(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