(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,a,g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

avg1(s(s(T23)), T26, T25) :- avg1(T23, s(s(T26)), T25).
avg1(s(T42), s(s(T45)), s(T44)) :- avg1(s(T42), T45, T44).
avg1(s(0), 0, 0).
avg1(s(0), s(0), s(0)).
avg1(s(T61), s(s(s(T64))), s(T63)) :- avg1(s(s(T61)), T64, T63).
avg1(T95, s(s(s(T98))), s(T97)) :- avg1(T95, s(T98), T97).
avg1(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) :- avg1(s(s(T114)), T117, T116).
avg1(0, s(s(0)), s(0)).
avg1(0, 0, 0).
avg1(0, s(0), 0).
avg1(0, s(0), 0).
avg1(0, s(s(0)), s(0)).

Queries:

avg1(g,a,g).

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

avg1_in_gag(s(s(T23)), T26, T25) → U1_gag(T23, T26, T25, avg1_in_gag(T23, s(s(T26)), T25))
avg1_in_gag(s(T42), s(s(T45)), s(T44)) → U2_gag(T42, T45, T44, avg1_in_gag(s(T42), T45, T44))
avg1_in_gag(s(0), 0, 0) → avg1_out_gag(s(0), 0, 0)
avg1_in_gag(s(0), s(0), s(0)) → avg1_out_gag(s(0), s(0), s(0))
avg1_in_gag(s(T61), s(s(s(T64))), s(T63)) → U3_gag(T61, T64, T63, avg1_in_gag(s(s(T61)), T64, T63))
avg1_in_gag(T95, s(s(s(T98))), s(T97)) → U4_gag(T95, T98, T97, avg1_in_gag(T95, s(T98), T97))
avg1_in_gag(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → U5_gag(T114, T117, T116, avg1_in_gag(s(s(T114)), T117, T116))
avg1_in_gag(0, s(s(0)), s(0)) → avg1_out_gag(0, s(s(0)), s(0))
avg1_in_gag(0, 0, 0) → avg1_out_gag(0, 0, 0)
avg1_in_gag(0, s(0), 0) → avg1_out_gag(0, s(0), 0)
U5_gag(T114, T117, T116, avg1_out_gag(s(s(T114)), T117, T116)) → avg1_out_gag(T114, s(s(s(s(s(s(T117)))))), s(s(T116)))
U4_gag(T95, T98, T97, avg1_out_gag(T95, s(T98), T97)) → avg1_out_gag(T95, s(s(s(T98))), s(T97))
U3_gag(T61, T64, T63, avg1_out_gag(s(s(T61)), T64, T63)) → avg1_out_gag(s(T61), s(s(s(T64))), s(T63))
U2_gag(T42, T45, T44, avg1_out_gag(s(T42), T45, T44)) → avg1_out_gag(s(T42), s(s(T45)), s(T44))
U1_gag(T23, T26, T25, avg1_out_gag(T23, s(s(T26)), T25)) → avg1_out_gag(s(s(T23)), T26, T25)

The argument filtering Pi contains the following mapping:
avg1_in_gag(x1, x2, x3)  =  avg1_in_gag(x1, x3)
s(x1)  =  s(x1)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x4)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
0  =  0
avg1_out_gag(x1, x2, x3)  =  avg1_out_gag(x2)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
U5_gag(x1, x2, x3, x4)  =  U5_gag(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_gag(s(s(T23)), T26, T25) → U1_gag(T23, T26, T25, avg1_in_gag(T23, s(s(T26)), T25))
avg1_in_gag(s(T42), s(s(T45)), s(T44)) → U2_gag(T42, T45, T44, avg1_in_gag(s(T42), T45, T44))
avg1_in_gag(s(0), 0, 0) → avg1_out_gag(s(0), 0, 0)
avg1_in_gag(s(0), s(0), s(0)) → avg1_out_gag(s(0), s(0), s(0))
avg1_in_gag(s(T61), s(s(s(T64))), s(T63)) → U3_gag(T61, T64, T63, avg1_in_gag(s(s(T61)), T64, T63))
avg1_in_gag(T95, s(s(s(T98))), s(T97)) → U4_gag(T95, T98, T97, avg1_in_gag(T95, s(T98), T97))
avg1_in_gag(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → U5_gag(T114, T117, T116, avg1_in_gag(s(s(T114)), T117, T116))
avg1_in_gag(0, s(s(0)), s(0)) → avg1_out_gag(0, s(s(0)), s(0))
avg1_in_gag(0, 0, 0) → avg1_out_gag(0, 0, 0)
avg1_in_gag(0, s(0), 0) → avg1_out_gag(0, s(0), 0)
U5_gag(T114, T117, T116, avg1_out_gag(s(s(T114)), T117, T116)) → avg1_out_gag(T114, s(s(s(s(s(s(T117)))))), s(s(T116)))
U4_gag(T95, T98, T97, avg1_out_gag(T95, s(T98), T97)) → avg1_out_gag(T95, s(s(s(T98))), s(T97))
U3_gag(T61, T64, T63, avg1_out_gag(s(s(T61)), T64, T63)) → avg1_out_gag(s(T61), s(s(s(T64))), s(T63))
U2_gag(T42, T45, T44, avg1_out_gag(s(T42), T45, T44)) → avg1_out_gag(s(T42), s(s(T45)), s(T44))
U1_gag(T23, T26, T25, avg1_out_gag(T23, s(s(T26)), T25)) → avg1_out_gag(s(s(T23)), T26, T25)

The argument filtering Pi contains the following mapping:
avg1_in_gag(x1, x2, x3)  =  avg1_in_gag(x1, x3)
s(x1)  =  s(x1)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x4)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
0  =  0
avg1_out_gag(x1, x2, x3)  =  avg1_out_gag(x2)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
U5_gag(x1, x2, x3, x4)  =  U5_gag(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_GAG(s(s(T23)), T26, T25) → U1_GAG(T23, T26, T25, avg1_in_gag(T23, s(s(T26)), T25))
AVG1_IN_GAG(s(s(T23)), T26, T25) → AVG1_IN_GAG(T23, s(s(T26)), T25)
AVG1_IN_GAG(s(T42), s(s(T45)), s(T44)) → U2_GAG(T42, T45, T44, avg1_in_gag(s(T42), T45, T44))
AVG1_IN_GAG(s(T42), s(s(T45)), s(T44)) → AVG1_IN_GAG(s(T42), T45, T44)
AVG1_IN_GAG(s(T61), s(s(s(T64))), s(T63)) → U3_GAG(T61, T64, T63, avg1_in_gag(s(s(T61)), T64, T63))
AVG1_IN_GAG(s(T61), s(s(s(T64))), s(T63)) → AVG1_IN_GAG(s(s(T61)), T64, T63)
AVG1_IN_GAG(T95, s(s(s(T98))), s(T97)) → U4_GAG(T95, T98, T97, avg1_in_gag(T95, s(T98), T97))
AVG1_IN_GAG(T95, s(s(s(T98))), s(T97)) → AVG1_IN_GAG(T95, s(T98), T97)
AVG1_IN_GAG(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → U5_GAG(T114, T117, T116, avg1_in_gag(s(s(T114)), T117, T116))
AVG1_IN_GAG(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → AVG1_IN_GAG(s(s(T114)), T117, T116)

The TRS R consists of the following rules:

avg1_in_gag(s(s(T23)), T26, T25) → U1_gag(T23, T26, T25, avg1_in_gag(T23, s(s(T26)), T25))
avg1_in_gag(s(T42), s(s(T45)), s(T44)) → U2_gag(T42, T45, T44, avg1_in_gag(s(T42), T45, T44))
avg1_in_gag(s(0), 0, 0) → avg1_out_gag(s(0), 0, 0)
avg1_in_gag(s(0), s(0), s(0)) → avg1_out_gag(s(0), s(0), s(0))
avg1_in_gag(s(T61), s(s(s(T64))), s(T63)) → U3_gag(T61, T64, T63, avg1_in_gag(s(s(T61)), T64, T63))
avg1_in_gag(T95, s(s(s(T98))), s(T97)) → U4_gag(T95, T98, T97, avg1_in_gag(T95, s(T98), T97))
avg1_in_gag(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → U5_gag(T114, T117, T116, avg1_in_gag(s(s(T114)), T117, T116))
avg1_in_gag(0, s(s(0)), s(0)) → avg1_out_gag(0, s(s(0)), s(0))
avg1_in_gag(0, 0, 0) → avg1_out_gag(0, 0, 0)
avg1_in_gag(0, s(0), 0) → avg1_out_gag(0, s(0), 0)
U5_gag(T114, T117, T116, avg1_out_gag(s(s(T114)), T117, T116)) → avg1_out_gag(T114, s(s(s(s(s(s(T117)))))), s(s(T116)))
U4_gag(T95, T98, T97, avg1_out_gag(T95, s(T98), T97)) → avg1_out_gag(T95, s(s(s(T98))), s(T97))
U3_gag(T61, T64, T63, avg1_out_gag(s(s(T61)), T64, T63)) → avg1_out_gag(s(T61), s(s(s(T64))), s(T63))
U2_gag(T42, T45, T44, avg1_out_gag(s(T42), T45, T44)) → avg1_out_gag(s(T42), s(s(T45)), s(T44))
U1_gag(T23, T26, T25, avg1_out_gag(T23, s(s(T26)), T25)) → avg1_out_gag(s(s(T23)), T26, T25)

The argument filtering Pi contains the following mapping:
avg1_in_gag(x1, x2, x3)  =  avg1_in_gag(x1, x3)
s(x1)  =  s(x1)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x4)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
0  =  0
avg1_out_gag(x1, x2, x3)  =  avg1_out_gag(x2)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
U5_gag(x1, x2, x3, x4)  =  U5_gag(x4)
AVG1_IN_GAG(x1, x2, x3)  =  AVG1_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4)  =  U1_GAG(x4)
U2_GAG(x1, x2, x3, x4)  =  U2_GAG(x4)
U3_GAG(x1, x2, x3, x4)  =  U3_GAG(x4)
U4_GAG(x1, x2, x3, x4)  =  U4_GAG(x4)
U5_GAG(x1, x2, x3, x4)  =  U5_GAG(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_GAG(s(s(T23)), T26, T25) → U1_GAG(T23, T26, T25, avg1_in_gag(T23, s(s(T26)), T25))
AVG1_IN_GAG(s(s(T23)), T26, T25) → AVG1_IN_GAG(T23, s(s(T26)), T25)
AVG1_IN_GAG(s(T42), s(s(T45)), s(T44)) → U2_GAG(T42, T45, T44, avg1_in_gag(s(T42), T45, T44))
AVG1_IN_GAG(s(T42), s(s(T45)), s(T44)) → AVG1_IN_GAG(s(T42), T45, T44)
AVG1_IN_GAG(s(T61), s(s(s(T64))), s(T63)) → U3_GAG(T61, T64, T63, avg1_in_gag(s(s(T61)), T64, T63))
AVG1_IN_GAG(s(T61), s(s(s(T64))), s(T63)) → AVG1_IN_GAG(s(s(T61)), T64, T63)
AVG1_IN_GAG(T95, s(s(s(T98))), s(T97)) → U4_GAG(T95, T98, T97, avg1_in_gag(T95, s(T98), T97))
AVG1_IN_GAG(T95, s(s(s(T98))), s(T97)) → AVG1_IN_GAG(T95, s(T98), T97)
AVG1_IN_GAG(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → U5_GAG(T114, T117, T116, avg1_in_gag(s(s(T114)), T117, T116))
AVG1_IN_GAG(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → AVG1_IN_GAG(s(s(T114)), T117, T116)

The TRS R consists of the following rules:

avg1_in_gag(s(s(T23)), T26, T25) → U1_gag(T23, T26, T25, avg1_in_gag(T23, s(s(T26)), T25))
avg1_in_gag(s(T42), s(s(T45)), s(T44)) → U2_gag(T42, T45, T44, avg1_in_gag(s(T42), T45, T44))
avg1_in_gag(s(0), 0, 0) → avg1_out_gag(s(0), 0, 0)
avg1_in_gag(s(0), s(0), s(0)) → avg1_out_gag(s(0), s(0), s(0))
avg1_in_gag(s(T61), s(s(s(T64))), s(T63)) → U3_gag(T61, T64, T63, avg1_in_gag(s(s(T61)), T64, T63))
avg1_in_gag(T95, s(s(s(T98))), s(T97)) → U4_gag(T95, T98, T97, avg1_in_gag(T95, s(T98), T97))
avg1_in_gag(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → U5_gag(T114, T117, T116, avg1_in_gag(s(s(T114)), T117, T116))
avg1_in_gag(0, s(s(0)), s(0)) → avg1_out_gag(0, s(s(0)), s(0))
avg1_in_gag(0, 0, 0) → avg1_out_gag(0, 0, 0)
avg1_in_gag(0, s(0), 0) → avg1_out_gag(0, s(0), 0)
U5_gag(T114, T117, T116, avg1_out_gag(s(s(T114)), T117, T116)) → avg1_out_gag(T114, s(s(s(s(s(s(T117)))))), s(s(T116)))
U4_gag(T95, T98, T97, avg1_out_gag(T95, s(T98), T97)) → avg1_out_gag(T95, s(s(s(T98))), s(T97))
U3_gag(T61, T64, T63, avg1_out_gag(s(s(T61)), T64, T63)) → avg1_out_gag(s(T61), s(s(s(T64))), s(T63))
U2_gag(T42, T45, T44, avg1_out_gag(s(T42), T45, T44)) → avg1_out_gag(s(T42), s(s(T45)), s(T44))
U1_gag(T23, T26, T25, avg1_out_gag(T23, s(s(T26)), T25)) → avg1_out_gag(s(s(T23)), T26, T25)

The argument filtering Pi contains the following mapping:
avg1_in_gag(x1, x2, x3)  =  avg1_in_gag(x1, x3)
s(x1)  =  s(x1)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x4)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
0  =  0
avg1_out_gag(x1, x2, x3)  =  avg1_out_gag(x2)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
U5_gag(x1, x2, x3, x4)  =  U5_gag(x4)
AVG1_IN_GAG(x1, x2, x3)  =  AVG1_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4)  =  U1_GAG(x4)
U2_GAG(x1, x2, x3, x4)  =  U2_GAG(x4)
U3_GAG(x1, x2, x3, x4)  =  U3_GAG(x4)
U4_GAG(x1, x2, x3, x4)  =  U4_GAG(x4)
U5_GAG(x1, x2, x3, x4)  =  U5_GAG(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_GAG(s(T42), s(s(T45)), s(T44)) → AVG1_IN_GAG(s(T42), T45, T44)
AVG1_IN_GAG(s(s(T23)), T26, T25) → AVG1_IN_GAG(T23, s(s(T26)), T25)
AVG1_IN_GAG(s(T61), s(s(s(T64))), s(T63)) → AVG1_IN_GAG(s(s(T61)), T64, T63)
AVG1_IN_GAG(T95, s(s(s(T98))), s(T97)) → AVG1_IN_GAG(T95, s(T98), T97)
AVG1_IN_GAG(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → AVG1_IN_GAG(s(s(T114)), T117, T116)

The TRS R consists of the following rules:

avg1_in_gag(s(s(T23)), T26, T25) → U1_gag(T23, T26, T25, avg1_in_gag(T23, s(s(T26)), T25))
avg1_in_gag(s(T42), s(s(T45)), s(T44)) → U2_gag(T42, T45, T44, avg1_in_gag(s(T42), T45, T44))
avg1_in_gag(s(0), 0, 0) → avg1_out_gag(s(0), 0, 0)
avg1_in_gag(s(0), s(0), s(0)) → avg1_out_gag(s(0), s(0), s(0))
avg1_in_gag(s(T61), s(s(s(T64))), s(T63)) → U3_gag(T61, T64, T63, avg1_in_gag(s(s(T61)), T64, T63))
avg1_in_gag(T95, s(s(s(T98))), s(T97)) → U4_gag(T95, T98, T97, avg1_in_gag(T95, s(T98), T97))
avg1_in_gag(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → U5_gag(T114, T117, T116, avg1_in_gag(s(s(T114)), T117, T116))
avg1_in_gag(0, s(s(0)), s(0)) → avg1_out_gag(0, s(s(0)), s(0))
avg1_in_gag(0, 0, 0) → avg1_out_gag(0, 0, 0)
avg1_in_gag(0, s(0), 0) → avg1_out_gag(0, s(0), 0)
U5_gag(T114, T117, T116, avg1_out_gag(s(s(T114)), T117, T116)) → avg1_out_gag(T114, s(s(s(s(s(s(T117)))))), s(s(T116)))
U4_gag(T95, T98, T97, avg1_out_gag(T95, s(T98), T97)) → avg1_out_gag(T95, s(s(s(T98))), s(T97))
U3_gag(T61, T64, T63, avg1_out_gag(s(s(T61)), T64, T63)) → avg1_out_gag(s(T61), s(s(s(T64))), s(T63))
U2_gag(T42, T45, T44, avg1_out_gag(s(T42), T45, T44)) → avg1_out_gag(s(T42), s(s(T45)), s(T44))
U1_gag(T23, T26, T25, avg1_out_gag(T23, s(s(T26)), T25)) → avg1_out_gag(s(s(T23)), T26, T25)

The argument filtering Pi contains the following mapping:
avg1_in_gag(x1, x2, x3)  =  avg1_in_gag(x1, x3)
s(x1)  =  s(x1)
U1_gag(x1, x2, x3, x4)  =  U1_gag(x4)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
0  =  0
avg1_out_gag(x1, x2, x3)  =  avg1_out_gag(x2)
U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
U5_gag(x1, x2, x3, x4)  =  U5_gag(x4)
AVG1_IN_GAG(x1, x2, x3)  =  AVG1_IN_GAG(x1, x3)

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_GAG(s(T42), s(s(T45)), s(T44)) → AVG1_IN_GAG(s(T42), T45, T44)
AVG1_IN_GAG(s(s(T23)), T26, T25) → AVG1_IN_GAG(T23, s(s(T26)), T25)
AVG1_IN_GAG(s(T61), s(s(s(T64))), s(T63)) → AVG1_IN_GAG(s(s(T61)), T64, T63)
AVG1_IN_GAG(T95, s(s(s(T98))), s(T97)) → AVG1_IN_GAG(T95, s(T98), T97)
AVG1_IN_GAG(T114, s(s(s(s(s(s(T117)))))), s(s(T116))) → AVG1_IN_GAG(s(s(T114)), T117, T116)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
AVG1_IN_GAG(x1, x2, x3)  =  AVG1_IN_GAG(x1, x3)

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_GAG(s(T42), s(T44)) → AVG1_IN_GAG(s(T42), T44)
AVG1_IN_GAG(s(s(T23)), T25) → AVG1_IN_GAG(T23, T25)
AVG1_IN_GAG(s(T61), s(T63)) → AVG1_IN_GAG(s(s(T61)), T63)
AVG1_IN_GAG(T95, s(T97)) → AVG1_IN_GAG(T95, T97)
AVG1_IN_GAG(T114, s(s(T116))) → AVG1_IN_GAG(s(s(T114)), T116)

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

(13) 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_GAG(s(T42), s(T44)) → AVG1_IN_GAG(s(T42), T44)
    The graph contains the following edges 1 >= 1, 2 > 2

  • AVG1_IN_GAG(s(s(T23)), T25) → AVG1_IN_GAG(T23, T25)
    The graph contains the following edges 1 > 1, 2 >= 2

  • AVG1_IN_GAG(s(T61), s(T63)) → AVG1_IN_GAG(s(s(T61)), T63)
    The graph contains the following edges 2 > 2

  • AVG1_IN_GAG(T95, s(T97)) → AVG1_IN_GAG(T95, T97)
    The graph contains the following edges 1 >= 1, 2 > 2

  • AVG1_IN_GAG(T114, s(s(T116))) → AVG1_IN_GAG(s(s(T114)), T116)
    The graph contains the following edges 2 > 2

(14) YES