(0) Obligation:
Clauses:member(X, .(Y, Xs)) :- member(X, Xs).
member(X, .(X, Xs)).
subset(.(X, Xs), Ys) :- ','(member(X, Ys), subset(Xs, Ys)).
subset([], Ys).
member1(X, .(Y, Xs)) :- member1(X, Xs).
member1(X, .(X, Xs)).
subset1(.(X, Xs), Ys) :- ','(member1(X, Ys), subset1(Xs, Ys)).
subset1([], Ys).
Queries:
subset(g,g).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:member10(T42, .(T43, T44)) :- member10(T42, T44).
subset11(.(T21, T7), .(T22, T23)) :- member10(T21, T23).
subset11(.(T21, T7), .(T22, T23)) :- ','(memberc10(T21, T23), subset11(T7, .(T22, T23))).
subset11(.(T66, T7), .(T66, T67)) :- subset11(T7, .(T66, T67)).
Clauses:
subsetc1(.(T21, T7), .(T22, T23)) :- ','(memberc10(T21, T23), subsetc1(T7, .(T22, T23))).
subsetc1(.(T66, T7), .(T66, T67)) :- subsetc1(T7, .(T66, T67)).
subsetc1([], T73).
memberc10(T42, .(T43, T44)) :- memberc10(T42, T44).
memberc10(T52, .(T52, T53)).
Afs:
subset11(x1, x2) = subset11(x1, x2)
(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:subset11_in: (b,b)
member10_in: (b,b)
memberc10_in: (b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
SUBSET11_IN_GG(.(T21, T7), .(T22, T23)) → U2_GG(T21, T7, T22, T23, member10_in_gg(T21, T23))
SUBSET11_IN_GG(.(T21, T7), .(T22, T23)) → MEMBER10_IN_GG(T21, T23)
MEMBER10_IN_GG(T42, .(T43, T44)) → U1_GG(T42, T43, T44, member10_in_gg(T42, T44))
MEMBER10_IN_GG(T42, .(T43, T44)) → MEMBER10_IN_GG(T42, T44)
SUBSET11_IN_GG(.(T21, T7), .(T22, T23)) → U3_GG(T21, T7, T22, T23, memberc10_in_gg(T21, T23))
U3_GG(T21, T7, T22, T23, memberc10_out_gg(T21, T23)) → U4_GG(T21, T7, T22, T23, subset11_in_gg(T7, .(T22, T23)))
U3_GG(T21, T7, T22, T23, memberc10_out_gg(T21, T23)) → SUBSET11_IN_GG(T7, .(T22, T23))
SUBSET11_IN_GG(.(T66, T7), .(T66, T67)) → U5_GG(T66, T7, T67, subset11_in_gg(T7, .(T66, T67)))
SUBSET11_IN_GG(.(T66, T7), .(T66, T67)) → SUBSET11_IN_GG(T7, .(T66, T67))
The TRS R consists of the following rules:
memberc10_in_gg(T42, .(T43, T44)) → U10_gg(T42, T43, T44, memberc10_in_gg(T42, T44))
memberc10_in_gg(T52, .(T52, T53)) → memberc10_out_gg(T52, .(T52, T53))
U10_gg(T42, T43, T44, memberc10_out_gg(T42, T44)) → memberc10_out_gg(T42, .(T43, T44))
Pi is empty.
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:
SUBSET11_IN_GG(.(T21, T7), .(T22, T23)) → U2_GG(T21, T7, T22, T23, member10_in_gg(T21, T23))
SUBSET11_IN_GG(.(T21, T7), .(T22, T23)) → MEMBER10_IN_GG(T21, T23)
MEMBER10_IN_GG(T42, .(T43, T44)) → U1_GG(T42, T43, T44, member10_in_gg(T42, T44))
MEMBER10_IN_GG(T42, .(T43, T44)) → MEMBER10_IN_GG(T42, T44)
SUBSET11_IN_GG(.(T21, T7), .(T22, T23)) → U3_GG(T21, T7, T22, T23, memberc10_in_gg(T21, T23))
U3_GG(T21, T7, T22, T23, memberc10_out_gg(T21, T23)) → U4_GG(T21, T7, T22, T23, subset11_in_gg(T7, .(T22, T23)))
U3_GG(T21, T7, T22, T23, memberc10_out_gg(T21, T23)) → SUBSET11_IN_GG(T7, .(T22, T23))
SUBSET11_IN_GG(.(T66, T7), .(T66, T67)) → U5_GG(T66, T7, T67, subset11_in_gg(T7, .(T66, T67)))
SUBSET11_IN_GG(.(T66, T7), .(T66, T67)) → SUBSET11_IN_GG(T7, .(T66, T67))
The TRS R consists of the following rules:
memberc10_in_gg(T42, .(T43, T44)) → U10_gg(T42, T43, T44, memberc10_in_gg(T42, T44))
memberc10_in_gg(T52, .(T52, T53)) → memberc10_out_gg(T52, .(T52, T53))
U10_gg(T42, T43, T44, memberc10_out_gg(T42, T44)) → memberc10_out_gg(T42, .(T43, T44))
Pi is empty.
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.(6) Complex Obligation (AND)
(7) Obligation:
Pi DP problem:The TRS P consists of the following rules:
MEMBER10_IN_GG(T42, .(T43, T44)) → MEMBER10_IN_GG(T42, T44)
The TRS R consists of the following rules:
memberc10_in_gg(T42, .(T43, T44)) → U10_gg(T42, T43, T44, memberc10_in_gg(T42, T44))
memberc10_in_gg(T52, .(T52, T53)) → memberc10_out_gg(T52, .(T52, T53))
U10_gg(T42, T43, T44, memberc10_out_gg(T42, T44)) → memberc10_out_gg(T42, .(T43, T44))
Pi is empty.
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:
MEMBER10_IN_GG(T42, .(T43, T44)) → MEMBER10_IN_GG(T42, T44)
R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
(10) PiDPToQDPProof (EQUIVALENT 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:
MEMBER10_IN_GG(T42, .(T43, T44)) → MEMBER10_IN_GG(T42, T44)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(12) 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:
- MEMBER10_IN_GG(T42, .(T43, T44)) → MEMBER10_IN_GG(T42, T44)
The graph contains the following edges 1 >= 1, 2 > 2
(13) YES
(14) Obligation:
Pi DP problem:The TRS P consists of the following rules:
SUBSET11_IN_GG(.(T21, T7), .(T22, T23)) → U3_GG(T21, T7, T22, T23, memberc10_in_gg(T21, T23))
U3_GG(T21, T7, T22, T23, memberc10_out_gg(T21, T23)) → SUBSET11_IN_GG(T7, .(T22, T23))
SUBSET11_IN_GG(.(T66, T7), .(T66, T67)) → SUBSET11_IN_GG(T7, .(T66, T67))
The TRS R consists of the following rules:
memberc10_in_gg(T42, .(T43, T44)) → U10_gg(T42, T43, T44, memberc10_in_gg(T42, T44))
memberc10_in_gg(T52, .(T52, T53)) → memberc10_out_gg(T52, .(T52, T53))
U10_gg(T42, T43, T44, memberc10_out_gg(T42, T44)) → memberc10_out_gg(T42, .(T43, T44))
Pi is empty.
We have to consider all (P,R,Pi)-chains
(15) PiDPToQDPProof (EQUIVALENT transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(16) Obligation:
Q DP problem:The TRS P consists of the following rules:
SUBSET11_IN_GG(.(T21, T7), .(T22, T23)) → U3_GG(T21, T7, T22, T23, memberc10_in_gg(T21, T23))
U3_GG(T21, T7, T22, T23, memberc10_out_gg(T21, T23)) → SUBSET11_IN_GG(T7, .(T22, T23))
SUBSET11_IN_GG(.(T66, T7), .(T66, T67)) → SUBSET11_IN_GG(T7, .(T66, T67))
The TRS R consists of the following rules:
memberc10_in_gg(T42, .(T43, T44)) → U10_gg(T42, T43, T44, memberc10_in_gg(T42, T44))
memberc10_in_gg(T52, .(T52, T53)) → memberc10_out_gg(T52, .(T52, T53))
U10_gg(T42, T43, T44, memberc10_out_gg(T42, T44)) → memberc10_out_gg(T42, .(T43, T44))
The set Q consists of the following terms:
memberc10_in_gg(x0, x1)
U10_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(17) 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:
- U3_GG(T21, T7, T22, T23, memberc10_out_gg(T21, T23)) → SUBSET11_IN_GG(T7, .(T22, T23))
The graph contains the following edges 2 >= 1 - SUBSET11_IN_GG(.(T66, T7), .(T66, T67)) → SUBSET11_IN_GG(T7, .(T66, T67))
The graph contains the following edges 1 > 1, 2 >= 2 - SUBSET11_IN_GG(.(T21, T7), .(T22, T23)) → U3_GG(T21, T7, T22, T23, memberc10_in_gg(T21, T23))
The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 2 > 4