(0) Obligation:

Clauses:

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

Queries:

avg(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

avg1(s(s(T23)), T24, T26) :- avg1(T23, s(s(T24)), T26).
avg1(s(T42), s(s(T43)), s(T45)) :- avg1(s(T42), T43, T45).
avg1(s(0), 0, 0).
avg1(s(0), s(0), s(0)).
avg1(s(T61), s(s(s(T62))), s(T64)) :- avg1(s(s(T61)), T62, T64).
avg1(T95, s(s(s(T96))), s(T98)) :- avg1(T95, s(T96), T98).
avg1(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) :- avg1(s(s(T114)), T115, T117).
avg1(0, 0, 0).
avg1(0, s(0), 0).
avg1(0, s(s(0)), s(0)).

Queries:

avg1(g,g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
avg1_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

avg1_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, avg1_in_gga(T23, s(s(T24)), T26))
avg1_in_gga(s(T42), s(s(T43)), s(T45)) → U2_gga(T42, T43, T45, avg1_in_gga(s(T42), T43, T45))
avg1_in_gga(s(0), 0, 0) → avg1_out_gga(s(0), 0, 0)
avg1_in_gga(s(0), s(0), s(0)) → avg1_out_gga(s(0), s(0), s(0))
avg1_in_gga(s(T61), s(s(s(T62))), s(T64)) → U3_gga(T61, T62, T64, avg1_in_gga(s(s(T61)), T62, T64))
avg1_in_gga(T95, s(s(s(T96))), s(T98)) → U4_gga(T95, T96, T98, avg1_in_gga(T95, s(T96), T98))
avg1_in_gga(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → U5_gga(T114, T115, T117, avg1_in_gga(s(s(T114)), T115, T117))
avg1_in_gga(0, 0, 0) → avg1_out_gga(0, 0, 0)
avg1_in_gga(0, s(0), 0) → avg1_out_gga(0, s(0), 0)
avg1_in_gga(0, s(s(0)), s(0)) → avg1_out_gga(0, s(s(0)), s(0))
U5_gga(T114, T115, T117, avg1_out_gga(s(s(T114)), T115, T117)) → avg1_out_gga(T114, s(s(s(s(s(s(T115)))))), s(s(T117)))
U4_gga(T95, T96, T98, avg1_out_gga(T95, s(T96), T98)) → avg1_out_gga(T95, s(s(s(T96))), s(T98))
U3_gga(T61, T62, T64, avg1_out_gga(s(s(T61)), T62, T64)) → avg1_out_gga(s(T61), s(s(s(T62))), s(T64))
U2_gga(T42, T43, T45, avg1_out_gga(s(T42), T43, T45)) → avg1_out_gga(s(T42), s(s(T43)), s(T45))
U1_gga(T23, T24, T26, avg1_out_gga(T23, s(s(T24)), T26)) → avg1_out_gga(s(s(T23)), T24, T26)

The argument filtering Pi contains the following mapping:
avg1_in_gga(x1, x2, x3)  =  avg1_in_gga(x1, x2)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
0  =  0
avg1_out_gga(x1, x2, x3)  =  avg1_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

avg1_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, avg1_in_gga(T23, s(s(T24)), T26))
avg1_in_gga(s(T42), s(s(T43)), s(T45)) → U2_gga(T42, T43, T45, avg1_in_gga(s(T42), T43, T45))
avg1_in_gga(s(0), 0, 0) → avg1_out_gga(s(0), 0, 0)
avg1_in_gga(s(0), s(0), s(0)) → avg1_out_gga(s(0), s(0), s(0))
avg1_in_gga(s(T61), s(s(s(T62))), s(T64)) → U3_gga(T61, T62, T64, avg1_in_gga(s(s(T61)), T62, T64))
avg1_in_gga(T95, s(s(s(T96))), s(T98)) → U4_gga(T95, T96, T98, avg1_in_gga(T95, s(T96), T98))
avg1_in_gga(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → U5_gga(T114, T115, T117, avg1_in_gga(s(s(T114)), T115, T117))
avg1_in_gga(0, 0, 0) → avg1_out_gga(0, 0, 0)
avg1_in_gga(0, s(0), 0) → avg1_out_gga(0, s(0), 0)
avg1_in_gga(0, s(s(0)), s(0)) → avg1_out_gga(0, s(s(0)), s(0))
U5_gga(T114, T115, T117, avg1_out_gga(s(s(T114)), T115, T117)) → avg1_out_gga(T114, s(s(s(s(s(s(T115)))))), s(s(T117)))
U4_gga(T95, T96, T98, avg1_out_gga(T95, s(T96), T98)) → avg1_out_gga(T95, s(s(s(T96))), s(T98))
U3_gga(T61, T62, T64, avg1_out_gga(s(s(T61)), T62, T64)) → avg1_out_gga(s(T61), s(s(s(T62))), s(T64))
U2_gga(T42, T43, T45, avg1_out_gga(s(T42), T43, T45)) → avg1_out_gga(s(T42), s(s(T43)), s(T45))
U1_gga(T23, T24, T26, avg1_out_gga(T23, s(s(T24)), T26)) → avg1_out_gga(s(s(T23)), T24, T26)

The argument filtering Pi contains the following mapping:
avg1_in_gga(x1, x2, x3)  =  avg1_in_gga(x1, x2)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
0  =  0
avg1_out_gga(x1, x2, x3)  =  avg1_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)

(5) 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:

AVG1_IN_GGA(s(s(T23)), T24, T26) → U1_GGA(T23, T24, T26, avg1_in_gga(T23, s(s(T24)), T26))
AVG1_IN_GGA(s(s(T23)), T24, T26) → AVG1_IN_GGA(T23, s(s(T24)), T26)
AVG1_IN_GGA(s(T42), s(s(T43)), s(T45)) → U2_GGA(T42, T43, T45, avg1_in_gga(s(T42), T43, T45))
AVG1_IN_GGA(s(T42), s(s(T43)), s(T45)) → AVG1_IN_GGA(s(T42), T43, T45)
AVG1_IN_GGA(s(T61), s(s(s(T62))), s(T64)) → U3_GGA(T61, T62, T64, avg1_in_gga(s(s(T61)), T62, T64))
AVG1_IN_GGA(s(T61), s(s(s(T62))), s(T64)) → AVG1_IN_GGA(s(s(T61)), T62, T64)
AVG1_IN_GGA(T95, s(s(s(T96))), s(T98)) → U4_GGA(T95, T96, T98, avg1_in_gga(T95, s(T96), T98))
AVG1_IN_GGA(T95, s(s(s(T96))), s(T98)) → AVG1_IN_GGA(T95, s(T96), T98)
AVG1_IN_GGA(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → U5_GGA(T114, T115, T117, avg1_in_gga(s(s(T114)), T115, T117))
AVG1_IN_GGA(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → AVG1_IN_GGA(s(s(T114)), T115, T117)

The TRS R consists of the following rules:

avg1_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, avg1_in_gga(T23, s(s(T24)), T26))
avg1_in_gga(s(T42), s(s(T43)), s(T45)) → U2_gga(T42, T43, T45, avg1_in_gga(s(T42), T43, T45))
avg1_in_gga(s(0), 0, 0) → avg1_out_gga(s(0), 0, 0)
avg1_in_gga(s(0), s(0), s(0)) → avg1_out_gga(s(0), s(0), s(0))
avg1_in_gga(s(T61), s(s(s(T62))), s(T64)) → U3_gga(T61, T62, T64, avg1_in_gga(s(s(T61)), T62, T64))
avg1_in_gga(T95, s(s(s(T96))), s(T98)) → U4_gga(T95, T96, T98, avg1_in_gga(T95, s(T96), T98))
avg1_in_gga(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → U5_gga(T114, T115, T117, avg1_in_gga(s(s(T114)), T115, T117))
avg1_in_gga(0, 0, 0) → avg1_out_gga(0, 0, 0)
avg1_in_gga(0, s(0), 0) → avg1_out_gga(0, s(0), 0)
avg1_in_gga(0, s(s(0)), s(0)) → avg1_out_gga(0, s(s(0)), s(0))
U5_gga(T114, T115, T117, avg1_out_gga(s(s(T114)), T115, T117)) → avg1_out_gga(T114, s(s(s(s(s(s(T115)))))), s(s(T117)))
U4_gga(T95, T96, T98, avg1_out_gga(T95, s(T96), T98)) → avg1_out_gga(T95, s(s(s(T96))), s(T98))
U3_gga(T61, T62, T64, avg1_out_gga(s(s(T61)), T62, T64)) → avg1_out_gga(s(T61), s(s(s(T62))), s(T64))
U2_gga(T42, T43, T45, avg1_out_gga(s(T42), T43, T45)) → avg1_out_gga(s(T42), s(s(T43)), s(T45))
U1_gga(T23, T24, T26, avg1_out_gga(T23, s(s(T24)), T26)) → avg1_out_gga(s(s(T23)), T24, T26)

The argument filtering Pi contains the following mapping:
avg1_in_gga(x1, x2, x3)  =  avg1_in_gga(x1, x2)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
0  =  0
avg1_out_gga(x1, x2, x3)  =  avg1_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
AVG1_IN_GGA(x1, x2, x3)  =  AVG1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)

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

(6) Obligation:

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

AVG1_IN_GGA(s(s(T23)), T24, T26) → U1_GGA(T23, T24, T26, avg1_in_gga(T23, s(s(T24)), T26))
AVG1_IN_GGA(s(s(T23)), T24, T26) → AVG1_IN_GGA(T23, s(s(T24)), T26)
AVG1_IN_GGA(s(T42), s(s(T43)), s(T45)) → U2_GGA(T42, T43, T45, avg1_in_gga(s(T42), T43, T45))
AVG1_IN_GGA(s(T42), s(s(T43)), s(T45)) → AVG1_IN_GGA(s(T42), T43, T45)
AVG1_IN_GGA(s(T61), s(s(s(T62))), s(T64)) → U3_GGA(T61, T62, T64, avg1_in_gga(s(s(T61)), T62, T64))
AVG1_IN_GGA(s(T61), s(s(s(T62))), s(T64)) → AVG1_IN_GGA(s(s(T61)), T62, T64)
AVG1_IN_GGA(T95, s(s(s(T96))), s(T98)) → U4_GGA(T95, T96, T98, avg1_in_gga(T95, s(T96), T98))
AVG1_IN_GGA(T95, s(s(s(T96))), s(T98)) → AVG1_IN_GGA(T95, s(T96), T98)
AVG1_IN_GGA(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → U5_GGA(T114, T115, T117, avg1_in_gga(s(s(T114)), T115, T117))
AVG1_IN_GGA(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → AVG1_IN_GGA(s(s(T114)), T115, T117)

The TRS R consists of the following rules:

avg1_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, avg1_in_gga(T23, s(s(T24)), T26))
avg1_in_gga(s(T42), s(s(T43)), s(T45)) → U2_gga(T42, T43, T45, avg1_in_gga(s(T42), T43, T45))
avg1_in_gga(s(0), 0, 0) → avg1_out_gga(s(0), 0, 0)
avg1_in_gga(s(0), s(0), s(0)) → avg1_out_gga(s(0), s(0), s(0))
avg1_in_gga(s(T61), s(s(s(T62))), s(T64)) → U3_gga(T61, T62, T64, avg1_in_gga(s(s(T61)), T62, T64))
avg1_in_gga(T95, s(s(s(T96))), s(T98)) → U4_gga(T95, T96, T98, avg1_in_gga(T95, s(T96), T98))
avg1_in_gga(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → U5_gga(T114, T115, T117, avg1_in_gga(s(s(T114)), T115, T117))
avg1_in_gga(0, 0, 0) → avg1_out_gga(0, 0, 0)
avg1_in_gga(0, s(0), 0) → avg1_out_gga(0, s(0), 0)
avg1_in_gga(0, s(s(0)), s(0)) → avg1_out_gga(0, s(s(0)), s(0))
U5_gga(T114, T115, T117, avg1_out_gga(s(s(T114)), T115, T117)) → avg1_out_gga(T114, s(s(s(s(s(s(T115)))))), s(s(T117)))
U4_gga(T95, T96, T98, avg1_out_gga(T95, s(T96), T98)) → avg1_out_gga(T95, s(s(s(T96))), s(T98))
U3_gga(T61, T62, T64, avg1_out_gga(s(s(T61)), T62, T64)) → avg1_out_gga(s(T61), s(s(s(T62))), s(T64))
U2_gga(T42, T43, T45, avg1_out_gga(s(T42), T43, T45)) → avg1_out_gga(s(T42), s(s(T43)), s(T45))
U1_gga(T23, T24, T26, avg1_out_gga(T23, s(s(T24)), T26)) → avg1_out_gga(s(s(T23)), T24, T26)

The argument filtering Pi contains the following mapping:
avg1_in_gga(x1, x2, x3)  =  avg1_in_gga(x1, x2)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
0  =  0
avg1_out_gga(x1, x2, x3)  =  avg1_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
AVG1_IN_GGA(x1, x2, x3)  =  AVG1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 5 less nodes.

(8) Obligation:

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

AVG1_IN_GGA(s(T42), s(s(T43)), s(T45)) → AVG1_IN_GGA(s(T42), T43, T45)
AVG1_IN_GGA(s(s(T23)), T24, T26) → AVG1_IN_GGA(T23, s(s(T24)), T26)
AVG1_IN_GGA(s(T61), s(s(s(T62))), s(T64)) → AVG1_IN_GGA(s(s(T61)), T62, T64)
AVG1_IN_GGA(T95, s(s(s(T96))), s(T98)) → AVG1_IN_GGA(T95, s(T96), T98)
AVG1_IN_GGA(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → AVG1_IN_GGA(s(s(T114)), T115, T117)

The TRS R consists of the following rules:

avg1_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, avg1_in_gga(T23, s(s(T24)), T26))
avg1_in_gga(s(T42), s(s(T43)), s(T45)) → U2_gga(T42, T43, T45, avg1_in_gga(s(T42), T43, T45))
avg1_in_gga(s(0), 0, 0) → avg1_out_gga(s(0), 0, 0)
avg1_in_gga(s(0), s(0), s(0)) → avg1_out_gga(s(0), s(0), s(0))
avg1_in_gga(s(T61), s(s(s(T62))), s(T64)) → U3_gga(T61, T62, T64, avg1_in_gga(s(s(T61)), T62, T64))
avg1_in_gga(T95, s(s(s(T96))), s(T98)) → U4_gga(T95, T96, T98, avg1_in_gga(T95, s(T96), T98))
avg1_in_gga(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → U5_gga(T114, T115, T117, avg1_in_gga(s(s(T114)), T115, T117))
avg1_in_gga(0, 0, 0) → avg1_out_gga(0, 0, 0)
avg1_in_gga(0, s(0), 0) → avg1_out_gga(0, s(0), 0)
avg1_in_gga(0, s(s(0)), s(0)) → avg1_out_gga(0, s(s(0)), s(0))
U5_gga(T114, T115, T117, avg1_out_gga(s(s(T114)), T115, T117)) → avg1_out_gga(T114, s(s(s(s(s(s(T115)))))), s(s(T117)))
U4_gga(T95, T96, T98, avg1_out_gga(T95, s(T96), T98)) → avg1_out_gga(T95, s(s(s(T96))), s(T98))
U3_gga(T61, T62, T64, avg1_out_gga(s(s(T61)), T62, T64)) → avg1_out_gga(s(T61), s(s(s(T62))), s(T64))
U2_gga(T42, T43, T45, avg1_out_gga(s(T42), T43, T45)) → avg1_out_gga(s(T42), s(s(T43)), s(T45))
U1_gga(T23, T24, T26, avg1_out_gga(T23, s(s(T24)), T26)) → avg1_out_gga(s(s(T23)), T24, T26)

The argument filtering Pi contains the following mapping:
avg1_in_gga(x1, x2, x3)  =  avg1_in_gga(x1, x2)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
0  =  0
avg1_out_gga(x1, x2, x3)  =  avg1_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
AVG1_IN_GGA(x1, x2, x3)  =  AVG1_IN_GGA(x1, x2)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

AVG1_IN_GGA(s(T42), s(s(T43)), s(T45)) → AVG1_IN_GGA(s(T42), T43, T45)
AVG1_IN_GGA(s(s(T23)), T24, T26) → AVG1_IN_GGA(T23, s(s(T24)), T26)
AVG1_IN_GGA(s(T61), s(s(s(T62))), s(T64)) → AVG1_IN_GGA(s(s(T61)), T62, T64)
AVG1_IN_GGA(T95, s(s(s(T96))), s(T98)) → AVG1_IN_GGA(T95, s(T96), T98)
AVG1_IN_GGA(T114, s(s(s(s(s(s(T115)))))), s(s(T117))) → AVG1_IN_GGA(s(s(T114)), T115, T117)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
AVG1_IN_GGA(x1, x2, x3)  =  AVG1_IN_GGA(x1, x2)

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

AVG1_IN_GGA(s(T42), s(s(T43))) → AVG1_IN_GGA(s(T42), T43)
AVG1_IN_GGA(s(s(T23)), T24) → AVG1_IN_GGA(T23, s(s(T24)))
AVG1_IN_GGA(s(T61), s(s(s(T62)))) → AVG1_IN_GGA(s(s(T61)), T62)
AVG1_IN_GGA(T95, s(s(s(T96)))) → AVG1_IN_GGA(T95, s(T96))
AVG1_IN_GGA(T114, s(s(s(s(s(s(T115))))))) → AVG1_IN_GGA(s(s(T114)), T115)

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

(13) 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:

AVG1_IN_GGA(s(T42), s(s(T43))) → AVG1_IN_GGA(s(T42), T43)
AVG1_IN_GGA(s(T61), s(s(s(T62)))) → AVG1_IN_GGA(s(s(T61)), T62)
AVG1_IN_GGA(T95, s(s(s(T96)))) → AVG1_IN_GGA(T95, s(T96))
AVG1_IN_GGA(T114, s(s(s(s(s(s(T115))))))) → AVG1_IN_GGA(s(s(T114)), T115)


Used ordering: Polynomial interpretation [POLO]:

POL(AVG1_IN_GGA(x1, x2)) = x1 + x2   
POL(s(x1)) = 1 + x1   

(14) Obligation:

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

AVG1_IN_GGA(s(s(T23)), T24) → AVG1_IN_GGA(T23, s(s(T24)))

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

(15) 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:

  • AVG1_IN_GGA(s(s(T23)), T24) → AVG1_IN_GGA(T23, s(s(T24)))
    The graph contains the following edges 1 > 1

(16) YES