Left Termination of the query pattern count_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 AND
      ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇐)
        ↳11 QDP
        ↳12 UsableRulesReductionPairsProof (⇔)
        ↳13 QDP
        ↳14 PisEmptyProof (⇔)
        ↳15 TRUE
      ↳16 PiDP
        ↳17 UsableRulesProof (⇔)
        ↳18 PiDP
        ↳19 PiDPToQDPProof (⇐)
        ↳20 QDP
        ↳21 UsableRulesReductionPairsProof (⇔)
        ↳22 QDP
        ↳23 QReductionProof (⇔)
        ↳24 QDP
        ↳25 MRRProof (⇔)
        ↳26 QDP
        ↳27 PisEmptyProof (⇔)
        ↳28 TRUE
  ↳29 PrologToPiTRSProof (⇐)
    ↳30 PiTRS
    ↳31 DependencyPairsProof (⇔)
    ↳32 PiDP
    ↳33 DependencyGraphProof (⇔)
    ↳34 AND
      ↳35 PiDP
        ↳36 UsableRulesProof (⇔)
        ↳37 PiDP
        ↳38 PiDPToQDPProof (⇐)
        ↳39 QDP
        ↳40 UsableRulesReductionPairsProof (⇔)
        ↳41 QDP
        ↳42 PisEmptyProof (⇔)
        ↳43 TRUE
      ↳44 PiDP
        ↳45 UsableRulesProof (⇔)
        ↳46 PiDP
        ↳47 PiDPToQDPProof (⇐)
        ↳48 QDP

(0) Obligation:

Clauses:

flatten(atom(X), .(X, [])).
flatten(cons(atom(X), U), .(X, Y)) :- flatten(U, Y).
flatten(cons(cons(U, V), W), X) :- flatten(cons(U, cons(V, W)), X).
count(atom(X), s(0)).
count(cons(atom(X), Y), s(Z)) :- count(Y, Z).
count(cons(cons(U, V), W), Z) :- ','(flatten(cons(cons(U, V), W), X), count(X, Z)).

Queries:

count(g,a).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
count_in: (b,f)
flatten_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

count_in_ga(atom(X), s(0)) → count_out_ga(atom(X), s(0))
count_in_ga(cons(atom(X), Y), s(Z)) → U3_ga(X, Y, Z, count_in_ga(Y, Z))
count_in_ga(cons(cons(U, V), W), Z) → U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_ga(U, V, W, Z, count_in_ga(X, Z))
U5_ga(U, V, W, Z, count_out_ga(X, Z)) → count_out_ga(cons(cons(U, V), W), Z)
U3_ga(X, Y, Z, count_out_ga(Y, Z)) → count_out_ga(cons(atom(X), Y), s(Z))

The argument filtering Pi contains the following mapping:
count_in_ga(x1, x2)  =  count_in_ga(x1)
atom(x1)  =  atom(x1)
count_out_ga(x1, x2)  =  count_out_ga(x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

count_in_ga(atom(X), s(0)) → count_out_ga(atom(X), s(0))
count_in_ga(cons(atom(X), Y), s(Z)) → U3_ga(X, Y, Z, count_in_ga(Y, Z))
count_in_ga(cons(cons(U, V), W), Z) → U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_ga(U, V, W, Z, count_in_ga(X, Z))
U5_ga(U, V, W, Z, count_out_ga(X, Z)) → count_out_ga(cons(cons(U, V), W), Z)
U3_ga(X, Y, Z, count_out_ga(Y, Z)) → count_out_ga(cons(atom(X), Y), s(Z))

The argument filtering Pi contains the following mapping:
count_in_ga(x1, x2)  =  count_in_ga(x1)
atom(x1)  =  atom(x1)
count_out_ga(x1, x2)  =  count_out_ga(x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)

(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:

COUNT_IN_GA(cons(atom(X), Y), s(Z)) → U3_GA(X, Y, Z, count_in_ga(Y, Z))
COUNT_IN_GA(cons(atom(X), Y), s(Z)) → COUNT_IN_GA(Y, Z)
COUNT_IN_GA(cons(cons(U, V), W), Z) → U4_GA(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
COUNT_IN_GA(cons(cons(U, V), W), Z) → FLATTEN_IN_GA(cons(cons(U, V), W), X)
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → U1_GA(X, U, Y, flatten_in_ga(U, Y))
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → FLATTEN_IN_GA(U, Y)
FLATTEN_IN_GA(cons(cons(U, V), W), X) → U2_GA(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
FLATTEN_IN_GA(cons(cons(U, V), W), X) → FLATTEN_IN_GA(cons(U, cons(V, W)), X)
U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_GA(U, V, W, Z, count_in_ga(X, Z))
U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → COUNT_IN_GA(X, Z)

The TRS R consists of the following rules:

count_in_ga(atom(X), s(0)) → count_out_ga(atom(X), s(0))
count_in_ga(cons(atom(X), Y), s(Z)) → U3_ga(X, Y, Z, count_in_ga(Y, Z))
count_in_ga(cons(cons(U, V), W), Z) → U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_ga(U, V, W, Z, count_in_ga(X, Z))
U5_ga(U, V, W, Z, count_out_ga(X, Z)) → count_out_ga(cons(cons(U, V), W), Z)
U3_ga(X, Y, Z, count_out_ga(Y, Z)) → count_out_ga(cons(atom(X), Y), s(Z))

The argument filtering Pi contains the following mapping:
count_in_ga(x1, x2)  =  count_in_ga(x1)
atom(x1)  =  atom(x1)
count_out_ga(x1, x2)  =  count_out_ga(x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
COUNT_IN_GA(x1, x2)  =  COUNT_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)
FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x5)

We have to consider all (P,R,Pi)-chains

(4) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

COUNT_IN_GA(cons(atom(X), Y), s(Z)) → U3_GA(X, Y, Z, count_in_ga(Y, Z))
COUNT_IN_GA(cons(atom(X), Y), s(Z)) → COUNT_IN_GA(Y, Z)
COUNT_IN_GA(cons(cons(U, V), W), Z) → U4_GA(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
COUNT_IN_GA(cons(cons(U, V), W), Z) → FLATTEN_IN_GA(cons(cons(U, V), W), X)
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → U1_GA(X, U, Y, flatten_in_ga(U, Y))
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → FLATTEN_IN_GA(U, Y)
FLATTEN_IN_GA(cons(cons(U, V), W), X) → U2_GA(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
FLATTEN_IN_GA(cons(cons(U, V), W), X) → FLATTEN_IN_GA(cons(U, cons(V, W)), X)
U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_GA(U, V, W, Z, count_in_ga(X, Z))
U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → COUNT_IN_GA(X, Z)

The TRS R consists of the following rules:

count_in_ga(atom(X), s(0)) → count_out_ga(atom(X), s(0))
count_in_ga(cons(atom(X), Y), s(Z)) → U3_ga(X, Y, Z, count_in_ga(Y, Z))
count_in_ga(cons(cons(U, V), W), Z) → U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_ga(U, V, W, Z, count_in_ga(X, Z))
U5_ga(U, V, W, Z, count_out_ga(X, Z)) → count_out_ga(cons(cons(U, V), W), Z)
U3_ga(X, Y, Z, count_out_ga(Y, Z)) → count_out_ga(cons(atom(X), Y), s(Z))

The argument filtering Pi contains the following mapping:
count_in_ga(x1, x2)  =  count_in_ga(x1)
atom(x1)  =  atom(x1)
count_out_ga(x1, x2)  =  count_out_ga(x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
COUNT_IN_GA(x1, x2)  =  COUNT_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)
FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x5)

We have to consider all (P,R,Pi)-chains

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

FLATTEN_IN_GA(cons(cons(U, V), W), X) → FLATTEN_IN_GA(cons(U, cons(V, W)), X)
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → FLATTEN_IN_GA(U, Y)

The TRS R consists of the following rules:

count_in_ga(atom(X), s(0)) → count_out_ga(atom(X), s(0))
count_in_ga(cons(atom(X), Y), s(Z)) → U3_ga(X, Y, Z, count_in_ga(Y, Z))
count_in_ga(cons(cons(U, V), W), Z) → U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_ga(U, V, W, Z, count_in_ga(X, Z))
U5_ga(U, V, W, Z, count_out_ga(X, Z)) → count_out_ga(cons(cons(U, V), W), Z)
U3_ga(X, Y, Z, count_out_ga(Y, Z)) → count_out_ga(cons(atom(X), Y), s(Z))

The argument filtering Pi contains the following mapping:
count_in_ga(x1, x2)  =  count_in_ga(x1)
atom(x1)  =  atom(x1)
count_out_ga(x1, x2)  =  count_out_ga(x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)

We have to consider all (P,R,Pi)-chains

(8) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(9) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

FLATTEN_IN_GA(cons(cons(U, V), W), X) → FLATTEN_IN_GA(cons(U, cons(V, W)), X)
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → FLATTEN_IN_GA(U, Y)

R is empty.
The argument filtering Pi contains the following mapping:
atom(x1)  =  atom(x1)
cons(x1, x2)  =  cons(x1, x2)
.(x1, x2)  =  .(x1, x2)
FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)

We have to consider all (P,R,Pi)-chains

(10) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(11) Obligation:

Q DP problem:
The TRS P consists of the following rules:

FLATTEN_IN_GA(cons(cons(U, V), W)) → FLATTEN_IN_GA(cons(U, cons(V, W)))
FLATTEN_IN_GA(cons(atom(X), U)) → FLATTEN_IN_GA(U)

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

(12) 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:

FLATTEN_IN_GA(cons(cons(U, V), W)) → FLATTEN_IN_GA(cons(U, cons(V, W)))
FLATTEN_IN_GA(cons(atom(X), U)) → FLATTEN_IN_GA(U)
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(FLATTEN_IN_GA(x1)) = 2·x1   
POL(atom(x1)) = x1   
POL(cons(x1, x2)) = 2 + 2·x1 + x2   

(13) Obligation:

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

(14) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(15) TRUE

(16) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

COUNT_IN_GA(cons(cons(U, V), W), Z) → U4_GA(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → COUNT_IN_GA(X, Z)
COUNT_IN_GA(cons(atom(X), Y), s(Z)) → COUNT_IN_GA(Y, Z)

The TRS R consists of the following rules:

count_in_ga(atom(X), s(0)) → count_out_ga(atom(X), s(0))
count_in_ga(cons(atom(X), Y), s(Z)) → U3_ga(X, Y, Z, count_in_ga(Y, Z))
count_in_ga(cons(cons(U, V), W), Z) → U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_ga(U, V, W, Z, count_in_ga(X, Z))
U5_ga(U, V, W, Z, count_out_ga(X, Z)) → count_out_ga(cons(cons(U, V), W), Z)
U3_ga(X, Y, Z, count_out_ga(Y, Z)) → count_out_ga(cons(atom(X), Y), s(Z))

The argument filtering Pi contains the following mapping:
count_in_ga(x1, x2)  =  count_in_ga(x1)
atom(x1)  =  atom(x1)
count_out_ga(x1, x2)  =  count_out_ga(x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
COUNT_IN_GA(x1, x2)  =  COUNT_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)

We have to consider all (P,R,Pi)-chains

(17) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(18) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

COUNT_IN_GA(cons(cons(U, V), W), Z) → U4_GA(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → COUNT_IN_GA(X, Z)
COUNT_IN_GA(cons(atom(X), Y), s(Z)) → COUNT_IN_GA(Y, Z)

The TRS R consists of the following rules:

flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))

The argument filtering Pi contains the following mapping:
atom(x1)  =  atom(x1)
cons(x1, x2)  =  cons(x1, x2)
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
COUNT_IN_GA(x1, x2)  =  COUNT_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)

We have to consider all (P,R,Pi)-chains

(19) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

COUNT_IN_GA(cons(cons(U, V), W)) → U4_GA(flatten_in_ga(cons(cons(U, V), W)))
U4_GA(flatten_out_ga(X)) → COUNT_IN_GA(X)
COUNT_IN_GA(cons(atom(X), Y)) → COUNT_IN_GA(Y)

The TRS R consists of the following rules:

flatten_in_ga(cons(cons(U, V), W)) → U2_ga(flatten_in_ga(cons(U, cons(V, W))))
U2_ga(flatten_out_ga(X)) → flatten_out_ga(X)
flatten_in_ga(cons(atom(X), U)) → U1_ga(X, flatten_in_ga(U))
U1_ga(X, flatten_out_ga(Y)) → flatten_out_ga(.(X, Y))
flatten_in_ga(atom(X)) → flatten_out_ga(.(X, []))

The set Q consists of the following terms:

flatten_in_ga(x0)
U2_ga(x0)
U1_ga(x0, x1)

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

(21) 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:

COUNT_IN_GA(cons(atom(X), Y)) → COUNT_IN_GA(Y)
The following rules are removed from R:

flatten_in_ga(cons(atom(X), U)) → U1_ga(X, flatten_in_ga(U))
flatten_in_ga(atom(X)) → flatten_out_ga(.(X, []))
U1_ga(X, flatten_out_ga(Y)) → flatten_out_ga(.(X, Y))
Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x1, x2)) = x1 + x2   
POL(COUNT_IN_GA(x1)) = x1   
POL(U1_ga(x1, x2)) = 1 + 2·x1 + x2   
POL(U2_ga(x1)) = x1   
POL(U4_GA(x1)) = x1   
POL([]) = 0   
POL(atom(x1)) = 1 + 2·x1   
POL(cons(x1, x2)) = 2·x1 + x2   
POL(flatten_in_ga(x1)) = x1   
POL(flatten_out_ga(x1)) = 2·x1   

(22) Obligation:

Q DP problem:
The TRS P consists of the following rules:

COUNT_IN_GA(cons(cons(U, V), W)) → U4_GA(flatten_in_ga(cons(cons(U, V), W)))
U4_GA(flatten_out_ga(X)) → COUNT_IN_GA(X)

The TRS R consists of the following rules:

flatten_in_ga(cons(cons(U, V), W)) → U2_ga(flatten_in_ga(cons(U, cons(V, W))))
U2_ga(flatten_out_ga(X)) → flatten_out_ga(X)

The set Q consists of the following terms:

flatten_in_ga(x0)
U2_ga(x0)
U1_ga(x0, x1)

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

(23) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

U1_ga(x0, x1)

(24) Obligation:

Q DP problem:
The TRS P consists of the following rules:

COUNT_IN_GA(cons(cons(U, V), W)) → U4_GA(flatten_in_ga(cons(cons(U, V), W)))
U4_GA(flatten_out_ga(X)) → COUNT_IN_GA(X)

The TRS R consists of the following rules:

flatten_in_ga(cons(cons(U, V), W)) → U2_ga(flatten_in_ga(cons(U, cons(V, W))))
U2_ga(flatten_out_ga(X)) → flatten_out_ga(X)

The set Q consists of the following terms:

flatten_in_ga(x0)
U2_ga(x0)

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

(25) 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:

COUNT_IN_GA(cons(cons(U, V), W)) → U4_GA(flatten_in_ga(cons(cons(U, V), W)))
U4_GA(flatten_out_ga(X)) → COUNT_IN_GA(X)


Used ordering: Polynomial interpretation [POLO]:

POL(COUNT_IN_GA(x1)) = 1 + x1   
POL(U2_ga(x1)) = x1   
POL(U4_GA(x1)) = x1   
POL(cons(x1, x2)) = x1 + x2   
POL(flatten_in_ga(x1)) = x1   
POL(flatten_out_ga(x1)) = 2 + x1   

(26) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

flatten_in_ga(cons(cons(U, V), W)) → U2_ga(flatten_in_ga(cons(U, cons(V, W))))
U2_ga(flatten_out_ga(X)) → flatten_out_ga(X)

The set Q consists of the following terms:

flatten_in_ga(x0)
U2_ga(x0)

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

(27) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(28) TRUE

(29) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
count_in: (b,f)
flatten_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

count_in_ga(atom(X), s(0)) → count_out_ga(atom(X), s(0))
count_in_ga(cons(atom(X), Y), s(Z)) → U3_ga(X, Y, Z, count_in_ga(Y, Z))
count_in_ga(cons(cons(U, V), W), Z) → U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_ga(U, V, W, Z, count_in_ga(X, Z))
U5_ga(U, V, W, Z, count_out_ga(X, Z)) → count_out_ga(cons(cons(U, V), W), Z)
U3_ga(X, Y, Z, count_out_ga(Y, Z)) → count_out_ga(cons(atom(X), Y), s(Z))

The argument filtering Pi contains the following mapping:
count_in_ga(x1, x2)  =  count_in_ga(x1)
atom(x1)  =  atom(x1)
count_out_ga(x1, x2)  =  count_out_ga(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x3, x5)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(30) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

count_in_ga(atom(X), s(0)) → count_out_ga(atom(X), s(0))
count_in_ga(cons(atom(X), Y), s(Z)) → U3_ga(X, Y, Z, count_in_ga(Y, Z))
count_in_ga(cons(cons(U, V), W), Z) → U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_ga(U, V, W, Z, count_in_ga(X, Z))
U5_ga(U, V, W, Z, count_out_ga(X, Z)) → count_out_ga(cons(cons(U, V), W), Z)
U3_ga(X, Y, Z, count_out_ga(Y, Z)) → count_out_ga(cons(atom(X), Y), s(Z))

The argument filtering Pi contains the following mapping:
count_in_ga(x1, x2)  =  count_in_ga(x1)
atom(x1)  =  atom(x1)
count_out_ga(x1, x2)  =  count_out_ga(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x3, x5)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)

(31) 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:

COUNT_IN_GA(cons(atom(X), Y), s(Z)) → U3_GA(X, Y, Z, count_in_ga(Y, Z))
COUNT_IN_GA(cons(atom(X), Y), s(Z)) → COUNT_IN_GA(Y, Z)
COUNT_IN_GA(cons(cons(U, V), W), Z) → U4_GA(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
COUNT_IN_GA(cons(cons(U, V), W), Z) → FLATTEN_IN_GA(cons(cons(U, V), W), X)
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → U1_GA(X, U, Y, flatten_in_ga(U, Y))
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → FLATTEN_IN_GA(U, Y)
FLATTEN_IN_GA(cons(cons(U, V), W), X) → U2_GA(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
FLATTEN_IN_GA(cons(cons(U, V), W), X) → FLATTEN_IN_GA(cons(U, cons(V, W)), X)
U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_GA(U, V, W, Z, count_in_ga(X, Z))
U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → COUNT_IN_GA(X, Z)

The TRS R consists of the following rules:

count_in_ga(atom(X), s(0)) → count_out_ga(atom(X), s(0))
count_in_ga(cons(atom(X), Y), s(Z)) → U3_ga(X, Y, Z, count_in_ga(Y, Z))
count_in_ga(cons(cons(U, V), W), Z) → U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_ga(U, V, W, Z, count_in_ga(X, Z))
U5_ga(U, V, W, Z, count_out_ga(X, Z)) → count_out_ga(cons(cons(U, V), W), Z)
U3_ga(X, Y, Z, count_out_ga(Y, Z)) → count_out_ga(cons(atom(X), Y), s(Z))

The argument filtering Pi contains the following mapping:
count_in_ga(x1, x2)  =  count_in_ga(x1)
atom(x1)  =  atom(x1)
count_out_ga(x1, x2)  =  count_out_ga(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x3, x5)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
COUNT_IN_GA(x1, x2)  =  COUNT_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x2, x4)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x3, x5)
FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x3, x5)

We have to consider all (P,R,Pi)-chains

(32) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

COUNT_IN_GA(cons(atom(X), Y), s(Z)) → U3_GA(X, Y, Z, count_in_ga(Y, Z))
COUNT_IN_GA(cons(atom(X), Y), s(Z)) → COUNT_IN_GA(Y, Z)
COUNT_IN_GA(cons(cons(U, V), W), Z) → U4_GA(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
COUNT_IN_GA(cons(cons(U, V), W), Z) → FLATTEN_IN_GA(cons(cons(U, V), W), X)
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → U1_GA(X, U, Y, flatten_in_ga(U, Y))
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → FLATTEN_IN_GA(U, Y)
FLATTEN_IN_GA(cons(cons(U, V), W), X) → U2_GA(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
FLATTEN_IN_GA(cons(cons(U, V), W), X) → FLATTEN_IN_GA(cons(U, cons(V, W)), X)
U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_GA(U, V, W, Z, count_in_ga(X, Z))
U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → COUNT_IN_GA(X, Z)

The TRS R consists of the following rules:

count_in_ga(atom(X), s(0)) → count_out_ga(atom(X), s(0))
count_in_ga(cons(atom(X), Y), s(Z)) → U3_ga(X, Y, Z, count_in_ga(Y, Z))
count_in_ga(cons(cons(U, V), W), Z) → U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_ga(U, V, W, Z, count_in_ga(X, Z))
U5_ga(U, V, W, Z, count_out_ga(X, Z)) → count_out_ga(cons(cons(U, V), W), Z)
U3_ga(X, Y, Z, count_out_ga(Y, Z)) → count_out_ga(cons(atom(X), Y), s(Z))

The argument filtering Pi contains the following mapping:
count_in_ga(x1, x2)  =  count_in_ga(x1)
atom(x1)  =  atom(x1)
count_out_ga(x1, x2)  =  count_out_ga(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x3, x5)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
COUNT_IN_GA(x1, x2)  =  COUNT_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x2, x4)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x3, x5)
FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x3, x5)

We have to consider all (P,R,Pi)-chains

(33) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.

(34) Complex Obligation (AND)

(35) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

FLATTEN_IN_GA(cons(cons(U, V), W), X) → FLATTEN_IN_GA(cons(U, cons(V, W)), X)
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → FLATTEN_IN_GA(U, Y)

The TRS R consists of the following rules:

count_in_ga(atom(X), s(0)) → count_out_ga(atom(X), s(0))
count_in_ga(cons(atom(X), Y), s(Z)) → U3_ga(X, Y, Z, count_in_ga(Y, Z))
count_in_ga(cons(cons(U, V), W), Z) → U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_ga(U, V, W, Z, count_in_ga(X, Z))
U5_ga(U, V, W, Z, count_out_ga(X, Z)) → count_out_ga(cons(cons(U, V), W), Z)
U3_ga(X, Y, Z, count_out_ga(Y, Z)) → count_out_ga(cons(atom(X), Y), s(Z))

The argument filtering Pi contains the following mapping:
count_in_ga(x1, x2)  =  count_in_ga(x1)
atom(x1)  =  atom(x1)
count_out_ga(x1, x2)  =  count_out_ga(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x3, x5)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)

We have to consider all (P,R,Pi)-chains

(36) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(37) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

FLATTEN_IN_GA(cons(cons(U, V), W), X) → FLATTEN_IN_GA(cons(U, cons(V, W)), X)
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → FLATTEN_IN_GA(U, Y)

R is empty.
The argument filtering Pi contains the following mapping:
atom(x1)  =  atom(x1)
cons(x1, x2)  =  cons(x1, x2)
.(x1, x2)  =  .(x1, x2)
FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)

We have to consider all (P,R,Pi)-chains

(38) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(39) Obligation:

Q DP problem:
The TRS P consists of the following rules:

FLATTEN_IN_GA(cons(cons(U, V), W)) → FLATTEN_IN_GA(cons(U, cons(V, W)))
FLATTEN_IN_GA(cons(atom(X), U)) → FLATTEN_IN_GA(U)

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

(40) 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:

FLATTEN_IN_GA(cons(cons(U, V), W)) → FLATTEN_IN_GA(cons(U, cons(V, W)))
FLATTEN_IN_GA(cons(atom(X), U)) → FLATTEN_IN_GA(U)
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(FLATTEN_IN_GA(x1)) = 2·x1   
POL(atom(x1)) = x1   
POL(cons(x1, x2)) = 2 + 2·x1 + x2   

(41) Obligation:

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

(42) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(43) TRUE

(44) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

COUNT_IN_GA(cons(cons(U, V), W), Z) → U4_GA(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → COUNT_IN_GA(X, Z)
COUNT_IN_GA(cons(atom(X), Y), s(Z)) → COUNT_IN_GA(Y, Z)

The TRS R consists of the following rules:

count_in_ga(atom(X), s(0)) → count_out_ga(atom(X), s(0))
count_in_ga(cons(atom(X), Y), s(Z)) → U3_ga(X, Y, Z, count_in_ga(Y, Z))
count_in_ga(cons(cons(U, V), W), Z) → U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_ga(U, V, W, Z, count_in_ga(X, Z))
U5_ga(U, V, W, Z, count_out_ga(X, Z)) → count_out_ga(cons(cons(U, V), W), Z)
U3_ga(X, Y, Z, count_out_ga(Y, Z)) → count_out_ga(cons(atom(X), Y), s(Z))

The argument filtering Pi contains the following mapping:
count_in_ga(x1, x2)  =  count_in_ga(x1)
atom(x1)  =  atom(x1)
count_out_ga(x1, x2)  =  count_out_ga(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x3, x5)
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x3, x5)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
COUNT_IN_GA(x1, x2)  =  COUNT_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x3, x5)

We have to consider all (P,R,Pi)-chains

(45) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(46) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

COUNT_IN_GA(cons(cons(U, V), W), Z) → U4_GA(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → COUNT_IN_GA(X, Z)
COUNT_IN_GA(cons(atom(X), Y), s(Z)) → COUNT_IN_GA(Y, Z)

The TRS R consists of the following rules:

flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))

The argument filtering Pi contains the following mapping:
atom(x1)  =  atom(x1)
cons(x1, x2)  =  cons(x1, x2)
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
COUNT_IN_GA(x1, x2)  =  COUNT_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x3, x5)

We have to consider all (P,R,Pi)-chains

(47) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(48) Obligation:

Q DP problem:
The TRS P consists of the following rules:

COUNT_IN_GA(cons(cons(U, V), W)) → U4_GA(U, V, W, flatten_in_ga(cons(cons(U, V), W)))
U4_GA(U, V, W, flatten_out_ga(cons(cons(U, V), W), X)) → COUNT_IN_GA(X)
COUNT_IN_GA(cons(atom(X), Y)) → COUNT_IN_GA(Y)

The TRS R consists of the following rules:

flatten_in_ga(cons(cons(U, V), W)) → U2_ga(U, V, W, flatten_in_ga(cons(U, cons(V, W))))
U2_ga(U, V, W, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
flatten_in_ga(cons(atom(X), U)) → U1_ga(X, U, flatten_in_ga(U))
U1_ga(X, U, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
flatten_in_ga(atom(X)) → flatten_out_ga(atom(X), .(X, []))

The set Q consists of the following terms:

flatten_in_ga(x0)
U2_ga(x0, x1, x2, x3)
U1_ga(x0, x1, x2)

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