4_aa(X, Y, half_in_aa(X, Y))
U4_aa(X, Y, half_out_aa(X, Y)) → half_out_aa(s(s(X)), s(Y))
U2_agg(X, I, Y, half_out_aa(s(s(X)), X1)) → U3_agg(X, I, Y, log2_in_ggg(X1, s(I), Y))
log2_in_ggg(0, I, I) → log2_out_ggg(0, I, I)
log2_in_ggg(s(0), I, I) → log2_out_ggg(s(0), I, I)
log2_in_ggg(s(s(X)), I, Y) → U2_ggg(X, I, Y, half_in_ga(s(s(X)), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(0), 0) → half_out_ga(s(0), 0)
half_in_ga(s(s(X)), s(Y)) → U4_ga(X, Y, half_in_ga(X, Y))
U4_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U2_ggg(X, I, Y, half_out_ga(s(s(X)), X1)) → U3_ggg(X, I, Y, log2_in_ggg(X1, s(I), Y))
U3_ggg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_ggg(s(s(X)), I, Y)
U3_agg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_agg(s(s(X)), I, Y)
U1_ag(X, Y, log2_out_agg(X, 0, Y)) → log2_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
log2_in_ag(x1, x2)  =  log2_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
log2_in_agg(x1, x2, x3)  =  log2_in_agg(x2, x3)
log2_out_agg(x1, x2, x3)  =  log2_out_agg(x1)
U2_agg(x1, x2, x3, x4)  =  U2_agg(x2, x3, x4)
half_in_aa(x1, x2)  =  half_in_aa
half_out_aa(x1, x2)  =  half_out_aa(x1, x2)
U4_aa(x1, x2, x3)  =  U4_aa(x3)
s(x1)  =  s(x1)
U3_agg(x1, x2, x3, x4)  =  U3_agg(x1, x4)
log2_in_ggg(x1, x2, x3)  =  log2_in_ggg(x1, x2, x3)
0  =  0
log2_out_ggg(x1, x2, x3)  =  log2_out_ggg
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x2)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
log2_out_ag(x1, x2)  =  log2_out_ag(x1)
HALF_IN_AA(x1, x2)  =  HALF_IN_AA

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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
  ↳ PrologToPiTRSProof

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

HALF_IN_AA(s(s(X)), s(Y)) → HALF_IN_AA(X, Y)

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

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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof
  ↳ PrologToPiTRSProof

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

HALF_IN_AA → HALF_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

HALF_IN_AA → HALF_IN_AA

The TRS R consists of the following rules:none


s = HALF_IN_AA evaluates to t =HALF_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from HALF_IN_AA to HALF_IN_AA.




We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
log2_in: (f,b)
log2_in: (f,b,b) (b,b,b)
half_in: (f,f) (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

log2_in_ag(X, Y) → U1_ag(X, Y, log2_in_agg(X, 0, Y))
log2_in_agg(0, I, I) → log2_out_agg(0, I, I)
log2_in_agg(s(0), I, I) → log2_out_agg(s(0), I, I)
log2_in_agg(s(s(X)), I, Y) → U2_agg(X, I, Y, half_in_aa(s(s(X)), X1))
half_in_aa(0, 0) → half_out_aa(0, 0)
half_in_aa(s(0), 0) → half_out_aa(s(0), 0)
half_in_aa(s(s(X)), s(Y)) → U4_aa(X, Y, half_in_aa(X, Y))
U4_aa(X, Y, half_out_aa(X, Y)) → half_out_aa(s(s(X)), s(Y))
U2_agg(X, I, Y, half_out_aa(s(s(X)), X1)) → U3_agg(X, I, Y, log2_in_ggg(X1, s(I), Y))
log2_in_ggg(0, I, I) → log2_out_ggg(0, I, I)
log2_in_ggg(s(0), I, I) → log2_out_ggg(s(0), I, I)
log2_in_ggg(s(s(X)), I, Y) → U2_ggg(X, I, Y, half_in_ga(s(s(X)), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(0), 0) → half_out_ga(s(0), 0)
half_in_ga(s(s(X)), s(Y)) → U4_ga(X, Y, half_in_ga(X, Y))
U4_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U2_ggg(X, I, Y, half_out_ga(s(s(X)), X1)) → U3_ggg(X, I, Y, log2_in_ggg(X1, s(I), Y))
U3_ggg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_ggg(s(s(X)), I, Y)
U3_agg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_agg(s(s(X)), I, Y)
U1_ag(X, Y, log2_out_agg(X, 0, Y)) → log2_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
log2_in_ag(x1, x2)  =  log2_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
log2_in_agg(x1, x2, x3)  =  log2_in_agg(x2, x3)
log2_out_agg(x1, x2, x3)  =  log2_out_agg(x1, x2, x3)
U2_agg(x1, x2, x3, x4)  =  U2_agg(x2, x3, x4)
half_in_aa(x1, x2)  =  half_in_aa
half_out_aa(x1, x2)  =  half_out_aa(x1, x2)
U4_aa(x1, x2, x3)  =  U4_aa(x3)
s(x1)  =  s(x1)
U3_agg(x1, x2, x3, x4)  =  U3_agg(x1, x2, x3, x4)
log2_in_ggg(x1, x2, x3)  =  log2_in_ggg(x1, x2, x3)
0  =  0
log2_out_ggg(x1, x2, x3)  =  log2_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x1, x2, x3, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x1, x2)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x1, x2, x3, x4)
log2_out_ag(x1, x2)  =  log2_out_ag(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

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

log2_in_ag(X, Y) → U1_ag(X, Y, log2_in_agg(X, 0, Y))
log2_in_agg(0, I, I) → log2_out_agg(0, I, I)
log2_in_agg(s(0), I, I) → log2_out_agg(s(0), I, I)
log2_in_agg(s(s(X)), I, Y) → U2_agg(X, I, Y, half_in_aa(s(s(X)), X1))
half_in_aa(0, 0) → half_out_aa(0, 0)
half_in_aa(s(0), 0) → half_out_aa(s(0), 0)
half_in_aa(s(s(X)), s(Y)) → U4_aa(X, Y, half_in_aa(X, Y))
U4_aa(X, Y, half_out_aa(X, Y)) → half_out_aa(s(s(X)), s(Y))
U2_agg(X, I, Y, half_out_aa(s(s(X)), X1)) → U3_agg(X, I, Y, log2_in_ggg(X1, s(I), Y))
log2_in_ggg(0, I, I) → log2_out_ggg(0, I, I)
log2_in_ggg(s(0), I, I) → log2_out_ggg(s(0), I, I)
log2_in_ggg(s(s(X)), I, Y) → U2_ggg(X, I, Y, half_in_ga(s(s(X)), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(0), 0) → half_out_ga(s(0), 0)
half_in_ga(s(s(X)), s(Y)) → U4_ga(X, Y, half_in_ga(X, Y))
U4_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U2_ggg(X, I, Y, half_out_ga(s(s(X)), X1)) → U3_ggg(X, I, Y, log2_in_ggg(X1, s(I), Y))
U3_ggg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_ggg(s(s(X)), I, Y)
U3_agg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_agg(s(s(X)), I, Y)
U1_ag(X, Y, log2_out_agg(X, 0, Y)) → log2_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
log2_in_ag(x1, x2)  =  log2_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
log2_in_agg(x1, x2, x3)  =  log2_in_agg(x2, x3)
log2_out_agg(x1, x2, x3)  =  log2_out_agg(x1, x2, x3)
U2_agg(x1, x2, x3, x4)  =  U2_agg(x2, x3, x4)
half_in_aa(x1, x2)  =  half_in_aa
half_out_aa(x1, x2)  =  half_out_aa(x1, x2)
U4_aa(x1, x2, x3)  =  U4_aa(x3)
s(x1)  =  s(x1)
U3_agg(x1, x2, x3, x4)  =  U3_agg(x1, x2, x3, x4)
log2_in_ggg(x1, x2, x3)  =  log2_in_ggg(x1, x2, x3)
0  =  0
log2_out_ggg(x1, x2, x3)  =  log2_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x1, x2, x3, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x1, x2)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x1, x2, x3, x4)
log2_out_ag(x1, x2)  =  log2_out_ag(x1, 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_AG(X, Y) → U1_AG(X, Y, log2_in_agg(X, 0, Y))
LOG2_IN_AG(X, Y) → LOG2_IN_AGG(X, 0, Y)
LOG2_IN_AGG(s(s(X)), I, Y) → U2_AGG(X, I, Y, half_in_aa(s(s(X)), X1))
LOG2_IN_AGG(s(s(X)), I, Y) → HALF_IN_AA(s(s(X)), X1)
HALF_IN_AA(s(s(X)), s(Y)) → U4_AA(X, Y, half_in_aa(X, Y))
HALF_IN_AA(s(s(X)), s(Y)) → HALF_IN_AA(X, Y)
U2_AGG(X, I, Y, half_out_aa(s(s(X)), X1)) → U3_AGG(X, I, Y, log2_in_ggg(X1, s(I), Y))
U2_AGG(X, I, Y, half_out_aa(s(s(X)), X1)) → LOG2_IN_GGG(X1, s(I), Y)
LOG2_IN_GGG(s(s(X)), I, Y) → U2_GGG(X, I, Y, half_in_ga(s(s(X)), X1))
LOG2_IN_GGG(s(s(X)), I, Y) → HALF_IN_GA(s(s(X)), X1)
HALF_IN_GA(s(s(X)), s(Y)) → U4_GA(X, Y, half_in_ga(X, Y))
HALF_IN_GA(s(s(X)), s(Y)) → HALF_IN_GA(X, Y)
U2_GGG(X, I, Y, half_out_ga(s(s(X)), X1)) → U3_GGG(X, I, Y, log2_in_ggg(X1, s(I), Y))
U2_GGG(X, I, Y, half_out_ga(s(s(X)), X1)) → LOG2_IN_GGG(X1, s(I), Y)

The TRS R consists of the following rules:

log2_in_ag(X, Y) → U1_ag(X, Y, log2_in_agg(X, 0, Y))
log2_in_agg(0, I, I) → log2_out_agg(0, I, I)
log2_in_agg(s(0), I, I) → log2_out_agg(s(0), I, I)
log2_in_agg(s(s(X)), I, Y) → U2_agg(X, I, Y, half_in_aa(s(s(X)), X1))
half_in_aa(0, 0) → half_out_aa(0, 0)
half_in_aa(s(0), 0) → half_out_aa(s(0), 0)
half_in_aa(s(s(X)), s(Y)) → U4_aa(X, Y, half_in_aa(X, Y))
U4_aa(X, Y, half_out_aa(X, Y)) → half_out_aa(s(s(X)), s(Y))
U2_agg(X, I, Y, half_out_aa(s(s(X)), X1)) → U3_agg(X, I, Y, log2_in_ggg(X1, s(I), Y))
log2_in_ggg(0, I, I) → log2_out_ggg(0, I, I)
log2_in_ggg(s(0), I, I) → log2_out_ggg(s(0), I, I)
log2_in_ggg(s(s(X)), I, Y) → U2_ggg(X, I, Y, half_in_ga(s(s(X)), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(0), 0) → half_out_ga(s(0), 0)
half_in_ga(s(s(X)), s(Y)) → U4_ga(X, Y, half_in_ga(X, Y))
U4_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U2_ggg(X, I, Y, half_out_ga(s(s(X)), X1)) → U3_ggg(X, I, Y, log2_in_ggg(X1, s(I), Y))
U3_ggg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_ggg(s(s(X)), I, Y)
U3_agg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_agg(s(s(X)), I, Y)
U1_ag(X, Y, log2_out_agg(X, 0, Y)) → log2_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
log2_in_ag(x1, x2)  =  log2_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
log2_in_agg(x1, x2, x3)  =  log2_in_agg(x2, x3)
log2_out_agg(x1, x2, x3)  =  log2_out_agg(x1, x2, x3)
U2_agg(x1, x2, x3, x4)  =  U2_agg(x2, x3, x4)
half_in_aa(x1, x2)  =  half_in_aa
half_out_aa(x1, x2)  =  half_out_aa(x1, x2)
U4_aa(x1, x2, x3)  =  U4_aa(x3)
s(x1)  =  s(x1)
U3_agg(x1, x2, x3, x4)  =  U3_agg(x1, x2, x3, x4)
log2_in_ggg(x1, x2, x3)  =  log2_in_ggg(x1, x2, x3)
0  =  0
log2_out_ggg(x1, x2, x3)  =  log2_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x1, x2, x3, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x1, x2)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x1, x2, x3, x4)
log2_out_ag(x1, x2)  =  log2_out_ag(x1, x2)
U3_AGG(x1, x2, x3, x4)  =  U3_AGG(x1, x2, x3, x4)
LOG2_IN_AGG(x1, x2, x3)  =  LOG2_IN_AGG(x2, x3)
LOG2_IN_GGG(x1, x2, x3)  =  LOG2_IN_GGG(x1, x2, x3)
U3_GGG(x1, x2, x3, x4)  =  U3_GGG(x1, x2, x3, x4)
HALF_IN_GA(x1, x2)  =  HALF_IN_GA(x1)
U2_AGG(x1, x2, x3, x4)  =  U2_AGG(x2, x3, x4)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x1, x2, x3, x4)
HALF_IN_AA(x1, x2)  =  HALF_IN_AA
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
U4_AA(x1, x2, x3)  =  U4_AA(x3)
LOG2_IN_AG(x1, x2)  =  LOG2_IN_AG(x2)

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

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

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

LOG2_IN_AG(X, Y) → U1_AG(X, Y, log2_in_agg(X, 0, Y))
LOG2_IN_AG(X, Y) → LOG2_IN_AGG(X, 0, Y)
LOG2_IN_AGG(s(s(X)), I, Y) → U2_AGG(X, I, Y, half_in_aa(s(s(X)), X1))
LOG2_IN_AGG(s(s(X)), I, Y) → HALF_IN_AA(s(s(X)), X1)
HALF_IN_AA(s(s(X)), s(Y)) → U4_AA(X, Y, half_in_aa(X, Y))
HALF_IN_AA(s(s(X)), s(Y)) → HALF_IN_AA(X, Y)
U2_AGG(X, I, Y, half_out_aa(s(s(X)), X1)) → U3_AGG(X, I, Y, log2_in_ggg(X1, s(I), Y))
U2_AGG(X, I, Y, half_out_aa(s(s(X)), X1)) → LOG2_IN_GGG(X1, s(I), Y)
LOG2_IN_GGG(s(s(X)), I, Y) → U2_GGG(X, I, Y, half_in_ga(s(s(X)), X1))
LOG2_IN_GGG(s(s(X)), I, Y) → HALF_IN_GA(s(s(X)), X1)
HALF_IN_GA(s(s(X)), s(Y)) → U4_GA(X, Y, half_in_ga(X, Y))
HALF_IN_GA(s(s(X)), s(Y)) → HALF_IN_GA(X, Y)
U2_GGG(X, I, Y, half_out_ga(s(s(X)), X1)) → U3_GGG(X, I, Y, log2_in_ggg(X1, s(I), Y))
U2_GGG(X, I, Y, half_out_ga(s(s(X)), X1)) → LOG2_IN_GGG(X1, s(I), Y)

The TRS R consists of the following rules:

log2_in_ag(X, Y) → U1_ag(X, Y, log2_in_agg(X, 0, Y))
log2_in_agg(0, I, I) → log2_out_agg(0, I, I)
log2_in_agg(s(0), I, I) → log2_out_agg(s(0), I, I)
log2_in_agg(s(s(X)), I, Y) → U2_agg(X, I, Y, half_in_aa(s(s(X)), X1))
half_in_aa(0, 0) → half_out_aa(0, 0)
half_in_aa(s(0), 0) → half_out_aa(s(0), 0)
half_in_aa(s(s(X)), s(Y)) → U4_aa(X, Y, half_in_aa(X, Y))
U4_aa(X, Y, half_out_aa(X, Y)) → half_out_aa(s(s(X)), s(Y))
U2_agg(X, I, Y, half_out_aa(s(s(X)), X1)) → U3_agg(X, I, Y, log2_in_ggg(X1, s(I), Y))
log2_in_ggg(0, I, I) → log2_out_ggg(0, I, I)
log2_in_ggg(s(0), I, I) → log2_out_ggg(s(0), I, I)
log2_in_ggg(s(s(X)), I, Y) → U2_ggg(X, I, Y, half_in_ga(s(s(X)), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(0), 0) → half_out_ga(s(0), 0)
half_in_ga(s(s(X)), s(Y)) → U4_ga(X, Y, half_in_ga(X, Y))
U4_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U2_ggg(X, I, Y, half_out_ga(s(s(X)), X1)) → U3_ggg(X, I, Y, log2_in_ggg(X1, s(I), Y))
U3_ggg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_ggg(s(s(X)), I, Y)
U3_agg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_agg(s(s(X)), I, Y)
U1_ag(X, Y, log2_out_agg(X, 0, Y)) → log2_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
log2_in_ag(x1, x2)  =  log2_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
log2_in_agg(x1, x2, x3)  =  log2_in_agg(x2, x3)
log2_out_agg(x1, x2, x3)  =  log2_out_agg(x1, x2, x3)
U2_agg(x1, x2, x3, x4)  =  U2_agg(x2, x3, x4)
half_in_aa(x1, x2)  =  half_in_aa
half_out_aa(x1, x2)  =  half_out_aa(x1, x2)
U4_aa(x1, x2, x3)  =  U4_aa(x3)
s(x1)  =  s(x1)
U3_agg(x1, x2, x3, x4)  =  U3_agg(x1, x2, x3, x4)
log2_in_ggg(x1, x2, x3)  =  log2_in_ggg(x1, x2, x3)
0  =  0
log2_out_ggg(x1, x2, x3)  =  log2_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x1, x2, x3, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x1, x2)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x1, x2, x3, x4)
log2_out_ag(x1, x2)  =  log2_out_ag(x1, x2)
U3_AGG(x1, x2, x3, x4)  =  U3_AGG(x1, x2, x3, x4)
LOG2_IN_AGG(x1, x2, x3)  =  LOG2_IN_AGG(x2, x3)
LOG2_IN_GGG(x1, x2, x3)  =  LOG2_IN_GGG(x1, x2, x3)
U3_GGG(x1, x2, x3, x4)  =  U3_GGG(x1, x2, x3, x4)
HALF_IN_GA(x1, x2)  =  HALF_IN_GA(x1)
U2_AGG(x1, x2, x3, x4)  =  U2_AGG(x2, x3, x4)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x1, x2, x3, x4)
HALF_IN_AA(x1, x2)  =  HALF_IN_AA
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
U4_AA(x1, x2, x3)  =  U4_AA(x3)
LOG2_IN_AG(x1, x2)  =  LOG2_IN_AG(x2)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 3 SCCs with 10 less nodes.

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

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

HALF_IN_GA(s(s(X)), s(Y)) → HALF_IN_GA(X, Y)

The TRS R consists of the following rules:

log2_in_ag(X, Y) → U1_ag(X, Y, log2_in_agg(X, 0, Y))
log2_in_agg(0, I, I) → log2_out_agg(0, I, I)
log2_in_agg(s(0), I, I) → log2_out_agg(s(0), I, I)
log2_in_agg(s(s(X)), I, Y) → U2_agg(X, I, Y, half_in_aa(s(s(X)), X1))
half_in_aa(0, 0) → half_out_aa(0, 0)
half_in_aa(s(0), 0) → half_out_aa(s(0), 0)
half_in_aa(s(s(X)), s(Y)) → U4_aa(X, Y, half_in_aa(X, Y))
U4_aa(X, Y, half_out_aa(X, Y)) → half_out_aa(s(s(X)), s(Y))
U2_agg(X, I, Y, half_out_aa(s(s(X)), X1)) → U3_agg(X, I, Y, log2_in_ggg(X1, s(I), Y))
log2_in_ggg(0, I, I) → log2_out_ggg(0, I, I)
log2_in_ggg(s(0), I, I) → log2_out_ggg(s(0), I, I)
log2_in_ggg(s(s(X)), I, Y) → U2_ggg(X, I, Y, half_in_ga(s(s(X)), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(0), 0) → half_out_ga(s(0), 0)
half_in_ga(s(s(X)), s(Y)) → U4_ga(X, Y, half_in_ga(X, Y))
U4_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U2_ggg(X, I, Y, half_out_ga(s(s(X)), X1)) → U3_ggg(X, I, Y, log2_in_ggg(X1, s(I), Y))
U3_ggg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_ggg(s(s(X)), I, Y)
U3_agg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_agg(s(s(X)), I, Y)
U1_ag(X, Y, log2_out_agg(X, 0, Y)) → log2_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
log2_in_ag(x1, x2)  =  log2_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
log2_in_agg(x1, x2, x3)  =  log2_in_agg(x2, x3)
log2_out_agg(x1, x2, x3)  =  log2_out_agg(x1, x2, x3)
U2_agg(x1, x2, x3, x4)  =  U2_agg(x2, x3, x4)
half_in_aa(x1, x2)  =  half_in_aa
half_out_aa(x1, x2)  =  half_out_aa(x1, x2)
U4_aa(x1, x2, x3)  =  U4_aa(x3)
s(x1)  =  s(x1)
U3_agg(x1, x2, x3, x4)  =  U3_agg(x1, x2, x3, x4)
log2_in_ggg(x1, x2, x3)  =  log2_in_ggg(x1, x2, x3)
0  =  0
log2_out_ggg(x1, x2, x3)  =  log2_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x1, x2, x3, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x1, x2)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x1, x2, x3, x4)
log2_out_ag(x1, x2)  =  log2_out_ag(x1, x2)
HALF_IN_GA(x1, x2)  =  HALF_IN_GA(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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP

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

HALF_IN_GA(s(s(X)), s(Y)) → HALF_IN_GA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
HALF_IN_GA(x1, x2)  =  HALF_IN_GA(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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP

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

HALF_IN_GA(s(s(X))) → HALF_IN_GA(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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP

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

LOG2_IN_GGG(s(s(X)), I, Y) → U2_GGG(X, I, Y, half_in_ga(s(s(X)), X1))
U2_GGG(X, I, Y, half_out_ga(s(s(X)), X1)) → LOG2_IN_GGG(X1, s(I), Y)

The TRS R consists of the following rules:

log2_in_ag(X, Y) → U1_ag(X, Y, log2_in_agg(X, 0, Y))
log2_in_agg(0, I, I) → log2_out_agg(0, I, I)
log2_in_agg(s(0), I, I) → log2_out_agg(s(0), I, I)
log2_in_agg(s(s(X)), I, Y) → U2_agg(X, I, Y, half_in_aa(s(s(X)), X1))
half_in_aa(0, 0) → half_out_aa(0, 0)
half_in_aa(s(0), 0) → half_out_aa(s(0), 0)
half_in_aa(s(s(X)), s(Y)) → U4_aa(X, Y, half_in_aa(X, Y))
U4_aa(X, Y, half_out_aa(X, Y)) → half_out_aa(s(s(X)), s(Y))
U2_agg(X, I, Y, half_out_aa(s(s(X)), X1)) → U3_agg(X, I, Y, log2_in_ggg(X1, s(I), Y))
log2_in_ggg(0, I, I) → log2_out_ggg(0, I, I)
log2_in_ggg(s(0), I, I) → log2_out_ggg(s(0), I, I)
log2_in_ggg(s(s(X)), I, Y) → U2_ggg(X, I, Y, half_in_ga(s(s(X)), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(0), 0) → half_out_ga(s(0), 0)
half_in_ga(s(s(X)), s(Y)) → U4_ga(X, Y, half_in_ga(X, Y))
U4_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U2_ggg(X, I, Y, half_out_ga(s(s(X)), X1)) → U3_ggg(X, I, Y, log2_in_ggg(X1, s(I), Y))
U3_ggg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_ggg(s(s(X)), I, Y)
U3_agg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_agg(s(s(X)), I, Y)
U1_ag(X, Y, log2_out_agg(X, 0, Y)) → log2_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
log2_in_ag(x1, x2)  =  log2_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
log2_in_agg(x1, x2, x3)  =  log2_in_agg(x2, x3)
log2_out_agg(x1, x2, x3)  =  log2_out_agg(x1, x2, x3)
U2_agg(x1, x2, x3, x4)  =  U2_agg(x2, x3, x4)
half_in_aa(x1, x2)  =  half_in_aa
half_out_aa(x1, x2)  =  half_out_aa(x1, x2)
U4_aa(x1, x2, x3)  =  U4_aa(x3)
s(x1)  =  s(x1)
U3_agg(x1, x2, x3, x4)  =  U3_agg(x1, x2, x3, x4)
log2_in_ggg(x1, x2, x3)  =  log2_in_ggg(x1, x2, x3)
0  =  0
log2_out_ggg(x1, x2, x3)  =  log2_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x1, x2, x3, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x1, x2)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x1, x2, x3, x4)
log2_out_ag(x1, x2)  =  log2_out_ag(x1, x2)
LOG2_IN_GGG(x1, x2, x3)  =  LOG2_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x1, x2, x3, x4)

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

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

LOG2_IN_GGG(s(s(X)), I, Y) → U2_GGG(X, I, Y, half_in_ga(s(s(X)), X1))
U2_GGG(X, I, Y, half_out_ga(s(s(X)), X1)) → LOG2_IN_GGG(X1, s(I), Y)

The TRS R consists of the following rules:

half_in_ga(s(s(X)), s(Y)) → U4_ga(X, Y, half_in_ga(X, Y))
U4_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(0), 0) → half_out_ga(s(0), 0)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x1, x2)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
LOG2_IN_GGG(x1, x2, x3)  =  LOG2_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x1, x2, x3, x4)

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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Rewriting
              ↳ PiDP

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

U2_GGG(X, I, Y, half_out_ga(s(s(X)), X1)) → LOG2_IN_GGG(X1, s(I), Y)
LOG2_IN_GGG(s(s(X)), I, Y) → U2_GGG(X, I, Y, half_in_ga(s(s(X))))

The TRS R consists of the following rules:

half_in_ga(s(s(X))) → U4_ga(X, half_in_ga(X))
U4_ga(X, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
half_in_ga(0) → half_out_ga(0, 0)
half_in_ga(s(0)) → half_out_ga(s(0), 0)

The set Q consists of the following terms:

half_in_ga(x0)
U4_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
By rewriting [15] the rule LOG2_IN_GGG(s(s(X)), I, Y) → U2_GGG(X, I, Y, half_in_ga(s(s(X)))) at position [3] we obtained the following new rules:

LOG2_IN_GGG(s(s(X)), I, Y) → U2_GGG(X, I, Y, U4_ga(X, half_in_ga(X)))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Rewriting
                          ↳ QDP
                            ↳ QDPOrderProof
              ↳ PiDP

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

LOG2_IN_GGG(s(s(X)), I, Y) → U2_GGG(X, I, Y, U4_ga(X, half_in_ga(X)))
U2_GGG(X, I, Y, half_out_ga(s(s(X)), X1)) → LOG2_IN_GGG(X1, s(I), Y)

The TRS R consists of the following rules:

half_in_ga(s(s(X))) → U4_ga(X, half_in_ga(X))
U4_ga(X, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
half_in_ga(0) → half_out_ga(0, 0)
half_in_ga(s(0)) → half_out_ga(s(0), 0)

The set Q consists of the following terms:

half_in_ga(x0)
U4_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U2_GGG(X, I, Y, half_out_ga(s(s(X)), X1)) → LOG2_IN_GGG(X1, s(I), Y)
The remaining pairs can at least be oriented weakly.

LOG2_IN_GGG(s(s(X)), I, Y) → U2_GGG(X, I, Y, U4_ga(X, half_in_ga(X)))
Used ordering: Combined order from the following AFS and order.
LOG2_IN_GGG(x1, x2, x3)  =  LOG2_IN_GGG(x1, x2, x3)
s(x1)  =  s(x1)
U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x2, x3, x4)
U4_ga(x1, x2)  =  U4_ga(x2)
half_in_ga(x1)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x2)
0  =  0

Recursive path order with status [2].
Quasi-Precedence:
[LOG2INGGG3, U2GGG3] > [s1, U4ga1, halfinga1, halfoutga1] > 0

Status:
U2GGG3: [3,1,2]
U4ga1: [1]
LOG2INGGG3: [1,2,3]
halfoutga1: [1]
halfinga1: [1]
0: multiset
s1: [1]


The following usable rules [17] were oriented:

half_in_ga(s(0)) → half_out_ga(s(0), 0)
U4_ga(X, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
half_in_ga(0) → half_out_ga(0, 0)
half_in_ga(s(s(X))) → U4_ga(X, half_in_ga(X))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Rewriting
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
              ↳ PiDP

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

LOG2_IN_GGG(s(s(X)), I, Y) → U2_GGG(X, I, Y, U4_ga(X, half_in_ga(X)))

The TRS R consists of the following rules:

half_in_ga(s(s(X))) → U4_ga(X, half_in_ga(X))
U4_ga(X, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
half_in_ga(0) → half_out_ga(0, 0)
half_in_ga(s(0)) → half_out_ga(s(0), 0)

The set Q consists of the following terms:

half_in_ga(x0)
U4_ga(x0, x1)

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.

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

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

HALF_IN_AA(s(s(X)), s(Y)) → HALF_IN_AA(X, Y)

The TRS R consists of the following rules:

log2_in_ag(X, Y) → U1_ag(X, Y, log2_in_agg(X, 0, Y))
log2_in_agg(0, I, I) → log2_out_agg(0, I, I)
log2_in_agg(s(0), I, I) → log2_out_agg(s(0), I, I)
log2_in_agg(s(s(X)), I, Y) → U2_agg(X, I, Y, half_in_aa(s(s(X)), X1))
half_in_aa(0, 0) → half_out_aa(0, 0)
half_in_aa(s(0), 0) → half_out_aa(s(0), 0)
half_in_aa(s(s(X)), s(Y)) → U4_aa(X, Y, half_in_aa(X, Y))
U4_aa(X, Y, half_out_aa(X, Y)) → half_out_aa(s(s(X)), s(Y))
U2_agg(X, I, Y, half_out_aa(s(s(X)), X1)) → U3_agg(X, I, Y, log2_in_ggg(X1, s(I), Y))
log2_in_ggg(0, I, I) → log2_out_ggg(0, I, I)
log2_in_ggg(s(0), I, I) → log2_out_ggg(s(0), I, I)
log2_in_ggg(s(s(X)), I, Y) → U2_ggg(X, I, Y, half_in_ga(s(s(X)), X1))
half_in_ga(0, 0) → half_out_ga(0, 0)
half_in_ga(s(0), 0) → half_out_ga(s(0), 0)
half_in_ga(s(s(X)), s(Y)) → U4_ga(X, Y, half_in_ga(X, Y))
U4_ga(X, Y, half_out_ga(X, Y)) → half_out_ga(s(s(X)), s(Y))
U2_ggg(X, I, Y, half_out_ga(s(s(X)), X1)) → U3_ggg(X, I, Y, log2_in_ggg(X1, s(I), Y))
U3_ggg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_ggg(s(s(X)), I, Y)
U3_agg(X, I, Y, log2_out_ggg(X1, s(I), Y)) → log2_out_agg(s(s(X)), I, Y)
U1_ag(X, Y, log2_out_agg(X, 0, Y)) → log2_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
log2_in_ag(x1, x2)  =  log2_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
log2_in_agg(x1, x2, x3)  =  log2_in_agg(x2, x3)
log2_out_agg(x1, x2, x3)  =  log2_out_agg(x1, x2, x3)
U2_agg(x1, x2, x3, x4)  =  U2_agg(x2, x3, x4)
half_in_aa(x1, x2)  =  half_in_aa
half_out_aa(x1, x2)  =  half_out_aa(x1, x2)
U4_aa(x1, x2, x3)  =  U4_aa(x3)
s(x1)  =  s(x1)
U3_agg(x1, x2, x3, x4)  =  U3_agg(x1, x2, x3, x4)
log2_in_ggg(x1, x2, x3)  =  log2_in_ggg(x1, x2, x3)
0  =  0
log2_out_ggg(x1, x2, x3)  =  log2_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x1, x2, x3, x4)
half_in_ga(x1, x2)  =  half_in_ga(x1)
half_out_ga(x1, x2)  =  half_out_ga(x1, x2)
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x1, x2, x3, x4)
log2_out_ag(x1, x2)  =  log2_out_ag(x1, x2)
HALF_IN_AA(x1, x2)  =  HALF_IN_AA

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

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

HALF_IN_AA(s(s(X)), s(Y)) → HALF_IN_AA(X, Y)

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

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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof

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

HALF_IN_AA → HALF_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

HALF_IN_AA → HALF_IN_AA

The TRS R consists of the following rules:none


s = HALF_IN_AA evaluates to t =HALF_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from HALF_IN_AA to HALF_IN_AA.