Left Termination of the query pattern p_in_1(g) w.r.t. the given Prolog program could not be shown:

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 NonTerminationProof (⇔)
        ↳13 FALSE
      ↳14 PiDP
        ↳15 UsableRulesProof (⇔)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇔)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔)
        ↳20 TRUE
  ↳21 PrologToPiTRSProof (⇐)
    ↳22 PiTRS
    ↳23 DependencyPairsProof (⇔)
    ↳24 PiDP
    ↳25 DependencyGraphProof (⇔)
    ↳26 AND
      ↳27 PiDP
        ↳28 UsableRulesProof (⇔)
        ↳29 PiDP
        ↳30 PiDPToQDPProof (⇐)
        ↳31 QDP
        ↳32 NonTerminationProof (⇔)
        ↳33 FALSE
      ↳34 PiDP
        ↳35 UsableRulesProof (⇔)
        ↳36 PiDP
        ↳37 PiDPToQDPProof (⇔)
        ↳38 QDP
        ↳39 QDPSizeChangeProof (⇔)
        ↳40 TRUE

(0) Obligation:

Clauses:

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

Queries:

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

p_in_g(X) → U1_g(X, q_in_a(f(Y)))
q_in_a(g(Y)) → q_out_a(g(Y))
U1_g(X, q_out_a(f(Y))) → U2_g(X, p_in_a(Y))
p_in_a(X) → U1_a(X, q_in_a(f(Y)))
U1_a(X, q_out_a(f(Y))) → U2_a(X, p_in_a(Y))
p_in_a(g(X)) → U3_a(X, p_in_a(X))
U3_a(X, p_out_a(X)) → p_out_a(g(X))
U2_a(X, p_out_a(Y)) → p_out_a(X)
U2_g(X, p_out_a(Y)) → p_out_g(X)
p_in_g(g(X)) → U3_g(X, p_in_g(X))
U3_g(X, p_out_g(X)) → p_out_g(g(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a
U2_g(x1, x2)  =  U2_g(x2)
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
U3_a(x1, x2)  =  U3_a(x2)
p_out_a(x1)  =  p_out_a
p_out_g(x1)  =  p_out_g
g(x1)  =  g(x1)
U3_g(x1, x2)  =  U3_g(x2)

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_g(X) → U1_g(X, q_in_a(f(Y)))
q_in_a(g(Y)) → q_out_a(g(Y))
U1_g(X, q_out_a(f(Y))) → U2_g(X, p_in_a(Y))
p_in_a(X) → U1_a(X, q_in_a(f(Y)))
U1_a(X, q_out_a(f(Y))) → U2_a(X, p_in_a(Y))
p_in_a(g(X)) → U3_a(X, p_in_a(X))
U3_a(X, p_out_a(X)) → p_out_a(g(X))
U2_a(X, p_out_a(Y)) → p_out_a(X)
U2_g(X, p_out_a(Y)) → p_out_g(X)
p_in_g(g(X)) → U3_g(X, p_in_g(X))
U3_g(X, p_out_g(X)) → p_out_g(g(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a
U2_g(x1, x2)  =  U2_g(x2)
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
U3_a(x1, x2)  =  U3_a(x2)
p_out_a(x1)  =  p_out_a
p_out_g(x1)  =  p_out_g
g(x1)  =  g(x1)
U3_g(x1, x2)  =  U3_g(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:

P_IN_G(X) → U1_G(X, q_in_a(f(Y)))
P_IN_G(X) → Q_IN_A(f(Y))
U1_G(X, q_out_a(f(Y))) → U2_G(X, p_in_a(Y))
U1_G(X, q_out_a(f(Y))) → P_IN_A(Y)
P_IN_A(X) → U1_A(X, q_in_a(f(Y)))
P_IN_A(X) → Q_IN_A(f(Y))
U1_A(X, q_out_a(f(Y))) → U2_A(X, p_in_a(Y))
U1_A(X, q_out_a(f(Y))) → P_IN_A(Y)
P_IN_A(g(X)) → U3_A(X, p_in_a(X))
P_IN_A(g(X)) → P_IN_A(X)
P_IN_G(g(X)) → U3_G(X, p_in_g(X))
P_IN_G(g(X)) → P_IN_G(X)

The TRS R consists of the following rules:

p_in_g(X) → U1_g(X, q_in_a(f(Y)))
q_in_a(g(Y)) → q_out_a(g(Y))
U1_g(X, q_out_a(f(Y))) → U2_g(X, p_in_a(Y))
p_in_a(X) → U1_a(X, q_in_a(f(Y)))
U1_a(X, q_out_a(f(Y))) → U2_a(X, p_in_a(Y))
p_in_a(g(X)) → U3_a(X, p_in_a(X))
U3_a(X, p_out_a(X)) → p_out_a(g(X))
U2_a(X, p_out_a(Y)) → p_out_a(X)
U2_g(X, p_out_a(Y)) → p_out_g(X)
p_in_g(g(X)) → U3_g(X, p_in_g(X))
U3_g(X, p_out_g(X)) → p_out_g(g(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a
U2_g(x1, x2)  =  U2_g(x2)
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
U3_a(x1, x2)  =  U3_a(x2)
p_out_a(x1)  =  p_out_a
p_out_g(x1)  =  p_out_g
g(x1)  =  g(x1)
U3_g(x1, x2)  =  U3_g(x2)
P_IN_G(x1)  =  P_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
Q_IN_A(x1)  =  Q_IN_A
U2_G(x1, x2)  =  U2_G(x2)
P_IN_A(x1)  =  P_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U2_A(x1, x2)  =  U2_A(x2)
U3_A(x1, x2)  =  U3_A(x2)
U3_G(x1, x2)  =  U3_G(x2)

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

(4) Obligation:

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

P_IN_G(X) → U1_G(X, q_in_a(f(Y)))
P_IN_G(X) → Q_IN_A(f(Y))
U1_G(X, q_out_a(f(Y))) → U2_G(X, p_in_a(Y))
U1_G(X, q_out_a(f(Y))) → P_IN_A(Y)
P_IN_A(X) → U1_A(X, q_in_a(f(Y)))
P_IN_A(X) → Q_IN_A(f(Y))
U1_A(X, q_out_a(f(Y))) → U2_A(X, p_in_a(Y))
U1_A(X, q_out_a(f(Y))) → P_IN_A(Y)
P_IN_A(g(X)) → U3_A(X, p_in_a(X))
P_IN_A(g(X)) → P_IN_A(X)
P_IN_G(g(X)) → U3_G(X, p_in_g(X))
P_IN_G(g(X)) → P_IN_G(X)

The TRS R consists of the following rules:

p_in_g(X) → U1_g(X, q_in_a(f(Y)))
q_in_a(g(Y)) → q_out_a(g(Y))
U1_g(X, q_out_a(f(Y))) → U2_g(X, p_in_a(Y))
p_in_a(X) → U1_a(X, q_in_a(f(Y)))
U1_a(X, q_out_a(f(Y))) → U2_a(X, p_in_a(Y))
p_in_a(g(X)) → U3_a(X, p_in_a(X))
U3_a(X, p_out_a(X)) → p_out_a(g(X))
U2_a(X, p_out_a(Y)) → p_out_a(X)
U2_g(X, p_out_a(Y)) → p_out_g(X)
p_in_g(g(X)) → U3_g(X, p_in_g(X))
U3_g(X, p_out_g(X)) → p_out_g(g(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a
U2_g(x1, x2)  =  U2_g(x2)
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
U3_a(x1, x2)  =  U3_a(x2)
p_out_a(x1)  =  p_out_a
p_out_g(x1)  =  p_out_g
g(x1)  =  g(x1)
U3_g(x1, x2)  =  U3_g(x2)
P_IN_G(x1)  =  P_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
Q_IN_A(x1)  =  Q_IN_A
U2_G(x1, x2)  =  U2_G(x2)
P_IN_A(x1)  =  P_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U2_A(x1, x2)  =  U2_A(x2)
U3_A(x1, x2)  =  U3_A(x2)
U3_G(x1, x2)  =  U3_G(x2)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 10 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

P_IN_A(g(X)) → P_IN_A(X)

The TRS R consists of the following rules:

p_in_g(X) → U1_g(X, q_in_a(f(Y)))
q_in_a(g(Y)) → q_out_a(g(Y))
U1_g(X, q_out_a(f(Y))) → U2_g(X, p_in_a(Y))
p_in_a(X) → U1_a(X, q_in_a(f(Y)))
U1_a(X, q_out_a(f(Y))) → U2_a(X, p_in_a(Y))
p_in_a(g(X)) → U3_a(X, p_in_a(X))
U3_a(X, p_out_a(X)) → p_out_a(g(X))
U2_a(X, p_out_a(Y)) → p_out_a(X)
U2_g(X, p_out_a(Y)) → p_out_g(X)
p_in_g(g(X)) → U3_g(X, p_in_g(X))
U3_g(X, p_out_g(X)) → p_out_g(g(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a
U2_g(x1, x2)  =  U2_g(x2)
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
U3_a(x1, x2)  =  U3_a(x2)
p_out_a(x1)  =  p_out_a
p_out_g(x1)  =  p_out_g
g(x1)  =  g(x1)
U3_g(x1, x2)  =  U3_g(x2)
P_IN_A(x1)  =  P_IN_A

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:

P_IN_A(g(X)) → P_IN_A(X)

R is empty.
The argument filtering Pi contains the following mapping:
g(x1)  =  g(x1)
P_IN_A(x1)  =  P_IN_A

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:

P_IN_A → P_IN_A

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

(12) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = P_IN_A evaluates to t =P_IN_A

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Matcher: [ ]
  • Semiunifier: [ ]



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from P_IN_A to P_IN_A.



(13) FALSE

(14) Obligation:

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

P_IN_G(g(X)) → P_IN_G(X)

The TRS R consists of the following rules:

p_in_g(X) → U1_g(X, q_in_a(f(Y)))
q_in_a(g(Y)) → q_out_a(g(Y))
U1_g(X, q_out_a(f(Y))) → U2_g(X, p_in_a(Y))
p_in_a(X) → U1_a(X, q_in_a(f(Y)))
U1_a(X, q_out_a(f(Y))) → U2_a(X, p_in_a(Y))
p_in_a(g(X)) → U3_a(X, p_in_a(X))
U3_a(X, p_out_a(X)) → p_out_a(g(X))
U2_a(X, p_out_a(Y)) → p_out_a(X)
U2_g(X, p_out_a(Y)) → p_out_g(X)
p_in_g(g(X)) → U3_g(X, p_in_g(X))
U3_g(X, p_out_g(X)) → p_out_g(g(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a
U2_g(x1, x2)  =  U2_g(x2)
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
U3_a(x1, x2)  =  U3_a(x2)
p_out_a(x1)  =  p_out_a
p_out_g(x1)  =  p_out_g
g(x1)  =  g(x1)
U3_g(x1, x2)  =  U3_g(x2)
P_IN_G(x1)  =  P_IN_G(x1)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

P_IN_G(g(X)) → P_IN_G(X)

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

(17) PiDPToQDPProof (EQUIVALENT transformation)

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

(18) Obligation:

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

P_IN_G(g(X)) → P_IN_G(X)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • P_IN_G(g(X)) → P_IN_G(X)
    The graph contains the following edges 1 > 1

(20) TRUE

(21) 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)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p_in_g(X) → U1_g(X, q_in_a(f(Y)))
q_in_a(g(Y)) → q_out_a(g(Y))
U1_g(X, q_out_a(f(Y))) → U2_g(X, p_in_a(Y))
p_in_a(X) → U1_a(X, q_in_a(f(Y)))
U1_a(X, q_out_a(f(Y))) → U2_a(X, p_in_a(Y))
p_in_a(g(X)) → U3_a(X, p_in_a(X))
U3_a(X, p_out_a(X)) → p_out_a(g(X))
U2_a(X, p_out_a(Y)) → p_out_a(X)
U2_g(X, p_out_a(Y)) → p_out_g(X)
p_in_g(g(X)) → U3_g(X, p_in_g(X))
U3_g(X, p_out_g(X)) → p_out_g(g(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a
U2_g(x1, x2)  =  U2_g(x1, x2)
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
U3_a(x1, x2)  =  U3_a(x2)
p_out_a(x1)  =  p_out_a
p_out_g(x1)  =  p_out_g(x1)
g(x1)  =  g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(22) Obligation:

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

p_in_g(X) → U1_g(X, q_in_a(f(Y)))
q_in_a(g(Y)) → q_out_a(g(Y))
U1_g(X, q_out_a(f(Y))) → U2_g(X, p_in_a(Y))
p_in_a(X) → U1_a(X, q_in_a(f(Y)))
U1_a(X, q_out_a(f(Y))) → U2_a(X, p_in_a(Y))
p_in_a(g(X)) → U3_a(X, p_in_a(X))
U3_a(X, p_out_a(X)) → p_out_a(g(X))
U2_a(X, p_out_a(Y)) → p_out_a(X)
U2_g(X, p_out_a(Y)) → p_out_g(X)
p_in_g(g(X)) → U3_g(X, p_in_g(X))
U3_g(X, p_out_g(X)) → p_out_g(g(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a
U2_g(x1, x2)  =  U2_g(x1, x2)
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
U3_a(x1, x2)  =  U3_a(x2)
p_out_a(x1)  =  p_out_a
p_out_g(x1)  =  p_out_g(x1)
g(x1)  =  g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)

(23) 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_G(X) → U1_G(X, q_in_a(f(Y)))
P_IN_G(X) → Q_IN_A(f(Y))
U1_G(X, q_out_a(f(Y))) → U2_G(X, p_in_a(Y))
U1_G(X, q_out_a(f(Y))) → P_IN_A(Y)
P_IN_A(X) → U1_A(X, q_in_a(f(Y)))
P_IN_A(X) → Q_IN_A(f(Y))
U1_A(X, q_out_a(f(Y))) → U2_A(X, p_in_a(Y))
U1_A(X, q_out_a(f(Y))) → P_IN_A(Y)
P_IN_A(g(X)) → U3_A(X, p_in_a(X))
P_IN_A(g(X)) → P_IN_A(X)
P_IN_G(g(X)) → U3_G(X, p_in_g(X))
P_IN_G(g(X)) → P_IN_G(X)

The TRS R consists of the following rules:

p_in_g(X) → U1_g(X, q_in_a(f(Y)))
q_in_a(g(Y)) → q_out_a(g(Y))
U1_g(X, q_out_a(f(Y))) → U2_g(X, p_in_a(Y))
p_in_a(X) → U1_a(X, q_in_a(f(Y)))
U1_a(X, q_out_a(f(Y))) → U2_a(X, p_in_a(Y))
p_in_a(g(X)) → U3_a(X, p_in_a(X))
U3_a(X, p_out_a(X)) → p_out_a(g(X))
U2_a(X, p_out_a(Y)) → p_out_a(X)
U2_g(X, p_out_a(Y)) → p_out_g(X)
p_in_g(g(X)) → U3_g(X, p_in_g(X))
U3_g(X, p_out_g(X)) → p_out_g(g(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a
U2_g(x1, x2)  =  U2_g(x1, x2)
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
U3_a(x1, x2)  =  U3_a(x2)
p_out_a(x1)  =  p_out_a
p_out_g(x1)  =  p_out_g(x1)
g(x1)  =  g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
P_IN_G(x1)  =  P_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
Q_IN_A(x1)  =  Q_IN_A
U2_G(x1, x2)  =  U2_G(x1, x2)
P_IN_A(x1)  =  P_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U2_A(x1, x2)  =  U2_A(x2)
U3_A(x1, x2)  =  U3_A(x2)
U3_G(x1, x2)  =  U3_G(x1, x2)

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

(24) Obligation:

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

P_IN_G(X) → U1_G(X, q_in_a(f(Y)))
P_IN_G(X) → Q_IN_A(f(Y))
U1_G(X, q_out_a(f(Y))) → U2_G(X, p_in_a(Y))
U1_G(X, q_out_a(f(Y))) → P_IN_A(Y)
P_IN_A(X) → U1_A(X, q_in_a(f(Y)))
P_IN_A(X) → Q_IN_A(f(Y))
U1_A(X, q_out_a(f(Y))) → U2_A(X, p_in_a(Y))
U1_A(X, q_out_a(f(Y))) → P_IN_A(Y)
P_IN_A(g(X)) → U3_A(X, p_in_a(X))
P_IN_A(g(X)) → P_IN_A(X)
P_IN_G(g(X)) → U3_G(X, p_in_g(X))
P_IN_G(g(X)) → P_IN_G(X)

The TRS R consists of the following rules:

p_in_g(X) → U1_g(X, q_in_a(f(Y)))
q_in_a(g(Y)) → q_out_a(g(Y))
U1_g(X, q_out_a(f(Y))) → U2_g(X, p_in_a(Y))
p_in_a(X) → U1_a(X, q_in_a(f(Y)))
U1_a(X, q_out_a(f(Y))) → U2_a(X, p_in_a(Y))
p_in_a(g(X)) → U3_a(X, p_in_a(X))
U3_a(X, p_out_a(X)) → p_out_a(g(X))
U2_a(X, p_out_a(Y)) → p_out_a(X)
U2_g(X, p_out_a(Y)) → p_out_g(X)
p_in_g(g(X)) → U3_g(X, p_in_g(X))
U3_g(X, p_out_g(X)) → p_out_g(g(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a
U2_g(x1, x2)  =  U2_g(x1, x2)
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
U3_a(x1, x2)  =  U3_a(x2)
p_out_a(x1)  =  p_out_a
p_out_g(x1)  =  p_out_g(x1)
g(x1)  =  g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
P_IN_G(x1)  =  P_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
Q_IN_A(x1)  =  Q_IN_A
U2_G(x1, x2)  =  U2_G(x1, x2)
P_IN_A(x1)  =  P_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U2_A(x1, x2)  =  U2_A(x2)
U3_A(x1, x2)  =  U3_A(x2)
U3_G(x1, x2)  =  U3_G(x1, x2)

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

(25) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 10 less nodes.

(26) Complex Obligation (AND)

(27) Obligation:

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

P_IN_A(g(X)) → P_IN_A(X)

The TRS R consists of the following rules:

p_in_g(X) → U1_g(X, q_in_a(f(Y)))
q_in_a(g(Y)) → q_out_a(g(Y))
U1_g(X, q_out_a(f(Y))) → U2_g(X, p_in_a(Y))
p_in_a(X) → U1_a(X, q_in_a(f(Y)))
U1_a(X, q_out_a(f(Y))) → U2_a(X, p_in_a(Y))
p_in_a(g(X)) → U3_a(X, p_in_a(X))
U3_a(X, p_out_a(X)) → p_out_a(g(X))
U2_a(X, p_out_a(Y)) → p_out_a(X)
U2_g(X, p_out_a(Y)) → p_out_g(X)
p_in_g(g(X)) → U3_g(X, p_in_g(X))
U3_g(X, p_out_g(X)) → p_out_g(g(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a
U2_g(x1, x2)  =  U2_g(x1, x2)
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
U3_a(x1, x2)  =  U3_a(x2)
p_out_a(x1)  =  p_out_a
p_out_g(x1)  =  p_out_g(x1)
g(x1)  =  g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
P_IN_A(x1)  =  P_IN_A

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

(28) UsableRulesProof (EQUIVALENT transformation)

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

(29) Obligation:

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

P_IN_A(g(X)) → P_IN_A(X)

R is empty.
The argument filtering Pi contains the following mapping:
g(x1)  =  g(x1)
P_IN_A(x1)  =  P_IN_A

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

(30) PiDPToQDPProof (SOUND transformation)

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

(31) Obligation:

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

P_IN_A → P_IN_A

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

(32) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = P_IN_A evaluates to t =P_IN_A

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Matcher: [ ]
  • Semiunifier: [ ]



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from P_IN_A to P_IN_A.



(33) FALSE

(34) Obligation:

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

P_IN_G(g(X)) → P_IN_G(X)

The TRS R consists of the following rules:

p_in_g(X) → U1_g(X, q_in_a(f(Y)))
q_in_a(g(Y)) → q_out_a(g(Y))
U1_g(X, q_out_a(f(Y))) → U2_g(X, p_in_a(Y))
p_in_a(X) → U1_a(X, q_in_a(f(Y)))
U1_a(X, q_out_a(f(Y))) → U2_a(X, p_in_a(Y))
p_in_a(g(X)) → U3_a(X, p_in_a(X))
U3_a(X, p_out_a(X)) → p_out_a(g(X))
U2_a(X, p_out_a(Y)) → p_out_a(X)
U2_g(X, p_out_a(Y)) → p_out_g(X)
p_in_g(g(X)) → U3_g(X, p_in_g(X))
U3_g(X, p_out_g(X)) → p_out_g(g(X))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
q_in_a(x1)  =  q_in_a
q_out_a(x1)  =  q_out_a
U2_g(x1, x2)  =  U2_g(x1, x2)
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
U2_a(x1, x2)  =  U2_a(x2)
U3_a(x1, x2)  =  U3_a(x2)
p_out_a(x1)  =  p_out_a
p_out_g(x1)  =  p_out_g(x1)
g(x1)  =  g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
P_IN_G(x1)  =  P_IN_G(x1)

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

(35) UsableRulesProof (EQUIVALENT transformation)

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

(36) Obligation:

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

P_IN_G(g(X)) → P_IN_G(X)

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

(37) PiDPToQDPProof (EQUIVALENT transformation)

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

(38) Obligation:

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

P_IN_G(g(X)) → P_IN_G(X)

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

(39) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • P_IN_G(g(X)) → P_IN_G(X)
    The graph contains the following edges 1 > 1

(40) TRUE