Left Termination proof of /tmp/tmppZrvbV/log2a-oi.pl

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

0 Prolog

  ↳1 PrologToPiTRSProof (⇐)

    ↳2 PiTRS

    ↳3 DependencyPairsProof (⇔)

    ↳4 PiDP

    ↳5 DependencyGraphProof (⇔)

    ↳6 AND

      ↳7 PiDP

        ↳8 UsableRulesProof (⇔)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇐)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔)

        ↳13 TRUE

      ↳14 PiDP

        ↳15 UsableRulesProof (⇔)

        ↳16 PiDP

        ↳17 PiDPToQDPProof (⇐)

        ↳18 QDP

        ↳19 MRRProof (⇔)

        ↳20 QDP

        ↳21 DependencyGraphProof (⇔)

        ↳22 TRUE

      ↳23 PiDP

        ↳24 UsableRulesProof (⇔)

        ↳25 PiDP

        ↳26 PiDPToQDPProof (⇐)

        ↳27 QDP

        ↳28 NonTerminationProof (⇔)

        ↳29 FALSE

  ↳30 PrologToPiTRSProof (⇐)

    ↳31 PiTRS

    ↳32 DependencyPairsProof (⇔)

    ↳33 PiDP

    ↳34 DependencyGraphProof (⇔)

    ↳35 AND

      ↳36 PiDP

        ↳37 UsableRulesProof (⇔)

        ↳38 PiDP

        ↳39 PiDPToQDPProof (⇐)

        ↳40 QDP

        ↳41 QDPSizeChangeProof (⇔)

        ↳42 TRUE

      ↳43 PiDP

        ↳44 UsableRulesProof (⇔)

        ↳45 PiDP

        ↳46 PiDPToQDPProof (⇐)

        ↳47 QDP

        ↳48 QDPOrderProof (⇔)

        ↳49 QDP

        ↳50 DependencyGraphProof (⇔)

        ↳51 TRUE

      ↳52 PiDP

        ↳53 UsableRulesProof (⇔)

        ↳54 PiDP

        ↳55 PiDPToQDPProof (⇐)

        ↳56 QDP

        ↳57 NonTerminationProof (⇔)

        ↳58 FALSE

(0) Obligation:

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).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. 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)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] 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)
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)
U2_AGG(x1, x2, x3, x4) = U2_AGG(x2, x3, x4)
HALF_IN_AA(x1, x2) = HALF_IN_AA
U4_AA(x1, x2, x3) = U4_AA(x3)
U3_AGG(x1, x2, x3, x4) = U3_AGG(x1, x4)
LOG2_IN_GGG(x1, x2, x3) = LOG2_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4) = U2_GGG(x2, x3, x4)
HALF_IN_GA(x1, x2) = HALF_IN_GA(x1)
U4_GA(x1, x2, x3) = U4_GA(x3)
U3_GGG(x1, x2, x3, x4) = U3_GGG(x4)

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

(4) Obligation:

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)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 10 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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)

(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:

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

(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:

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.

(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:

HALF_IN_GA(s(s(X))) → HALF_IN_GA(X)
The graph contains the following edges 1 > 1

(13) TRUE

(14) Obligation:

Pi 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)), X1))

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

(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:

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)), X1))

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

(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:

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

(19) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

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

Strictly oriented rules of the TRS R:

half_in_ga(s(0)) → half_out_ga(0)

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(LOG2_IN_GGG(x₁, x₂, x₃)) = 2·x₁ + x₂ + x₃
POL(U2_GGG(x₁, x₂, x₃)) = 1 + 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

(20) Obligation:

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

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)

The set Q consists of the following terms:

half_in_ga(x0)
U4_ga(x0)

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

(21) DependencyGraphProof (EQUIVALENT transformation)

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

(22) TRUE

(23) Obligation:

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)

(24) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(25) Obligation:

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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.

(28) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

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:

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

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

(29) FALSE

(30) PrologToPiTRSProof (SOUND transformation)

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)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(31) Obligation:

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)

(32) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] 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)
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)
U2_AGG(x1, x2, x3, x4) = U2_AGG(x2, x3, x4)
HALF_IN_AA(x1, x2) = HALF_IN_AA
U4_AA(x1, x2, x3) = U4_AA(x3)
U3_AGG(x1, x2, x3, x4) = U3_AGG(x1, x2, x3, x4)
LOG2_IN_GGG(x1, x2, x3) = LOG2_IN_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)
U4_GA(x1, x2, x3) = U4_GA(x1, x3)
U3_GGG(x1, x2, x3, x4) = U3_GGG(x1, x2, x3, x4)

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

(33) Obligation:

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)

(34) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 10 less nodes.

(35) Complex Obligation (AND)

(36) Obligation:

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)

(37) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(38) Obligation:

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

(39) PiDPToQDPProof (SOUND transformation)

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

(40) Obligation:

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.

(41) 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:

HALF_IN_GA(s(s(X))) → HALF_IN_GA(X)
The graph contains the following edges 1 > 1

(42) TRUE

(43) Obligation:

Pi 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)), X1))

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

(44) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(45) Obligation:

Pi 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)), X1))

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

(46) PiDPToQDPProof (SOUND transformation)

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

(47) Obligation:

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.

(48) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

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.
Used ordering: Matrix interpretation [MATRO]:

POL(U2_GGG(x₁, x₂, x₃, x₄)) = 1 +
[ 0, 0 ]

· x₁ +
[ 0, 0 ]

· x₂ +
[ 0, 0 ]

· x₃ +
[ 0, 1 ]

· x₄

POL(half_out_ga(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 1 0 /

· x₂

POL(s(x₁)) =
/ 1 \

\ 0 /

+
/ 1 1 \

\ 1 1 /

· x₁

POL(LOG2_IN_GGG(x₁, x₂, x₃)) = 0 +
[ 1, 0 ]

· x₁ +
[ 0, 0 ]

· x₂ +
[ 0, 0 ]

· x₃

POL(half_in_ga(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 1 /

· x₁

POL(U4_ga(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 1 1 /

· x₁ +
/ 1 1 \

\ 1 1 /

· x₂

POL(0) =
/ 0 \

\ 0 /

The following usable rules [FROCOS05] were oriented:

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

(49) Obligation:

Q 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))))

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.

(50) DependencyGraphProof (EQUIVALENT transformation)

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

(51) TRUE

(52) Obligation:

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)

(53) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(54) Obligation:

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

(55) PiDPToQDPProof (SOUND transformation)

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

(56) Obligation:

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.

(57) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

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

(0) Obligation:

(1) PrologToPiTRSProof (SOUND transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) PiDPToQDPProof (SOUND transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) TRUE

(14) Obligation:

(15) UsableRulesProof (EQUIVALENT transformation)

(16) Obligation:

(17) PiDPToQDPProof (SOUND transformation)

(18) Obligation:

(19) MRRProof (EQUIVALENT transformation)

(20) Obligation:

(21) DependencyGraphProof (EQUIVALENT transformation)

(22) TRUE

(23) Obligation:

(24) UsableRulesProof (EQUIVALENT transformation)

(25) Obligation:

(26) PiDPToQDPProof (SOUND transformation)

(27) Obligation:

(28) NonTerminationProof (EQUIVALENT transformation)

(29) FALSE

(30) PrologToPiTRSProof (SOUND transformation)

(31) Obligation:

(32) DependencyPairsProof (EQUIVALENT transformation)

(33) Obligation:

(34) DependencyGraphProof (EQUIVALENT transformation)

(35) Complex Obligation (AND)

(36) Obligation:

(37) UsableRulesProof (EQUIVALENT transformation)

(38) Obligation:

(39) PiDPToQDPProof (SOUND transformation)

(40) Obligation:

(41) QDPSizeChangeProof (EQUIVALENT transformation)

(42) TRUE

(43) Obligation:

(44) UsableRulesProof (EQUIVALENT transformation)

(45) Obligation:

(46) PiDPToQDPProof (SOUND transformation)

(47) Obligation:

(48) QDPOrderProof (EQUIVALENT transformation)

(49) Obligation:

(50) DependencyGraphProof (EQUIVALENT transformation)

(51) TRUE

(52) Obligation:

(53) UsableRulesProof (EQUIVALENT transformation)

(54) Obligation:

(55) PiDPToQDPProof (SOUND transformation)

(56) Obligation:

(57) NonTerminationProof (EQUIVALENT transformation)

(58) FALSE