(0) Obligation:
Clauses:p(X, X).
p(f(X), g(Y)) :- ','(p(f(X), f(Z)), p(Z, g(W))).
Queries:
p(g,a).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:p6(T21, T21, X9) :- p11(T21, X9).
p11(f(T38), X41) :- p6(T38, X39, X40).
p1(f(T7), g(T8)) :- p6(T7, X8, X9).
p1(f(T57), g(T44)) :- p11(T57, X55).
Clauses:
qc6(T21, T21, X9) :- pc11(T21, X9).
pc11(g(T35), T35).
pc11(f(T38), X41) :- qc6(T38, X39, X40).
Afs:
p1(x1, x2) = p1(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:p1_in: (b,f)
p6_in: (b,f,f)
p11_in: (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
P1_IN_GA(f(T7), g(T8)) → U3_GA(T7, T8, p6_in_gaa(T7, X8, X9))
P1_IN_GA(f(T7), g(T8)) → P6_IN_GAA(T7, X8, X9)
P6_IN_GAA(T21, T21, X9) → U1_GAA(T21, X9, p11_in_ga(T21, X9))
P6_IN_GAA(T21, T21, X9) → P11_IN_GA(T21, X9)
P11_IN_GA(f(T38), X41) → U2_GA(T38, X41, p6_in_gaa(T38, X39, X40))
P11_IN_GA(f(T38), X41) → P6_IN_GAA(T38, X39, X40)
P1_IN_GA(f(T57), g(T44)) → U4_GA(T57, T44, p11_in_ga(T57, X55))
P1_IN_GA(f(T57), g(T44)) → P11_IN_GA(T57, X55)
R is empty.
The argument filtering Pi contains the following mapping:
f(x1) = f(x1)
p6_in_gaa(x1, x2, x3) = p6_in_gaa(x1)
p11_in_ga(x1, x2) = p11_in_ga(x1)
g(x1) = g
P1_IN_GA(x1, x2) = P1_IN_GA(x1)
U3_GA(x1, x2, x3) = U3_GA(x1, x3)
P6_IN_GAA(x1, x2, x3) = P6_IN_GAA(x1)
U1_GAA(x1, x2, x3) = U1_GAA(x1, x3)
P11_IN_GA(x1, x2) = P11_IN_GA(x1)
U2_GA(x1, x2, x3) = U2_GA(x1, x3)
U4_GA(x1, x2, x3) = U4_GA(x1, x3)
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:
P1_IN_GA(f(T7), g(T8)) → U3_GA(T7, T8, p6_in_gaa(T7, X8, X9))
P1_IN_GA(f(T7), g(T8)) → P6_IN_GAA(T7, X8, X9)
P6_IN_GAA(T21, T21, X9) → U1_GAA(T21, X9, p11_in_ga(T21, X9))
P6_IN_GAA(T21, T21, X9) → P11_IN_GA(T21, X9)
P11_IN_GA(f(T38), X41) → U2_GA(T38, X41, p6_in_gaa(T38, X39, X40))
P11_IN_GA(f(T38), X41) → P6_IN_GAA(T38, X39, X40)
P1_IN_GA(f(T57), g(T44)) → U4_GA(T57, T44, p11_in_ga(T57, X55))
P1_IN_GA(f(T57), g(T44)) → P11_IN_GA(T57, X55)
R is empty.
The argument filtering Pi contains the following mapping:
f(x1) = f(x1)
p6_in_gaa(x1, x2, x3) = p6_in_gaa(x1)
p11_in_ga(x1, x2) = p11_in_ga(x1)
g(x1) = g
P1_IN_GA(x1, x2) = P1_IN_GA(x1)
U3_GA(x1, x2, x3) = U3_GA(x1, x3)
P6_IN_GAA(x1, x2, x3) = P6_IN_GAA(x1)
U1_GAA(x1, x2, x3) = U1_GAA(x1, x3)
P11_IN_GA(x1, x2) = P11_IN_GA(x1)
U2_GA(x1, x2, x3) = U2_GA(x1, x3)
U4_GA(x1, x2, x3) = U4_GA(x1, x3)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 6 less nodes.(6) Obligation:
Pi DP problem:The TRS P consists of the following rules:
P6_IN_GAA(T21, T21, X9) → P11_IN_GA(T21, X9)
P11_IN_GA(f(T38), X41) → P6_IN_GAA(T38, X39, X40)
R is empty.
The argument filtering Pi contains the following mapping:
f(x1) = f(x1)
P6_IN_GAA(x1, x2, x3) = P6_IN_GAA(x1)
P11_IN_GA(x1, x2) = P11_IN_GA(x1)
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:
P6_IN_GAA(T21) → P11_IN_GA(T21)
P11_IN_GA(f(T38)) → P6_IN_GAA(T38)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(9) 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:
- P11_IN_GA(f(T38)) → P6_IN_GAA(T38)
The graph contains the following edges 1 > 1 - P6_IN_GAA(T21) → P11_IN_GA(T21)
The graph contains the following edges 1 >= 1