(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) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

average1(0, s(s(s(T17))), s(0)) :- average1(0, s(T17), 0).
average1(s(s(T38)), T41, T40) :- average1(T38, s(s(T41)), T40).
average1(s(T54), s(s(T57)), s(T56)) :- average1(s(T54), T57, T56).
average1(s(T67), s(s(s(T70))), s(T69)) :- average1(s(s(T67)), T70, T69).
average1(T95, s(s(s(T98))), s(T97)) :- average1(T95, s(T98), T97).
average1(T105, s(s(s(s(s(s(T108)))))), s(s(T107))) :- average1(s(s(T105)), T108, T107).

Clauses:

averagec1(0, 0, 0).
averagec1(0, s(0), 0).
averagec1(0, s(0), 0).
averagec1(0, s(s(0)), s(0)).
averagec1(0, s(s(s(T17))), s(0)) :- averagec1(0, s(T17), 0).
averagec1(s(0), 0, 0).
averagec1(s(0), s(0), s(0)).
averagec1(s(s(T38)), T41, T40) :- averagec1(T38, s(s(T41)), T40).
averagec1(s(T54), s(s(T57)), s(T56)) :- averagec1(s(T54), T57, T56).
averagec1(s(T67), s(s(s(T70))), s(T69)) :- averagec1(s(s(T67)), T70, T69).
averagec1(T95, s(s(s(T98))), s(T97)) :- averagec1(T95, s(T98), T97).
averagec1(T105, s(s(s(s(s(s(T108)))))), s(s(T107))) :- averagec1(s(s(T105)), T108, T107).

Afs:

average1(x1, x2, x3)  =  average1(x1, x3)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
average1_in: (b,f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

AVERAGE1_IN_GAG(0, s(s(s(T17))), s(0)) → U1_GAG(T17, average1_in_gag(0, s(T17), 0))
AVERAGE1_IN_GAG(0, s(s(s(T17))), s(0)) → AVERAGE1_IN_GAG(0, s(T17), 0)
AVERAGE1_IN_GAG(s(s(T38)), T41, T40) → U2_GAG(T38, T41, T40, average1_in_gag(T38, s(s(T41)), T40))
AVERAGE1_IN_GAG(s(s(T38)), T41, T40) → AVERAGE1_IN_GAG(T38, s(s(T41)), T40)
AVERAGE1_IN_GAG(s(T54), s(s(T57)), s(T56)) → U3_GAG(T54, T57, T56, average1_in_gag(s(T54), T57, T56))
AVERAGE1_IN_GAG(s(T54), s(s(T57)), s(T56)) → AVERAGE1_IN_GAG(s(T54), T57, T56)
AVERAGE1_IN_GAG(s(T67), s(s(s(T70))), s(T69)) → U4_GAG(T67, T70, T69, average1_in_gag(s(s(T67)), T70, T69))
AVERAGE1_IN_GAG(s(T67), s(s(s(T70))), s(T69)) → AVERAGE1_IN_GAG(s(s(T67)), T70, T69)
AVERAGE1_IN_GAG(T95, s(s(s(T98))), s(T97)) → U5_GAG(T95, T98, T97, average1_in_gag(T95, s(T98), T97))
AVERAGE1_IN_GAG(T95, s(s(s(T98))), s(T97)) → AVERAGE1_IN_GAG(T95, s(T98), T97)
AVERAGE1_IN_GAG(T105, s(s(s(s(s(s(T108)))))), s(s(T107))) → U6_GAG(T105, T108, T107, average1_in_gag(s(s(T105)), T108, T107))
AVERAGE1_IN_GAG(T105, s(s(s(s(s(s(T108)))))), s(s(T107))) → AVERAGE1_IN_GAG(s(s(T105)), T108, T107)

R is empty.
The argument filtering Pi contains the following mapping:
average1_in_gag(x1, x2, x3)  =  average1_in_gag(x1, x3)
0  =  0
s(x1)  =  s(x1)
AVERAGE1_IN_GAG(x1, x2, x3)  =  AVERAGE1_IN_GAG(x1, x3)
U1_GAG(x1, x2)  =  U1_GAG(x2)
U2_GAG(x1, x2, x3, x4)  =  U2_GAG(x1, x3, x4)
U3_GAG(x1, x2, x3, x4)  =  U3_GAG(x1, x3, x4)
U4_GAG(x1, x2, x3, x4)  =  U4_GAG(x1, x3, x4)
U5_GAG(x1, x2, x3, x4)  =  U5_GAG(x1, x3, x4)
U6_GAG(x1, x2, x3, x4)  =  U6_GAG(x1, x3, x4)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

AVERAGE1_IN_GAG(0, s(s(s(T17))), s(0)) → U1_GAG(T17, average1_in_gag(0, s(T17), 0))
AVERAGE1_IN_GAG(0, s(s(s(T17))), s(0)) → AVERAGE1_IN_GAG(0, s(T17), 0)
AVERAGE1_IN_GAG(s(s(T38)), T41, T40) → U2_GAG(T38, T41, T40, average1_in_gag(T38, s(s(T41)), T40))
AVERAGE1_IN_GAG(s(s(T38)), T41, T40) → AVERAGE1_IN_GAG(T38, s(s(T41)), T40)
AVERAGE1_IN_GAG(s(T54), s(s(T57)), s(T56)) → U3_GAG(T54, T57, T56, average1_in_gag(s(T54), T57, T56))
AVERAGE1_IN_GAG(s(T54), s(s(T57)), s(T56)) → AVERAGE1_IN_GAG(s(T54), T57, T56)
AVERAGE1_IN_GAG(s(T67), s(s(s(T70))), s(T69)) → U4_GAG(T67, T70, T69, average1_in_gag(s(s(T67)), T70, T69))
AVERAGE1_IN_GAG(s(T67), s(s(s(T70))), s(T69)) → AVERAGE1_IN_GAG(s(s(T67)), T70, T69)
AVERAGE1_IN_GAG(T95, s(s(s(T98))), s(T97)) → U5_GAG(T95, T98, T97, average1_in_gag(T95, s(T98), T97))
AVERAGE1_IN_GAG(T95, s(s(s(T98))), s(T97)) → AVERAGE1_IN_GAG(T95, s(T98), T97)
AVERAGE1_IN_GAG(T105, s(s(s(s(s(s(T108)))))), s(s(T107))) → U6_GAG(T105, T108, T107, average1_in_gag(s(s(T105)), T108, T107))
AVERAGE1_IN_GAG(T105, s(s(s(s(s(s(T108)))))), s(s(T107))) → AVERAGE1_IN_GAG(s(s(T105)), T108, T107)

R is empty.
The argument filtering Pi contains the following mapping:
average1_in_gag(x1, x2, x3)  =  average1_in_gag(x1, x3)
0  =  0
s(x1)  =  s(x1)
AVERAGE1_IN_GAG(x1, x2, x3)  =  AVERAGE1_IN_GAG(x1, x3)
U1_GAG(x1, x2)  =  U1_GAG(x2)
U2_GAG(x1, x2, x3, x4)  =  U2_GAG(x1, x3, x4)
U3_GAG(x1, x2, x3, x4)  =  U3_GAG(x1, x3, x4)
U4_GAG(x1, x2, x3, x4)  =  U4_GAG(x1, x3, x4)
U5_GAG(x1, x2, x3, x4)  =  U5_GAG(x1, x3, x4)
U6_GAG(x1, x2, x3, x4)  =  U6_GAG(x1, x3, 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 7 less nodes.

(6) Obligation:

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

AVERAGE1_IN_GAG(s(T54), s(s(T57)), s(T56)) → AVERAGE1_IN_GAG(s(T54), T57, T56)
AVERAGE1_IN_GAG(s(s(T38)), T41, T40) → AVERAGE1_IN_GAG(T38, s(s(T41)), T40)
AVERAGE1_IN_GAG(s(T67), s(s(s(T70))), s(T69)) → AVERAGE1_IN_GAG(s(s(T67)), T70, T69)
AVERAGE1_IN_GAG(T95, s(s(s(T98))), s(T97)) → AVERAGE1_IN_GAG(T95, s(T98), T97)
AVERAGE1_IN_GAG(T105, s(s(s(s(s(s(T108)))))), s(s(T107))) → AVERAGE1_IN_GAG(s(s(T105)), T108, T107)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
AVERAGE1_IN_GAG(x1, x2, x3)  =  AVERAGE1_IN_GAG(x1, x3)

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

(7) PiDPToQDPProof (SOUND transformation)

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

(8) Obligation:

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

AVERAGE1_IN_GAG(s(T54), s(T56)) → AVERAGE1_IN_GAG(s(T54), T56)
AVERAGE1_IN_GAG(s(s(T38)), T40) → AVERAGE1_IN_GAG(T38, T40)
AVERAGE1_IN_GAG(s(T67), s(T69)) → AVERAGE1_IN_GAG(s(s(T67)), T69)
AVERAGE1_IN_GAG(T95, s(T97)) → AVERAGE1_IN_GAG(T95, T97)
AVERAGE1_IN_GAG(T105, s(s(T107))) → AVERAGE1_IN_GAG(s(s(T105)), T107)

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

(9) 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:

  • AVERAGE1_IN_GAG(s(T54), s(T56)) → AVERAGE1_IN_GAG(s(T54), T56)
    The graph contains the following edges 1 >= 1, 2 > 2

  • AVERAGE1_IN_GAG(s(s(T38)), T40) → AVERAGE1_IN_GAG(T38, T40)
    The graph contains the following edges 1 > 1, 2 >= 2

  • AVERAGE1_IN_GAG(s(T67), s(T69)) → AVERAGE1_IN_GAG(s(s(T67)), T69)
    The graph contains the following edges 2 > 2

  • AVERAGE1_IN_GAG(T95, s(T97)) → AVERAGE1_IN_GAG(T95, T97)
    The graph contains the following edges 1 >= 1, 2 > 2

  • AVERAGE1_IN_GAG(T105, s(s(T107))) → AVERAGE1_IN_GAG(s(s(T105)), T107)
    The graph contains the following edges 2 > 2

(10) YES