(0) Obligation:
Clauses:
right(tree(X, XS1, XS2), XS2).
flat(niltree, nil).
flat(tree(X, niltree, XS), cons(X, YS)) :- ','(right(tree(X, niltree, XS), ZS), flat(ZS, YS)).
flat(tree(X, tree(Y, YS1, YS2), XS), ZS) :- flat(tree(Y, YS1, tree(X, YS2, XS)), ZS).
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(niltree, nil) → flatA_out_ga(niltree, nil)
flatA_in_ga(tree(T18, niltree, T19), cons(T18, T9)) → U1_ga(T18, T19, T9, flatA_in_ga(T19, T9))
flatA_in_ga(tree(T69, tree(T68, niltree, T70), T71), cons(T68, T59)) → U2_ga(T69, T68, T70, T71, T59, flatA_in_ga(tree(T69, T70, T71), T59))
flatA_in_ga(tree(T92, tree(T88, tree(T89, T90, T91), T93), T94), T96) → U3_ga(T92, T88, T89, T90, T91, T93, T94, T96, flatA_in_ga(tree(T89, T90, tree(T88, T91, tree(T92, T93, T94))), T96))
U3_ga(T92, T88, T89, T90, T91, T93, T94, T96, flatA_out_ga(tree(T89, T90, tree(T88, T91, tree(T92, T93, T94))), T96)) → flatA_out_ga(tree(T92, tree(T88, tree(T89, T90, T91), T93), T94), T96)
U2_ga(T69, T68, T70, T71, T59, flatA_out_ga(tree(T69, T70, T71), T59)) → flatA_out_ga(tree(T69, tree(T68, niltree, T70), T71), cons(T68, T59))
U1_ga(T18, T19, T9, flatA_out_ga(T19, T9)) → flatA_out_ga(tree(T18, niltree, T19), cons(T18, T9))
The argument filtering Pi contains the following mapping:
flatA_in_ga(
x1,
x2) =
flatA_in_ga(
x1)
niltree =
niltree
flatA_out_ga(
x1,
x2) =
flatA_out_ga(
x1,
x2)
tree(
x1,
x2,
x3) =
tree(
x1,
x2,
x3)
U1_ga(
x1,
x2,
x3,
x4) =
U1_ga(
x1,
x2,
x4)
U2_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_ga(
x1,
x2,
x3,
x4,
x6)
U3_ga(
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9) =
U3_ga(
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x9)
cons(
x1,
x2) =
cons(
x1,
x2)
(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(tree(T18, niltree, T19), cons(T18, T9)) → U1_GA(T18, T19, T9, flatA_in_ga(T19, T9))
FLATA_IN_GA(tree(T18, niltree, T19), cons(T18, T9)) → FLATA_IN_GA(T19, T9)
FLATA_IN_GA(tree(T69, tree(T68, niltree, T70), T71), cons(T68, T59)) → U2_GA(T69, T68, T70, T71, T59, flatA_in_ga(tree(T69, T70, T71), T59))
FLATA_IN_GA(tree(T69, tree(T68, niltree, T70), T71), cons(T68, T59)) → FLATA_IN_GA(tree(T69, T70, T71), T59)
FLATA_IN_GA(tree(T92, tree(T88, tree(T89, T90, T91), T93), T94), T96) → U3_GA(T92, T88, T89, T90, T91, T93, T94, T96, flatA_in_ga(tree(T89, T90, tree(T88, T91, tree(T92, T93, T94))), T96))
FLATA_IN_GA(tree(T92, tree(T88, tree(T89, T90, T91), T93), T94), T96) → FLATA_IN_GA(tree(T89, T90, tree(T88, T91, tree(T92, T93, T94))), T96)
The TRS R consists of the following rules:
flatA_in_ga(niltree, nil) → flatA_out_ga(niltree, nil)
flatA_in_ga(tree(T18, niltree, T19), cons(T18, T9)) → U1_ga(T18, T19, T9, flatA_in_ga(T19, T9))
flatA_in_ga(tree(T69, tree(T68, niltree, T70), T71), cons(T68, T59)) → U2_ga(T69, T68, T70, T71, T59, flatA_in_ga(tree(T69, T70, T71), T59))
flatA_in_ga(tree(T92, tree(T88, tree(T89, T90, T91), T93), T94), T96) → U3_ga(T92, T88, T89, T90, T91, T93, T94, T96, flatA_in_ga(tree(T89, T90, tree(T88, T91, tree(T92, T93, T94))), T96))
U3_ga(T92, T88, T89, T90, T91, T93, T94, T96, flatA_out_ga(tree(T89, T90, tree(T88, T91, tree(T92, T93, T94))), T96)) → flatA_out_ga(tree(T92, tree(T88, tree(T89, T90, T91), T93), T94), T96)
U2_ga(T69, T68, T70, T71, T59, flatA_out_ga(tree(T69, T70, T71), T59)) → flatA_out_ga(tree(T69, tree(T68, niltree, T70), T71), cons(T68, T59))
U1_ga(T18, T19, T9, flatA_out_ga(T19, T9)) → flatA_out_ga(tree(T18, niltree, T19), cons(T18, T9))
The argument filtering Pi contains the following mapping:
flatA_in_ga(
x1,
x2) =
flatA_in_ga(
x1)
niltree =
niltree
flatA_out_ga(
x1,
x2) =
flatA_out_ga(
x1,
x2)
tree(
x1,
x2,
x3) =
tree(
x1,
x2,
x3)
U1_ga(
x1,
x2,
x3,
x4) =
U1_ga(
x1,
x2,
x4)
U2_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_ga(
x1,
x2,
x3,
x4,
x6)
U3_ga(
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9) =
U3_ga(
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x9)
cons(
x1,
x2) =
cons(
x1,
x2)
FLATA_IN_GA(
x1,
x2) =
FLATA_IN_GA(
x1)
U1_GA(
x1,
x2,
x3,
x4) =
U1_GA(
x1,
x2,
x4)
U2_GA(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_GA(
x1,
x2,
x3,
x4,
x6)
U3_GA(
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9) =
U3_GA(
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x9)
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(tree(T18, niltree, T19), cons(T18, T9)) → U1_GA(T18, T19, T9, flatA_in_ga(T19, T9))
FLATA_IN_GA(tree(T18, niltree, T19), cons(T18, T9)) → FLATA_IN_GA(T19, T9)
FLATA_IN_GA(tree(T69, tree(T68, niltree, T70), T71), cons(T68, T59)) → U2_GA(T69, T68, T70, T71, T59, flatA_in_ga(tree(T69, T70, T71), T59))
FLATA_IN_GA(tree(T69, tree(T68, niltree, T70), T71), cons(T68, T59)) → FLATA_IN_GA(tree(T69, T70, T71), T59)
FLATA_IN_GA(tree(T92, tree(T88, tree(T89, T90, T91), T93), T94), T96) → U3_GA(T92, T88, T89, T90, T91, T93, T94, T96, flatA_in_ga(tree(T89, T90, tree(T88, T91, tree(T92, T93, T94))), T96))
FLATA_IN_GA(tree(T92, tree(T88, tree(T89, T90, T91), T93), T94), T96) → FLATA_IN_GA(tree(T89, T90, tree(T88, T91, tree(T92, T93, T94))), T96)
The TRS R consists of the following rules:
flatA_in_ga(niltree, nil) → flatA_out_ga(niltree, nil)
flatA_in_ga(tree(T18, niltree, T19), cons(T18, T9)) → U1_ga(T18, T19, T9, flatA_in_ga(T19, T9))
flatA_in_ga(tree(T69, tree(T68, niltree, T70), T71), cons(T68, T59)) → U2_ga(T69, T68, T70, T71, T59, flatA_in_ga(tree(T69, T70, T71), T59))
flatA_in_ga(tree(T92, tree(T88, tree(T89, T90, T91), T93), T94), T96) → U3_ga(T92, T88, T89, T90, T91, T93, T94, T96, flatA_in_ga(tree(T89, T90, tree(T88, T91, tree(T92, T93, T94))), T96))
U3_ga(T92, T88, T89, T90, T91, T93, T94, T96, flatA_out_ga(tree(T89, T90, tree(T88, T91, tree(T92, T93, T94))), T96)) → flatA_out_ga(tree(T92, tree(T88, tree(T89, T90, T91), T93), T94), T96)
U2_ga(T69, T68, T70, T71, T59, flatA_out_ga(tree(T69, T70, T71), T59)) → flatA_out_ga(tree(T69, tree(T68, niltree, T70), T71), cons(T68, T59))
U1_ga(T18, T19, T9, flatA_out_ga(T19, T9)) → flatA_out_ga(tree(T18, niltree, T19), cons(T18, T9))
The argument filtering Pi contains the following mapping:
flatA_in_ga(
x1,
x2) =
flatA_in_ga(
x1)
niltree =
niltree
flatA_out_ga(
x1,
x2) =
flatA_out_ga(
x1,
x2)
tree(
x1,
x2,
x3) =
tree(
x1,
x2,
x3)
U1_ga(
x1,
x2,
x3,
x4) =
U1_ga(
x1,
x2,
x4)
U2_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_ga(
x1,
x2,
x3,
x4,
x6)
U3_ga(
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9) =
U3_ga(
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x9)
cons(
x1,
x2) =
cons(
x1,
x2)
FLATA_IN_GA(
x1,
x2) =
FLATA_IN_GA(
x1)
U1_GA(
x1,
x2,
x3,
x4) =
U1_GA(
x1,
x2,
x4)
U2_GA(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_GA(
x1,
x2,
x3,
x4,
x6)
U3_GA(
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9) =
U3_GA(
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x9)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
FLATA_IN_GA(tree(T69, tree(T68, niltree, T70), T71), cons(T68, T59)) → FLATA_IN_GA(tree(T69, T70, T71), T59)
FLATA_IN_GA(tree(T18, niltree, T19), cons(T18, T9)) → FLATA_IN_GA(T19, T9)
FLATA_IN_GA(tree(T92, tree(T88, tree(T89, T90, T91), T93), T94), T96) → FLATA_IN_GA(tree(T89, T90, tree(T88, T91, tree(T92, T93, T94))), T96)
The TRS R consists of the following rules:
flatA_in_ga(niltree, nil) → flatA_out_ga(niltree, nil)
flatA_in_ga(tree(T18, niltree, T19), cons(T18, T9)) → U1_ga(T18, T19, T9, flatA_in_ga(T19, T9))
flatA_in_ga(tree(T69, tree(T68, niltree, T70), T71), cons(T68, T59)) → U2_ga(T69, T68, T70, T71, T59, flatA_in_ga(tree(T69, T70, T71), T59))
flatA_in_ga(tree(T92, tree(T88, tree(T89, T90, T91), T93), T94), T96) → U3_ga(T92, T88, T89, T90, T91, T93, T94, T96, flatA_in_ga(tree(T89, T90, tree(T88, T91, tree(T92, T93, T94))), T96))
U3_ga(T92, T88, T89, T90, T91, T93, T94, T96, flatA_out_ga(tree(T89, T90, tree(T88, T91, tree(T92, T93, T94))), T96)) → flatA_out_ga(tree(T92, tree(T88, tree(T89, T90, T91), T93), T94), T96)
U2_ga(T69, T68, T70, T71, T59, flatA_out_ga(tree(T69, T70, T71), T59)) → flatA_out_ga(tree(T69, tree(T68, niltree, T70), T71), cons(T68, T59))
U1_ga(T18, T19, T9, flatA_out_ga(T19, T9)) → flatA_out_ga(tree(T18, niltree, T19), cons(T18, T9))
The argument filtering Pi contains the following mapping:
flatA_in_ga(
x1,
x2) =
flatA_in_ga(
x1)
niltree =
niltree
flatA_out_ga(
x1,
x2) =
flatA_out_ga(
x1,
x2)
tree(
x1,
x2,
x3) =
tree(
x1,
x2,
x3)
U1_ga(
x1,
x2,
x3,
x4) =
U1_ga(
x1,
x2,
x4)
U2_ga(
x1,
x2,
x3,
x4,
x5,
x6) =
U2_ga(
x1,
x2,
x3,
x4,
x6)
U3_ga(
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9) =
U3_ga(
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x9)
cons(
x1,
x2) =
cons(
x1,
x2)
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(tree(T69, tree(T68, niltree, T70), T71), cons(T68, T59)) → FLATA_IN_GA(tree(T69, T70, T71), T59)
FLATA_IN_GA(tree(T18, niltree, T19), cons(T18, T9)) → FLATA_IN_GA(T19, T9)
FLATA_IN_GA(tree(T92, tree(T88, tree(T89, T90, T91), T93), T94), T96) → FLATA_IN_GA(tree(T89, T90, tree(T88, T91, tree(T92, T93, T94))), T96)
R is empty.
The argument filtering Pi contains the following mapping:
niltree =
niltree
tree(
x1,
x2,
x3) =
tree(
x1,
x2,
x3)
cons(
x1,
x2) =
cons(
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(tree(T69, tree(T68, niltree, T70), T71)) → FLATA_IN_GA(tree(T69, T70, T71))
FLATA_IN_GA(tree(T18, niltree, T19)) → FLATA_IN_GA(T19)
FLATA_IN_GA(tree(T92, tree(T88, tree(T89, T90, T91), T93), T94)) → FLATA_IN_GA(tree(T89, T90, tree(T88, T91, tree(T92, T93, T94))))
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(tree(T69, tree(T68, niltree, T70), T71)) → FLATA_IN_GA(tree(T69, T70, T71))
FLATA_IN_GA(tree(T18, niltree, T19)) → FLATA_IN_GA(T19)
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(FLATA_IN_GA(x1)) = 2·x1
POL(niltree) = 0
POL(tree(x1, x2, x3)) = 2·x1 + 2·x2 + x3
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
FLATA_IN_GA(tree(T92, tree(T88, tree(T89, T90, T91), T93), T94)) → FLATA_IN_GA(tree(T89, T90, tree(T88, T91, tree(T92, T93, T94))))
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(tree(T92, tree(T88, tree(T89, T90, T91), T93), T94)) → FLATA_IN_GA(tree(T89, T90, tree(T88, T91, tree(T92, T93, T94))))
Used ordering: Polynomial interpretation [POLO]:
POL(FLATA_IN_GA(x1)) = 2·x1
POL(tree(x1, x2, x3)) = 2 + 2·x1 + 2·x2 + x3
(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