Left Termination of the query pattern p_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 PiDP
    ↳7 UsableRulesProof (⇔)
    ↳8 PiDP
    ↳9 PiDPToQDPProof (⇐)
    ↳10 QDP
    ↳11 MRRProof (⇔)
    ↳12 QDP
    ↳13 PisEmptyProof (⇔)
    ↳14 TRUE
  ↳15 PrologToPiTRSProof (⇐)
    ↳16 PiTRS
    ↳17 DependencyPairsProof (⇔)
    ↳18 PiDP
    ↳19 DependencyGraphProof (⇔)
    ↳20 PiDP
    ↳21 UsableRulesProof (⇔)
    ↳22 PiDP
    ↳23 PiDPToQDPProof (⇐)
    ↳24 QDP

(0) Obligation:

Clauses:

p(X, X).
p(f(X), g(Y)) :- ','(p(f(X), f(Z)), p(Z, g(W))).

Queries:

p(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:
p_in: (b,f) (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p_in_ga(X, X) → p_out_ga(X, X)
p_in_ga(f(X), g(Y)) → U1_ga(X, Y, p_in_ga(f(X), f(Z)))
U1_ga(X, Y, p_out_ga(f(X), f(Z))) → U2_ga(X, Y, Z, p_in_gg(Z, g(W)))
p_in_gg(X, X) → p_out_gg(X, X)
p_in_gg(f(X), g(Y)) → U1_gg(X, Y, p_in_ga(f(X), f(Z)))
U1_gg(X, Y, p_out_ga(f(X), f(Z))) → U2_gg(X, Y, Z, p_in_gg(Z, g(W)))
U2_gg(X, Y, Z, p_out_gg(Z, g(W))) → p_out_gg(f(X), g(Y))
U2_ga(X, Y, Z, p_out_gg(Z, g(W))) → p_out_ga(f(X), g(Y))

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
f(x1)  =  f(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
g(x1)  =  g
p_out_gg(x1, x2)  =  p_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

p_in_ga(X, X) → p_out_ga(X, X)
p_in_ga(f(X), g(Y)) → U1_ga(X, Y, p_in_ga(f(X), f(Z)))
U1_ga(X, Y, p_out_ga(f(X), f(Z))) → U2_ga(X, Y, Z, p_in_gg(Z, g(W)))
p_in_gg(X, X) → p_out_gg(X, X)
p_in_gg(f(X), g(Y)) → U1_gg(X, Y, p_in_ga(f(X), f(Z)))
U1_gg(X, Y, p_out_ga(f(X), f(Z))) → U2_gg(X, Y, Z, p_in_gg(Z, g(W)))
U2_gg(X, Y, Z, p_out_gg(Z, g(W))) → p_out_gg(f(X), g(Y))
U2_ga(X, Y, Z, p_out_gg(Z, g(W))) → p_out_ga(f(X), g(Y))

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
f(x1)  =  f(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
g(x1)  =  g
p_out_gg(x1, x2)  =  p_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U2_gg(x1, x2, x3, x4)  =  U2_gg(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:

P_IN_GA(f(X), g(Y)) → U1_GA(X, Y, p_in_ga(f(X), f(Z)))
P_IN_GA(f(X), g(Y)) → P_IN_GA(f(X), f(Z))
U1_GA(X, Y, p_out_ga(f(X), f(Z))) → U2_GA(X, Y, Z, p_in_gg(Z, g(W)))
U1_GA(X, Y, p_out_ga(f(X), f(Z))) → P_IN_GG(Z, g(W))
P_IN_GG(f(X), g(Y)) → U1_GG(X, Y, p_in_ga(f(X), f(Z)))
P_IN_GG(f(X), g(Y)) → P_IN_GA(f(X), f(Z))
U1_GG(X, Y, p_out_ga(f(X), f(Z))) → U2_GG(X, Y, Z, p_in_gg(Z, g(W)))
U1_GG(X, Y, p_out_ga(f(X), f(Z))) → P_IN_GG(Z, g(W))

The TRS R consists of the following rules:

p_in_ga(X, X) → p_out_ga(X, X)
p_in_ga(f(X), g(Y)) → U1_ga(X, Y, p_in_ga(f(X), f(Z)))
U1_ga(X, Y, p_out_ga(f(X), f(Z))) → U2_ga(X, Y, Z, p_in_gg(Z, g(W)))
p_in_gg(X, X) → p_out_gg(X, X)
p_in_gg(f(X), g(Y)) → U1_gg(X, Y, p_in_ga(f(X), f(Z)))
U1_gg(X, Y, p_out_ga(f(X), f(Z))) → U2_gg(X, Y, Z, p_in_gg(Z, g(W)))
U2_gg(X, Y, Z, p_out_gg(Z, g(W))) → p_out_gg(f(X), g(Y))
U2_ga(X, Y, Z, p_out_gg(Z, g(W))) → p_out_ga(f(X), g(Y))

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
f(x1)  =  f(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
g(x1)  =  g
p_out_gg(x1, x2)  =  p_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
P_IN_GG(x1, x2)  =  P_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x4)

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

(4) Obligation:

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

P_IN_GA(f(X), g(Y)) → U1_GA(X, Y, p_in_ga(f(X), f(Z)))
P_IN_GA(f(X), g(Y)) → P_IN_GA(f(X), f(Z))
U1_GA(X, Y, p_out_ga(f(X), f(Z))) → U2_GA(X, Y, Z, p_in_gg(Z, g(W)))
U1_GA(X, Y, p_out_ga(f(X), f(Z))) → P_IN_GG(Z, g(W))
P_IN_GG(f(X), g(Y)) → U1_GG(X, Y, p_in_ga(f(X), f(Z)))
P_IN_GG(f(X), g(Y)) → P_IN_GA(f(X), f(Z))
U1_GG(X, Y, p_out_ga(f(X), f(Z))) → U2_GG(X, Y, Z, p_in_gg(Z, g(W)))
U1_GG(X, Y, p_out_ga(f(X), f(Z))) → P_IN_GG(Z, g(W))

The TRS R consists of the following rules:

p_in_ga(X, X) → p_out_ga(X, X)
p_in_ga(f(X), g(Y)) → U1_ga(X, Y, p_in_ga(f(X), f(Z)))
U1_ga(X, Y, p_out_ga(f(X), f(Z))) → U2_ga(X, Y, Z, p_in_gg(Z, g(W)))
p_in_gg(X, X) → p_out_gg(X, X)
p_in_gg(f(X), g(Y)) → U1_gg(X, Y, p_in_ga(f(X), f(Z)))
U1_gg(X, Y, p_out_ga(f(X), f(Z))) → U2_gg(X, Y, Z, p_in_gg(Z, g(W)))
U2_gg(X, Y, Z, p_out_gg(Z, g(W))) → p_out_gg(f(X), g(Y))
U2_ga(X, Y, Z, p_out_gg(Z, g(W))) → p_out_ga(f(X), g(Y))

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
f(x1)  =  f(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
g(x1)  =  g
p_out_gg(x1, x2)  =  p_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
P_IN_GG(x1, x2)  =  P_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)
U2_GG(x1, x2, x3, x4)  =  U2_GG(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 6 less nodes.

(6) Obligation:

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

U1_GG(X, Y, p_out_ga(f(X), f(Z))) → P_IN_GG(Z, g(W))
P_IN_GG(f(X), g(Y)) → U1_GG(X, Y, p_in_ga(f(X), f(Z)))

The TRS R consists of the following rules:

p_in_ga(X, X) → p_out_ga(X, X)
p_in_ga(f(X), g(Y)) → U1_ga(X, Y, p_in_ga(f(X), f(Z)))
U1_ga(X, Y, p_out_ga(f(X), f(Z))) → U2_ga(X, Y, Z, p_in_gg(Z, g(W)))
p_in_gg(X, X) → p_out_gg(X, X)
p_in_gg(f(X), g(Y)) → U1_gg(X, Y, p_in_ga(f(X), f(Z)))
U1_gg(X, Y, p_out_ga(f(X), f(Z))) → U2_gg(X, Y, Z, p_in_gg(Z, g(W)))
U2_gg(X, Y, Z, p_out_gg(Z, g(W))) → p_out_gg(f(X), g(Y))
U2_ga(X, Y, Z, p_out_gg(Z, g(W))) → p_out_ga(f(X), g(Y))

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
f(x1)  =  f(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
g(x1)  =  g
p_out_gg(x1, x2)  =  p_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
P_IN_GG(x1, x2)  =  P_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)

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:

U1_GG(X, Y, p_out_ga(f(X), f(Z))) → P_IN_GG(Z, g(W))
P_IN_GG(f(X), g(Y)) → U1_GG(X, Y, p_in_ga(f(X), f(Z)))

The TRS R consists of the following rules:

p_in_ga(X, X) → p_out_ga(X, X)

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
f(x1)  =  f(x1)
g(x1)  =  g
P_IN_GG(x1, x2)  =  P_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)

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:

U1_GG(p_out_ga(f(Z))) → P_IN_GG(Z, g)
P_IN_GG(f(X), g) → U1_GG(p_in_ga(f(X)))

The TRS R consists of the following rules:

p_in_ga(X) → p_out_ga(X)

The set Q consists of the following terms:

p_in_ga(x0)

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:

U1_GG(p_out_ga(f(Z))) → P_IN_GG(Z, g)
P_IN_GG(f(X), g) → U1_GG(p_in_ga(f(X)))

Strictly oriented rules of the TRS R:

p_in_ga(X) → p_out_ga(X)

Used ordering: Polynomial interpretation [POLO]:

POL(P_IN_GG(x1, x2)) = 2 + x1 + x2   
POL(U1_GG(x1)) = x1   
POL(f(x1)) = 3 + x1   
POL(g) = 0   
POL(p_in_ga(x1)) = 1 + x1   
POL(p_out_ga(x1)) = x1   

(12) Obligation:

Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:

p_in_ga(x0)

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) TRUE

(15) PrologToPiTRSProof (SOUND transformation)

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

p_in_ga(X, X) → p_out_ga(X, X)
p_in_ga(f(X), g(Y)) → U1_ga(X, Y, p_in_ga(f(X), f(Z)))
U1_ga(X, Y, p_out_ga(f(X), f(Z))) → U2_ga(X, Y, Z, p_in_gg(Z, g(W)))
p_in_gg(X, X) → p_out_gg(X, X)
p_in_gg(f(X), g(Y)) → U1_gg(X, Y, p_in_ga(f(X), f(Z)))
U1_gg(X, Y, p_out_ga(f(X), f(Z))) → U2_gg(X, Y, Z, p_in_gg(Z, g(W)))
U2_gg(X, Y, Z, p_out_gg(Z, g(W))) → p_out_gg(f(X), g(Y))
U2_ga(X, Y, Z, p_out_gg(Z, g(W))) → p_out_ga(f(X), g(Y))

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
f(x1)  =  f(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
g(x1)  =  g
p_out_gg(x1, x2)  =  p_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x3)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(16) Obligation:

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

p_in_ga(X, X) → p_out_ga(X, X)
p_in_ga(f(X), g(Y)) → U1_ga(X, Y, p_in_ga(f(X), f(Z)))
U1_ga(X, Y, p_out_ga(f(X), f(Z))) → U2_ga(X, Y, Z, p_in_gg(Z, g(W)))
p_in_gg(X, X) → p_out_gg(X, X)
p_in_gg(f(X), g(Y)) → U1_gg(X, Y, p_in_ga(f(X), f(Z)))
U1_gg(X, Y, p_out_ga(f(X), f(Z))) → U2_gg(X, Y, Z, p_in_gg(Z, g(W)))
U2_gg(X, Y, Z, p_out_gg(Z, g(W))) → p_out_gg(f(X), g(Y))
U2_ga(X, Y, Z, p_out_gg(Z, g(W))) → p_out_ga(f(X), g(Y))

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
f(x1)  =  f(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
g(x1)  =  g
p_out_gg(x1, x2)  =  p_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x3)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x4)

(17) 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:

P_IN_GA(f(X), g(Y)) → U1_GA(X, Y, p_in_ga(f(X), f(Z)))
P_IN_GA(f(X), g(Y)) → P_IN_GA(f(X), f(Z))
U1_GA(X, Y, p_out_ga(f(X), f(Z))) → U2_GA(X, Y, Z, p_in_gg(Z, g(W)))
U1_GA(X, Y, p_out_ga(f(X), f(Z))) → P_IN_GG(Z, g(W))
P_IN_GG(f(X), g(Y)) → U1_GG(X, Y, p_in_ga(f(X), f(Z)))
P_IN_GG(f(X), g(Y)) → P_IN_GA(f(X), f(Z))
U1_GG(X, Y, p_out_ga(f(X), f(Z))) → U2_GG(X, Y, Z, p_in_gg(Z, g(W)))
U1_GG(X, Y, p_out_ga(f(X), f(Z))) → P_IN_GG(Z, g(W))

The TRS R consists of the following rules:

p_in_ga(X, X) → p_out_ga(X, X)
p_in_ga(f(X), g(Y)) → U1_ga(X, Y, p_in_ga(f(X), f(Z)))
U1_ga(X, Y, p_out_ga(f(X), f(Z))) → U2_ga(X, Y, Z, p_in_gg(Z, g(W)))
p_in_gg(X, X) → p_out_gg(X, X)
p_in_gg(f(X), g(Y)) → U1_gg(X, Y, p_in_ga(f(X), f(Z)))
U1_gg(X, Y, p_out_ga(f(X), f(Z))) → U2_gg(X, Y, Z, p_in_gg(Z, g(W)))
U2_gg(X, Y, Z, p_out_gg(Z, g(W))) → p_out_gg(f(X), g(Y))
U2_ga(X, Y, Z, p_out_gg(Z, g(W))) → p_out_ga(f(X), g(Y))

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
f(x1)  =  f(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
g(x1)  =  g
p_out_gg(x1, x2)  =  p_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x3)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
P_IN_GG(x1, x2)  =  P_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x3)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x1, x4)

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

(18) Obligation:

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

P_IN_GA(f(X), g(Y)) → U1_GA(X, Y, p_in_ga(f(X), f(Z)))
P_IN_GA(f(X), g(Y)) → P_IN_GA(f(X), f(Z))
U1_GA(X, Y, p_out_ga(f(X), f(Z))) → U2_GA(X, Y, Z, p_in_gg(Z, g(W)))
U1_GA(X, Y, p_out_ga(f(X), f(Z))) → P_IN_GG(Z, g(W))
P_IN_GG(f(X), g(Y)) → U1_GG(X, Y, p_in_ga(f(X), f(Z)))
P_IN_GG(f(X), g(Y)) → P_IN_GA(f(X), f(Z))
U1_GG(X, Y, p_out_ga(f(X), f(Z))) → U2_GG(X, Y, Z, p_in_gg(Z, g(W)))
U1_GG(X, Y, p_out_ga(f(X), f(Z))) → P_IN_GG(Z, g(W))

The TRS R consists of the following rules:

p_in_ga(X, X) → p_out_ga(X, X)
p_in_ga(f(X), g(Y)) → U1_ga(X, Y, p_in_ga(f(X), f(Z)))
U1_ga(X, Y, p_out_ga(f(X), f(Z))) → U2_ga(X, Y, Z, p_in_gg(Z, g(W)))
p_in_gg(X, X) → p_out_gg(X, X)
p_in_gg(f(X), g(Y)) → U1_gg(X, Y, p_in_ga(f(X), f(Z)))
U1_gg(X, Y, p_out_ga(f(X), f(Z))) → U2_gg(X, Y, Z, p_in_gg(Z, g(W)))
U2_gg(X, Y, Z, p_out_gg(Z, g(W))) → p_out_gg(f(X), g(Y))
U2_ga(X, Y, Z, p_out_gg(Z, g(W))) → p_out_ga(f(X), g(Y))

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
f(x1)  =  f(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
g(x1)  =  g
p_out_gg(x1, x2)  =  p_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x3)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
P_IN_GG(x1, x2)  =  P_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x3)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x1, x4)

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

(19) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 6 less nodes.

(20) Obligation:

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

U1_GG(X, Y, p_out_ga(f(X), f(Z))) → P_IN_GG(Z, g(W))
P_IN_GG(f(X), g(Y)) → U1_GG(X, Y, p_in_ga(f(X), f(Z)))

The TRS R consists of the following rules:

p_in_ga(X, X) → p_out_ga(X, X)
p_in_ga(f(X), g(Y)) → U1_ga(X, Y, p_in_ga(f(X), f(Z)))
U1_ga(X, Y, p_out_ga(f(X), f(Z))) → U2_ga(X, Y, Z, p_in_gg(Z, g(W)))
p_in_gg(X, X) → p_out_gg(X, X)
p_in_gg(f(X), g(Y)) → U1_gg(X, Y, p_in_ga(f(X), f(Z)))
U1_gg(X, Y, p_out_ga(f(X), f(Z))) → U2_gg(X, Y, Z, p_in_gg(Z, g(W)))
U2_gg(X, Y, Z, p_out_gg(Z, g(W))) → p_out_gg(f(X), g(Y))
U2_ga(X, Y, Z, p_out_gg(Z, g(W))) → p_out_ga(f(X), g(Y))

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
f(x1)  =  f(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
g(x1)  =  g
p_out_gg(x1, x2)  =  p_out_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x3)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x1, x4)
P_IN_GG(x1, x2)  =  P_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x3)

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

(21) UsableRulesProof (EQUIVALENT transformation)

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

(22) Obligation:

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

U1_GG(X, Y, p_out_ga(f(X), f(Z))) → P_IN_GG(Z, g(W))
P_IN_GG(f(X), g(Y)) → U1_GG(X, Y, p_in_ga(f(X), f(Z)))

The TRS R consists of the following rules:

p_in_ga(X, X) → p_out_ga(X, X)

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
f(x1)  =  f(x1)
g(x1)  =  g
P_IN_GG(x1, x2)  =  P_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x3)

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

(23) PiDPToQDPProof (SOUND transformation)

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

(24) Obligation:

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

U1_GG(X, p_out_ga(f(X), f(Z))) → P_IN_GG(Z, g)
P_IN_GG(f(X), g) → U1_GG(X, p_in_ga(f(X)))

The TRS R consists of the following rules:

p_in_ga(X) → p_out_ga(X, X)

The set Q consists of the following terms:

p_in_ga(x0)

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