(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)).
Query: avg(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:
avgA_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, avgA_in_gga(T23, s(s(T24)), T26))
avgA_in_gga(s(T42), s(s(T43)), s(T45)) → U2_gga(T42, T43, T45, avgA_in_gga(s(T42), T43, T45))
avgA_in_gga(s(0), 0, 0) → avgA_out_gga(s(0), 0, 0)
avgA_in_gga(s(0), s(0), s(0)) → avgA_out_gga(s(0), s(0), s(0))
avgA_in_gga(s(T61), s(s(s(T62))), s(T64)) → U3_gga(T61, T62, T64, avgA_in_gga(s(s(T61)), T62, T64))
avgA_in_gga(T101, s(s(s(T102))), s(T104)) → U4_gga(T101, T102, T104, avgA_in_gga(T101, s(T102), T104))
avgA_in_gga(T123, s(s(s(s(s(s(T124)))))), s(s(T126))) → U5_gga(T123, T124, T126, avgA_in_gga(s(s(T123)), T124, T126))
avgA_in_gga(0, 0, 0) → avgA_out_gga(0, 0, 0)
avgA_in_gga(0, s(0), 0) → avgA_out_gga(0, s(0), 0)
avgA_in_gga(0, s(s(0)), s(0)) → avgA_out_gga(0, s(s(0)), s(0))
U5_gga(T123, T124, T126, avgA_out_gga(s(s(T123)), T124, T126)) → avgA_out_gga(T123, s(s(s(s(s(s(T124)))))), s(s(T126)))
U4_gga(T101, T102, T104, avgA_out_gga(T101, s(T102), T104)) → avgA_out_gga(T101, s(s(s(T102))), s(T104))
U3_gga(T61, T62, T64, avgA_out_gga(s(s(T61)), T62, T64)) → avgA_out_gga(s(T61), s(s(s(T62))), s(T64))
U2_gga(T42, T43, T45, avgA_out_gga(s(T42), T43, T45)) → avgA_out_gga(s(T42), s(s(T43)), s(T45))
U1_gga(T23, T24, T26, avgA_out_gga(T23, s(s(T24)), T26)) → avgA_out_gga(s(s(T23)), T24, T26)
The argument filtering Pi contains the following mapping:
avgA_in_gga(
x1,
x2,
x3) =
avgA_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
avgA_out_gga(
x1,
x2,
x3) =
avgA_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)
(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:
AVGA_IN_GGA(s(s(T23)), T24, T26) → U1_GGA(T23, T24, T26, avgA_in_gga(T23, s(s(T24)), T26))
AVGA_IN_GGA(s(s(T23)), T24, T26) → AVGA_IN_GGA(T23, s(s(T24)), T26)
AVGA_IN_GGA(s(T42), s(s(T43)), s(T45)) → U2_GGA(T42, T43, T45, avgA_in_gga(s(T42), T43, T45))
AVGA_IN_GGA(s(T42), s(s(T43)), s(T45)) → AVGA_IN_GGA(s(T42), T43, T45)
AVGA_IN_GGA(s(T61), s(s(s(T62))), s(T64)) → U3_GGA(T61, T62, T64, avgA_in_gga(s(s(T61)), T62, T64))
AVGA_IN_GGA(s(T61), s(s(s(T62))), s(T64)) → AVGA_IN_GGA(s(s(T61)), T62, T64)
AVGA_IN_GGA(T101, s(s(s(T102))), s(T104)) → U4_GGA(T101, T102, T104, avgA_in_gga(T101, s(T102), T104))
AVGA_IN_GGA(T101, s(s(s(T102))), s(T104)) → AVGA_IN_GGA(T101, s(T102), T104)
AVGA_IN_GGA(T123, s(s(s(s(s(s(T124)))))), s(s(T126))) → U5_GGA(T123, T124, T126, avgA_in_gga(s(s(T123)), T124, T126))
AVGA_IN_GGA(T123, s(s(s(s(s(s(T124)))))), s(s(T126))) → AVGA_IN_GGA(s(s(T123)), T124, T126)
The TRS R consists of the following rules:
avgA_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, avgA_in_gga(T23, s(s(T24)), T26))
avgA_in_gga(s(T42), s(s(T43)), s(T45)) → U2_gga(T42, T43, T45, avgA_in_gga(s(T42), T43, T45))
avgA_in_gga(s(0), 0, 0) → avgA_out_gga(s(0), 0, 0)
avgA_in_gga(s(0), s(0), s(0)) → avgA_out_gga(s(0), s(0), s(0))
avgA_in_gga(s(T61), s(s(s(T62))), s(T64)) → U3_gga(T61, T62, T64, avgA_in_gga(s(s(T61)), T62, T64))
avgA_in_gga(T101, s(s(s(T102))), s(T104)) → U4_gga(T101, T102, T104, avgA_in_gga(T101, s(T102), T104))
avgA_in_gga(T123, s(s(s(s(s(s(T124)))))), s(s(T126))) → U5_gga(T123, T124, T126, avgA_in_gga(s(s(T123)), T124, T126))
avgA_in_gga(0, 0, 0) → avgA_out_gga(0, 0, 0)
avgA_in_gga(0, s(0), 0) → avgA_out_gga(0, s(0), 0)
avgA_in_gga(0, s(s(0)), s(0)) → avgA_out_gga(0, s(s(0)), s(0))
U5_gga(T123, T124, T126, avgA_out_gga(s(s(T123)), T124, T126)) → avgA_out_gga(T123, s(s(s(s(s(s(T124)))))), s(s(T126)))
U4_gga(T101, T102, T104, avgA_out_gga(T101, s(T102), T104)) → avgA_out_gga(T101, s(s(s(T102))), s(T104))
U3_gga(T61, T62, T64, avgA_out_gga(s(s(T61)), T62, T64)) → avgA_out_gga(s(T61), s(s(s(T62))), s(T64))
U2_gga(T42, T43, T45, avgA_out_gga(s(T42), T43, T45)) → avgA_out_gga(s(T42), s(s(T43)), s(T45))
U1_gga(T23, T24, T26, avgA_out_gga(T23, s(s(T24)), T26)) → avgA_out_gga(s(s(T23)), T24, T26)
The argument filtering Pi contains the following mapping:
avgA_in_gga(
x1,
x2,
x3) =
avgA_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
avgA_out_gga(
x1,
x2,
x3) =
avgA_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)
AVGA_IN_GGA(
x1,
x2,
x3) =
AVGA_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:
AVGA_IN_GGA(s(s(T23)), T24, T26) → U1_GGA(T23, T24, T26, avgA_in_gga(T23, s(s(T24)), T26))
AVGA_IN_GGA(s(s(T23)), T24, T26) → AVGA_IN_GGA(T23, s(s(T24)), T26)
AVGA_IN_GGA(s(T42), s(s(T43)), s(T45)) → U2_GGA(T42, T43, T45, avgA_in_gga(s(T42), T43, T45))
AVGA_IN_GGA(s(T42), s(s(T43)), s(T45)) → AVGA_IN_GGA(s(T42), T43, T45)
AVGA_IN_GGA(s(T61), s(s(s(T62))), s(T64)) → U3_GGA(T61, T62, T64, avgA_in_gga(s(s(T61)), T62, T64))
AVGA_IN_GGA(s(T61), s(s(s(T62))), s(T64)) → AVGA_IN_GGA(s(s(T61)), T62, T64)
AVGA_IN_GGA(T101, s(s(s(T102))), s(T104)) → U4_GGA(T101, T102, T104, avgA_in_gga(T101, s(T102), T104))
AVGA_IN_GGA(T101, s(s(s(T102))), s(T104)) → AVGA_IN_GGA(T101, s(T102), T104)
AVGA_IN_GGA(T123, s(s(s(s(s(s(T124)))))), s(s(T126))) → U5_GGA(T123, T124, T126, avgA_in_gga(s(s(T123)), T124, T126))
AVGA_IN_GGA(T123, s(s(s(s(s(s(T124)))))), s(s(T126))) → AVGA_IN_GGA(s(s(T123)), T124, T126)
The TRS R consists of the following rules:
avgA_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, avgA_in_gga(T23, s(s(T24)), T26))
avgA_in_gga(s(T42), s(s(T43)), s(T45)) → U2_gga(T42, T43, T45, avgA_in_gga(s(T42), T43, T45))
avgA_in_gga(s(0), 0, 0) → avgA_out_gga(s(0), 0, 0)
avgA_in_gga(s(0), s(0), s(0)) → avgA_out_gga(s(0), s(0), s(0))
avgA_in_gga(s(T61), s(s(s(T62))), s(T64)) → U3_gga(T61, T62, T64, avgA_in_gga(s(s(T61)), T62, T64))
avgA_in_gga(T101, s(s(s(T102))), s(T104)) → U4_gga(T101, T102, T104, avgA_in_gga(T101, s(T102), T104))
avgA_in_gga(T123, s(s(s(s(s(s(T124)))))), s(s(T126))) → U5_gga(T123, T124, T126, avgA_in_gga(s(s(T123)), T124, T126))
avgA_in_gga(0, 0, 0) → avgA_out_gga(0, 0, 0)
avgA_in_gga(0, s(0), 0) → avgA_out_gga(0, s(0), 0)
avgA_in_gga(0, s(s(0)), s(0)) → avgA_out_gga(0, s(s(0)), s(0))
U5_gga(T123, T124, T126, avgA_out_gga(s(s(T123)), T124, T126)) → avgA_out_gga(T123, s(s(s(s(s(s(T124)))))), s(s(T126)))
U4_gga(T101, T102, T104, avgA_out_gga(T101, s(T102), T104)) → avgA_out_gga(T101, s(s(s(T102))), s(T104))
U3_gga(T61, T62, T64, avgA_out_gga(s(s(T61)), T62, T64)) → avgA_out_gga(s(T61), s(s(s(T62))), s(T64))
U2_gga(T42, T43, T45, avgA_out_gga(s(T42), T43, T45)) → avgA_out_gga(s(T42), s(s(T43)), s(T45))
U1_gga(T23, T24, T26, avgA_out_gga(T23, s(s(T24)), T26)) → avgA_out_gga(s(s(T23)), T24, T26)
The argument filtering Pi contains the following mapping:
avgA_in_gga(
x1,
x2,
x3) =
avgA_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
avgA_out_gga(
x1,
x2,
x3) =
avgA_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)
AVGA_IN_GGA(
x1,
x2,
x3) =
AVGA_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:
AVGA_IN_GGA(s(T42), s(s(T43)), s(T45)) → AVGA_IN_GGA(s(T42), T43, T45)
AVGA_IN_GGA(s(s(T23)), T24, T26) → AVGA_IN_GGA(T23, s(s(T24)), T26)
AVGA_IN_GGA(s(T61), s(s(s(T62))), s(T64)) → AVGA_IN_GGA(s(s(T61)), T62, T64)
AVGA_IN_GGA(T101, s(s(s(T102))), s(T104)) → AVGA_IN_GGA(T101, s(T102), T104)
AVGA_IN_GGA(T123, s(s(s(s(s(s(T124)))))), s(s(T126))) → AVGA_IN_GGA(s(s(T123)), T124, T126)
The TRS R consists of the following rules:
avgA_in_gga(s(s(T23)), T24, T26) → U1_gga(T23, T24, T26, avgA_in_gga(T23, s(s(T24)), T26))
avgA_in_gga(s(T42), s(s(T43)), s(T45)) → U2_gga(T42, T43, T45, avgA_in_gga(s(T42), T43, T45))
avgA_in_gga(s(0), 0, 0) → avgA_out_gga(s(0), 0, 0)
avgA_in_gga(s(0), s(0), s(0)) → avgA_out_gga(s(0), s(0), s(0))
avgA_in_gga(s(T61), s(s(s(T62))), s(T64)) → U3_gga(T61, T62, T64, avgA_in_gga(s(s(T61)), T62, T64))
avgA_in_gga(T101, s(s(s(T102))), s(T104)) → U4_gga(T101, T102, T104, avgA_in_gga(T101, s(T102), T104))
avgA_in_gga(T123, s(s(s(s(s(s(T124)))))), s(s(T126))) → U5_gga(T123, T124, T126, avgA_in_gga(s(s(T123)), T124, T126))
avgA_in_gga(0, 0, 0) → avgA_out_gga(0, 0, 0)
avgA_in_gga(0, s(0), 0) → avgA_out_gga(0, s(0), 0)
avgA_in_gga(0, s(s(0)), s(0)) → avgA_out_gga(0, s(s(0)), s(0))
U5_gga(T123, T124, T126, avgA_out_gga(s(s(T123)), T124, T126)) → avgA_out_gga(T123, s(s(s(s(s(s(T124)))))), s(s(T126)))
U4_gga(T101, T102, T104, avgA_out_gga(T101, s(T102), T104)) → avgA_out_gga(T101, s(s(s(T102))), s(T104))
U3_gga(T61, T62, T64, avgA_out_gga(s(s(T61)), T62, T64)) → avgA_out_gga(s(T61), s(s(s(T62))), s(T64))
U2_gga(T42, T43, T45, avgA_out_gga(s(T42), T43, T45)) → avgA_out_gga(s(T42), s(s(T43)), s(T45))
U1_gga(T23, T24, T26, avgA_out_gga(T23, s(s(T24)), T26)) → avgA_out_gga(s(s(T23)), T24, T26)
The argument filtering Pi contains the following mapping:
avgA_in_gga(
x1,
x2,
x3) =
avgA_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
avgA_out_gga(
x1,
x2,
x3) =
avgA_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)
AVGA_IN_GGA(
x1,
x2,
x3) =
AVGA_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:
AVGA_IN_GGA(s(T42), s(s(T43)), s(T45)) → AVGA_IN_GGA(s(T42), T43, T45)
AVGA_IN_GGA(s(s(T23)), T24, T26) → AVGA_IN_GGA(T23, s(s(T24)), T26)
AVGA_IN_GGA(s(T61), s(s(s(T62))), s(T64)) → AVGA_IN_GGA(s(s(T61)), T62, T64)
AVGA_IN_GGA(T101, s(s(s(T102))), s(T104)) → AVGA_IN_GGA(T101, s(T102), T104)
AVGA_IN_GGA(T123, s(s(s(s(s(s(T124)))))), s(s(T126))) → AVGA_IN_GGA(s(s(T123)), T124, T126)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
AVGA_IN_GGA(
x1,
x2,
x3) =
AVGA_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:
AVGA_IN_GGA(s(T42), s(s(T43))) → AVGA_IN_GGA(s(T42), T43)
AVGA_IN_GGA(s(s(T23)), T24) → AVGA_IN_GGA(T23, s(s(T24)))
AVGA_IN_GGA(s(T61), s(s(s(T62)))) → AVGA_IN_GGA(s(s(T61)), T62)
AVGA_IN_GGA(T101, s(s(s(T102)))) → AVGA_IN_GGA(T101, s(T102))
AVGA_IN_GGA(T123, s(s(s(s(s(s(T124))))))) → AVGA_IN_GGA(s(s(T123)), T124)
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:
AVGA_IN_GGA(s(T42), s(s(T43))) → AVGA_IN_GGA(s(T42), T43)
AVGA_IN_GGA(s(s(T23)), T24) → AVGA_IN_GGA(T23, s(s(T24)))
AVGA_IN_GGA(s(T61), s(s(s(T62)))) → AVGA_IN_GGA(s(s(T61)), T62)
AVGA_IN_GGA(T101, s(s(s(T102)))) → AVGA_IN_GGA(T101, s(T102))
AVGA_IN_GGA(T123, s(s(s(s(s(s(T124))))))) → AVGA_IN_GGA(s(s(T123)), T124)
Used ordering: Knuth-Bendix order [KBO] with precedence:
s1 > AVGAINGGA2
and weight map:
s_1=1
AVGA_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