(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).
Query: average(g,g,a)
(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)
Transformed Prolog program to (Pi-)TRS.
(2) Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:
averageA_in_gga(0, 0, 0) → averageA_out_gga(0, 0, 0)
averageA_in_gga(0, s(0), 0) → averageA_out_gga(0, s(0), 0)
averageA_in_gga(0, s(s(0)), s(0)) → averageA_out_gga(0, s(s(0)), s(0))
averageA_in_gga(s(0), 0, 0) → averageA_out_gga(s(0), 0, 0)
averageA_in_gga(s(0), s(0), s(0)) → averageA_out_gga(s(0), s(0), s(0))
averageA_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, averageA_in_gga(T23, s(s(T24)), T26))
averageA_in_gga(s(T39), s(s(T40)), s(T42)) → U2_gga(T39, T40, T42, averageA_in_gga(s(T39), T40, T42))
averageA_in_gga(s(T49), s(s(s(T50))), s(T52)) → U3_gga(T49, T50, T52, averageA_in_gga(s(s(T49)), T50, T52))
averageA_in_gga(T74, s(s(s(T75))), s(T77)) → U4_gga(T74, T75, T77, averageA_in_gga(T74, s(T75), T77))
averageA_in_gga(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) → U5_gga(T84, T85, T87, averageA_in_gga(s(s(T84)), T85, T87))
U5_gga(T84, T85, T87, averageA_out_gga(s(s(T84)), T85, T87)) → averageA_out_gga(T84, s(s(s(s(s(s(T85)))))), s(s(T87)))
U4_gga(T74, T75, T77, averageA_out_gga(T74, s(T75), T77)) → averageA_out_gga(T74, s(s(s(T75))), s(T77))
U3_gga(T49, T50, T52, averageA_out_gga(s(s(T49)), T50, T52)) → averageA_out_gga(s(T49), s(s(s(T50))), s(T52))
U2_gga(T39, T40, T42, averageA_out_gga(s(T39), T40, T42)) → averageA_out_gga(s(T39), s(s(T40)), s(T42))
U1_gga(T23, T24, T26, averageA_out_gga(T23, s(s(T24)), T26)) → averageA_out_gga(s(s(T23)), T24, T26)
The argument filtering Pi contains the following mapping:
averageA_in_gga(
x1,
x2,
x3) =
averageA_in_gga(
x1,
x2)
0 =
0
averageA_out_gga(
x1,
x2,
x3) =
averageA_out_gga(
x1,
x2,
x3)
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)
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)
(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:
AVERAGEA_IN_GGA(s(s(T23)), T24, T26) → U1_GGA(T23, T24, T26, averageA_in_gga(T23, s(s(T24)), T26))
AVERAGEA_IN_GGA(s(s(T23)), T24, T26) → AVERAGEA_IN_GGA(T23, s(s(T24)), T26)
AVERAGEA_IN_GGA(s(T39), s(s(T40)), s(T42)) → U2_GGA(T39, T40, T42, averageA_in_gga(s(T39), T40, T42))
AVERAGEA_IN_GGA(s(T39), s(s(T40)), s(T42)) → AVERAGEA_IN_GGA(s(T39), T40, T42)
AVERAGEA_IN_GGA(s(T49), s(s(s(T50))), s(T52)) → U3_GGA(T49, T50, T52, averageA_in_gga(s(s(T49)), T50, T52))
AVERAGEA_IN_GGA(s(T49), s(s(s(T50))), s(T52)) → AVERAGEA_IN_GGA(s(s(T49)), T50, T52)
AVERAGEA_IN_GGA(T74, s(s(s(T75))), s(T77)) → U4_GGA(T74, T75, T77, averageA_in_gga(T74, s(T75), T77))
AVERAGEA_IN_GGA(T74, s(s(s(T75))), s(T77)) → AVERAGEA_IN_GGA(T74, s(T75), T77)
AVERAGEA_IN_GGA(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) → U5_GGA(T84, T85, T87, averageA_in_gga(s(s(T84)), T85, T87))
AVERAGEA_IN_GGA(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) → AVERAGEA_IN_GGA(s(s(T84)), T85, T87)
The TRS R consists of the following rules:
averageA_in_gga(0, 0, 0) → averageA_out_gga(0, 0, 0)
averageA_in_gga(0, s(0), 0) → averageA_out_gga(0, s(0), 0)
averageA_in_gga(0, s(s(0)), s(0)) → averageA_out_gga(0, s(s(0)), s(0))
averageA_in_gga(s(0), 0, 0) → averageA_out_gga(s(0), 0, 0)
averageA_in_gga(s(0), s(0), s(0)) → averageA_out_gga(s(0), s(0), s(0))
averageA_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, averageA_in_gga(T23, s(s(T24)), T26))
averageA_in_gga(s(T39), s(s(T40)), s(T42)) → U2_gga(T39, T40, T42, averageA_in_gga(s(T39), T40, T42))
averageA_in_gga(s(T49), s(s(s(T50))), s(T52)) → U3_gga(T49, T50, T52, averageA_in_gga(s(s(T49)), T50, T52))
averageA_in_gga(T74, s(s(s(T75))), s(T77)) → U4_gga(T74, T75, T77, averageA_in_gga(T74, s(T75), T77))
averageA_in_gga(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) → U5_gga(T84, T85, T87, averageA_in_gga(s(s(T84)), T85, T87))
U5_gga(T84, T85, T87, averageA_out_gga(s(s(T84)), T85, T87)) → averageA_out_gga(T84, s(s(s(s(s(s(T85)))))), s(s(T87)))
U4_gga(T74, T75, T77, averageA_out_gga(T74, s(T75), T77)) → averageA_out_gga(T74, s(s(s(T75))), s(T77))
U3_gga(T49, T50, T52, averageA_out_gga(s(s(T49)), T50, T52)) → averageA_out_gga(s(T49), s(s(s(T50))), s(T52))
U2_gga(T39, T40, T42, averageA_out_gga(s(T39), T40, T42)) → averageA_out_gga(s(T39), s(s(T40)), s(T42))
U1_gga(T23, T24, T26, averageA_out_gga(T23, s(s(T24)), T26)) → averageA_out_gga(s(s(T23)), T24, T26)
The argument filtering Pi contains the following mapping:
averageA_in_gga(
x1,
x2,
x3) =
averageA_in_gga(
x1,
x2)
0 =
0
averageA_out_gga(
x1,
x2,
x3) =
averageA_out_gga(
x1,
x2,
x3)
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)
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)
AVERAGEA_IN_GGA(
x1,
x2,
x3) =
AVERAGEA_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
(4) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
AVERAGEA_IN_GGA(s(s(T23)), T24, T26) → U1_GGA(T23, T24, T26, averageA_in_gga(T23, s(s(T24)), T26))
AVERAGEA_IN_GGA(s(s(T23)), T24, T26) → AVERAGEA_IN_GGA(T23, s(s(T24)), T26)
AVERAGEA_IN_GGA(s(T39), s(s(T40)), s(T42)) → U2_GGA(T39, T40, T42, averageA_in_gga(s(T39), T40, T42))
AVERAGEA_IN_GGA(s(T39), s(s(T40)), s(T42)) → AVERAGEA_IN_GGA(s(T39), T40, T42)
AVERAGEA_IN_GGA(s(T49), s(s(s(T50))), s(T52)) → U3_GGA(T49, T50, T52, averageA_in_gga(s(s(T49)), T50, T52))
AVERAGEA_IN_GGA(s(T49), s(s(s(T50))), s(T52)) → AVERAGEA_IN_GGA(s(s(T49)), T50, T52)
AVERAGEA_IN_GGA(T74, s(s(s(T75))), s(T77)) → U4_GGA(T74, T75, T77, averageA_in_gga(T74, s(T75), T77))
AVERAGEA_IN_GGA(T74, s(s(s(T75))), s(T77)) → AVERAGEA_IN_GGA(T74, s(T75), T77)
AVERAGEA_IN_GGA(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) → U5_GGA(T84, T85, T87, averageA_in_gga(s(s(T84)), T85, T87))
AVERAGEA_IN_GGA(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) → AVERAGEA_IN_GGA(s(s(T84)), T85, T87)
The TRS R consists of the following rules:
averageA_in_gga(0, 0, 0) → averageA_out_gga(0, 0, 0)
averageA_in_gga(0, s(0), 0) → averageA_out_gga(0, s(0), 0)
averageA_in_gga(0, s(s(0)), s(0)) → averageA_out_gga(0, s(s(0)), s(0))
averageA_in_gga(s(0), 0, 0) → averageA_out_gga(s(0), 0, 0)
averageA_in_gga(s(0), s(0), s(0)) → averageA_out_gga(s(0), s(0), s(0))
averageA_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, averageA_in_gga(T23, s(s(T24)), T26))
averageA_in_gga(s(T39), s(s(T40)), s(T42)) → U2_gga(T39, T40, T42, averageA_in_gga(s(T39), T40, T42))
averageA_in_gga(s(T49), s(s(s(T50))), s(T52)) → U3_gga(T49, T50, T52, averageA_in_gga(s(s(T49)), T50, T52))
averageA_in_gga(T74, s(s(s(T75))), s(T77)) → U4_gga(T74, T75, T77, averageA_in_gga(T74, s(T75), T77))
averageA_in_gga(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) → U5_gga(T84, T85, T87, averageA_in_gga(s(s(T84)), T85, T87))
U5_gga(T84, T85, T87, averageA_out_gga(s(s(T84)), T85, T87)) → averageA_out_gga(T84, s(s(s(s(s(s(T85)))))), s(s(T87)))
U4_gga(T74, T75, T77, averageA_out_gga(T74, s(T75), T77)) → averageA_out_gga(T74, s(s(s(T75))), s(T77))
U3_gga(T49, T50, T52, averageA_out_gga(s(s(T49)), T50, T52)) → averageA_out_gga(s(T49), s(s(s(T50))), s(T52))
U2_gga(T39, T40, T42, averageA_out_gga(s(T39), T40, T42)) → averageA_out_gga(s(T39), s(s(T40)), s(T42))
U1_gga(T23, T24, T26, averageA_out_gga(T23, s(s(T24)), T26)) → averageA_out_gga(s(s(T23)), T24, T26)
The argument filtering Pi contains the following mapping:
averageA_in_gga(
x1,
x2,
x3) =
averageA_in_gga(
x1,
x2)
0 =
0
averageA_out_gga(
x1,
x2,
x3) =
averageA_out_gga(
x1,
x2,
x3)
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)
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)
AVERAGEA_IN_GGA(
x1,
x2,
x3) =
AVERAGEA_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
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 5 less nodes.
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
AVERAGEA_IN_GGA(s(T39), s(s(T40)), s(T42)) → AVERAGEA_IN_GGA(s(T39), T40, T42)
AVERAGEA_IN_GGA(s(s(T23)), T24, T26) → AVERAGEA_IN_GGA(T23, s(s(T24)), T26)
AVERAGEA_IN_GGA(s(T49), s(s(s(T50))), s(T52)) → AVERAGEA_IN_GGA(s(s(T49)), T50, T52)
AVERAGEA_IN_GGA(T74, s(s(s(T75))), s(T77)) → AVERAGEA_IN_GGA(T74, s(T75), T77)
AVERAGEA_IN_GGA(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) → AVERAGEA_IN_GGA(s(s(T84)), T85, T87)
The TRS R consists of the following rules:
averageA_in_gga(0, 0, 0) → averageA_out_gga(0, 0, 0)
averageA_in_gga(0, s(0), 0) → averageA_out_gga(0, s(0), 0)
averageA_in_gga(0, s(s(0)), s(0)) → averageA_out_gga(0, s(s(0)), s(0))
averageA_in_gga(s(0), 0, 0) → averageA_out_gga(s(0), 0, 0)
averageA_in_gga(s(0), s(0), s(0)) → averageA_out_gga(s(0), s(0), s(0))
averageA_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, averageA_in_gga(T23, s(s(T24)), T26))
averageA_in_gga(s(T39), s(s(T40)), s(T42)) → U2_gga(T39, T40, T42, averageA_in_gga(s(T39), T40, T42))
averageA_in_gga(s(T49), s(s(s(T50))), s(T52)) → U3_gga(T49, T50, T52, averageA_in_gga(s(s(T49)), T50, T52))
averageA_in_gga(T74, s(s(s(T75))), s(T77)) → U4_gga(T74, T75, T77, averageA_in_gga(T74, s(T75), T77))
averageA_in_gga(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) → U5_gga(T84, T85, T87, averageA_in_gga(s(s(T84)), T85, T87))
U5_gga(T84, T85, T87, averageA_out_gga(s(s(T84)), T85, T87)) → averageA_out_gga(T84, s(s(s(s(s(s(T85)))))), s(s(T87)))
U4_gga(T74, T75, T77, averageA_out_gga(T74, s(T75), T77)) → averageA_out_gga(T74, s(s(s(T75))), s(T77))
U3_gga(T49, T50, T52, averageA_out_gga(s(s(T49)), T50, T52)) → averageA_out_gga(s(T49), s(s(s(T50))), s(T52))
U2_gga(T39, T40, T42, averageA_out_gga(s(T39), T40, T42)) → averageA_out_gga(s(T39), s(s(T40)), s(T42))
U1_gga(T23, T24, T26, averageA_out_gga(T23, s(s(T24)), T26)) → averageA_out_gga(s(s(T23)), T24, T26)
The argument filtering Pi contains the following mapping:
averageA_in_gga(
x1,
x2,
x3) =
averageA_in_gga(
x1,
x2)
0 =
0
averageA_out_gga(
x1,
x2,
x3) =
averageA_out_gga(
x1,
x2,
x3)
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)
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)
AVERAGEA_IN_GGA(
x1,
x2,
x3) =
AVERAGEA_IN_GGA(
x1,
x2)
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:
AVERAGEA_IN_GGA(s(T39), s(s(T40)), s(T42)) → AVERAGEA_IN_GGA(s(T39), T40, T42)
AVERAGEA_IN_GGA(s(s(T23)), T24, T26) → AVERAGEA_IN_GGA(T23, s(s(T24)), T26)
AVERAGEA_IN_GGA(s(T49), s(s(s(T50))), s(T52)) → AVERAGEA_IN_GGA(s(s(T49)), T50, T52)
AVERAGEA_IN_GGA(T74, s(s(s(T75))), s(T77)) → AVERAGEA_IN_GGA(T74, s(T75), T77)
AVERAGEA_IN_GGA(T84, s(s(s(s(s(s(T85)))))), s(s(T87))) → AVERAGEA_IN_GGA(s(s(T84)), T85, T87)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
AVERAGEA_IN_GGA(
x1,
x2,
x3) =
AVERAGEA_IN_GGA(
x1,
x2)
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:
AVERAGEA_IN_GGA(s(T39), s(s(T40))) → AVERAGEA_IN_GGA(s(T39), T40)
AVERAGEA_IN_GGA(s(s(T23)), T24) → AVERAGEA_IN_GGA(T23, s(s(T24)))
AVERAGEA_IN_GGA(s(T49), s(s(s(T50)))) → AVERAGEA_IN_GGA(s(s(T49)), T50)
AVERAGEA_IN_GGA(T74, s(s(s(T75)))) → AVERAGEA_IN_GGA(T74, s(T75))
AVERAGEA_IN_GGA(T84, s(s(s(s(s(s(T85))))))) → AVERAGEA_IN_GGA(s(s(T84)), T85)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(11) 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:
AVERAGEA_IN_GGA(s(T39), s(s(T40))) → AVERAGEA_IN_GGA(s(T39), T40)
AVERAGEA_IN_GGA(s(s(T23)), T24) → AVERAGEA_IN_GGA(T23, s(s(T24)))
AVERAGEA_IN_GGA(s(T49), s(s(s(T50)))) → AVERAGEA_IN_GGA(s(s(T49)), T50)
AVERAGEA_IN_GGA(T74, s(s(s(T75)))) → AVERAGEA_IN_GGA(T74, s(T75))
AVERAGEA_IN_GGA(T84, s(s(s(s(s(s(T85))))))) → AVERAGEA_IN_GGA(s(s(T84)), T85)
Used ordering: Knuth-Bendix order [KBO] with precedence:
s1 > AVERAGEAINGGA2
and weight map:
s_1=1
AVERAGEA_IN_GGA_2=0
The variable weight is 1
(12) Obligation:
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(13) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(14) YES