(0) Obligation:
Clauses:list([]).
list(.(X, XS)) :- list(XS).
s2l(s(X), .(Y, Xs)) :- s2l(X, Xs).
s2l(0, []).
goal(X) :- ','(s2l(X, XS), list(XS)).
Queries:
goal(g).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:s2l9(s(T16), .(X65, X66)) :- s2l9(T16, X66).
list21(.(T28, T30)) :- list21(T30).
goal1(s(T8)) :- s2l9(T8, X28).
goal1(s(T8)) :- ','(s2lc9(T8, T23), list21(T23)).
Clauses:
s2lc9(s(T16), .(X65, X66)) :- s2lc9(T16, X66).
s2lc9(0, []).
listc21([]).
listc21(.(T28, T30)) :- listc21(T30).
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)
s2l9_in: (b,f)
s2lc9_in: (b,f)
list21_in: (b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
GOAL1_IN_G(s(T8)) → U3_G(T8, s2l9_in_ga(T8, X28))
GOAL1_IN_G(s(T8)) → S2L9_IN_GA(T8, X28)
S2L9_IN_GA(s(T16), .(X65, X66)) → U1_GA(T16, X65, X66, s2l9_in_ga(T16, X66))
S2L9_IN_GA(s(T16), .(X65, X66)) → S2L9_IN_GA(T16, X66)
GOAL1_IN_G(s(T8)) → U4_G(T8, s2lc9_in_ga(T8, T23))
U4_G(T8, s2lc9_out_ga(T8, T23)) → U5_G(T8, list21_in_g(T23))
U4_G(T8, s2lc9_out_ga(T8, T23)) → LIST21_IN_G(T23)
LIST21_IN_G(.(T28, T30)) → U2_G(T28, T30, list21_in_g(T30))
LIST21_IN_G(.(T28, T30)) → LIST21_IN_G(T30)
The TRS R consists of the following rules:
s2lc9_in_ga(s(T16), .(X65, X66)) → U7_ga(T16, X65, X66, s2lc9_in_ga(T16, X66))
s2lc9_in_ga(0, []) → s2lc9_out_ga(0, [])
U7_ga(T16, X65, X66, s2lc9_out_ga(T16, X66)) → s2lc9_out_ga(s(T16), .(X65, X66))
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
s2l9_in_ga(x1, x2) = s2l9_in_ga(x1)
.(x1, x2) = .(x2)
s2lc9_in_ga(x1, x2) = s2lc9_in_ga(x1)
U7_ga(x1, x2, x3, x4) = U7_ga(x1, x4)
0 = 0
s2lc9_out_ga(x1, x2) = s2lc9_out_ga(x1, x2)
list21_in_g(x1) = list21_in_g(x1)
GOAL1_IN_G(x1) = GOAL1_IN_G(x1)
U3_G(x1, x2) = U3_G(x1, x2)
S2L9_IN_GA(x1, x2) = S2L9_IN_GA(x1)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4)
U4_G(x1, x2) = U4_G(x1, x2)
U5_G(x1, x2) = U5_G(x1, x2)
LIST21_IN_G(x1) = LIST21_IN_G(x1)
U2_G(x1, x2, x3) = U2_G(x2, 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:
GOAL1_IN_G(s(T8)) → U3_G(T8, s2l9_in_ga(T8, X28))
GOAL1_IN_G(s(T8)) → S2L9_IN_GA(T8, X28)
S2L9_IN_GA(s(T16), .(X65, X66)) → U1_GA(T16, X65, X66, s2l9_in_ga(T16, X66))
S2L9_IN_GA(s(T16), .(X65, X66)) → S2L9_IN_GA(T16, X66)
GOAL1_IN_G(s(T8)) → U4_G(T8, s2lc9_in_ga(T8, T23))
U4_G(T8, s2lc9_out_ga(T8, T23)) → U5_G(T8, list21_in_g(T23))
U4_G(T8, s2lc9_out_ga(T8, T23)) → LIST21_IN_G(T23)
LIST21_IN_G(.(T28, T30)) → U2_G(T28, T30, list21_in_g(T30))
LIST21_IN_G(.(T28, T30)) → LIST21_IN_G(T30)
The TRS R consists of the following rules:
s2lc9_in_ga(s(T16), .(X65, X66)) → U7_ga(T16, X65, X66, s2lc9_in_ga(T16, X66))
s2lc9_in_ga(0, []) → s2lc9_out_ga(0, [])
U7_ga(T16, X65, X66, s2lc9_out_ga(T16, X66)) → s2lc9_out_ga(s(T16), .(X65, X66))
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
s2l9_in_ga(x1, x2) = s2l9_in_ga(x1)
.(x1, x2) = .(x2)
s2lc9_in_ga(x1, x2) = s2lc9_in_ga(x1)
U7_ga(x1, x2, x3, x4) = U7_ga(x1, x4)
0 = 0
s2lc9_out_ga(x1, x2) = s2lc9_out_ga(x1, x2)
list21_in_g(x1) = list21_in_g(x1)
GOAL1_IN_G(x1) = GOAL1_IN_G(x1)
U3_G(x1, x2) = U3_G(x1, x2)
S2L9_IN_GA(x1, x2) = S2L9_IN_GA(x1)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4)
U4_G(x1, x2) = U4_G(x1, x2)
U5_G(x1, x2) = U5_G(x1, x2)
LIST21_IN_G(x1) = LIST21_IN_G(x1)
U2_G(x1, x2, x3) = U2_G(x2, x3)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 7 less nodes.(6) Complex Obligation (AND)
(7) Obligation:
Pi DP problem:The TRS P consists of the following rules:
LIST21_IN_G(.(T28, T30)) → LIST21_IN_G(T30)
The TRS R consists of the following rules:
s2lc9_in_ga(s(T16), .(X65, X66)) → U7_ga(T16, X65, X66, s2lc9_in_ga(T16, X66))
s2lc9_in_ga(0, []) → s2lc9_out_ga(0, [])
U7_ga(T16, X65, X66, s2lc9_out_ga(T16, X66)) → s2lc9_out_ga(s(T16), .(X65, X66))
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
.(x1, x2) = .(x2)
s2lc9_in_ga(x1, x2) = s2lc9_in_ga(x1)
U7_ga(x1, x2, x3, x4) = U7_ga(x1, x4)
0 = 0
s2lc9_out_ga(x1, x2) = s2lc9_out_ga(x1, x2)
LIST21_IN_G(x1) = LIST21_IN_G(x1)
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:
LIST21_IN_G(.(T28, T30)) → LIST21_IN_G(T30)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
LIST21_IN_G(x1) = LIST21_IN_G(x1)
We have to consider all (P,R,Pi)-chains
(10) PiDPToQDPProof (SOUND 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:
LIST21_IN_G(.(T30)) → LIST21_IN_G(T30)
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:
- LIST21_IN_G(.(T30)) → LIST21_IN_G(T30)
The graph contains the following edges 1 > 1
(13) YES
(14) Obligation:
Pi DP problem:The TRS P consists of the following rules:
S2L9_IN_GA(s(T16), .(X65, X66)) → S2L9_IN_GA(T16, X66)
The TRS R consists of the following rules:
s2lc9_in_ga(s(T16), .(X65, X66)) → U7_ga(T16, X65, X66, s2lc9_in_ga(T16, X66))
s2lc9_in_ga(0, []) → s2lc9_out_ga(0, [])
U7_ga(T16, X65, X66, s2lc9_out_ga(T16, X66)) → s2lc9_out_ga(s(T16), .(X65, X66))
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
.(x1, x2) = .(x2)
s2lc9_in_ga(x1, x2) = s2lc9_in_ga(x1)
U7_ga(x1, x2, x3, x4) = U7_ga(x1, x4)
0 = 0
s2lc9_out_ga(x1, x2) = s2lc9_out_ga(x1, x2)
S2L9_IN_GA(x1, x2) = S2L9_IN_GA(x1)
We have to consider all (P,R,Pi)-chains
(15) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(16) Obligation:
Pi DP problem:The TRS P consists of the following rules:
S2L9_IN_GA(s(T16), .(X65, X66)) → S2L9_IN_GA(T16, X66)
R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
.(x1, x2) = .(x2)
S2L9_IN_GA(x1, x2) = S2L9_IN_GA(x1)
We have to consider all (P,R,Pi)-chains
(17) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(18) Obligation:
Q DP problem:The TRS P consists of the following rules:
S2L9_IN_GA(s(T16)) → S2L9_IN_GA(T16)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(19) 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:
- S2L9_IN_GA(s(T16)) → S2L9_IN_GA(T16)
The graph contains the following edges 1 > 1