(0) Obligation:
Clauses:
p(a).
p(X) :- p(Y).
q(b).
goal(X) :- ','(p(X), q(X)).
Queries:
goal(g).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
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(2) Obligation:
Triples:
p12(X15) :- p12(X14).
goal1(T6) :- p12(X7).
Clauses:
pc12(a).
pc12(X15) :- pc12(X14).
Afs:
goal1(x1) = goal1(x1)
(3) TriplesToPiDPProof (SOUND transformation)
We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal1_in: (b)
p12_in: (f)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
GOAL1_IN_G(T6) → U2_G(T6, p12_in_a(X7))
GOAL1_IN_G(T6) → P12_IN_A(X7)
P12_IN_A(X15) → U1_A(X15, p12_in_a(X14))
P12_IN_A(X15) → P12_IN_A(X14)
R is empty.
The argument filtering Pi contains the following mapping:
p12_in_a(
x1) =
p12_in_a
GOAL1_IN_G(
x1) =
GOAL1_IN_G(
x1)
U2_G(
x1,
x2) =
U2_G(
x1,
x2)
P12_IN_A(
x1) =
P12_IN_A
U1_A(
x1,
x2) =
U1_A(
x2)
We have to consider all (P,R,Pi)-chains
Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
(4) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
GOAL1_IN_G(T6) → U2_G(T6, p12_in_a(X7))
GOAL1_IN_G(T6) → P12_IN_A(X7)
P12_IN_A(X15) → U1_A(X15, p12_in_a(X14))
P12_IN_A(X15) → P12_IN_A(X14)
R is empty.
The argument filtering Pi contains the following mapping:
p12_in_a(
x1) =
p12_in_a
GOAL1_IN_G(
x1) =
GOAL1_IN_G(
x1)
U2_G(
x1,
x2) =
U2_G(
x1,
x2)
P12_IN_A(
x1) =
P12_IN_A
U1_A(
x1,
x2) =
U1_A(
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 3 less nodes.
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
P12_IN_A(X15) → P12_IN_A(X14)
R is empty.
The argument filtering Pi contains the following mapping:
P12_IN_A(
x1) =
P12_IN_A
We have to consider all (P,R,Pi)-chains
(7) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
P12_IN_A → P12_IN_A
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(9) 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 =
P12_IN_A evaluates to t =
P12_IN_AThus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequenceThe DP semiunifies directly so there is only one rewrite step from P12_IN_A to P12_IN_A.
(10) NO