(0) Obligation:

Clauses:

p1(f(X)) :- p1(X).
p2(f(X)) :- p2(X).

Queries:

p1(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:
p1_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p1_in_g(f(X)) → U1_g(X, p1_in_g(X))
U1_g(X, p1_out_g(X)) → p1_out_g(f(X))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
f(x1)  =  f(x1)
U1_g(x1, x2)  =  U1_g(x2)
p1_out_g(x1)  =  p1_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:

p1_in_g(f(X)) → U1_g(X, p1_in_g(X))
U1_g(X, p1_out_g(X)) → p1_out_g(f(X))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
f(x1)  =  f(x1)
U1_g(x1, x2)  =  U1_g(x2)
p1_out_g(x1)  =  p1_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:

P1_IN_G(f(X)) → U1_G(X, p1_in_g(X))
P1_IN_G(f(X)) → P1_IN_G(X)

The TRS R consists of the following rules:

p1_in_g(f(X)) → U1_g(X, p1_in_g(X))
U1_g(X, p1_out_g(X)) → p1_out_g(f(X))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
f(x1)  =  f(x1)
U1_g(x1, x2)  =  U1_g(x2)
p1_out_g(x1)  =  p1_out_g
P1_IN_G(x1)  =  P1_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)

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

(4) Obligation:

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

P1_IN_G(f(X)) → U1_G(X, p1_in_g(X))
P1_IN_G(f(X)) → P1_IN_G(X)

The TRS R consists of the following rules:

p1_in_g(f(X)) → U1_g(X, p1_in_g(X))
U1_g(X, p1_out_g(X)) → p1_out_g(f(X))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
f(x1)  =  f(x1)
U1_g(x1, x2)  =  U1_g(x2)
p1_out_g(x1)  =  p1_out_g
P1_IN_G(x1)  =  P1_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

P1_IN_G(f(X)) → P1_IN_G(X)

The TRS R consists of the following rules:

p1_in_g(f(X)) → U1_g(X, p1_in_g(X))
U1_g(X, p1_out_g(X)) → p1_out_g(f(X))

The argument filtering Pi contains the following mapping:
p1_in_g(x1)  =  p1_in_g(x1)
f(x1)  =  f(x1)
U1_g(x1, x2)  =  U1_g(x2)
p1_out_g(x1)  =  p1_out_g
P1_IN_G(x1)  =  P1_IN_G(x1)

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

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

P1_IN_G(f(X)) → P1_IN_G(X)

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

(9) PiDPToQDPProof (EQUIVALENT 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:

P1_IN_G(f(X)) → P1_IN_G(X)

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

(11) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
p1_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p1_in_g(f(X)) → U1_g(X, p1_in_g(X))
U1_g(X, p1_out_g(X)) → p1_out_g(f(X))

Pi is empty.

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(12) Obligation:

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

p1_in_g(f(X)) → U1_g(X, p1_in_g(X))
U1_g(X, p1_out_g(X)) → p1_out_g(f(X))

Pi is empty.

(13) 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:

P1_IN_G(f(X)) → U1_G(X, p1_in_g(X))
P1_IN_G(f(X)) → P1_IN_G(X)

The TRS R consists of the following rules:

p1_in_g(f(X)) → U1_g(X, p1_in_g(X))
U1_g(X, p1_out_g(X)) → p1_out_g(f(X))

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

(14) Obligation:

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

P1_IN_G(f(X)) → U1_G(X, p1_in_g(X))
P1_IN_G(f(X)) → P1_IN_G(X)

The TRS R consists of the following rules:

p1_in_g(f(X)) → U1_g(X, p1_in_g(X))
U1_g(X, p1_out_g(X)) → p1_out_g(f(X))

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

(15) DependencyGraphProof (EQUIVALENT transformation)

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

(16) Obligation:

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

P1_IN_G(f(X)) → P1_IN_G(X)

The TRS R consists of the following rules:

p1_in_g(f(X)) → U1_g(X, p1_in_g(X))
U1_g(X, p1_out_g(X)) → p1_out_g(f(X))

Pi is empty.
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:

P1_IN_G(f(X)) → P1_IN_G(X)

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:

P1_IN_G(f(X)) → P1_IN_G(X)

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

(21) 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:

  • P1_IN_G(f(X)) → P1_IN_G(X)
    The graph contains the following edges 1 > 1

(22) TRUE