(0) Obligation:
Clauses:
flat([], []).
flat(.([], T), R) :- flat(T, R).
flat(.(.(H, T), TT), .(H, R)) :- flat(.(T, TT), R).
Query: flat(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:
flatA_in_ga([], []) → flatA_out_ga([], [])
flatA_in_ga(.([], []), []) → flatA_out_ga(.([], []), [])
flatA_in_ga(.([], .([], T16)), T18) → U1_ga(T16, T18, flatA_in_ga(T16, T18))
flatA_in_ga(.([], .(.(T35, T36), T37)), .(T35, T39)) → U2_ga(T35, T36, T37, T39, flatA_in_ga(.(T36, T37), T39))
flatA_in_ga(.(.(T44, []), T57), .(T44, T59)) → U3_ga(T44, T57, T59, flatA_in_ga(T57, T59))
flatA_in_ga(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) → U4_ga(T44, T68, T69, T70, T72, flatA_in_ga(.(T69, T70), T72))
U4_ga(T44, T68, T69, T70, T72, flatA_out_ga(.(T69, T70), T72)) → flatA_out_ga(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72)))
U3_ga(T44, T57, T59, flatA_out_ga(T57, T59)) → flatA_out_ga(.(.(T44, []), T57), .(T44, T59))
U2_ga(T35, T36, T37, T39, flatA_out_ga(.(T36, T37), T39)) → flatA_out_ga(.([], .(.(T35, T36), T37)), .(T35, T39))
U1_ga(T16, T18, flatA_out_ga(T16, T18)) → flatA_out_ga(.([], .([], T16)), T18)
The argument filtering Pi contains the following mapping:
flatA_in_ga(
x1,
x2) =
flatA_in_ga(
x1)
[] =
[]
flatA_out_ga(
x1,
x2) =
flatA_out_ga(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ga(
x1,
x2,
x3) =
U1_ga(
x1,
x3)
U2_ga(
x1,
x2,
x3,
x4,
x5) =
U2_ga(
x1,
x2,
x3,
x5)
U3_ga(
x1,
x2,
x3,
x4) =
U3_ga(
x1,
x2,
x4)
U4_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U4_ga(
x1,
x2,
x3,
x4,
x6)
(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:
FLATA_IN_GA(.([], .([], T16)), T18) → U1_GA(T16, T18, flatA_in_ga(T16, T18))
FLATA_IN_GA(.([], .([], T16)), T18) → FLATA_IN_GA(T16, T18)
FLATA_IN_GA(.([], .(.(T35, T36), T37)), .(T35, T39)) → U2_GA(T35, T36, T37, T39, flatA_in_ga(.(T36, T37), T39))
FLATA_IN_GA(.([], .(.(T35, T36), T37)), .(T35, T39)) → FLATA_IN_GA(.(T36, T37), T39)
FLATA_IN_GA(.(.(T44, []), T57), .(T44, T59)) → U3_GA(T44, T57, T59, flatA_in_ga(T57, T59))
FLATA_IN_GA(.(.(T44, []), T57), .(T44, T59)) → FLATA_IN_GA(T57, T59)
FLATA_IN_GA(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) → U4_GA(T44, T68, T69, T70, T72, flatA_in_ga(.(T69, T70), T72))
FLATA_IN_GA(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) → FLATA_IN_GA(.(T69, T70), T72)
The TRS R consists of the following rules:
flatA_in_ga([], []) → flatA_out_ga([], [])
flatA_in_ga(.([], []), []) → flatA_out_ga(.([], []), [])
flatA_in_ga(.([], .([], T16)), T18) → U1_ga(T16, T18, flatA_in_ga(T16, T18))
flatA_in_ga(.([], .(.(T35, T36), T37)), .(T35, T39)) → U2_ga(T35, T36, T37, T39, flatA_in_ga(.(T36, T37), T39))
flatA_in_ga(.(.(T44, []), T57), .(T44, T59)) → U3_ga(T44, T57, T59, flatA_in_ga(T57, T59))
flatA_in_ga(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) → U4_ga(T44, T68, T69, T70, T72, flatA_in_ga(.(T69, T70), T72))
U4_ga(T44, T68, T69, T70, T72, flatA_out_ga(.(T69, T70), T72)) → flatA_out_ga(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72)))
U3_ga(T44, T57, T59, flatA_out_ga(T57, T59)) → flatA_out_ga(.(.(T44, []), T57), .(T44, T59))
U2_ga(T35, T36, T37, T39, flatA_out_ga(.(T36, T37), T39)) → flatA_out_ga(.([], .(.(T35, T36), T37)), .(T35, T39))
U1_ga(T16, T18, flatA_out_ga(T16, T18)) → flatA_out_ga(.([], .([], T16)), T18)
The argument filtering Pi contains the following mapping:
flatA_in_ga(
x1,
x2) =
flatA_in_ga(
x1)
[] =
[]
flatA_out_ga(
x1,
x2) =
flatA_out_ga(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ga(
x1,
x2,
x3) =
U1_ga(
x1,
x3)
U2_ga(
x1,
x2,
x3,
x4,
x5) =
U2_ga(
x1,
x2,
x3,
x5)
U3_ga(
x1,
x2,
x3,
x4) =
U3_ga(
x1,
x2,
x4)
U4_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U4_ga(
x1,
x2,
x3,
x4,
x6)
FLATA_IN_GA(
x1,
x2) =
FLATA_IN_GA(
x1)
U1_GA(
x1,
x2,
x3) =
U1_GA(
x1,
x3)
U2_GA(
x1,
x2,
x3,
x4,
x5) =
U2_GA(
x1,
x2,
x3,
x5)
U3_GA(
x1,
x2,
x3,
x4) =
U3_GA(
x1,
x2,
x4)
U4_GA(
x1,
x2,
x3,
x4,
x5,
x6) =
U4_GA(
x1,
x2,
x3,
x4,
x6)
We have to consider all (P,R,Pi)-chains
(4) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
FLATA_IN_GA(.([], .([], T16)), T18) → U1_GA(T16, T18, flatA_in_ga(T16, T18))
FLATA_IN_GA(.([], .([], T16)), T18) → FLATA_IN_GA(T16, T18)
FLATA_IN_GA(.([], .(.(T35, T36), T37)), .(T35, T39)) → U2_GA(T35, T36, T37, T39, flatA_in_ga(.(T36, T37), T39))
FLATA_IN_GA(.([], .(.(T35, T36), T37)), .(T35, T39)) → FLATA_IN_GA(.(T36, T37), T39)
FLATA_IN_GA(.(.(T44, []), T57), .(T44, T59)) → U3_GA(T44, T57, T59, flatA_in_ga(T57, T59))
FLATA_IN_GA(.(.(T44, []), T57), .(T44, T59)) → FLATA_IN_GA(T57, T59)
FLATA_IN_GA(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) → U4_GA(T44, T68, T69, T70, T72, flatA_in_ga(.(T69, T70), T72))
FLATA_IN_GA(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) → FLATA_IN_GA(.(T69, T70), T72)
The TRS R consists of the following rules:
flatA_in_ga([], []) → flatA_out_ga([], [])
flatA_in_ga(.([], []), []) → flatA_out_ga(.([], []), [])
flatA_in_ga(.([], .([], T16)), T18) → U1_ga(T16, T18, flatA_in_ga(T16, T18))
flatA_in_ga(.([], .(.(T35, T36), T37)), .(T35, T39)) → U2_ga(T35, T36, T37, T39, flatA_in_ga(.(T36, T37), T39))
flatA_in_ga(.(.(T44, []), T57), .(T44, T59)) → U3_ga(T44, T57, T59, flatA_in_ga(T57, T59))
flatA_in_ga(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) → U4_ga(T44, T68, T69, T70, T72, flatA_in_ga(.(T69, T70), T72))
U4_ga(T44, T68, T69, T70, T72, flatA_out_ga(.(T69, T70), T72)) → flatA_out_ga(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72)))
U3_ga(T44, T57, T59, flatA_out_ga(T57, T59)) → flatA_out_ga(.(.(T44, []), T57), .(T44, T59))
U2_ga(T35, T36, T37, T39, flatA_out_ga(.(T36, T37), T39)) → flatA_out_ga(.([], .(.(T35, T36), T37)), .(T35, T39))
U1_ga(T16, T18, flatA_out_ga(T16, T18)) → flatA_out_ga(.([], .([], T16)), T18)
The argument filtering Pi contains the following mapping:
flatA_in_ga(
x1,
x2) =
flatA_in_ga(
x1)
[] =
[]
flatA_out_ga(
x1,
x2) =
flatA_out_ga(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ga(
x1,
x2,
x3) =
U1_ga(
x1,
x3)
U2_ga(
x1,
x2,
x3,
x4,
x5) =
U2_ga(
x1,
x2,
x3,
x5)
U3_ga(
x1,
x2,
x3,
x4) =
U3_ga(
x1,
x2,
x4)
U4_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U4_ga(
x1,
x2,
x3,
x4,
x6)
FLATA_IN_GA(
x1,
x2) =
FLATA_IN_GA(
x1)
U1_GA(
x1,
x2,
x3) =
U1_GA(
x1,
x3)
U2_GA(
x1,
x2,
x3,
x4,
x5) =
U2_GA(
x1,
x2,
x3,
x5)
U3_GA(
x1,
x2,
x3,
x4) =
U3_GA(
x1,
x2,
x4)
U4_GA(
x1,
x2,
x3,
x4,
x5,
x6) =
U4_GA(
x1,
x2,
x3,
x4,
x6)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 4 less nodes.
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
FLATA_IN_GA(.([], .(.(T35, T36), T37)), .(T35, T39)) → FLATA_IN_GA(.(T36, T37), T39)
FLATA_IN_GA(.([], .([], T16)), T18) → FLATA_IN_GA(T16, T18)
FLATA_IN_GA(.(.(T44, []), T57), .(T44, T59)) → FLATA_IN_GA(T57, T59)
FLATA_IN_GA(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) → FLATA_IN_GA(.(T69, T70), T72)
The TRS R consists of the following rules:
flatA_in_ga([], []) → flatA_out_ga([], [])
flatA_in_ga(.([], []), []) → flatA_out_ga(.([], []), [])
flatA_in_ga(.([], .([], T16)), T18) → U1_ga(T16, T18, flatA_in_ga(T16, T18))
flatA_in_ga(.([], .(.(T35, T36), T37)), .(T35, T39)) → U2_ga(T35, T36, T37, T39, flatA_in_ga(.(T36, T37), T39))
flatA_in_ga(.(.(T44, []), T57), .(T44, T59)) → U3_ga(T44, T57, T59, flatA_in_ga(T57, T59))
flatA_in_ga(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) → U4_ga(T44, T68, T69, T70, T72, flatA_in_ga(.(T69, T70), T72))
U4_ga(T44, T68, T69, T70, T72, flatA_out_ga(.(T69, T70), T72)) → flatA_out_ga(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72)))
U3_ga(T44, T57, T59, flatA_out_ga(T57, T59)) → flatA_out_ga(.(.(T44, []), T57), .(T44, T59))
U2_ga(T35, T36, T37, T39, flatA_out_ga(.(T36, T37), T39)) → flatA_out_ga(.([], .(.(T35, T36), T37)), .(T35, T39))
U1_ga(T16, T18, flatA_out_ga(T16, T18)) → flatA_out_ga(.([], .([], T16)), T18)
The argument filtering Pi contains the following mapping:
flatA_in_ga(
x1,
x2) =
flatA_in_ga(
x1)
[] =
[]
flatA_out_ga(
x1,
x2) =
flatA_out_ga(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ga(
x1,
x2,
x3) =
U1_ga(
x1,
x3)
U2_ga(
x1,
x2,
x3,
x4,
x5) =
U2_ga(
x1,
x2,
x3,
x5)
U3_ga(
x1,
x2,
x3,
x4) =
U3_ga(
x1,
x2,
x4)
U4_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U4_ga(
x1,
x2,
x3,
x4,
x6)
FLATA_IN_GA(
x1,
x2) =
FLATA_IN_GA(
x1)
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:
FLATA_IN_GA(.([], .(.(T35, T36), T37)), .(T35, T39)) → FLATA_IN_GA(.(T36, T37), T39)
FLATA_IN_GA(.([], .([], T16)), T18) → FLATA_IN_GA(T16, T18)
FLATA_IN_GA(.(.(T44, []), T57), .(T44, T59)) → FLATA_IN_GA(T57, T59)
FLATA_IN_GA(.(.(T44, .(T68, T69)), T70), .(T44, .(T68, T72))) → FLATA_IN_GA(.(T69, T70), T72)
R is empty.
The argument filtering Pi contains the following mapping:
[] =
[]
.(
x1,
x2) =
.(
x1,
x2)
FLATA_IN_GA(
x1,
x2) =
FLATA_IN_GA(
x1)
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:
FLATA_IN_GA(.([], .(.(T35, T36), T37))) → FLATA_IN_GA(.(T36, T37))
FLATA_IN_GA(.([], .([], T16))) → FLATA_IN_GA(T16)
FLATA_IN_GA(.(.(T44, []), T57)) → FLATA_IN_GA(T57)
FLATA_IN_GA(.(.(T44, .(T68, T69)), T70)) → FLATA_IN_GA(.(T69, T70))
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(11) UsableRulesReductionPairsProof (EQUIVALENT transformation)
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
The following dependency pairs can be deleted:
FLATA_IN_GA(.([], .(.(T35, T36), T37))) → FLATA_IN_GA(.(T36, T37))
FLATA_IN_GA(.([], .([], T16))) → FLATA_IN_GA(T16)
FLATA_IN_GA(.(.(T44, []), T57)) → FLATA_IN_GA(T57)
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(.(x1, x2)) = 2·x1 + 2·x2
POL(FLATA_IN_GA(x1)) = 2·x1
POL([]) = 0
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
FLATA_IN_GA(.(.(T44, .(T68, T69)), T70)) → FLATA_IN_GA(.(T69, T70))
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:
FLATA_IN_GA(.(.(T44, .(T68, T69)), T70)) → FLATA_IN_GA(.(T69, T70))
Used ordering: Knuth-Bendix order [KBO] with precedence:
.2 > FLATAINGA1
and weight map:
FLATA_IN_GA_1=1
._2=0
The variable weight is 1
(14) Obligation:
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(15) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(16) YES