Left Termination of the query pattern h_in_1(g) 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 MRRProof (⇔)
        ↳13 QDP
        ↳14 PisEmptyProof (⇔)
        ↳15 TRUE
      ↳16 PiDP
        ↳17 UsableRulesProof (⇔)
        ↳18 PiDP
        ↳19 PiDPToQDPProof (⇔)
        ↳20 QDP
        ↳21 MRRProof (⇔)
        ↳22 QDP
        ↳23 PisEmptyProof (⇔)
        ↳24 TRUE
  ↳25 PrologToPiTRSProof (⇐)
    ↳26 PiTRS
    ↳27 DependencyPairsProof (⇔)
    ↳28 PiDP
    ↳29 DependencyGraphProof (⇔)
    ↳30 AND
      ↳31 PiDP
        ↳32 UsableRulesProof (⇔)
        ↳33 PiDP
        ↳34 PiDPToQDPProof (⇔)
        ↳35 QDP
        ↳36 MRRProof (⇔)
        ↳37 QDP
        ↳38 PisEmptyProof (⇔)
        ↳39 TRUE
      ↳40 PiDP
        ↳41 UsableRulesProof (⇔)
        ↳42 PiDP
        ↳43 PiDPToQDPProof (⇔)
        ↳44 QDP

(0) Obligation:

Clauses:

f(c(s(X), Y)) :- f(c(X, s(Y))).
g(c(X, s(Y))) :- g(c(s(X), Y)).
h(X) :- ','(f(X), g(X)).

Queries:

h(g).

(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:
h_in: (b)
f_in: (b)
g_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

h_in_g(X) → U3_g(X, f_in_g(X))
f_in_g(c(s(X), Y)) → U1_g(X, Y, f_in_g(c(X, s(Y))))
U1_g(X, Y, f_out_g(c(X, s(Y)))) → f_out_g(c(s(X), Y))
U3_g(X, f_out_g(X)) → U4_g(X, g_in_g(X))
g_in_g(c(X, s(Y))) → U2_g(X, Y, g_in_g(c(s(X), Y)))
U2_g(X, Y, g_out_g(c(s(X), Y))) → g_out_g(c(X, s(Y)))
U4_g(X, g_out_g(X)) → h_out_g(X)

The argument filtering Pi contains the following mapping:
h_in_g(x1)  =  h_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f_in_g(x1)  =  f_in_g(x1)
c(x1, x2)  =  c(x1, x2)
s(x1)  =  s(x1)
U1_g(x1, x2, x3)  =  U1_g(x3)
f_out_g(x1)  =  f_out_g
U4_g(x1, x2)  =  U4_g(x2)
g_in_g(x1)  =  g_in_g(x1)
U2_g(x1, x2, x3)  =  U2_g(x3)
g_out_g(x1)  =  g_out_g
h_out_g(x1)  =  h_out_g

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

h_in_g(X) → U3_g(X, f_in_g(X))
f_in_g(c(s(X), Y)) → U1_g(X, Y, f_in_g(c(X, s(Y))))
U1_g(X, Y, f_out_g(c(X, s(Y)))) → f_out_g(c(s(X), Y))
U3_g(X, f_out_g(X)) → U4_g(X, g_in_g(X))
g_in_g(c(X, s(Y))) → U2_g(X, Y, g_in_g(c(s(X), Y)))
U2_g(X, Y, g_out_g(c(s(X), Y))) → g_out_g(c(X, s(Y)))
U4_g(X, g_out_g(X)) → h_out_g(X)

The argument filtering Pi contains the following mapping:
h_in_g(x1)  =  h_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f_in_g(x1)  =  f_in_g(x1)
c(x1, x2)  =  c(x1, x2)
s(x1)  =  s(x1)
U1_g(x1, x2, x3)  =  U1_g(x3)
f_out_g(x1)  =  f_out_g
U4_g(x1, x2)  =  U4_g(x2)
g_in_g(x1)  =  g_in_g(x1)
U2_g(x1, x2, x3)  =  U2_g(x3)
g_out_g(x1)  =  g_out_g
h_out_g(x1)  =  h_out_g

(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:

H_IN_G(X) → U3_G(X, f_in_g(X))
H_IN_G(X) → F_IN_G(X)
F_IN_G(c(s(X), Y)) → U1_G(X, Y, f_in_g(c(X, s(Y))))
F_IN_G(c(s(X), Y)) → F_IN_G(c(X, s(Y)))
U3_G(X, f_out_g(X)) → U4_G(X, g_in_g(X))
U3_G(X, f_out_g(X)) → G_IN_G(X)
G_IN_G(c(X, s(Y))) → U2_G(X, Y, g_in_g(c(s(X), Y)))
G_IN_G(c(X, s(Y))) → G_IN_G(c(s(X), Y))

The TRS R consists of the following rules:

h_in_g(X) → U3_g(X, f_in_g(X))
f_in_g(c(s(X), Y)) → U1_g(X, Y, f_in_g(c(X, s(Y))))
U1_g(X, Y, f_out_g(c(X, s(Y)))) → f_out_g(c(s(X), Y))
U3_g(X, f_out_g(X)) → U4_g(X, g_in_g(X))
g_in_g(c(X, s(Y))) → U2_g(X, Y, g_in_g(c(s(X), Y)))
U2_g(X, Y, g_out_g(c(s(X), Y))) → g_out_g(c(X, s(Y)))
U4_g(X, g_out_g(X)) → h_out_g(X)

The argument filtering Pi contains the following mapping:
h_in_g(x1)  =  h_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f_in_g(x1)  =  f_in_g(x1)
c(x1, x2)  =  c(x1, x2)
s(x1)  =  s(x1)
U1_g(x1, x2, x3)  =  U1_g(x3)
f_out_g(x1)  =  f_out_g
U4_g(x1, x2)  =  U4_g(x2)
g_in_g(x1)  =  g_in_g(x1)
U2_g(x1, x2, x3)  =  U2_g(x3)
g_out_g(x1)  =  g_out_g
h_out_g(x1)  =  h_out_g
H_IN_G(x1)  =  H_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x1, x2)
F_IN_G(x1)  =  F_IN_G(x1)
U1_G(x1, x2, x3)  =  U1_G(x3)
U4_G(x1, x2)  =  U4_G(x2)
G_IN_G(x1)  =  G_IN_G(x1)
U2_G(x1, x2, x3)  =  U2_G(x3)

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

(4) Obligation:

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

H_IN_G(X) → U3_G(X, f_in_g(X))
H_IN_G(X) → F_IN_G(X)
F_IN_G(c(s(X), Y)) → U1_G(X, Y, f_in_g(c(X, s(Y))))
F_IN_G(c(s(X), Y)) → F_IN_G(c(X, s(Y)))
U3_G(X, f_out_g(X)) → U4_G(X, g_in_g(X))
U3_G(X, f_out_g(X)) → G_IN_G(X)
G_IN_G(c(X, s(Y))) → U2_G(X, Y, g_in_g(c(s(X), Y)))
G_IN_G(c(X, s(Y))) → G_IN_G(c(s(X), Y))

The TRS R consists of the following rules:

h_in_g(X) → U3_g(X, f_in_g(X))
f_in_g(c(s(X), Y)) → U1_g(X, Y, f_in_g(c(X, s(Y))))
U1_g(X, Y, f_out_g(c(X, s(Y)))) → f_out_g(c(s(X), Y))
U3_g(X, f_out_g(X)) → U4_g(X, g_in_g(X))
g_in_g(c(X, s(Y))) → U2_g(X, Y, g_in_g(c(s(X), Y)))
U2_g(X, Y, g_out_g(c(s(X), Y))) → g_out_g(c(X, s(Y)))
U4_g(X, g_out_g(X)) → h_out_g(X)

The argument filtering Pi contains the following mapping:
h_in_g(x1)  =  h_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f_in_g(x1)  =  f_in_g(x1)
c(x1, x2)  =  c(x1, x2)
s(x1)  =  s(x1)
U1_g(x1, x2, x3)  =  U1_g(x3)
f_out_g(x1)  =  f_out_g
U4_g(x1, x2)  =  U4_g(x2)
g_in_g(x1)  =  g_in_g(x1)
U2_g(x1, x2, x3)  =  U2_g(x3)
g_out_g(x1)  =  g_out_g
h_out_g(x1)  =  h_out_g
H_IN_G(x1)  =  H_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x1, x2)
F_IN_G(x1)  =  F_IN_G(x1)
U1_G(x1, x2, x3)  =  U1_G(x3)
U4_G(x1, x2)  =  U4_G(x2)
G_IN_G(x1)  =  G_IN_G(x1)
U2_G(x1, x2, x3)  =  U2_G(x3)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

G_IN_G(c(X, s(Y))) → G_IN_G(c(s(X), Y))

The TRS R consists of the following rules:

h_in_g(X) → U3_g(X, f_in_g(X))
f_in_g(c(s(X), Y)) → U1_g(X, Y, f_in_g(c(X, s(Y))))
U1_g(X, Y, f_out_g(c(X, s(Y)))) → f_out_g(c(s(X), Y))
U3_g(X, f_out_g(X)) → U4_g(X, g_in_g(X))
g_in_g(c(X, s(Y))) → U2_g(X, Y, g_in_g(c(s(X), Y)))
U2_g(X, Y, g_out_g(c(s(X), Y))) → g_out_g(c(X, s(Y)))
U4_g(X, g_out_g(X)) → h_out_g(X)

The argument filtering Pi contains the following mapping:
h_in_g(x1)  =  h_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f_in_g(x1)  =  f_in_g(x1)
c(x1, x2)  =  c(x1, x2)
s(x1)  =  s(x1)
U1_g(x1, x2, x3)  =  U1_g(x3)
f_out_g(x1)  =  f_out_g
U4_g(x1, x2)  =  U4_g(x2)
g_in_g(x1)  =  g_in_g(x1)
U2_g(x1, x2, x3)  =  U2_g(x3)
g_out_g(x1)  =  g_out_g
h_out_g(x1)  =  h_out_g
G_IN_G(x1)  =  G_IN_G(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:

G_IN_G(c(X, s(Y))) → G_IN_G(c(s(X), Y))

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

(10) PiDPToQDPProof (EQUIVALENT 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:

G_IN_G(c(X, s(Y))) → G_IN_G(c(s(X), Y))

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

(12) 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:

G_IN_G(c(X, s(Y))) → G_IN_G(c(s(X), Y))


Used ordering: Polynomial interpretation [POLO]:

POL(G_IN_G(x1)) = 2·x1   
POL(c(x1, x2)) = x1 + 2·x2   
POL(s(x1)) = 1 + x1   

(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:

F_IN_G(c(s(X), Y)) → F_IN_G(c(X, s(Y)))

The TRS R consists of the following rules:

h_in_g(X) → U3_g(X, f_in_g(X))
f_in_g(c(s(X), Y)) → U1_g(X, Y, f_in_g(c(X, s(Y))))
U1_g(X, Y, f_out_g(c(X, s(Y)))) → f_out_g(c(s(X), Y))
U3_g(X, f_out_g(X)) → U4_g(X, g_in_g(X))
g_in_g(c(X, s(Y))) → U2_g(X, Y, g_in_g(c(s(X), Y)))
U2_g(X, Y, g_out_g(c(s(X), Y))) → g_out_g(c(X, s(Y)))
U4_g(X, g_out_g(X)) → h_out_g(X)

The argument filtering Pi contains the following mapping:
h_in_g(x1)  =  h_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
f_in_g(x1)  =  f_in_g(x1)
c(x1, x2)  =  c(x1, x2)
s(x1)  =  s(x1)
U1_g(x1, x2, x3)  =  U1_g(x3)
f_out_g(x1)  =  f_out_g
U4_g(x1, x2)  =  U4_g(x2)
g_in_g(x1)  =  g_in_g(x1)
U2_g(x1, x2, x3)  =  U2_g(x3)
g_out_g(x1)  =  g_out_g
h_out_g(x1)  =  h_out_g
F_IN_G(x1)  =  F_IN_G(x1)

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:

F_IN_G(c(s(X), Y)) → F_IN_G(c(X, s(Y)))

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

(19) PiDPToQDPProof (EQUIVALENT 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:

F_IN_G(c(s(X), Y)) → F_IN_G(c(X, s(Y)))

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

(21) 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:

F_IN_G(c(s(X), Y)) → F_IN_G(c(X, s(Y)))


Used ordering: Polynomial interpretation [POLO]:

POL(F_IN_G(x1)) = 2·x1   
POL(c(x1, x2)) = 2·x1 + x2   
POL(s(x1)) = 1 + x1   

(22) Obligation:

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

(23) PisEmptyProof (EQUIVALENT transformation)

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

(24) TRUE

(25) PrologToPiTRSProof (SOUND transformation)

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

h_in_g(X) → U3_g(X, f_in_g(X))
f_in_g(c(s(X), Y)) → U1_g(X, Y, f_in_g(c(X, s(Y))))
U1_g(X, Y, f_out_g(c(X, s(Y)))) → f_out_g(c(s(X), Y))
U3_g(X, f_out_g(X)) → U4_g(X, g_in_g(X))
g_in_g(c(X, s(Y))) → U2_g(X, Y, g_in_g(c(s(X), Y)))
U2_g(X, Y, g_out_g(c(s(X), Y))) → g_out_g(c(X, s(Y)))
U4_g(X, g_out_g(X)) → h_out_g(X)

Pi is empty.

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(26) Obligation:

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

h_in_g(X) → U3_g(X, f_in_g(X))
f_in_g(c(s(X), Y)) → U1_g(X, Y, f_in_g(c(X, s(Y))))
U1_g(X, Y, f_out_g(c(X, s(Y)))) → f_out_g(c(s(X), Y))
U3_g(X, f_out_g(X)) → U4_g(X, g_in_g(X))
g_in_g(c(X, s(Y))) → U2_g(X, Y, g_in_g(c(s(X), Y)))
U2_g(X, Y, g_out_g(c(s(X), Y))) → g_out_g(c(X, s(Y)))
U4_g(X, g_out_g(X)) → h_out_g(X)

Pi is empty.

(27) 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:

H_IN_G(X) → U3_G(X, f_in_g(X))
H_IN_G(X) → F_IN_G(X)
F_IN_G(c(s(X), Y)) → U1_G(X, Y, f_in_g(c(X, s(Y))))
F_IN_G(c(s(X), Y)) → F_IN_G(c(X, s(Y)))
U3_G(X, f_out_g(X)) → U4_G(X, g_in_g(X))
U3_G(X, f_out_g(X)) → G_IN_G(X)
G_IN_G(c(X, s(Y))) → U2_G(X, Y, g_in_g(c(s(X), Y)))
G_IN_G(c(X, s(Y))) → G_IN_G(c(s(X), Y))

The TRS R consists of the following rules:

h_in_g(X) → U3_g(X, f_in_g(X))
f_in_g(c(s(X), Y)) → U1_g(X, Y, f_in_g(c(X, s(Y))))
U1_g(X, Y, f_out_g(c(X, s(Y)))) → f_out_g(c(s(X), Y))
U3_g(X, f_out_g(X)) → U4_g(X, g_in_g(X))
g_in_g(c(X, s(Y))) → U2_g(X, Y, g_in_g(c(s(X), Y)))
U2_g(X, Y, g_out_g(c(s(X), Y))) → g_out_g(c(X, s(Y)))
U4_g(X, g_out_g(X)) → h_out_g(X)

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

(28) Obligation:

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

H_IN_G(X) → U3_G(X, f_in_g(X))
H_IN_G(X) → F_IN_G(X)
F_IN_G(c(s(X), Y)) → U1_G(X, Y, f_in_g(c(X, s(Y))))
F_IN_G(c(s(X), Y)) → F_IN_G(c(X, s(Y)))
U3_G(X, f_out_g(X)) → U4_G(X, g_in_g(X))
U3_G(X, f_out_g(X)) → G_IN_G(X)
G_IN_G(c(X, s(Y))) → U2_G(X, Y, g_in_g(c(s(X), Y)))
G_IN_G(c(X, s(Y))) → G_IN_G(c(s(X), Y))

The TRS R consists of the following rules:

h_in_g(X) → U3_g(X, f_in_g(X))
f_in_g(c(s(X), Y)) → U1_g(X, Y, f_in_g(c(X, s(Y))))
U1_g(X, Y, f_out_g(c(X, s(Y)))) → f_out_g(c(s(X), Y))
U3_g(X, f_out_g(X)) → U4_g(X, g_in_g(X))
g_in_g(c(X, s(Y))) → U2_g(X, Y, g_in_g(c(s(X), Y)))
U2_g(X, Y, g_out_g(c(s(X), Y))) → g_out_g(c(X, s(Y)))
U4_g(X, g_out_g(X)) → h_out_g(X)

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

(29) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(30) Complex Obligation (AND)

(31) Obligation:

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

G_IN_G(c(X, s(Y))) → G_IN_G(c(s(X), Y))

The TRS R consists of the following rules:

h_in_g(X) → U3_g(X, f_in_g(X))
f_in_g(c(s(X), Y)) → U1_g(X, Y, f_in_g(c(X, s(Y))))
U1_g(X, Y, f_out_g(c(X, s(Y)))) → f_out_g(c(s(X), Y))
U3_g(X, f_out_g(X)) → U4_g(X, g_in_g(X))
g_in_g(c(X, s(Y))) → U2_g(X, Y, g_in_g(c(s(X), Y)))
U2_g(X, Y, g_out_g(c(s(X), Y))) → g_out_g(c(X, s(Y)))
U4_g(X, g_out_g(X)) → h_out_g(X)

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

(32) UsableRulesProof (EQUIVALENT transformation)

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

(33) Obligation:

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

G_IN_G(c(X, s(Y))) → G_IN_G(c(s(X), Y))

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

(34) PiDPToQDPProof (EQUIVALENT transformation)

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

(35) Obligation:

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

G_IN_G(c(X, s(Y))) → G_IN_G(c(s(X), Y))

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

(36) 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:

G_IN_G(c(X, s(Y))) → G_IN_G(c(s(X), Y))


Used ordering: Polynomial interpretation [POLO]:

POL(G_IN_G(x1)) = 2·x1   
POL(c(x1, x2)) = x1 + 2·x2   
POL(s(x1)) = 1 + x1   

(37) Obligation:

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

(38) PisEmptyProof (EQUIVALENT transformation)

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

(39) TRUE

(40) Obligation:

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

F_IN_G(c(s(X), Y)) → F_IN_G(c(X, s(Y)))

The TRS R consists of the following rules:

h_in_g(X) → U3_g(X, f_in_g(X))
f_in_g(c(s(X), Y)) → U1_g(X, Y, f_in_g(c(X, s(Y))))
U1_g(X, Y, f_out_g(c(X, s(Y)))) → f_out_g(c(s(X), Y))
U3_g(X, f_out_g(X)) → U4_g(X, g_in_g(X))
g_in_g(c(X, s(Y))) → U2_g(X, Y, g_in_g(c(s(X), Y)))
U2_g(X, Y, g_out_g(c(s(X), Y))) → g_out_g(c(X, s(Y)))
U4_g(X, g_out_g(X)) → h_out_g(X)

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

(41) UsableRulesProof (EQUIVALENT transformation)

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

(42) Obligation:

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

F_IN_G(c(s(X), Y)) → F_IN_G(c(X, s(Y)))

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

(43) PiDPToQDPProof (EQUIVALENT transformation)

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

(44) Obligation:

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

F_IN_G(c(s(X), Y)) → F_IN_G(c(X, s(Y)))

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