Left Termination of the query pattern average_in_3(g, a, g) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 PiDP
    ↳7 UsableRulesProof (⇔)
    ↳8 PiDP
    ↳9 PiDPToQDPProof (⇐)
    ↳10 QDP
    ↳11 QDPSizeChangeProof (⇔)
    ↳12 TRUE
  ↳13 PrologToPiTRSProof (⇐)
    ↳14 PiTRS
    ↳15 DependencyPairsProof (⇔)
    ↳16 PiDP
    ↳17 DependencyGraphProof (⇔)
    ↳18 PiDP
    ↳19 UsableRulesProof (⇔)
    ↳20 PiDP
    ↳21 PiDPToQDPProof (⇐)
    ↳22 QDP

(0) Obligation:

Clauses:

average(0, 0, 0).
average(0, s(0), 0).
average(0, s(s(0)), s(0)).
average(s(X), Y, Z) :- average(X, s(Y), Z).
average(X, s(s(s(Y))), s(Z)) :- average(s(X), Y, Z).

Queries:

average(g,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:
average_in: (b,f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

average_in_gag(0, 0, 0) → average_out_gag(0, 0, 0)
average_in_gag(0, s(0), 0) → average_out_gag(0, s(0), 0)
average_in_gag(0, s(s(0)), s(0)) → average_out_gag(0, s(s(0)), s(0))
average_in_gag(s(X), Y, Z) → U1_gag(X, Y, Z, average_in_gag(X, s(Y), Z))
average_in_gag(X, s(s(s(Y))), s(Z)) → U2_gag(X, Y, Z, average_in_gag(s(X), Y, Z))
U2_gag(X, Y, Z, average_out_gag(s(X), Y, Z)) → average_out_gag(X, s(s(s(Y))), s(Z))
U1_gag(X, Y, Z, average_out_gag(X, s(Y), Z)) → average_out_gag(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
average_in_gag(x1, x2, x3)  =  average_in_gag(x1, x3)
0  =  0
average_out_gag(x1, x2, x3)  =  average_out_gag(x2)
s(x1)  =  s(x1)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x4)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

average_in_gag(0, 0, 0) → average_out_gag(0, 0, 0)
average_in_gag(0, s(0), 0) → average_out_gag(0, s(0), 0)
average_in_gag(0, s(s(0)), s(0)) → average_out_gag(0, s(s(0)), s(0))
average_in_gag(s(X), Y, Z) → U1_gag(X, Y, Z, average_in_gag(X, s(Y), Z))
average_in_gag(X, s(s(s(Y))), s(Z)) → U2_gag(X, Y, Z, average_in_gag(s(X), Y, Z))
U2_gag(X, Y, Z, average_out_gag(s(X), Y, Z)) → average_out_gag(X, s(s(s(Y))), s(Z))
U1_gag(X, Y, Z, average_out_gag(X, s(Y), Z)) → average_out_gag(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
average_in_gag(x1, x2, x3)  =  average_in_gag(x1, x3)
0  =  0
average_out_gag(x1, x2, x3)  =  average_out_gag(x2)
s(x1)  =  s(x1)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x4)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)

(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:

AVERAGE_IN_GAG(s(X), Y, Z) → U1_GAG(X, Y, Z, average_in_gag(X, s(Y), Z))
AVERAGE_IN_GAG(s(X), Y, Z) → AVERAGE_IN_GAG(X, s(Y), Z)
AVERAGE_IN_GAG(X, s(s(s(Y))), s(Z)) → U2_GAG(X, Y, Z, average_in_gag(s(X), Y, Z))
AVERAGE_IN_GAG(X, s(s(s(Y))), s(Z)) → AVERAGE_IN_GAG(s(X), Y, Z)

The TRS R consists of the following rules:

average_in_gag(0, 0, 0) → average_out_gag(0, 0, 0)
average_in_gag(0, s(0), 0) → average_out_gag(0, s(0), 0)
average_in_gag(0, s(s(0)), s(0)) → average_out_gag(0, s(s(0)), s(0))
average_in_gag(s(X), Y, Z) → U1_gag(X, Y, Z, average_in_gag(X, s(Y), Z))
average_in_gag(X, s(s(s(Y))), s(Z)) → U2_gag(X, Y, Z, average_in_gag(s(X), Y, Z))
U2_gag(X, Y, Z, average_out_gag(s(X), Y, Z)) → average_out_gag(X, s(s(s(Y))), s(Z))
U1_gag(X, Y, Z, average_out_gag(X, s(Y), Z)) → average_out_gag(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
average_in_gag(x1, x2, x3)  =  average_in_gag(x1, x3)
0  =  0
average_out_gag(x1, x2, x3)  =  average_out_gag(x2)
s(x1)  =  s(x1)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x4)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
AVERAGE_IN_GAG(x1, x2, x3)  =  AVERAGE_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4)  =  U1_GAG(x4)
U2_GAG(x1, x2, x3, x4)  =  U2_GAG(x4)

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

(4) Obligation:

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

AVERAGE_IN_GAG(s(X), Y, Z) → U1_GAG(X, Y, Z, average_in_gag(X, s(Y), Z))
AVERAGE_IN_GAG(s(X), Y, Z) → AVERAGE_IN_GAG(X, s(Y), Z)
AVERAGE_IN_GAG(X, s(s(s(Y))), s(Z)) → U2_GAG(X, Y, Z, average_in_gag(s(X), Y, Z))
AVERAGE_IN_GAG(X, s(s(s(Y))), s(Z)) → AVERAGE_IN_GAG(s(X), Y, Z)

The TRS R consists of the following rules:

average_in_gag(0, 0, 0) → average_out_gag(0, 0, 0)
average_in_gag(0, s(0), 0) → average_out_gag(0, s(0), 0)
average_in_gag(0, s(s(0)), s(0)) → average_out_gag(0, s(s(0)), s(0))
average_in_gag(s(X), Y, Z) → U1_gag(X, Y, Z, average_in_gag(X, s(Y), Z))
average_in_gag(X, s(s(s(Y))), s(Z)) → U2_gag(X, Y, Z, average_in_gag(s(X), Y, Z))
U2_gag(X, Y, Z, average_out_gag(s(X), Y, Z)) → average_out_gag(X, s(s(s(Y))), s(Z))
U1_gag(X, Y, Z, average_out_gag(X, s(Y), Z)) → average_out_gag(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
average_in_gag(x1, x2, x3)  =  average_in_gag(x1, x3)
0  =  0
average_out_gag(x1, x2, x3)  =  average_out_gag(x2)
s(x1)  =  s(x1)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x4)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
AVERAGE_IN_GAG(x1, x2, x3)  =  AVERAGE_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4)  =  U1_GAG(x4)
U2_GAG(x1, x2, x3, x4)  =  U2_GAG(x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(6) Obligation:

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

AVERAGE_IN_GAG(X, s(s(s(Y))), s(Z)) → AVERAGE_IN_GAG(s(X), Y, Z)
AVERAGE_IN_GAG(s(X), Y, Z) → AVERAGE_IN_GAG(X, s(Y), Z)

The TRS R consists of the following rules:

average_in_gag(0, 0, 0) → average_out_gag(0, 0, 0)
average_in_gag(0, s(0), 0) → average_out_gag(0, s(0), 0)
average_in_gag(0, s(s(0)), s(0)) → average_out_gag(0, s(s(0)), s(0))
average_in_gag(s(X), Y, Z) → U1_gag(X, Y, Z, average_in_gag(X, s(Y), Z))
average_in_gag(X, s(s(s(Y))), s(Z)) → U2_gag(X, Y, Z, average_in_gag(s(X), Y, Z))
U2_gag(X, Y, Z, average_out_gag(s(X), Y, Z)) → average_out_gag(X, s(s(s(Y))), s(Z))
U1_gag(X, Y, Z, average_out_gag(X, s(Y), Z)) → average_out_gag(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
average_in_gag(x1, x2, x3)  =  average_in_gag(x1, x3)
0  =  0
average_out_gag(x1, x2, x3)  =  average_out_gag(x2)
s(x1)  =  s(x1)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x4)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
AVERAGE_IN_GAG(x1, x2, x3)  =  AVERAGE_IN_GAG(x1, x3)

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

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

AVERAGE_IN_GAG(X, s(s(s(Y))), s(Z)) → AVERAGE_IN_GAG(s(X), Y, Z)
AVERAGE_IN_GAG(s(X), Y, Z) → AVERAGE_IN_GAG(X, s(Y), Z)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
AVERAGE_IN_GAG(x1, x2, x3)  =  AVERAGE_IN_GAG(x1, x3)

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

(9) PiDPToQDPProof (SOUND transformation)

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

(10) Obligation:

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

AVERAGE_IN_GAG(X, s(Z)) → AVERAGE_IN_GAG(s(X), Z)
AVERAGE_IN_GAG(s(X), Z) → AVERAGE_IN_GAG(X, Z)

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

(11) 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:

  • AVERAGE_IN_GAG(X, s(Z)) → AVERAGE_IN_GAG(s(X), Z)
    The graph contains the following edges 2 > 2

  • AVERAGE_IN_GAG(s(X), Z) → AVERAGE_IN_GAG(X, Z)
    The graph contains the following edges 1 > 1, 2 >= 2

(12) TRUE

(13) PrologToPiTRSProof (SOUND transformation)

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

average_in_gag(0, 0, 0) → average_out_gag(0, 0, 0)
average_in_gag(0, s(0), 0) → average_out_gag(0, s(0), 0)
average_in_gag(0, s(s(0)), s(0)) → average_out_gag(0, s(s(0)), s(0))
average_in_gag(s(X), Y, Z) → U1_gag(X, Y, Z, average_in_gag(X, s(Y), Z))
average_in_gag(X, s(s(s(Y))), s(Z)) → U2_gag(X, Y, Z, average_in_gag(s(X), Y, Z))
U2_gag(X, Y, Z, average_out_gag(s(X), Y, Z)) → average_out_gag(X, s(s(s(Y))), s(Z))
U1_gag(X, Y, Z, average_out_gag(X, s(Y), Z)) → average_out_gag(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
average_in_gag(x1, x2, x3)  =  average_in_gag(x1, x3)
0  =  0
average_out_gag(x1, x2, x3)  =  average_out_gag(x1, x2, x3)
s(x1)  =  s(x1)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x1, x3, x4)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x1, x3, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(14) Obligation:

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

average_in_gag(0, 0, 0) → average_out_gag(0, 0, 0)
average_in_gag(0, s(0), 0) → average_out_gag(0, s(0), 0)
average_in_gag(0, s(s(0)), s(0)) → average_out_gag(0, s(s(0)), s(0))
average_in_gag(s(X), Y, Z) → U1_gag(X, Y, Z, average_in_gag(X, s(Y), Z))
average_in_gag(X, s(s(s(Y))), s(Z)) → U2_gag(X, Y, Z, average_in_gag(s(X), Y, Z))
U2_gag(X, Y, Z, average_out_gag(s(X), Y, Z)) → average_out_gag(X, s(s(s(Y))), s(Z))
U1_gag(X, Y, Z, average_out_gag(X, s(Y), Z)) → average_out_gag(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
average_in_gag(x1, x2, x3)  =  average_in_gag(x1, x3)
0  =  0
average_out_gag(x1, x2, x3)  =  average_out_gag(x1, x2, x3)
s(x1)  =  s(x1)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x1, x3, x4)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x1, x3, x4)

(15) 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:

AVERAGE_IN_GAG(s(X), Y, Z) → U1_GAG(X, Y, Z, average_in_gag(X, s(Y), Z))
AVERAGE_IN_GAG(s(X), Y, Z) → AVERAGE_IN_GAG(X, s(Y), Z)
AVERAGE_IN_GAG(X, s(s(s(Y))), s(Z)) → U2_GAG(X, Y, Z, average_in_gag(s(X), Y, Z))
AVERAGE_IN_GAG(X, s(s(s(Y))), s(Z)) → AVERAGE_IN_GAG(s(X), Y, Z)

The TRS R consists of the following rules:

average_in_gag(0, 0, 0) → average_out_gag(0, 0, 0)
average_in_gag(0, s(0), 0) → average_out_gag(0, s(0), 0)
average_in_gag(0, s(s(0)), s(0)) → average_out_gag(0, s(s(0)), s(0))
average_in_gag(s(X), Y, Z) → U1_gag(X, Y, Z, average_in_gag(X, s(Y), Z))
average_in_gag(X, s(s(s(Y))), s(Z)) → U2_gag(X, Y, Z, average_in_gag(s(X), Y, Z))
U2_gag(X, Y, Z, average_out_gag(s(X), Y, Z)) → average_out_gag(X, s(s(s(Y))), s(Z))
U1_gag(X, Y, Z, average_out_gag(X, s(Y), Z)) → average_out_gag(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
average_in_gag(x1, x2, x3)  =  average_in_gag(x1, x3)
0  =  0
average_out_gag(x1, x2, x3)  =  average_out_gag(x1, x2, x3)
s(x1)  =  s(x1)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x1, x3, x4)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x1, x3, x4)
AVERAGE_IN_GAG(x1, x2, x3)  =  AVERAGE_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4)  =  U1_GAG(x1, x3, x4)
U2_GAG(x1, x2, x3, x4)  =  U2_GAG(x1, x3, x4)

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

(16) Obligation:

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

AVERAGE_IN_GAG(s(X), Y, Z) → U1_GAG(X, Y, Z, average_in_gag(X, s(Y), Z))
AVERAGE_IN_GAG(s(X), Y, Z) → AVERAGE_IN_GAG(X, s(Y), Z)
AVERAGE_IN_GAG(X, s(s(s(Y))), s(Z)) → U2_GAG(X, Y, Z, average_in_gag(s(X), Y, Z))
AVERAGE_IN_GAG(X, s(s(s(Y))), s(Z)) → AVERAGE_IN_GAG(s(X), Y, Z)

The TRS R consists of the following rules:

average_in_gag(0, 0, 0) → average_out_gag(0, 0, 0)
average_in_gag(0, s(0), 0) → average_out_gag(0, s(0), 0)
average_in_gag(0, s(s(0)), s(0)) → average_out_gag(0, s(s(0)), s(0))
average_in_gag(s(X), Y, Z) → U1_gag(X, Y, Z, average_in_gag(X, s(Y), Z))
average_in_gag(X, s(s(s(Y))), s(Z)) → U2_gag(X, Y, Z, average_in_gag(s(X), Y, Z))
U2_gag(X, Y, Z, average_out_gag(s(X), Y, Z)) → average_out_gag(X, s(s(s(Y))), s(Z))
U1_gag(X, Y, Z, average_out_gag(X, s(Y), Z)) → average_out_gag(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
average_in_gag(x1, x2, x3)  =  average_in_gag(x1, x3)
0  =  0
average_out_gag(x1, x2, x3)  =  average_out_gag(x1, x2, x3)
s(x1)  =  s(x1)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x1, x3, x4)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x1, x3, x4)
AVERAGE_IN_GAG(x1, x2, x3)  =  AVERAGE_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4)  =  U1_GAG(x1, x3, x4)
U2_GAG(x1, x2, x3, x4)  =  U2_GAG(x1, x3, x4)

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

(17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(18) Obligation:

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

AVERAGE_IN_GAG(X, s(s(s(Y))), s(Z)) → AVERAGE_IN_GAG(s(X), Y, Z)
AVERAGE_IN_GAG(s(X), Y, Z) → AVERAGE_IN_GAG(X, s(Y), Z)

The TRS R consists of the following rules:

average_in_gag(0, 0, 0) → average_out_gag(0, 0, 0)
average_in_gag(0, s(0), 0) → average_out_gag(0, s(0), 0)
average_in_gag(0, s(s(0)), s(0)) → average_out_gag(0, s(s(0)), s(0))
average_in_gag(s(X), Y, Z) → U1_gag(X, Y, Z, average_in_gag(X, s(Y), Z))
average_in_gag(X, s(s(s(Y))), s(Z)) → U2_gag(X, Y, Z, average_in_gag(s(X), Y, Z))
U2_gag(X, Y, Z, average_out_gag(s(X), Y, Z)) → average_out_gag(X, s(s(s(Y))), s(Z))
U1_gag(X, Y, Z, average_out_gag(X, s(Y), Z)) → average_out_gag(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
average_in_gag(x1, x2, x3)  =  average_in_gag(x1, x3)
0  =  0
average_out_gag(x1, x2, x3)  =  average_out_gag(x1, x2, x3)
s(x1)  =  s(x1)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x1, x3, x4)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x1, x3, x4)
AVERAGE_IN_GAG(x1, x2, x3)  =  AVERAGE_IN_GAG(x1, x3)

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

(19) UsableRulesProof (EQUIVALENT transformation)

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

(20) Obligation:

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

AVERAGE_IN_GAG(X, s(s(s(Y))), s(Z)) → AVERAGE_IN_GAG(s(X), Y, Z)
AVERAGE_IN_GAG(s(X), Y, Z) → AVERAGE_IN_GAG(X, s(Y), Z)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
AVERAGE_IN_GAG(x1, x2, x3)  =  AVERAGE_IN_GAG(x1, x3)

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

(21) PiDPToQDPProof (SOUND transformation)

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

(22) Obligation:

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

AVERAGE_IN_GAG(X, s(Z)) → AVERAGE_IN_GAG(s(X), Z)
AVERAGE_IN_GAG(s(X), Z) → AVERAGE_IN_GAG(X, Z)

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